iv

ACKNOWLEDGMENTS

In the name of Allah, the most beneficent and the most merciful. All the praises and thanks be to Him for providing me the strength and the blessing to complete this thesis. I want to express my genuine gratitude to my supervisor, Prof. Hassan Noura for his support and guidance from the start and up to the end of this journey. Without his encouragement, I will not even start this difficult mission, where his flexibility and professionalism made everything easy for me. I would like also to show my sincere appreciation to my supervisors at UPJV, Prof.

Ahmad El Hajjaji and Dr. Abdelhamid Rabhi, for their help and support throughout my study. Personally, many thanks go to Dr. Abdelhamid Rabhi for being as a brother to me during my stay in France, where I felt like home because of his caring and kindness. Also, I would like to thank Prof. Ahmad El Hajjaji for remarkable comments and advises throughout my work. The words are not to express my sincere thanks and appreciation for both of you. Finally, I would like to thank my lovely wife, Eng. Nedaa, on her caring and encouragement during my study and her understanding of being away spending the days and nights doing the research in the lab. And for Lamar my sweetest daughter, I regret the time that I could not spend with her because of being super busy on my study and work, but I hope she will understand this one day when she grows up. v Résumé Les principaux travaux développés dans cette thèse traitent de la commande, du diagnostic et de la tolérance aux fautes utilisant le concept de la commande sans modèle récemment proposé dans la littérature. Les méthodes développées sont testées en temps réel et validées sur un quadrotor.

Dans un premier temps, le modèle du quadrotor est présenté et analysé dans l'objectif de concevoir et implanter une loi de commande robuste aux perturbations et aux incertitudes de modèle. Pour cela, le concept de commande sans modèle est utilisé.

Ce concept est basé sur la détermination d'une loi de commande calculée à partir de modèles ultra-locaux ajoutée à la commande calculée par des régulateurs classiques. Une technique utilisant le régulateur linéaire quadratique ainsi que la technique non linéaire de Backstepping avec intégrateur augmenté de ce concept de commande sans modèle ont été proposées. Ces deux approches ont été testées et validées en les appliquant en temps-réel au drone considéré. Une étude comparative des lois de commandes avec et sans l'utilisation du concept de commande sans modèle a été proposée mettant en évidence les avantages de l'utilisation de ce concept.

Le diagnostic de défauts est une étape importante pour la commande tolérante aux fautes. Les techniques de diagnostic développées dans cette thèse sont basées sur la génération de résidus issus de la comparaison entre les mesures réelles et les estimations de ces mesures obtenues par des estimateurs. La génération de résidus dépend de la qualité du modèle utilisé ainsi que des perturbations qui peuvent conduire à des fausses alarmes ou des non détections. Une nouvelle technique « d'estimation intelligente » inspirée du concept de commande sans modèle a été proposée et implantée afin de rendre la génération de résidus insensible aux incertitudes de modèle et aux perturbations et ainsi améliorer la décision quant à la présence de défauts. vi Deux "estimateurs intelligents" ont été proposés en rajoutant à l'observateur d'état classique et l'observateur de Thau un concept inspiré de la technique de commande sans modèle.

L'estimation des sorties du système obtenue par l'estimateur intelligent est utilisée afin d'estimer l'amplitude des défauts d'actionneurs et de capteurs. L'estimation des défauts d'actionneurs est basée sur des modèles ultra-locaux. Quant aux défauts de capteurs, un algorithme structure a été proposé pour estimer leur amplitude en utilisant les résidus générés à partir de cet estimateur.

Les résultats de détection et estimation de défauts ainsi obtenus sont ensuite utilisés pour compenser l'effet des défauts sur les performances du drone. La commande tolérante aux fautes mise en oeuvre permet de modifier la loi de commande par rapport à l'estimation de l'amplitude du défaut d'actionneur, alors que lorsqu'un défaut de capteur est détecté et estimé, la trajectoire de référence est régénérée afin de compenser l'effet du défaut et maintenir le drone sur la trajectoire initialement définie. The FDD is a very important step towards the development of FTC techniques. The FDD approach developed in this thesis is based on the residual generation between the measured outputs and the estimated outputs obtained using observers/estimators.

Residuals are expected to be close to zero in the fault free case and deviate from zero in the presence of a fault or failure. However, as the residuals are generated using models, they highly depend on the quality of the model used and on the presence of disturbances which may lead to false alarms or to non-detections. A novel "intelligent estimator" viii inspired from the MF concept has been developed and used in order to improve the residual generation and the fault diagnostic.

Two intelligent estimators have been designed by integrating the MF scheme with the state and Thau observers for Multi-Input-Multi-Output (MIMO) systems, where the intelligent Output-Estimator (iOE) represents the integration between the MF technique with the state observer, and the intelligent Thau Output-Estimator (iTOE) represents the augmentation of the MF technique with the Thau Observer.

The estimation of the system outputs obtained using the iOE are then used to estimate the actuator and sensor faults. The estimation of the actuator faults is improved by using the ultra-local models. A structured algorithm is then developed and implemented in order to estimate sensor fault magnitudes using the residuals generated by the intelligent estimator.

The results obtained from the fault detection and estimation are then used to compensate for the fault effect on the flight control performance. The implemented faulttolerant control technique compensates for the actuator faults by adjusting the control law based on the fault estimation. In case a sensor fault is detected and estimated, the desired path is regenerated according to the estimated fault magnitude in order to compensate for the fault effect. As the control technology is so handy and helpful, as it could cause severe problems if it runs improperly, where any fatal error in its operation could lead to a catastrophic failure that harms not only the humans but also the living creatures in this planet.

Some of the examples that caused disasters because of a malfunction in the control system are presented next:

 In 1967, the X-15 Flight 191 crashed because of a degraded performance of the aircraft's control system that was affected by an abrupt electrical disturbance [START_REF] Orr | A Comprehensive Analysis of the X-15 Flight 3-65 Accident[END_REF].

 Thousands of people killed because of a large toxic gas leak in Bhopal, India, 1984. The gas leak would be prevented if the temperature control on the tanks is turned on [2].

 Another accident happened when a blocked pitot tube sensor led to a fatal error in the autopilot system and caused the crash of the Birgenair Flight 301 in the sea 26 km of Puerto Plata, Dominican Republic, 1996 [3].

 One of the theories that cause the crash of the Germanwings Airbus A320 flight in France ( 2015) is the ice-up of the angle-of-attack sensor causing the aircraft to descend rapidly [4].

(a) (b) (c) Figure 1.1. Disasters caused by a fault in the control system. (a) X-15 crash site [START_REF] Younes | Intelligent Output-Estimator Based on Model-Free Algorithm: Application to a Quadrotor UAV[END_REF], (b) Bhopal gas disaster [6], (c) Germanwings Airbus A320 crash site [4].

Therefore the need for developing strategies to accommodate or reduce the effects of the fault will not only protect the system or the machine but also save thousands of lives on this earth.

The control systems in most of the applications are running without any intervention from the users, where the safety and the reliability of its operation are highly demanded.

Nowadays, the complexity of the systems is potentially increased to provide smart and intelligent services to the users. If the automatic control system complexity is increased then the chances of having faults that interfere its operation will be higher. As a result, the researchers all around the world are impatiently studying the problems that can face any control system to find the suitable remedies that can solve or at least prevent the failure of the system or even prevent the disaster that could happen if not handled properly. Hence, tolerating and anticipating the faults in a system are preferable; costly and safety wise. As the drones are newly introduced in the different aspects of life, the fault accommodation is still in a developing stage. Therefore, in this work the quadrotor vehicle will be used to develop, design and implement various FDD and FTC strategies to overcome the effect of the different faults that can hit the system.

Background and Literature Review

In this section, the background of selected FDD/FTC approaches will be presented based on previous published works. The classification of the different Fault diagnosis approaches will be presented in Section 1.2.1, then different fault scenarios and their representations will be addressed in Section 1.2.2. In Sections 1.2.3 and 1.2.4, an overview of previous works in FDD and FTC will be discussed and presented, respectively. In Section 1.2.5, different FDD and FTC algorithms applied on the quadrotor system will be summarized. Finally, the quadrotor system used in this thesis and the different control strategies applied on it will be described in Sections 1.2.6 and 1.2.7, respectively.

Classification of Fault Diagnosis (FD) Approaches

The objective of the FD is to produce signals that reflect the discrepancy between faultfree and faulty system operations [7]. Several FD approaches for actuator faults have been used, such as the signal processing-based, knowledge-based, and analytical/modelbased approaches [8].

Based on the signal data of a system, the signal processing-based approach uses different methodologies to analyze the system's behavior by determining the signal characteristics when the fault occurs. This method can be applied to both linear and nonlinear systems. Wavelet Transform [9], Time Domain Analysis [10], Time-Frequency Domain Analysis [11], and Bicoherence Analysis [12], are examples of the different methods used in the signal processing-based approach for unmanned system.

In the knowledge-based approach, a wide-range of information that is related to the fault diagnosis will be collected and processed through systematic methodologies to be used in the FD strategy. The mathematical model in this approach is not required, where enough information about the system behavior will provide expert decisions about the system in general. Neural Network [13], Expert system [START_REF] Liu | Design and implementation of fault diagnosis expert system for helicopter[END_REF], Granular Computing [START_REF] Wang | A method for rule extraction based on granular computing: application in the fault diagnosis of a helicopter transmission system[END_REF], and Genetic Algorithms [START_REF] Firpi | Genetically programmed-based artificial features extraction applied to fault detection[END_REF], are examples of the knowledge-based approach used in the literature for unmanned helicopter systems.

In the analytical or model-based approach, mathematical model of the system is considered to diagnose the fault in real-time flights. In this study, the concentration will be focused on the model-based approach, where the analytical representation of the system has to be acquired and processed accordingly.

Generally, based on the modelling structure of a system, the model can be considered one of these types: Linear Time-Invariant (LTI) model, Linear Parameter Varying (LPV) model and nonlinear model. In the LTI system, generally the model will be constructed from differential and/or algebraic equations using the following state space representation:

{ 𝑥(𝑡) = 𝐴𝑥(𝑡) + 𝐵𝑢(𝑡) 𝑦(𝑡) = 𝐶𝑥(𝑡) + 𝐷𝑢(𝑡) (1.1)
where 𝑥(𝑡) ∈ ℝ 𝑛 is the state vector, 𝑢(𝑡) ∈ ℝ 𝑚 is the control input vector, which controls the system actuators, 𝑦(𝑡) ∈ ℝ 𝑞 is the system outputs vector that represents the measured variables. A, B, C, and D are the matrices of a continuous-time system that are constructed based on the differential and/or the algebraic equations of the system.

The uncontrolled inputs can be augmented in the system representation, as shown in (1.2), and can be considered as external disturbances or modelling uncertainties which have similar effects on the system [START_REF] Blanke | Diagnosis and fault-tolerant control[END_REF], also the fault can be included as shown in following system representation:

{ 𝑥(𝑡) = 𝐴𝑥(𝑡) + 𝐵𝑢(𝑡) + 𝑅 1 𝑑(𝑡) + 𝐹 1 𝑓(𝑡) 𝑦(𝑡) = 𝐶𝑥(𝑡) + 𝐷𝑢(𝑡) + 𝑅 2 𝑑(𝑡) + 𝐹 2 𝑓(𝑡) (1.2)
where 𝑑(𝑡) ∈ ℝ 𝑑 is the uncontrolled input vector, which could represent also the external disturbances or the modelling uncertainties of the system. 𝑅 1 and 𝑅 2 are the uncontrolled input system matrices. 𝑓(𝑡) ∈ ℝ 𝑓 is the faults vector, and 𝐹 1 and 𝐹 2 are the fault matrices with appropriate dimensions.

It is important to differentiate between the uncertainty/disturbance terms presented in (1.2) and the system's fault represented by 𝐹 1 𝑓(𝑡) and 𝐹 2 𝑓(𝑡) [START_REF] Shi | Observer based active fault tolerant control of descriptor systems[END_REF]. The effect of the uncertainty or the disturbance can be compensated by designing a robust controller while the fault has to be accommodated using systematic techniques that include the FDD and FTC designs [START_REF] Blanke | Diagnosis and fault-tolerant control[END_REF].

In the nonlinear model representation, the sophisticated relations between the inputs and the outputs have to be considered. A general representation of a nonlinear model is as follows:

{ 𝑔(𝑥(𝑡), 𝑥(𝑡), 𝑢(𝑡), 𝑑(𝑡), 𝑓(𝑡), 𝑡) = 0 ℎ( 𝑦(𝑡), 𝑥(𝑡), 𝑢(𝑡), 𝑑(𝑡), 𝑓(𝑡), 𝑡) = 0

(1.3)
where 𝑔(•) and ℎ(•) are the nonlinear functions that represent the relation between the different variables of the system.

A special state-space representation of a nonlinear system can be represented in (1.4). A more refined version of this model will be extracted later that matches with the setup used in this project.

{ 𝑥(𝑡) = 𝑓(𝑥(𝑡)) + 𝑔(𝑥(𝑡))𝑢(𝑡) 𝑦(𝑡) = ℎ(𝑥(𝑡)) (1.4)
The complexity of designing controller, FDD and FTC strategies for nonlinear systems are addressed in many research studies, but extracting the exact model and implementing the different techniques on it are quite challenging tasks. To reduce the complexity of the nonlinear model, the linearization about an operating point is considered as a solution if the system is running about that operating point. Any divergence could lead to unwanted behavior, as the techniques will not be able anymore to accommodate for the variations in the system's dynamics after working around another operating point.

The linearized system can be adjusted to accommodate the change in the dynamics of the nonlinear system if it runs in a different operating point. where 𝐴(𝑝), 𝐵(𝑝), 𝐶(𝑝) and 𝐷(𝑝) are the time-varying matrices parameterized by the scheduling parameter vector p.

In this study the LTI and nonlinear systems will be considered for the FD analysis.

The overall representation of the system considering faults will be discussed and addressed next.

Fault Scenarios

In general, different electronic sensors and devices reduce the reliability of the systems and cause unwanted sensor faults/failures scenarios, specifically on the UAV systems as the work in this thesis is about the quadrotor system. On the other hand, the actuator or the component faults or failures could lead to a catastrophic situation. Defining the type of the fault is important to anticipate its effect on the system, also this will help to identify the best way of dealing with the fault, and to ensure the safety of the system and its surrounding.

In this work, the actuator and sensor faults are considered, where the representation of both faults will be presented next.

Representation of Actuator Faults

The actuator fault is not related only to the variation of the control input 𝑢 around an operating point, but also to the variation of the global control input 𝑈 applied to the system [START_REF] Noura | Fault-tolerant control systems: Design and practical applications[END_REF], where the global faulty control input can be represented as follows:

𝑈 𝑓 = ℱ𝑈 + 𝑈 𝑓0 (1.6)
Where 𝑈 𝑓 is the global faulty control input,

ℱ = 𝑑𝑖𝑎𝑔(𝛼), 𝛼 = [ 𝛼 1 … 𝛼 𝑖 … 𝛼 𝑚] 𝑇 ,
ℱ𝑈 corresponds to the multiplicative actuator faults,

𝑈 𝑓0 = [ 𝑢 𝑓01 … 𝑢 𝑓0𝑖 … 𝑢 𝑓0] 𝑇
represents the additive actuator faults.

The relation between the control input around an operating point 𝑈 0 and the global control input can be represented by 𝑢 = 𝑈 -𝑈 0 , and similarly for the faulty control input

𝑢 𝑓 = 𝑈 𝑓 -𝑈 0 .
The occurrence of a fault in the 𝑖 𝑡ℎ actuator could be represented by 𝛼 𝑖 ≠ 1 or 𝑢 𝑓0𝑖 ≠ 0. Different scenarios of actuator faults are presented in Table 1-1 for certain values of 𝛼 𝑖 and 𝑢 𝑓0𝑖 .

Out of order

Actuator Block

The linear representation of a system presented in (1.1) with the consideration of the actuator faults can be re-written as follows:

{ 𝑥(𝑡) = 𝐴𝑥(𝑡) + 𝐵𝑢(𝑡) + 𝐵[(ℱ -𝐼)𝑈(𝑡) + 𝑈 𝑓0 ] 𝑦(𝑡) = 𝐶𝑥(𝑡) (1.7)
By considering an unknown input vector that represents the magnitude of actuator fault 𝑓 𝑎 (𝑡), the system can be represented as:

{ 𝑥(𝑡) = 𝐴𝑥(𝑡) + 𝐵𝑢(𝑡) + 𝐹 𝑎 𝑓 𝑎 (𝑡) 𝑦(𝑡) = 𝐶𝑥(𝑡) (1.8)
where 𝐹 𝑎 is considered to be equal to matrix 𝐵 as presented in most of the studies, and 𝑓 𝑎 (𝑡) = (ℱ -𝐼)𝑈(𝑡) + 𝑈 𝑓0 corresponds to the magnitude of the fault.

Similarly, the representation of the nonlinear system given in (1.4) with the presence of the actuator fault can be described as follows:

{ 𝑥(𝑡) = 𝑓(𝑥(𝑡)) + 𝑔(𝑥(𝑡))𝑢(𝑡) + 𝐹 𝑎 𝑓 𝑎 (𝑡) 𝑦(𝑡) = ℎ(𝑥(𝑡)) (1.9)

Representation of Sensor Faults

The sensor fault can be related to a degradation in the sensor performance while the sensor failure means the inability of the sensor to perform its task. e.g. stopping from providing a measurement. In the fault scenarios, the control system can be reconfigured or adjusted to overcome the degradation of the measured output from the faulty sensor, while the sensor failure could lead to a catastrophic or unsafe effect on the system.

In general, the sensing devices can fail (fault/failure) in different ways. The fault/failure types could be in one of the following forms: [START_REF] Heredia | Sensor and actuator fault detection in small autonomous helicopters[END_REF]  Total sensor failure. It happens when the sensor stops measuring the data and gives a zero output. This is considered to be an unmanageable failure of the sensor readings, and the reason could be due to electronics/electrical problems.

 Stuck with constant reading failure. This type of failure occurs when the sensor measurement is getting stuck on a certain output reading (not equal to zero). Also, this type of failure could lead to unwanted behavior from the system.

 Additive-type sensor fault (Sensor Bias). This is a very common sensor fault, which could be caused by the environmental changes that affect the calibration of the sensor. In this case, the sensor will measure the output with a constant offset above the actual value. This type of sensor fault will be considered in this work.

 Multiplicative-type sensor fault (Sensor Drift). The drift from the actual reading will be caused when a multiplicative factor is presented on the nominal sensor reading.

In the additive and multiplicative-type sensor faults, an accurate estimation of the fault could lead to a desired compensation of the fault by using one of the active faulttolerant techniques, in which the system will show an adaptation of the performance despite the presence of the fault.

Similar to the actuator fault, the system representations in (1.8) and (1.9) with the consideration of an unknown sensor fault 𝑓 𝑎 (𝑡) that affects the measured output of the system can be represented in the linear and nonlinear system representation as follows: where matrix 𝐹 𝑠 describes the relationship between the fault vector and the output, and 𝑓 𝑠 (𝑡) corresponds to the magnitude of the fault.

{ 𝑥̇(𝑡) = 𝐴𝑥(𝑡) + 𝐵𝑢(𝑡) 𝑦(𝑡) = 𝐶𝑥(𝑡) + 𝐹 𝑠 𝑓 𝑠 (𝑡) (1.10)

Overall System Representation

From (1.8), (1.10) and (1.9), (1.11) where the matrices 𝐹 𝑎 and 𝐹 𝑠 are assumed to be known and 𝑓 𝑎 (𝑡) and 𝑓 𝑠 (𝑡) represent the unknown values of the actuator and sensor faults that need to be estimated for later use in the Fault-tolerant control.

Fault Detection and Diagnosis (FDD)

The malfunction specifically on a flying machine, could lead to a catastrophic failure, where the importance of introducing different FDD and FTC strategies are highlighted by the researchers to avoid such a case.

FDD involves the process of detecting, isolating and identifying or estimating the faults [8]. Numerous FDD approaches for different types of faults have been used by the researchers. The concentration in this work will be on the model-based approach. The model-based approach considers mathematical model of the system to diagnose the fault in real-time flights.

The FDD scheme is composed of the Fault-Detection-and-Isolation (FDI) and Fault-Estimation (FE) processes. In some studies, FE is separated from FDD, but in this study the FE will be considered the last stage of the FDD system. The general procedure of the FDD goes through the following stages [START_REF] Noura | Fault-tolerant control systems: Design and practical applications[END_REF]:

1. Estimating the system's outputs by implementing an appropriate estimator design based on the system complexity, where a linear or nonlinear outputestimator can be applied to estimate the values of the variable outputs continuously. A proper knowledge of the system's dynamics will make the implementation process an easier task, where an accurate system model will lead to a better estimation of the system's outputs.

2.

Residual generation, which is the difference between the measured and the estimated output pair that generates the residual needed to detect the fault. The error can be computed by (𝑟 = 𝑒 𝑦 = 𝑦 -𝑦 ̂). In the fault-free situations the output-estimation errors will be close to zero, while in the presence of the fault the corresponding error values will deviate from zero to indicate the existence of the fault.

3.

Residual evaluation the process of producing symptoms 𝑆(𝑟) that are generated by comparing the residuals r to some pre-defined thresholds. The thresholds are set based on empirical or mathematical study of the system behavior against the faults.

4. Decision-Making Unit (DMU), which is the process that identifies which element is faulty by comparing the fault symptoms 𝑆(𝑟) to a certain pattern according to a constructed Fault Signature Table (FTS).

5. Fault estimation is the process that computes the magnitude of the fault based on different estimation techniques and according to the fault types.

The stages from 1 to 4 are summarizing the FDI process. A general FDI scheme is presented and shown in Figure 1.2. In this figure, the residuals (𝑟) will be generated from the output-estimation error (𝑒 𝑦 ) of a model-based observer. The output-estimation errors are calculated from the difference between the actual outputs (𝑦) and the estimated outputs (𝑦 ̂) from the observer, where 𝑟 = 𝑒 𝑦 = 𝑦 -𝑦 ̂. Then output-estimation errors will be compared to a pre-defined thresholds to generate symptoms 𝑆(𝑟) that will be fed to the Decision Making Unit (DMU) with the information needed for fault isolation process. Different observers for linear systems are introduced in [START_REF] Kreisselmeier | Adaptive observers with exponential rate of convergence[END_REF][START_REF] Reilly | Observers for linear systems[END_REF][START_REF] Luenberger | Observers for multivariable systems[END_REF]. For nonlinear systems, the design of a stable observer is a very hard task, where some approaches have been analyzed for sets of nonlinear systems [START_REF] Frank | Nonlinear observers for fault detection and isolation[END_REF][START_REF] Frank | On-line fault detection in uncertain nonlinear systems using diagnostic observers: a survey[END_REF].

Thau introduced an asymptotic stable observer that estimates the states of a nonlinear system, and Raghavan continued Thau work by formulating a procedure to obtain the observer gain matrix by solving the Riccati equation [START_REF] Pettersen | Nonlinear Control Approach to Helicopter Autonomy[END_REF]. Thau observer structure will be described in Chapter 4, Section 4.1.3.

The complexity of extracting the exact model, which imitates the system dynamics, is one of the worrying and challenging tasks of using the model-based approach. In some cases of sensor or actuator failures, the model-based approach will fail to provide the correct data to the active FTC unit, which could lead to a catastrophic failure of the whole system.

One of the contributions of this work is focused on implementing a novel outputestimator that can cope and compensate for the unmodeled dynamics and modeling errors of the quadrotor vehicle. The proposed work is presented and applied on the quadrotor system in Chapters 4 and 5.

On the other hand the residual generations have been investigated in the literature.

The Dedicated Observer Scheme (DOS) and the Generalized Observer Scheme (GOS) are examples of the techniques used to generate structured residuals for detecting and isolating the faulty elements [START_REF] Hsu | Diagnosis of multiple sensor and actuator failures in automotive engines[END_REF][START_REF] Frisk | Model-based fault diagnosis applied to an SI-Engine[END_REF].

An overview of the main published works about the model-based FDD approaches will be summarized and presented in the following subsections.

FDI using Unknown Input Observer (UIO)

One of the effective approaches to detect and isolate the faulty element is the UIO. The unknown observer concept is introduced in 1982 to FDI applications by Watanabe and Himmelblau [START_REF] Watanabe | Instrument fault detection in systems with uncertainties[END_REF], and then studied and suggested as a robust FDI tool for different applications by Frank [START_REF] Frank | Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy: A survey and some new results[END_REF], and Chen and Patton in [START_REF] Chen | Active fault-tolerant flight control systems design using the linear matrix inequality method[END_REF]. Based on the presentation done by Noura et al. in [START_REF] Noura | Fault-tolerant control systems: Design and practical applications[END_REF], the observer is designed as follows:

The general formulation of the system that considers the unknown input vector is presented in [START_REF] Noura | Fault-tolerant control systems: Design and practical applications[END_REF] and shown below:

{ 𝑥(𝑡) = 𝐴𝑥(𝑡) + 𝐵𝑢(𝑡) + 𝐹 𝑑 𝑓 𝑑 (𝑡) 𝑦(𝑡) = 𝐶𝑥(𝑡) (1.14)
where 𝐹 𝑑 is the coefficient matrix of the unknown input vector 𝑓 𝑑 (𝑡).

{ 𝓌̇(𝑡) = 𝐸𝓌(𝑡) + 𝑇𝐵𝑢(𝑡) + 𝐾𝑦(𝑡) 𝑥 ̂(𝑡) = 𝓌(𝑡) + 𝐻𝑦(𝑡) (1.15) where 𝑥 ̂(𝑡) is the estimated state-vector, 𝓌(𝑡) is the state of the UIO full-order observer, and matrices 𝐸, 𝑇, 𝐾, and 𝐻 are designed to achieve the decoupling between the unknown inputs. The design of the observer is obtained by solving the following equations:

(𝐻𝐶 -𝐼)𝐹 𝑑 = 0 𝑇 = 𝐼 -𝐻𝐶 𝐸 = 𝐴 -𝐻𝐶𝐴 -𝐾 1 𝐶 𝐾 2 = 𝐸𝐻 𝐾 = 𝐾 1 + 𝐾 2 (1.16)
The convergence of the state error estimation is attained by having a stable matrix E.

The system in (1.15) can be considered as an unknown input observer for the system given in (1.14) if the following conditions are satisfied:

 Rank(𝐶𝐹 𝑑 ) = rank(𝐹 𝑑 )  (𝐶, 𝐴 1
) is a detectable pair, where 𝐴 1 = 𝐸 + 𝐾 1 𝐶

Then the estimation of the state vector can be used to generate residuals that are independent to the unknown input vector 𝑓 𝑑 (𝑡).

The reliability of UIO in FDI is justifiable because of its capability in tolerating some of the model uncertainties [7].

FDI using Nonlinear Identity Observer

The nonlinear identity observer has been introduced by Hengy and Frank in [START_REF] Hengy | Component failure detection via nonlinear state observers[END_REF]. The structure of nonlinear identity observer approach is suitable for the quadrotor nonlinear model. In this work, this type of observer is utilized side by side with the robust GOS scheme and presented in details in Appendix-A.

FDI, FE and FTC using Sliding-Mode Observer (SMO)

As an FDI approach, the sliding mode observer is introduced by Sreedhar in [START_REF] Sreedhar | Robust fault detection in nonlinear systems using sliding mode observers[END_REF]. The residual generated by an SMO will have information about the fault if the sliding motion is disturbed at the occurrence time of the fault. Also, the SMO could be used as an FTC tool.

As an FDI approach, the sliding mode observer is introduced by Sreedhar in [START_REF] Sreedhar | Robust fault detection in nonlinear systems using sliding mode observers[END_REF]. The residual generated by an SMO will have information about the fault if the sliding motion is disturbed at the occurrence time of the fault. Also, the SMO could be used as an FTC tool.

After that the sliding mode approach witnessed further developments in terms of the advancement of the design method [START_REF] Edwards | Sliding mode control: theory and applications[END_REF] and considering the modelling uncertainty in [START_REF] Tan | Sliding mode observers for robust detection and reconstruction of actuator and sensor faults[END_REF]. The different developments in the sliding mode approach provide not only the FDI information but also identification about the fault for the use of the controller reconfiguration [7]. Progressive works using the sliding mode FTC scheme in aerospace applications were published by Alwi and Edwards in [START_REF] Alwi | Fault tolerant control of a civil aircraft using a sliding mode based scheme[END_REF][START_REF] Alwi | Fault tolerant control of a large civil aircraft using a sliding mode based scheme[END_REF][START_REF] Alwi | Application of fault tolerant control using sliding modes with on-line control allocation on a large civil aircraft[END_REF]. Different FDD and FTC studies applied on the quadrotor system using the sliding-mode approach are presented in Section 1.2.5.

FDD and FTC based on Linear Parameter Varying (LPV) approach

Reconfiguring the controller parameters according to a tabulated gain scheduling is studied widely especially in the field of flight control systems [START_REF] Shamma | Analysis of gain scheduled control for nonlinear plants[END_REF][START_REF] Shamma | Gain Scheduling: Polenlial Hazards and Possible Remedies[END_REF]. At each operating point of the nonlinear system, a linear technique will be designed to ensure not only the stability but also the best performance of the system's response on that operating point.

One of the drawbacks of using the gain scheduling approach is the global stability when the transition between the operating points occurs, where a slow variation is required to ensure the stability of the system [START_REF] Shamma | Gain Scheduling: Polenlial Hazards and Possible Remedies[END_REF].

The LPV approach is used for FDD designs in previous works such as [START_REF] Szabó | Detection filter design for LPV systems-a geometric approach[END_REF][START_REF] Hallouzi | Fault detection and identification of actuator faults using linear parameter varying models[END_REF][START_REF] Zolghadri | Fault diagnosis for LPV systems[END_REF]. And used as an FTC strategy in [START_REF] Patton | An LPV pole-placement approach to friction compensation as an FTC problem[END_REF]. Recently, the LPV approach is applied on the quadrotor system, where a FDI using bank of LPV observers is presented in [START_REF] López-Estrada | LPV Model-Based Tracking Control and Robust Sensor Fault Diagnosis for a Quadrotor UAV[END_REF] and a FTC using robust LPV framework is proposed in [START_REF] Rotondo | Fault tolerant control design for polytopic uncertain LPV systems: application to a quadrotor[END_REF].

Fault Estimation using Singular Value Decomposition (SVD)

The SVD method for actuator and sensor fault estimation is described in the discrete-time domain in [START_REF] Noura | Fault-tolerant control systems: Design and practical applications[END_REF] and will be presented next.

This method is applied to estimate the actuator fault on the quadrotor system and compared to the proposed methods in this thesis.

The system presented in (1.8) The estimation of the fault magnitude can be extracted from the last component of 𝑋 ̅ 𝑎 𝑘 . A script is written using Matlab ® software to compute the coefficient matrices presented in the SVD technique, and then used in the Simulink environment to do the online estimation of the actuator fault magnitude. The comparison results between the SVD technique and the proposed algorithm are presented and discussed in Section 5.5.3.

Fault-Tolerant Control (FTC)

Many published works on FTC methods have been recently proposed in the presence of the different faults, which can be categorized into Passive FTC (PFTC) and Active FTC (AFTC).

PFTC techniques can compensate for a predefined set of faults by using a fixed robust controller design. The PFTCs presented in [START_REF] Berbra | A multi-observer switching strategy for fault-tolerant control of a quadrotor helicopter[END_REF][START_REF] Khebbache | Robust Stabilization of a Quadrotor Aerial Vehicle in Presence of Actuator Faults[END_REF][START_REF] Li | Fault tolerant control applied to a quadrotor unmanned helicopter[END_REF][START_REF] De Oca | Fault-tolerant control strategy for actuator faults using LPV techniques: Application to a two degree of freedom helicopter[END_REF][START_REF] Khelassi | Reconfigurability analysis for reliable fault-tolerant control design[END_REF][START_REF] Theilliol | Actuator fault tolerant control design based on a reconfigurable reference input[END_REF][START_REF] Boussaid | Performance evaluation based fault tolerant control with actuator saturation avoidance[END_REF] used different control methods, such as sliding-mode control, backstepping control, and model adaptive control to compensate not only for the disturbances but also for special types of faults under certain limitations. In this work, robust control techniques will be introduced in Chapter 3 and their ability to tolerate low-magnitude actuator faults will be tested and presented in Chapter 5. However, the disadvantage of relying on the PFTC techniques is that only specific faults with certain limitations can be tolerated.

On the other hand, active FTCs are compensating for the faults by the online reconfiguration of the controller or the desired references. The reconfigurable control law will give the flexibility to compensate for larger and maybe more critical types of faults that cannot be compensated by the PFTC methods. Some works have dealt with AFTC of quadrotors [START_REF] Ranjbaran | Fault recovery of an under-actuated quadrotor aerial vehicle[END_REF][START_REF] Izadi | Fault tolerant model predictive control of quad-rotor helicopters with actuator fault estimation[END_REF][START_REF] Sadeghzadeh | Fault-tolerant trajectory tracking control of a quadrotor helicopter using gain-scheduled PID and model reference adaptive control[END_REF][START_REF] Yang | Supervisory fault tolerant control with integrated fault detection and isolation: a switched system approach[END_REF][START_REF] Fang | Fault monitoring and fault recovery control for positionmoored vessels[END_REF][START_REF] Edwards | Sliding mode methods for fault detection and fault tolerant control with application to aerospace systems[END_REF]. In these AFTC approaches, the fault is diagnosed first using FDD methodologies and then the control laws are adjusted based on the information from the FDD and the fault estimation magnitude to compensate for the fault. The activation of the AFTC process is so crucial where it has to be triggered when the fault occurs.

