Y Alaouchiche
email: yasmine.alaouchiche@utt.fr

Y Ouazene

F Yalaoui
email: farouk.yalaoui@utt.fr

Energetic and Economic Performance Evaluation of Production Systems: Perspective Analysis

Keywords: Energy efficiency, production lines, throughput evaluation, machine performance evaluation, Equivalent Machine Method

Improving the economic as well as the energetic performances of manufacturing systems has become a real challenge for both researchers and industrials. Due to the economic, legislative and environmental pressure, enhancing both economic and energetic performances arouses interest and attracts further efforts. The objective of this paper is to discuss current work done on this subject, analyze the results and propose further perspectives for research in this area towards sustainability. Furthermore, an introduction for an economic and energetic evaluation method for serial production lines is also presented.

INTRODUCTION

The industrial sector represents more than 31% of total world energy consumption according to statistics from the International Energy Agency IEA (2019). Thus, to face these serious situations of increasing energy prices, depletion of global resources and global warming, improving energy efficiency in manufacturing is becoming crucial. The industrial sector is therefore increasingly interested in the issue of energy efficiency for financial, legislative and environmental concerns. Unfortunately, only few work has been done when it comes to study energetic combined with economic efficiency of industrial production lines. Indeed, many research results have been reached in the field of improving energy efficiency for machine tools but there is a significant lack of work in the case of serial production lines.

The aim of this paper is to discuss and analyze current work and progress concerning the study of energetic and economic performances of manufacturing systems, as well as to suggest further perspectives of research in this area. An approach for evaluating both economic and energetic performances for serial production lines is also introduced.

LITERATURE REVUE

Improving the economic efficiency of serial production lines has been a major concern since decades. This problem has been widely treated in the literature from the smallest systems to the largest ones. Exact solutions for small systems with two machines and a single buffer were obtained [START_REF] Dallery | Manufacturing flow line systems: a review of models and analytical results[END_REF][START_REF] Li | Comparisons of two-machine line models in throughput analysis[END_REF]. For larger systems, the complexity being more consequent, approximate solutions are proposed, either based on aggregation (Li andMeerkov, 2003, 2008;Jacobs and Meerkov, 1995b,a), decomposition approaches (Gershwin, 1987a(Gershwin, ,b, 1989;;[START_REF] Choong | A decomposition method for the approximate evaluation of capacitated transfer lines with unreliable machines and random processing times[END_REF][START_REF] Dallery | An efficient algorithm for analysis of transfer lines with unreliable machines and finite buffers[END_REF][START_REF] Dallery | Approximate analysis of transfer lines with unreliable machines and finite buffers[END_REF], or other proposed analytical methods such as the Equivalent Machine Method [START_REF] Ouazene | Equivalent machine method for approximate evaluation of buffered unreliable production lines[END_REF]. The EMM is an approached analytical method that evaluates the system throughput of a buffered serial production line of unreliable machines with exponentially distributed parameters. The study is based on the analysis of the different states of each buffer using birth-death Markov processes. Thereafter, an equivalent machine replaces each original machine, taking into account the probabilities of blockage and starvation. This allows the evaluation of the throughput of the production line, defined as the bottleneck between the effective production rates of the equivalent machines. This method has been tested on a large benchmark and its relevance has been demonstrated. Indeed, it reduces significantly the state space cardinality of the Markov chain representation of the system and consequently the computational times.

Nevertheless, the current circumstances are pushing researchers and industrials for further performance enhancement. The concern in no more just improving the performances of manufacturing systems economically but also energetically. Further efforts are therefore settled to bring solutions for this problem. However, work done towards the evaluation and improvement of energetic performances of serial production lines is still insufficient, and requires more efforts and advances, according to our non exhaustive literature review. In the following, we summarize our literature review according to several axes of focus to which energy performance evaluation studies have been adressed.

Generic methods

From a general perspective, [START_REF] Gutowski | Thermodynamic analysis of resources used in manufacturing processes[END_REF] used a thermodynamic framework to characterize the material and energy resources used in manufacturing processes. The analysis of different processes allowed to illustrate the relevance of thermodynamics in manufacturing processes analysis. The authors used exergy analysis to identify loss in resources aiming to design more efficient processes. The energy consumption of a manufacturing system is formulated as shown in (1) where Ẇ is total power used by the process equipment, Ẇ0 the idle power for the equipment in the ready position, ṁ the rate of material processing in (mass/time), and k a constant (J/mass)).

Ẇ = Ẇ0 + k ṁ (1)
On another hand, a novel generic method is proposed by [START_REF] Dietmair | A generic energy consumption model for decision making and energy efficiency optimisation in manufacturing[END_REF] in order to model the energy consumption behavior of machines and plants based on a statistical discrete event formulation. Authors introduced applications in real-time, tactical and strategic decision making processes for energy consumption minimization of production systems using the modeling framework.

2.2 Energy efficiency improvement for Bernoulli serial production lines [START_REF] Su | Improving energy efficiency in bernoulli serial lines: an integrated model[END_REF] studied the improvement of energetic performance of Bernoulli serial production lines. They developed an integrated model for improving energy efficiency for buffered lines composed of two machines. The energy consumption of the line was formulated as shown in (2). W i is the total electrical power consumed, W Oi represents the set-up power needed for the machine to reach 'ready' status, and k i ρ i the additional power needed to process parts, where k i is a constant, and ρ i the processing rate of machine i.

W i = 2 i=1 W Oi + 2 i=1 k i ρ i (2)
Recently, [START_REF] Su | Reducing energy consumption in serial production lines with bernoulli reliability machines[END_REF] extended the study for larger systems with unreliable machines and finite buffers. An integrated model for minimizing energy consumption under a desired production rate was presented (3) and ( 4), where ρ i is the machine's processing rate and P R d is the desired production rate, only the processing energy required to produce is considered in this study. Exact analysis to optimally allocate production capacity for small systems were developed, an aggregation is used for medium systems and finally a heuristic algorithm for larger systems.

min E = K i=1 k i ρ i (3) s.t. P R ≥ P R d (4)

Energy performance improvement for serial production lines using real time data

Methods for the improvement of energetic performance of serial production lines using real time date are also treated. [START_REF] Brundage | Energy savings opportunities and energy efficiency performance indicators for a serial production line[END_REF] developed new energy savings opportunity strategies to maximize energy savings for a serial production line. The method is based on the evaluation of the opportunity window from on-line production data in order to turn off certain machines for energy savings without negatively affecting throughput. Using real time production data, energy efficiency performance indicators are evaluated to identify the least energy efficient machine on the line. The Energy Efficiency Performance Indicator is developed (5), where W i,1 is the static part of the energy consumption for machine i and W i,2 the dynamic part attributed to random disruptions on the line. This indicator is used within the energy savings opportunity strategy to allow to take the opportunity window for the least energy efficient machine at opportune times, in order to make improvements to the machine towards the increase of the overall energy efficiency of the line.

