Ingkarat Rak-Amnouykit

Ana Milanova

Guillaume Baudart

Martin Hirzel

Julian Dolby

ICLR 2021 Workshop on Security and Safety in Machine Learning Systems EXTRACTING HYPERPARAMETER CONSTRAINTS FROM CODE

Machine-learning operators often have correctness constraints that cut across multiple hyperparameters and/or data. Violating these constraints causes runtime exceptions, but they are usually documented only informally or not at all. This paper presents a weakest precondition analysis for Python code. We demonstrate our analysis by extracting hyperparameter constraints for 45 sklearn operators. Our analysis is a step towards safer and more robust machine learning.

INTRODUCTION

To use machine-learning (ML) operators, data scientists must configure their hyperparameters, usually in the form of constructor arguments. For example, in sklearn [START_REF] Buitinck | API design for machine learning software: Experiences from the scikit-learn project[END_REF], the StandardScaler operator has hyperparameters with_mean and with_std, and the LogisticRegression operator has hyperparameters dual, solver, penalty, etc. Incorrect hyperparameter configurations cause crashes, slowdowns, or sub-optimal accuracy. But configuring hyperparameters correctly is often not easy due to hyperparameter constraints. For example, StandardScaler does not allow with_mean to be true if the input data is sparse, and LogisticRegression does not allow dual to be true unless solver=="liblinear" and penalty=="l2". We need a reliable formal specification of these constraints for dynamic precondition checks, static verifiers, or pruning the search spaces of automated hyperparameter tuning.

Unfortunately, it is difficult to find a reliable formal specification of hyperparameter constraints. Type annotations are insufficient: putting aside the fact that types are not yet widely adopted in Python and often wrong [START_REF] Rak-Amnouykit | Python 3 types in the wild: A tale of two type systems[END_REF], they are also not expressive enough for constraints with logical implications across multiple hyperparameters, or across hyperparameters and data. Hyperparameter tuning tools, such as auto-sklearn [START_REF] Feurer | Efficient and robust automated machine learning[END_REF] or hyperopt-sklearn [START_REF] Komer | Hyperopt-sklearn: Automatic hyperparameter configuration for scikit-learn[END_REF], come with search space specifications. But writing those specifications by hand is tedious and error-prone: for example, they take up 25 KLOC of Python in auto-pandas [START_REF] Bavishi | AutoPandas: Neuralbacked generators for program synthesis[END_REF]. Therefore, these search space specifications often cut corners, making under-approximations (e.g., specifying only one of the types of a union) and over-approximations (e.g., missing constraints). This may be tolerable for search but is unacceptable for verification.

One might be tempted to turn to natural-language documentation for hyperparameter constraints (Baudart et al., 2020b). But even though popular packages like sklearn have high-quality documentation, this is at most semi-formal and not always reliable. The code may raise an undocumented exception. For example, using the techniques in this paper, we found that sklearn's ExtraTreesClassifier raises an exception if bootstrap==False and oob_score==True. But the documentation did not mention this constraint; we submitted a pull request (github.com/scikitlearn/scikit-learn/pull/19444), which the developers confirmed and merged. As another example, the documentation for GradientBoostingRegressor describes the alpha hyperparameter with "Only if loss=='huber' or loss=='quantile'." One might think this is a correctness constraint, but in fact, the code does not raise an exception for it. A reliable formal specification can help debug the documentation.

This paper presents a static analysis for extracting hyperparameter constraints from code implementing machine-learning operators. We focus on Python and sklearn [START_REF] Buitinck | API design for machine learning software: Experiences from the scikit-learn project[END_REF], since sklearn is the most widely-used ML framework today. Our analysis returns logic formulas with constraints over at least two hyperparameters or hyperparameters and data. We can use these as preconditions for dynamic checking at the interface, which is friendlier than raising an exception from deep within the implementation. For better error messages, the formulas are factored to be easily associated with individual exceptions. We can also use these preconditions to prune search spaces for hyperparameter tuning. Moreover, we can envision using them for static verifiers of client code.

We ran our analysis on 45 ML operators from sklearn. On the 30 operators with non-trivial constraints, it achieved 82.9% precision and 99.8% recall. Overall, we hope that our interface specifications make machine-learning libraries more reliable and explainable, and serve as a step towards better formal verification of machine learning systems.

WEAKEST PRECONDITION ANALYSIS

The analysis is standard backwards reasoning, also known as weakest precondition reasoning [START_REF] Hoare | An axiomatic basis for computer programming[END_REF]Leino, 2005;[START_REF] Barnett | Boogie: A modular reusable verifier for object-oriented programs[END_REF], adapted for Python and extended with soundness reasoning about the generated precondition. The analysis starts from a raise exception statement in the code, then computes the precondition that must hold at the start of the enclosing function to prevent the exception being raised. At each step of backward reasoning, our analysis computes WP(stmt, Q, S) → (Q , S ): given a Python statement, a postcondition Q, and a soundness flag S, our analysis returns a pair (Q , S ) of a precondition Q and its soundness flag S . WP(raise E, Q, S) handles Raise statements:

return (False, True)

At a blank raise statement, an exception is certain, so the precondition for not raising it is Q = False and its soundness flag is S = True.

WP(x=RHS, Q, S) handles Assignment statements:

Q ← Q[RHS/x] return (Q , S ∧ mod (RHS) ∩ read (Q ) = ∅)
The precondition Q results from the substitution of left-hand side x with RHS. If the set of locations modified by RHS and the set of locations read by Q are disjoint, then Q is sound, meaning that Q evaluates to true iff after the execution of x=RHS, Q evaluates to true, similarly to O'Hearn et al. (2001). Otherwise, S is false, meaning that precondition Q is potentially unsound.

WP(if E: Seq1 else: Seq2, Q, S) handles If statements:

(Q 1 , S 1 ) ← WP(Seq1, Q, S) (Q 2 , S 2 ) ← WP(Seq2, Q, S) Q ← (E ⇒ Q 1 ) ∧ (not E ⇒ Q 2 ) return (Q , S 1 ∧ S 2 ∧ mod (E)∩read (Q ) = ∅)
The precondition formula is standard. Other statements are Python statements that do not match the syntax of the core subset specified above. These include while, delete, try, and the rest of the Statement nodes specified by the Python AST, as well as Assignments whose left-hand-side is not a variable or an attribute of self, e.g., self.bootstrap. The code for Other propagates Q as is, however, it sets the soundness flag to False, since the statement may have affected Q. Our implementation does propagate exceptions raised from within for loops; when propagating backwards past a for loop, it is treated as Other.

We illustrate with the example from StandardScaler we mention in the introduction. Our analysis infers the sound precondition issparse(X) ⇒ not with_mean:

1 {(( issparse (X) ⇒ not with mean) and (not issparse(X) ⇒ True), True)} equiv. to {(issparse(X) ⇒ not with mean, True)} 2 if issparse (X): There are two sources of unsoundness: 1) the disjointness check fails and 2) Other Python statements such as while loops. Currently we assume that the disjointness check always returns true, which usually holds in practice. (We plan to implement known static analyses that verify disjointness.) A large percentage of preconditions are reported sound meaning that our handling of a core subset of Python is sufficient for this problem. As a concrete example, the analysis infers eight preconditions in ExtraTreesClassifier. All are reported sound. We make some additional assumptions. First, we infer the set of hyperparameters by examining constructor arguments, and assume that the code follows sklearn conventions and stores them as fields of the same name in the operator object. We treat syntactic occurrences of X and y as the only references to the input data. Our results indicate that, like the disjointness check, these assumptions hold true for sklearn. Second, we examine only exceptions that occur in the operator file, which our experiments indicate is insufficient. There were a number of runtime exceptions that our analysis did not catch as they were raised by imported code. In future work, we will extend the analysis with interprocedural and inter-module analysis.

EXPERIMENTAL RESULTS

Methodology. For each operator, we start with carefully crafted schemas for individual hyperparameters that do not involve other hyperparameters or data. Sampling from the domain of these schemas, we generate 1,000 random hyperparameters configurations based on hyperparameters that are relevant to hyperparameter optimization. Then, we create a trial by calling the operator's __init__ function with the hyperparameter configuration, then calling its fit function and checking for dynamic exceptions. We experiment with two kinds of datasets, dense and sparse, resulting in a total of 2,000 trials for each operator. A trial fails if an exception is thrown and it passes otherwise.

The results from the dynamic exceptions are our ground truth. It is possible that some exceptions might not be covered. We translate weakest precondition constraints from the analysis into JSON schema [START_REF] Pezoa | Foundations of JSON schema[END_REF] format and use schema validation to check the hyperparameter configuration against the schema. For example, a trial is a false positive if it fails but its hyperparameter configuration is valid against the schema from the analysis. As another example, a trial is a true negative if it fails and it is invalid against the schema as well.

Research Question 1: How well does our weakest precondition analysis work? We evaluate our analysis on 45 sklearn operators. Table 1 shows performances on 30 operators that have failed trials, excluding 15 operators where all trials succeeded (for those 15 operators, the analysis has 100% precision and 100% recall). For example, the constraints of RandomForestClassifier precisely reject 558 hyperparameter configurations from the failed trials. Currently, the analysis is intraprocedural and it misses exceptions from imported modules. In LogisticRegression, the analysis correctly identifies 594 true positives and 1,368 true negatives across 6 weakest precondition constraints. The remaining 38 trials are false positives where the runtime exceptions are raised by imported code. Overall, our analysis outperforms other methods, with an average F-score of 88.6.

Research Question 2: How effective are hand-written constraints extracted from the documentation? We also validate hyperparameter configurations against the hand-written constraints from the Lale open-source project (Baudart et al., 2020a). The hand-written constraints are extracted by careful examination of the documentation. Out of 30 operators with failed trials, 12 have logical hand-written constraints (for the 18 operators in the table with no such constraints, schema validation always passes.) Table 1 shows that performance of the hand-written constraints is slightly worse. This is mainly because they leave out constraints that appear in the code as exceptions but are missing from documentation. Similarly to the weakest precondition, hand-written constraints miss exceptions in imported modules. On a rare occasion, hand-written constraints reject hyperparameter configurations that are specified in the documentation but do not exist in the code. For example, sklearn's LinearSVC states that the combination of penalty='l1' and loss='hinge' is not supported. However, no exception exists in the source code, resulting in 492 false negatives. Automatic weakest precondition analysis can be used to extract precise hyperparameter constraints and improve documentation.

Research Question 3: How does our approach compare against related work ? Baudart et al. (2020b) extract constraints for sklearn from natural language documentation. While the technique works well for constraints on a single hyperparameter, it does not work as well for logical constraints. Of the 30 operators, 27 operators have the natural-language documentation, but the technique only successfully extracts logical constraints for 7 operators. The remaining 23 operators do not raise exceptions and all their trials are either true positives or false positives. Our weakest precondition analysis outperforms the technique from (Baudart et al., 2020b) as shown Table 1.

RELATED WORK

While backward reasoning and, more generally, verification condition generation have a long history, e.g., [START_REF] Cormac Flanagan | Extended static checking for java[END_REF][START_REF] Barnett | Boogie: A modular reusable verifier for object-oriented programs[END_REF] among other works, as far as we know, we are the first to apply these techniques on Python, whose rich dynamic semantics notoriously complicate static analysis. We posit that our novel "soundness" reasoning is a stride towards effective analysis of Python; while it is unrealistic to model every single construct, one can handle a core subset, assume that the rest of constructs have no effect, but still mark a constraint if affected by those other constructs. The programmer can manually verify soundness of marked constraints. In general, work on static analysis for Python is scarce. Ariadne [START_REF] Dolby | Ariadne: Analysis for machine learning programs[END_REF] explores static analysis of machine-learning libraries and outlines challenges to traditional static analysis techniques and [START_REF] Monat | Static Type Analysis by Abstract Interpretation of Python Programs[END_REF] present type analysis via abstract interpretation; we focus on the specific problem of extracting hyperparameter constraints. Finally, iComment for C [START_REF] Tan | /* iComment: Bugs or bad comments? *[END_REF] and jDoctor for Java [START_REF] Blasi | Translating code comments to procedure specifications[END_REF] have similar goal to ours -reconciling documentation with code and identifying issues with either of them. Our analysis outputs JSON schemas; [START_REF] Habib | Finding data compatibility bugs with JSON subschema checking[END_REF] describe a sub-schema checker for JSON.