In some methodologies the estimation process will give enough data to detect and isolate the fault. Keep tracking the estimation magnitude of the fault will identify the fault information and acts to the fault accordingly.

It is essential to consider the fault-tolerant capacity of different control approaches based on the fault estimation value, and then apply the FTC accordingly.

Fault-parking is derived from a recently proposed concept of safety-parking in the research area of industry safety and reliability [START_REF] Du | An integrated fault detection and isolation and safe-parking framework for networked process systems[END_REF][START_REF] Du | A safe-parking and safe-switching framework for faulttolerant control of switched nonlinear systems[END_REF][START_REF] Gandhi | Safe-parking of nonlinear process systems[END_REF][START_REF] Gandhi | A safe-parking framework for plant-wide faulttolerant control[END_REF]. It means that safety measures, in an economical way, are advised to be taken in the case that absolute recovery is impossible. The normal operation mode or profile will be replaced by a relatively safe but simple one. Especially for aircraft flight control applications, it is strongly suggested to let the aircraft land and park when the fault is not recoverable or the FTC also could not handle it.

FDD and FTC applied on the quadrotor vehicle

Different FDD and FTC algorithms are applied on the quadrotor system, Table 12showing some of the methodologies used for FDD and/or FTC for specific fault type and the way each method was tested and validated (whether it's based on simulation or experimental procedure). A collection of different algorithms proposed by Y. Zhang and his team in [START_REF] Zhang | Fault tolerant flight control techniques with application to a Quadrotor UAV testbed[END_REF] is applied on the Qball-X4 quadrotor system to compensate for the actuator faults. A summary of the comprehensive study of the FTC methods and their simulation and realtime results applied on the quadrotor vehicle are provided in Table 1-3. For the real damage of the actuator test, the tip of the propeller is sliced during the flight test.

AFTC × × √ √ Sliding-Mode Control PFTC × √ √ √ Backstepping Control PFTC × √ × × Model Predictive Control AFTC √ √ × × Flatness-based Trajectory Planning / Re-planning AFTC √ √ √ ×
In case of the actuator failure, the new unstable quadrotor dynamics will make it impossible to accommodate for the actuator failure, unless a changed in the quadrotor dynamics is attained very quickly (e.g. changing the CoG of the quadrotor). Other rotorcraft designs, such as the hexarotor or octorotor have the capabilities to accommodate the failure in one of the actuators, where different FDD and FTC studies are proposed to tolerate the failure as presented in [START_REF] Du | Controllability Analysis and Degraded Control for a Class of Hexacopters Subject to Rotor Failures[END_REF][START_REF] Schneider | Fault-tolerant control allocation for multirotor helicopters using parametric programming[END_REF][START_REF] Raabe | Failure-Tolerant Control and Vision-Based Navigation for Hexacopters[END_REF][START_REF] Saied | Fault diagnosis and fault-tolerant control strategy for rotor failure in an octorotor[END_REF][START_REF] Saied | Actuator Fault Diagnosis in an Octorotor UAV Using Sliding Modes Technique: Theory and Experimentation[END_REF][START_REF] Saied | Controllability analysis and motors failures symmetry in a coaxial octorotor[END_REF].

Quadrotor System

Nowadays, many researches and studies are conducted in the area of the Vertical Take

Off and Landing vehicles (VTOL). The high power to weight ratio and marvelous maneuverability give the VTOL crafts the advantage over the other types of the Unmanned Aerial Vehicles (UAV). Besides that, the cutting edge technology in electronics, sensor devices, and processing units encourage the researchers to design control methodologies that deal with the nonlinearities in such system as the quadrotor vehicle.

The quadrotor vehicle is a type of VTOL rotorcraft that has four rotors; each generates a thrust force due to the rotation of the propellers. Two rotors rotate clockwise and two counter-clockwise. The difference in the thrust between the four rotors gives the vehicle the ability to move in six degrees of freedom (6DOF). The 6DOF are yaw, pitch, roll (attitude angles), x, y, and z (altitude). Figure 1.3 illustrates the possible motions caused by this variation [START_REF] Al-Younes | Establishing Autonomous AUS-Quadrotor[END_REF].

The symmetry of the design allows for a centralization of control systems and payload. The four rotors of a quadrotor provide a larger amount of thrust than a typical helicopter which allows for larger payloads and computing platforms, especially important in UAV applications [START_REF] Al-Younes | Establishing Autonomous AUS-Quadrotor[END_REF]. Several quadrotor setups such as: AscTec Pelican [START_REF] Younes | Quadrotor Position Control Using Cascaded Adaptive Integral Backstepping Controllers[END_REF], Q-ball X4 [START_REF] Zhou | Design of feedback linearization control and reconfigurable control allocation with application to a quadrotor UAV[END_REF], X4-flyer [START_REF] Guenard | A practical visual servo control for an unmanned aerial vehicle[END_REF], OS4 [START_REF] Bouabdallah | Towards intelligent miniature flying robots[END_REF], STARMAC [START_REF] Hoffmann | The Stanford testbed of autonomous rotorcraft for multi agent control (STARMAC)[END_REF], Pixhawk [START_REF] Meier | Pixhawk: A system for autonomous flight using onboard computer vision[END_REF] and other well-known systems had been designed for various purposes including academic, commercial and military purposes.

These quadrotors were the stars for researchers in their works, where different papers have been published based on the experimental data obtained from them.

Recently, several countries are studying the feasibility of deploying quadrotors for the application of food and parcel delivery. The quadrotors are relatively cheap and easy to fly, thus making them the best choice when it comes to testing different control strategies on a UAV. Next, surveys about different control algorithms are presented.

Quadrotor Control

Different control algorithms have been applied to the unmanned quadrotor vehicles. The challenging nature of the system attracted the researchers to implement various control strategies to deal with the nonlinearities within the system dynamics. During the last decade, the designers proposed many linear and nonlinear control techniques to control the attitude and/or the position of the quadrotor. Various classical and optimal linear control techniques and nonlinear control algorithms have been implemented in many references [START_REF] Hoffmann | The Stanford testbed of autonomous rotorcraft for multi agent control (STARMAC)[END_REF][START_REF] Bouabdallah | Design and control of quadrotors with application to autonomous flying[END_REF][START_REF] Bouadi | Sliding Mode Control based on Backstepping Approach for an UAV Type-Quadrotor[END_REF][START_REF] Fowers | Stabilization and control of a quad-rotor micro-UAV using vision sensors[END_REF][START_REF] Petersen | Autonomous Hovering with a Quadrotor Helicopter[END_REF][START_REF] Raffo | An integral predictive/nonlinear H∞ control structure for a quadrotor helicopter[END_REF].

The authors in [START_REF] Hoffmann | The Stanford testbed of autonomous rotorcraft for multi agent control (STARMAC)[END_REF][START_REF] Bouabdallah | Design and control of quadrotors with application to autonomous flying[END_REF][START_REF] Petersen | Autonomous Hovering with a Quadrotor Helicopter[END_REF][START_REF] Fowers | Stabilization and Control of A Quad-Rotor Micro-UAV Using Vision Sensors[END_REF], controlled various quadrotor vehicles using the classical control algorithms, such as PID and LQR techniques. In the case of quadrotor hovering and slow velocity flights, the linearization of the nonlinear system is acceptable and reasonable results are attained.

The linearization of a highly nonlinear model degrades the controlling performance, and in such a situation, the linear control algorithms fail to control the vehicle at points rather than the operating point. Specifically, when the quadrotor asked to do aggressive maneuvers, e.g. high velocity flights and heave flight actions (aerobatic moves, rescue missions, food and parcel delivery, … etc.), then the aerodynamic effects will have an influence on the quadrotor dynamics, and this will lead to a degraded control performance or even instabilities [START_REF] Zhang | A Survey of Modelling and Identification of Quadrotor Robot[END_REF]. A study about the aerodynamics influence on the quadrotor model and control is conducted by [START_REF] Huang | Aerodynamics and control of autonomous quadrotor helicopters in aggressive maneuvering[END_REF]. It is shown that the existing techniques achieved inaccurate trajectory tracking performance by considering the aerodynamic effects.

This pushes toward developing alternative methodologies to control the nonlinear model of the quadrotor while maneuvering aggressively and flying in harsh environments. The nonlinear control techniques, such as Backstepping and Sliding Mode algorithms, illustrate a great deal in stabilizing the quadrotor system in the presence of perturbations [START_REF] Bouabdallah | Design and control of quadrotors with application to autonomous flying[END_REF]. Others used various nonlinear control techniques to control the quadrotor, such as: Hybrid Backstepping and Sliding-Mode control [START_REF] Bouadi | Sliding Mode Control based on Backstepping Approach for an UAV Type-Quadrotor[END_REF][START_REF] Colorado | Miniquadrotor attitude control based on Hybrid Backstepping & Frenet-Serret theory[END_REF]. However, implementing the nonlinear control techniques requires a solid and complete study of the system nonlinearities, where extracting the exact model of the system is a crucial and almost an impossible task to be achieved. In addition, faults and external disturbances could lead to partial or complete changes in the system's dynamics.

Complex mathematical models are needed to fully account for the nonlinearities of a quadrotor vehicle. This problem, combined with the unmodeled uncertainties and disturbances of such system, prove to be a challenging task when trying to achieve robustness of a controller. The authors of [START_REF] Hoffmann | The Stanford testbed of autonomous rotorcraft for multi agent control (STARMAC)[END_REF][START_REF] Bouabdallah | Design and control of quadrotors with application to autonomous flying[END_REF][START_REF] Bouadi | Sliding Mode Control based on Backstepping Approach for an UAV Type-Quadrotor[END_REF][START_REF] Fowers | Stabilization and Control of A Quad-Rotor Micro-UAV Using Vision Sensors[END_REF] discuss several nonlinear techniques that target this challenge, such as H-infinity control, model predictive control and nonlinear feedback control. Even though these techniques yield reasonable outcomes, they are still considered challenging.

Recently, M. Fliess and C. Join [START_REF] Fliess | Model-free control and intelligent PID controllers: towards a possible trivialization of nonlinear control?[END_REF] introduced a remedy to overcome the system uncertainties and the modeling errors in control for a class of Single-Input-Single-Output (SISO) systems. To achieve this task, they proposed the Model-Free Control (MFC) algorithm. The continuous updating nature of the input-output behavior, characterized by the ultra-local-model, anticipates the unmodeled dynamics and system uncertainties, which makes the control possible even in the presence of disturbances and some actuator faults of low magnitudes.

MFC is a complement control algorithm that works with the system feedback controller. In [START_REF] Fliess | Model-free control and intelligent PID controllers: towards a possible trivialization of nonlinear control?[END_REF] the intelligent-PID controller (iPID) is presented as the ultimate desired combination between the classical PID controller and the MFC to handle the system uncertainties. Therefore, the system should not be considered as a black box, where a general knowledge about the system dynamics should be provided to implement the feedback controller with its suitable gains. MFC is used in many solid applications such as shape memory alloys [START_REF] Gédouin | A new control strategy for shape memory alloys actuators[END_REF], DC/DC converters [START_REF] Michel | Model-free control of dc/dc converters[END_REF], an active magnetic bearing [START_REF] De Miras | Active magnetic bearing: A new step for model-free control[END_REF] and a two-dimensional planar manipulator [START_REF] Madonski | Model-free control of a two-dimensional system based on uncertainty reconstruction and attenuation[END_REF].

In the literature, the use of MFC is linked to the iPID control technique, where the MFC is always related to the linear PID controller. The use of iPID showed a significant improvement in control performance compared to the classical PID controller, and this is due to the capability of MFC in estimating the system uncertainties and the modeling errors [START_REF] Fliess | Model-free control[END_REF].

MFC algorithm is utilized in this work and augmented with the other control strategies such as LQR and Backstepping control techniques. The work is proposed and applied on the quadrotor system in Chapter 3.

Problem Statement

The goal of this work is to design and implement novel Fault-Detection-and-Diagnosis techniques and Active Fault-Tolerant-Control strategies to accommodate for different actuator and sensor fault scenarios that can affect the performance of the quadrotor vehicle or cause its crash. Moreover, robust control strategies are introduced in this thesis by proposing algorithms capable to reject the disturbances and to passively accommodate for the faults under certain limitations.

Thesis Contributions

The contributions of this thesis are summarized as follows:

Quadrotor control:

1. The Model-Free Control technique (MFC) is used in the quadrotor's control, where different combinations between the MFC with linear and nonlinear control algorithms are proposed to compensate for the unmodeled dynamics and the uncertainties of the quadrotor vehicle. The reason of presenting the methods of ( 6) and (7) in Appendix A and B, respectively, is that the results are obtained using different quadrotor platform (AscTec Pelican quadrotor). Therefore, the confusion between the different quadrotors' setups will be avoided, where the Qball-X4 is the base platform that is used in this thesis.

Fault Estimation:

1. The estimation process for the actuator faults is obtained by proposing a new methodology that helps in providing a better estimation of the fault magnitude based on the MF technique used in the intelligent estimators' design.

2. The estimation of the proposed method is compared with the fault estimation achieved by SVD approach through experimental data. 4. The sensor fault estimation is done through a proposed structured algorithm that senses the maximum error deviation between the output estimations from the proposed estimator and the measured outputs.

Fault-Tolerant Control:

1. Different FTC schemes are implemented to compensate for sensor and actuator faults, where the estimation of the fault magnitude is integrated into the control law to compensate for the actuator fault, while the estimation of the sensor bias is compensated by re-generating the desired trajectory of the quadrotor.

2. Different video clips that show the real-time implementation of the proposed algorithms are captured and uploaded in following YouTube link: (Flight Tests Link)

3. An FDD and FTC strategies are presented based on the hardware redundancy of three altitude sensors used on the quadrotor vehicle. The strategy is verified experimentally through real-time flight results and presented in Appendix-C.

Thesis Structure

After introducing the objectives and the motivations of this work in Chapter 1 and presenting the related work done in the control, FDD, and FTC fields, a breakdown of the thesis structure that contains another five chapters will be presented next.

Before presenting the different methodologies in this thesis, a closer look at the nature of the flying machine (quadrotor vehicle) and its dynamics are presented in Chapter 2. This chapter aims to extract the quadrotor model, where the transformations between the coordinate systems, the dynamics and the kinematics of the system are presented in its nonlinear and linearized formats. Specifically, the Qball-X4 quadrotor model with its assumption and information are extracted to be the basic flying machine that is used in the real-time implementations of the proposed methodologies in this thesis.

The need of having a robust and reliable control algorithm is the main objective of In this chapter, a set of nonlinear equations that represents the motion of the quadrotor vehicle is extracted. The derivation is based on the force-moment dynamics and kinematics. This derivation has been successfully studied and applied in various quadrotor control designs [START_REF] Zhang | A Survey of Modelling and Identification of Quadrotor Robot[END_REF].

The following assumptions are used in modeling the quadrotor [START_REF] Bresciani | Modelling, identification and control of a quadrotor helicopter[END_REF]:

 The structure of the quadrotor is considered to be rigid.

 The quadrotor frame is symmetrical.

 The CoG is coincided with the origin of the body-fixed frame.

In this chapter, the frames and notations used in the quadrotor model will be defined, and the transformations between the different frames will be presented in section 2.1. In section 2.2, the kinematics and dynamics of the quadrotor are presented, and then the nonlinear equations of motion and the linearized equations of the quadrotor model are extracted in sections 2.3 and 2.4, respectively. The actuator dynamics is considered in this work and presented in section 2.5. In section 2.6, the overall structure of the quadrotor model is depicted and discussed, and finally the Qball-X4 quadrotor model used in this projected is presented in section 2.7.

Frames and Transformations

In this section, the frames and notations used in the quadrotor model will be defined, and the transformations between the defined frames will be presented. The description of various frames is important for the following reasons [START_REF] Al-Younes | Establishing Autonomous AUS-Quadrotor[END_REF]:

Reference Frames

 Aerodynamic forces and torques are applied in the BF.

 The measurements of the rate gyros and accelerometers are with respect to the BF.

 GPS and Magnetometer measurements are correlated to the EF.

 Path trajectories are provided in the EF.

Therefore, the transformation between the coordinate systems will be represented next.

Transformations

Rotation matrix

Forces and moments on the quadrotor vehicle are represented in the BF. Controlling the vehicle is performed with respect to the EF. Therefore, any vector in BF should be transformed to the EF by using the rotation matrix (𝑅 𝑏 𝑒 ) .

X e = 𝑅 𝑏 𝑒 X b (2.1)
where X b = [𝑥 𝑏 , 𝑦 𝑏 , 𝑧 𝑏 ] 𝑇 and X e = [𝑥 𝑒 , 𝑦 𝑒 , 𝑧 𝑒 ] 𝑇 are the position vectors in BF and EF respectively.

The rotation matrix can be found using Euler angles (𝜙, 𝜃, 𝜓) with rotation arranged in 3-2-1 order [START_REF] Pettersen | Nonlinear Control Approach to Helicopter Autonomy[END_REF].

The sequence of rotation begins with the rotation about z-axis, y-axis, and x-axis, respectively, as follows:

𝑅 𝑏 𝑒 (𝑧) = [ cos(𝜓) -sin (𝜓) 0 sin (𝜓) cos(𝜓) 0 0 0 1 ] 𝑅 𝑏 𝑒 (𝑦) = [ cos(𝜃) 0 sin (𝜃) 0 1 0 -sin (𝜃) 0 cos(𝜃) ] 𝑅 𝑏 𝑒 (𝑥) = [ 1 0 0 0 cos(𝜙) -sin (𝜙) 0 sin (𝜙) cos(𝜙) ] (2.2)
Because of the orthonormality between the rotations given in (2.2), the translational transformation matrix (𝑅 𝑏 𝑒 ) can be extracted as follows: where sec(𝜃) = 1/ cos(𝜃).

𝑅 𝑏 𝑒 = 𝑅 𝑏 𝑒 (𝑥) 𝑅 𝑏 𝑒 (𝑦) 𝑅 𝑏 𝑒 (𝑧) (2.3) = [ c (

Quadrotor Dynamics and Kinematics

Force Dynamics and Kinematics of the Translational System

First of all, an overview of the forces applied on the quadrotor will be represented to understand their influence on the system:

Rotor Lift Force

As the propellers that are mounted on the rotors start to spin, the air will be pushed downward, and this will generate a vertical aerodynamic lift force on the z-axis of the BF (𝑧 𝑏 ).

The total lifting forces from the four rotors are responsible for the motion of the quadrotor on the different axes.

Gravity

The weight of the quadrotor is the down force that is passing the Center of Gravity (CoG)

and point towards the ground in the direction of the negative z-axis of the EF (-𝑧 𝑒 ).

Ground Effect

The lift force generated by the rotor near the ground will experience a better reaction from the ground. The variation in the lift force with respect to the ground is called the ground effect. In this work, the ground effect is neglected where the altitude of the quadrotor is vary within a small range.

The forces applied on the quadrotor by the actuators and the gravity according to EF can be represented as follows [START_REF] Craig | Introduction to robotics: mechanics and control: Pearson/Prentice Hall[END_REF]:

𝐹 𝑒 = 𝑅 𝑏 𝑒 . ∑ ( 4 𝑖=1 𝐹 𝑖 ) -𝑚 𝑔 𝑒 (2.6)
where 𝐹 𝑖 is the magnitude of the lift force from the i th rotor with respect to the BF, 𝑔 𝑒 = [0 0 9.81] 𝑇 is the gravitational acceleration and m is the quadrotor's mass.

According to the Newton's second law:

𝐹 𝑒 = 𝑚 𝑎 𝑒 (2.7)
where;

𝑎 𝑒 = [𝑥̈𝑒, 𝑦̈𝑒, 𝑧̈𝑒] 𝑇 (2.8)
From equations (2.5) and (2.6), the translational equations of motion on the EF:

𝑎 𝑒 = 1 𝑚 𝑅 𝑏 𝑒 . ∑ ( 4 𝑖=1 𝐹 𝑖 ) -𝑔 𝑒 (2.9)
Equation (2.9) can be rewritten in the three axes of the EF. Therefore, the dynamics of the translational motions are: -𝑚𝑔 ]

(2.10)

By using the kinematics of a rigid body, the velocity and position can be described by integrating the linear acceleration in (2.8), as follows: Hence, equation (2.12) can be written as:

𝑣 𝑒 = [𝑥̇𝑒, 𝑦̇𝑒, 𝑧̇𝑒] 𝑇 = ∫ 𝑎 𝑒 𝑑𝑡 𝑋 𝑒 = [𝑥 𝑒 , 𝑦 𝑒 , 𝑧 𝑒 ] 𝑇 = ∬ 𝑎 𝑒 . 𝑑𝑡 = ∫ 𝑣 𝑒 𝑑𝑡 (2.11)

Moment Dynamics and Kinematics of the Rotational System

𝑀 = Iω̇b + ω b × I. ω b Iω̇b = 𝑀 -ω b × I. ω b (2.15)
where ω̇b = [ṗ q̇ r] T .

Because of the symmetry of the quadrotor structure [START_REF] Bouabdallah | Design and control of quadrotors with application to autonomous flying[END_REF] and the small values of the non-diagonal inertias in I, the inertia matrix in (2.14) can be written as follows:

I = [ I xx 0 0 0 I yy 0 0 0 I zz ] (2.16)
Therefore, the rotational equations of motion can be written as follows: where 𝐽 is the rotor's moment of inertia, and

I xx 𝑝̇=
(Ω r = Ω 2 + Ω 4 -Ω 1 -Ω 3 ) .
The kinematics of the rotational system can be obtained by integrating the angular acceleration. Where, the angular velocities about BF axes are:

ω b = ∫ ω̇b 𝑑𝑡 (2.19)
The Euler rates about EF axes can be found using the transformation matrix, in equations (2.3) and (2.5).

Consequently, the integration of the Euler rates will yield to the Euler angles:

Θ = ∫ Θ ̇ 𝑑𝑡 (2.20)

Nonlinear Equations of Motion

All in all, the equations that represent the motion of the quadrotor are: (2.21)

The inputs of the system can be denoted as follows:

𝑢 1 = ∑ 𝐹 𝑖 4 𝑖=1 𝑢 2 = 𝑀 𝑥 = 𝑙(𝐹 4 -𝐹 2 ) 𝑢 3 = 𝑀 𝑦 = 𝑙(𝐹 1 -𝐹 3 ) 𝑢 4 = 𝑀 𝑧 = 𝑑(𝐹 2 + 𝐹 4 -𝐹 1 -𝐹 3 ) (2.22)
where l is the arm length, and d is the rotor's reaction torque constant.

In addition to the work done in [START_REF] Al-Younes | Linear vs. nonlinear control techniques for a quadrotor vehicle[END_REF], the translational and rotational drag coefficients are considered as presented in [START_REF] Bouadi | Sliding Mode Control based on Backstepping Approach for an UAV Type-Quadrotor[END_REF]. In general, the equations that represent the motion of the quadrotor are given below: where 𝑐 ≡ cosine, 𝑠 ≡ sine, 𝑘 𝑑(𝑥,𝑦,𝑧) represent the translational drag coefficients, 𝑘 𝑑(𝑝,𝑞,𝑟) represent the rotational drag coefficients, and (x, y, z) = (𝑥 𝑒 , 𝑦 𝑒 , 𝑧 𝑒 ).

Linearization of the Quadrotor Model

Beside the nonlinear algorithms used in this work, different linear algorithms are presented in control, fault detection and diagnostic and fault-tolerant control. Therefore, the nonlinear model will be linearized around an operating point to apply the different algorithms as required.

The equations of motion in (2.23) can be simplified and linearized, as shown in (2.24), for small perturbations around the hovering point, where the change in angles are considered to be small [START_REF] Zhang | Fault tolerant flight control techniques with application to a Quadrotor UAV testbed[END_REF].

[ 𝜙 θ ψ ẍÿ z ̈] = [ 𝑢 2 I xx ⁄ 𝑢 3 I yy ⁄ 𝑢 4 𝐼 𝑧𝑧 ⁄ (-𝑘 𝑑x ẋ+ 𝜃 𝑢 1 ) 𝑚 ⁄ (-𝑘 𝑑y ẏ-𝜙 𝑢 1 ) 𝑚 ⁄ (-𝑘 𝑑z ż-𝑚𝑔 + 𝑢 1 ) 𝑚 ⁄ ]
(2.24)

Actuator Dynamics

Each rotor is composed of an Electronic Speed Controller (ESC), brushless motor and a Propeller.

ESCs are operated by Pulse Width Modulation (PWM) signals from the controller.

The PWM signal for a corresponding rotor (𝑖 = 1 … 4) is denoted by 𝑢 𝑖 * . Experimentally, the thrust generated by each rotor is modelled using the first-order equation that relates the thrust to the PWM signal provided by a given rotor:

𝐹 𝑖 = 𝜅 𝑎 𝑤 𝑠+𝑤 𝑢 𝑖 * (2.25)
In (2.25), 𝑤 is the actuator bandwidth and 𝜅 𝑎 is a positive actuator gain.

The relation between the thrust generated by each rotor and the propeller's angular speed is discussed in [START_REF] Pounds | Towards dynamically-favourable quad-rotor aerial robots[END_REF], where it can be presented in the following relation:

𝐹 𝑖 = 𝑏Ω 𝑖 2 (2.26)
where b is the proportional constant between the thrust and the angular speed of the rotors. Then the equation in (2.22) can be written as follows:

𝑢 1 = ∑ 𝐹 𝑖 4 𝑖=1 = 𝑏. ∑ Ω 𝑖 2 4 𝑖=1 𝑢 2 = 𝑙(𝐹 4 -𝐹 2 ) = 𝑏𝑙(Ω 4 2 -Ω 2 2 ) 𝑢 3 = 𝑙(𝐹 1 -𝐹 3 ) = 𝑏𝑙(Ω 1 2 -Ω 3 2 ) 𝑢 4 = 𝑑(𝐹 2 + 𝐹 4 -𝐹 1 -𝐹 3 ) = 𝑏𝑑(Ω 2 2 + Ω 4 2 -Ω 1 2 -Ω 3 2 ) (2.27)
where l is the quadrotor's arm length, and d is the rotor's reaction torque constant.

The relation between the rotor's angular speeds (Ω 𝑖 ) and the system inputs (𝑢 𝑖 ) is needed to compute the propeller gyroscopic effect in (2.18). From (2.27) the rotor's angular speeds can be written as a function of the system inputs as shown below:

[

Ω 1 2 Ω 2 2 Ω 3 2 Ω 4 2 ] = [ 1 4𝑏 0 1 2𝑏𝑙 - 1 4𝑏𝑑 1 4𝑏 - 1 2𝑏𝑙 0 1 4𝑏𝑑 1 4𝑏 0 - 1 2𝑏𝑙 - 1 4𝑏𝑑 1 4𝑏 1 2𝑏𝑙 0 1 4𝑏𝑑 ] [ 𝑢 1 𝑢 2 𝑢 3 𝑢 4 ]
(2.28)

Overall Structure of the Quadrotor Model

The overall structure of the quadrotor model is illustrated in Figure 2.3 [START_REF] Zhang | Fault tolerant flight control techniques with application to a Quadrotor UAV testbed[END_REF]. It is consist of three parts:

 Actuators Dynamics that are commanded by PWM signals (𝑢 𝑖 * ) and produce proper thrusts (𝐹 𝑖 ) accordingly.

 Geometry that represents the relation between the thrusts generated by the rotors and the system inputs (moments and lift force) described in (2.22) and (2.27).

 Quadrotor Dynamics & Kinematics that describe the 6-DOF motion of the quadrotor based on the system inputs.

Qball-X4 Quadrotor Model

It is pertinent to discuss the dynamics of the Qball-X4 quadrotor system from the practical/implementation point of view, where it considers some of the terms that are neglected and omitted in the theoretical derivations (e,g, actuator dynamics, PWM signals).

A study of the linearized system model has been provided in [107] and illustrated in this section. The reference frame notations used in the Qball-X4 system are different from the one presented before, as can be seen in Figure 2.4.a. Therefore, the notations will be adjusted to be consistent with the previous presentation, and the equations derived next will be according to the adjusted frame notations as depicted in The values of quadrotor parameters were provided by the Qball-X4 designers [START_REF] Quanser | QUARC® Real-Time Control Software[END_REF],

and can be seen in Table 2-1. The thrust generated by each rotor in (2.25) can be represented as follows:

𝐹 = 𝜅 𝑎 𝜚 (2.29)
The use of a state variable, 𝜚, will be needed to represent the actuator dynamics and is defined as follows:

𝜚 = 𝑤 𝑠+𝑤 𝑢 * (2.30)
The actuator input signal is 𝑢 * , which is the PWM signal, while 𝑤 is the actuator bandwidth and 𝜅 𝑎 is a positive actuator gain.

For the purpose of controlling the quadrotor, a decomposed linearized model is presented by the Qball-X4 developer. In this model, the actuator dynamics and the integral terms of the states are augmented and considered to be used in the feedback controller as discussed in chapter 3.

The control inputs could be written as follows:

𝑢 1 = 𝜅 𝑎 w s+w (u 1 * + u 2 * + u 3 * + u 4 * ) = 𝜅 𝑎 w s+w u t * = 𝜅 𝑎 𝜚 t 𝑢 2 = 𝑙(𝐹 4 -𝐹 2 ) = 𝑙 𝜅 𝑎 w s+w ∆u 2 * = 𝑙 𝜅 𝑎 𝜚 𝜙 𝑢 3 = 𝑙(𝐹 1 -𝐹 3 ) = 𝑙 𝜅 𝑎 w s+w ∆u 3 * = 𝑙 𝜅 𝑎 𝜚 𝜃 (2.31)
The following state space representation describes the dynamics of the roll/pitch variables:

[ 𝜙 φ ρ∅ ] = [ 0 1 0 0 0 𝜅 𝑎 𝑙 I xx 0 0 -𝑤 ] [ 𝜙 𝜙 ρ∅ ] + [ 0 0 𝑤 ] ∆𝑢 2 *
(2.32)

[ 𝜃 θ ρθ ] = [ 0 1 0 0 0 𝜅 𝑎 𝑙 I yy 0 0 -𝑤 ] [ 𝜃 𝜃 ρ𝜃 ] + [ 0 0 𝑤 ] ∆𝑢 3 *
where ∆u 2 * and ∆u 3 * are the difference in the PWM control inputs of the two opposite rotors on x-and y-axes, respectively.

The height model dynamics equation that includes the actuator dynamics can be written as follows:

[ zż ρ 𝑡 ̇] = [ 0 1 0 0 0 𝜅 𝑎 𝑚 0 0 -𝑤 ] [ z zρ 𝑡 ] + [ 0 0 𝑤 ] 𝑢 𝑡 * + [ 0 -𝑔 0 ] (2.33)
where 𝑢 𝑡 * is the PWM signals of total thrust from all rotors. The linear state space equations for the x and y positions are:

[ xẋ ρ ̇𝑡] = [ 0 1 0 0 0 𝜅 𝑎 𝑚 𝜃 0 0 -𝑤 ] [ x xρ 𝑡 ] + [ 0 0 𝑤 ] 𝑢 𝑡 * (2.34) [ yẏ ρ̇𝑡 ] = [ 0 1 0 0 0 -𝜅 𝑎 𝑚 𝜙 0 0 -𝑤 ] [ y yρ 𝑡 ] + [ 0 0 𝑤 ] 𝑢 𝑡 *
The relationship between the torque generated by each motor, τ, and the PWM input signal can be expressed as:

τ = 𝜅 𝑦 𝑢 * (2.35)
where 𝜅 𝑦 is a positive gain and its value can be seen in Table 2-1. Then the yaw motion can be modeled as:

I zz 𝜓 ̈= ∆τ (2.36)
where ∆τ = τ 4 + τ 2 -τ 3 -τ 1 . The yaw dynamics can be represented as follows:

[ 𝜓 ψ ̈] = [ 0 1 0 0 ] [ 𝜓 𝜓 ̇] + [ 0 𝜅 𝑦 I zz ] ∆τ (2.37)
The system dynamics in the state-space representation showed in (2.32) can be rewritten as follows:

[ 𝜙 φ ρ∅ 𝑠 ∅ ̇] = [ 0 1 0 0 0 0 𝜅 𝑎 𝑙 I xx 0 0 0 -𝑤 0 1 0 0 0 ] [ 𝜙 𝜙 ρ∅ 𝑠 ∅ ] + [ 0 0 𝑤 0 ] ∆𝑢 2 * (2.38) [ 𝜃 θ ρθ 𝑠 𝜃 ̇] = [ 0 1 0 0 0 0 𝜅 𝑎 𝑙 I yy 0 0 0 -𝑤 0 1 0 0 0 ] [ 𝜃 𝜃 ρ𝜃 𝑠 𝜃 ] + [ 0 0 𝑤 0 ] ∆𝑢 3 *
The integrator (e.g. 𝑠 𝜃 ̇= 𝜃) will be used in the feedback controller, where the fourth state is added to the state vector, as shown above in (2.38).

Similarly, for the other state dynamics the state-space representation given in (2.33),

(2.34) and (2.37) can be re-written with the augmented integral states.