EEP I i = W i,1 W i,1 + W i,2 (5)
Later, Brundage et al. ( 2014) incorporated the warm-up time of each machine in the analysis, allowing for more accurate results. [START_REF] Bajpai | Modeling, analysis, and improvement of integrated productivity and energy consumption in a serial manufacturing system[END_REF] extended previous studies to include economic concerns within the energetic performance evaluation. Performance indices for measuring energy efficiency and also productivity using available sensor data from the production line are developed. These indices capture dynamic nature of manufacturing systems. Authors analyze the energy structure and effects of downtime events on the production line in order to improve the system's performances. They proposed methods for measuring efficiency and highlighting the areas of inefficiency in the system.

Serial production lines with geometric machines were studied by [START_REF] Chen | Transient performance analysis of serial production lines with geometric machines[END_REF]. Transient analysis for energetic performance were presented and discussed.

Further energy performance improvements for serial production lines

Further solutions for the improvement of serial production lines energetic performances were also proposed. [START_REF] Chen | Feedback control of machine startup for energy-efficient manufacturing in bernoulli serial lines[END_REF] addressed energy consumption reduction for the simplest Bernoulli system of a buffered two machines line and introduced feedback control of machine start-up with buffer depletion at the end of each shift. Transient analysis were performed in order to achieve performance evaluation formulations and propose effective controllers for machine startup schedule.

Larger systems with warm-up time were considered by [START_REF] Jia | Performance analysis for serial production lines with bernoulli machines and real-time wip-based machine switch-on/off control[END_REF]. The later extended the previous study and achieved results in both transients and steady state.

Furthermore, idle time reduction for energetic performance improvement was studied by [START_REF] Zhang | Idle duration prediction for manufacturing system using a gaussian mixture model integrated neural network for energy efficiency improvement[END_REF] who developed a Gaussian mixture model to predict machine idle periods duration for manufacturing systems. The idle time prediction allows to suggest optimal actions considering energy savings while respecting throughput constraint. 2.5 Energy performance evaluation and improvement for machine tools

Although the issue of improving both economic and energetic performances requires further efforts in the case of serial production lines, this concern was widely studied for machines tools, considering models based on machine components or machines tools in a global way.

A comprehensive literature revue concerning energy consumption models and energy efficiency of machine tools was presented by [START_REF] Zhou | Energy consumption model and energy efficiency of machine tools: a comprehensive literature review[END_REF]. Energy efficiency formulations were presented and discussed. The design, scheduling, optimization and assessment based on energy efficiency of machine tools were also introduced. The authors presented several perspectives for energy consumption modeling and decomposition for machine tools according to several axes (see Fig. 1). Classification and prediction methods of energy consumption were presented and over-viewed by [START_REF] Zhao | Energy consumption in machining: Classification, prediction, and reduction strategy[END_REF]. They also treated strategies for energy consumption reduction in machining processes. [START_REF] Schudeleit | The total energy efficiency index for machine tools[END_REF] worked on the development of a metric able to quantify the design of machine tools regarding energy efficiency. Therefore, the total energy efficiency index was developed based on the respective assembly of components. [START_REF] Tuo | Key performance indicators for assessing inherent energy performance of machine tools in industries[END_REF] developed key performance indicators referred to as 'inherent energy performance' (IEP) indexes focusing on the design and selection of machine tools with the consideration of the main process controls in the usage phase and their interaction. The developed method is based on simplified measurements of basic data in order to calculate the IEP indexes. These indexes are of two types: Energyconsumption function indexes (such as Standby power and idling power function) and Equivalent energy-consumption indexes (such as Equivalent standby power and Equivalent idling power). Each equivalent energy index is represented by the weighted average of powers for different cases times the probability or frequency of each case (see the example of Equivalent standby power formulation in ( 6)). The main characteristic of these indexes in that of taking into consideration the various process control distributions such as the probability of each spindle revolving speed along with the specific energy demand for each machine tool activity.

EP sb = C(P sb,i )P sb,i (6)
Word done on sustainability for machine tools has not been resumed to energy efficiency studies only, green performances as a whole were studied by [START_REF] Lv | A composite evaluation model of sustainable manufacturing in machining process for typical machine tools[END_REF] for CNC (Computer Numerical Control) machine tools. The authors established a model for the evaluation of energy efficiency, carbon efficiency and green degree. Energy efficiency was defined by the ratio of the energy required for the cutting process (material removal energy) and the total energy consumed for the machining process (7).

η = E cutting E process (7)
The approach for energy consumption evaluation of CNC machine tools was to decompose it by machine states. The later are: start-up state, standby state, no-load state (in which the machine is working without load) and load state (in which the machine is effectively processing). The state energy is defined as the corresponding state power times each state's duration (8,9).

E processus = E start + E standby + E no-load + E load (8) E processus = P start × t start + P standby × t standby + P no-load × t no-load + P load × t load E cutting = P cut × t load (9)
The energy efficiency as defined in ( 7) is calculated as a function of machine and process parameters.

Wang et al. ( 2019) studied the optimization of energy efficiency in machining process and presented a model based on workingstep energy calculation method in Standard for the Exchange of Product model STEP-NC. For the resolution, the authors developed an improved ant colony optimization algorithm.

Analysis

The conducted literature revue reveals the necessity of developing a system of evaluation in which both economic and energetic performances of serial production lines are considered. On one hand, economic efficiency of serial production lines has been treated over and over in the literature since decades and satisfying results have been reached. However, on the other hand, the issue of energy efficiency evaluation and optimization for serial production lines requires further work. Although, this problem has been largely studied and considerable results have been reached in the case of machine tools, efforts are still necessary for the study, evaluation and optimization of energy efficiency for serial production lines. Therefore, the need for the development of an analytical method that allows to evaluate energy consumption and efficiency of serial production lines taking into account their economic performance.

In the next section, the problem formulation is introduced towards the introduction of our evaluation approach.

PROBLEM FORMULATION

The objective is to develop an analytical and computational method that allows the evaluation of both economic The system under consideration consists of K machines (workstations) connected by intermediate storage areas (K -1) buffers ( as illustrated by Fig. 2 ). The system is subject to non-availabilities due to the limited capacity of the buffers and / or the failure and repair rates of the machines. The following assumptions which are frequently used in the literature, address the machines, the buffers, the energy consumption and their mutual interactions:

• The failure state of the machines depends on the operations. A machine cannot fail if it is starved or blocked.

• The first machine cannot be starved and the last machine cannot be blocked. • The failure and repair times are independent and distributed according to an exponential law. • Buffer B i , has a finite capacity N and cannot be down. The transition times between machines and buffers are zero. • Energy consumption of each buffer B i is neglected.