CONCLUSIONS

This paper presents a static analysis for Python for extracting weakest precondition constraints. While the core analysis is standard, we extend it with a soundness flag and describe heuristics that make it effective on sklearn code. We automatically transform the analysis results to JSON schemas suitable for validation and search space pruning. In future work, we plan to make our analysis interprocedural.

  3 {((with mean ⇒ False) and (not with mean ⇒ True), True)} equiv. to {(not with mean, True)} 4 if with mean: 5 {(False,True)} 6 raise ValueError("Cannot center sparse matrices : pass 'with mean=False' instead . ") 7 {(True,True)} 8 {(True,True)}

Table 1 :

 1 Summary of results on operators that have failed trials.

		Weakest Precondition	Hand-Written Constraints	NL Docstrings
		F-score precision recall	F-score precision recall	F-score precision recall
	StandardScaler	100.0	100.0 100.0	100.0	100.0 100.0	85.7	75.0 100.0
	RobustScaler	100.0	100.0 100.0	85.7	75.0 100.0	85.7	75.0 100.0
	ExtraTreesRegressor	100.0	100.0 100.0	84.1	72.6 100.0	84.1	72.6 100.0
	ExtraTreesClassifier	100.0	100.0 100.0	84.0	72.4 100.0	84.0	72.4 100.0
	RandomForestRegressor	100.0	100.0 100.0	84.0	72.4 100.0	84.0	72.4 100.0
	RandomForestClassifier	100.0	100.0 100.0	83.8	72.1 100.0	83.8	72.1 100.0
	SGDRegressor	99.7	99.5 100.0	99.7	99.5 100.0	56.1	99.1	39.1
	GradientBoostingRegressor	99.6	99.3 100.0	99.6	99.3 100.0	68.3	99.2	52.1
	QuantileTransformer	99.1	98.3 100.0	99.1	98.3 100.0	99.1	98.3 100.0
	SGDClassifier	98.3	96.7 100.0	98.3	96.7 100.0	30.1	94.5	17.9
	Ridge	98.1	100.0	96.4	90.4	82.4 100.0	90.4	82.4 100.0
	RidgeClassifier	98.1	100.0	96.4	90.4	82.4 100.0	90.4	82.4 100.0
	KNeighborsClassifier	97.8	95.6 100.0	47.7	93.6	32.0	97.8	95.6 100.0
	KNeighborsRegressor	97.8	95.6 100.0	47.7	93.6	32.0	97.8	95.6 100.0
	BaggingClassifier	97.1	94.3 100.0	75.1	60.2 100.0	-	-	-
	LogisticRegression	96.9	94.0 100.0	92.4	85.8 100.0	45.8	29.7 100.0
	RFE	96.5	93.3 100.0	96.5	93.3 100.0	-	-	-
	PassiveAggressiveClassifier	95.5	91.3 100.0	95.5	91.3 100.0	60.2	87.4	45.9
	Nystroem	93.9	88.5 100.0	93.9	88.5 100.0	93.9	88.5 100.0
	FunctionTransformer	93.8	88.3 100.0	75.5	83.9	68.6	93.8	88.3 100.0
	DecisionTreeRegressor	90.0	81.8 100.0	90.0	81.8 100.0	90.0	81.8 100.0
	LinearSVC	86.2	75.8 100.0	80.6	100.0	67.5	86.2	75.8 100.0
	NMF	83.7	71.9 100.0	83.7	71.9 100.0	83.7	71.9 100.0
	PCA	75.0	60.0 100.0	9.8	100.0	5.2	75.0	60.0 100.0
	GaussianNB	66.7	50.0 100.0	66.7	50.0 100.0	66.7	50.0 100.0
	MinMaxScaler	66.7	50.0 100.0	66.7	50.0 100.0	66.7	50.0 100.0
	QuadraticDiscriminantAnalysis	66.7	50.0 100.0	66.7	50.0 100.0	66.7	50.0 100.0
	MissingIndicator	65.8	49.0 100.0	65.8	49.0 100.0	65.8	49.0 100.0
	GradientBoostingClassifier	64.4	47.5 100.0	64.4	47.5 100.0	64.4	47.5 100.0
	FeatureAgglomeration	30.1	17.7 100.0	30.1	17.7 100.0	-	-	-
	average	88.6	82.9	99.8	78.3	77.7	90.2	77.6	74.7	90.9


Zhong Liu

Didier Theilliol
email: didier.theilliol@univ-lorraine.fr

Liying Yang

Yuqing He

Jianda Han

Observer-based linear parameter varying control design with unmeasurable varying parameters under sensor faults for quad-tilt rotor unmanned aerial vehicle

Keywords: Tilt rotor, UAV, LPV system, unmeasurable varying parameter, sensor fault

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

= mass, kg O ⋆ x ⋆ y ⋆ z ⋆ = coordinate system, subscript ⋆ =b, e, or w for different coordinate systems q, q = pitch rate and its estimation, rad/s R = rotor radius, m R vu (•, •), R -1 vu (•, •) = algebraic relation between applied and virtual control inputs, its inverse function s = wing area, m 2 u(t), ũ(t) = applied control input vector and control inputs from inverse procedure V , V = flight velocity and its estimation, m/s V ṗ = a polytope of the vertexes of ṗ v(t) = virtual control input vector v V , v q , v α = virtual control inputs for V , q, and α W 1 (p), W 2 (p) = parameter-dependent and invertible matrices for K(p) and L(p) W 1 , W 2 = constant versions of W 1 (p) and W 2 (p) x r = longitudinal distance from the rotor to center of gravity, m x(t) = an augmented vector about system states X(p), Y (p) = parameter-dependent and positive definite matrices for stability analysis X, Y = constant versions of X(p) and Y (p) α, α = angle of attack and its estimation, rad γ, γ = tracking angle and its estimation, rad δ e = deflection of elevator, rad ζ 1 (t), ζ 2 (t), ξ(t)

= intermediate vectors for stability analysis θ = pitch angle, rad ρ = air density, kg/m 3 τ = the upper bound of H ∞ disturbance attenuation ϕ 1 , ϕ 2 , ϕ x = scalars for controller and observer design Ψ 1 (t), Ψ 2 (t), Ψ x (t) = zero vectors, which introduce some inherent relationships for stability analysis Ω B , Ω F = backward and forward rotor speeds, rpm Superscript -1, T = inverse and transpose of a matrix -T = transpose and inverse or inverse and transpose of a matrix ˆ= estimation of corresponding variable or vector Subscript b, e, w = body-axis, north-east-down, and wind-axis coordinate systems

Introduction

As representations of unmanned aerial vehicles (UAVs) with conventional structures, fixed-wing UAVs (FWUAVs) and rotorcraft UAVs (RUAVs) have played irreplaceable roles for a long time in the fields of commerce, agriculture, and military [START_REF] Liu | Control techniques of tilt rotor unmanned aerial vehicle systems: A review[END_REF]. However, FWUAVs are affected by the take-off environment and cannot hover at a fixed position. The cruise speed and flight endurance of RUAVs are lower compared with FWUAVs. To break the above limitations and combine advantages from conventional structures, many hybrid UAVs that could hover at a fixed position and cruise at a high speed have been investigated and designed [START_REF] Saeed | A review on the platform design, dynamic modeling and control of hybrid UAVs[END_REF]. Tilt rotor UAVs (TRUAVs) that rely on wings and rotors to generate lift together are included therein.

In contrast to tail-sitter UAVs [START_REF] Saeed | A review on the platform design, dynamic modeling and control of hybrid UAVs[END_REF] and convertiplane UAVs [START_REF] Tyan | Comprehensive preliminary sizing/resizing method for a fixed wing-VTOL electric UAV[END_REF], TRUAVs are equipped with tiltable rotors. According to the rotor-tilt angle, TRUAVs have two primary flight modes called helicopter mode and airplane mode [START_REF] Liu | Control techniques of tilt rotor unmanned aerial vehicle systems: A review[END_REF], in which the directions of rotor thrusts are vertical and horizontal, respectively. During the transition procedure between these two modes, the rotors could tilt in order to provide acceleration or deceleration. This means TRUAV is accompanied by varying structure and dynamics. Due to the development of TRUAVs with progressively symmetric structures, flight control is easier with fixed rotor-tilt angles compared with transformable cases. For example, [START_REF] Sato | Flight controller design and demonstration of quad-tilt-wing unmanned aerial vehicle[END_REF] and [START_REF] Wang | Mathematical modeling and control of a tilt-rotor aircraft[END_REF] applied the classical proportional-integral-derivative (PID) method and feedback linearization method for quad-TRUAV control in helicopter and airplane modes. However, flight control during the transition procedure with varying rotor-tilt angles is still challenging [START_REF] Liu | Control techniques of tilt rotor unmanned aerial vehicle systems: A review[END_REF] and always requires complex controller structures.

Due to linear control methods could only ensure the local stability under nonlinear controlled plants, linear controllers cannot exhibit performances throughout the transition procedure that is accompanied by varying dynamics. Moreover, typical nonlinear control methods, such as backstepping and feedback linearization, usually require strict-feedback or affine nonlinear form of controlled plant, but coupling of control effects from rotors and aerodynamics complicates the nonlinear form, which limits direct application [START_REF] Wang | Mathematical modeling and control of a tilt-rotor aircraft[END_REF] of these methods. To deal with such varying structure and dynamics, the gain scheduling (GS) control structure was classically considered for TRUAV 4

The blue statements without special labels are mainly to modify the typos and expressions For Reviewer #1 Q4 transition control [START_REF] Sato | Flight controller design and demonstration of quad-tilt-wing unmanned aerial vehicle[END_REF][START_REF] Zhao | Robust transition control of a martian coaxial tiltrotor aerobot[END_REF][START_REF] Muraoka | Transition flight of quad tilt wing VTOL UAV[END_REF][START_REF] Chen | Control and flight test of a tilt-rotor unmanned aerial vehicle[END_REF]. In [START_REF] Sato | Flight controller design and demonstration of quad-tilt-wing unmanned aerial vehicle[END_REF] and [START_REF] Muraoka | Transition flight of quad tilt wing VTOL UAV[END_REF], two sets of controllers were designed for the helicopter and airplane modes, and a smooth-weighted control structure with these was formed for the transition procedure, where [START_REF] Sato | Flight controller design and demonstration of quad-tilt-wing unmanned aerial vehicle[END_REF] displayed sufficient experimental results to illustrate the control performance. The divide and conquer GS method was applied in [START_REF] Zhao | Robust transition control of a martian coaxial tiltrotor aerobot[END_REF] and [START_REF] Chen | Control and flight test of a tilt-rotor unmanned aerial vehicle[END_REF], in which several linear controllers were directly scheduled and switched depending on flight velocity, and [START_REF] Chen | Control and flight test of a tilt-rotor unmanned aerial vehicle[END_REF] conducted flight experiments with the Pixhawk flight controller. The multi-model adaptive control structure in [START_REF] Liu | Transition control of tilt rotor unmanned aerial vehicle based on multi-model adaptive method[END_REF] could be regarded as an extension of the GS idea, which determines the weight values of the smooth-weighted control structure adaptively rather than based on heuristic experience, as was the case in previous GS methods. The GS control structure is applied effectively in practical applications [START_REF] Sato | Flight controller design and demonstration of quad-tilt-wing unmanned aerial vehicle[END_REF][START_REF] Muraoka | Transition flight of quad tilt wing VTOL UAV[END_REF][START_REF] Chen | Control and flight test of a tilt-rotor unmanned aerial vehicle[END_REF]. However, without references that consider stability conditions of GS methods for TRUAV transition control, the transition procedure must be limited by a special flight envelope with respect to the rotor-tilt angle and flight velocity, which is called the tilt corridor [START_REF] Liu | Control techniques of tilt rotor unmanned aerial vehicle systems: A review[END_REF][START_REF] Liu | Transition control of tilt rotor unmanned aerial vehicle based on multi-model adaptive method[END_REF] or conversion corridor [START_REF] Droandi | ProprotorCwing aerodynamic interaction in the first stages of conversion from helicopter to aeroplane mode[END_REF]. To ensure the transition procedure remains asymptotically stable, [START_REF] Liu | Mode transition and fault tolerant control under rotor-tilt axle stuck fault of quad-TRUAV[END_REF] considered nonlinear control methods and introduced virtual control inputs that include coupling of control effects from the rotors and aerodynamics. Nonlinear controllers could be designed easily for these virtual values [START_REF] Wang | Mathematical modeling and control of a tilt-rotor aircraft[END_REF][START_REF] Liu | Mode transition and fault tolerant control under rotor-tilt axle stuck fault of quad-TRUAV[END_REF]. Based on the relationship between virtual and applied control inputs, an inverse procedure for the applied values was designed further in [START_REF] Liu | Mode transition and fault tolerant control under rotor-tilt axle stuck fault of quad-TRUAV[END_REF]. Nevertheless, the nonlinear observer for state feedback control was not designed or considered in [START_REF] Liu | Mode transition and fault tolerant control under rotor-tilt axle stuck fault of quad-TRUAV[END_REF].