[

xẋ ρ 𝑡 ṡx ̇] = [ 0 1 0 0 0 0 𝜅 𝑎 𝑚 𝜃 0 0 0 -𝑤 0 1 0 0 0 ] [ x xρ 𝑡 𝑠 x ] + [ 0 0 𝑤 0 ] 𝑢 𝑡 * [ yẏ ρ 𝑡 ṡy ̇] = [ 0 1 0 0 0 0 -𝜅 𝑎 𝑚 𝜙 0 0 0 -𝑤 0 1 0 0 0 ] [ y yρ 𝑡 𝑠 y ] + [ 0 0 𝑤 0 ] 𝑢 𝑡 * [ zż ρ 𝑡 ṡz ̇] = [ 0 1 0 0 0 0 𝜅 𝑎 𝑚 0 0 0 -𝑤 0 1 0 0 0 ] [ z zρ 𝑡 𝑠 z ] + [ 0 0 𝑤 0 ] 𝑢 𝑡 * + [ 0 -𝑔 0 0 ] [ 𝜓 ψ 𝑠 𝜓 ̇̈] = [ 0 1 0 0 0 0 1 0 0 ] [ 𝜓 𝜓 𝑠 𝜓 ̇] + [ 0 𝜅 𝑦 I zz 0 ] ∆τ (2.39)
In the next chapter, the augmented linearized model will be used to control the Qball-X4 quadrotor using a linear control algorithm, which is presented by the developer, and then it will be compared to the proposed nonlinear control algorithms that utilize the nonlinear model presented in (2.23). Also both models (nonlinear and linear models) will be used to design different output-estimator structures in Chapter 4 for fault detection and diagnostic purposes.

Summary

In this chapter, the nonlinear quadrotor model is derived based on the force-moment dynamics and kinematics, which includes the actuator dynamics as well. The model and its linearized version will be applied to various quadrotor control strategies and output estimators' designs in Chapter 3 and Chapter 4.

In particular, the Qball-X4 quadrotor model with its assumption and information provided by the developers are presented in this chapter, where the system will be utilized in the upcoming research work.

Chapter 3

QUADROTOR CONTROL

In this chapter different linear and nonlinear control algorithms are proposed and applied to the Quadrotor system. The Qball-X4 setup will be utilized to implement the algorithms and compare their performance experimentally. The best technique will be used for the entire work of this project.

Recently, the Model-Free Control (MFC) is introduced to compensate for the system uncertainties in control. Introducing the MFC on our control algorithms will be the main contribution in this chapter. The augmentation of the MFC with other controllers yields significant results, and this will be illustrated and presented in this chapter.

In this work, four control algorithms will be applied and tested on the Qball-X4

system. The optimal Linear-Quadratic Regulator (LQR) control, which is provided by the developer of the Qball-X4 system, will be utilized as the baseline controller for comparison purposes. The effect of augmenting the Model-Free Control (MFC) on the LQR control, under the name of LQR-MFC, will be investigated and discussed.

Furthermore, the Nonlinear Integral-Backstepping (NIB) algorithm will be derived and implemented on the system, and finally the NIB will be boosted by adding the MFC to deal with the inaccuracies of the quadrotor model. The latter will be called NIB-MFC.

Verifying the system stability by using the different control algorithms are presented and implemented on the Qball-X4 quadrotor vehicle, and a comparison between LQR, NIB, LQR-MFC and NIB-MFC techniques is provided as well.

The Qball-X4 linear model represented in Section 2.7 will be used to develop and test the linear control algorithms (e.g. LQR and LQR-MFC), and the nonlinear model derived in Section 2.3 will be utilized to synthesize and implement the nonlinear control techniques (e.g. NIB and NIB-MFC).

Quadrotor Model for Control

For control purposes, the state variables vector and the control inputs vector will be defined first:

State variables: the state variables vector will be defined as follows:

𝑥 = [ 𝜙 𝜙 ̇𝜃 𝜃 ̇𝜓 𝜓 ̇x ẋy ẏz z] 𝑇

Control inputs: The control inputs were presented previously in (2.26)

Then the nonlinear model given in (2.23) can be written as:

[ 𝑥̇1 𝑥2 𝑥3 𝑥4 𝑥̇5 𝑥̇6 𝑥̇7 𝑥8 𝑥9 𝑥1 0 𝑥1 1 𝑥1 2 ] = [ 𝑥 2 (-
) c(𝑥 1 ) -c(𝑥 5 ) s(𝑥 1 )) 𝑢 1 ] 𝑚 ⁄ 𝑥 12 [-𝑘 𝑑𝑧 𝑥 12 -𝑚𝑔 + (c(𝑥 3 ) c(𝑥 1 )) 𝑢 1 ] 𝑚 ⁄ ] (3.1)
As shown in (3.1), the general representation of the nonlinear model is 𝑥̇= 𝑓(𝑥, 𝑢).

If the assumption of small perturbation and change in the angles are considered (sin 𝜃 ≈ 𝜃, cos 𝜃 ≈ 1), then the system could be represented in the form of

[ 𝑥̇1 𝑥̇2 𝑥̇3 𝑥̇4 𝑥̇5 𝑥̇6 𝑥̇7 𝑥̇8 𝑥̇9 𝑥̇1 0 𝑥̇1 1 𝑥̇1 2 ] = [ 𝑥 2 [(-
(3.2)
where 𝑥 1𝑑 and 𝑥 3𝑑 are the desired pitch and roll angles, respectively. In the state-space representation in (3.2), the quadrotor system is decomposed into six subsystems: roll, pitch, yaw, x, y and z. Each steered by a control input. The control inputs from the position controllers in x and y will be the desired commanding angles of the pitch and roll, respectively, and for small perturbations in angles the total thrust will be almost the same.

In general, any assumptions could be considered as modeling errors in the system, where the MFC in the proposed algorithm will compensate for these errors.

For the linearized model given in (2.24), the aerodynamics effects could be neglected if the quadrotor moves with slow velocities, which is the case in this study, then the model can be written as follows:

[ 𝑥1 𝑥̇2 𝑥3 𝑥4 𝑥5 𝑥6 𝑥7 𝑥8 𝑥9 𝑥1 0 𝑥1 1 𝑥1 2 ] = [ 𝑥 2 0 𝑥 4 0 𝑥 6 0 𝑥 8 0 𝑥 10 0 𝑥 12 0 ] + [ 0 (1 I xx ⁄ ) 𝑢 2 0 (1 I yy ⁄ ) 𝑢 3 0 (1 I zz ⁄ ) 𝑢 4 0 (𝑢 1 𝑚 ⁄ ) 𝑥 3𝑑 0 (-𝑢 1 𝑚 ⁄ ) 𝑥 1𝑑 0 (1 𝑚 ⁄ ) 𝑢 1 ] + [ 0 0 0 0 0 0 0 0 0 0 0 -𝑔] (3.3)
For the sake of applying the linear feedback controller, the system is decomposed into multi-SISO subsystems. The representations of the Qball-X4 subsystems that consider the rotor dynamics and the integrator of the state variables are presented in Section 2.7

For instance, by comparing the roll angles and its rates (𝑥

2 = 𝜙 ̇, 𝑥2 = 𝜙 ̈) in (3.3)
with the first equation in (2.31), both have the same structure but the later considers the dynamics of the rotors.

The first two rows in (3.3) are:

[ 𝑥̇1 𝑥2 ] = [ 𝑥 2 0 ] + [ 0 (1 I xx ⁄ ) 𝑢 2 ] ; 𝑢 2 = 𝑙(𝐹 4 -𝐹 2 ) (3.4)
The first state-space representation in (2.31) is: For the nonlinear model, the rotors' dynamics will be included in the control inputs, where:

[ ∅ ∅ ρ∅ ] = [ 0 1 0 0 0 𝜅 𝑎 𝑙 I xx 0 0 -𝑤 ] [ ∅ ∅ ρ ∅ ] + [ 0 0 𝑤 ] ∆𝑢 2 * (3.
𝑢 1 = ∑ 𝐹 𝑖 4 𝑖=1 = 𝜅 𝑎 𝜚 𝑡 𝑢 2 = 𝑙(𝐹 4 -𝐹 2 ) = 𝜅 𝑎 𝑙 𝜚 𝜙 𝑢 3 = 𝑙(𝐹 1 -𝐹 3 ) = 𝜅 𝑎 𝑙 𝜚 𝜃 𝑢 4 = 𝑑(𝐹 2 + 𝐹 4 -𝐹 1 -𝐹 3 ) = 𝜅 𝑦 ∆τ (3.10)
Next, in this work the linear control algorithms will be developed based on the Qball-X4 sub-models, and the nonlinear control techniques will be synthesized according to the nonlinear model.

Linear-Quadratic Regulator (LQR) control

In this section, an LQR controller will be implemented for each Qball-X4 sub-model.

Each sub-model can be described by the following LTI equation:

𝑥̇= 𝐴𝑥 + 𝐵𝑢 (3.11)
The equation of the state variable feedback regulator is;

𝑢 𝐿𝑄𝑅 = -𝐾𝑥 (3.12)
where K is the state feedback gain matrix. The optimization procedure involves the determination of the control input u that minimizes the performance criterion or the cost function 𝐽 𝐿𝑄𝑅 . The performance characteristic and the control input optimization will be determined by the cost function 𝐽 𝐿𝑄𝑅 [START_REF] Cen | Robust Fault Diagnosis for Quadrotor UAVs Using Adaptive Thau Observer[END_REF].

𝐽 𝐿𝑄𝑅 = ∫ (𝑥 𝑇 𝑄𝑥 + 𝑢 𝑇 𝑅𝑢) 𝑑𝑡 ∞ 0 (3.13)
The state feedback gain matrix is defined as:

𝐾 = 𝑅 -1 𝐵 𝑇 𝑃 (3.14)
where the matrix P should satisfy the Riccati equation:

𝐴 𝑇 𝑃 + 𝑃𝐴 -𝑃𝐵𝑅 𝑇 𝐵 𝑇 𝑃 + 𝑄 = 0 (3.15)
The presence of the system uncertainties in the actual quadrotor vehicle could lead to a steady-state error. The steady-state error can be compensated by adding the integral action to the LQR control technique. This is can be done by augmenting the model with a state that represents the integral term as shown in (3.9).

Given the augmented system, the normal design of LQR control can be applied to find the state feedback gain matrix that contains the integral gain.

Nonlinear Integral Backstepping (NIB) control

In this part, the derivation of the Nonlinear Integral Backstepping control technique will be presented. The system stability using the NIB will be investigated as well.

The nonlinear model shown in (3.2) can be decomposed into six SISO subsystems,

where each represents a state variable of the 6-DOF states. For instance, the angular acceleration of the roll angle (𝜙 ̈) can be represented by the following nonlinear system:

𝑥̇= 𝑓(𝑥) + 𝑏𝑢 (3.16)
The system for a certain state variable (e.g. 𝜙) can be described as follows:

𝑥 1 = 𝑥 𝑥 2 = 𝑥̇1 (3.17)
The error between the actual and the desired value of that state variable is defined as:

𝑒 1 = 𝑥 1 -𝑥 𝑑 (3.18)
The recursive nature of the backstepping theory ensures, not only the position tracking error convergence to a certain state variable but also the velocity-tracking-error convergence of that state as well. To guarantee the convergence of the errors and the stability of the nonlinear system, the Lyapunov function will be utilized for this purpose.

At the beginning, the Lyapunov function 𝑉(𝑒 1 ) will be selected to be positive definite around the desired position:

𝑉 1 (𝑒 1 ) = 1 2 𝑒 1 2 (3.19)
The derivative of the Lyapunov function can be written as:

𝑉 ̇1(𝑒 1 ) = 𝑒 1 𝑒1 = 𝑒 1 (𝑥 2 -𝑥ḋ) (3.20)
Since, 𝑥 2 is not our control input, there will be a dynamic error between it and its desired value 𝑥 2 𝑑 . Therefore, the velocity tracking error is presented to compensate for the dynamics error:

𝑒 2 = 𝑥 2 -𝑥 2 𝑑 𝑒 2 = 𝑥 2 -𝑥ḋ + 𝑘 1 𝑒 1 (3.21)
The convergence of the error to zero can be fulfilled, if and only if the derivative of the Lyapunov function is semi-negative definite, 𝑉 ̇1(𝑒 1 ) ≤ 0.

For that purpose, the virtual input 𝑥 2 will be chosen as:

𝑥 2 𝑑 = 𝑥ḋ -𝑘 1 𝑒 1 (3.22)

The Integral Action

The steady state error that caused by the modeling error and system uncertainties can be eliminated by augmenting the integral term to the system as following:

𝑥 2 𝑑 = 𝑥ḋ -𝑘 1 𝑒 1 -𝑘 3 ∫ 𝑒 1 𝑑𝑡 𝑒 2 = 𝑥 2 -𝑥̇𝑑 + 𝑘 1 𝑒 1 + 𝑘 3 ∫ 𝑒 1 𝑑𝑡 (3.23)
Consequently, the derivative of the position and velocity tracking errors can be extracted:

𝑒1 = 𝑥1 -𝑥ḋ = 𝑒 2 + 𝑥 * -𝑥ḋ = 𝑒 2 -𝑘 1 𝑒 1 -𝑘 3 ∫ 𝑒 1 𝑑𝑡 𝑒̇2 = 𝑥̇2 -𝑥̈𝑑 + 𝑘 1 𝑒̇1 -𝑘 3 𝑒 1 = 𝑥̇2 -𝑥̈𝑑 + 𝑘 1 𝑒 2 -𝑘 1 2 𝑒 1 -𝑘 1 𝑘 3 ∫ 𝑒 1 𝑑𝑡 -𝑘 3 𝑒 1 (3.24)
The augmented Lyapunov functions and their derivatives, for the position and velocity tracking errors, will be derived as follows:

𝑉 1 (𝑒 1 , ∫ 𝑒 1 𝑑𝑡) = 1 2 𝑒 1 2 + 𝑘 3 2 (∫ 𝑒 1 𝑑𝑡) 2 𝑉 ̇1(𝑒 1 ) = 𝑒 1 𝑒1 + 𝑘 3 𝑒 1 ∫ 𝑒 1 𝑑𝑡 = 𝑒 1 𝑒 2 -𝑘 1 𝑒 1 2 𝑉 2 (𝑒 1 , 𝑒 2 , ∫ 𝑒 1 𝑑𝑡) = 1 2 𝑒 1 2 + 1 2 𝑒 2 2 + 𝑘 3 2 (∫ 𝑒 1 𝑑𝑡) 2 𝑉 ̇2(𝑒 1 , 𝑒 2 ) = 𝑒 1 𝑒1 + 𝑒 2 𝑒2 + 𝑘 3 𝑒 1 ∫ 𝑒 1 𝑑𝑡 = 𝑒 1 𝑒 2 -𝑘 1 𝑒 1 2 + 𝑒 2 (𝑥̇2 -𝑥̈𝑑 + 𝑘 1 𝑒 2 -𝑘 1 2 𝑒 1 -𝑘 1 𝑘 3 ∫ 𝑒 1 𝑑𝑡 -𝑘 3 𝑒 1 ) (3.25)
To ensure the convergence of the velocity tracking error to zero, the derivative of the Lyapunov function should be semi-negative definite, 𝑉 ̇2(𝑒 1 , 𝑒 2 ) ≤ 0. This is can be fulfilled if the following equivalency is achieved:

𝑒 1 𝑒 2 + 𝑒 2 (𝑥2 -𝑥d + 𝑘 1 𝑒 2 -𝑘 1 2 𝑒 1 -𝑘 1 𝑘 3 ∫ 𝑒 1 𝑑𝑡 -𝑘 3 𝑒 1 ) = -𝑘 2 𝑒 2 2 (3.26)
Therefore, 𝑥2 will be chosen to assure the convergence as following:

𝑥2 = 𝑥d + (𝑘 1 2 + 𝑘 3 -1)𝑒 1 -(𝑘 1 + 𝑘 2 )𝑒 2 + 𝑘 1 𝑘 3 ∫ 𝑒 1 𝑑𝑡 (3.27)
The control input for a certain state variable could be extracted by substituting ( It's worth to mention that the global stability for the system will be ensured if and only if all the control subsystems are stable.

Next, in Section 3.4 the Model-Free Control concept will be presented, and in Section 3.5 the MFC will be augmented to the LQR and NIB control algorithms.

Model-Free Control (MFC)

Consider the following n th order nonlinear SISO system:

𝑦 (𝑛) = 𝑓( 𝑦, 𝑦̇, … , 𝑦 (𝑛) ) + 𝑏𝑢 (3.30)
Where, 𝑓(•) is the modeled system dynamics, 𝑢 is the system input, and 𝑏 is unknown input factor.

The unmodeled dynamics and uncertainties with the unknown input factor could be represented in the system as 𝑓 𝑒 (•):

𝑓 𝑒 (•) = Model Uncertainties + (b -𝛽)𝑢 (3.31)
Thus, the system could be written as follows:

𝑦 (𝑣) = 𝑓(•) + 𝑓 𝑒 (•) + 𝛽𝑢 (3.32)
Where 𝑣 is representing the order of the anticipated model, and 𝛽 is the estimate of the unknown scaling factor 𝑏 that is going to be determined by the operator to ensure a certain control performance [START_REF] Madonski | Model-free control of a two-dimensional system based on uncertainty reconstruction and attenuation[END_REF] .

The input-output relation could be represented by an ultra-local model that is continuously restructured:

𝑦 (𝑣) = 𝐹 + 𝛽𝑢 (3.33)
Where, F is a continuously updated value that represents the overall time-varying dynamics of the system (𝑓(•) + 𝑓 𝑒 (•)), and it could be approximated to attenuate the noise produced from the derivative 𝑦 (𝑣) . [START_REF] Fliess | Model-free control[END_REF] 𝐹 = 𝑦 (𝑣) -𝛽𝑢 (3.34)

It is worth mentioning that the value of estimation is valid for a short period of time, and it should be continuously updated [START_REF] Fliess | Model-free control[END_REF]. In general, the model-free control input could be written as follows:

𝑢 = - 𝐹-𝑦 𝑑 (𝑣) +𝑢 𝑐 𝛽 (3.35)
Where, 𝑦 𝑑 (𝑣) is the 𝑣 𝑡ℎ derivative of the desired trajectory, and 𝑢 𝑐 is the control input of the feedback controller.

By substituting (3.35) in (3.34),

𝑦 (𝑣) = 𝐹 + 𝛽 (- 𝐹-𝑦 𝑑 (𝑣) +𝑢 𝑐 𝛽 ) = 𝑦 𝑑 (𝑣) -𝑢 𝑐 (3.36)
This will yield to, 𝑦 (𝑣) -𝑦 𝑑 (𝑣) + 𝑢 𝑐 = 0 (3.37)

𝑒 (𝑣) + 𝑢 𝑐 = 0 (3.38)
Where, 𝑒 (𝑣) is the 𝑣 𝑡ℎ derivative of (𝑒 = 𝑦 -𝑦 𝑑 ), and 𝑢 𝑐 should be selected to lead to a linear differential equation that is asymptotically converged to a desired trajectory.

[101]

The operator has to choose the proper value of 𝑣 according to the system stability and the type of the feedback controller used in the system. For instance, iPID and iPD would use a second order system with 𝑣 = 2, and both iPI and iP will suit a system with 𝑣 = 1.

To make this statement clear, assume 𝑣 = 1, then (3.38) will be written as:

𝑒̇+ 𝑢 𝑐 = 0 (3.39)
A second order differential equation could be extracted, if P or PI controller is implemented:

𝑒̇+ 𝐾 𝑃 𝑒 + 𝐾 𝐼 ∫ 𝑒 𝑑𝑡 = 0 (3.40)
Similarly, if 𝑣 = 2, then a third order differential equation will be expressed, if PD or PID controller is used:

𝑒̈+

Augmentation of MFC in LQR and NIB Controllers

The purpose of this section is to design two controllers that rely on the MFC algorithm in compensating the system uncertainties. Some of these uncertainties are presented because of the assumptions made on the system's model. Furthermore, the modelling errors and the other external disturbances make the control performance of the LQR and NIB controllers inefficient as it will be presented in Section 3.7. Therefore, the use of the MFC technique will be a paramount solution to resolve the control performance issues.

Hence, in this section the MFC will be merged with the LQR control technique and this new combination MFC-LQR will be evaluated later and compared to the LQR controller. On the other hand, a novel approach is proposed to implement the MFC with NIB technique rather than using it with the classical linear controllers. It is true that the NIB needs a model to be designed and implemented, however, the parameters of the model are not accurate, and some modeling errors are expected in the nonlinear model.

Therefore, the use of the MFC-NIB will allow to compensate for these uncertainties as well as for other disturbances and low magnitude faults.

The structure of the MFC works for SISO systems. Decomposing the MIMO system to a group of interconnected SISO systems will solve this issue as presented in Section 2.7. Therefore, the system in this work will be decomposed accordingly to apply the MFC with the nonlinear control algorithm.

Linear Quadratic Regulator ̶ Model-Free Control (LQR-MFC)

After computing the control inputs with the integral terms in (3.29), the structure of LQR controller is suitable to be used with the MFC. Hence, MFCs with 𝑣 = 2 can be directly implemented to the existing LQR controllers and the stability will be satisfied according to equation (3.38) for LQR controller with a reference tracking input and integral action.

Nonlinear Integral Backstepping ̶ Model-Free Control (NIB-MFC)

To describe the dynamics of one state variable in the quadrotor model, equation (3.30) could be written as follows:

𝑥 (𝑣) = 𝑓( 𝑥) + 𝑏𝑢 (3.42)
Where 𝑓(𝑥) is the modeled system dynamics and 𝑢 is the system input. Then equation (3.42) will be:

𝑥 (𝑣) = 𝑓(𝑥) + 𝑓 𝑒 (•) + 𝛽𝑢 (3.43)
To minimize the prediction of the MFC for the unmodeled system dynamics, the known and modeled nonlinear dynamics will be considered in the ultra-local model:

𝑥 (𝑣) = 𝑓(𝑥) + 𝐹 + 𝛽𝑢 𝐹 = 𝑥 (𝑣) -𝑓(𝑥) -𝛽𝑢 (3.44)
The general formulation of the model-free control input that will be used in NIB can be described as:

𝑢 = - 𝐹-𝑥 𝑑 (𝑣) -𝑢 𝑐 𝛽 (3.45)
Where 𝑢 𝑐 is the control input of the nonlinear controller.

+ 𝑘 3 -1)𝑒 1 -(𝑘 1 + 𝑘 2 )𝑒 2 + 𝑘 1 𝑘 3 ∫ 𝑒 1 𝑑𝑡 𝑢 𝑐 = [(𝑘 1 2 + 𝑘 3 -1)𝑒 1 -(𝑘 1 + 𝑘 2 )𝑒 2 + 𝑘 1 𝑘 3 ∫ 𝑒 1 𝑑𝑡] -𝑓(𝑥) (3.48)
To optimize the control input effort, the known input factor 𝑏 could be considered in the computation, where 𝑢 𝑐 = 𝑏𝑢. By that, the resulting control input will be similar to (3.29) and this will lead to the desired convergence of the error to zero and then the required stability of the system.

Figure 3.2 depicts the NIB-MFC scheme that is used for a SISO system. The values of Qball-X4 quadrotor parameters were presented in Table 2-1.

LQR Controller

QUANSER provides a ready-made controller to be downloaded onto the Qball-X4 upon first flight. The Linear-Quadratic Regulator (LQR) controller is mainly suitable for linear systems or linearized systems around a specific operating point. However, the tuning of the LQR control law proves to be challenging and highly dependent on the model of the system [START_REF] Cowling | Direct method based control system for an autonomous quadrotor[END_REF].

Based on the state representation in (3.9), the developers of the Qball-X4 system chose Q and R matrices that achieve optimal control performance and avoid the saturation of the actuators.

For pitch & roll gain matrix

𝑄 𝜃,𝜙 = [ 100 0 0 0 0 1x10 -3 0 0 0 0 22x10 3 0 0 0 0 10 ] , 𝑅 𝜃,𝜙 = [30000]
Then using Matlab™ software, the gain matrix is computed:

 𝐾 𝜃,𝜙 = [0.0616 0.0127 1.1066 0.0183]
Similarly, the gain matrix for position control is found.  𝐾 𝑧 = [0.0067 0.0066 0.1445 0.0022]

For position (x & y)gain matrix

For yaw gain matrix

𝑄 𝑦𝑎𝑤 = [ 1 0 0 0 0.1 0 0 0 0.1 ] , 𝑅 𝑦𝑎𝑤 = [1x103]
 𝐾 𝑦𝑎𝑤 = [0.0362 0.0157 0.0100]

LQR-MFC Controller

In this work, the Qball-X4 quadrotor system is decomposed into multi-SISO subsystems that are linked to each other. Six MFCs are used, each designated to control a certain state in the 6-DOF quadrotor system. The gain 𝛽 for each controller is identified empirically.

Three MFCs are designed to control the altitude (z) and the planar position (x, y) of the quadrotor. Cascaded scheme of the controllers between the translational and rotational motion will be implemented. The output of the planar position controllers will feed two more MFCs for roll and pitch stabilization. In addition, another MFC is designed for yaw control. The control architecture of the quadrotor system is depicted in Figure 3.6a.

NIB Controller

The system is tested in many flight situations until the gains of equation (3.29) have been chosen.

NIB-MFC Controller

Similar to the structure of LQR-MFC, cascaded scheme of the planar position controllers (x, y) and rotational motion controllers (pitch, roll) are implemented as shown in Figure -3.6b, in addition to the yaw and height controllers. Also, the gain 𝛽 for each controller is identified empirically.

Tests and Validation

The purpose of the following tests is to validate the effectiveness of MFC algorithm that is represented by NIB-MFC and LQR-MFC controllers. The performance of the proposed controllers will be tested under normal flight conditions, and also the robustness will be 64 checked when the system is affected by a disturbance. As well as, the behavior of the MFC will be checked when the nominal control performance is degraded.

Normal Flight Conditions Test (Infinity-Shape Path)

In this test, the quadrotor will be tracking a path with an infinity shape as shown in Figure 3.7, where it completes the shape every 20 seconds. Two videos of the infinity-shape flight tests were implemented and uploaded in following YouTube link: (Flight Tests Link), under the names "1. Infinity_Shape_2D" and "2.IInfinity_Shape_3D". The one that considered in this study is the "2.IInfinity_Shape_3D", which includes the variation on the altitude as well.

The Root Mean Square (RMS) of the tracking error between the actual and the desired path is utilized in this work in order to investigate the performance of the different controllers numerically. Table 3-1 shows that adding the MFC algorithm on LQR and NIB controllers has obvious effect in reducing the RMS values of the quadrotor position, which means that MFC controllers (LQR-MFC and NIB-MFC) achieved better tracking performance and were the closest to the reference path compared to the other controllers. The purpose of this test is to examine the robustness of the proposed controllers by this work against the disturbances.

In this test, the quadrotor is commanded to follow a circular path and then a disturbance is injected when the vehicle is at (x = 0, y = 0.5) m. The disturbance pushes the quadrotor toward the center of the circle. A complete circular loop takes 40 seconds, and the disturbance is injected after 10 seconds from the start of the second loop (at t = 80 sec).

The quadrotor will be given 30 seconds to take-off and hover at (x = -0.5, y = 0, z = 0.95) m. There are many reasons to give this pre-flight time, where the gradual take-off needs some time, and the switching between the built-in controller and the proposed controllers will be done where enough time should be provided to hover in the specified location.

For NIB and LQR, both controllers could not handle the disturbance effect. The simulation time presented in Figures 3.11 and 3.13 is between (30-81) seconds, where the first loop and 11 seconds from the second loop are plotted to check the trackability performance before losing the control in the second loop (after injecting the disturbance).

For NIB-MFC and LQR-MFC, the second loop is plotted only between seconds, where both controllers managed to reject the disturbance and continue tracking the path. The reason of showing the second loop only in Figures 3.12 and 3.14 is to provide clear plotting of the system response by avoiding the overlap of the lines from both loops (if considered).

For each control strategy, the closed-loop control performance is tested in the presence of the disturbance. Figures 3.11

LQR Controller:

The disturbance is injected in the second loop at t = 80 sec, where the trackability on the circular path is tested in the first loop. LQR shows a lose control performance and a juggling motion while following the path in the first circular loop. In the second loop and after injecting the disturbance, the controller could not handle it and then an uncontrollable behavior is occurred as shown in the plots of Figure 3.11.

NIB-MFC Controller:

As an ultimate solution, NIB-MFC shows the best performance among the presented control methodologies in terms of robustness, stability and trackability performance. By considering the system nonlinearities within the MFC structure, the system response using NIB-MFC is slightly enhanced compared to the LQR-MFC control strategy. The plots in Figure 3.14 provide an insight to the closed-loop system performance using the NIB-MFC technique.

Control is lost

Control is lost shows the quadrotor following a circular loop in normal condition (without injecting the disturbance), while in the video "4. Circular path with disturbance" the disturbance is injected and compensated as can be watched. Both videos can be seen in the link: (Flight Tests Link).

Degraded Controller Performance Test (Square-Shape Path)

The objective of the experiment is to check the performance of the quadrotor in following a pre-generated square path for LQR and LQR-MFC controllers. The added value of this test compared to the previous tests is to check the effect of augmenting the MFC on a degraded LQR control performance [START_REF] Younes | Model-free control of a quadrotor vehicle[END_REF].

The square path extends from -0.5 m to +0.5 m on the x-axis and -0.5m to 0.5m on the y-axis. The z-axis (height) is kept constant at 0.95m. The quadrotor flies for 10 sec on each side of the square path, and stays 5 sec on each corner. The total real flight time from taking off till landing is 75 sec.

Next, the nominal LQR and LQR-MFC controllers will be implemented and verified.

After that, the degraded performance of LQR controller will be tested and then the effect of adding the MFC on the degraded LQR controller will be examined.

A. LQR Controller:

The system performance of the ready-made optimal controller (LQR) is depicted in

B. LQR-MFC Controller:

-0. As shown above in Figure 3.17, augmenting the MFC in the LQR controller revealed a significant improvement in control performance and trajectory tracking due to the capability of MFC in estimating the system uncertainties.

C. Degraded LQR Controller:

The objective of this experiment is to improve the performance of a badly tuned closedloop system by adding the MFC. A bad selection of the Q and R matrices could lead to unwanted closed-loop system response. To degrade the closed-loop system performance, quantitatively, the gain matrices Ks presented in Section 3.61 are reduced by a given factor (35%), and this led to a poorer performance as shown in Figure 3.18.

D. Degraded LQR+MFC Controller:

The remedy of the degraded controller performance is done by adding the MFC. As shown in Figure 3.19, the MFC compensates for the poor performance of the degraded controller and shows an excellent tracking performance. A comparison between the different controller strategies is depicted in Figure 3.20, where the improvement of adding the MFC can be seen in the 2D-drawing. Despite the fact that the quadrotor system is a highly MIMO nonlinear system, the performance of the controlled system using the MFC is outstanding, this is due to the powerful nature of MFC in compensating the system uncertainties and disturbances. The structured way of implementing the MFC in this work has a crucial effect on the system performance.

Summary

In this chapter, different combinations between the MFC with linear and nonlinear control algorithms are proposed to compensate for the unmodeled dynamics and system uncertainties of the Qball-X4 quadrotor. The stability analysis of using the nonlinear algorithm with the MFC is analytically proven and then verified by real flight tests. The results show the MFC robustness in the presence of disturbances, where the linear and nonlinear controllers (LQR & NIB) failed to compensate and interact with the injected disturbance. Also the MFC is tested by augmenting it to a degraded nominal controller, where it compensates for the system uncertainties resulted from the degraded control performance.

Finally, it can be concluded that NIB-MFC controller shows superior performance among the other control algorithms tested in terms of trackability of the path, stability, and robustness in the presence of the disturbances. To conclude the work of this chapter, the best technique, which is the NIB-MFC, will be used for the entire work of this project.

A comparative study for the presented control strategies against low-magnitude actuator fault will be implemented and presented in Chapter 5. The limits for the fault rejection capabilities of the MFC controllers to larger magnitudes will be investigated in Chapter 5 as well, by proposing an active FTC algorithm that can remove or reduce the effect of the actuator fault on the overall performance of the system. 80 Chapter 4

INTELLIGENT OUTPUT ESTIMATORS' DESIGN

The importance of using different Fault Detection and Diagnosis (FDD) techniques gain the researchers' interest as a prevented solution to protect the real systems from a catastrophic failure. Based on the fault severity level, some of the faults can be tolerable if and only if an action is taken within a short period of time, where detecting, isolating, estimating, and accommodating the fault are crucial processes to prevent the failure of the system.

As discussed in Section 1.2.3, the FDD involves the process of detecting, isolating and estimating the faults. Several FDD approaches for actuator faults have been used, such as the model-based, signal processing-based, and knowledge-based approaches. A model-based observer considers a mathematical model that represents the system dynamics. The real-time computations of the mathematical model will provide an estimation of the actual outputs of the real system. The performance and reliability of the observer depends on how accurate the mathematical model in representing the system dynamics. In reality, extracting an ideal model that imitates the system dynamics is very often hard and challenging task. Even when models exist, the system parameters are never accurate and one has to deal with model uncertainties.

One of the FDD techniques is based on residual generation, where the residuals are the difference between the real measurements and their estimations. In order to detect and isolate a fault, residuals between the measured and the estimated outputs are generated.

Based on a certain pattern of the residuals' symptoms, represented by a Fault Signature Table (FST), a Decision-Making Unit (DMU) is built to decide which component, sensor or actuator is faulty [START_REF] Younes | Sensor fault detection and isolation in the quadrotor vehicle using nonlinear identity observer approach[END_REF] The main objective in this chapter is to propose output estimators that will be used to generate residuals needed for FDD and FTC processes.

In this work, the research is steered towards introducing a technique that is going to be augmented to a linear and nonlinear observers' designs to deal with the inaccuracy found in the mathematical model.