EVALUATION APPROACH AND RESULTS

The objective is to develop an analytical and computational method able to assess both economic and energetic performance of a buffered serial production line of unreliable exponential machines.

The approach is to evaluate the economic performance using the Equivalent Machine Method [START_REF] Ouazene | Equivalent machine method for approximate evaluation of buffered unreliable production lines[END_REF], and then assess energetic performance by developing a model based on machine states. The energy evaluation model uses machines and buffers parameters as well as output data from the economic evaluation. Eventually, the method evaluates an economic performance indicator which is the throughput ψ of the line, as well as energy performance indicators which are the energy consumption E and the energy efficiency η of the serial production line. Both performances and therefore indicators are also evaluated for each machine of the line (ρ i , E i , η i ) / ∀i = 1...K.

The chosen approach to evaluate energy consumption is to assess it per part. Several segmentation methods have been introduced by [START_REF] Zhou | Energy consumption model and energy efficiency of machine tools: a comprehensive literature review[END_REF] as summarized in Fig. 1. Depending on the needs of the study, the energy consumption of a machine can be evaluated according to several axes. For the conducted study, we have chosen to assess energy consumption according to the machine states.

The problem is then formulated using a discrete time and states Markov chain. The set of states S consists of the different states in which a machine M i could be: On, Down, Starved, Blocked and Starved&Blocked. S = {O, D, S, B, SB} Transition probabilities and then steady state probabilities (P O,i , P D,i , P S,i , P B,i , P SB,i ) for each machine M i were evaluated using machines and buffers parameters as well as output data from the economic evaluation. Next, energy consumption E i for each machine is formulated. It is evaluated on an horizon using specific energy consumption for each state (10).

E i = P D,i E down,i + (P S,i + P B,i + P SB,i )E no-load,i + P O,i E Cload,i + e op,i ρ i (10)

E down,i is the energy consumed in the down state in units of energy, E no-load,i the energy consumed in the idle states (starved, blocked or both) in units of energy, E Cload,i the constant part of the energy consumed in the processing state 'On' (which is assumed equal to E no-load,i in our tests) in units of energy, as well as e op,i which is the specific operating energy consumed per part manufactured in units of energy per part manufactured. These input data are balanced by each state's steady probability at the exception of the specific operating energy e op,i which is multiplied by total number of produced parts that corresponds to the production rate ρ i of each machine, in order to acquire the energy consumption E i of each machine M i . Finally, the total energy consumption E of the production line is evaluated (11) as well as the energy efficiency η.

E = K i=1 E i (11)
The particularity of the method is its ability to assess both economic and energetic performance as well as to highlight the possible interactions and trade-off between both performances. Conducted tests on a large benchmark from the literature allowed to extract many analysis and findings regarding the behavior of the economic and energetic performance and there mutual interactions. The correlation proved to be not as intuitive as it could be expected. Indeed, although energy parameters were kept constant among the tests due to the absence of adequate energy benchmarks in the literature, the throughout was not directly correlated with neither energy consumption nor energy efficiency. Producing more does not imply consuming more energy. For the tests, the energy parameters were fixed as follows:

E down,i = 1 Unit of energy. E no-load,i = E Cload,i = 10 Units of energy. e op,i = 8 Units of energy/part manufactured.

Figure . 3 as an illustrative case, represents the standardized production rate and energy consumption of each machine to the respective maximums. The corresponding tested instance is composed of five machines and four intermediate buffers whose parameters (failure rate λ i , repair rate µ i , production capacity ω i for each machine M i , and buffer size N i for each buffer B i ) are given in table 1.

We can effectively notice from that example that although the first machine produces less than the third one, it consumes more energy. The proposed evaluation method allows to conduct several analysis on the behavior of the economic and energetic performances of serial production lines as well as the corresponding interactions.

CONCLUSION

This paper gives an overall review on energy evaluation in both production lines and machining systems. Economic performance evaluation for serial production lines has revealed being largely studied and its results sufficient.

As of the energetic performances, this axes remains in need for further work. Indeed, although this issue has been largely studied for machining systems, more efforts are required for the evaluation and optimization of energetic performance of serial production lines. Especially, taking into consideration the economic performance.

Therefore, to fill the gap, it is crucial to develop a method that would evaluate both economic and energetic performance of serial production lines, in order to highlight areas of correlation and the mutual interactions between both axes of performance. The approach for the development of the method for a buffered serial production line with unreliable exponential machines is therefore introduced. This proposed solution allows the evaluation of key performance indicators able to asses performances in terms of throughput, energy consumption and energy efficiency of the production line. In addition, the numerical tests brought-out important findings regarding the tradeoff between economic and energetic performance of serial production lines.

In the future, the focus will be on the improvement of the method and its extension upon different production system types. Furthermore, practical case studies aiming for the development of an energy benchmark would allow to bring more accurate analysis especially considering energy parameters influence.

Fig. 1 .

 1 Fig. 1. Energy consumption decomposition[START_REF] Zhou | Energy consumption model and energy efficiency of machine tools: a comprehensive literature review[END_REF]

Fig. 2 .

 2 Fig. 2. Serial production line and energetic performance of a serial production line.The system under consideration consists of K machines (workstations) connected by intermediate storage areas (K -1) buffers ( as illustrated by Fig.2). The system is subject to non-availabilities due to the limited capacity of the buffers and / or the failure and repair rates of the machines. The following assumptions which are frequently used in the literature, address the machines, the buffers, the energy consumption and their mutual interactions:

Fig. 3 .

 3 Fig. 3. Production rate and energy consumption for machines of a serial production line (illustrative case)

Table 1 .

 1 Tested instance parameters

	Machine	λ i	µ i	ω i	N i
	M1	0.01 0.45 0.2	5
	M2	0.2	0.55 0.5	20
	M3	0.5	0.3	2	6
	M4	0.4	0.22 2.5	8
	M5	0.3	0.4	1	//

ACKNOWLEDGEMENTS

This research was supported by Chair Connected Innovation. We would like to show our gratitude to its support without which this paper and the research behind it would not have been possible.


Manh-Hung Do
email: manh-hung.do@gipsa-lab.grenoble-inp.fr

Damien Koenig
email: damien.koenig@gipsa-lab.grenoble-inp.fr

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

Fault estimation methods in descriptor system with partially decoupled disturbances

Keywords: Unknown Input Observer, H ∞ synthesis, Descriptor system, Fault estimation, Frequency-shaping filter

The main contribution of this paper consists of the development of two methods for actuator fault estimation in dealing with the partially decoupled disturbances of the descriptor system, which is divided into decoupled and non-decoupled unknown inputs (UI). Based on the conventional UI observer, both of the solutions decouples the fault estimation with the first group of UI, while the second UI group is handled differently by each method. Finally, a numerical example with comparisons points out the performance of each approach.