To design the nonlinear observer and state feedback controller simultaneously, the linear parameter-varying (LPV) concept could be considered for observer-based control [START_REF] Koroglu | Improved conditions for observer-based LPV control with guaranteed L2-gain performance[END_REF]. An LPV system contains state-space matrices that depend on varying parameters [START_REF] Rugh | Research on gain scheduling[END_REF], thus it reserves nonlinearity to a degree, and its linear form is beneficial for further controller design with the development of linear matrix inequality (LMI) optimization. With these advantages, LPV modeling and controller design are very popular in aerial vehicles [START_REF] Al-Jiboory | LPV modeling of a flexible wing aircraft using modal alignment and adaptive gridding methods[END_REF][START_REF] Huang | Non-fragile switching tracking control for a flexible air-breathing hypersonic vehicle based on polytopic LPV model[END_REF] for capturing the varying nature of the system dynamics, particularly the longitudinal dynamics. In most of the existing references [START_REF] Cao | A descriptor system approach to robust stability analysis and controller synthesis[END_REF][START_REF] Aouani | H2 analysis for LPV systems by parameter-dependent Lyapunov functions[END_REF][START_REF] Lu | In-flight adaptive modeling using polynomial LPV approach for turbofan engine dynamic behavior[END_REF][START_REF] Liu | State feedback controller design for affine parameter-dependent LPV systems[END_REF], the varying parameters in LPV systems are all assumed to be measurable. Regarding LPV models for aerial vehicles [START_REF] Al-Jiboory | LPV modeling of a flexible wing aircraft using modal alignment and adaptive gridding methods[END_REF][START_REF] Huang | Non-fragile switching tracking control for a flexible air-breathing hypersonic vehicle based on polytopic LPV model[END_REF], they have varying parameters that depend on system states, which are called quasi-LPV systems. Strictly speaking, all varying parameters in quasi-LPV systems are unmeasurable. This means that an LPV controller for TRUAV could not be scheduled by the varying parameters directly. With the method in [START_REF] Ichalal | How to cope with unmeasurable premise variables in Takagi-Sugeno observer design: Dynamic extension approach[END_REF], some varying parameters of quasi-LPV systems might be measurable depending on the measurement outputs. However, if measurement noises or sensor faults are present, an LPV observer would still be required to estimate the states and unmeasurable varying parameters for state feedback control.

Regarding the topic of LPV systems with unmeasurable varying parameters, observer design is primarily focused on [START_REF] Ichalal | Advances in observer design for Takagi-Sugeno systems with unmeasurable premise variables[END_REF][START_REF] Bergsten | Observers for Takagi-Sugeno fuzzy systems[END_REF][START_REF] Theilliol | Design of LPV observers with immeasurable gain scheduling variable under sensor faults[END_REF][START_REF] Lopez-Estrada | Observer synthesis for a class of Takagi-Sugeno descriptor system with unmeasurable premise variable. Application to fault diagnosis[END_REF]. For further controller design, observer-based LPV control methods are used logically for state estimations and control inputs simultaneously [START_REF] Heemels | Observer-based control of discrete-time LPV systems with uncertain parameters[END_REF][START_REF] Moodi | On observer-based controller design for Sugeno systems with unmeasurable premise variables[END_REF]. To address varying parameter uncertainties represented by differences between varying parameters and their estimations, [START_REF] Heemels | Observer-based control of discrete-time LPV systems with uncertain parameters[END_REF] applied the input-to-state stability concept to ensure states and estimation errors converge into an appropriate set close to zero, and this idea was also considered by [START_REF] Ichalal | Advances in observer design for Takagi-Sugeno systems with unmeasurable premise variables[END_REF] for observer design. Similar to [START_REF] Bergsten | Observers for Takagi-Sugeno fuzzy systems[END_REF], where these uncertainties were regarded as vanishing perturbations, [START_REF] Moodi | On observer-based controller design for Sugeno systems with unmeasurable premise variables[END_REF] also assumed Lipschitz conditions with respect to estimation errors in order to limit the quadratic magnitude of these uncertainties. Moreover, the bounded condition was applied in [START_REF] Theilliol | Design of LPV observers with immeasurable gain scheduling variable under sensor faults[END_REF] and [START_REF] Lopez-Estrada | Observer synthesis for a class of Takagi-Sugeno descriptor system with unmeasurable premise variable. Application to fault diagnosis[END_REF] for LPV observer design, and the closed-loop stability could be ensured with respect to a range of varying parameter uncertainties. However, few references focus on controller design based on this idea.

Due to the unstable dynamics, underlying faults threaten flight reliability and safety of TRUAVs, similar to RUAVs. To ensure the stability or acceptable control performance under different faults, research on fault-tolerant control (FTC) for aerial vehicles and other systems has been conducted for a long time, and various passive and active FTC methods have been presented. For example, [START_REF] Hu | Smooth finite-time fault-tolerant attitude tracking control for rigid spacecraft[END_REF] and [START_REF] Qi | A review on fault diagnosis and fault tolerant control methods for single-rotor aerial vehicles[END_REF] focused on FTC methods of different unmanned systems, and [START_REF] Zhang | Bibliographical review on reconfigurable faulttolerant control systems[END_REF] presented a general review about passive and active FTC methods. However, an FTC of TRUAV is seldom considered because of its complex coupling and varying dynamics.

This paper focuses on the FTC of a quad-TRUAV during the transition procedure, where the main challenge is to ensure the closed-loop stability of the controlled plant with the LPV method under bias sensor faults. As an extension of [START_REF] Liu | Mode transition and fault tolerant control under rotor-tilt axle stuck fault of quad-TRUAV[END_REF], virtual control inputs are introduced into the quad-TRUAV nonlinear model, and an equivalent quasi-LPV system is formulated with the varying parameters depending on flight velocity. With such unmeasurable varying parameters under sensor faults, the study in this paper considers a controlled plant with a polytopic LPV form, and the observer-based LPV controller is designed for virtual control inputs. The bounded condition in [START_REF] Theilliol | Design of LPV observers with immeasurable gain scheduling variable under sensor faults[END_REF] and [START_REF] Lopez-Estrada | Observer synthesis for a class of Takagi-Sugeno descriptor system with unmeasurable premise variable. Application to fault diagnosis[END_REF] is used here to address varying parameter uncertainties. For fault tolerance against bias sensor faults, low-pass filters are used to transform them into actuator fault formulations according to [START_REF] Sami | Active fault tolerant control for nonlinear systems with simultaneous actuator and sensor faults[END_REF]. The proportional integral observer (PIO) used in [START_REF] Ichalal | Observer design and fault tolerant control of Takagi-Sugeno nonlinear systems with unmeasurable premise variables[END_REF] is designed to estimate states and fault magnitudes jointly under the observer-based control structure. One should note that virtual control inputs for the quad-TRUAV are obtained from above controller. For applied control inputs, a nonlinear programming problem is formed as an inverse procedure based on the algebraic relation between virtual and applied control inputs. The continuity of the control inputs and actuator constraints would receive particular focus.

In the proposed observer-based LPV control method, an improved idea for varying parameter uncertainties based on the bounded condition is represented. Compared with [START_REF] Theilliol | Design of LPV observers with immeasurable gain scheduling variable under sensor faults[END_REF] and [START_REF] Lopez-Estrada | Observer synthesis for a class of Takagi-Sugeno descriptor system with unmeasurable premise variable. Application to fault diagnosis[END_REF], where large ranges for varying parameter uncertainties were assigned, the improved idea regards these ranges as solvable variables for less conservatism. In addition, to reduce conservatism from the perspective of a common Lyapunov method [START_REF] Bergsten | Observers for Takagi-Sugeno fuzzy systems[END_REF], a parameter-dependent Lyapunov function [START_REF] Cao | A descriptor system approach to robust stability analysis and controller synthesis[END_REF][START_REF] Aouani | H2 analysis for LPV systems by parameter-dependent Lyapunov functions[END_REF][START_REF] Ichalal | Advances in observer design for Takagi-Sugeno systems with unmeasurable premise variables[END_REF][START_REF] Moodi | On observer-based controller design for Sugeno systems with unmeasurable premise variables[END_REF] is considered for stability analysis, where varying parameter derivatives are introduced into the stability conditions. In contrast to most references, where the derivatives are evolved in a polytope only [START_REF] Ichalal | Advances in observer design for Takagi-Sugeno systems with unmeasurable premise variables[END_REF][START_REF] Moodi | On observer-based controller design for Sugeno systems with unmeasurable premise variables[END_REF], or some references, where new representations of the derivatives are considered [START_REF] Cao | A descriptor system approach to robust stability analysis and controller synthesis[END_REF][START_REF] Aouani | H2 analysis for LPV systems by parameter-dependent Lyapunov functions[END_REF], the relationship between varying parameter derivatives is introduced into the stability conditions for a less conservative controller. [START_REF] Mozelli | Reducing conservativeness in recent stability conditions of TS fuzzy systems[END_REF] and [START_REF] Montagner | Stability analysis and gain-scheduled state feedback control for continuous-time systems with bounded parameter variations[END_REF] also considered this relationship for stability analysis. Compared with the proposed method, [START_REF] Mozelli | Reducing conservativeness in recent stability conditions of TS fuzzy systems[END_REF] ignored the convexly dependent form of varying parameter derivatives. [START_REF] Montagner | Stability analysis and gain-scheduled state feedback control for continuous-time systems with bounded parameter variations[END_REF] applied this convex dependence, but only to special derivatives combined convexly by given vertexes.

The main contributions of this paper are as follows:

1) The LPV method is used to ensure the stability of a quad-TRUAV during the transition procedure with virtual control inputs and an inverse procedure, in which the observer-based LPV controller with bounded condition is designed to deal with unmeasurable varying parameters.

2) The PIO is designed under an observer-based control structure for fault-tolerant ability of an LPV controller under bias sensor faults.

3) Parameter-dependent Lyapunov matrices are used for stability analysis. The relationship between varying parameter derivatives is introduced, and the common Lyapunov method is analyzed as a special case.

The remaining parts of this paper are organized as follows. In Section 2, a longitudinal model of the quad-TRUAV is introduced and reformulated into an affine parameter-dependent LPV system. The polytopic LPV form of the controlled plant and the relationship between varying parameter derivatives are represented in Section 3. The FTC problem of this polytopic LPV system with unmeasurable varying parameters under bias sensor faults is analyzed immediately. The observer-based LPV control design for virtual control inputs is introduced in Section 4. Parameter-dependent Lyapunov matrices are used for stability analysis with LMI conditions, and the common Lyapunov method is proven as a special case. In Section 5, a nonlinear programming problem for applied control inputs with the above virtual values and a specific algebraic relation is formulated. Some numerical results showing the control effectiveness for the quad-TRUAV under bias sensor faults are represented in Section 6. Conclusions are offered in Section 7.

Quad-TRUAV modeling

Nonlinear modeling of quad-TRUAV

The quad-TRUAV, whose prototype is show in Fig. 1(a), is the object of focus in this paper. This TRUAV is equipped with two pairs of rotors that could tilt simultaneously. With a rotor-tilt angle i n = π/2 rad in helicopter mode, the quad-TRUAV could hover at a fixed position, and four rotor thrusts are regulated by corresponding motors for necessary flight control; with i n = 0 in airplane mode, the quad-TRUAV could cruise with a high speed like FWUAVs, in which the ailerons, elevator, and rudder provide additional aerodynamic forces and moments with control effects; during the transition procedure between these two flight modes, tilt servos adjust the directions of the rotor thrusts, and the control effects from rotors and aerodynamics coexist. The Pixhawk flight controller is equipped with the platform, where a Pitot tube and gyro are included to measure the flight velocity and attitude rates.