To overcome the presence of model uncertainties when estimating the system outputs, two output-estimator designs will be presented in this work based on the Model-Free Control (MFC) technique developed by M. Fliess and C. Join [START_REF] Fliess | Model-free control[END_REF]. The continuous updating nature of the ultra-local model in the MF scheme makes from it a suitable solution to compensate for the system uncertainties in modelling (unmodeled dynamics, modelling errors, inaccurate system parameters, noises, etc.). In the ultra-local model, the objective is to force the output to track a predefined reference signal despite modeling uncertainties and disturbances.

By recalling what is presented in Chapter 3, MF algorithm had been used in control and applied to SISO systems. The Model-Free Control (MFC) is a complement control algorithm that works with the system feedback controller. In [START_REF] Fliess | Model-free control and intelligent PID controllers: towards a possible trivialization of nonlinear control?[END_REF] the intelligent-PID controller (iPID) is presented as the ultimate desired combination between the classical PID controller and the MFC to handle the system uncertainties.

In this chapter, the MF scheme will be utilized to enhance the estimation of the model-based observer designs. On the one hand, the integrated synthesis of the MF scheme with the state observer will be proposed for MIMO systems. The new formulation will be called, intelligent Output-Estimator (iOE). On the other hand, the intelligent Thau Output-Estimator (iTOE) will be introduced by augmenting the MF technique in Thau observer design. Unlike the observers, the suggested designs of the estimators are used for output estimation and not for state estimation, where the purpose here is to produce the required residuals for FDD.

A comparative study of the different observer/estimator structures will be presented to estimate the outputs of a quadrotor system, where two real-time flight tests will be implemented to validate the proposed algorithms under fault-free and actuator fault conditions.

Before proceeding to the estimator design, recall to the comprehensive study of the MFC is required, where it is presented in details in section 3.2.

Output Estimators' Design

The state vector x of a system model characterizes the future behavior of the model from the past information [START_REF] Oppenheim | Signals, systems, and inference[END_REF]. Predicting the future behavior by estimating the outputs of a system is one of the most important stages for Fault Detection and Diagnostic (FDD).

For a Linear-Time-Invariant (LTI) system model, the real-time simulation will express the behavior of the estimated outputs, but any initial erroneous condition on the states and even slight model uncertainties on the system dynamics may lead to unstable behavior of the simulated model. Therefore, the need of the observer that relies not only on the inputs but also on the outputs is highly demanded for state and output estimations.

Different observer/estimator designs will be presented and evaluated experimentally for this project. Based on MF methodology, an integrated scheme of MF with linear and nonlinear observer structures will be proposed. Where, the state observer design of a MIMO linear system will be presented, and then the iOE will be introduced. Also, for a set of MIMO nonlinear systems, the nonlinear Thau observer design and iTOE will be developed. The four observer/estimator structures will be presented next.

State Observer

A well-known technique used for state estimation of linear systems is the state observer. This model-based observer uses the input values and the output values to formulate the estimation of the model states [START_REF] Oppenheim | Signals, systems, and inference[END_REF].

The LTI can be expressed as:

{ 𝑥(𝑡) = 𝐴𝑥(𝑡) + 𝐵𝑢(𝑡) 𝑦(𝑡) = 𝐶𝑥(𝑡) (4.1)
where 𝑥(𝑡) ∈ ℝ 𝑛 is the state vector, 𝑢(𝑡) ∈ ℝ 𝑚 is the control input vector, 𝑦(𝑡) ∈ ℝ 𝑞 is the system output vector and A, B, and C are the matrices of the continuous system of appropriate dimensions. The time-domain representation (𝑡) will be omitted next for simplicity.

A linear state observer can be designed to estimate the system states and outputs as follows:

{ 𝑥 ̂̇= 𝐴𝑥 ̂+ 𝐵𝑢 + 𝐿(𝑦 -𝐶𝑥 ̂) 𝑦 ̂= 𝐶𝑥 ̂ (4.2)
where 𝑥 ̂∈ ℝ 𝑛 is the estimated state vector, 𝑦 ̂∈ ℝ 𝑞 is the estimated output vector and 𝐿 ∈ ℝ 𝑛×𝑞 is the observer gain matrix.

By comparing the observer scheme in (4.2) to the system in (4.1), the observer mainly consists of two essential parts: the first part simulates the dynamics of the system in (4.1) and the second part feeds the correction between the measured and the estimated outputs (𝑦 -𝑦 ̂) by the amount specified by the observer gain matrix (L).

The error of the states and its derivative can be shown as: The uncertainties in the dynamic model may cause unobservable modes, and this will affect the output prediction of the error. Therefore, it will be hard to extract an optimal observer gain matrix. The problem arises when the unobservable modes are unstable, and this could lead to a divergence of the estimated output. The robustness of the observer is very crucial to avoid false alarms or non-detection in fault-free situation, where this could lead to unwanted behavior or even catastrophic failure for the system.

𝑒 𝑥 = 𝑥 -𝑥 ̂ 𝑒̇𝑥 = 𝑥̇-𝑥 ̂̇ (4.
The purpose of this work is to design a robust output-estimator against the unobservable system dynamics. The MF scheme developed previously will be utilized next to achieve this purpose.

intelligent Output-Estimator (iOE)

The objective is to design a novel output-estimator for a MIMO system that is able to provide an accurate estimation of the system outputs even in the presence of model uncertainties. The proposed technique is inspired from the MFC technique. The novel estimator structure will be as follows:

{ 𝑥 ̂̇= 𝐴𝑥 ̂+ 𝐵𝑢 + 𝐿(𝑦 -𝐶𝑥 ̂) + 𝑢 𝑖𝑂𝐸 𝑦 ̂= 𝐶𝑥 ̂ (4.5)
where 𝑢 𝑖𝑂𝐸 ∈ ℝ 𝑛 is the feedback term from the MF structure that is responsible in estimating the varying system uncertainties.

In this estimator an assumption is made so that all the system's states are measured, and thus matrix C is an identity matrix.

Model-Free Scheme:

The model uncertainties can be expressed by a continuously updated ultra-local model that represents the relation between the input and the output of the estimator. The structure of the ultra-local model will include the system outputs as shown in (4.6) for two reasons: to prove the convergence of the error and to help us choosing the constants of MF matrix Γ using rigorous techniques, while in the literature, for MFC techniques, the MF gain was chosen empirically by the practitioner.

Again this work proposes and shapes the MF scheme for MIMO systems and not only SISO systems as presented in (3.23). Therefore the structure and the parameters of the MF technique will be in matrices form.

The ultra-local model that represents the output-input relationship can be constructed as follows:

𝑦 ̂̇+ Γ𝑦 ̂= 𝐹 𝑖𝑂𝐸 + Γ 𝑢 𝑖𝑂𝐸 (4.6) where Γ ∈ ℝ 𝑛×𝑛 is the Model-Free gain matrix, where here all the system states are considered to be measured, and 𝐹 𝑖𝑂𝐸 ∈ ℝ 𝑛 is a continuously updated matrix that represents the poorly known parts of the system, the unmodeled system dynamics and the various types of disturbances, in another word it represents the overall time-varying dynamics and disturbances on the system [START_REF] Fliess | Model-free control[END_REF].

Due to the continuous updating nature of matrix 𝐹 𝑖𝑂𝐸 , the values of this matrix will suffer from strong fluctuations. Therefore, the need of filtering the values of matrix F using a filter design is necessary to produce the desired estimation of the outputs. A simple low-pass filter or differential filter can be used to attenuate the high oscillation of the updated values of 𝐹 𝑖𝑂𝐸 .

The reason for adding the term Γ𝑦 ̂ in (4.6) is to prove the convergence of the output error to zero when the MF scheme is used for the MIMO systems as it will be shown next.

From the ultra-local model (4.6), matrix 𝐹 𝑖𝑂𝐸 can be written as follows:

𝐹 𝑖𝑂𝐸 = 𝑦 ̂̇+ Γ𝑦 ̂-Γ 𝑢 𝑖𝑂𝐸 (4.7)
Based on the MF scheme, the iOE control input can be written as follows:

𝑢 𝑖𝑂𝐸 = Γ -1 (-𝐹 𝑖𝑂𝐸 + 𝑦̇+ Γ𝑦 -𝑢 𝑠𝑜 ) (4.8)
where 𝑦 is considered our desired trajectory, which is the measured outputs from the actual system, and 𝑢 𝑠𝑜 is the input from the output estimator based on the SO definition: Then, the error convergence will be achieved by choosing the MF gain matrix Γ such that the eigenvalues of (𝐿 -Γ𝐼) are in the left-hand side of the complex plane.

𝑢 𝑠𝑜 = 𝐿(𝑦 -𝐶𝑥 ̂) (4.
In this case, the estimation error is equal to the state estimation error, where this derivation shows that all the states are measured, and matrix C is an identity matrix. This assumption (𝐶 = 𝐼) can be stepped over with slight changes to the structure of the MF scheme to achieve the consistency of the matrices in the derivation, where a relaxed version will be presented next, in case that not all the states are measured.

In this relaxed assumption, the measured outputs will be related to some but not all the states. Therefore a new matrix will be introduced to map the measured outputs to the related states (𝜇 ∈ ℝ q×n ). For example, if a system has four states 𝑥 = [𝑥 1 𝑥 2 𝑥 3 𝑥 4 ] 𝑇 , where two of them are measured, which are 𝑦 1 = 𝑥 1 and 𝑦 2 = 𝑥 3 , then a matrix 𝜇 will be introduced with elements zeros and ones to map the measured outputs to the corresponding states, hence matrix 𝜇 = [ 1 0 0 0 0 0 1 0 ]. The difference between the matrix 𝜇 and C is that matrix 𝜇 will only map the measured outputs to the corresponding states by elements with values "1", where this could not be the case for the C matrix.

Similarly, the mapping between the states that relate to the measured outputs can be achieved by considering the matrix transpose of 𝜇 to avoid the complication resulted from the pseudo inverse operation, and to ensure the mapping between the states and the respective measured outputs.

From that, the estimator structure will be modified as follows:

{ 𝑥 ̂̇= 𝐴𝑥 ̂+ 𝐵𝑢 + 𝐿(𝑦 -𝐶𝑥 ̂) + 𝜇 𝑇 𝑢 𝑖𝑂𝐸 𝑦 ̂= 𝐶𝑥 ̂ (4.12)
where 𝑢 𝑖𝑂𝐸 ∈ ℝ 𝑞 . The Model-Free gain matrix also will have the following dimension Γ ∈ ℝ 𝑞×𝑞 , where similar structure of the ultra-local model given in (4.7) will be used in this assumption.

Based on the MF scheme, the iOE control will have the following structure:

𝑢 𝑖𝑂𝐸 = Γ -1 (-𝐹 𝑖𝑂𝐸 + 𝑦̇+ Γ𝑦 -𝜇𝑢 𝑠𝑜 ) (4.13)
By substituting equation (4.12) and (4.9) into (4.7):

𝐹 𝑖𝑂𝐸 = 𝑦 ̂̇+ Γ𝑦 ̂-Γ Γ -1 (-𝐹 𝑖𝑂𝐸 + 𝑦̇+ Γ𝑦 -𝜇𝑢 𝑠𝑜 ) 𝐹 𝑖𝑂𝐸 = 𝑦 ̂̇+ Γ𝑦 ̂+ 𝐹 𝑖𝑂𝐸 -𝑦̇-Γ𝑦 + 𝜇𝑢 𝑠𝑜 𝑦̇-𝑦 ̂̇= -Γ(𝑦 -𝑦 ̂) + 𝜇𝐿(𝑦 -𝑦 ̂) 𝑦̇-𝑦 ̂̇= (𝜇𝐿 -Γ)(𝑦 -𝑦 ̂) (4.14)
And by defining the estimation error as:

𝑒 𝑦 = 𝑦 -𝑦 ̂ 𝑒ẏ = 𝑦̇-𝑦 ̂̇ 𝑒ẏ = (𝜇𝐿 -Γ)𝑒 𝑦 (4.15)
Then, the error convergence will be achieved by choosing the MF gain matrix Γ such that the eigenvalues of (𝜇𝐿 -Γ) are in the left-hand side of the complex plane.

The new structure can be seen in Figure 4.4 below.

Thau Observer

Thau introduced an asymptotic stable observer that observes the states of a nonlinear system. The nonlinear terms of a plant can be utilized in the observer structure to improve the estimation of the system's states. In this work a design of TO will be presented and then the MF technique will be integrated to enhance the estimation provided by the nonlinear observer.

The nonlinear system considered in this work is represented as follows:

{ 𝑥̇= 𝐴𝑥 + 𝑓(𝑥, 𝑢) + 𝐵𝑢 𝑦 = 𝐶𝑥 (4.16)
As defined in (4.1), 𝑥 ∈ ℝ 𝑛 is the state vector, 𝑢 ∈ ℝ 𝑚 is the control input vector, 𝑦 ∈ ℝ 𝑞 is the system output vector, and 𝑓(𝑥, 𝑢) ∈ ℝ 𝑛 corresponds to the nonlinear terms in the system.

For Thau observer design, the following conditions must be satisfied: the pair (C, A) is observable and the non-linear function 𝑓(𝑥, 𝑢) must be continuously differentiable, which is locally Lipschitz with constant 𝛾 [START_REF] Cen | Robust Fault Diagnosis for Quadrotor UAVs Using Adaptive Thau Observer[END_REF],

‖𝑓(𝑥 1 , 𝑢) -𝑓(𝑥 2 , 𝑢)‖ ≤ 𝛾‖𝑥 1 -𝑥 2 ‖ (4.17)
Then the Thau observer can be represented as follows:

{ 𝑥 ̂̇= 𝐴𝑥 ̂+ 𝑓(𝑥 ̂, 𝑢) + 𝐵𝑢 + 𝐻(𝑦 -𝑦 ̂) 𝑦 ̂= 𝐶𝑥 ̂ (4.18)
where 𝑥 ̂ is the observer state vector, 𝑦 ̂ is the observer output vector. H is the observer feedback gain matrix, which can be computed from equation (4.19),

𝐻 = 𝑃 𝜃 -1 𝐶 𝑇 (4.19)
where 𝑃 𝜃 is the solution of equation (4.20),

𝐴 𝑇 𝑃 𝜃 + 𝑃 𝜃 𝐴 -𝐶 𝑇 𝐶 + 𝜃𝑃 𝜃 = 0 (4.20)

with 𝜃 > 0 such that 𝑃 𝜃 is a positive definite matrix, then the observer states are the estimation of the system states, as proofed by [START_REF] Chen | Robust Model-Based Fault Diagnosis for Dynamic SystemsKluwer Academic[END_REF] where:

intelligent Thau Output-Estimator (iTOE)

For output estimation, neglecting the nonlinear terms by using a linear observer could lead to undesirable estimated values. MF technique in iOE presented a solution to tackle this problem by improving the estimation with the presence of different model uncertainties, while the nonlinear terms are considered within this uncertainty bubble.

Next, the MF technique will be used with the nonlinear TO in iTOE. The nonlinear terms of the system model will be considered in the iTOE structure, and then the estimation will consider the modeling errors resulted from the linearization of the nonlinear system. The question will be raised whether the estimation of iTOE is better than iOE, or the extra computational analysis in iTOE is unnecessary? The answer to this question will be discussed in the implementation section on the quadrotor vehicle.

iTO is a combination of the nonlinear Thau observer and MF technique. The MF is an innovative technique that will be utilized to keep the Thau observer tracking the system states with the presence of the modeling errors in the model-based observer dynamics. Therefore, iTOE will ensure the convergence of the error between the actual and the estimated state to zero.

Similar to the assumptions done in the iOE, first all the states are assumed to be measured, which is the case in the quadrotor system, then the assumption will be relaxed by mapping the measured outputs to the corresponding states.

Based on the observer equation in Thau's form as shown in (4.18), MF algorithm will be augmented so that the output estimator can be written in the following form: where 𝑢 𝑖𝑇𝑂𝐸 ∈ ℝ 𝑛 is the estimation input of the iTOE generated from the MF algorithm.

From Section 3.4, by recalling the ultra-local model given in (3.32), the estimation of the overall time-varying dynamics represented by the continuous updating value F in (3.33) can be minimized by considering the nonlinear terms, so that the ultra-local model in (3.33) can be written as:

𝑦 (𝑣) = 𝐹 + 𝑓(•) + 𝛽𝑢 (4.23)
For estimator design and similar to (4.6), the first order ultra-local model that represents the continuous-updated relation between the input and the output is designed as follows:

𝑦 ̂̇+ Λ 𝑦 ̂= 𝐹 𝑖𝑇𝑂𝐸 + 𝑓(𝑥 ̂, 𝑢) + Λ 𝑢 𝑖𝑇𝑂𝐸 (4.24) where, 𝐹 𝑖𝑇𝑂𝐸 ∈ ℝ 𝑛 is a continuously updated value that represents the model and unmodeled system dynamics in the iTOE, and Λ ∈ ℝ 𝑛×𝑛 is the iTOE gain matrix. 𝐹 𝑖𝑇𝑂𝐸 value in (4.24) can be written as:

𝐹 𝑖𝑇𝑂𝐸 = 𝑦 ̂̇+ Λ 𝑦 ̂-𝑓(𝑥 ̂, 𝑢) -Λ 𝑢 𝑖𝑇𝑂𝐸 (4.25)
According to the MF technique, the estimation input, 𝑢 𝑖𝑇𝑂𝐸 , could be written as follows:

𝑢 𝑖𝑇𝑂𝐸 = Λ -1 (-𝐹 𝑖𝑇𝑂𝐸 + 𝑦̇+ Λ𝑦 -𝑓(𝑥, 𝑢) -𝑢 𝑇𝑂 ) (4.26)
where 𝑢 𝑇𝑂 = 𝐻 (𝑦 -𝑦 ̂) which is the feedback term of Thau observer in (4.18) for a certain state variable.

From equations (4.25) and (4.26):

𝐹 𝑖𝑇𝑂𝐸 = 𝑦 ̂̇+ Λ 𝑦 ̂-𝑓(𝑥 ̂, 𝑢) -Λ Λ -1 (-𝐹 𝑖𝑇𝑂𝐸 + 𝑦̇+ Λ𝑦 -𝑓(𝑥, 𝑢) -𝑢 𝑇𝑂 ) ⇒ 𝑦̇-𝑦 ̂̇= (𝐻 -Λ)(𝑦 -𝑦 ̂) + (𝑓(𝑥, 𝑢) -𝑓(𝑥 ̂, 𝑢)) (4.27)
And then the outputs estimation error will be: 𝑒̇𝑦 = (𝐻 -Λ𝐼)𝑒 𝑦 + (𝑓(𝑥, 𝑢) -𝑓(𝑥 ̂, 𝑢)) (4.28)

If iTOE gain matrix, Λ, is chosen so that (𝐻 -Λ𝐼) is stable and the nonlinear term

𝑓(𝑥, 𝑢) is Lipschitz with constant 𝜂 such that, ‖𝑓(𝑥 1 , 𝑢) -𝑓(𝑥 2 , 𝑢)‖ ≤ 𝜂‖𝑥 1 -𝑥 2 ‖ (4.29)
Then there exists positive definite matrices (P, Q) such that:

(𝐻 -Λ𝐼) 𝑇 𝑃 + 𝑃(𝐻 -Λ𝐼) = -𝑄 (4.30)
According to Lyapunov theorem, the following Lyapunov function will be defined and its derivative will be extracted [START_REF] Hedrick | Nonlinear Observers[END_REF], The condition presented in (4.34) can be maximized by choosing 𝑄 = 𝐼, so that 𝜂 < 𝐼 2𝜆 𝑚𝑎𝑥 (𝑃) .

𝑉
The values in matrix H for all the states will be determined first by designing the TO, and then iTOE will be implemented to adjust the modeling errors between the actual and the observer. iTOE estimator matrix gain is computed to satisfy the convergence of the error in (4.33). The iTOE structure is depicted in Figure 4.5. As discussed in Section 4.1.2, the assumption will be relaxed by introducing a matrix 𝜇 so that the measured outputs will be mapped to the corresponding states. In contrary, the set of the states that are related to the measured outputs will be mapped by taking the transpose of the matrix 𝜇, where only the mapping is needed here.

Therefore, the estimator structure will be modified as follows:

{ 𝑥 ̂̇= 𝐴𝑥 ̂+ 𝑓(𝑥 ̂, 𝑢) + 𝐵𝑢 + 𝐻(𝑦 -𝑦 ̂) + 𝜇 𝑇 𝑢 𝑖𝑇𝑂𝐸 𝑦 ̂= 𝐶𝑥 ̂ (4.35)
where 𝑢 𝑖𝑇𝑂𝐸 ∈ ℝ 𝑞 and 𝜇 ∈ ℝ 𝑞×𝑛 in this case.

The ultra-local model structure shown in (4.24) will be modified as follows:

𝑦 ̂̇+ Λ 𝑦 ̂= 𝐹 𝑖𝑇𝑂𝐸 + 𝜇𝑓(𝑥 ̂, 𝑢) + Λ 𝑢 𝑖𝑇𝑂𝐸 (4.36) where the dimension of continuous updated matrix is 𝐹 𝑖𝑇𝑂𝐸 ∈ ℝ 𝑞 , and the iTOE gain matrix will have the size of Λ ∈ ℝ 𝑞×𝑞 .

On the other hand, the estimation input, 𝑢 𝑖𝑇𝑂𝐸 , from the MF technique could be written as follows:

𝑢 𝑖𝑇𝑂𝐸 = Λ -1 (-𝐹 𝑖𝑇𝑂𝐸 + 𝑦̇+ Λ𝑦 -𝜇𝑓(𝑥, 𝑢) -𝜇𝑢 𝑇𝑂 ) (4.37)

From equations (4.36) and (4.37):

𝐹 𝑖𝑇𝑂𝐸 = 𝑦 ̂̇+ Λ 𝑦 ̂-𝜇𝑓(𝑥 ̂, 𝑢) -Λ Λ -1 (-𝐹 𝑖𝑇𝑂𝐸 + 𝑦̇+ Λ𝑦 -𝜇𝑓(𝑥, 𝑢) -𝜇𝑢 𝑇𝑂 ) ⇒ 𝑦̇-𝑦 ̂̇= (𝜇𝐻 -Λ)(𝑦 -𝑦 ̂) + (𝜇𝑓(𝑥, 𝑢) -𝜇𝑓(𝑥 ̂, 𝑢)) (4.38)
And then the outputs estimation error will be: Then the system will be asymptotically stable if 𝑉 ̇≤ 0 when,

𝑒̇𝑦 = (
𝜂 < 𝜆 𝑚𝑖𝑛 (𝑄) 2𝜆 𝑚𝑎𝑥 (𝑃) (4.45)
The modified structure can be depicted in Figure 4.6 below. For the quadrotor system, all the states are measured therefore the first assumption when (𝐶 = 𝐼) will be used next for both intelligent estimator designs (iOE and iTOE).

Estimator Implementation on the Quadrotor Platform

In this Section, the different observer/estimator designs will be applied to the Qball-X4

Quadrotor Vehicle.

Two flight missions are implemented in this work. In the first flight test, the Quadrotor will be tracking a path with an infinity shape as shown in Figure 4.7.a, where it completes the shape every 20 seconds. The data between t = 50s and t = 70s will be recorded to simulate a complete infinity-shape path. The path varies between (-0.5, 0. During the flight and at t = 75s, a 15% Loss-of-Effectiveness (LoE) actuator fault will be injected into the front rotor. This test will check how the estimators will respond to the fault presence on the system.

Flight Mission-1 (Fault-Free Case)

For a complete infinity loop, between t = 50s and t = 70s, the results in Figures 4. 4.9 show better estimation performance to the actual outputs for the intelligent estimators (iOE and iTOE) compared to SO and TO. The ability of the intelligent estimators in compensating the system uncertainties improve the estimation and anticipate the oscillatory motion of the quadrotor while following the infinity path. The video that shows the quadrotor tracking the infinity-shape path is "2.IInfinity_Shape_3D", (Flight Tests Link). As a result, enhancing the linear and nonlinear observers by augmenting the MF scheme will improve the estimation performance, and this can be numerically compared by calculating the Root-Mean-Square (RMS) error as shown in Table 4-1.

On the other hand, by comparing the RMS values of the intelligent estimators, iTOE shows a slight better performance than iOE at the expense of the computational analysis needed for iTOE. The influence of the system nonlinearities on the quadrotor dynamics was minimal because of the slow tracking motion of the quadrotor. iTOE would be recommended to be used when the quadrotor makes aggressive maneuvers during flight.

Again, one of the objectives of using the MF scheme is to compensate for the system uncertainties including the system nonlinearities. And both iOE and iTOE will do the job with a difference that the iOE has to compensate for all the uncertainties including the known nonlinearities, while the iTOE will consider them. Practically for the quadrotor vehicle, the effects of the noises and external disturbances on the quadrotor dynamics will be dominant over the effect of the system nonlinearities, especially with the slow motion of the rotor craft, and therefore iOE with its simple form is a better solution to be used in this case.

Flight Mission 2 (Actuator Fault Case)

In this test, the intelligent estimators' performance will be validated in the presence of 15% Loss-of-Effectiveness (LoE) actuator fault.

As shown in Figures 4.10 In the middle of the third side of the desired path, in particular at t = 75 sec, a 15%

LoE actuator fault will be injected to check how the intelligent estimators will react in the presence of this type of fault. The fault will be on the front actuator and injected online through Matlab-Simulink.

The NIB-MFC robustness managed to accommodate for the effect of the actuator fault on the system and to return the vehicle to the desired path. A comprehensive comparison between the different control algorithms in the presence of a low-magnitude actuator fault will be presented in Chapter 5.

After injecting the fault, the estimated states from the observers/estimators will be disturbed, but for this special fault case, the observers/estimators will manage to track the actual position states again but an offset of the residuals will be produced for the angular outputs and this is due to the difference between the control inputs before and after the actuator fault. For the front actuator fault, the effect is going to be on the pitch angle, where the residual of the pitch angle will be estimated as shown in Figures 4.10

. FDI scheme

Estimating the system's outputs by implementing an appropriate observer/estimator design.

Residual generation, which is the difference between the measured and the estimated output pair that generates the residual needed to detect the fault.

Residual evaluation: the process of producing symptoms S(r) that are generated by comparing the residuals r to some pre-defined thresholds.

Decision-Making Unit (DMU), which is the process that decides which element is faulty by comparing the fault symptoms to a certain pattern according to a constructed Fault Signature Table .   Fault estimation is the process that computes the magnitude of the fault based on different techniques according to the fault type.

Controllers' Performance against Low-Magnitude Actuator Faults

In this section, an experimental comparison study between the different control algorithms proposed in Chapter 3, will be done against low-magnitude actuator fault. This comparison aims to test the behavior of each control algorithm when a 15% LoE actuator fault is injected to the front rotor, where no Active Fault-Tolerant Control (AFTC) algorithm is used in this section. The objective of this comparison is to select a base controller which shows the best performance in turns of trackability of the desired path and the stability of the quadrotor in the presence of low magnitude actuator faults.

On the other hand, the AFTC for different types and severity levels of actuator faults will be proposed and tested later in Section 5.5.

In this experiment, the quadrotor will be tested to follow a pre-generated square path as shown in Figure 5.2. The total flight time from take-off till landing is 105s. The quadrotor will be given 35s to take-off and hover at (x = 0, y = 0, z = 0.85) m, then the square-shape path will start and last for 70 seconds. For comparison purposes, the data before 35s and after 105s will be omitted in the following figures. As depicted in Figure 5.4, the quadrotor will fly for 10s on each side and will stop 5s on each corner of the desired path. Actuator Fault @ t = 75s [START_REF] Younes | Sensor fault detection and isolation in the quadrotor vehicle using nonlinear identity observer approach[END_REF] The actuator fault will be injected at t = 75s. This fault corresponds to 15% Loss-of Effectiveness (LoE) in the front rotor. The arrangement of the rotors in the Qball-X4 quadrotor is presented in Figure 2.4.

As shown in Figure 5.5, the LQR and NIB controllers have acceptable tracking performance before the fault occurrence. Once the fault happens at t=75s, the quadrotor becomes unstable and crashes. However, the use of the MFC without retuning the parameters of LQR and NIB will allow to compensate for both parameter uncertainties and for the fault and keep the quadrotor flying as it will be shown later. The addition of the MFC to the LQR controller and the NIB controller by keeping the same gain matrices for these controllers allow to drastically improve the tracking results before the fault occurs and keeps the quadrotor flying and get back to its reference after the occurrence of the fault, which shows a kind of fault tolerating capabilities due to the anticipation abilities of the MFC algorithm. It is true that the magnitude of the fault is low, but as said earlier, it made the quadrotor unstable when the LQR and NIB controller are used. In general, the augmented MFC controllers represented by LQR-MFC, and NIB-MFC were able to compensate for the actuator fault unlike their predecessors.

One of the main advantages of using the proposed NIB-MFC algorithm can be seen in Figure 5.6. The actuator fault directly affects the pitch angle and as a consequence the position in the x-axis is affected. The coupling between the x-and y-axes in the quadrotor dynamics causes the vehicle to drift on the y-axis, where the LQR-MFC controller was not quick enough to compensate for it as shown in Figure 5.6b. The effect of the fault can be observed also from the height of the quadrotor, where losing 15% of the efficiency will lead to an adjustment in the total thrust to prevent sliding the vehicle away from the path, this adjustment will descend the quadrotor. As shown in Figure 5.7d, the response of NIB-MFC controller to the change in the height is better than the LQR-MFC controller, where a smoother transition with less oscillation is achieved. NIB-MFC controller shows superior performance among the other control algorithms in terms of trackability of the path, stability, and robustness with the presence of the fault. Handling the modeled nonlinear terms by the NIB and the unmodeled dynamics and modeling errors through the MFC are the reasons behind the outstanding performance of the proposed algorithm. Figure 5.8 shows the 2D and 3D system response using the NIB-MFC controller alone. To compare the performance of the different controllers, RMS errors are calculated between 35s and just before the occurrence of the fault at 75s. As seen in Table 5-1, it is clear that NIB-MFC controller has the lowest values of RMS error compared to the other controllers. Numerically, this implies that the NIB-MFC controller achieved the best tracking performance and was the closest to the reference path. The reliability of the NIB-MFC towards 18% Loss-of-Effectiveness actuator fault is tested, where this percentage found to be the limit to handle the fault using the NIB-MFC. A video record of the flight test can be watched in the YouTube link: (Flight Tests Link) under the name "5. Hover flight _ Actuator fault (Front) _ No fault compensation", also a video of the square-shape flight test is uploaded which has the title "6. Square path _ Actuator fault (Front) _ No fault compensation".

It is true that the NIB-MFC shows superior performance against low-magnitude actuator fault compared to the other controllers, but how the system will react if the fault severity level is increased? And what is the effect of the fault compensation on the system response using the proposed AFTC, which is presented in Section 5.5, compared to the capability shown by NIB-MFC in tolerating the 15% LoE actuator fault. All of these questions will be presented and discussed in Section 5.5.

Anyway, based on the NIB-MFC performance showed in this section, the NIB-MFC will be utilized in this work and used in the experimental procedures next.

Actuator Fault Applied on the Quadrotor

In this section, the effect of the LoE of an actuator will be analyzed and an AFTC technique will be implemented and tested to compensate for constant and variable LoE faults.

In this thesis, two approaches are used to perform the FDI process. The first approach, which presented in Figure 5.2, will be applied to detect and isolate the sensor fault as it will be presented in Section 5.5. This method can be utilized for actuator faults as well, but because of the following reasons, another approach will be used:

 The fault takes some time to be detected as some of the residuals take longer time to be generated than others, where the DMU unit has to wait until the complete set of the residuals' pattern is acquired.

 Unlike the sensor fault, the LoE actuator fault will be tried to be compensated (if possible) by the closed-loop control system, where the NIB-MFC controller shows some tolerating capabilities against the actuator fault within a certain limitation of the its severity level, as presented and tested in Section 5.4. Therefore, the FDD method will not have the time to provide the FTC strategy with the information about the fault, and the controller will deal with the fault in a much shorter time before processing the faulty data. The rapidness of the controller behavior and its effect on the fault compensation will be presented through the real-time flight tests in this section.

The second method directly extracts the information about the actuator fault from the online estimation process presented by the proposed intelligent estimators in Chapter 4. The fault estimation will be fed into the control law directly to compensate for the fault as it will be presented and implemented in this section. This method will be used in the experimental procedures for both fault types. The fault estimation and compensation processes used for the actuator faults are presented next.

Fault Estimation & Compensation for Actuator Faults

The linear system representation that considers an actuator fault is presented in (5.1) and recalled below:

{ 𝑥̇(𝑡) = 𝐴𝑥(𝑡) + 𝐵𝑢(𝑡) + 𝐹 𝑎 𝑓 𝑎 𝑦(𝑡) = 𝐶𝑥(𝑡) (5.3)
where 𝑥(𝑡) ∈ ℝ 𝑛 is the state vector, 𝑦(𝑡) ∈ ℝ 𝑞 is the system output vector, 𝑢(𝑡) ∈ ℝ 𝑚 is the control input vector, 𝐹 𝑎 corresponds to the faulty actuator that is represented by the i th column of matrix B. In most of the cases 𝐹 𝑎 is equal to matrix B and the magnitude of the fault on the i th actuator can be determined by the estimation of fault vector, 𝑓 𝑎 (𝑡). It is worth to mention that the pair (A, B) is controllable and the pair (C, A) is observable.