INTRODUCTION

Nowadays, the descriptor system, i.e. singular system, plays an important role in both theoretical and practical aspects as it can be used to model a wide range of chemical, mineral, electrical and economic systems (see [START_REF] Dai | Singular control systems[END_REF]). That leads to a great interest dedicated to the analysis, design, and especially fault detection and diagnosis (FDD) techniques, which not only identify faults but also estimate their magnitudes and shapes under the existence of disturbances. The FDD process in the descriptor system is inspired by state estimator, which was realized through the works of [START_REF] Darouach | Design of observers for descriptor systems[END_REF]; [START_REF] Hou | Observer design for descriptor systems[END_REF]. Then questions concerning the solution for system perturbed by unknown inputs (UI) are placed, which promotes the development of the well-known unknown input (UI) observer in [START_REF] Darouach | Reduced-order observer design for descriptor systems with unknown inputs[END_REF]; [START_REF] Chen | Design of unknown input observers and robust fault detection filters[END_REF]; [START_REF] Koenig | Unknown input observers for switched nonlinear discrete time descriptor systems[END_REF]. By choosing properly a group of parametric matrices that decouple the UI disturbances in state estimation, the dynamics of estimation error is asymptotically stable, i.e. the convergence towards 0. Regarding FDD purpose, a proportional multi-integral UI observer has been introduced in [START_REF] Koenig | Unknown input proportional multipleintegral observer design for linear descriptor systems: application to state and fault estimation[END_REF] by considering the high-order polynomial fault and its derivatives as states of an augmented system. However, one notable drawback associated with this kind of UI observer design is the satisfaction of decoupling and detectability conditions (see Lemma 2 in [START_REF] Koenig | Unknown input proportional multipleintegral observer design for linear descriptor systems: application to state and fault estimation[END_REF]). Consequently, the design of UI observer with partially decoupled disturbances becomes an interesting topic in the research community.

To the best of authors' knowledge, few works such as that of [START_REF] Bezzaoucha | On the unknown input observer design: a decoupling class approach[END_REF] were conducted to handle the above problems in linear systems with partially decoupled disturbances by using the UI observer. In terms of FDD, [START_REF] Xu | A novel design of unknown input observers using set-theoretic methods for robust fault detection[END_REF] and [START_REF] Gao | Unknown input observer-based robust fault estimation for systems corrupted by partially decoupled disturbances[END_REF] have presented a novelty of UI-observer. In that design, the disturbances are divided into two main groups: one can be decoupled by choosing the appropriate matrices, and the other contains all non-decoupled disturbances. As a result, the estimation error is now only affected by the group of the second one. In [START_REF] Xu | A novel design of unknown input observers using set-theoretic methods for robust fault detection[END_REF], the fault detection has been realized by using the set-theorem to deal with this non-decoupled group; while for 2 nd -order polynomial fault estimation in [START_REF] Gao | Unknown input observer-based robust fault estimation for systems corrupted by partially decoupled disturbances[END_REF] the gains of UI observer are calculated to ensure the stability of estimation error dynamics as well as its insensitivity to disturbances by using the H ∞ synthesis. Despite its performance, those papers are only focused on the special case of the singular matrix (E = I) and the H ∞ performance can be affected when tackling a great amount of non-decoupled UIs. Inspired by this strategy, [START_REF] Liu | State estimation and fault reconstruction with integral measurements under partially decoupled disturbances[END_REF] introduced an UI observer for the descriptor system generated from system state and fault. However, this solution is implemented only if the initial system is non-singular, not to mention that its existence conditions only concern the new augmented descriptor system instead of the original one. Hence, there is a need to complete this kind of observer design for the descriptor system framework.

For such above reasons in UI observer-based design, authors are motivated to make the following contributions for the FDD process in the UI descriptor system:

• Development of the global H ∞ approach as an extension result of [START_REF] Gao | Unknown input observer-based robust fault estimation for systems corrupted by partially decoupled disturbances[END_REF] with its existence conditions depending directly on the parameters of the initial descriptor system.

• A novel approach to overcome the limitation of global H ∞ solution in handling the non-decoupled UIs, which is resulted from the combination of frequencyshaping filter and the H ∞ synthesis. Additionally, a numerical example is presented to demonstrate the approaches. Through the frequency analysis and time simulation, the performance of methods is highlighted.

The paper is organized as follows. Firstly, Section 2 introduces the system representation. To deal with partially decoupled disturbances, the observer design with the two approaches is defined in Section 3 where the proof of detectability condition is provided in the Appendix. Then, a numerical example with comparisons in Section 4 illustrates the performance of each solution. A general discussion on existence conditions of observer and frequencyshaping filter is mentioned in Section 5. Finally, Section 6 concludes the paper.

Notations: R n and R m×n respectively represent the ndimensional Euclidean space and the set of all m × n real matrices; X T is the transpose of the matrix X; 0 and I denote, respectively, the zero and the identity matrix with appropriate dimensions; X † is the Moore-Penrose inverse of X; the symbol ( * ) denotes the transposed block in the symmetric position; R(x) is the real part of the complex number x; eig(X) presents all eigenvalues of matrix X; and we denote He{A} = A + A T .

PROBLEM FORMULATION

Consider the following descriptor system with faulty actuator:

E ẋ = Ax + Bu + D w w + Bf y = Cx , (1)
where:

• x ∈ R nx is the state vector; y ∈ R ny is the measurement output vector; u ∈ R nu is the input vector

• w ∈ R nw = w T 1 w T 2 w T 3 T is the disturbance vector.
In which,

• w 1 ∈ R nw1 is the UI satisfying decoupling condition in UI observer design. • w 2 ∈ R nw2 is the bounded UI with known bandwidth [f w2 , f w2 ] and does not satisfy decoupling condition in UI observer design. • w 3 ∈ R nw3 is the non-decoupled disturbance with unknown bandwidth. • f ∈ R nu is the actuator fault vector to be estimated, which can be presented as a polynomial to address a wide range of faults, such as abrupt faults ( ḟ = 0) and incipient faults ( f = 0) (see [START_REF] Ding | Model-based fault diagnosis techniques: design schemes, algorithms, and tools[END_REF]), or even the degradation (see [START_REF] Do | Robust H 2 observer design for actuator degradation: Application to suspension system[END_REF]).

f (t) = α 0 + α 1 t + . . . + α n-1 t n-1 + α n t n , (2)
where the (n+1) th derivative of f is null (i.e, f (n+1) = 0) and α i (i = 0, 1, . . . , n) is unknown constant vector.

• Matrices E, A, B, C, D w = [D w1 D w2 D w3 ] are con-
stant matrices with appropriate dimension.