Because the longitudinal dynamics of TRUAV changes significantly with varying structure and flight velocity, only the longitudinal model for this quad-TRUAV is considered here, similar to modeling of fixed-wing aerial vehicles [START_REF] Huang | Non-fragile switching tracking control for a flexible air-breathing hypersonic vehicle based on polytopic LPV model[END_REF]. Based on the coordinate systems and states shown in could be formulated as follows according to [START_REF] Liu | Mode transition and fault tolerant control under rotor-tilt axle stuck fault of quad-TRUAV[END_REF] and [START_REF] Huang | Non-fragile switching tracking control for a flexible air-breathing hypersonic vehicle based on polytopic LPV model[END_REF]:

V = F wx m , ḣ = V sin(θ -α), α = F wz mV + q, q = M by I y , θ = q, ( 1
)
where V is the flight velocity, h is the flight height, α is the angle of attack, q is the pitch rate, θ is the pitch angle, and some other variables are all defined in the Nomenclature. F wx and F wz are the resultant forces along the xand z-directions of the wind-axis coordinate system, respectively, which are generated by rotors, aerodynamics, and gravity. M by is the relevant moment in the y-direction of the body-axis coordinate system generated by rotors and aerodynamics.

The rotors in the quad-TRUAV platform shown in Fig. 1(a) are driven by brushless direct current motors. The generated thrusts are proportional to the squares of the rotor speeds [START_REF] Pounds | Modelling and control of a large quadrotor robot[END_REF]. These thrusts are further transformed into the wind-axis coordinate system using the angle of attack and rotor-tilt angle, and the following component forces and moment are obtained:

[ F wxr F wzr ] = [ cos α sin α -sin α cos α ] [ cos i n -sin i n ] 2C t ρAR 2 (Ω 2 F + Ω 2 B ), M byr = 2C t ρAR 2 x r sin i n (Ω 2 F -Ω 2 B ).
(

) 2
The aerodynamic forces and moment are obtained as follows according to [START_REF] Zhao | Robust transition control of a martian coaxial tiltrotor aerobot[END_REF] and [START_REF] Huang | Non-fragile switching tracking control for a flexible air-breathing hypersonic vehicle based on polytopic LPV model[END_REF]:

F wxa = - 1 2 ρV 2 s • C D = - 1 2 ρV 2 s • (C D0 + C D1 α + C D2 α 2 + C Dδ δ e ), F wza = - 1 2 ρV 2 s • C L = - 1 2 ρV 2 s • (C L0 + C L1 α + c 2V C Lq q + C Lδ δ e ), M bya = 1 2 ρV 2 sc • C m = 1 2 ρV 2 sc • (C m0 + C m1 α + c 2V C mq q + C mδ δ e ). (3)
Gravity only generates component forces as follows:

[ F wxg F wzg ] = [ cos α sin α -sin α cos α ] [ -mg sin θ mg cos θ ] = [ -mg sin(θ -α) mg cos(θ -α) ] . ( 4
)
The variables in the above equations are all defined in the Nomenclature, including i n as the rotor-tilt angle, Ω F and Ω B as the forward and backward rotor speeds, respectively, and δ e as the deflection of the elevator.

With the above forces and moments, F wx = F wxr + F wxa + F wxg , F wz = F wzr +F wza +F wzg , and M by = M byr +M bya are introduced into (1) to formulate the nonlinear model for the quad-TRUAV, where the kinematic equation about h is ignored for the following LPV form. There are

V = F wxr + F wxa + F wxg m , γ = - F wzr + F wza + F wzg mV , α = q -γ, q = M byr + M bya I y , ( 5
)
where the new state γ = θ -α is considered, and the control for h could be achieved by regarding γ as the virtual control input [START_REF] Liu | Mode transition and fault tolerant control under rotor-tilt axle stuck fault of quad-TRUAV[END_REF]. With the above equations, a typical nonlinear model ẋ

(t) = f (x(t), u(t)) is established with state vector x = [V γ α q] T and control input vector u = [i n Ω F Ω B δ e ] T .
In the following contents without special declarations, "control inputs" refer to the applied values conventionally. For state estimation and tracking control, define y = x = [V γ α q] T as the measurement output vector, where V and q could be measured with the Pitot tube and gyro equipped with the quad-TRUAV. Due to the lack of sensors for α and γ on this real platform, assume these two values are obtained directly from an ideal navigation algorithm without any fault.

Equivalent quasi-LPV form

From the forms of F wx , F wz , and M by , the coupling of control effects from the rotors and aerodynamics is obvious. To avoid considering this complex coupling directly in the controller design, the following virtual control inputs are introduced into the nonlinear model for the quad-TRUAV:

v V = [ 2ρAR 2 m C t • cos(i n + α)(Ω 2 F + Ω 2 B ) - ρs 2m C Dδ • V 2 δ e ] -g sin γ- ρs 2m C D1 • V 2 α - ρs 2m C D2 • V 2 α 2 , ( 6
)
v α = [ - 2ρAR 2 mV C t • sin(i n + α)(Ω 2 F + Ω 2 B ) - ρs 2m C Lδ • V δ e ] + g cos γ V , (7)
v q = [ 2ρAR 2 x r I y C t • sin i n (Ω 2 F -Ω 2 B ) + ρsc 2I y C mδ • V 2 δ e ] + ρsc 2I y C m1 • V 2 α, ( 8
)
where the virtual control inputs are functions of the states and applied control inputs. Consequently, the nonlinear model in the previous subsection is equivalent to the following quasi-LPV system, where the flight velocity V is the varying parameter:

    V γ α q     =            0 0 0 0 a 21 0 0 a 24 a 31 0 0 a 34 0 0 0 0     + V     -ρs 2m C D0 0 0 0 0 0 a 23 0 0 0 a 33 0 ρsc 2Iy C m0 0 0 ρsc 2 4Iy C mq                V γ α q     +     1 0 0 0 -1 0 0 1 0 0 0 1       v V v α v q   . ( 9
)
where a 21 = ρs 2m C L0 , a 24 = ρsc 4m C Lq , a 23 = ρs 2m C L1 , a 31 = -a 21 , a 34 = 1 -a 24 , and a 33 = -a 23 . Obviously, the above equation is an affine parameterdependent LPV system [START_REF] Liu | State feedback controller design for affine parameter-dependent LPV systems[END_REF] with a constant input matrix as following typical form:

ẋ(t) = A a (p a )x(t) + B a v(t), ( 10
)
y(t) = C a x(t), (11)
with x(t) ∈ R nx , v(t) ∈ R nv as a virtual control input vector, y(t) ∈ R ny , and varying parameters p a = [p a1 . . . p ana ], where [START_REF] Liu | Transition control of tilt rotor unmanned aerial vehicle based on multi-model adaptive method[END_REF]. The system matrix A a (p a ) = A a0 + p a1 A a1 + • • • + p ana A ana is affinely parameter-dependent. The input matrix B a and output matrix C a are all constant. In ( 10) and ( 11), the subscript "a" is used to distinguish the above affine parameter-dependent LPV system from following equivalent polytopic form, which is the controlled plant for closed-loop analysis in next section.

n x = 4, n v = 3, n y = 4, n a = 1, and v = [v V v α v q ] T for

Control problem analysis

Facing a quasi-LPV system like that in [START_REF] Liu | Transition control of tilt rotor unmanned aerial vehicle based on multi-model adaptive method[END_REF], the varying parameters therein are unmeasurable absolutely due to measurement noises and sensor faults. To address this LPV system with unmeasurable varying parameters, a polytopic LPV controlled plant would be formed based on [START_REF] Droandi | ProprotorCwing aerodynamic interaction in the first stages of conversion from helicopter to aeroplane mode[END_REF] and [START_REF] Liu | Mode transition and fault tolerant control under rotor-tilt axle stuck fault of quad-TRUAV[END_REF] in this section for bounded varying parameters. Then the closed-loop system with an observer-based LPV controller under bias sensor faults would be formed to analyze the control problem for the quad-TRUAV.

Polytopic LPV controlled plant

For some nonlinear systems and affine parameter-dependent LPV systems like that in [START_REF] Droandi | ProprotorCwing aerodynamic interaction in the first stages of conversion from helicopter to aeroplane mode[END_REF], the sector nonlinearity method [START_REF] Tanaka | Takagi-Sugeno fuzzy model and parallel distributed compensation[END_REF] could be used for the equivalent polytopic LPV forms. Consider the following polytopic LPV system from the quasi-LPV model in [START_REF] Droandi | ProprotorCwing aerodynamic interaction in the first stages of conversion from helicopter to aeroplane mode[END_REF] and [START_REF] Liu | Mode transition and fault tolerant control under rotor-tilt axle stuck fault of quad-TRUAV[END_REF] for the quad-TRUAV with disturbance inputs w(t) and bias sensor faults f (t):

ẋ(t) = A(p)x(t) + Bv(t) + B w w(t), ( 12
)
y(t) = Cx(t) + Df (t) + D w w(t), (13)
where

w(t) ∈ R nw , f (t) ∈ R n f , A(p) = ∑ n i=1 p i A i , B = B a , and C = C a .
Because the sector nonlinearity method considers 2 na vertexes of the original varying parameters p a and applies normalization, A i could be regarded as a vertex of the original matrix A a (p a ), and varying parameter p i interpolates these vertex matrices with ∑ n i=1 p i = 1, 0 ≤ p i ≤ 1, and n = 2 na . The derivatives of varying parameters satisfy ∑ n i=1 ṗi = 0 according to [START_REF] Mozelli | Reducing conservativeness in recent stability conditions of TS fuzzy systems[END_REF]. Note that p is the simplified representation of p(x(t)). Sensor faults f (t) and measurement noises w(t) make p unmeasurable in the above equations, even when C is an identity matrix.

For the LPV model of the quad-TRUAV, the control input vector v(t) is virtual and depends on states like ( 6)- [START_REF] Chen | Control and flight test of a tilt-rotor unmanned aerial vehicle[END_REF]. There would be an algebraic relation

v(t) = R vu (x(t), u(t)) ( 14
)
that describes the relationship between the applied control inputs u(t) and the virtual control inputs v(t). When the states are not available, given state estimations x(t) and assumed inverse function

R -1 vu (•, •), only ũ(t) = R -1 vu (x(t), v(t)) ( 15
)
is available in an application. By introducing the above applied control input vector into ( 12) with ( 14), the following equation could be formulated:

ẋ(t) = A(p)x(t) + B • R vu (x(t), ũ(t)) + B w w(t) = A(p)x(t) + Bṽ(t) + B w w(t), ( 16
)
where ṽ(t) is defined as

R vu (x(t), ũ(t)). The relationships u(t) = ũ(t) = R -1 vu (x(t), v(t)
) and ṽ(t) -v(t) = 0 are satisfied only when x(t) = x(t). To avoid considering ṽ(t) directly, δ(t) = ṽ(t)-v(t) is defined as an unstructured disturbance vector. With this definition, the polytopic LPV controlled plant for the analysis in next subsection is reformulated as follows:

ẋ(t) = A(p)x(t) + Bv(t) + B w w(t) + Bδ(t), ( 17
)
y(t) = Cx(t) + Df (t) + D w w(t). ( 18
)
With the polytopic form of the LPV system, varying parameters are bounded within [0, 1], and varying parameter uncertainties due to unmeasurable varying parameters would also be bounded. This is beneficial for stability analysis. Moreover, the sum of the varying parameter derivatives is equal to zero. This relationship would be used to further reduce control conservatism in the parameter-dependent Lyapunov method.

FTC design and analysis with unmeasurable varying parameters

Due to measurement noises and sensor faults, the varying parameters p in the polytopic LPV controlled plant [START_REF] Aouani | H2 analysis for LPV systems by parameter-dependent Lyapunov functions[END_REF] and ( 18) are absolutely unmeasurable. To deal with this problem, this subsection introduces p = [p 1 . . . pn ] T as estimations of p = [p 1 . . . p n ] T , where p is the simplified representation of p(x(t)) based on the state estimation and has the properties ∑ n i=1 pi = 1 13 and 0 ≤ pi ≤ 1. LPV observer should be designed not only for the FTC with x(t) and f (t) but also for p. Consequently, the following observer-based LPV control structure is considered. Firstly, to ensure the partially fault-free measurement outputs C r [y(t) -Df (t)] track a reference signal r(t) ∈ R nr for LPV control with unmeasurable varying parameters, a tracking error integration [START_REF] Lopez-Estrada | LPV model-based tracking control and robust sensor fault diagnosis for a quadrotor UAV[END_REF] is defined as

e(t) = ∫ { r(t) -C r [ y(t) -D f (t)
]} dt, and

ė(t) = -C r Cx(t) + C r De f (t) + r(t) -C r D w w(t), ( 19
)
where e f (t) = f (t) -f (t). Augmenting [START_REF] Aouani | H2 analysis for LPV systems by parameter-dependent Lyapunov functions[END_REF] with the above equation yields the following new polytopic LPV equation for controller design:

[ ė(t) ẋ(t) ] = [ 0 -C r C 0 A(p) ] [ e(t) x(t) ] + [ 0 B ] v(t) + [ C r D 0 ] e f (t)+ [ r(t) -C r D w w(t) B w w(t) + Bδ(t)
] .