Next, the analysis of the fault estimation and compensation will be presented using the intelligent-Output-Estimator (iOE) design. By recalling the structure of iOE and by considering all the variable states are measured (n=q), which is the case of the quadrotor system, then:

{ 𝑥 ̂(𝑡) = 𝐴𝑥 ̂(𝑡) + 𝐵𝑢(𝑡) + 𝐿(𝑦(𝑡) -𝐶𝑥 ̂(𝑡)) + 𝑢 𝑖𝑂𝐸 (𝑡) 𝑦 ̂(𝑡) = 𝐶𝑥 ̂(𝑡) (5.4)
𝑢 𝑎 (𝑡) = -𝐵 -1 𝐹 𝑎 𝑓 ̂𝑎(𝑡) (

where 𝐾 𝑎 = 𝐵 -1 𝐹 𝑎 . This compensation strategy will be used to remove the effect of dynamic actuator fault resulted from the LoE actuator faults.

Remark: Matrix B should be a square invertible matrix. In case B is a non-square matrix then equation (5.10) will modified as follows:

𝑢 𝑎 (𝑡) = -𝐵 + 𝐹 𝑎 𝑓 ̂𝑎(𝑡) (5.11)
where 𝐵 + is the pseudo-inverse of matrix 𝐵, where the condition (𝐵𝐵 + = 𝐼) should be satisfied.

If both conditions are not satisfied and matrix B is not of full row rank, where the number of system's states is more than the number of the system's inputs, then the system variables will be prioritized by choosing a set of priority outputs that are related directly to the control inputs and it determines the other secondary outputs. The details of this decomposition is presented in [START_REF] Noura | Fault-tolerant control systems: Design and practical applications[END_REF].

As postulated in the literature, the fault estimation 𝑓 ̂𝑎(𝑡) can be determined from the integral of the output estimation error as follows [7]:

𝑓 ̂𝑎(𝑡) = ∫ 𝐿 𝑒 (𝑦(𝑡) -𝐶𝑥 ̂(𝑡)) 𝑑𝑡 (5.12)

where 𝐿 𝑒 ∈ ℝ 𝑚×𝑞 is fault-estimation gains matrix, which needs to be designed as represented next.

In this study intelligent estimator will be formulated within the structure of the fault estimation given in (5.12) as it has the ability to provide an estimation of the values in the vector 𝑓 ̂𝑎(𝑡) from the estimation term 𝑢 𝑖𝑂𝐸 . This is can be done by augmenting the estimations of iOE in (5.12) as follow:

𝑓 ̂𝑎(𝑡) = ∫ 𝐿 𝑒 (𝑦(𝑡) -𝐶𝑥 ̂(𝑡)) 𝑑𝑡 + 𝐿 𝑒 𝑢 𝑖𝑂𝐸 (𝑡) (5.13)
And by considering; 𝑓 ̂𝑎1 (𝑡) = ∫ 𝐿 𝑒 (𝑦(𝑡) -𝐶𝑥 ̂(𝑡)) 𝑑𝑡, and 𝑓 ̂𝑎2 (𝑡) = 𝐿 𝑒 𝑢 𝑖𝑂𝐸 (𝑡), then the fault estimation 𝑓 ̂𝑎(𝑡) can be written in this format:

[ 𝑒ẋ(𝑡) 𝑓 ̂𝑎̇1 (𝑡) ] = [ 𝐴 -𝐿𝐶 𝐵𝐾 𝑎 𝐿 𝑒 𝐶 0 ] [ 𝑒 𝑥 (𝑡) 𝑓 ̂𝑎1 (𝑡) ] + [ 𝐹 𝑎 0 ] 𝑓 𝑎 (𝑡) (5.19) [ 𝑒ẋ(𝑡) 𝑓 ̂𝑎̇1 (𝑡) ] = ([ 𝐴 𝐵𝐾 𝑎 0 0 ] -[ 𝐿 -𝐿 𝑒 ] [ 𝐶 0 ]) [ 𝑒 𝑥 (𝑡) 𝑓 ̂𝑎1 (𝑡) ] + [ 𝐹 𝑎 0 ] 𝑓 𝑎 (𝑡)
The system can be represented by the following notations: it enters and remains within the bounded set 𝜛, where the set 𝜛 is a small neighborhood of the origin [START_REF] Edwards | Sliding mode control: theory and applications[END_REF].

𝑒(𝑡) = [ 𝑒ẋ (
If the pair (𝐶 ̃, 𝐴 ̃) is observable, and if matrix 𝐿 ̃ is chosen such that there exists a symmetrical positive definite matrix P satisfying the following Riccati equation:

𝑃(𝐴 ̃-𝐿 ̃𝐶 ̃) + (𝐴 ̃-𝐿 ̃𝐶 ̃)𝑇 𝑃 = -𝜃𝜉𝐼 (5.21)
where 𝜃 is a positive number and 𝜉 ≥ ‖𝑓 𝑎 (𝑡)‖, then 𝑒(𝑡) will be within a bounded region around the origin of the equilibrium points 𝑒 𝑥 (0), 𝑥 ̂(0) and 𝑓 𝑎 (0).

For the stability analysis, the following Lyapunov function and its derivative is The compensation strategy presented in this section is used to remove the effect of the actuator fault resulted from the LoE of an actuator in the quadrotor vehicle. Different actuator fault scenarios are implemented and presented next.

Loss of Effectiveness Actuator Fault (Constant % Degradation)

In the following flight tests, different severity levels of the constant percentage LoE actuator faults (15% and 25%) will be injected in the front rotor. As presented in Section 5.4, MFC controllers managed to compensate for the low-magnitude actuator fault represented by 15% LoE of the front rotor. The same fault severity level will be tested on the quadrotor but using the AFTC to check how the active compensation process will improve and boost the system's performance. In the second test a higher LoE fault severity level of 25% will be examined with and without the fault compensation using the AFTC.

In both tests, the FDD and AFTC algorithms will be tested on the quadrotor while it is under hover flight condition and when it follows a square path similar to the one presented in Section 5.4.

As the performance of front rotor will be affected by the fault, the quadrotor will pitch down suddenly causing it to move forward in the x-axis and to lose from its altitude. The controller used in the quadrotor, NIB-MFC, shows robustness against the disturbances and some tolerating capabilities towards the actuator fault but up to a certain extent.

Despite the quick response of the quadrotor vehicle and its closed-loop control system, an active FTC strategy is needed to compensate for higher fault magnitudes (e.g.

25% LoE) and also to boost the overall performance of the quadrotor (e.g. 15% LoE) as presented next.

Hover Flight Test with 15% LoE Actuator Fault:

In this test, the quadrotor will hover at the origin and will fly at a height of (z = 0.95 m).

At t = 30 sec, a 15% LoE actuator fault on the front rotor will be injected. The fault is introduced by adding a step input block using the Simulink® software. The step input is multiplied by the PWM signal that goes to the front rotor and will change its value from 1 to 0.85 at t = 30 sec, as depicted in Figure 5.9 below. Hence, a sliding mode robust differentiator will be introduced next to attenuate the noise level and achieve better estimation for the actuator fault.

Sliding-Mode Robust Differentiator (SMRD)

Different techniques can be used to attenuate the noise produced by the continuously updated regime of the ultra-local model in the MF technique. Low-Pass Filters and numerical differentiators were used in the literature for the noise attenuation process as presented in [START_REF] De Miras | Active magnetic bearing: A new step for model-free control[END_REF][START_REF] Madonski | Model-free control of a two-dimensional system based on uncertainty reconstruction and attenuation[END_REF][START_REF] Fliess | Model-free control[END_REF].

A robust noise reduction technique using sliding mode will be utilized in this work, and it known as robust exact differentiation [START_REF] Rabhi | Robust fuzzy control for stabilization of a quadrotor[END_REF].

Consider the following sliding surface: The FDD algorithm will estimate the control-input values of the system instead of the thrust values of the motors. Therefore, the LoE of the front actuator will be mainly reflected in the control input value of the pitch angle (𝑢 3 ). The degradation will change the difference in the actuators' thrust values between the front and the back rotors as shown in Figure 5.14. The difference can be calculated by subtracting the PWM signal of the front motor from the back motor as presented in (5.33).

PWM Front -PWM Back = 2 𝑢 3 (5.33) Under fault-free conditions the difference between the PWM signals will be almost equal to zero (not exactly zero because of the asymmetry found in the quadrotor structure and the difference in the motors' performance), therefore the control input 𝑢 3 will be fluctuating around zero.

The magnitude of the PWM signals for each rotor is found to be about 0.095.

Degrading the performance of the front motor will decrease its magnitude by 15%. That's mean the new control input 𝑢 3 can be calculated as follows:

0.85(~0.095) -(~0.095) = 2 𝑢 3 𝑢 3 ≈ -0.00713 (5.34)
The value of the real fault magnitude is computed in (5.34) for a 15% LoE front motor fault and can be seen in Figure 5.14. The time evolution of the estimated fault magnitude shows an expected delay because of the following reasons:

 The first reason is that the de-noising process will take some time according to the noise intensity level. Despite that the multi-stage SMRD-LPF shows better performance, with regards to the convergence time and the noise level, but the delay lag for an accepted de-nosing level will be present, as shown in Figure 5.14.

On the other hand, decreasing the convergence time will be at the expense of the noise, where a noisy estimation will cause fluctuation in the control law which may lead to system instability.

 The second reason is that the FDD process will take some time where the estimation will not be fed-back to the control law unless it exceeds a certain threshold, then the compensation of the actuator fault will take place to accommodate for the effect of the fault.

To compensate for the degradation value of the actuator fault presented in (5.34) the controller will adjust the control input by a value equals to the fault magnitude value with a negative sign (~ + 0.00713). Two different flight tests are applied to analyze the control input for the pitch angle (𝑢 3 ) with and without fault compensation as shown in Figure 5.15. Based on the fault compensation technique detailed in (5.13), 𝑢 𝑎 (𝑡) will be computed and then added to the control law as presented in (5.6). When the fault compensation occurs, the control inputs from the controller will converge back to its zero value. For more severe LoE actuator faults and for the ramp-type LoE actuator faults (as it will be presented in Section 5.5.2), the closed-loop system without the AFTC will not be able to accommodate for the fault effect. In those cases the compensation process will work to accommodate for the fault until further reaction is taken by the user or the system for a safe shutdown.

Hover Flight Test with 25% LoE Actuator Fault:

In this test, the effect of the fault compensation is investigated by increasing the severity level of the actuator fault. Another test is conducted by injecting a 25% LoE actuator fault (front rotor) at t = 30s at hover state. At this LoE level the quadrotor lost control without the fault compensation process, while it managed to overcome the effect of the fault when the value of the fault estimation is accommodated. It is important to note that the bias sensor fault in the position of the quadrotor will be seen as a change in the desired reference, where the misleading measurement of the position sensor will be understood by the system as a negative change in the desired output from the current one. Hence, the controller will take the new error and provide an action to follow the incorrect-position.

As well as, the abrupt change in the desired output could be understood as a fault, where a discrepancy between the actual and the estimated output will be introduced and understood by the FDI algorithm as a sensor fault.

Therefore, the structured algorithm will try to avoid such misleading information by checking the desired reference changes within a past short time span from the moment that the sensor fault occurred. If the change is happened and if it is caused by a change in the reference value of the quadrotor position, then a false alarm is avoided and no action is required in this case. Also, a reset signal will be sent to the memory that finds the maximum error as shown in the "Running maximum error" block in Figure 5.29.

estimation performance and the computational analysis needed to do the output estimation for MIMO systems, such as the quadrotor vehicle. According to the fault symptoms and the fault signature table represented in Table 5-3, an abrupt bias sensor fault in the y-position reading is detected and isolated.

𝒇 𝒙 1 0 0 0 1 0 𝒇 𝒚 0 1 0 1 0 0 𝒇 𝒛 0 0 1 0 0 0
After the FDI process, the estimation of the fault magnitude will be presented next.

Sensor Fault Estimation

As discussed in Section 5.6.1, the "Running maximum error" and the "Desired reference change" algorithms will be used to estimate the magnitude of the fault. In this experiment, the "Running maximum error" algorithm calculated a maximum fault of 0.304 m in the past-short-time window from the fault occurrence time. A confirmation from the "Desired reference change" block comes to ensure that no change in the desired value of y-axis is happened at the time of the maximum error, and therefore the estimation is acknowledged and sent to the fault compensation algorithm. It is worth to mention again that the value of estimation will be acquired from the "Running maximum error" algorithm once the fault is detected and isolated. From the conducted flight test, it is found that the FDD algorithm detects, isolates and estimates the bias value after 265ms from the fault occurrence time, which is relatively short. The rapidness of the FDD process will give the system the ability to respond to the fault within a short period of time, so the effect of the fault will be minimal as it is shown in the FTC method next.

Compensation for Sensor Fault

The bias sensor fault can be seen by the controller as a change in the desired reference.

Therefore, adjusting the desired reference at the time the fault occurs will be a suitable solution to compensate for the effect of the fault on the system. This will be done by the online regeneration of the desired path.

Without fault compensation, the quadrotor will be misled by a +0.3 m faulty sensor measurement in the y-axis position. The controller will directly sense the change and tricked to move to the origin again. The measurement will converge to zero, but the actual output will be displaced by 0.3 m in the negative direction as shown in Figure 5.33. The compensation of the fault by the value of the estimation can be seen in Figure 5.34. At the time the fault is estimated, the online regeneration of the desired path is achieved to compensate for the fault before it has the full effect on the system. In 265 ms the quadrotor moved about 5 cm, then the compensation process took place to return the quadrotor to its original position. It is important to mention that the sensor will keep measuring the position with the bias value, but the FTC will do the required compensation to remove the effect of the fault on the actual reading as shown in Figure 5.34. The flight test results validate the proposed algorithms and accommodate for the sensor and actuator faults that are considered in this study. Also a comparative study for the fault estimation values produced from the proposed algorithm in iOE and the SVD approach is conducted and presented for the LoE actuator faults.

For the sensor fault, the iOE estimator design proposed in Chapter 4 is utilized to detect and isolate the faults in the FDI strategy. A systematic procedure is encompassed in the FDI method, which includes the outputs' estimation, residual generation, residual evaluation and the DMU unit. The estimation of the fault magnitude is done through a proposed structured algorithm that senses the maximum error deviation between the output estimations and the measured outputs. Then the fault compensation is achieved by regenerating the desired path.

Chapter 2 is intended to give a closer look to the nature of this flying machine by defining its parameters and the coordinate frames that represent its motion. The dynamics of the quadrotor system is described by deriving the nonlinear model based on the forcemoment dynamics and kinematics study, which includes the actuator dynamics as well.

The extracted nonlinear model and its linearized version have been successfully applied to various quadrotor control strategies and estimators' designs in the following chapters.

In particular, the Qball-X4 quadrotor model with its assumption and information provided by the developers are also presented in Chapter 2, where the system is used in the real-time implementations of the proposed methodologies.

The necessity of proposing a reliable control algorithm to be applied on the For sensor faults (bias-type in the position), the iOE estimator design is utilized to detect and isolate the faults in the FDI strategy. The FDI includes the outputs' estimation, residuals generation, residuals evaluation and the DMU unit. The estimation of the fault magnitude is done through a proposed structured algorithm that senses the maximum error deviation between the output estimations and the measured outputs. Then the fault compensation is achieved by regenerating the desired path.

Future Work

The following recommendations for this work can be carried out in the future:

1. Design an MFC control technique for MIMO systems.

2. Compare the MFC algorithms with other robust control algorithms. e.g. sliding mode control, 𝐻 ∞ , … etc.

3. Compare the power consumption and the computational analysis needed by each control algorithm.

4. Investigate the use of the intelligent estimators and its estimation in compensating for different types of faults that are not presented in this study, where the intelligent estimator could be utilized in estimating not only the actuator fault but also the sensor and the component faults.

Investigate and apply the intelligent estimators' design in different engineering

applications.

Résumé en Français

Le modèle de Qball-X4, qui est le quadrirotor utilisé dans ce projet, est dérivé et décomposé en sous-systèmes tels que présentés ci-dessous:

La formulation générale de l'entrée de commande sans modèle qui est utilisée dans NIB peut être décrite comme:

u = - F-ẍd+u c β ( 7
)
où 𝑢 𝑐 est l'entrée de commande du dispositif de commande non linéaire.

La figure 2 représente le schéma NIB-MFC qui est utilisé pour un système SISO.

Basée sur le schéma MF, l'entrée de commande de l'iES est écrite comme suit:

𝑢 𝑖𝑂𝐸 = Γ -1 (-𝐹 𝑖𝑂𝐸 + 𝑦̇+ Γ𝑦 -𝑢 𝑠𝑜 ) ( 10
La figure 6 montre la structure de l'iES. L'hypothèse (C = I) est franchie avec de légères modifications de la structure du régime MF pour obtenir la consistance des matrices dans la dérivation, où une version relaxée de l'OIE est présentée, dans le cas où tous les états sont mesurés. Par conséquent, une nouvelle matrice μ, est introduite.

La structure de l'estimateur est modifiée comme suit:

{ 𝑥 ̂̇= 𝐴𝑥 ̂+ 𝐵𝑢 + 𝐿(𝑦 -𝐶𝑥 ̂) + 𝜇 𝑇 𝑢 𝑖𝑂𝐸 𝑦 ̂= 𝐶𝑥 ̂ (11)
Le modèle ultra-local et l'entrée de commande de l'iES sont sous les formes suivantes:

𝑦 ̂̇+ Γ𝑦 ̂= 𝐹 𝑖𝑂𝐸 + Γ 𝑢 𝑖𝑂𝐸 (12)
𝑢 𝑖𝑂𝐸 = Γ -1 (-𝐹 𝑖𝑂𝐸 + 𝑦̇+ Γ𝑦 -𝜇𝑢 𝑠𝑜 )

La nouvelle structure est donnée sur la figure 7 ci-dessous :

𝑢 𝑖𝑇𝑂𝐸 = Λ -1 (-𝐹 𝑖𝑇𝑂𝐸 + 𝑦̇+ Λ𝑦 -𝑓(𝑥, 𝑢) -𝑢 𝑇𝑂 )

La structure de l'iETS est représentée par la Figure 8 :

𝑢 𝑖𝑇𝑂𝐸 = Λ -1 (-𝐹 𝑖𝑇𝑂𝐸 + 𝑦̇+ Λ𝑦 -𝜇𝑓(𝑥, 𝑢) -𝜇𝑢 𝑇𝑂 )

La structure modifiée peut être représentée par la figure 9 ci-dessous. From that the estimated output error 𝜀 can be computed. The residual will be extracted from the estimated output error with the fault presence:

𝑟 = 𝐶𝑒 -𝑓 𝑠 (A.20)
It can be noted that the residual reflects the fault indirectly by means of the state error. Despite the presence of the fault, the system will be asymptotically stable, where the stability conditions of the observer are satisfied [START_REF] Adjallah | Non-linear observer-based fault detection[END_REF].

The fault will be detected by relating the residuals to a specific threshold. More work about the fault estimation and its robustness will be investigated in future work.

A.3 Fault Detection and Isolation (FDI)

Structured Residuals: A bank of structured residuals will be designed based on Generalized Observer Scheme (GOS) that introduced by [START_REF] Frank | Fault diagnosis in dynamic systems via state estimation-a survey[END_REF]. In GOS, a set of observers will be designed, in which each observer is fed by all sensor's measurements except one. In contrary, the Dedicated Observer Scheme (DOS) fed only one output measurement into each observer. Each technique has pros and cons, where in DOS-bank multi faults can be detected at a certain time but at the expense of the reliability, where a residual misfire could occur in a free-fault case, and then a wrong fault decision will be taken.

On the other hand, the robustness of GOS-bank due to the structured decisionmaking of the residuals, makes it more reliable than DOS. In GOS, a set of residuals should be fired from different observers at the same time to decide which sensor is faulty.

The decision-making logic will be explained later in the fault signature table. The price of the robustness is paid by detecting a single fault in the GOS scheme, but installing redundant sensors will make it possible to detect and isolate the occurrence of different faults at a time [START_REF] Frank | Nonlinear observers for fault detection and isolation[END_REF].

A bank of GOS is depicted in

A.4 Implementation on the Quadrotor Model

In this section the quadrotor model is considered and the observer will be driven according to the nonlinear identity observer. Due to the variety of sensors used to measure the states of the quadrotor, all states will be measured and observed in the fault detection and isolation technique.

Nonlinear Identity Observer: The Jacobian matrix 𝜕 𝑥 𝑓 can be found as follows:

𝜕 𝑥 𝑓 = [ 0 1 0 0 0 0 0 0 0 0 0 0 0 Θ 1 0 Θ 4 0 Θ 7 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 Θ 2 0 Θ 5 0 Θ 8 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 Θ 3 0 Θ 6 0 Θ 9 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 𝜒 1 0 𝜒 4 0 𝜒 7 0 0 𝑏 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 𝜒 2 0 𝜒 5 0 𝜒 8 0 0 0 0 𝑏 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 𝜒 3 0 𝜒 6 0 0 0 0 0 0 0 0 𝑏 3 ] To render the convergence of the state error and to determine matrix P, a matrix K that represents the null space of C should be defined. According to GOS, all output will be considered but one, then matrix C will consider all the states except one (i.e. except The final step is to compute 𝐹(𝑥 ̂, 𝑢) according to (A.13) and matrix Q that satisfies 𝐶 𝑇 𝑄𝐶 -𝐼 ≥ 0.

Due to the complexity of writing matrix 𝐹(𝑥 ̂, 𝑢), its result will be reflected in the gain matrix ℎ(𝑥 ̂, 𝑢) as is shown later.

Recall the observer equation (A.17), the estimated values will be computed for all sensors except one. Hence, the same procedure is followed with slight modification for the next GOS structure to accommodate all the sensors readings except one.

Summing up the different structures of the GOS, the dynamic gain matrix ℎ(𝑥 ̂, 𝑢) is computed, where it includes all the states. where 𝑥 ̂(𝑡) ∈ 𝑅 𝑛 is the observer state vector, 𝑦 ̂(𝑡) ∈ 𝑅 𝑝 is the observer output vector. K is the observer feedback gain matrix, and is designed according to conditions in Theorem 1. Proof. The details of poof can be seen in [START_REF] Chen | Robust Model-Based Fault Diagnosis for Dynamic SystemsKluwer Academic[END_REF].

In case a fault occurs, the observer will not track the states of system again, the residual 𝑒(𝑡) = 𝑦(𝑡) -𝑦 ̂(𝑡) will deviate from zero indicating the presence of the fault. where the fault estimation error is denoted as 𝑒 𝑓 (𝑡) = 𝑓(𝑡) -𝑓 ̂(𝑡).

203

The purpose of the proposed adaptive Thau observer is not only to detect faults, but also to estimate the fault parameters, which can be used for fault accommodation. where 𝛽 = 𝜆 𝑚𝑎𝑥 (𝛤 -1 ) • 𝑓 2 + 𝜎𝑓 2 , 𝜆 𝑚𝑖𝑛 (•) is the minimum eigenvalue of the matrix.

If the appropriate parameters satisfy 𝜎 -𝜆 𝑚𝑎𝑥 (𝛤 -1 ) > 0, then can be derived as follows:

𝑉 ̇≤ -min[𝜆 𝑚𝑖𝑛 (𝑄), 𝜎 -𝜆 𝑚𝑎𝑥 (𝛤 - This complete the proof.

B.2 Simulation Results

In order to show the performance and effectiveness of the proposed method, the nonlinear model of a quadrotor UAV is simulated in Matlab/Simulink environment. The quadrotor model, which parameters are identified from a AscTec Pelican quadrotor UAV available in the control lab at UAE University, is employed for simulation.

The parameters of the quadrotor can be seen in Table B-1: Also, the observer matrices and parameters can be seen below:

  Also, I would like to thank all of my colleagues in the UAEU_UVL lab at United Arab Emirates University and in the MIS lab at UPJV, (Eng. Ahmad Drak, Eng. Mohammad Muflehi, Eng. Bagus, Dr. Cen, Dr. Merdad, Dr. Dahmani Hamid, and Eng. Ahmed) who have helped me out in my research and created remarkable memories during my study. I also admire the spiritual support of my parents and my sisters and brothers. I really love you all.

  Les algorithmes proposés ont été implantés et testés sur un quadrotor Qball-X4. Les résultats de vol en temps-réel ont été analysés et ont permis de valider les techniques de commande et de tolérance aux fautes de capteurs et d'actionneurs. Des vidéos illustrant différentes expérimentations sont disponibles en ligne. vii ABSTRACT The main objectives of this thesis consist of developing Control, Fault Detection and Diagnosis (FDD) and Fault-Tolerant Control (FTC) techniques based on a the Model-Free (MF) concept recently introduced in the literature. The proposed approaches are implemented, tested and validated on a quadrotor platform. The first step of this work consisted of the modelling of the quadrotor, and then analyzing, designing and implementing new robust control strategies based on the Model-Free Control (MFC) technique recently developed in the literature. The MFC algorithm helps compensating for disturbances and model uncertainties. The advantage of this recent concept is in the simplicity of the design of the controller by adding a control law using ultra-local models to the classical control techniques. To test and validate this new approach, the Linear-Quadratic-Regulator (LQR) and the Nonlinear-Integral-Backstepping (NIB) controllers have been considered by integrating the MFC concept to design a (LQR-MFC) and a (NIB-MFC), respectively. Both algorithms are validated through analytical and experimental procedures and the robustness checked and compared with respect to the initial controllers in the presence of disturbances and model uncertainties.

Figure 1 . 1 . 2 Figure 1 . 2 . 11 Figure 1 . 3 . 21 Figure 2 . 1 . 30 Figure 2 . 2 . 34 Figure 2 . 3 . 39 Figure 2 . 4 . 39 Figure 3 . 1 . 55 Figure 3 . 2 . 58 Figure 3 . 3 . 59 Figure 3 . 4 . 60 Figure 3 . 5 . 61 Figure 3 . 6 . 63 Figure 3 . 7 . 64 Figure 3 . 8 . 65 Figure 3 . 9 . 66 Figure 3 . 66 Figure 3 .Figure 3 .

 11212111321213022342339243931553258335934603561366337643865396636633 Figure 1.1. Disasters caused by a fault in the control system. ............................................ 2 Figure 1.2. FDI Scheme .................................................................................................... 11 Figure 1.3. (a) Total Thrust, (b) Yaw Motion, and (c) Roll/Pitch Motion........................ 21 Figure 2.1. Reference Coordinate Systems. ...................................................................... 30 Figure 2.2. Moments about the axes. ................................................................................ 34 Figure 2.3. Block Diagram of the Quadrotor Model. ....................................................... 39 Figure 2.4. (a) Reference Frame used by the developers (b) adjusted reference frame ... 39 Figure 3.1. MFC scheme ................................................................................................... 55 Figure 3.2. NIB-MFC scheme .......................................................................................... 58 Figure 3.3. Qball-X4 Workspace ...................................................................................... 59 Figure 3.4. (a) Qball-X4 quadrotor, (b) Optitrack camera ................................................ 60 Figure 3.5. Quanser system ............................................................................................... 61 Figure 3.6. System control architecture. ........................................................................... 63 Figure 3.7. Infinity-shape desired path ............................................................................. 64 Figure 3.8. 3D system response ....................................................................................... 65 Figure 3.9. 2D system response ....................................................................................... 66 Figure 3.10. 2D system response (LQR-MFC vs NIB-MFC) ......................................... 66 Figure 3.11. LQR control system performance ................................................................ 69

Figure 3 .

 3 Figure 3.13. NIB control system performance.................................................................

Figure 3 .

 3 Figure 3.14. NIB-MFC control system performance .......................................................

Figure 3 .

 3 Figure 3.15. LQR vs NIB .................................................................................................

Figure 3 .

 3 Figure 3.16. LQR-MFC vs NIB-MFC .............................................................................

Figure 3 . 17 .

 317 Figure 3.17. System response (LQR controller). .............................................................

Figure 3 . 18 .

 318 Figure 3.18. System response (LQR-MFC controller). ...................................................

Figure 3 . 19 .

 319 Figure 3.19. System response (degraded LQR controller). .............................................

Figure 3 . 20 .

 320 Figure 3.20. System response (degraded LQR-MFC controller). ....................................

Figure 3 . 21 .

 321 Figure 3.21. Comparison between the different control techniques. ...............................

Figure 4 . 1 .

 41 Figure 4.1. FDI structure ...................................................................................................

Figure 4 . 2 .

 42 Figure 4.2. Schematic Diagram of State Observer. ..........................................................

Figure 4 . 3 .

 43 Figure 4.3. iOE structure with the assumption (𝐶 = 𝐼) ....................................................

Figure 4 . 4 .

 44 Figure 4.4. iOE structure with the relaxed assumption when 𝐶 ≠ 𝐼 .................................

Figure 4 . 5 .

 45 Figure 4.5. iTOE architecture with the assumption (𝐶 = 𝐼) ............................................

Figure 4 . 6 .

 46 Figure 4.6. iTOE modified structure based on the assumption (𝐶 ≠ 𝐼) ..........................

Figure 4 . 7 .

 47 Figure 4.7. (a) Infinity shape path. (b) square-shape path with Actuator Fault. ...............

Figure 4 . 8 .

 48 Figure 4.8. SO vs. iOE (Fault free condition). ................................................................

Figure 4 . 9 .

 49 Figure 4.9. TO vs. iTOE (Fault free condition). .............................................................

Figure 4 .

 4 Figure 4.10. SO vs. iOE (Actuator Fault condition). ......................................................

Figure 4 .

 4 Figure 4.11. TO vs. iTOE (Actuator Fault condition). ...................................................

Figure 5 . 1 .

 51 Figure 5.1. FDD scheme .................................................................................................

Figure 5 . 2 .

 52 Figure 5.2. FDI scheme ...................................................................................................

Figure 5 . 3 .

 53 Figure 5.3. Active FTC Scheme .....................................................................................

Figure 5 . 4 .

 54 Figure 5.4. Desired Path.................................................................................................

Figure 5 .

 5 Figure 5.05. LQR vs NIB Controllers ............................................................................

Figure 5 . 6 .

 56 Figure 5.6. (a) LQR vs NIB Controllers, (b) LQR-MFC vs NIB-MFC Controllers.......

Figure 5 . 7 .

 57 Figure 5.7. LQR-MFC vs NIB-MFC Controllers ...........................................................

Figure 5 . 8 .

 58 Figure 5.8. 2D and 3D response of the NIB-MFC Controllers ......................................

Figure 5 . 9 .

 59 Figure 5.9. LoE actuator fault injection on the front motor ............................................

Figure 5 .Figure 5 . 11 .

 5511 Figure 5.10. Fault Estimation..........................................................................................

Figure 5 . 12 .

 512 Figure 5.12. Estimated signal from FDD and Filtered signal using SMRD-LPF ...........

Figure 5 . 13 .

 513 Figure 5.13. Actual signal vs. Filtered signal using different filters...............................

Figure 5 . 14 .

 514 Figure 5.14. Actuator fault magnitude estimation ..........................................................

Figure 5 . 15 .

 515 Figure 5.15. Control input 𝒖𝟑 from the controller for 15% LoE actuator fault .............

Figure 5 . 16 .

 516 Figure 5.16. Effect of 15% actuator fault compensation (on z-and x-axes) .................

Figure 5 . 17 .

 517 Figure 5.17. Effect of the actuator fault compensation ...................................................

Figure 5 . 18 .

 518 Figure 5.18. Effect of the actuator fault compensation ...................................................

Figure 5 . 19 .

 519 Figure 5.19. Effect of 25% actuator fault (compensation on z-and x-axes) .................

Figure 5 . 20 .

 520 Figure 5.20. Effect of 25% actuator fault (3D plot) ........................................................

Figure 5 . 21 .

 521 Figure 5.21. Effect of 25% LoE on both tests; with and without fault compensation. ...

Figure 5 .

 5 Figure 5.22. Control input (u3) for 25% LoE actuator fault. (a) fault estimation magnitude, (b) control input form the controller, and (c) total control input from AFTC. ....

Figure 5 . 23 .

 523 Figure 5.23. Actuator fault magnitude estimation ..........................................................

Figure 5 . 24 .

 524 Figure 5.24. Estimated signal from FDD and Filtered signal using SMRD-LPF ...........

Figure 5 . 25 .

 525 Figure 5.25. Control input (u2 = uroll) with and without fault compensation ..................

Figure 5 . 26 .

 526 Figure 5.26. Effect of the actuator fault compensation on the quadrotor altitude ..........

Figure 5 .

 5 Figure 5.27. FE approaches (iOE vs. SVD) for 15% LoE actuator fault ........................

Figure 5 .

 5 Figure 5.28. FE approaches (iOE vs. SVD) for ramp-type actuator fault ......................

Figure 5 .

 5 Figure 5.29. FDD and FTC architecture for the bias sensor fault ..................................

Figure 5 . 30 .

 530 Figure 5.30. Measured and estimated outputs of the quadrotor .....................................

Figure 5 . 31 .Figure 5 . 32 .

 531532 Figure 5.31. Residuals generation: errors between the measured and estimated output Figure 5.32. Symptoms generated from the residuals....................................................

Figure 5 . 33 .

 533 Figure 5.33. Actual and measured outputs without fault compensation ........................

Figure 5 . 34 .

 534 Figure 5.34. Actual and measured outputs with fault compensation .............................

Figure 5 . 35 .

 535 Figure 5.35. Actual output with and without fault compensation ..................................

Figure 5 . 36 .

 536 Figure 5.36. Quadrotor response for sensor bias fault (a) with fault compensation (b) with and without fault compensation ...................................................

Figure A. 1 .

 1 Figure A.1. Nonlinear Observer scheme .........................................................................

Figure A. 2 .

 2 Figure A.2. Generalized Observer Scheme .....................................................................

Figure A. 3 .

 3 Figure A.3. Bias Fault on x position ..............................................................................

Figure B. 1 .

 1 Figure B.1. FDD scheme ................................................................................................

Figure B. 2 .

 2 Figure B.2. Fault-Free control trajectory for the attitude and altitude of the quadrotor .

Figure B. 3 .