By considering the derivatives of f as extended states, an augmented system is obtained:

E a ẋa = A a x a + B a u + D wa w y = C a x a , (3)
where

x a = x T f T f (1)T . . . f (n-1)T f (n)T T ∈ R nx a , n xa = n x + (n + 1)n u , C a = [C 0 0 . . . 0 0], E a =        E 0 0 . . . 0 0 0 I 0 . . . 0 0 0 0 I . . . 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 . . . I 0 0 0 0 . . . 0 I        , A a =        A B 0 . . . 0 0 0 0 I . . . 0 0 0 0 0 . . . 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 . . . 0 I 0 0 0 0 . . . 0        , B a =        B 0 0 . . . 0 0       
, and

D wa =        D w 0 0 . . . 0 0        = [D w1a D w2a D w3a ].
The observer design to estimate fault f under the existence of UI w is presented in the next section.

OBSERVER DESIGN

In this section, the UI w 1 of w is decoupled from estimation process by the conventional UI observer-based design, whereas the impact of the UI w 23 = w T 2 w T

3

T on fault estimation are studied in the two following approaches:

Approach 1: Global H ∞ attenuation for both nondecoupled UIs w 2 and w 3 , similar to the approach suggested by [START_REF] Gao | Unknown input observer-based robust fault estimation for systems corrupted by partially decoupled disturbances[END_REF]. In which, the existence conditions are derived from the initial descriptor system and observer parameters satisfy the following objectives.

• For w 23 = 0, the estimation error is asymptotically stable. • For w 23 = 0, attenuation of exogenous input w 23 on the fault estimation error e f is achieved by minimizing γ 23 such that:

e f 2 w 23 2 ≤ γ 23 , (4)
where e f = f -f is fault estimation error and f is the estimated fault.

Approach 2: Combination of the frequency-shaping filter and H ∞ attenuation, where UIs w 2 and w 3 are handled separately:

• UI w 2 is attenuated by a frequency-shaping filter corresponding to its known bandwidth. • UI w 3 is attenuated by H ∞ optimization as mentioned in Approach 1. For w 3 = 0, attenuation of exogenous input w 3 on the fault estimation error e f is achieved by minimizing γ 3 such that:

e f 2 w 3 2 ≤ γ 3 . ( 5
)
Remark 1: Comparing to Approach 1, this method relaxes the number of elements in non-decoupled UI vector used for H ∞ optimization.

The details on observer design for each approach are presented in subsections 3.1 and 3.2, respectively.

For the existence of observer, the following assumptions are considered:

(A.1) rank E D w1 C 0 = n x + n w1 . (6) (A.2) rank (sE -A) -B D w1 0 sI 0 C 0 0 = n x + n w1 , ∀R(s) ≥ 0. ( 7
)
It is noted that the condition (A.1) corresponds to not only the impulse-free condition of the singular system but also the UI-decoupling condition. Meanwhile, the assumption (A.2) is the condition for R-detectability.

Approach 1: Global H ∞ attenuation

The UI observer has the structure (as illustrated in Fig. 1):

   ż = F z + Gu + Ly xa = z + N y f = C af xa , (8)
where xa = xT f T f (1)T . . . f (n-1)T f (n)T T is the estimated state of x a in (3); f is the estimated fault; and

C af = 0 nu×nx I nu 0 nu,nx a -nx-nu . f u y f ż = N z + Gu + Ly xa = z + T 2 y Plant w = w T 1 w T 2 w T 3 T x

Fig. 1. General scheme of UI observer

Choose e = x a -xa as the estimation error and suppose that there exists T such that T E a + N C a = I, (9) we obtain:

e = x a -z -N y = T E a x a -z. (10)
Then, its dynamics is presented as:

ė = T E a ẋa -ż (11) = F e + (T A a -F T E a -LC a )x a + (T B a -G)u + T D w1a w 1 + T D w23a w 23 , (12)
where

w 23 = w T 2 w T 3 T ∈ R nw23 , n w23 = n w2 + n w3 and D w23a = [D w2a D w3a ].
In order for e to be stabilized and decoupled from the UI w 1 , the following conditions have to be satisfied:

F is Hurwitz, i.e. R(eig(F )) < 0, (13) T A a -F T E a -LC a = 0, (14) G = T B a , ( 15
) T D w1a = 0. ( 16
)
From ( 14), by replacing T E a = I -N C a and then choosing K = L -F N , it follows that:

T A a -KC a -F = 0 (17)
By combining the three conditions which are T E a +N C a = I, ( 16) and ( 17), we obtain:

[T N K F ] Θ = Ω, (18)
where

Ω = I nx a 0 nx a ×(nx a +nw1) , Θ =    E a A a D w1a C a 0 0 0 -C a 0 0 -I nx a 0   
The solution of (18) exists if and only if rank Θ Ω = rank(Θ), which is equivalent that Θ is a full-column rank matrix (see [START_REF] Koenig | Unknown input proportional multipleintegral observer design for linear descriptor systems: application to state and fault estimation[END_REF]), i.e. rank(Θ) = 2n xa + n w1

⇔ rank

E a D w1a C a 0 = n xa + n w1 . ( 19
)
Replacing the definition of E a , C a , and D w1a with matrices of original system (1), a condition which is equivalent to assumption (A.1) is obtained.

Under (A.1), the generalized solution of ( 18) is given as:

[T N K F ] = ΩΘ † -ZΘ ⊥ , (20)
where Θ ⊥ = (I -ΘΘ † ) and Z is an arbitrary matrix.

From ( 12), ( 17) and ( 20), the influence of w 23 on estimation error e f = f -f is expressed as:

ė = (T A a -KC a )e + T D w23a w 23 e f = C af e (21)
In other words,

ė = (ΩΘ † φ 1 -ZΘ ⊥ φ 1 )e + (ΩΘ † φ 2 -ZΘ ⊥ φ 2 )w 23 e f = C af e (22)
where

φ 1 =    A a 0 ny×nx a -C a 0 nx a ×nx a    and φ 2 = D w23a 0 (2ny+nx a )×nw23
.

The above error dynamics can be stabilized thanks to the detectability of the pair (ΩΘ † φ 1 , Θ ⊥ φ 1 ) given as the condition below:

rank sI -ΩΘ † φ 1 Θ ⊥ φ 1 = n xa ∀ R(s) ≥ 0, (23)
which is equivalent to condition (A.2) (proof is easily derived from Appendix of [START_REF] Koenig | Unknown input observers for switched nonlinear discrete time descriptor systems[END_REF]). Then, the gain Z satisfying the objective (4) can be found from the following theorem.