(

) 20
With LPV controller gains K 1 (p) ∈ R nv×nr and K 2 (p) ∈ R nv×nx scheduled by p rather than p, one could introduce

v(t) = [ K 1 (p) K 2 (p) ] [ e(t) x(t) ] (21)
into [START_REF] Ichalal | How to cope with unmeasurable premise variables in Takagi-Sugeno observer design: Dynamic extension approach[END_REF], and the following closed-loop system could be formulated:

[ ė(t) ẋ(t) ] = {[ 0 -C r C 0 A(p) ] + [ 0 B ] [ K 1 (p) K 2 (p) ] - [ 0 0 0 A(∆p) ]} [ e(t) x(t) ] + [ 0 B ] [ K 1 (p) K 2 (p) ] [ 0 e x (t) ] + [ C r D 0 ] e f (t) + [ r(t) -C r D w w(t) B w w(t) + Bδ(t) ] , (22)
where e x (t) = x(t) -x(t), ∆p = p -p represents varying parameter uncertainties, and

A(∆p) = A(p) -A(p) = ∑ n i=1 ∆p i A i .
With C r as an identity matrix and no sensor faults, ( 21) is similar to the tracking controller in [START_REF] Lopez-Estrada | LPV model-based tracking control and robust sensor fault diagnosis for a quadrotor UAV[END_REF]. Only V , γ, and α in y(t) must be tracked to control the quad-TRUAV, thus it is unnecessary to consider C r as an identity matrix here.

Secondly, to design an LPV observer for x(t) and f (t) simultaneously, the measurement output vector could be separated into two parts as follows:

y(t) = [ y 1 (t) y 2 (t) ] = [ C 1 C 2 ] x(t) + [ 0 I ] f (t) + [ D w1 D w2 ] w(t), ( 23
)
where y 1 (t) ∈ R ny-n f is a fault-free output vector, and y 2 (t) ∈ R n f is a faulty output vector. To address bias sensor faults f (t), a filter could be introduced to transform the sensor faults into actuator faults [START_REF] Sami | Active fault tolerant control for nonlinear systems with simultaneous actuator and sensor faults[END_REF]:

ẋf (t) = -A f x f (t) + A f y 2 (t) = A f C 2 x(t) -A f x f (t) + A f f (t) + A f D w2 w(t), ( 24
)
where x f (t) ∈ R n f is an augmented state vector, and -A f ∈ R n f ×n f is a Hurwitz matrix. An augmented LPV system could be formulated as follows:

[ ẋ(t) ẋf (t) ] = [ A(p) 0 A f C 2 -A f ] [ x(t) x f (t) ] + [ B 0 ] v(t) + [ 0 A f ] f (t)+ [ B w w(t) + Bδ(t) A f D w2 w(t) ] , [ y 1 (t) y f (t) ] = [ C 1 0 0 I ] [ x(t) x f (t) ] + [ D w1 w(t) 0 ] . ( 25
)
An LPV PIO could be designed for x(t) and f (t) with observer gains L 1 (p) ∈ R (nx+n f )×ny and L 2 (p) ∈ R n f ×ny that are scheduled by p rather than p as follows:

[ ẋ(t) ẋf (t) ] = [ A(p) 0 A f C 2 -A f ] [ x(t) xf (t) ] + [ B 0 ] v(t) + [ 0 A f ] f (t)+ L 1 (p) {[ C 1 0 0 I ] [ x(t) xf (t) ] - [ y 1 (t) y f (t) ]} , ḟ (t) =L 2 (p) {[ C 1 0 0 I ] [ x(t) xf (t) ] - [ y 1 (t) y f (t) ]} . ( 26
)
With e x (t) = x(t) -x(t), e xf (t) = xf (t) -x f (t), and e f (t) = f (t) -f (t), there are

[ ėx (t) ėxf (t) ] = [ A(∆p) 0 ] x(t)+ {[ A(p) 0 A f C 2 -A f ] +L 1 (p) [ C 1 0 0 I ]} [ e x (t) e xf (t) ] + [ 0 A f ] e f (t)+ [ -B w w(t) -Bδ(t) -A f D w2 w(t) ] +L 1 (p) [ -D w1 0 ] w(t), ėf (t) =L 2 (p) [ C 1 0 0 I n f ] [ e x (t) e xf (t) ] -ḟ (t) + L 2 (p) [ -D w1 0 ] w(t). ( 27
)
Assuming ḟ (t) is bounded and ḟ (+∞) = 0 for limited energy, the closedloop systems [START_REF] Bergsten | Observers for Takagi-Sugeno fuzzy systems[END_REF] and ( 27) could be formulated in following typical form:

[ ẋ(t)

ė(t) ] = [ Ã(p) + BK(p) -Ãu (∆p) BK(p) Ẽ + D A u (∆p) A(p) + L(p)C ] [ x(t) e(t) ] + [ Bw B w + L(p)D w ] ω(t), z(t) = [ 0 C e ] [ x(t) e(t) ] + F w ω(t), (28)
where

x(t) = [ e(t) x(t) ] , e(t) =   e x (t) e xf (t) e f (t)   , Ã(p) = [ 0 -C r C 0 A(p) ] , B = [ 0 B ] , Ãu (∆p) = [ 0 0 0 A(∆p) ] , Ẽ = [ 0 0 0 I 0 0 ] , D = [ 0 0 C r D 0 0 0 ] , A u (∆p) =   0 A(∆p) 0 0 0 0   , A(p) =   A(p) 0 0 A f C 2 -A f A f 0 0 0   , C = [ C 1 0 0 0 I 0 ] , Bw = [ I -C r D w 0 0 0 B w B 0 ] , B w =   0 -B w -B 0 0 -A f D w2 0 0 0 0 0 -I   , D w = [ 0 -D w1 0 0 0 0 0 0 ] , ω(t) =     r(t) w(t) δ(t) ḟ (t)     ,
and z(t) ∈ R nx+2n f is the performance output vector, matrices C e and F w could be set for special performance outputs.

K(p) = [ K 1 (p) K 2 (p) ] and L(p) = [ L T 1 (p) L T 2 (p)
] T are LPV controller gain and observer gain.

Varying parameter uncertainties ∆p are included in Ãu (∆p) and A u (∆p). These two terms could be reformulated as follows:

Ãu (∆p) = [ 0 nr×nx I nx ] A(∆p) [ 0 nx×nr I nx ] LA(∆p) R and A u (∆p) =   I nx 0 n f ×nx 0 n f ×nx   A(∆p) [ 0 nx×nr I nx ] LA(∆p)R. ( 29
)
Figure 2: Observer-based LPV control structure for quad-TRUAV.

According to [START_REF] Theilliol | Design of LPV observers with immeasurable gain scheduling variable under sensor faults[END_REF] and [START_REF] Lopez-Estrada | Observer synthesis for a class of Takagi-Sugeno descriptor system with unmeasurable premise variable. Application to fault diagnosis[END_REF], A(∆p) could be represented as

A(∆p) = [ I nx . . . I nx ]    ∆p 1 I nx . . . 0 . . . . . . . . . 0 . . . ∆p n I nx       A 1 . . . A n    I t FA = [ A 1 . . . A n ]    ∆p 1 I nx . . . 0 . . . . . . . . . 0 . . . ∆p n I nx       I nx . . . I nx    A t FI, (30)
where I t FI = 0 nx×nx , I t = I T , and LT = R = R.

Based on above analysis with unmeasurable varying parameters under bias sensor faults, the FTC for the quad-TRUAV requires an LPV controller [START_REF] Ichalal | Advances in observer design for Takagi-Sugeno systems with unmeasurable premise variables[END_REF] and LPV PIO [START_REF] Moodi | On observer-based controller design for Sugeno systems with unmeasurable premise variables[END_REF] to ensure the controller gain K(p) and observer gain L(p) keep the asymptotic stability of closed-loop system [START_REF] Qi | A review on fault diagnosis and fault tolerant control methods for single-rotor aerial vehicles[END_REF].

In general, Fig. 2 shows the control structure for the quad-TRUAV. With this structure, the control design of TRUAV first focuses on the observerbased LPV controller with unmeasurable varying parameters under bias sensor faults to ensure asymptotic stability of (28) with virtual control inputs v(t), which would be considered in Section 4. Secondly, with virtual control inputs and the algebraic relation [START_REF] Al-Jiboory | LPV modeling of a flexible wing aircraft using modal alignment and adaptive gridding methods[END_REF], an inverse procedure is necessary for further applied control inputs ũ(t) in the quad-TRUAV, which would be considered in Section 5.

Observer-based LPV controller design

Observer-based LPV control methods for virtual control inputs in the quad-TRUAV would be synthesized in this section. Thanks to the polytopic form of the controlled plant, the bounded condition would be used to ensure the closed-loop system (28) remains stable within bounded ranges of varying parameter uncertainties. Before presenting the main results, some useful lemmas are introduced as follows:

Lemma 1. (bounded condition) [START_REF] Xie | Output feedback H ∞ control of systems with parameter uncertainty[END_REF] Given matrices Q = Q T , R, S, and

H = H T > 0 of appropriate dimensions, Q + RG(t)S + S T G T (t)R T < 0 for all G(t) satisfying G T (t)G(t) ≤ H, if and only if there exists a scalar ϵ > 0 such that Q + ϵ -1 RR T + ϵS T HS < 0.
Lemma 2. [START_REF] Tuan | Parameterized linear matrix inequality techniques in fuzzy control system design[END_REF] If following conditions are true:

T ii < 0, 1 ≤ i ≤ n, 1 n -1 T ii + 1 2 (T ij + T ji ) < 0, 1 ≤ i ̸ = j ≤ n,
then the following inequality would also be tenable:

n ∑ i=1 n ∑ j=1 β i β j T ij < 0,
where ∑ n i=1 β i = 1 and 0 ≤ β i ≤ 1 (i = 1, ..., n). In the following theorems, the assumed uncertainty range for (30) is as follows:

-

F m ≤ F ≤ F m , (31)
where

F m =    ∆p m1 I nx . . . 0 . . . . . . . . . 0 . . . ∆p mn I nx   
, and a varying parameter uncertainty ∆p i is bounded by ∆p mi with -∆p mi ≤ ∆p i ≤ ∆p mi (i = 1, ..., n). Further assume the vertexes of ṗ as the following polytope:

V ṗ = { [µ 1 ... µ n ] T |µ k ∈ { ṗk , ṗk }, k = 1, ..., n } , ( 32
)
where ṗk and ṗk represent the minimum and maximum values of ṗk , respectively. This polytope also includes vertexes of ṗ, and the relationship between varying parameter derivatives is listed according to the definition of previous polytopic LPV system [START_REF] Koroglu | Improved conditions for observer-based LPV control with guaranteed L2-gain performance[END_REF] and ( 13):

n ∑ k=1 ṗk = n ∑ k=1 ṗk = 0. ( 33
)

Parameter-dependent Lyapunov method

Parameter-dependent Lyapunov matrices are considered here for stability analysis of [START_REF] Qi | A review on fault diagnosis and fault tolerant control methods for single-rotor aerial vehicles[END_REF]. The relationship between varying parameter derivatives is introduced into the stability condition for less conservatism.