 3 Figure B.3. Fault Estimation for f1=1/5 Hz. ...................................................................

Figure B. 4 .

 4 Figure B.4. Fault Estimation for f2=1/50 Hz. .................................................................

Figure B. 5 .

 5 Figure B.5. System outputs in case of time-varying LoE fault for f2=1/50 Hz..............

Figure B. 6 .

 6 Figure B.6. Controller outputs in case of time-varying LoE fault for f2=1/50 Hz. ........

Figure B. 7 .

 7 Figure B.7. Motor outputs in case of time-varying LoE fault for f2=1/50 Hz. ...............

Figure B. 8 .

 8 Figure B.8. Fault Estimation residuals for the rotors ......................................................

Figure B. 9 .

 9 Figure B.9. Comparison between fault estimation and fault offset signal ......................

Figure C. 1 .

 1 Figure C.1. Sensor fault-tolerant control scheme ...........................................................

Figure C. 2 .

 2 Figure C.2. Sonar sensors mounted on quadrotor ...........................................................

Figure C. 3 .

 3 Figure C.3. Hardware connection of the sonar sensors ..................................................

Figure C. 4 .

 4 Figure C.4. Path taken by quadrotor ...............................................................................

Figure C. 5 .

 5 Figure C.5. Fault indicators ............................................................................................

Figure C. 6 .

 6 Figure C.6. Main sensor indicators .................................................................................

Figure C. 7 .

 7 Figure C.7. Quadrotor position under presence of fault .................................................

- 3 :

 3 Different algorithms applied on the Qball-X4 quadrotor vehicle ................... 19 Table 2-1: Qball-X4 Parameters ....................................................................................... 40 Table 3-1: RMS error in cm .............................................................................................. 67

  Re-configurable control, self-repairing control or what is known recently by the Fault Detection and Diagnosis (FDD) and Fault-Tolerant Control (FTC) systems are developed and applied on different applications in the twentieth century. A historical review of the FDD and FTC systems will be presented in Section 1.2.

  Representing the nonlinear system by a linear state-space model whose dynamics are time-varying based on parameters' scheduling is called Linear Parameter Varying (LPV) model. In this representation the system's coefficient matrices in (1.2) will be parameters' dependent and can be represented as follows: { 𝑥̇(𝑡) = 𝐴(𝑝)𝑥(𝑡) + 𝐵(𝑝)𝑢(𝑡) + 𝑅 1 𝑑(𝑡) + 𝐹 1 𝑓(𝑡) 𝑦(𝑡) = 𝐶(𝑝)𝑥(𝑡) + 𝐷(𝑝)𝑢(𝑡) + 𝑅 2 𝑑(𝑡) + 𝐹 2 𝑓(𝑡) (1.5)

{

  𝑥̇(𝑡) = 𝑓(𝑥(𝑡)) + 𝑔(𝑥(𝑡))𝑢(𝑡) 𝑦(𝑡) = ℎ(𝑥(𝑡)) + 𝐹 𝑠 𝑓 𝑠 (𝑡) (1.11)

Figure 1 .

 1 Figure 1.2. FDI Scheme

Figure 1 . 3 .

 13 Figure 1.3. (a) Total Thrust, (b) Yaw Motion, and (c) Roll/Pitch Motion

2 . 3 . 4 .Fault Detection and Isolation: 1 . 4 .

 23414 The MFC is augmented to the LQR controller in (LQR-MFC) and to the Nonlinear Integral Backstepping (NIB) controller in (NIB-MFC). The integrated formulations in both are implemented on the quadrotor vehicle and compared to the classical control techniques. The stability analysis of LQR-MFC and NIB-MFC are analytically proven and then verified by real flight tests. The robustness of the proposed algorithms is tested in the presence of disturbances and low-magnitude actuator faults. Novel output estimators' designs are introduced by integrating the Model-Free (MF) concept with a linear and nonlinear observers designs for output estimation purposes. The new formulations of this integration are named intelligent output estimators, where iOE denotes for the integration between the MF technique with the State observer (SO), and iTOE for the combination between the MF algorithm with the Thau observer (TO).2. iOE and iTOE are derived for Multi-Input-Multi-Output (MIMO) systems, in general, and then applied to the quadrotor system. It is to be noted that in the literature the Model-Free technique is presented for Single-Input-Single-Output (SISO) systems and offered as a control strategy.3. Real-time flights are implemented to compare the performance of the proposed intelligent estimators (iOE and iTOE) to the SO and TO, respectively, under fault-free and actuator fault conditions. The iOE estimator is used over the iTOE in the FDI strategy as it provides a balanced solution between the estimation performance and the computational analysis needed to do the outputs estimation for MIMO systems. 5. The fault estimation based on the intelligent estimators are used in the FDI process, where the residual evaluation and generation processes are performed through statistical analysis of the fault extracted from the estimation. 6. An FDI strategy based on Nonlinear Identity Observer design and Generalized Observer Structure (GOS) is proposed and simulated towards different sensor fault scenarios in Appendix-A. 7. The Adaptive Thau Observer (ATO) is proposed to detect, isolate and estimate the actuator fault magnitudes of the quadrotor system and presented through different flight test scenarios applied to the AscTec Pelican quadrotor, in Appendix-B.

3 .

 3 The noisy estimation of actuator fault magnitude is undertaken by proposing the multi-stage Sliding-Mode Robust Differentiator -Low-Pass-Filter (SMRD-LPF) technique to get rid of the noise within an acceptable convergence time. Real-time experiments were conducted to compare the performance of this technique with the other filter designs to become the paramount solution of the signal noise reduction in this work.

Chapter 3 .In Chapter 4 ,

 34 quadrotor through various real-time flight results. A comparison of the presented and

Figure 2 .

 2 Figure 2.1 depicts the two coordinate systems used on the quadrotor: the Body Frame (BF) with constant inertia that is represented by the basis vector (𝑥 𝑏 , 𝑦 𝑏 , 𝑧 𝑏 ), and the Earth Frame (EF) which is the non-accelerated frame (𝑥 𝑒 , 𝑦 𝑒 , 𝑧 𝑒 ) that is used to describe the position and the translational motion of the quadrotor body.

Figure 2 . 1 .

 21 Figure 2.1. Reference Coordinate Systems.

  𝜃) c(𝜓) c(𝜓) s(𝜃) s(𝜙) -s(𝜓) 𝑐(𝜙) 𝑐(𝜓) s(𝜃) c(𝜙) + s(𝜓) sin(𝜙) c(𝜃) 𝑠(𝜓) s(𝜓) s(𝜃) s(𝜙) + c(𝜓) c(𝜙) s(𝜓) s(𝜃) c(𝜙) -c(𝜓) sin(𝜙) -s(𝜃) c(𝜃) s(𝜙) c(𝜃) c(𝜙) ] Euler rates ω b = [𝑝 𝑞 𝑟] T represents the body angular rates vector about BF. While the Euler rates Θ ̇= [𝜙 ̇ 𝜃 ̇ 𝜓 ̇]T are the angular rates about the EF. Using the rotational transformation matrix 𝑇 𝑏 𝑒 , the Euler angles in EF can be represented by the body angular rates in BF as follows [103]: 𝜃) sin(𝜙) tan(𝜃) cos(𝜙) 0 cos(𝜙) -sin(𝜙) 0 sec(𝜃) sin(𝜙) sec(𝜃) cos(𝜙) ] (2.5)

  𝜓) s(𝜃) c(𝜙) + s(𝜓) s(𝜙)) . ∑ 𝐹 𝑖 4 𝑖=1 (s(𝜓) s(𝜃) c(𝜙) -c(𝜓) s(𝜙)) . ∑ 𝐹 𝑖 4 𝑖=1 c(𝜃) c(𝜙) ∑ 𝐹 𝑖 4 𝑖=1

Figure 2 .

 2 Figure 2.2 below shows the moments generated by the lifting forces and the reaction moments from the actuators.

Figure 2 . 2 .

 22 Figure 2.2. Moments about the axes. The derivation of the angular momentum equation of the quadrotor body about the CoG is given as:

  (𝜙) 𝑡(𝜃) 𝑞 + 𝑐(𝜙) 𝑡(𝜃) 𝑟 𝑐(𝜙) 𝑞 -𝑠(𝜙) 𝑟 (𝑠(𝜙) 𝑐(𝜃) ⁄ ) 𝑞 + (𝑐(𝜙) 𝑐(𝜃) ⁄ ) 𝑟 (𝑀 𝑥 + (I yy -I zz ) 𝑞 𝑟 -𝐽Ω 𝑟 𝑞) I xx ⁄ (𝑀 𝑦 + (I zz -I xx ) 𝑝 𝑟 + 𝐽Ω 𝑟 𝑝) I yy ⁄ (𝑀 𝑧 + (I xx -I yy ) 𝑝 𝑞) I zz ⁄ (𝑐(𝜓) 𝑠(𝜃) 𝑐(𝜙) + 𝑠(𝜓) 𝑠(𝜙)) . ∑ 𝐹 𝑖 4 𝑖=1 𝑚 ⁄ (𝑠(𝜓) 𝑠(𝜃) 𝑐(𝜙) -𝑐(𝜓) 𝑠(𝜙)) . ∑ 𝐹 𝑖 4 𝑖=1 𝑚 ⁄ -𝑔 + (𝑐(𝜃) 𝑐(𝜙)).

  𝑑𝑝 𝜙 ̇2 + (I yy -I zz ) 𝜃 ̇ 𝜓 ̇-𝐽𝛺 𝑟 𝜃 ̇+ 𝑢 2 ) I xx ⁄ (-𝑘 𝑑𝑞 𝜃 ̇2 + (I zz -I xx ) 𝜙 ̇ 𝜓 ̇+ 𝐽𝛺 𝑟 𝜙 ̇+ 𝑢 3 ) I yy ⁄ (-𝑘 𝑑𝑟 𝜓 ̇2 + (I xx -I yy ) 𝜙 ̇ 𝜃 ̇+ 𝑢 4 ) I zz ⁄ (-𝑘 𝑑x ẋ+ (𝑐(𝜓) 𝑠(𝜃) 𝑐(𝜙) + 𝑠(𝜓) 𝑠(𝜙)) 𝑢 1 ) 𝑚 ⁄ (-𝑘 𝑑y ẏ+ (𝑠(𝜓) 𝑠(𝜃) 𝑐(𝜙) -𝑐(𝜓) 𝑠(𝜙)) 𝑢 1 ) 𝑚 ⁄ (-𝑘 𝑑z ż-𝑚𝑔 + (𝑐(𝜃) 𝑐(𝜙)) 𝑢 1 ) 𝑚 ⁄ ](2.23)

Figure 2 . 3 .

 23 Figure 2.3. Block Diagram of the Quadrotor Model.

  Figure 2.4.b.

  Figure 2.4. (a) Reference Frame used by the developers (b) adjusted reference frame 40

5 ) 8 )

 58 According to(2.29) and (2.30), which are; 𝐹 = 𝜅 𝑎 𝜚 and 𝜚 = 𝑤 𝑠+𝑤 𝑢 * , respectively. The relation between the control input 𝑢 2 and the actuator dynamics 𝜚 𝜙 can be extracted as follows: 𝑢 2 = 𝑙(𝐹 4 -𝐹 2 ) = 𝑙(𝜅 𝑎 𝜚 4 -𝜅 𝑎 𝜚 2 ) = 𝜅 𝑎 𝑙 𝜚 𝜙 (3second row of the state-space representation in (3.5). Now, for the derivative of the rotor dynamics; 𝜚∅ = -𝑤𝜚 ∅ + 𝑤∆𝑢 2 * = -𝑤 ( represents the derivative of the rotor dynamics term. It can be concluded that both representations are the same but the Qball-X4 model includes the rotors' dynamics, also the integrator terms are included in (2.38) and (2.39). Similarly, the comparison can be done for the other variable states, where the submodels of the Qball-X4 given in (2.38) and (2.39) are extracted from the linearized model in (3.3), whereas the rotor dynamics and the integrator terms are considered in the Qball-X4 sub-models. By recalling the Qball-X4 sub-models in (2.38) and (2.39), the state-space representations of the subsystems are:

  Fig. 1 depicts the general MFC scheme for a SISO system, where 𝑣 = 2.

Figure 3 .

 3 Figure 3.1. MFC scheme

By substituting ( 3 .

 3 45) into (3.44), 𝑥 (𝑣) = 𝑓(𝑥) + 𝑥 𝑑 (𝑣) + 𝑢 𝑐 (3.46) If the system is considered a second order system then: 58 𝑥̈= 𝑓(𝑥) + 𝑥d + 𝑢 𝑐 (3.47) As a MIMO system, the quadrotor dynamics could be decomposed into multi-SISO systems. Each controlled by a designated nonlinear controller. Then the augmentation of the MFC can be utilized by relating equations (3.47) with (3.27). 𝑥2 = 𝑓(𝑥) + 𝑥d + 𝑢 𝑐 = 𝑥d + (𝑘 1 2

Figure 3 . 2 .

 32 Figure 3.2. NIB-MFC scheme

Figure 3 Figure 3 Figure 3 . 5 .

 3335 Figure 3.3. Qball-X4 Workspace

1 ]

 1 , 𝑅 𝑥,𝑦 = [50]  𝐾 𝑥,𝑦 = [0.3663 0.3825 0.4391 0.0447] For height-position (z) gain matrix

Figure 3 . 6 .

 36 Figure 3.6. System control architecture.

Figure 3 . 7 .

 37 Figure 3.7. Infinity-shape desired path

Figure 3 . 8 .Figure 3 . 9 .Figure 3 . 10 .

 3839310 Figure 3.8. 3D system response

  , 3.12, 3.13 and 3.14 show the control performance of the system outputs for LQR, NIB, LQR-MFC and NIB-MFC controllers, respectively. Each figure has four plots that depict the position, angles, 2D and 3D responses.

Figure 3 .

 3 Figure 3.12. LQR-MFC control system performance

Figure 3 .

 3 Figure 3.13. NIB control system performance

Figure 3 .

 3 Figure 3.14. NIB-MFC control system performance

Figure 3 .Figure 3 .

 33 Figure 3.15. LQR vs NIB

Figure 5 . 16 .

 516 Figure 5.16. The quadrotor shows some oscillation and offset from desired trajectory.

Figure 3 . 17 .

 317 Figure 3.17. System response (LQR controller).

Figure 3 . 18 .

 318 Figure 3.18. System response (LQR-MFC controller).

Figure 3 . 19 .

 319 Figure 3.19. System response (degraded LQR controller).

Figure 3 . 20 .

 320 Figure 3.20. System response (degraded LQR-MFC controller).

Figure 3 . 21 .

 321 Figure 3.21. Comparison between the different control techniques.

  . The Fault Detection and Isolation (FDI) structure is presented in Section 1.2.3 and depicted in Figure 4.1.

Figure 4 .

 4 Figure 4.1. FDI structure

3 )

 3 By substituting (4.1) and (4.2) in (4.3), the error can be expressed by: 𝑒ẋ = (𝐴 -𝐿𝐶)𝑒 (4.4) The convergence of the error to zero can be achieved if the eigenvalues of term (𝐴 -𝐿𝐶) are in the left-hand side of the complex plane. The pole placement technique can be used to satisfy the convergence of the error. The block diagram representation of the state observer is shown in Figure 4.2.

Figure 4 . 2 .

 42 Figure 4.2. Schematic Diagram of State Observer.

Figure 4 .

 4 Figure 4.2 shows that the observer consists of two essential parts: the first part simulates the dynamics of the system by the linear model, and the second part feeds the correction between the measured and the estimated outputs (𝑦 -𝑦 ̂) by the amount specified by the observer gain matrix (L).

9 )

 9 By substituting equation (4.8) and (4.9) into (4.7):𝐹 𝑖𝑂𝐸 = 𝑦 ̂̇+ Γ𝑦 ̂-Γ Γ -1 (-𝐹 𝑖𝑂𝐸 + 𝑦̇+ Γ𝑦 -𝑢 𝑠𝑜 ) 𝐹 𝑖𝑂𝐸 = 𝑦 ̂̇+ Γ𝑦 ̂+ 𝐹 𝑖𝑂𝐸 -𝑦̇-Γ𝑦 + 𝑢 𝑠𝑜 𝑦̇-𝑦 ̂̇= -Γ(𝑦 -𝑦 ̂) + 𝐿(𝑦 -𝑦 ̂) 𝑦̇-𝑦 ̂̇= (𝐿 -Γ)(𝑦 -𝑦 ̂)(4.10)And by defining the estimation error as:𝑒 𝑦 = 𝑦 -𝑦 ̂ 𝑒̇y = 𝑦̇-𝑦 ̂̇ 𝑒ẏ = (𝐿 -Γ𝐼)𝑒 𝑦(4.11)

Figure 4 .

 4 Figure 4.3 shows the iOE structure.

Figure 4 . 3 .

 43 Figure 4.3. iOE structure with the assumption (𝐶 = 𝐼)

Figure 4 . 4 .

 44 Figure 4.4. iOE structure with the relaxed assumption when 𝐶 ≠ 𝐼

{

  𝑥 ̂̇= 𝐴𝑥 ̂+ 𝑓(𝑥 ̂, 𝑢) + 𝐵𝑢 + 𝐻(𝑦 -𝑦 ̂) + 𝑢 𝑖𝑇𝑂𝐸 𝑦 ̂= 𝐶𝑥 ̂ (4.22)

Figure 4 . 5 .

 45 Figure 4.5. iTOE architecture with the assumption (𝐶 = 𝐼)

Figure 4 . 6 .

 46 Figure 4.6. iTOE modified structure based on the assumption (𝐶 ≠ 𝐼)

  5)m in x-and y-axes and (0.6, 1.3)m in z-axis. The path makes one complete cycle (f = 0.05 Hz) about the x-axis and two complete cycles (f = 0.1 Hz) about y-and z-axes. The purpose of this test is to excite the different dynamics of the Quadrotor to check the estimator capabilities in estimating the measured outputs.In the second test the Quadrotor will be commanded to follow a desired square-shape path that covers an area of 1-m 2 . The total flight in the second test is 105 sec. The flight path is shown in Figure4.7.b. The pre-flight time is provided to give the quadrotor the time needed for the gradual taking off and for the observers to follow the actual path.

Figure 4 . 7 .

 47 Figure 4.7. (a) Infinity shape path. (b) square-shape path with Actuator Fault.

8

 8

  and

Figures 4 .

 4 Figures 4.8.b, 4.8.d, 4.9.b and 4.9.d show that the convergence of the error for the estimated outputs in the intelligent estimators is superior to the output estimations of SO and TO. The 3D plots in Figures 4.8.e and 4.9.e show excellent matching of the output estimations with respect to the position outputs (x, y and z) for the intelligent estimators, unlike the non-intelligent observers.

  and 4.11, iOE and iTOE show better convergence and good tracking performance to the actual outputs, while SO and TO show noticeable errors and off-track oscillation between the actual and the estimated outputs. Even with the high fluctuations of the quadrotor angles, the intelligent estimators managed to estimate the angles while the state and Thau observer failed to follow the actual outputs as depicted in Figures 4.10.c, 4.10.d, 4.11.c, and 4.1.d.

.d and 4 .

 4 11.d. Similar to flight mission 1, the 3D plots in Figures 4.10.e and 4.11.e show better estimations to the actual outputs of the position states (x, y and z) for the intelligent estimators compared to SO and TO.

Figure 4 . 8 .Figure 4 . 9 .

 4849 Figure 4.8. SO vs. iOE (Fault free condition).(a) and (c) shows the actual and estimated outputs of the position and angles for both observer designs. (b) and (d) depicts the estimation errors from SO and iOE for the position and angular outputs, respectively. (e) shows a 3D plot of the position outputs that compares the SO and iOE with respect to the actual path.

  (a) and (c) shows the actual and estimated outputs of the position and angles for both observer designs. (b) and (d) depicts the estimation errors from TO and iTOE for the position and angular outputs, respectively. (e) shows a 3D plot of the position outputs that compares the TO and iTOE with respect to the

Figure 4 .

 4 Figure 4.10. SO vs. iOE (Actuator Fault condition).

  (a) and (c) shows the actual and estimated outputs of the position and angles for both observer designs. (b) and (d) depicts the estimation errors from SO and iOE for the position and angular outputs, respectively. (e) shows a 3D plot of the position outputs that compares the SO and iOE with respect to the actual path.

Figure 4 .

 4 Figure 4.11. TO vs. iTOE (Actuator Fault condition).

  (a) and (c) shows the actual and estimated outputs of the position and angles for both observer designs. (b) and (d) depicts the estimation errors from TO and iTOE for the position and angular outputs, respectively. (e) shows a 3D plot of the position outputs that compares the TO and iTOE with respect to the actual path.

Figure 5 .

 5 Figure 5.1. FDD scheme

Figure 5 . 4 .

 54 Figure 5.4. Desired Path.

Figure 5 .

 5 5a illustrates the tracking performance of the nominal controllers (LQR and NIB) before and after the occurrence of the actuator fault. Figures 5.5b

  Figure 5.05. LQR vs NIB Controllers

Figure 5 . 6 .Figure 5 . 7 .

 5657 Figure 5.6. (a) LQR vs NIB Controllers, (b) LQR-MFC vs NIB-MFC Controllers

Figure 5 . 8 .

 58 Figure 5.8. 2D and 3D response of the NIB-MFC Controllers

Furthermore, a comparison

  between the controllers that completed the whole mission, which are LQR-MFC and NIB-MFC controllers, had been performed from the start until the end of the flight time. The robustness of NIB-MFC gives an advantage compared to LQR-MFC in terms of trackability as shown from Table5-2.

22 )

 22 𝑉 ̇= 𝑒̃𝑇 (𝑃(𝐴 ̃-𝐿 ̃𝐶 ̃) + (𝐴 ̃-𝐿 ̃𝐶 ̃)𝑇 𝑃) 𝑒̃+ 2𝑒̃𝑇𝑃𝐹 ̃𝑓𝑎 = 𝑒̃𝑇(-𝜃𝜉𝐼)𝑒̃+ 2𝑒̃𝑇𝑃𝐹 ̃𝑓𝑎 123 By considering the condition 𝜉 ≥ ‖𝑓 𝑎 ‖: 𝑉 ̇≤ -𝜃𝜉‖𝑒‖ 2 + 2 𝜉‖𝑒̃𝑇𝑃𝐹 ̃‖ (5.23) The term ‖𝑒̃𝑇𝑃𝐹 ̃‖ can be expressed as: ‖𝑒̃𝑇𝑃𝐹 ̃‖ = √ 𝑒̃𝑇𝑃𝐹 ̃𝐹 ̃𝑇𝑃𝑒̃ (5.24) Using the largest eigenvalue 𝜆 𝑚𝑎𝑥 (𝑃𝐹 ̃𝐹 ̃𝑇𝑃), and based on Caushy-Schwarz inequality [118], (5.24) can be expressed as: ‖𝑒̃𝑇𝑃𝐹 ̃‖ ≤ √𝜆 𝑚𝑎𝑥 (𝑃𝐹 ̃𝐹 ̃𝑇𝑃) ‖𝑒‖ (5.25) Substituting (5.25) into (5.23) results in: 𝑉 ̇≤ -𝜃𝜉‖𝑒‖ 2 + 2 𝜉√𝜆 𝑚𝑎𝑥 (𝑃𝐹 ̃𝐹 ̃𝑇𝑃) ‖𝑒‖ (𝑃𝐹 ̃𝐹 ̃𝑇𝑃). It can be concluded that 𝑉 ̇≤ 0 for all values of ‖𝑒‖ ≥ 𝜎 and 𝑒̃ is bounded with respect to the region { 𝑒: ‖𝑒‖ < 𝜎}.

Figure 5 . 9 .

 59 Figure 5.9. LoE actuator fault injection on the front motor

Figure 5 .

 5 Figure 5.10. Fault Estimation

  [N.m] a multi-stage design. The filtering process using the multi-stage SMRD-LPF structure produce a faster converging and smoother signal compared to other filters.The fault estimation signal depicted in Figure5.10 is attenuated using SMRD-LPF as presented in Figures 5.12 and 5.13. The signal resulted from the multi-stage SMRD-LPF shows better filtration performance compared to the LPF filters, even with a longer constant time, as shown in Figure5.13. Therefore, the multi-stage SMRD-LPF will be utilized in the fault compensation process of the actuator fault, where reducing the noise is crucial when the fault estimation is fed-back and included in the control law as illustrated next.

Figure 5 . 12 .Figure 5 . 13 .

 512513 Figure 5.12. Estimated signal from FDD and Filtered signal using SMRD-LPF

[N.m] 129

 129

Figure 5 . 14 .

 514 Figure 5.14. Actuator fault magnitude estimation

Figure 5 . 18 .

 518 Figure 5.18. Effect of the actuator fault compensation

Figures 5 .

 5 19 and 5.20 show the effect of the fault compensation on the system response.

Figure 5 . 19 .Figure 5 . 20 .Figure 5 . 24 .

 519520524 Figure 5.19. Effect of 25% actuator fault (compensation on z-and x-axes)

Figures 5 .Figure 5 .

 55 Figures 5.30 shows the measured and estimated outputs of the quadrotor, and Figure 5.31 shows the residuals of the outputs estimation and the sensor readings of the quadrotor position and angles, where a discrepancy in the y-position and the roll angle is detected at t = 40s. The errors then are evaluated by generating symptoms for the residuals, 𝑆(𝑟) based on a pre-defined thresholds of the error values. The fault symptoms are identified as shown in Figure 5.32.

Figure 5 . 32 .

 532 Figure 5.32. Symptoms generated from the residuals

Figure 5 . 31 .Figure 5

 5315 Figure 5.31. Residuals generation: errors between the measured and estimated output

Figure 5 . 33 .

 533 Figure 5.33. Actual and measured outputs without fault compensation

Figure 5 . 34 .Figure 5 .

 5345 Figure 5.34. Actual and measured outputs with fault compensation Figure 5.35 below is comparing the actual value of the quadrotor y-axis position with and without the compensation process.

Figure 5 . 35 .

 535 Figure 5.35. Actual output with and without fault compensation

Figure 5 . 5 Figure 5 . 36 . 7 Summary

 555367 Figure 5.36-a shows the 3D plot of the Qball-X4 response after the compensation process. It can be seen that the quadrotor returned to the hover state position after it moved approximately -5 cm in the y-axis. The system responses with and without the fault compensation process are compared in Figure 5.36-b, where the response without fault compensation process is misled and moved to -0.3 m, while it returns to the origin after the fault compensation is achieved as shown in the figure and enlarged in Figure 5.36-a.

  quadrotor vehicle was the main mission of Chapter 3. The Model-Free Control technique is utilized in the quadrotor's control, where different combinations between the MFC with linear and nonlinear control algorithms are proposed to compensate for the unmodeled dynamics and system uncertainties of the Qball-X4 quadrotor. The LQR controller with an integral action and the Nonlinear Integral Backstepping (NIB) controller are used in this study and compared to the control performance by augmenting the MFC technique on both of them. The stability analysis of using the linear and nonlinear control algorithms with the MFC is analytically proven and then verified by real flight tests.

Figure 2 .Figure 3 .

 23 Figure 2. Régime NIB-MFC

Figure 4 .

 4 Figure 4. Réponse système lorsqu'une perturbation est injectée dans le système. (a) LQR, (b) NIB, (c) LQR-MFC, et (d) NIB-MFC contrôleur

Figure 5 . 1 -

 51 Figure 5. La réponse du système pour le troisième test

Figure 6 .

 6 Figure 6. Structure de l'iES avec l'hypothèse (C = I)

Figure 7 .

 7 Figure 7. Structure de l'OIE avec l'hypothèse de C ≠ I

Figure 8 .

 8 Figure 8. Architecture de l'iETS avec l'hypothèse (C = I)

Figure 9 .

 9 Figure 9. la structure modifiée de l'IETS sur la base de l'hypothèse (C ≠ I)

Figure 10 .

 10 Figure 10. Comparaison de résultats des estimateurs SO et OIF pour compenser les incertitudes.

Figure 11 .

 11 Figure 11. Comparaison de résultats des estimateurs TO et iETS

Figure 12 .

 12 Figure 12. Défaut actionneur OIE.

Figure 13 .

 13 Figure 13. Défaut actionneur. (a) et (c) montrent les sorties réelles et estimées de la position et les angles pour les deux conceptions d'observateurs.

  Figure A.2:

Figure A. 2 .

 2 Figure A.2. Generalized Observer Scheme

2 Θ 7 = a 1 x 4 Θ 8 = -a 2 x 2 Θ 9 =

 274829 -2K dr x 6 I z ⁄ 𝜒 1 = 𝑈 1 (𝑐(𝑥 1 )𝑠(𝑥 5 ) -𝑠(𝑥 1 )𝑠(𝑥 3 )𝑐(𝑥 5 )) 𝑚 ⁄ 𝜒 2 = -𝑈 1 (𝑐(𝑥 1 )𝑐(𝑥 5 ) + 𝑠(𝑥 1 )𝑠(𝑥 3 )𝑠(𝑥 5 )) 𝑚 ⁄ 𝜒 3 = -𝑈 1 𝑠(𝑥 1 )𝑐(𝑥 3 ) 𝑚 ⁄ 𝜒 4 = 𝑈 1 𝑐(𝑥 1 )𝑐(𝑥 3 )𝑐(𝑥 5 ) 𝑚 ⁄ 𝜒 5 = 𝑈 1 𝑐(𝑥 1 )𝑐(𝑥 3 )𝑠(𝑥 5 ) 𝑚 ⁄ 𝜒 6 = -𝑈 1 𝑐(𝑥 1 )𝑠(𝑥 3 ) 𝑚 ⁄ 𝜒 7 = 𝑈 1 (𝑠(𝑥 1 )𝑐(𝑥 5 ) -𝑐(𝑥 1 )𝑠(𝑥 3 )𝑠(𝑥 5 )) 𝑚 ⁄ 𝜒 8 = 𝑈 1 (𝑠(𝑥 1 )𝑠(𝑥 5 ) + 𝑐(𝑥 1 )𝑠(𝑥 3 )𝑐(𝑥 5 )) 𝑚 ⁄

x12) .

 . The null space of C is: semi-definite matrix P will be assigned as:𝑃 = 𝑑𝑖𝑎𝑔( 𝑃 1 , 𝑃 2 , … , 𝑃 12 ) (A.23)The negativity of 𝑉 ̇(𝑒) in (A.2) can be then analyzed, where; 𝑉 ̇(𝑒) = 𝜀 𝑇 𝐾 𝑇 𝑃𝜕 𝑥 𝑓 𝐾𝜀 = -(𝐾 𝑑𝑥 𝑚 ⁄ )𝑃 12 𝜀 2 < 0 for any positive real value of 𝑃 12 .

ℎ 1 (

 1 Figure B.1. FDD scheme

Theorem 1 .

 1 if the gain matrix K in (B.3) satisfies: 𝐾 = 𝑃 𝜃 -1 𝐶 𝑇 (B.4) and matrix 𝑃 𝜃 is the solution of (B.5): 𝐴 𝑇 𝑃 𝜃 + 𝑃 𝜃 𝐴 -𝐶 𝑇 𝐶 + 𝜃𝐶 𝑇 𝑃 𝜃 = 0 (B.5) where 𝜃 is a positive parameter such that (B.5) has a positive definite solution, then the state of (B.3) is an asymptotic estimate of the system state described by (B.1), That is, lim 𝑡→∞ 𝑒(𝑡) = lim 𝑡→∞ (𝑥(𝑡) -𝑥 ̂(𝑡)) = 0 (B.6)

B. 1 . 2

 12 Fault estimation based on adaptive Thau observer Based on the fault detection result from the Thau observer, an adaptive Thau observer is proposed to estimate the fault severity. With reference to (B.2) and (B.4), a novel Thau observer can be constructed as: { 𝑥 ̂̇(𝑡) = 𝐴𝑥 ̂(𝑡) + 𝐵𝑢(𝑡) + 𝐻(𝑥 ̂(𝑡), 𝑢(𝑡)) + 𝐹𝑓 ̂(𝑡) + 𝐾(𝑦(𝑡) -𝑦 ̂(𝑡)) 𝑦 ̂(𝑡) = 𝐶𝑥 ̂(𝑡) (B.7) where 𝑓 ̂(𝑡) is an estimation of the fault 𝑓(𝑡). 𝐾 ∈ 𝑅 𝑛×𝑝 is the observer feedback gain. Denoting the estimation error as 𝑒 𝑥 (𝑡) = 𝑥(𝑡) -𝑥 ̂(𝑡), the error dynamics is described by 𝑒̇𝑥(𝑡) = (𝐴 -𝐾𝐶)𝑒 𝑥 (𝑡) + [𝐻(𝑥) -𝐻(𝑥 ̂)] + 𝐹𝑒 𝑓 (𝑡) (B.8)

Theorem 2 .