Theorem 1. Under (A.1) and (A.2), if there exist a symmetric positive-definite matrix P and a matrix Q which minimize γ 23 in (4) and satisfy that:

  Γ P ΩΘ † φ 2 + QΘ ⊥ φ 2 C T af ( * ) -γ 2 23 I 0 ( * ) ( * ) -I   < 0, ( 24
) with Γ = He{P ΩΘ † φ 1 + QΘ ⊥ φ 1 }, (25)
φ 1 = A T a 0 T ny×nx a -C T a 0 T nx a ×nx a T , (26)
φ 2 = D T w23a 0 T (2ny+nx a )×nw23 T , (27)
the estimation error in ( 22) satisfies the objectives in Approach 1 with the gain Z = -QP -1 .

Remark 2: If E = I and the columns of Z corresponding to parametric matrices T and N in (20) are null, the result in Gao et al. ( 2016) is re-obtained. According to [START_REF] Gao | Unknown input observer-based robust fault estimation for systems corrupted by partially decoupled disturbances[END_REF], the parameters T and N are only the basic results of generalized solution, i.e. without the tuning of the arbitrary matrix, thus limiting the freedom of the observer design comparing to that in Theorem 1.

Proof. The sufficient condition for the stability of ( 22) and attenuation objective (4) is that:

V + e T f e f -γ 2 23 w T 23 w 23 < 0. ( 28
)
By choosing the Lyapunov function V = e T P e and Q = -P Z, it follows that:

e T w T 23 Γ + C T af C af P ΩΘ † φ 2 + QΘ ⊥ φ 2 ( * ) -γ 2 23 I e w 23 < 0. ( 29
)
The above inequality holds ∀ e T w T 23

T = 0 if: Γ + C T af C af P ΩΘ † φ 2 + QΘ ⊥ φ 2 ( * ) -γ 2 23 I < 0. ( 30
)
Applying the Schur complement to above LMI, the condition ( 24) is obtained, which completes the proof.

The parameters [T N K F ] are calculated by replacing values of Z in (20), then L = K + F N and G = T B a .

Approach 2: Combination of the frequency-shaping filter and H ∞ attenuation

The aim of UI observer is to decouple the estimation error from disturbances, which is equivalent to a transmission zeros from UIs to the measurement (see [START_REF] Chen | Design of unknown input observers and robust fault detection filters[END_REF]). Based on this idea, to generate similarly the behavior of UI w 2 as that of a possibly-decoupled disturbance, a frequency-shaping filter is implemented to the output y, which characterizes the disturbance attenuation in the known bandwidth [f w2 , f w2 ] of UI w 2 . The output ȳ of the stable filter Q can be now considered as a new measurement not perturbed by UI w 2 . The design process is summarized in Fig.

2

Q ẋQ = A Q x Q + B Q y ȳ = C Q x Q + D Q y f u y ȳ f UI Observer Plant w = w T 1 w T 2 w T 3 T [f w2 , f w2 ]
x Fig. 2. Frequency-shaping filter implementation for UI observer

The stable filter Q can be expressed as:

Q : ẋQ = A Q x Q + B Q y ȳ = C Q x Q + D y (31)
where A Q is Hurwitz.

It yields an augmented system: Ēa ẋa = Āa xa + Ba u + Dwa w

ȳ = Ca xa (32) In which, xa = x a x Q ∈ R nx a , Ēa = E a 0 0 I , Āa = A a 0 B Q C a A Q , Ba = B a 0 , Dwa = D wa 0 , Ca = [D Q C a C Q ].
The observer design for the above system (32) has the same structure as that of conventional UI observer to decouple w 1 while integrating implicitly the frequencyshaping effect for non-decoupled UI w 2 :

   ż = F z + Ḡu + Lȳ xa = z + N ȳ f = Caf xa (33)
where xa is the estimated state of xa and Caf = 0 nu×nx I nu 0 nu,nx a -nx-nu .

As Q is a stable filter, the conditions for observer existence (A.1) and (A.2) are also those for augmented system (32). Similarly to Approach 1, the following results are obtained:

T Theorem 2. Under (A.1) and (A.2), if there exist a symmetric positive-definite matrix P and a matrix Q which minimize γ 3 in (5) and satisfy that:

N K F = Ω Θ † -Z Θ⊥ , (34)
L = K + F N , (35) Ḡ = T Ba , (36)
  Γ P Ω Θ † φ2 + Q Θ⊥ φ2 CT af ( * ) -γ 2 3 I 0 ( * ) ( * ) -I   < 0, ( 38
) with Γ = He{ P Ω Θ † φ1 + Q Θ⊥ φ1 }, ( 39
) φ1 = ĀT a 0 T ny×nx a -CT a 0 T nx a ×nx a T , ( 40
) φ2 = DT w3a 0 T (2ny+nx a )×nw3 T , (41)
the estimation error in (37) satisfies the objectives in Approach 2 with the gains Z = -Q P -1 .

This theorem is similar to Theorem 1, so the proof is omitted.

In the next section, the comparison between the two above approaches is conducted to illustrate the performance of each method.

NUMERICAL EXAMPLE

Model Example

The following example is modified from the descriptor system in [START_REF] Darouach | Reduced-order observer design for descriptor systems with unknown inputs[END_REF]:

• Distribution Matrices: E =    1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0   , A =    -1 1 0 0 -1 0 0 1 0 -1 -1 0 0 0 0 1   , B =    0 0 1 1   , D w1 =    -0.5 0 0 0   , D w2 =    0 0.4 0 0   , D w3 =    0 0.2 0.1 0   , and C = 1 0 0 0 0 0 -1 1 .
• Actuator fault: is supposed to be a 3 rd order polynomial (see Fig. 3), so n = 3 is chosen for observer design.

f = n=3 i=1 (-0.21) i i! (t) i . (42)
Remark 3: The order n of the estimated fault f chosen for observer synthesis must be greater or equal to the real order of the existing fault f . Consequently, the filter Q is designed as a stable 8 th order Butterworth-bandstop (see Fig. 4).

Frequency Analysis

In this part, the frequency behavior of the two approaches in Section 3 are compared. Solving the optimization problem in Theorems 1 and 2 by using Yalmip (see [START_REF] Lofberg | YALMIP: A toolbox for modeling and optimization in MATLAB[END_REF]) and Sedumi solver (see [START_REF] Sturm | Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones[END_REF]), the parametric matrices for observer designs in Approach 1 and Approach 2 are synthesized. The attenuation level for disturbances in both approaches are presented and compared in the Table 1:

Table 1. Disturbance attenuation Comparison Approach 1 Approach 2 w 2 γ 23 = -6.990 (dB) Characteristics of filter Q w 3 γ 3 = -13.978 (dB)
In Fig. 5, a sudden drop in frequency domain [10,30] (Hz) expresses the result of filter Q's implementation, as expected from the usage from the frequency-shaping approach.