Theorem 1. The closed-loop system (28) is asymptotically stable, and the H ∞ disturbance attenuation from ω(t) to z(t) is less than τ , i.e., sup

∥ω∥ 2 ̸ =0 ∥z∥ 2 ∥ω∥ 2 ≤ τ , if there exist positive scalars ϵ, ϕ 1 , ϕ 2 , and ϕ x , arbitrary scalars ϵ Xij and ϵ Y ij , parameter matrices E Xij and E Y ij (i, j = 1, ..., n), invertible matrices W 1 (p) = ∑ n j=1 pj W 1j and W 2 (p) = ∑ n j=1 pj W 2j , matrices X(p) = ∑ n j=1 pj X j , Y (p) = ∑ n j=1 pj Y j , M (p) = ∑ n i=1 pi M i ,

and N (p) =

∑ n i=1 pi N i , and a diagonal matrix F m that satisfy the following conditions under the limitation in [START_REF] Ichalal | Observer design and fault tolerant control of Takagi-Sugeno nonlinear systems with unmeasurable premise variables[END_REF] for ∀µ = [µ 1 ... µ n ] T ∈ V ṗ:

X i > 0, Y i > 0, 0 ≤ F m ≤ ϵI, ( 34
)
G ii < 0, 0 ≤ i ≤ n, ( 35
) 1 n -1 G ii + 1 2 (G ij + G ji ) < 0, 0 ≤ i ̸ = j ≤ n, (36)
where

G ij =             G ij(1,1) 0 G ij(1,3) 0 ϕ 1 Ẽ 0 0 * G ij(2,2) 0 0 G ij(2,5) N i D w 0 * * G ij(3,3) ϕ x X j D Bw 0 * * * -2ϕ x I 0 0 0 * * * * G ij(5,5) G ij(5,6) C T e * * * * * -τ I F T w * * * * * * -τ I Gij * -ϵI             and Gij =           0 0 0 0 0 0 0 0 X j RT A T + ϵ Xij LI t 0 -RT I t F m 0 0 0 0 R T I t F m 0 Y j LA t + ϵ Y ij LI t 0 0 0 0 0 0 0 0 0 0           with G ij(1,1) = -ϕ 1 He(W 1j ), G ij(2,2) = -ϕ 2 He(W 2j ), G ij(1,3) = M T i BT - ϕ 1 W 1j + ϕ 1 X j , G ij(2,5) = N i C -ϕ 2 W 2j + ϕ 2 Y j , G ij(3,3) = He( Ãi X j + BM i ) - X(µ) + ∑ n k=1 µ k E Xij , G ij(5,5) = He(Y j A i + N i C) + Y (µ) + ∑ n k=1 µ k E Y ij
,and G ij [START_REF] Wang | Mathematical modeling and control of a tilt-rotor aircraft[END_REF][START_REF] Zhao | Robust transition control of a martian coaxial tiltrotor aerobot[END_REF] = Y j B w +N i D w . The calculated LPV controller and observer gains are

K(p) = M (p)W -1 1 (p) and L(p) = W -T 2 (p)N (p), respectively. The feasible uncertainty range is F m = ϵ -1 F m .
Proof of Theorem 1. See Appendix A.

Remark 1.

In Theorem 1, X -1 (p) is the Lyapunov matrix of the LPV controller and Y (p) is the Lyapunov matrix of the observer. Parameterdependent forms of Lyapunov matrices introduce varying parameter derivatives into the stability condition (A.6). Although parameter-dependent Lyapunov matrices are beneficial to ensure a less conservative controller and observer, X(µ) in block G ij [START_REF] Tyan | Comprehensive preliminary sizing/resizing method for a fixed wing-VTOL electric UAV[END_REF][START_REF] Tyan | Comprehensive preliminary sizing/resizing method for a fixed wing-VTOL electric UAV[END_REF] and Y (µ) in block G ij [START_REF] Wang | Mathematical modeling and control of a tilt-rotor aircraft[END_REF][START_REF] Wang | Mathematical modeling and control of a tilt-rotor aircraft[END_REF] , which only consider the vertexes of ṗ, still introduce some conservatism compared with X( ṗ) and Y ( ṗ) [START_REF] Mozelli | Reducing conservativeness in recent stability conditions of TS fuzzy systems[END_REF]. Due to the special property of the polytopic LPV system, ( 33) is introduced into (A.11) using parameter matrices E X (p) and E Y (p). In this way, the matrices E Xij and E Y ij in blocks G ij [START_REF] Tyan | Comprehensive preliminary sizing/resizing method for a fixed wing-VTOL electric UAV[END_REF][START_REF] Tyan | Comprehensive preliminary sizing/resizing method for a fixed wing-VTOL electric UAV[END_REF] and G ij [START_REF] Wang | Mathematical modeling and control of a tilt-rotor aircraft[END_REF][START_REF] Wang | Mathematical modeling and control of a tilt-rotor aircraft[END_REF] could reduce conservatism for the LPV controller and observer design. Compared with [START_REF] Mozelli | Reducing conservativeness in recent stability conditions of TS fuzzy systems[END_REF], where [START_REF] Montagner | Stability analysis and gain-scheduled state feedback control for continuous-time systems with bounded parameter variations[END_REF] was introduced with a common parameter matrix, the parameter-dependent matrices E X (p) and E Y (p) in Theorem 1 are more general. Because [START_REF] Mozelli | Reducing conservativeness in recent stability conditions of TS fuzzy systems[END_REF] does not consider the convexly dependent form of varying parameter derivatives, the relationship between the Lyapunov matrix and the common parameter matrix must be considered.

Common Lyapunov method

The parameter-dependent Lyapunov method is less conservative, but too many LMI conditions must be considered in Theorem 1, which leads to high calculation load. With common Lyapunov matrices for stability analysis, a new theorem is proposed as follows, which could be regarded as a special case of the previous theorem with fewer LMI conditions. Theorem 2. The closed-loop system ( 28) is asymptotically stable, and the H ∞ disturbance attenuation from ω(t) to z(t) is less than τ , i.e., sup ∑ n i=1 pi N i , and a diagonal matrix F m that satisfy the following conditions under the limitation in [START_REF] Ichalal | Observer design and fault tolerant control of Takagi-Sugeno nonlinear systems with unmeasurable premise variables[END_REF]:

∥ω∥ 2 ̸ =0
X > 0, Y > 0, 0 ≤ F m ≤ ϵI, ( 37
)
H i < 0, 0 ≤ i ≤ n, ( 38
)
where Remark 2. According to the proof, Theorem 2 could be regarded as a special case of Theorem 1. Note that, without considering the relationship between varying parameter derivatives (33) by parameter matrices E Xij and E Y ij , Theorem 2 cannot be obtained directly from Theorem 1. Compared with Theorem 2, Theorem 1 has less conservatism for controller design due to the parameter-dependent Lyapunov matrices. However, the calculation load for solving the conditions in Theorem 2 is much less than that incurred with Theorem 1. According to conditions in these theorems, only n + 4 LMI conditions are required in Theorem 2, and n 2 + 2n + 2 conditions are required according to Theorem 1.

H i =             H i(1,1) 0 H i(1,3) 0 ϕ 1 Ẽ 0 0 * H i(2,2) 0 0 H i(2,5) N i D w 0 * * H i(3,3) ϕ x X D Bw 0 * * * -2ϕ x I 0 0 0 * * * * H i(5,5) H i(5,6) C T e * * * * * -τ I F T w * * * * * * -τ I Hi * -ϵI             and Hi =           0 0 0 0 0 0 0 0 X RT A T + ϵ Xi LI t 0 -RT I t F m 0 0 0 0 R T I t F m 0 Y LA t + ϵ Y i LI t 0 0 0 0 0 0 0 0 0 0           with H i(1,1) = -ϕ 1 He(W 1 ), H i(2,2) = -ϕ 2 He(W 2 ), H i(1,3) = M T i BT -ϕ 1 W 1 + ϕ 1 X, H i(2,5) = N i C -ϕ 2 W 2 + ϕ 2 Y , H i(3,3) = He( Ãi X + BM i ), H i(5,5) = He(Y A i + N i C),

Remark 3.

With Theorems 1 and 2, the following LMI minimization problem could be solved to optimize the H ∞ disturbance attenuation: min τ s.t.

(34), ( 35), [START_REF] Lopez-Estrada | LPV model-based tracking control and robust sensor fault diagnosis for a quadrotor UAV[END_REF] with Theorem 1 or ( 37), ( 38) with Theorem 2.

(

) 39
Compared with [START_REF] Theilliol | Design of LPV observers with immeasurable gain scheduling variable under sensor faults[END_REF] and [START_REF] Lopez-Estrada | Observer synthesis for a class of Takagi-Sugeno descriptor system with unmeasurable premise variable. Application to fault diagnosis[END_REF], where F m = I was tacitly approved directly, the proposed theorems that regard this range as a solvable variable with respect to ϵ and F m are more general. To magnify this range, the following LMI minimization problem could be considered:

min ϵ -trace(F m ) s.t.
(34), ( 35), [START_REF] Lopez-Estrada | LPV model-based tracking control and robust sensor fault diagnosis for a quadrotor UAV[END_REF] with Theorem 1 or ( 37), [START_REF] Tuan | Parameterized linear matrix inequality techniques in fuzzy control system design[END_REF] with Theorem 2.

(

) 40
In addition, the inherent relationship I t FI = 0 is introduced into (A.11) using the scalars ϵ X (p) and ϵ Y (p), which further reduces conservatism.

Control inputs design with inverse procedure

For the quad-TRUAV, the virtual control inputs from the above observerbased LPV controller are functions of states x(t) and the applied control inputs u(t), as shown in [START_REF] Al-Jiboory | LPV modeling of a flexible wing aircraft using modal alignment and adaptive gridding methods[END_REF], where

v = [v V v α v q ] T and u = [i n Ω F Ω B δ e ] T .
Considering the certain form of ( 6)-( 8), the procedure for the applied control inputs is actually an inverse procedure based on following equation:

  Ω 2 F +Ω 2 B Ω 2 F -Ω 2 B δ e   = R -1 (i n , V, α)•   v V +g sin γ + ρs 2m C D1 V 2 α+ ρs 2m C D2 V 2 α 2 v α -g cos γ V v q -ρsc 2Iy C m1 V 2 α   , ( 41
)
where

R(i n , V, α) =    2ρAR 2 m C t cos(i n + α) 0 -ρs 2m C Dδ V 2 -2ρAR 2 mV C t sin(i n + α) 0 -ρs 2m C Lδ V 0 2ρAR 2 xr Iy C t sin i n ρsc 2Iy C mδ V 2    . The determinant of R(i n , V, α) could be calculated as C • [C Dδ sin(i n + α)+ C Lδ cos(i n + α)] sin i n , where C = 2(ρAR 2 Ct) 2 ρsV xr m 2 Iy ̸ = 0.
Consequently, the basic equation ( 41) is solvable unless i n = 0 or i n +α = -arccos

( C Dδ √ C 2 Dδ +C 2

Lδ

) .

In the solvable case of ( 41), the dimension of u(t) is larger than the dimension of v(t). Therefore, when v(t) is obtained from the LPV controller, ( 14) is an indeterminate equation with respect to u(t). If the analytical form exists, the function u(t) = R -1 vu (x(t), v(t)) would be a set-valued map from v(t) to u(t). Assuming u(t) ∈ U set and v(t) ∈ V set , the following operator could be defined:

M(R -1 vu (x(t), •)) = { T ∈ R -1 vu (x(t), •) ∥T ∥ = min u∈R -1 vu (x(t),•) ∥u∥ } . ( 42
)
According to the set-valued analysis from [START_REF] Aubin | Selections and parametrization[END_REF], if 1) R -1 vu (x(t), •) is a lower semi-continuous set-valued map from V set to a Banach space U set , 2) the graph of the set-valued map

{ (v(t), u(t)) ∈ V set × U set u(t) ∈ R -1 vu ( x(t), v(t))} is closed, 3) R -1 vu (x(t), •) has nonempty closed convex images { R -1 vu (x(t), v(t)) v(t) ∈ V set }, and 4) M(R -1 vu (x(t), •)) is contained in a compact subset of U set , the minimum selection M(R -1 vu (x(t), •)
) would be a continuous and singlevalued function. With the above idea regarding minimum selection and the extended research in [START_REF] He | Robust and predictive control of nonlinear mechatronic systems[END_REF], if conditions 1)-4) are satisfied and there is a continuous guiding function η(t), the following programming problem with state estimations could yield a set of continuous and unique applied control inputs:

min ∥ũ(t) -η(t)∥ s.t. ũ(t) = R -1 vu (x(t), v(t)), ũ(t) ∈ U set , ( 43
)
where the actuator constraints are considered here using ũ(t) ∈ U set , and the guiding function η(t) could be regarded as the adjustable parameter to ensure the special control input tracks satisfy the expected control performance.