 2 For the available observer gain K in Theorem 1 and a defined matrix 𝑄 (𝑛×𝑛) > 0 and positive parameter 𝛾, if there exist two matrices 𝑃 𝑛×𝑛 and 𝐺 𝑟×𝑝 satisfying: 𝑃(𝐴 -𝐾𝐶) + (𝐴 -𝐾𝐶) 𝑇 𝑃 + 𝛾𝑃𝑃 -𝛾𝐼 = -𝑄 (B.9)𝑃𝐵 = 𝐶 𝑇 𝐺 𝑇 (B.10)Then the observer given in (B.11) with the adaptive fault estimation law[START_REF] Zhang | Adaptive observer-based fault diagnosis with application to satellite attitude control systems[END_REF][START_REF] Meng | Observer-based robust fault diagnosis for a class of uncertain nonlinear systems[END_REF]:𝑓 ̂(𝑡) = 𝛤𝐺(𝑦(𝑡) -𝑦 ̂(𝑡)) -𝜎𝛤𝑓 ̂(𝑡) (B.11) can lead to lim 𝑡→∞ 𝑒 𝑥 (𝑡) = 0 and lim 𝑡→∞ 𝑒 𝑓 (𝑡) = 0 , where 𝛤 = 𝛤 𝑇 > 0 is a weighting matrix; 𝜎 is a positive constant that satisfies 𝜎 -𝜆 𝑚𝑎𝑥 (Γ -1 ) > 0, 𝜆 𝑚𝑎𝑥 (•) is the maximum eigenvalue of the corresponding matrix. Proof. From 𝑒 𝑓 (𝑡) = 𝑓(𝑡) -𝑓 ̂(𝑡), the derivative 𝑒 𝑓 (𝑡) can be defined as 𝑒ḟ(𝑡) = 𝑓 (𝑡) -𝛤𝐺𝑒 𝑦 (𝑡) -𝜎𝛤𝑓(𝑡) + 𝜎𝛤𝑒 𝑓 (𝑡) (B.12)Considering the following Lyapunov function𝑉(𝑡) = 𝑒 𝑥 𝑇 𝑃𝑒 𝑥 + 𝑒 𝑓 𝑇 Γ -1 𝑒 𝑓 (B.13)Then its derivative is : 𝑉 ̇= 𝑒 𝑥 𝑇 [(𝐴 -𝐿𝐶) 𝑇 𝑃 + 𝑃(𝐴 -𝐿𝐶)]𝑒 𝑥 + 2𝑒 𝑥 𝑇 𝑃[𝐻(𝑥) -𝐻(𝑥 ̂)] + 2𝑒 𝑓 𝑇 𝛤 -1 𝑓 ̇+ 2𝜎𝑒 𝑓 𝑇 𝑓 -2𝜎𝑒 𝑓 𝑇 𝑒 𝑓 ≤ 𝑒 𝑥 𝑇 [(𝐴 -𝐿𝐶) 𝑇 𝑃 + 𝑃(𝐴 -𝐿𝐶) + 𝛾𝑃𝑃 + 𝛾𝐼]𝑒 𝑥 + 𝜆 𝑚𝑎𝑥 (𝛤 -1 ) • [‖𝑒 𝑓 ‖ 2 𝑚𝑖𝑛 (𝑄) • ‖𝑒 𝑥 ‖ 2 [𝜎 -𝜆 𝑚𝑎𝑥 (𝛤 -1 )] • ‖𝑒 𝑓 ‖ 2 + 𝛽 (B.14)

Remark1:Theorem 3 .

 3 )where 𝛼 = min[𝜆 𝑚𝑖𝑛 (𝑄),𝜎-𝜆 𝑚𝑎𝑥 (𝛤 -1 )] max[𝜆 𝑚𝑎𝑥(𝑃),𝜆 𝑚𝑎𝑥 (𝛤 -1 )] . The differential inequality (B.17) satisfies 𝛼𝑡 , as 𝑡 → ∞, 𝑉(𝑡) is uniformly ultimately bounded.Therefore, The adaptive Thau observer is asymptotically stable; 𝑒 𝑓 is also uniformly and ultimately bounded. The ultimate norm bound of 𝑒 𝑓 can be easily figured out: Theorem 2 is suitable for all faults with different time-varying natures theoretically. However, it should be pointed that the estimation convergence speed depends both on the fault time-varying nature and on the specified parameters of the Adaptive Thau Observer. Hereby as it can be seen from (B.12), suitable value of Γ and 𝜎 should be set for better estimation performance subjected to faults with different timevarying natures. If the fault vector 𝑓(𝑡) is constant or its derivative is close to zero, we can obtain another adaptive observer in a simplified form according to Theorem 2. If the fault 𝑓(𝑡) is constant and the adaptive fault estimation law is in the form of (B.[START_REF] Noura | Fault-tolerant control systems: Design and practical applications[END_REF]), 𝑉 ̇ < 0 will be satisfied and the proposed adaptive Thau observer is asymptotically stable based on Lyapunov theorem.𝑓 ̂(𝑡) = 𝛤𝐺(𝑦(𝑡) -𝑦 ̂(𝑡)) (B.19)Proof. Consider the same Lyapunov function of (B.15), then its derivative can be denoted:𝑉 ̇≤ -𝑒 𝑥 𝑇 𝑄𝑒 𝑥 ≤ -𝜆 𝑚𝑖𝑛 (𝑄)‖𝑒 𝑥 ‖ 2 ≤ 0 (B.20)

8 ,

 8 In order to validate the Fault Estimation performance of the proposed Adaptive Thau Observer for actuator partial Loss-of-Effectiveness (LoE), two fault scenarios were simulated. The fault-free control trajectory of attitude and altitude is shown in Figure B.2. The first fault corresponds to a sine wave offset fault signal with two different frequencies injected into the throttle input which is the control input for quadrotor altitude control. The other fault corresponds to a sine wave offset fault signal injected into one of the rotors output, which matches well with what the real fault derive from.

Figure B. 2 .B. 2 . 1 Fault Scenario 1 A 21 )

 221121 Figure B.2. Fault-Free control trajectory for the attitude and altitude of the quadrotor

Figure B. 5 ,Figure B. 3 .

 53 Figure B.5, Figure B.6 and Figure B.7 show comparisons between the fault-free case and the faulty case with time-varying Loss-of-Effectiveness fault f2=1/50Hz for the system output, control output, and rotor output. As it can be seen from Figure B.5 and Figure B.6, the altitude z and the corresponding control input U4 change at t=75s because the fault occurs while attitude state and corresponding control inputs keep normal. Figure B.7 shows that four rotor outputs F1-F4 change with same magnitude U4/4 because the fault is affecting U4.

Figure B. 4 .Figure B. 5 .Figure B. 6 .

 456 Figure B.4. Fault Estimation for f2=1/50 Hz.

Figure B. 7 .

 7 Figure B.7. Motor outputs in case of time-varying LoE fault for f2=1/50 Hz.

Figure C. 5 .

 5 Figure C.5. Fault indicators

Figure C. 7 .

 7 Figure C.7. Quadrotor position under presence of fault

Table 1 -

 1 1: Actuator Fault Scenarios ................................................................................... 7

Table 1 -

 1 2: Different FDD and FTC algorithms applied on the quadrotor vehicle ........... 18

Table 1

 1

Table 4 -

 4 1: RMS error values between the actual and the estimated outputs for Test 1 . 100

Table 4 -

 4 2: RMS error values between the actual and the estimated outputs Test 2 ....... 102

Table 5 -

 5 1: RMS error in cm (excluding the fault, 𝑡 ∈ 35,75s) ...................................... 116

Table 5 -

 5 2: RMS error in cm (including the fault, 𝑡 ∈ 35,105s) ..................................... 116

Table 5 -

 5 3: Fault Signature Table .................................................................................... 147 Table A-1: Fault Signature Table ................................................................................... 196 Table A-2: Residuals caused by drift fault in roll angle ∅ .............................................. 198 Table B-1: AscTec Pelican quadrotor Parameters .......................................................... 205 Table C-1: Fault Signature Table .................................................................................... 215 Body frame position vector, X b = [𝑥 𝑏 , 𝑦 𝑏 , 𝑧 𝑏 ] 𝑇 X e Earth frame position vector, X e = [𝑥 𝑒 , 𝑦 𝑒 , 𝑧 𝑒 ] 𝑇 𝑀 𝑦 , 𝑀 𝑧 Moments applied about 𝑥 𝑏 , 𝑦 𝑏 and 𝑧 𝑏 , respectively.

	xix

Θ

Euler angles vector Θ = [𝜙, 𝜃, 𝜓] 𝑇 , which are: roll, pitch and yaw.

𝑏

Magnitude of the lift force from the i th rotor with respect to the BF 𝑛 The feedback term of the MF structure in the iOE estimator Γ Model-Free gain matrix for iOE estimator 𝐹 𝑖𝑂𝐸 Continuously updated matrix in the MF structure for iOE estimator 𝑖𝑇𝑂𝐸 Continuously updated matrix in the MF structure for iTOE. 𝑝𝑜𝑠_𝑒𝑠𝑡 Filtered measurement from the real-time measurement 𝔣 𝑝𝑜𝑠 using the first SMRD filter 𝔣 𝑣𝑒𝑙_𝑒𝑠𝑡 Filtered measurement from the rate of the measurement 𝔣 𝑝𝑜𝑠 using a second SMRD filter 𝑢 𝑐 (𝑡) Control input of the controller in the fault-free case 𝑢 𝑎 (𝑡) Actuator-fault-control-input resulted from the fault estimation 𝐾 𝑎 Compensation gain matrix of the actuator fault PWMi Pulse width modulation signal for the i th motor ℎ(𝑥 ̂, 𝑢) Gain matrix in the Nonlinear Identity Observer 𝜀 the transformation of 𝑒 when 𝐾𝑒𝑟(𝐶) ≠ 0 𝐹(𝑥 ̂, 𝑢) Matrix that verifies the inequality: ‖𝐹(𝑥 ̂, 𝑢)‖ > 1 INTRODUCTION 1.1 Motivations and Scope of Work Modern control technologies are involved in many aspects of life. Control systems are invading our lives to help the human to be more productive and efficient. Control systems could be found in most of the home appliances, computers, cars, production lines in factories, chemical processes, robots, medical and aerospace applications … etc.

Table 1 -

 1

		1: Actuator Fault Scenarios
		Constant Offset 𝒖 𝒇𝟎𝒊 = 𝟎	Constant Offset 𝒖 𝒇𝟎𝒊 ≠ 𝟎
	𝜶 𝒊 = 𝟏	Fault-free case	Bias
	𝜶 𝒊 ∈ (𝟎, 𝟏) Loss of effectiveness (LoE) Loss of effectiveness (LoE)
	𝜶 𝒊 = 𝟎

  𝑥 is the state vector, 𝑧 ∈ ℝ 𝑝 is the integrator vector, 𝐶 1 is the coefficient matrix for tracked output vector, 𝑦 1 ∈ ℝ 𝑝 (𝑝 ≤ 𝑚), 𝑦 𝑟 ∈ ℝ 𝑝 is the reference input vector, 𝑇 𝑠 is the sampling time, and 𝐹 𝑎 is the coefficient matrix for actuator fault magnitude vector, 𝑓 𝑎 , which represents the 𝑖 𝑡ℎ faulty actuator in the 𝑖 𝑡ℎ column of matrix B.

	𝐸 ̅ 𝑎 = [	𝐼 𝑛 0 -𝐹 𝑎 0 𝐼 𝑝 0 𝐶 0 0	], 𝐴 ̅ 𝑎 = [	𝐴 -𝑇 𝑠 𝐶 1 𝐼 𝑝 0 0 0 0 0 0 ], 𝐵 ̅ 𝑎 = [ 𝐵 0 0 𝐼 𝑞 0 0	]
	𝐺 ̅ 𝑎 = [ 𝑇 𝑠 𝐼 𝑝 0 0	],	𝑋 ̅ 𝑎 𝑘 = [	𝑥 𝑘 𝑧 𝑘 𝑓 𝑎 𝑘-1	],	𝑈 ̅ 𝑘 = [ 𝑦 𝑘+1 𝑢 𝑘	]	(1.19)
	The SVD of matrix 𝐸 ̅ 𝑎 :
						𝐸 ̅ 𝑎 = 𝑇 [ 0 𝑆	] 𝑀 𝑇	(1.20)
	where 𝑇 = [𝑇 1 𝑇 2 ], and T and M matrices are orthonormal, where 𝑇𝑇 𝑇 = 𝐼, 𝑀𝑀 𝑇 = 𝐼
	and S is a diagonal nonsingular matrix.
	From (1.18) and (1.20), the system can be written as follows:
			𝑋 ̅ 𝑎 𝑘+1 = 𝐴 ̿ 𝑎 𝑋 ̅ 𝑎 𝑘 + 𝐵 ̿ 𝑎 𝑈 ̅ 𝑘 + 𝐺 ̿ 𝑎 𝑦 𝑟 𝑘
				0 = 𝐴 ̿ 0 𝑋 ̅ 𝑎 𝑘 + 𝐵 ̿ 0 𝑈 ̅ 𝑘 + 𝐺 ̿ 0 𝑦 𝑟 𝑘	(1.21)
	where,				, but in discrete-time domain, is augmented with an
	integrator and can be expressed by the following state-space representation:
	{ where, 𝐸 ̅ 𝑎 [ 𝑥 𝑘+1 𝑧 𝑘+1 + = 𝑀𝑆 -1 𝑇 1 ] = [ -𝑇 𝑠 𝐶 1 𝐴 𝐴 ̿ 𝑎 = 𝐸 ̅ 𝑎 + 𝐴 ̅ 𝑎 , 0 𝑛,𝑝 𝐼 𝑝 ] [ 𝑦 𝑘 = [𝐶 0 𝑞,𝑝 ] [ 𝑥 𝑘 𝑧 𝑘 ] + [ 𝐵 0 𝑝,𝑚 ] 𝑢 𝑘 + [ 𝐵 ̿ 𝑎 = 𝐸 ̅ 𝑎 + 𝐵 ̅ 𝑎 , 𝐺 ̿ 𝑎 = 𝐸 ̅ 𝑎 + 𝐺 ̅ 𝑎 0 𝑛,𝑝 𝑇 𝑠 𝐼 𝑝 ] 𝑦 𝑟 𝑘 + [ 𝐹 𝑎 0 𝐴 ̿ 0 = 𝑇 2 𝑇 𝐴 ̅ 𝑎 , 𝐵 ̿ 0 = 𝑇 2 𝑇 𝐵 ̅ 𝑎 , 𝐺 ̿ 0 = 𝑇 2 𝑇 𝐺 ̅ 𝑎 𝑥 𝑘 𝑧 𝑘 ] 𝑇 is the pseudo-inverse of matrix 𝐸 ̅ 𝑎 .	] 𝑓 𝑎 𝑘	(1.17)
	where, The estimation of the actuator fault 𝑓 ̂𝑎 can be computed by extracting the augmented
	state vector 𝑋 ̅ 𝑎 𝑘 from (1.17) as follows:
				𝐸 ̅ 𝑎 𝑋 ̅ 𝑎 𝑘+1 = 𝐴 ̅ 𝑎 𝑋 ̅ 𝑎 𝑘 + 𝐵 ̅ 𝑎 𝑈 ̅ 𝑘 + 𝐺 ̅ 𝑎 𝑦 𝑟 𝑘	(1.18)
	where

Table 1 -

 1 2: Different FDD and FTC algorithms applied on the quadrotor vehicle

	Methodology	Type of Fault	FDI/FDD	FTC	Simulation/ Experimental	Ref.
			Fault detection of
	Sliding mode approach	Actuator Fault	the actuator fault using a state	FTC using sliding mode approach	Simulation	[64]
			estimator
	Thau's observer	Actuator Fault	fault Thau observer detection using	N/A	Simulation	[65]
	Two-Stage Kalman Filter	Actuator Fault	FDD using Two-Filter Stage Kalman	N/A	Experimental	[66]
	Reduced order observers	Sensor Fault	FDI using bank observers of reduced order	N/A	Simulation	[67]
	Fuzzy Gain-Scheduled PID	Actuator Fault	N/A	FTC using fuzzy gains scheme to tune PID	Experimental	[68]
	Poly topic Uncertain LPV Systems	Actuator Fault	N/A	FTC using robust LPV framework	Simulation	[46]
	Takagi-Sugeno observers	Sensor Fault	FDI using bank observers of LPV	N/A	Simulation	[45]
				Learning-based FTC
	Learning-Based Fault Tolerant Tracking Control	Actuator Fault	N/A	using an extended feedback tracking Kalman filter to optimize fuzzy state-	Simulation	[69]
				controller.

Table 1 -

 1

	FTC Strategy	Type Require	Simulation Experiment Real Damage
			FDI
	PID -Gain Scheduling	AFTC	√	√	√	×
	Model reference adaptive
	control

3: Different algorithms applied on the Qball-X4 quadrotor vehicle

  𝑀 𝑥 + (I yy -I zz ) 𝑞 𝑟 I yy 𝑞̇= 𝑀 𝑦 + (I zz -I xx ) 𝑝 𝑟 I zz 𝑟̇= 𝑀 𝑧 + (I xx -I yy ) 𝑝 𝑞 (2.17) where 𝑀 𝑥 , 𝑀 𝑦 and 𝑀 𝑧 are the moments applied about 𝑥 𝑏 , 𝑦 𝑏 and 𝑧 𝑏 axes, respectively.

Furthermore, the propeller gyroscopic effect results from the propeller rotation coupled with the body rotation

[START_REF] Bouabdallah | Design and control of quadrotors with application to autonomous flying[END_REF]

can be augmented with the rotational equations (2.17) as follows: I xx 𝑝̇= 𝑀 𝑥 + (I yy -I zz ) 𝑞 𝑟 -𝐽Ω 𝑟 𝑞 I yy 𝑞̇= 𝑀 𝑦 + (I zz -I xx ) 𝑝 𝑟 + 𝐽Ω 𝑟 𝑝 I zz 𝑟̇= 𝑀 𝑧 + (I xx -I yy ) 𝑝 𝑞 (2.18)

Table 2 -

 2 1: Qball-X4 Parameters

	Parameter	Value
	𝜿 𝒂	120 N
	w	15 rad/s
	Ixx	0.03 kg.m 2
	Iyy	0.03 kg.m 2
	m	1.4 kg
	𝜿 𝒚	4 N.m
	Izz	0.04 kg.m 2
	l	0.2 m

  𝑘 𝑑𝑝 𝑥 2 2 + (I yy -I zz ) 𝑥 4 𝑥 6 -𝐽Ω 𝑟 𝑥 4 ) I xx

			⁄ ] + (1 I xx ⁄ ) 𝑢 2
			𝑥 4
	[(-𝑘 𝑑𝑞 𝑥 4	2 + (I zz -I xx ) 𝑥 2 𝑥 6 + 𝐽Ω 𝑟 𝑥 2 ) I yy ⁄ ] + (1 I yy ⁄ ) 𝑢 3
			𝑥 6
	[(-𝑘 𝑑𝑟 𝑥 6	2 + (I xx -I yy ) 𝑥 2 𝑥 4 ) I zz ⁄ ] + (1 I zz ⁄ ) 𝑢 4
			𝑥 8
	[(-𝑘 𝑑𝑥 𝑥 8 + (𝑥 5 𝑥 1 ) 𝑢 1 ) 𝑚 ⁄ ] + (𝑢 1 𝑚 ⁄ ) 𝑥 3𝑑
			𝑥 10
	[(-𝑘 𝑑𝑦 𝑥 10 + (𝑥 5 𝑥 3 ) 𝑢 1 ) 𝑚 ⁄ ] + (-𝑢 1 𝑚 ⁄ ) 𝑥 1𝑑
			𝑥 12
		[(-𝑘 𝑑𝑧 𝑥 12 -𝑚𝑔) 𝑚 ⁄ ] + (1 𝑚 ⁄ ) 𝑢 1	]

  𝒌 𝟑 -𝟏)𝒆 𝟏 -(𝒌 𝟏 + 𝒌 𝟐 )𝒆 𝟐 + 𝒌 𝟏 𝒌 𝟑 ∫ 𝒆 𝟏 𝒅𝒕 -𝒇(𝒙)] (3.28) 𝑒 3 -(𝑘 4 + 𝑘 5 )𝑒 4 + 𝑘 4 𝑘 6 ∫ 𝑒 3 𝑑𝑡] + 𝑘 𝑑𝑞 𝑥 4 2 -(I zz -I xx ) 𝑥 2 𝑥 6 -𝐽Ω 𝑟 𝑥 2 𝑢 4 = I zz [𝑥̈5 𝑑 + (𝑘 7 2 + 𝑘 9 -1)𝑒 5 -(𝑘 7 + 𝑘 8 )𝑒 6 + 𝑘 7 𝑘 9 ∫ 𝑒 5 𝑑𝑡] + 𝑘 𝑑𝑟 𝑥 6 𝑘 18 -1)𝑒 11 -(𝑘 16 + 𝑘 17 )𝑒 12 + 𝑘 16 𝑘 17 ∫ 𝑒 11 𝑑𝑡] + 𝑘 𝑑𝑧 𝑥 12 +

	3.27) 𝟐 + From (3.2) and (3.28), the control inputs for the subsystems will be written as into (3.16): 𝒖 = 𝟏 𝒃 [𝒙̈𝒅 + (𝒌 𝟏 follows: 𝑢 2 = I xx [𝑥1 𝑑 + (𝑘 1 2 + 𝑘 3 -1)𝑒 1 -(𝑘 1 + 𝑘 2 )𝑒 2 + 𝑘 1 𝑘 3 ∫ 𝑒 1 𝑑𝑡] + 𝑘 𝑑𝑝 𝑥 2 2 -(I yy -I zz ) 𝑥 4 𝑥 6 + 𝐽Ω 𝑟 𝑥 4 𝑢 3 = I yy [𝑥3 𝑑 + (𝑘 4 (I xx -I yy ) 𝑥 2 𝑥 4 𝑥 3𝑑 = 𝑚 𝑢 1 [𝑥7 𝑑 + (𝑘 10 2 + 𝑘 12 -1)𝑒 7 -(𝑘 10 + 𝑘 11 )𝑒 8 + 𝑘 10 𝑘 12 ∫ 𝑒 7 𝑑𝑡] + 𝑘 𝑑𝑥 𝑥 8 𝑢 1 -(𝑥 5 𝑥 1 ) 𝑥 1𝑑 = -𝑚 𝑢 1 [𝑥9 𝑑 + (𝑘 13 2 + 𝑘 15 -1)𝑒 9 -(𝑘 13 + 𝑘 14 )𝑒 10 + 𝑘 13 𝑘 15 ∫ 𝑒 9 𝑑𝑡] -𝑘 𝑑𝑦 𝑥 10 𝑢 1 + (𝑥 5 𝑥 3 ) 2 + 𝑘 6 -1)2 -𝑢 1 = 𝑚[𝑥1 1𝑑 + (𝑘 16 2 + 𝑚𝑔 (3.29)

Table 3

 3

		-1: RMS error in cm
		x-axis	y-axis	z-axis
	LQR	27.9	14.5	13.3
	LQR-MFC	17.8	9.99	4.93
	NIB	20.1	11.5	12.3
	NIB-MFC	13.0	8.00	5.62
	3.7.2 Disturbance Test (Circular-Shape Path)

Table 4 -

 4 1: RMS error values between the actual and the estimated outputs for Test 1

		x [cm]	y [cm]	z [cm]	roll [deg]	pitch [deg]	yaw [deg]
	SO	9.368	3.184	5.735	2.200	2.304	4.333
	TO	8.838	2.972	5.435	2.121	2.235	4.039

102 From Table 4 -

 From4 2, the RMS errors of iOE and iTOE are much less than the errors in SO and TO, respectively. And by comparing iOE and iTOE, the difference is negligible where both estimators have almost the same values of RMS errors.

Table 4 -

 4 2: RMS error values between the actual and the estimated outputs Test 2

		x [cm]	y [cm]	z [cm]	roll [deg]	pitch [deg] yaw [deg]
	SO	3.660	3.283	2.096	2.537	6.128	2.263
	TO	3.618	3.226	2.039	2.343	5.700	2.100
	𝒊𝑶𝑬	0.570	0.548	0.737	1.095	2.046	0.396
	𝒊𝑻𝑶𝑬	0.547	0.529	0.704	1.063	1.995	0.382

Table 5 -

 5

		x-axis	y-axis	z-axis
	LQR	7.60	8.87	3.00
	NIB	5.98	7.30	2.90
	LQR-MFC	4.39	4.88	2.69
	NIB-MFC	3.68	4.34	2.38

1: RMS error in cm (excluding the fault, 𝑡 ∈ [35,75]s)

Table 5

 5

	-2: RMS error in cm (including the fault, 𝑡 ∈ [35,105]s)
		x-axis	y-axis	z-axis
	LQR-MFC	7.25	5.81	6.28
	NIB-MFC	4.05	4.34	5.80

  If the actuator fault is bounded, then a positive number 𝜉 can be found such that 𝜉 ≥ ‖𝑓 𝑎 (𝑡)‖. By considering 𝜛 ⊂ ℝ 𝑛 is a bounded set, then the solution of 𝑒(𝑡) is considered to be bounded with respect to 𝜛 if it remains bounded for any finite time interval, and if

	𝑡) 𝑓 ̂𝑎̇1 (𝑡)	] , 𝑒(𝑡) = [ 𝑓 ̂𝑎1 (𝑡) 𝑒 𝑥 (𝑡)	] , 𝐹 ̃= [	𝐹 𝑎 0	], and (𝐴 ̃-𝐿 ̃𝐶 ̃) = ([	𝐴 𝐵𝐾 𝑎 0 0	] -[	𝐿 -𝐿 𝑒	] [𝐶 0])
	Then (5.19) can be expressed as:
		𝑒(𝑡) = (𝐴 ̃-𝐿 ̃𝐶 ̃)𝑒(𝑡) + 𝐹 ̃𝑓𝑎 (𝑡)				(5.20)

Table 5 -

 5 3: Fault Signature Table 𝑺(𝒓 𝒙 ) 𝑺(𝒓 𝒚 ) 𝑺(𝒓 𝒛 ) 𝑺(𝒓 𝝓 ) 𝑺(𝒓 𝜽 ) 𝑺(𝒓 𝝍 )

Schéma du modèle de quadrirotor

  Les principaux travaux développés dans cette thèse traitent de la commande, du diagnostic et de la tolérance aux fautes utilisant le concept de la commande sans modèle récemment proposé dans la littérature. Les méthodes développées sont testées en temps réel et validées sur un quadrotor.Dans un premier temps, le modèle du quadrotor est présenté. Dans cette partie, nous avons défini les repères, les paramètres et les différents mouvements du quadrotor. La dynamique du système est décrite en dérivant le modèle non linéaire en fonction de la dynamique des forces et des moments en tenant compte de la dynamique des actionneurs. Le modèle non linéaire extrait ensuite est linéarisé pour l'appliquer sur les différents algorithmes linéaires qui sont présentés dans ce travail.

	Le modèle non linéaire qui représente le mouvement de quadrirotor est présenté comme
	suit:
			𝜙 θ				𝑝 + 𝑠(𝜙) 𝑡(𝜃) 𝑞 + 𝑐(𝜙) 𝑡(𝜃) 𝑟
			ψ				𝑐(𝜙) 𝑞 -𝑠(𝜙) 𝑟
			ṗ				(𝑠(𝜙) 𝑐(𝜃) ⁄	)𝑞 + (𝑐(𝜙) 𝑐(𝜃) ⁄	)𝑟
	[	q ṙ 𝑦̈𝑒 𝑧̈𝑒 ] ẋ̈𝑒	=	[	(𝑀 𝑥 + (I yy -I zz )𝑞𝑟 -𝐽Ω 𝑟 𝑞) I xx ⁄ (𝑐(𝜓) 𝑠(𝜃) 𝑐(𝜙) + 𝑠(𝜓) 𝑠(𝜙)) . ∑ 𝐹 𝑖 4 𝑖=1 (𝑠(𝜓) 𝑠(𝜃) 𝑐(𝜙) -𝑐(𝜓) 𝑠(𝜙)) . ∑ 𝐹 𝑖 𝑖=1 -𝑔 + (𝑐(𝜃) 𝑐(𝜙)). ∑ 𝐹 𝑖 𝑖=1 𝑚 ⁄ 4 ⁄ 𝑚 4 𝑚 ⁄ (𝑀 𝑦 + (I zz -I xx )𝑝𝑟 + 𝐽Ω 𝑟 𝑝) I yy ⁄ (𝑀 𝑧 + (I xx -I yy )𝑝𝑞) I zz ⁄	] (1)
	Les équations du mouvement dans (1) peuvent être simplifiées et linéarisées, comme
	indiqué en (2).
			𝜙 θ				𝑢 2 I xx ⁄
			ψ				𝑢 3 I yy ⁄
			ẍÿ	=		𝑢 4 𝐼 𝑧𝑧 ⁄ (-𝑘 𝑑x ẋ+ 𝜃 𝑢 1 ) 𝑚 ⁄
			z				(-𝑘 𝑑y ẏ-𝜙 𝑢 1 ) 𝑚 ⁄
	[	̈]		[ (-𝑘 𝑑z ż-𝑚𝑔 + 𝑢 1 ) 𝑚 ⁄ ]

La structure générale du quadrirotor est illustrée par la figure 1. Il est constitué de trois parties principales, à savoir:  Les actionneurs qui sont commandés par des signaux MLI (𝑢 𝑖 * ) et produisent des orientations appropriées (𝐹 𝑖 ).  La géométrie qui représente la relation entre les efforts de poussée générés par les moteurs et les entrées du système.  La dynamique et la cinématique de quadrirotor qui décrivent le mouvement à 6 ddl du quadrirotor basée sur les entrées du système.

Figure 1.

  Après avoir présenté le modèle de quadrirotor, différentes méthodes de contrôle sont proposées. Ces nouvelles stratégies de contrôle robuste sont basées sur la technique de contrôle sans modèle (MFC).L'algorithme MFC aide à compenser les perturbations et les incertitudes des modèles. L'avantage de ce concept récent est la simplicité de la conception du dispositif de commande par l'ajout d'une loi de commande en utilisant des modèles ultra-locaux aux techniques de contrôle classiques.Les deux algorithmes sont vérifiés par des procédures analytiques et expérimentales. La robustesse est vérifiée et comparée en ce qui concerne les contrôleurs nominaux en présence de perturbations et des incertitudes de modèle.Basée sur la conception MFC, la relation d'entrée-sortie peut être représentée par un modèle ultra-local qui est continuellement restructuré:Pour la conception de LQR-MFC, la structure du contrôleur LQR avec son état intégrante est adaptée pour être utilisée avec la MFC. Par conséquent, le MFC avec v = 2 est directement mis en oeuvre pour les contrôleurs de LQR existants.

		𝑦 (𝑣) = 𝐹 + 𝛽𝑢	(4)
	Où, F est une valeur continuellement mise à jour et qui représente la dynamique
	d'ensemble variant dans le temps du système. Il est à noter que la valeur d'estimation
	est valable pour une courte période de temps, et elle doit être continuellement mise à
	jour. En général, l'entrée de commande sans modèle peut être écrite comme suit:
	u = -	F-y d (v) +u c β	(5)
	Pour tester et valider ces nouvelles approches, la commande Linéaire Quadratique(LQR)
	et non linéaire Integral-Backstepping (NIB) ont été considérées en intégrant le concept
	MFC pour concevoir un (LQR-MFC) et un (NIB-MFC), respectivement.

Pour la conception de NIB-MFC, la prédiction de la MFC pour la dynamique du système non modélisée en considérant la dynamique non linéaire connue et modélisée dans le modèle ultra-local: 𝑥̈= 𝑓(𝑥) + 𝐹 + 𝛽𝑢

  Deux vidéos de quadrirotor sont enregistrées pendant le vol en trajectoire circulaire utilisant le contrôleur NIB-MFC. La vidéo qui est nommée "Circular path without disturbance" montre le quadrirotor dans une boucle circulaire en un état normal (sans perturbation), tandis que dans la vidéo "4.Circular path with disturbance", la perturbation est injectée) (https://www.youtube.com/playlist?list=PLpwgGMTzOobanahVWaEvyOAvGllwn8MfC Dans un troisième test en vol, le comportement de la MFC est vérifié lorsque la performance de contrôleur nominal est dégradée. Le quadrirotor est programmé pour suivre une trajectoire carrée. Le résultat de contrôleur LQR, avec ses performances dégradées est comparé au contrôleur LQR-MFC.

	Control is lost	Control is lost
	(c)	(d)

  Comme on peut le voir sur la figure5, le MFC compense la mauvaise performance du contrôleur dégradé et montre une excellente performance de suivi. Cela est dû à la nature puissante de MFC à compenser les incertitudes et les perturbations du système.Pour conclure les travaux de la partie de commande, la meilleure technique, qui est la NIB-MFC, est considérée comme un algorithme de contrôle ultime pour l'ensemble des travaux de cette thèse.La détection et diagnostic de défaut (FDD) est une étape très importante vers le développement de techniques de la tolérance aux fautes (FTC). L'approche FDD développée dans cette thèse est basée sur la génération de résidus entre les résultats mesurés et les sorties estimées obtenues en utilisant des observateurs / estimateurs. Les résidus sont censés être proches de zéro dans le cas sans défaut et différents de zéro en présence d'un défaut ou d'une panne. Cependant, comme les résidus sont générés en utilisant des modèles, ils dépendent fortement de la qualité du modèle utilisé et de la présence de perturbations qui peuvent conduire à de fausses alarmes ou aux nondétections.