According to Table 1 andFigs. 5 and6, by relaxing the size of disturbance vector in H ∞ synthesis, Approach 2 gives better attenuation of disturbance influence on estimation error.

Test Conditions

• Simulation duration: 25 seconds.

• Disturbances:

• w 1 = 5sin(2πf w1 ) with f w1 = 10 (Hz);

• w 2 = 10sin(2πf w2 ) with f w2 = 17.5 (Hz) to illustrates clearly the difference in both approaches; • w 3 = 15sin(2πf w3 ) with f w3 = 35 (Hz).

• Control input: is chosen as a sinusoidal signal: u = 5sin(2π).

(43) • System output: is considered not to be perturbed by noise in this study, as described in (1). • Initial condition: x (0) = [0.001 0 0.0020 0]

T , x Q(0) = 0, and x(0) = 0. As observed, all fault estimations are converging to fault reference after about 10 seconds, i.e. the estimation error converges towards 0. However, Approach 1, i.e. global H ∞ attenuation method, is more likely to be affected by disturbances w 2 and w 3 comparing to Approach 2 due to its poor frequency behavior as discussed in subsection 4.2. To evaluate the accuracy of estimation, the root-meansquare value (RMS) of estimation errors is calculated in Table 2. From this comparison, when there is only w 1 , the decoupling between the UI w 1 and fault estimation error e f works correctly as designed in both cases; whereas the differences start to appear in the solutions coping with non-decoupled disturbances. It also proves the better performance of Approach 2 by treating separately UIs w 2 and w 3 , thus relaxing the amount of non-decoupled UIs implemented in H ∞ synthesis.

GENERAL DISCUSSION

Observer conditions

As noted in (A.1) and (A.2), these assumptions concern only the parameter of possibly-decoupled UI w 1 , i.e. D w1 , thereby being less restrictive than those of [START_REF] Darouach | Reduced-order observer design for descriptor systems with unknown inputs[END_REF]; [START_REF] Koenig | Unknown input proportional multipleintegral observer design for linear descriptor systems: application to state and fault estimation[END_REF] for all UIs in w, i.e. D w .

Frequency-shaping filter implementation

In Approach 1, the H ∞ performance can also be optimized for a specific bandwidth [f w2 , f w2 ] of w 2 by generating a fictive disturbance w2 through a weighting function F w , which is strictly stable and causal (see [START_REF] Koenig | Filtering and fault estimation of descriptor switched systems[END_REF]).

The weighting function F w can be displayed as:

F w : ẋw = A w x w + B w w2 w 2 = C w x w + D w w2 . ( 44
)
The process is summarized in Fig. 11:

As a result, the system can be rewritten as:

ẋF = A F x F + B F u + D F w y = C F x F , (45)
where

x F = x a x w , A F = A a D w2a C w 0 A w , B F = B a 0 , D F = D w1a D w2a D w D w3a 0 B w 0 , and
C F = [C a 0].
Similarly to Approach 1, the objective of observer design for (45) is derived from (4) and is displayed as: Consequently, the influence of the non-decoupled w 2 on fault estimation can be shaped by the choice of inversed filter F -1 w . Comparing to Approach 2, both inversed function F -1 w and frequency-shaping filter Q have the same functionality; however, filter Q only needs to be stable, instead of being simultaneously stable and causal as F w . Hence, this difference highlights the advantage of the frequencyshaping filter in Approach 2. For example, to achieve good attenuation in a narrow bandwidth as that in the numerical example of Section 4, a high-order weighting function F -1 w or frequency-shaping filter Q is required under the form of Butterworth bandstop. In this case, a stable filter Q can be easily designed, whereas the inverse of F -1 w , i.e. F w is not ensured to be stable.

CONCLUSION

In this study, two different approaches which are based on the conventional UI observer to deal with non-decoupled disturbances in the descriptor system have been introduced. The first of which considers the global H ∞ synthesis for UI impact attenuation, while the other handles the UIs based on the knowledge of their bandwidth. The comparison in fault estimation between the two solutions has highlighted the performance of the proposed separation strategy, where each non-decoupled disturbance is analyzed independently.

For future work, as the problem of partial disturbance in descriptor linear parameter-varying (D-LPV) system has not been broadly studied yet, an extension result of the two above approaches, as well as the impact of frequency-shaping filter on closed-loop performance, can be an interesting topic.

  = I nx a 0 0 .Due to the implementation of the output filter, the dynamics of observer can be reduced as follows: ė

•

  Fig. 3. Actuator fault f

Fig. 9 .

 9 Fig. 5. Sensitivity |e f /w 2 |

Fig. 10 .

 10 Fig. 10. Fault error under influence of all UIs (w 1 , w 2 , w 3 )

Table 2 .

 2 RMS of estimation error

	Scenarios	Approach 1	Approach 2
	w 1	3.881e-4	2.070e-4
	[w 1 , w 2 ]	7.008e-4	2.045e-4
	[w 1 , w 3 ]	18.656e-4	5.952e-4
	[w 1 , w 2 , w 3 ]	19.557e-4	5.941e-4

  Fig. 11. Weighting function implementation in H ∞ synthesis.

	[f w2 , f w2 ]
	w 2	F w	w 2	w 1	w 3	f
		u			Plant		y	UI Observer	x
										f
							e f 2 w23 2	≤ γ 23 ,	(46)
	where w23 = wT 2 w T 3		T .
	In other words,
	      	e f 2 w2 2 e f 2 w 3 2	≤ γ 23 ≤ γ 23	⇔	      		e f 2 w 2 2 w 3 2 e f 2	≤ γ 23 F w ≤ γ 23	-1	(47)

This work is supported by the ITEA3 European Project through EMPHYSIS under Grant 15016.


Kamrul Islam
email: kamrul-islam.shahin@univ-lorraine.fr

Christophe Simon
email: christophe.simon@univ-lorraine.fr

Philippe Weber
email: philippe.weber@univ-lorraine.fr

Bootstrap Confidence Interval on IOHMM Parameters for System Health Diagnostic Under Multiple Operating Conditions

Keywords: System health, PHM, Input Output Hidden Markov Model, Condition based diagnostic, Degradation design, Condition monitoring

The operating conditions have an important impact on system degradation. This paper uses the Input-Output Hidden Markov Model to represent the system degradation having multiple operating conditions. In this paper the bootstrap method is applied to estimate the model parameters and to diagnostic the system health. Parameters of the model are computed with 95% confidence intervals. The uncertainty about multiple data sequences and degradation speed is handled according to the operating conditions. A numerical application is given to explain the methodologies used to estimate the model parameters and diagnostic the system health.