With the algebraic relations ( 6)-( 8), conditions 1)-4) above are difficult to prove for set-valued map R -1 vu (x(t), •), but provide potential theory guarantee for continuity of the applied control inputs and actuator saturations. In the following numerical results with the quad-TRUAV, a nonlinear programming problem in the form of (43) could be used directly as the inverse procedure for applied control inputs ũ(t) signed by

[i n Ω F Ω B δ e ] T from virtual values v = [v V v α v q ] T : min ∥ũ(κT ) -ũ((κ -1)T )∥ s.t.   Ω 2 F (κT ) + Ω 2 B (κT ) Ω 2 F (κT ) -Ω 2 B (κT ) δ e (κT )   = R -1 ( i n (κT ), V (κT ), α(κT ) ) •    v V (κT )+g sin γ(κT )+ ρs 2m C D1 V 2 (κT )α(κT )+ ρs 2m C D2 V 2 (κT )α 2 (κT ) v α (κT ) -g cos γ(κT ) V (κT ) v q (κT ) -ρsc 2Iy C m1 V 2 (κT )α(κT )    , - 1 4 π ≤ i n (κT ) ≤ 3 4 π, 0 ≤ Ω 2 F (κT ) ≤ 4000 2 , 0 ≤ Ω 2 B (κT ) ≤ 4000 2 , - 1 6 π ≤ δ e (κT ) ≤ 1 6 π, (44)
where ũ(t -T ) with t = κT is a special guiding function for minimizing the rates of applied control inputs to prevent the discontinuity, and T and κ refer to the sampling period and number to approximate the above rates as discrete-time differences. Note that the estimated states V , α, and γ are used in (44), and additional actuator constraints in this nonlinear programming problem increase the difficulty of solvability at the same time. Consequently, inaccuracy might exist in the solutions from (44), and it is ũ(t) rather than exact u(t) that is calculated from this problem, as shown in [START_REF] Huang | Non-fragile switching tracking control for a flexible air-breathing hypersonic vehicle based on polytopic LPV model[END_REF]. This leads to unstructured disturbances δ(t) in [START_REF] Aouani | H2 analysis for LPV systems by parameter-dependent Lyapunov functions[END_REF], and the designed LPV controller has definite robustness against these disturbances. According to the form of v V shown in [START_REF] Zhao | Robust transition control of a martian coaxial tiltrotor aerobot[END_REF], longitudinal acceleration of the quad-TRUAV requires decreasing i n or α in order to provide extra force. This property based on TRUAV dynamics ensures a transition towards airplane mode and longitudinal acceleration simultaneously. However, airplane mode with i n = 0 induces the first unsolvable case of [START_REF] Fravolini | Data-driven scheme for robust fault detection of air data system sensors[END_REF]. To obtain the applied control inputs when i n = 0, α should approach a specific reference value to provide an extra degree of freedom, primarily to ensure solvability of the first two lines in [START_REF] Fravolini | Data-driven scheme for robust fault detection of air data system sensors[END_REF]. Based on the analysis in [START_REF] Liu | Mode transition and fault tolerant control under rotor-tilt axle stuck fault of quad-TRUAV[END_REF], the reference value of α could be formulated as follows:

α ref = { 0, i n > i n mean(N, i n + α), i n ≤ i n ( 45
)
where i n is a small and positive value, mean(N, •) is a mean filter with an N s time window to remove the underlying algebraic loop, and i n comes from the previous inverse procedure. According to (45), varying α ref near airplane mode (i n ≤ i n ) encourages α to replace i n and provide the dominant control effect, which ensures solvability of (41) and stabilization of i n near 0.

In addition, the second unsolvable case of ( 41) is practically impossible during regular flight. For common aerial vehicles, C Lδ > C Dδ > 0 is usually satisfied, and

C Dδ √ C 2 Dδ +C 2 Lδ = 1 √ 1+C 2 Lδ /C 2 Dδ < 1 √
2 could be obtained. However, it is difficult for i n + α to take a value less than -arccos( 1 √ 2 ) = -π 4 during regular flight. Therefore, this unsolvable case would not be considered here.

Numerical results

The values of parameters in the quad-TRUAV model ( 6)-( 9) are listed as follows for the numerical results, which are based on references [START_REF] Zhao | Robust transition control of a martian coaxial tiltrotor aerobot[END_REF] and [START_REF] Pounds | Modelling and control of a large quadrotor robot[END_REF]: The polytopic LPV model ( 17) and ( 18) in the fault-free case is formulated as follows: The varying parameters are p 1 = V -V V -V = 30-V 30-0 and p 2 = 1 -p 1 , where the flight velocity is assumed to vary from 0 and 30 m/s.

m = 2.
A 1 =     0
Bias sensor faults in the flight velocity V and pitch rate q are considered in the following numerical results:

f (t) = [ f V (t) f q (t) ] T , (46)
where

f V (t) = { 0, 0 ≤ t < 28s, -1, t ≥ 28s , f q (t) = { 0, 0 ≤ t < 8s, 0.5, t ≥ 8s , and D =     1 0 0 0 0 0 0 1     .
(47) Note that an abrupt flight velocity sensor fault could induce varying parameter uncertainties ∆p 1 and ∆p 2 . Due to the finite feasible uncertainty range represented by ∆p m1 and ∆p m2 in the designed LPV controller, much more severe flight velocity faults would not be considered in following contents to prevent dramatic values of ∆p 1 and ∆p 2 . The above nonlethal flight velocity fault might be caused by incorrect installation and positioning of the sensor [START_REF] Fravolini | Data-driven scheme for robust fault detection of air data system sensors[END_REF]. The pitch rate fault might be induced by gyro damage, which results in evident measurement bias.

The LPV controller gain K(p) and PIO gain L(p) for ( 21) and ( 26) are synthesized based on the LMI minimization problem [START_REF] Aubin | Selections and parametrization[END_REF] with Theorem 1. The results are calculated using MATLAB LMI Toolbox with minimum τ = 1.751×10 3 . For these results, the parameters in Theorem 1 are set as follows: F m is regarded as a solvable variable in the proposed theorems with respect to ϵ and F m , and the calculated value ∆p m1 = ∆p m2 = 0.0148 is maximized. Meanwhile, the performance about the H ∞ disturbance attenuation degenerates seriously. For problem [START_REF] Aubin | Selections and parametrization[END_REF], the values of ∆p m1 and ∆p m2 compromise the control performance and the varying parameter uncertainty range.

Note that, if the relationship [START_REF] Montagner | Stability analysis and gain-scheduled state feedback control for continuous-time systems with bounded parameter variations[END_REF] is not introduced into Theorem 1 using the parameter matrices E Xij and E Y ij (i, j = 1, ..., n), then problem (39) would be solved with minimum τ = 5.83 × 10 3 , which is larger than the value with [START_REF] Montagner | Stability analysis and gain-scheduled state feedback control for continuous-time systems with bounded parameter variations[END_REF]. This result indicates that introducing the inherent relationship between the varying parameter derivatives into the conditions could reduce the conservatism of the stability analysis. Moreover, solving problem [START_REF] Aubin | Selections and parametrization[END_REF] with Theorem 2 requires significantly less computing time (approximately 5 s) compared with Theorem 1 (more than 15 min). However, the minimum τ = 1.765 × 10 3 from Theorem 2 is larger than that from Theorem 1. These results further validate that Theorem 2 requires less calculation load, but is more conservative than Theorem 1.

For closed-loop simulation, the calculated LPV controller and observer gains from the above LMI minimization could be applied online, and the initial values are set as x(0) = [1.8 0 0 0] T , x(0) = [1.9 0 0 0] T , and f (0) = [0 0] T . Fig. 3 shows the estimation errors for all states, and Fig. 4 shows measurement outputs and their fault-free estimations from the estimated states, where T = 0.02 s, disturbance inputs w(t) = [0.1 sin 2πt 0.001 sin 2πt 0.001 sin 2πt 0.01 sin 2πt] T , initial and final references of V are 2 m/s and 22 m/s, respectively, and γ ref = 0. According to Fig. 3, the states of the quad-TRUAV are accurately estimated. The faulty output of V could not track the reference given the flight velocity fault shown in (46), as shown in Fig. 4. However, as its fault-free output estimation converges to V ref and the estimation error of V converges to 0, the fault-free output of V could track V ref without doubt. V ref is with fixed slope 1 m/s 2 to compromise the control performance of γ. Otherwise, the fast acceleration with decreased rotor-tilt angle would affect future flight height control. Under the pitch rate fault shown in (46), the faulty output of q is stabilized to 0.5 rad/s, but its fault-free output is stabilized at 0 according to the curve of its fault-free output estimation. Moreover, the bias sensor fault in the pitch rate induces large overshoots of γ and α, but their tracking control is still achieved when t → +∞.

Fig. 5 further shows estimations of the fault magnitudes. Tracking control of the quad-TRUAV under bias sensor faults is ensured as these estimations converge to fault magnitudes. f (t) appropriately represents the fault magnitudes in the fault-free and faulty cases. In addition, Fig. 6 shows varying parameter uncertainties. All values are within the available range in the form of ( 31), thus closed-loop stability of (28) could be ensured for tracking control and state estimation with unmeasurable varying parameters under bias sensor faults.

The control inputs are shown in Fig. 7. Although the four conditions in Section 5 are difficult to prove, all control inputs are continuous and within the actuator constraints with the inverse procedure (44). When the quad-TRUAV approaches airplane mode with small i n , the value of α ref begins to decrease according to (45) with N = 5 and i n = 5 180 π rad, which ensures the quad-TRUAV reaches airplane mode with i n = 0 and available control inputs. The transition from helicopter mode to airplane mode completes, even in the presence of sensor faults. What should be noted is that, the results calculated from (44) are sensitive to varying disturbances due to the inverse matrix, as shown in [START_REF] Fravolini | Data-driven scheme for robust fault detection of air data system sensors[END_REF], which increases the undesirable effects from disturbances. This is the reason why the final values of the control inputs include oscillations, which are caused by the disturbance inputs w(t).

The above observer-based LPV controller for virtual control inputs and the inverse procedure for applied control inputs provide the transition control of a quad-TRUAV. Compared with typical GS methods [START_REF] Sato | Flight controller design and demonstration of quad-tilt-wing unmanned aerial vehicle[END_REF][START_REF] Zhao | Robust transition control of a martian coaxial tiltrotor aerobot[END_REF][START_REF] Muraoka | Transition flight of quad tilt wing VTOL UAV[END_REF][START_REF] Chen | Control and flight test of a tilt-rotor unmanned aerial vehicle[END_REF] and previous research in [START_REF] Liu | Transition control of tilt rotor unmanned aerial vehicle based on multi-model adaptive method[END_REF], the proposed method in this paper ensures the stability throughout the transition procedure in theory. Superior to previous work in [START_REF] Liu | Mode transition and fault tolerant control under rotor-tilt axle stuck fault of quad-TRUAV[END_REF], this paper considers observer design under sensor faults and is more practical. Compared with some other nonlinear control methods [START_REF] Wang | Mathematical modeling and control of a tilt-rotor aircraft[END_REF][START_REF] Liu | Mode transition and fault tolerant control under rotor-tilt axle stuck fault of quad-TRUAV[END_REF], the LPV method in this paper is much easier to analyze the closed-loop stability with an observer-based control structure. Moreover, the computing time for the virtual and applied control inputs in the above simulation is approximately 0.0094 s, where the active-set algorithm is used to solve the nonlinear programming problem (44) with MATLAB installed on a laptop (operating system: Windows 7, CPU: Intel Core i5-5200U). This computing time makes proposed method attractive for future practical applications.

Conclusions

Observer-based LPV control of a quad-TRUAV with unmeasurable varying parameters under bias sensor faults was investigated in this paper. By introducing virtual control inputs, a nonlinear model for the quad-TRUAV was transformed into the polytopic LPV form for bounded varying parameters. To address unmeasurable varying parameters, the observer-based LPV control method was considered for the closed-loop system under bias sensor faults. The parameter-dependent Lyapunov method with bounded condition and relationship between varying parameter derivatives was used for stability analysis. The common Lyapunov method was analyzed further as a special case. The inverse procedure for applied control inputs ensured asymptotical stability of the quad-TRUAV during the transition procedure in theory. Numerical results showed the effectiveness of the proposed method in the presence of bias sensor faults.

A nonlinear programming problem was designed as an inverse procedure for applied control inputs, which relies on the accuracy of the model. Consequently, reasonable nonlinear modeling is an inevitable and important problem for TRUAV controller design. Moreover, solutions from the above inverse procedure are sensitive to varying disturbances due to the dependence on system states. These two problems should be investigated further to provide better applications of the proposed method in non-ideal conditions.