	Un nouveau "estimateur intelligent", inspiré du concept de la commande sans modèle a
	été développé et utilisé dans le but d'améliorer la génération de résidus et le diagnostic de
	défaut.
	Deux estimateurs intelligents des sorties du système ont été conçus en intégrant le
	système MF avec l'observateur d'état (iES) et le observateur de Thau (iETS).
	La nouvelle structure de l'iES est proposée comme suit:
	{	𝑥 ̂̇= 𝐴𝑥 ̂+ 𝐵𝑢 + 𝐿(𝑦 -𝐶𝑥 ̂) + 𝑢 𝑖𝑂𝐸 𝑦 ̂= 𝐶𝑥 ̂	(8)
	Dans cet estimateur une hypothèse est faite de telle sorte que tous les états du système
	sont mesurés, et donc la matrice C est une matrice d'identité.
	Le modèle ultra-local, qui représente la relation entrée-sortie, est réalisé comme suit:
		𝑦 ̂̇+ Γ𝑦 ̂= 𝐹 𝑖𝑂𝐸 + Γ 𝑢 𝑖𝑂𝐸

  Tableau 1: valeurs d'erreur de RMS entre le réel et les sorties estimées pour Test 1

	SO	3.660	3.283	2.096	2.537	6.128	2.263
	TO	3.618	3.226	2.039	2.343	5.700	2.100
	𝒊𝑶𝑬	0.570	0.548	0.737	1.095	2.046	0.396
	𝒊𝑻𝑶𝑬	0.547	0.529	0.704	1.063	1.995	0.382
		(a)				(b)
		(c)				(d)
		x [cm]	y [cm]	z [cm]	roll [rad]	pitch[rad]	yaw [rad]
	SO	9.368	3.184	5.735	2.200	2.304	4.333
	TO	8.838	2.972	5.435	2.121	2.235	4.039
	𝒊𝑶𝑬	0.649	0.487	0.654	2.127	2.221	0.152
	𝒊𝑻𝑶𝑬	0.631	0.462	0.650	2.140	2.195	0.141
	Tableau 2: valeurs d'erreur de RMS entre le réel et le sorties estimées Test 2
		x [cm]	y [cm]	z [cm]	roll [rad]	pitch[rad]	yaw [rad]
						(e
						)

  1 )] • (‖𝑒 𝑥 ‖ 2 + ‖𝑒 𝑓 ‖ 2 ) + 𝛽 (B.15) 204 where min(•) denotes the minimum value of a set of numbers. Moreover, based on equation (B.13), the following can be obtained: 𝑉 ≤ max[𝜆 𝑚𝑎𝑥 (𝑃), 𝜆 𝑚𝑎𝑥 (𝛤 -1 )] • (‖𝑒 𝑥 ‖ 2 + ‖𝑒 𝑓 ‖ 2 ) (B.16)where max(•) is the maximum one of the set. Thus

	𝑉 ̇≤ -𝛼𝑉 + 𝛽	(B.17

Table B -

 B 1: AscTec Pelican quadrotor Parameters

		Parameter	Value
			Ixx	0.0081 kg.m 2
			Iyy	0.0081 kg.m 2
			Izz	0.0142 kg.m 2
			m	1 kg
			l	0.24 m
	The dynamic model can be written as following:
	𝐴 = [	0 6×6 𝐼 6×6 0 6×6 0 6×6	] , 𝐵 = [	0 9×1 0 9×3 0 3×1 𝐼 3×3	] , 𝐶 = [𝐼 6×6 0 6×6 ]

Figure 5.16. Effect of 15% actuator fault compensation (on z-and x-axes)

Figure 5.17. Effect of the actuator fault compensation

LQR-MFC Controller:

The reliability of adding the MFC to the control algorithm and the need of having clear plots were the reasons of plotting the second loop only in this test. The boosting in the control performance that the MFC shows is due to the algorithm's capabilities in anticipating and compensating the disturbance. The robustness and the trackability performance of the LQR-MFC can be depicted from the plots provided by Figure 3.12.

Control is lost

Control is lost

Quadrotor Model

The linearized system model is required to design the state observer and iOE, and the nonlinear system model is necessary to build Thau observer and iTOE. The nonlinear model for Qball-X4 vehicle and its linearized version are described in chapter 2.

Quadrotor Control

The Nonlinear Integral Backstepping Model-Free Control (NIB-MFC) controller was proposed in [START_REF] Younes | Nonlinear Integral Backstepping ─Model-Free Control Applied to a Quadrotor System[END_REF] and explained in details previously in Chapter 3, Section 3.5.2, and it will be used to control the Qball-X4 quadrotor.

Quadrotor Setup

The Quadrotor platform used in the control chapter (Chapter 3) will be used in the observers and estimators implementation. For more details refer to Section 3.6.

Output Estimators Implementation

For iOE, the classical SO will be implemented, where SO gain matrix (L) can be computed by different methodologies. e.g. pole placement method or Linear-Matrix-Inequality (LMI) can be used to satisfy the convergence of the states' error.

Then based on SO gain matrix, the iOE gain matrix (Γ) will be chosen such that the eigenvalues of (𝐶𝐿 -Γ) are in the left-hand side of the complex plane.

For iTOE, the observer feedback gain matrix (𝐻), which computed by TO in (4. [START_REF] Noura | Fault-tolerant control systems: Design and practical applications[END_REF], is used in iTOE, while the estimator matrix gain (Λ) of iTOE is computed to satisfy the convergence of the error in (4.28).

Next, the testing and validation stage will be explored in details.

Flight-Test Missions

The purpose of the following tests is to evaluate the effectiveness of iOE and iTOE over SO and TO, respectively, in estimating the outputs of the quadrotor (angles and position outputs) under normal flight conditions and when an actuator fault occurs.

Summary

The main focus in this work is to develop a fault diagnostic and fault-tolerant control technique based on residuals generation. For that reason the output estimation is considered rather than the general state estimation, where the residuals are calculated from the difference between the real outputs measurements and the estimation.

Two output estimators are introduced in this work, where the Model-Free technique is used and implemented in a unique way to estimate the outputs of the quadrotor vehicle.

The integration of MF scheme with the state and Thau observers for a set of MIMO systems is derived and applied on the Qball-X4 quadrotor. The new formulations of this integration are designated by intelligent estimators, where iOE and iTOE represent the integration between the MF technique with the state observer and the Thau observer, respectively.

Real-flight results have compared and show superior performance of the intelligent estimators (iOE and iTOE) over SO and TO towards fault-free and actuator fault conditions. As an ultimate solution, iOE with its simple construction is preferred to be used over iTOE on the quadrotor system, where the results from both estimators are found to be very close due to the fast updating nature of the MF scheme that anticipates the system uncertainties including the nonlinearities of the model.

Chapter 5

FAULT DIAGNOSTIC AND FAULT TOLERANT CONTROL

The fault is a malfunction of a system element, where it could be a sensor fault, an actuator fault or a component fault (plant fault). The sensor is considered faulty when it gives measurements different from the true values, while if the actuator does not work properly, then the actuator is considered faulty. On the other hand, the fault on the system component happens when a malfunction affects the component and prevents it from working properly. Sensor and actuator faults of the quadrotor vehicle are considered in this study.

The necessity of designing an FTC system should be considered based on the fault severity that can be detected based on the FDD process. Some of the faults, such as the complete loss of the actuator, could lead to a complete system failure while others can be compensated with or without the degradation of the system's performance.

Before the fault compensation process using a FTC technique, the fault has to be detected and diagnosed using a sequenced methodology that goes through three main phases, which are: fault detection, fault isolation, and fault estimation. The FDI methodology has to be presented first before the estimation process in FDD.

The general schemes of FDD and FTC will be presented in sections 5.2 and 5.3 respectively.

The FDD and FTC methods described in this work are addressing special types of actuator and sensor faults. On the one hand, two types of actuator faults are considered here, which are: constant and variable Loss-of-Effectiveness (LoE) of the actuators. On the other hand, a bias of the sensor that measures the position data is addressed.

Fault Representation

The overall representations of the linear and nonlinear systems that consider the sensor and actuator faults are presented in Section 1.2.2. The state-space representations of the linear and nonlinear systems affected by an actuator and sensor faults are recalled here:

where 𝑓 𝑎 (𝑡) and 𝑓 𝑠 (𝑡) represent the unknown value of the actuator and sensor faults that need to be estimated for later use in the FTC.

Fault Detection & Diagnosis Scheme

The FDD scheme described in Section 1.2.3 is utilized in this work and recalled in Figure 5.1.

The general FDI scheme presented in Section 1.2.3 and recalled in Figure 5.2 will be used in this work to detect and isolate the sensor fault introduced to the quadrotor system.

The process of implementing the sensor fault is described in Section 5.4.

Fault-Tolerant Control Scheme

In this work, a complete active FTC scheme will be designed by reconfiguring the controller to overcome the faults caused by the loss of actuator effectiveness, while an active FTC structured algorithm will be synthesized to compensate for the sensor faults.

In the case of the actuator fault, the active FTC algorithm will reconfigure the control law based on the information provided by the FDI and FE units. The FDI aims to detect the fault and provide the information related to its nature and severity, and then the fault estimation will take place to provide the control law with the compensation value.

For the sensor fault, the active FTC structured algorithm will serve to compensate for the sensor fault based on its nature and scenario. The structured algorithm will obtain the information from the FDI module that determines the time and the location of the fault.

Based on that, the appropriate time to estimate the magnitude of fault will be decided.

The estimation will be used as redundant information about the system's variables, and the redundancy will be utilized to provide appropriate values to compensate for any misleading data from the sensors' readings by regenerating the desired path.

The active FTC scheme that considers both sensor and actuator faults is depicted in where 𝐿 ∈ ℝ 𝑛×𝑞 is the observer gain matrix and 𝑢 𝑖𝑂𝐸 ∈ ℝ 𝑛 is the feedback term from the MF structure. The state estimation error and its derivative can be written as follows:

𝑒 𝑥 (𝑡) = 𝑥(𝑡) -𝑥 ̂(𝑡) (5.5)

The goal here is to compensate for the actuator fault represented by the term 𝐹 𝑎 𝑓 𝑎 (𝑡). Augmenting the estimation of the actuator fault to the control law of the closedloop control system is an ideal solution to compensate for the actuator fault [START_REF] Noura | Fault-tolerant control systems: Design and practical applications[END_REF]. The actuator-fault-control-input 𝑢 𝑎 (𝑡) resulted from the fault estimation can be combined with the control input from the controller, and the new control law can be defined as [7]:

where 𝑢 𝑐 (𝑡) is the control input of the controller in the fault-free case. In the case of the fault, 𝑢 𝑎 (𝑡) will compensate for the actuator fault 𝑓 𝑎 (𝑡) that affects the output of the closed-loop control system.

The actuator-fault-control-input can be designed from the estimation of the actuator fault, 𝑓 ̂𝑎(𝑡), 𝑢 𝑎 (𝑡) = -𝐾 𝑎 𝑓 ̂𝑎(𝑡) (5.7)

where 𝐾 𝑎 ∈ ℝ 𝑚×𝑚 is the compensation gain matrix of the actuator fault. And the negative sign is considered to cancel the effect of the actuator fault.

Substituting (5.6) in (5.3) will lead to:

NIB-MFC control inputs were designed and the convergence of the system's outputs to the desired references was presented in Chapter 3. In order to remove the effect of the actuator fault, 𝑢 𝑎 (𝑡) must be designed as:

𝐵𝑢 𝑎 (𝑡) + 𝐹 𝑎 𝑓 𝑎 (𝑡) = 0 (5.9)

Therefore 𝑢 𝑎 (𝑡) can be calculated from the estimation of the actuator fault by: 𝑓 ̂𝑎(𝑡) = 𝑓 ̂𝑎1 (𝑡) + 𝑓 ̂𝑎2 (𝑡) (5.14) Again, in iOE analysis, all the variable states are considered to be measured (n=q), which is the case in the quadrotor platform.

By considering the fault estimation included in (5.8)

For the system stability analysis with the actuator fault, the intelligent estimator term can be omitted from the analysis by considering:

𝐵𝐾 𝑎 𝑓 ̂𝑎2 (𝑡) -𝑢 𝑖𝑂𝐸 (𝑡) = 0 (5.17)

𝐵𝐾 𝑎 𝐿 𝑒 𝑢 𝑖𝑂𝐸 (𝑡) -𝑢 𝑖𝑂𝐸 (𝑡) = 0

Then, this can be satisfied, if and only if, (𝐵𝐾 𝑎 𝐿 𝑒 = 𝐼). Then (5.16) can be written as follows:

The augmentation structure proposed in [7] will be used to satisfy the stability analysis of the system while an actuator fault is considered.

A combined system that considers the fault estimation 𝑓 ̂𝑎1 (𝑡) and (5.18) can be augmented in the following representation:

where 𝔣(𝑡) is a real-time noisy measurement that can be differentiated with a Lipschitz's constant 𝐶 > 0, and 𝓍 is an auxiliary input that its rate equals to the control law 𝔲, where, 𝓍̇= 𝔲 (5.28)

From (5.27), the derivative of the sliding surface can be written as:

The control law can be obtained based on the super twisting algorithm [START_REF] Levant | Robust exact differentiation via sliding mode technique[END_REF]:

where the 𝜔 and 𝜆 are strictly positive constants.

The convergence in a finite time (𝓍̇-𝔣 (𝑡) = 0 → 𝔲 = 𝔣 (𝑡)) will be satisfied under the following conditions [START_REF] Levant | Robust exact differentiation via sliding mode technique[END_REF]:

The SMRD architecture is depicted in Figure 5.11, where a smoothed signal can be estimated from the differentiator as shown below: The effect of the fault compensation on the quadrotor's response is depicted in Figure 5.16. Despite the delay found in the FDD and FTC algorithms, the compensation process works as a booster that helps the control algorithm to accommodate for the fault and then enhance the system performance as depicted in the figure below.

Fault occurrence

Fault occurrence

Square-path Test with 15% LoE Actuator Fault:

As illustrated in the hover test, the FTC algorithm will compensate for the 15% LoE actuator fault but this time when the quadrotor is asked to follow the square path presented in Section 5.4. The fault is injected at t = 75 sec on the front motor as well.

Two different flight tests are applied to follow the square path, one with fault compensation and the second without fault compensation as shown in Figures 5.17 The system performance resulted from the compensation can be seen at the altitude, where the compensation boosted the value of the PWM of the front rotor, and this the overall thrust of the quadrotor.

Fault occurrence

Square-path Test with 25% LoE Actuator Fault:

Two experiments are conducted for this test as the quadrotor is asked to follow a square-path, where a 25% LoE of the front actuator is injected at t = 75s. The first experiment is conducted without the fault compensation process while the second one considers the fault and compensates it. The 3D responses of the vehicle for both tests are depicted in Figure 5.21. From Figure 5.21, it can be seen that the fault compensation handled the effect of the 25% LoE actuator fault after it is injected at t = 75s, while the controller fails to compensate for the fault without the AFTC.

In the AFTC experiment, the fault magnitude computed from the fault estimation process approximately equals to (-0.012) and can be computed from equation (5.34). It can be noticed that the fault accommodation process takes some time to start. The effect of the fault compensation on the slow-response systems would be more tangible compared to the fast-response systems (e.g. the quadrotor vehicle). In general, the accommodation of higher severity faults using the AFTC proves the advantage of using it instead of relying only on the capabilities of rejecting the faults provided by the NIB-MFC without the use of the AFTC.

Loss of Effectiveness Actuator Fault (Ramp-Type)

The following flight test will examine the behavior of the quadrotor vehicle towards a ramp-type LoE actuator fault on the left motor.

In this test, the quadrotor will hover at the origin with an altitude of (z = 0.95 m). The left motor will lose its performance gradually by following a predefined declined slope.

The slope is chosen to be equal to (-0.005𝑡) and it starts at t = 30 sec.

The FDD algorithm will estimate the control inputs of the system. Therefore, the effect of the LoE actuator fault on the motor can be reflected on the control inputs.

Hence, the effect of the ramp-type fault on the left motor is related to the control input of the roll angle (𝑢 2 ), where the subtraction of the third and the fourth equations in (5.32) will yield to:

As discussed in Section 5.5.1, the opposite pair of the rotors will produce almost equivalent thrust, and therefore they need to be provided by the same voltage or PWM signals. Therefore, the effect of the declined slope on the control input (𝑢 2 ) can be calculated as follows:

The real fault magnitude value computed in (5.36) and the estimation resulted from the FDD can be seen in Figure 5.23. The delay in the estimation is caused mainly by the filtering process of the noise resulted from the estimation process, as shown in Figure 5.24, and also from the time needed for the FDD strategy as explained in details in Section 5.5.1.

The fault estimation 𝑓 ̂𝑎(𝑡) depicted in Figure 5.24 is computed from (5.13) using the fault estimation technique that is based on the iOE estimator, and the noise reduction of the fault estimation is processed using the multi-stage SMRD-LPF filter.

effects by producing a negative control law of the real fault value as depicted in Figure 5.25. On the other hand, based on the fault compensation technique obtained by (5.7), 𝑢 𝑎 (𝑡) will be computed and then added to the control law as presented in (5.6).  In the flight test that considers the fault compensation process, the control input stabilized around another operating point and did not converge back to zero, and this happened because of the delay resulted from the estimation process. On the other hand, the system lost the control after a certain time where the estimated value kept increasing and added to the control law until the voltage provided to the motor reached the maximum limit.

[N.m]

 The system ran for a longer period of time in the fault compensation case because of the saturation limits provided by the developer of the Qball-X4 on the altitude of the quadrotor, and this will be explained in details next.

The effect of the fault accommodation on the altitude of the quadrotor can be depicted in Figure 5.26. Under the non-compensation case, the saturation limits implemented on the control inputs will stop the quadrotor from keep descending even with the continuous decreasing on the thrust resulted from the actuator fault as shown in Figure 5.26. This will be at the expense of the other control inputs because of the coupling between them, therefore the system will lose the control before the expected time but this is not the case when the fault is accommodated.

In this type of actuator fault, the advantages of using the AFTC strategy is tangible and noticeable on the quadrotor response by returning it to its desired altitude as shown in the Figure 5.26 below.

Sensor Fault Applied on the Quadrotor

Sensor faults could affect the stability of the quadrotor's closed-loop system or provide misleading information about the system's mission.

In this work, a bias sensor fault on the localization data of the quadrotor position (x, y or z) will be considered. A structured algorithm will be introduced in this section to estimate the fault caused by the bias sensor fault. After detecting and isolating the sensor fault, the information will be fed to an algorithm to estimate the fault, and based on the estimation, the compensation process will be achieved. Figure 5.29 shows the breakdown architecture of the FDI and fault estimation and compensation for the bias sensor fault.

In order to detect and isolate the sensor fault, residuals between the measured and the estimated outputs are generated. Based on a certain pattern of the residuals' symptoms, represented by a pre-defined Fault Signature Table, the Decision Making Unit will decide which sensor is faulty. The FDI process can be seen in Figure 5.2 and also in the upper block of Figure 5.29.

When the FDI algorithm detects and isolates the sensor fault, an alarm will be sent to activate the fault estimation algorithm. The fault estimation process is represented by the structured algorithm that is illustrated in the middle block of Figure 5.29. The objective of the "Running maximum error" script is to continuously update the value of the maximum error during the flight mission after proper initialization. In case that the change happens in the desired reference under the fault-free condition, a reset command will be sent as mentioned previously, then the block will start to find the maximum error again.

On the other hand, if the desired reference value stayed the same without any changes within the past short time span, then the "Desired reference change" block will send an indication about the occurrence of the sensor fault and the fault estimation will be calculated as the maximum error value at that time.

The reason of scanning the past short time span is that the FDI process will detect the fault after some time from its occurrence time. This delay will provide wrong estimation of the fault value, where the convergence value between the time that the fault occurs and the time needed for the FDI process will not be calculated within the fault estimation value. The "Running maximum error" block will resolve this issue by providing the value of the maximum error and the time of this value, and by that a better estimation of the sensor fault will be extracted and the fault occurrence time from FDI will be adjusted.

After that the information about the fault estimation will be fed to the fault compensation algorithm, where the desired path will be re-generated to accommodate for the sensor fault, as it will be implemented and discussed in Section 5.6.3.

An experimental flight test of the quadrotor with a bias fault in the y-axis sensor data is done and presented in the following sub-sections. The bias has a magnitude of 0.3 m and is injected at t = 40s in the hover flight condition. The fault detection, isolation, estimation and compensation processes are achieved according to the methodology presented in this section and depicted in Figure 5.29.

Fault Detection and Isolation for Sensor Faults

The first step of the fault detection process is to obtain the system's outputs estimation from an observer/estimator design. In this experiment, the estimation of the outputs is extracted by using the iOE design presented in Chapter 4. As discussed in Section 4.3, the iOE was a compromised estimator design that provides a balanced solution between the Chapter 6

CONCLUSIONS AND RECOMMENDATIONS

The safety is always a priority of any work or operation. In machines, any fault could lead to a catastrophic failure not only to the machine but also it could threaten the life of its user(s). Therefore, the researchers are interested always to find a remedy for the main causes of the faults by designing different FDD and FTC strategies to cancel or reduce the fault effects. Further to the safety, the overall improvements in the system's availability and reliability are essential in the modern control system applications.

The research done in this thesis concerns about special types of actuator and sensor faults on the quadrotor system. Therefore, new strategies and techniques are proposed in the different fields (quadrotor control, FDD, and FTC) to accommodate for the effect of the fault on the system. The quadrotor vehicle is chosen to be the machine that is used in this study for different reasons that relate to the flexibility of operation, maneuverability, challenging nature of the system dynamics, availability and its wide-range civilian applications. Following to this, a summary of the thesis is presented next.

Thesis Summary

The methodologies presented and proposed in this thesis are applied on the quadrotor vehicle. Therefore, describing the machine by knowing the nature of its operation and extracting its dynamics are important tasks before proposing any algorithm related to it.

6. Study the effect of the estimation delay on the system performance by enhancing the filtering technique used in the estimation.

Investigate the estimation robustness of the Nonlinear Identity Observers presented in

Appendix A and the computational analysis management in implementation.

8. Apply the FDD and FTC techniques using the Adaptive Thau Observer, which is proposed by the UAEU-UVL team in Appendix B, and compare it with the FDD and FTC methodologies based on the intelligent estimator designs.

APPENDICES Appendix A

A FDD FOR SENSOR FAULT USING NONLINEAR IDENTITY OBSERVER

The nonlinear identity observer has been introduced by D. Hengy and P. M. Frank in [START_REF] Hengy | Component failure detection via nonlinear state observers[END_REF]. The structure of Nonlinear Identity Observer approach is suitable for the quadrotor nonlinear model. Therefore, this type of observer is utilized in this work side by side with the Generalized Observer Scheme (GOS).

In this research an FDI strategy based on the Nonlinear Idenity Observer is studied and simulated on the AscTec Pelican quadrotor.

A.1 Nonlinear Identity Observer Structure

The nonlinear system that matches the quadrotor model can be described as following:

where 𝑥 ∈ ℝ 𝑛 is the state vector, 𝑢 ∈ ℝ 𝑚 is the input vector, 𝑦 ∈ ℝ 𝑝 is the output vector of the quadrotor model. The identity observer structure will be defined as follows:

where the gain matrix ℎ(𝑥 ̂, 𝑢) is dynamic and its elements are adjusted according to the observed states.

At the beginning, the observer will be designed on the assumption that the system is in free-fault condition [START_REF] Frank | Nonlinear observers for fault detection and isolation[END_REF].

The state error and the output error can be described as following:

The derivative of the state error will be:

A Taylor series expansion of 𝑓(𝑥, 𝑢) around e = 0 can be written as following [START_REF] Frisk | Model-based fault diagnosis applied to an SI-Engine[END_REF]: The higher order terms (h.o.t) in the Taylor expansion will be neglected as shown in (A.6) [START_REF] Abid | Fault detection in nonlinear systems: An observer-based approach[END_REF]. Now, the state error derivative will be presented as:

To reduce the complexity of the written equations, let:

Then, 𝑒̇= (𝜕 𝑥 𝑓 -ℎ(𝑥 ̂, 𝑢)𝐶)𝑒 (A.9) ℎ(𝑥 ̂, 𝑢) should be designed so that the error will asymptotically converge to zero when 𝑡 → ∞ . To ensure the convergence of the state error, the Lyapunov methodology will be utilized. Therefore, the following positive definite Lyapunov function will be adopted:

where P is a positive definite matrix. The Lyapunov function derivative can be written as following:

The rate of the state error in (A.9) can be substituted on (A.11) and yields:

𝑉 ̇(𝑒) = 𝑒 𝑇 𝑃(𝜕 𝑥 𝑓 -ℎ(𝑥 ̂, 𝑢)𝐶)𝑒 (A.12)

The dynamic gain matrix should be designed so that (A.12) should be negative semidefinite. The author in [START_REF] Tsinias | Observer design for nonlinear systems[END_REF] proposed a methodology to calculate the gain matrix ℎ(𝑥 ̂, 𝑢)

based on 𝐾𝑒𝑟(𝐶) ≠ 0. This technique has been adopted and generalized by [START_REF] Adjallah | Non-linear observer-based fault detection[END_REF] in two steps: the first step is to determine matrix P and the second one is to find ℎ(𝑥 ̂, 𝑢) using P.

According to [START_REF] Adjallah | Non-linear observer-based fault detection[END_REF], the first step is summarized by finding matrix P from:

𝑉 ̇(𝑒) = 𝜀 𝑇 𝐾 𝑇 𝑃𝜕 𝑥 𝑓 𝐾𝜀 (A. 13) where K is right orthogonal to C, and 𝜀 is the transformation of 𝑒 when 𝐾𝑒𝑟(𝐶) ≠ 0.

Given that the dimension of 𝜀 is less than 𝑒, matrix P should be found to satisfy the negativity of 𝑉 ̇(𝑒) in (A.10).

In the second step, the value of P from the first step will be utilized, as an identity matrix, to determine ℎ(𝑥 ̂, 𝑢) that satisfies the convergence of Lyapunov function derivative:

Because P is positive semi-definite, the convergence will be verified if the term 𝜕 𝑥 𝑓 -ℎ(𝑥 ̂, 𝑢)𝐶 is negative semi-definite. This will be attained by choosing ℎ(𝑥 ̂, 𝑢) such that:

where 𝐹(𝑥 ̂, 𝑢) ∈ ℝ 𝑛×𝑛 is a matrix that verifies the inequality: ‖𝐹(𝑥 ̂, 𝑢)‖ > ‖𝑃𝜕 𝑥 𝑓 ‖, and 𝑄 ∈ ℝ 𝑝×𝑝 is a matrix satisfying 𝐶 𝑇 𝑄𝐶 -𝐼 ≥ 0.

The matrix valued function 𝐹(𝑥 ̂, 𝑢) is given by [START_REF] Frank | Nonlinear observers for fault detection and isolation[END_REF]:

where 𝜓 𝑖𝑗 is the 𝑖𝑗 𝑡ℎ element of 𝑃𝜕 𝑥 𝑓 .

All in all, the observer could be written and depicted in

A.2 Fault Estimation

In the presence of the sensor faults 𝑓 𝑠 ∈ ℝ 𝑝 , equation (A.1) could be written as:

Then the state estimation error equation becomes:

Structured Residuals: For the quadrotor vehicle, twelve observers are used; each estimates one state and observes the rest. Consequently, each observer will drop one observed measurement from the calculation of the gain matrix ℎ(𝑥 ̂, 𝑢). In its turn each residual will have eleven observed states and one estimated state. According to restriction of GOS, one fault is considered at a time. The decision-making logic is constructed empirically by means of observation and experimentation. This observation is summarized in the fault signature table (Table A-1).

1 1 𝒇 𝒙 0 0 0 0 0 0 1 0 0 0 0 𝒇 𝒚 0 0 0 0 0 0 0 0 1 0 0 𝒇 𝒛 0 0 0 0 0 0 0 0 0 0 1 𝒇 𝒙̇ 0 0 0 0 0 0 1 1 0 0 0 𝒇 𝒚̇ 0 0 0 0 0 0 0 0 1 1 0 𝒇 𝒛̇ 0 0 0 0 0 0 0 0 0 0 1

In the fault signature table, the alarm will be fired if a group of residuals becomes active as shown in table1. Each residual in the table is a vector of residuals for the same state (from all observers), and it will be activated according to a certain formation of that vector. A bias or drift fault is injected to generate the formations and the residuals shown in the fault signature table.

A.5 Simulation Results

The nonlinear model of the pelican quadrotor, control law and FDI are simulated using Matlab/Simulink environment.

Two simulation scenarios are tested and validated in this section. In the first scenario, a bias type fault is injected in the 𝑥 position, and in the second one a drift fault is inserted on the roll angle ∅.

A.5.1 Offset in the sensor reading of x-axis

An offset of 2-m is injected at t = 10 s. As depicted in Figure A.3, the fault will not be compensated by the controller, because the sensor feedback is biased by 2-m, where the green and the blue curves represent the desired and the actual trajectories respectively.

Hence, an estimation of the fault and a proper FTC system should be implemented to overcome the change caused by the faulty sensor. In this work, the FDI system detects and isolates the fault using the GOS of multiple nonlinear identity observers, and the FTC system will be addressed in future work. In this scenario, the position controller will compensate the faulty sensor reading for a certain time, until the drift becomes large enough so that the controller cannot handle it.

A.6 Summary

In this research, a fault detection and isolation method is presented for the sensors fault used in the quadrotor vehicle. The FDI is based on nonlinear identity observer to detect the fault and GOS to isolate it. The performance of the observer and its scheme is investigated through several simulation results to build the DMU. Various simulation results have validated the FDI method and shown the system performance towards the different scenarios of the sensors faults.

Two main drawbacks make it difficult to implement this work on a real quadrotor.

The first drawback is the huge amount of the computational analysis needed at each sample time (iteration), where the observer's equations are complicated and executing those equations on each observer found on the GOS structure make it even more difficult.

The second drawback is with the questionable estimation robustness of the Nonlinear Identity Observer, where it's not studied in the literature. Therefore and alternative observer design is proposed and implemented on this project.

Appendix B

B FDD for Actuator Fault using Adaptive-Thau-Observer

Thau introduced an asymptotic stable observer that observe the states of a nonlinear system, and Raghavan continued Thau work by formulating a procedure to obtain the observer gain matrix by solving the Riccati equation [START_REF] Raghavan | Observers and compensators for nonlinear systems, with application to flexible-joint robots[END_REF].

In this section, the Thau observer will be used to detect the faults based on residuals generation and evaluation, as well as, a novel scheme that uses the adaptation of the Thau observer will be introduced to estimate the fault based on the observer gain.

B.1 Fault Detection and Diagnosis

In

B.2.2 Fault Scenario 2

The actuators of the quadrotor are actually the four rotors installed in a cross configuration, but it is transformed into Roll, Pitch, Yaw and Throttle command inputs It can be seen that the estimation has a good convergence to the fault signal and relatively high accuracy for estimation. As it is not directly estimated and the estimation of the motor output F is derived from the estimation of system input U, it has a little error in the numerical calculation, but it is still suitable to provide a useful information for a Fault-Tolerant Control that can be used in later work.

Remarks:

The simulation results for fault scenario 1 and 2 demonstrate the fault estimation result for different fault injection ways with different time-varying natures.

Although the faults considered are additive faults, it is easy to be transformed as multiplicative faults by changing the fault matrix F with reference to matrix B. Actually, the fault considered here is a kind of multiplicative fault with 1/5 performance drop, but it is injected in the format of an additive fault. From both scenarios it can be seen that, the proposed Adaptive Thau Observer can estimate the fault offset parameters accurately that can be used in an Active fault-Tolerant Control strategy later.

B.3 Summary

The adaptive Thau observer is presented to estimate the quadrotor rotor fault magnitudes.

Different flight test scenarios showed the residual generation and the fault estimation based on the proposed method that is applied on the AscTec Pelican quadrotor.

As a co-author of this work, future work is planned with the UAEU_UVL team to implement this FDD algorithm on the Qball-X4 quadrotor and compare it with the intelligent observers that are proposed in this thesis.

The FTC algorithm proposed in the same research work by Cen in [START_REF] Cen | Systematic fault tolerant control based on adaptive Thau observer estimation for quadrotor UAVs[END_REF] proposed in this research work. Using additional sensors will make the hardware redundancy approach a paramount factor in safety and operational reliability. Even though the hardware redundancy approach proves impractical in certain applications, due to cost of installation and maintenance, but advancements in the cutting-edge electronics technology that provides low cost, low power consumption and small size sensing devices makes it a suitable approach in UAV systems [START_REF] Koren | Fault-tolerant systems[END_REF][START_REF] Sadeghzadeh | A review on fault-tolerant control for unmanned aerial vehicles (UAVs)[END_REF].

The proposed FTC will target the altitude control of a quadrotor vehicle. The algorithm makes use of three altitude sensors that provide three altitude measurements.

These measurements will serve a fault detection block which will provide indicators about fault occurrences by calculating residuals. The residuals in return will serve the fault isolation block which will isolate the faulty sensor, given it is used in the closedloop system. Automatic switching between sensors governs the redundancy of the flight of the quadrotor without possible crashing. To ensure the effectiveness of the proposed algorithm, the quadrotor will undergo a flight test along a predetermined path upon which a fault will be injected. The quadrotor used in this test is QUANSER's Qball-X4. The effectiveness of the proposed algorithm is demonstrated by means of real flight tests.

C.1 Fault Detection and Diagnostic and Fault-Tolerant Control

The methodology used for fault detection and fault-tolerant control for the quadrotor UAV is using three altitude sensors (𝑆 1 , 𝑆 2 , 𝑆 3 ) which generate three real-time altitude measurements (ℎ 1 , ℎ 2 , ℎ 3 ). The altitude measurements are obtained using Optitrack localization system and two sonar sensors (further discussed in the next section). The altitude measurements serve as inputs to the residual generation algorithm, which in return generates three residuals as follows

The residuals express a quantitative indication of the amount of error between the three altitude measurements. However, that is not sufficient for an indication of a fault. where the value of the threshold was chosen to be twice the standard deviation of the residuals.

C.2 Testing Platform and Hardware Setup

The testing platform and the hardware setup are presented in Chapter 3, Section 3. These sonar sensors prove to be the perfect choice for such an application, mainly due to their range of operation, ease of installation and low cost. They operate at 3.3v and