The necessary and sufficient condition for (A.6) is as follows:

He{ -ζ T 1 (t)ϕ 1 W 1 (p)ζ 1 (t) -ζ T 1 (t)ϕ 1 [W 1 (p) -X(p)]ξ(t) + ζ T 1 (t)ϕ 1 Ẽe(t) -ζ T 2 (t)ϕ 2 W 2 (p)ζ 2 (t) -ζ T 2 (t)ϕ 2 [W 2 (p) -Y (p)]e(t) + ζ T 2 (t)N (p)Ce(t) + ζ T 2 N (p)D w ω(t) + ξ T (t) BM (p)ζ 1 (t) + ξ T (t)[ Ã(p)X(p) + BM (p) - 1 2 X( ṗ) -Ãu (∆p)X(p)]ξ(t) + ξ T (t) De(t) + ξ T (t) Bw ω(t) + xT (t)ϕ x X(p)ξ(t) -xT (t)ϕ x x(t) + e T (t)Y (p)A u (∆p)x(t) + e T (t)[Y (p)A(p) + N (p)C + 1 2 Y ( ṗ)]e(t) + e T (t)[Y (p)B w + N (p)D w ]ω(t) + ω T (t)(-τ /2 • I)ω(t) + τ -1 z T (t)C e e(t) + τ -1 z T (t)F w ω(t) + τ -1 z T (t)(-τ /2 • I)τ -1 z(t)} < 0. (A.9) By defining σ(t) = [ζ T 1 (t) ζ T 2 (t) ξ T (t) xT (t) e T (t) ω T (t) τ -1 z T (t)] T ,
                                         0 0 0 0 X(p) RT A T + ϵ X (p) LI t 0 0 0 0 Y (p)LA t + ϵ Y (p)LI t 0 0 0 0               [ F 0 0 F ] [ 0 0 -I R 0 0 0 0 0 0 0 IR 0 0 0 ]                            .
(A.12)

According to Lemma 1, if F T F ≤ F T m F m with diagonal F m , then G(p, ṗ) + G(p, ∆p) < 0 is tenable if and only if there exists a scalar ϵ such that Finally, ( 35) and ( 36) could be established according to Lemma 2.

G(p, ṗ)+ ϵ -1               0 0 0 0 X(p) RT A T + ϵ X (p) LI t 0 0 0 0 Y (p)LA t +ϵ Y (p)LI t 0 0 0 0                             0 0 0 0 X(p) RT A T + ϵ X (p) LI t 0 0 0 0 Y (p)LA t +ϵ Y (p)LI t 0 0 0 0               T + ϵ -1           0 0 0 0 -ϵ RT I t F m 0 0 ϵR T I t F m 0 0 0 0 0 0                     0 0 0 0 -ϵ RT I t F m 0 0 ϵR T I t F m 0 0 0 0 0 0           T < 0,

Appendix B: Proof of Theorem 2

With the proof of Theorem 1, let

X 1 = • • • = X n = X, Y 1 = • • • = Y n = Y, (B.1) W 11 = • • • = W 1n = W 1 , W 21 = • • • = W 2n = W 2 , (B.2) ϵ Xi1 = • • • = ϵ Xin = ϵ Xi , ϵ Y i1 = • • • = ϵ Y in = ϵ Y i , (B.
3)

E Xij = X, E Y ij = -Y, (B.4)
conditions [START_REF] Xie | Output feedback H ∞ control of systems with parameter uncertainty[END_REF] and ( 38) could be derived according to [START_REF] Pounds | Modelling and control of a large quadrotor robot[END_REF] and [START_REF] Tanaka | Takagi-Sugeno fuzzy model and parallel distributed compensation[END_REF], and condition [START_REF] Tanaka | Takagi-Sugeno fuzzy model and parallel distributed compensation[END_REF] becomes the sufficient condition for [START_REF] Lopez-Estrada | LPV model-based tracking control and robust sensor fault diagnosis for a quadrotor UAV[END_REF]. In this way, the common Lyapunov function of closed-loop system (28) becomes V(x(t), e(t)) = xT (t)X -1 x(t) + e T (t)Y e(t), and Theorem 2 is established.

2 =

 2 Euclidean norm and L 2 norm * = transposed element in the symmetric position of a matrix 0, 0 n 1 ×n 2 = zero matrices with suitable dimension and n 1 × n 2 -dimension A = rotor area, m 2 A(p),B,B w ,C,D,D w = state-space matrices of the LPV system with p as varying parameter vector C D0 , C D1 , C D2 , C Dδ = aerodynamic coefficients of drag C L0 , C L1 , C Lq , C Lδ = aerodynamic coefficients of lift C m0 , C m1 , C mq , C mδ = aerodynamic coefficients of pitch moment C t = thrust coefficient of rotors c = average chord length of wing, m e(t) = an augmented vector about estimation errors F, F m = diagonal matrix containing varying parameter uncertainties and its bound F m , ϵ = parameters for controller design, F m = ϵF m g = acceleration of gravity, m/s 2 He([•]) = a shorthand notation for [•] + [•] T h = flight height, m I, I n = identity matrices with suitable dimension and n-dimension I y = moment of inertia in the y-direction, kg•m 2 i n = rotor-tilt angle, rad K(p), L(p) = LPV controller and observer gains M (p), N (p) = intermediate matrices for K(p) and L(p) m

Fig. 1 (

 1 b), dynamic and kinematic equations in the wind-axis coordinate system (a) Quad-TRUAV platform (b) Coordinate system definitions

Figure 1 :

 1 Figure 1: Quad-TRUAV and coordinate systems.

∥z∥ 2 ∥ω∥ 2

 2 ≤ τ , if there exist positive scalars ϵ, ϕ 1 , ϕ 2 , and ϕ x , arbitrary scalars ϵ Xi and ϵ Y i (i = 1, ..., n), invertible matrices W 1 and W 2 , matrices X, Y , M (p) = ∑ n i=1 pi M i , and N (p) =

1 1

 1 and H i(5,6) = Y B w + N i D w . The calculated LPV controller and observer gains are K(p) = M (p)W -and L(p) = W -T 2 N (p), respectively. The feasible uncertainty range F m = ϵ -1 F m . Proof of Theorem 2. See Appendix B.

  71 kg, I y = 0.0816 N•m, c = 0.2966 m, s = 0.452 m 2 , R = 0.12 m, A = R 2 π, x r = 0.1847 m, g = 9.8 m/s 2 , ρ = 1.225 kg/m 3 , C t = 0.0041, C D0 = 0.0111, C D1 = 2.6278 × 10 -4 , C D2 = 0.0035, C Dδ = 0.0061, C L0 = 0.1982, C L1 = 0.1823, C Lq = 1.5177, C Lδ = 0.0102, C m0 = 0.0189, C m1 = 4.333 × 10 -6 , C mq = -1.0365, C mδ = -0.1186.

  w = I, C = I, D = 0, D w = 0.1I, C e = I, F w = 0,

Figure 3 :

 3 Figure 3: Estimation errors for states.

ϕ 1 =

 1 70, ϕ 2 = 110, ϕ x = 0.05, A f = I, and ∆p m1 = ∆p m2 = 0.4/30 ≈ 0.0133,where suitable values of ϕ 1 , ϕ 2 , and ϕ x could be determined with line searches[START_REF] Koroglu | Improved conditions for observer-based LPV control with guaranteed L2-gain performance[END_REF], -A f should be a Hurwitz matrix, and the values of ∆p m1 and ∆p m2 affect the performance of the controller. If the LMI minimization problem (40) is considered using the same parameters, there would be ϵ = 38.78, F m = 0.5723I, and τ = 4.848 × 10 7 .

Figure 4 :

 4 Figure 4: Measurement outputs and fault-free estimations.

Figure 5 :

 5 Figure 5: Magnitudes and estimations of bias sensor faults.

Figure 6 :

 6 Figure 6: Varying parameter uncertainties.

Figure 7 :

 7 Figure 7: Control inputs of quad-TRUAV.

( 1 , 3 )

 13 the above equation is equivalent to σ T (t)G(p, ṗ, ∆p)σ(t) < 0 withG(p, ṗ, ∆p) = -ϕ 1 W 1 (p) + ϕ 1 X(p), G (2,5) = N (p)C -ϕ 2 W 2 (p) + ϕ 2 Y (p), G (3,3) = Ã(p)X(p) + BM (p) -1 2 X( ṗ) -Ãu (∆p)X(p), G (5,4) = Y (p)A u (∆p), G (5,5) = Y (p)A(p) + N (p)C + 1 2 Y ( ṗ),andG (5,6) = Y (p)B w + N (p)D w .According to the quadratic form in the LHS of (A.9), if G(p, ṗ, ∆p) < 0, LHS (A.6) ≤ 0 would be established always (LHS (A.6) = 0 only when σ(t) = 0).Because∑ n k=1 ṗk = 0 and I t FI = 0, the following LMI condition could be formulated as a sufficient condition of LHS (A.6) < 0 (σ(t) ̸ = 0) based on the form of G(p, ṗ, ∆p): , ṗ) + G(p, ∆p) < 0,(A.11) in which G (1,3) = M T (p) BT -ϕ 1 W 1 (p)+ϕ 1 X(p), G (2,5) = N (p)C -ϕ 2 W 2 (p)+ ϕ 2 Y (p), G (5,6) = Y (p)B w + N (p)D w , and G (3,3) = He[ Ã(p)X(p) + BM (p)] -X( ṗ)+ ∑ n k=1 ṗk E X (p), G (5,5) = He[Y (p)A(p)+N (p)C]+Y ( ṗ)+ ∑ n k=1 ṗk E Y (p), with E X (p) = ∑ n i=1 ∑ n j=1 pi pj E Xij and E Y (p) = ∑ n i=1 ∑ n j=1 pi pj E Y ijas parameter matrices. In addition, G(3,3) = -Ãu (∆p)X(p) -X(p) ÃT u (∆p) -2ϵ X (p) LI t FI R and G(5,4) = Y (p)A u (∆p) + ϵ Y (p)LI t FIR with ϵ X (p) = ∑ n i=1 ∑ n j=1 pi pj ϵ Xij and ϵ Y (p) = ∑ n i=1 ∑ n j=1 pi pj ϵ Y ij as arbitrary scalars. With the representations in (29) and (30), G(p, ∆p) = He

(A. 13 )

 13 and F T F ≤ F T m F m means -F m ≤ F ≤ F m . According to the Schur complement, the above LMI is equivalent to ) RT A T + ϵ X (p) LI t 0 -RT I t F m where F m = ϵF m . By introducing X( ṗ) = ∑ n k=1 ṗk X k and Y ( ṗ) = ∑ n k=1 ṗk Y k into (A.14),the LHS of (A.14) is affinely parameter-dependent on ṗk (k = 1, ..., n). All vertexes in (32) should be considered[START_REF] Liu | State feedback controller design for affine parameter-dependent LPV systems[END_REF]. Further introducingÃ(p) = ∑ n i=1 pi Ãi , A(p) = ∑ n i=1 pi A i , M (p) = ∑ n i=1 pi M i , N (p) = ∑ n i=1 pi N i , and W 1 (p) = ∑ n j=1 pj W 1j , W 2 (p) = ∑ n j=1 pj W 2j , X(p) = ∑ n j=1 pj X j , Y (p) = ∑ nj=1 pj Y j into (A.14) yields the following equivalent LMI:

For Reviewer #1 Q8

For Reviewer #1 Q12

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 61473282, 61503369, and U1508208).

Conflict of interest statement

There is no conflict of interest for this paper.

Appendix A: Proof of Theorem 1

For the Lyapunov function V(x(t), e(t)) = xT (t)X -1 (p)x(t)+e

could be introduced to prevent the bilinear matrix inequality (BMI) problem during stability analysis. Obviously,

, and the closed-loop system (28) becomes

The following condition on the H ∞ disturbance attenuation could be considered [START_REF] Koroglu | Improved conditions for observer-based LPV control with guaranteed L2-gain performance[END_REF] with the above closed-loop system to propose some LMI conditions:

) could be reformulated as dV(x(t),e(t))

dt

τ ≤ 0. Consequently, (A.6) is also the sufficient condition of asymptotical stability with ω(t) = 0. According to [START_REF] Amato | Finite-time stability of linear time-varying systems: analysis and controller design[END_REF], if there exist matrices 0

| t=t 0 , and xT (t 0 )X -1 0 x(t 0 ) + e T (t 0 )Y 0 e(t 0 ) ≤ 1, finite-time stability could be further ensured for t ∈ [t 0 , t 0 + T ] as follows:

xT (t)X -1 (p)x(t) + e T (t)Y (p)e(t) ≤ V(x(t), e(t)) ≤ V(x(t), e(t))| t=t 0 < xT (t 0 )X -1 0 x(t 0 ) + e T (t 0 )Y 0 e(t 0 ) ≤ 1. (A.7) Consider the left hand side (LHS) of (A.6):