INTRODUCTION

Diagnostics play a very important role in prognostic and health managing (PHM) of systems. Health diagnosis ensures the level of system degradation. Based on the estimated diagnosis, remaining useful life (RUL) of the system is evaluated which is an essential part of the next generation of maintenance. Maintenance is very crucial for trouble-free production, as the system must not be shut down during the operation. It is necessary to predict the health states of the system during the operation leading to the system for repairing or replacing before the unplanned shutdown. In this case, it is very important that the system is correctly diagnosed, otherwise, effective RUL prediction is not possible. Therefore, this will be worth the wrong estimate, and the system will be failed before maintenance. Thus, the production speed will decrease, and the production cost will increase. To solve this problem, many scientific works are proposed on diagnostic applications based on degradation identification. Degradation is not easy to assess because it is a hidden and unknown process. It is impossible to know the evolution of degradation until the system is stopped and observed directly inside, which is completely contrary to our goal. However, as the system ages, it gets damaged. If the system operates at higher pressures, it decomposes quickly, otherwise, the process is slow for the lower pressures. This means that the lifetime of the system very much depends on the operating conditions of the system. This document takes into account the impact of operating conditions on the degradation and offers a model for identifying the system's degradation under several operating conditions and assessing the state of the system (diagnostics). There are many models available in PHM applications that can be classified into three approaches: model-based, data-based, and hybrid. The purpose of this paper requires a parametric and non-supervised model, that is why the data-driven model is ideal for the solution. The aim is to have a stochastic system in which states are unknown and their evolution is random. In this case, the Hidden Markov Model (HMM) can be an excellent choice. HMM was first introduced by Baum in the early 1970s [START_REF] Baum | Statistical inference for probabilistic functions of finite state Markov chains[END_REF]], and [START_REF] Rabiner | A tutorial on hidden Markov models and selected applications in speech recognition[END_REF]] used it for the first time in an application for recognition of speech. It was later used in PHM challenges. For example [START_REF] Kumar | An HMM and polynomial regression-based approach for remaining useful life and health state estimation of cutting tools[END_REF][START_REF] Tobon-Mejia | Hidden Markov models for failure diagnostic and prognostic[END_REF][START_REF] Baruah | HMMs for diagnostics and prognostics in machining processes[END_REF] proposed HMM-based applications for diagnostics, but without regard to operating conditions. Some other researchers have used the "Hidden Semi-Markov models" [START_REF] Dong | A segmental hidden semi-Markov model (HSMM)-based diagnostics and prognostics framework and methodology[END_REF]], "Mixture of Gaussian Hidden Markov Models" [Tobon 2012], "Hidden Markovian Hierarchical Models" [START_REF] Camci | Hierarchical HMMs for autonomous diagnostics and prognostics[END_REF]] in various diagnostic applications. They have tried different versions of HMMs to produce better results, but no one integrates the operating conditions as inputs. These models cannot be used to consider the operating condition because HMMs mentioned above do not allow any input. However, the Input-Output Hidden Markov Model (IOHMM) which is another advanced version of HMM, an interesting model in which operating conditions can be considered as input conditions. Similar work is being done in [START_REF] Le | Multi-branch hidden semi-markov modeling for RUL prognosis[END_REF]], where a Multi-Branch HSMM (MBHSMM) is proposed. This is a motivating work, but there are also some limitations. The author divided the dataset into different parts for different models that correspond to the operating conditions. They fixed one operating condition while applying the matching model. In reality, it is not the same because operating conditions may change at any time during the operation. On the other hand, IOHMM trains several models simultaneously without separating the data sequence and allows us to switch the models at given input sequences. Thus, IOHMM assesses a more realistic degradation of the system with the effect of operating conditions. IOHMM was introduced in [START_REF] Bengio | An input output HMM architecture[END_REF] for the first time. Then, it is used in diagnostic and prognostic applications [START_REF] Shahin | Inputoutput hidden Markov model for diagnosis of complex system[END_REF][START_REF] Shahin | Estimating IOHMM parameters to compute remaining useful life of system[END_REF]]. First, a training approach to the model parameters has been proposed. Estimating model parameters from data through the algorithm proposed in [START_REF] Shahin | Estimating IOHMM parameters to compute remaining useful life of system[END_REF]] does not give the confidence intervals in results. To solve this problem the bootstrap-IOHMM method is applied to the model and estimated the model parameters with a 95% confidence interval. A statistical exposition of the bootstrap is given by [START_REF] Efron | A leisurely look at the bootstrap, the jackknife, and cross-validation[END_REF]. The bootstrap Hidden Markov Model (HMM) is used in several applications, such as clustering time series [START_REF] Oates | Using dynamic time warping to bootstrap HMM-based clustering of time series[END_REF]] and quality of synthetic speech [START_REF] Kim | Automatic segmentation combining an HMM-based approach and spectral boundary correction[END_REF]], etc. Inspired by this work we designed the bootstrap-IOHMM method to achieve the goal of this paper. The basic idea of the bootstrap involves the variability in an unknown distribution from which the data are drawn by resampling with replacement from the dataset [START_REF] Felsenstein | Confidence limits on phylogenies: an approach using the bootstrap[END_REF]]. It is most useful when a set of estimates has a distribution that approximates the distribution of the actual estimate. The bootstrap method is used to estimate model parameters with the confidence interval, average sample, standard error, lower limit, and upper limit. IOHMM uses three adapted algorithms: the Baum Welch algorithm which is a class of EM (Expectation-Maximization) algorithm based on the forward-backward algorithm in the training and adapted Viterbi algorithm in the diagnostic application. The adaptation of the algorithms for IOHMM is described in [START_REF] Shahin | Estimating IOHMM parameters to compute remaining useful life of system[END_REF]].

This paper is organized as follows: Section 2 defines the basis of model structure, training, and diagnostic methodologies. Then, a brief discussion of bootstrap and confidence intervals are given in section 3. After that, section 4 shows a numerical application and corresponding results. Finally, a conclusion and perspectives are developed in section 5.

PROPOSED MODEL

IOHMM is a class of HMM. It is a stochastic model where the state probability depends on its previous state, observation, and the input condition. This model can be used to represent systems with multiple operating conditions and multiple outputs. An IOHMM is presented (Fig1) as in Dynamic Bayesian Network (DBN) [START_REF] Salem | Dynamic Bayesian Networks in system reliability analysis[END_REF]]. Multiple transition matrices are represented as ,

where is the operating conditions number.

Emission probability

The set of possible emitted symbols of the output variable are assumed as . denotes the state emission probability matrix, where is the emission probabilities of state . The summation of for each state is 1.

Multiple output-observations are represented as ,

where is the number of outputs of the system.

Input Condition

In this model, the input sequence is presented as , and the transition probabilities are computed as follows . All the sequences are not length sensitive and sequences could come with any length. If the initial state probability distribution is and the model denoted by , then the triplet completely defines the model structure.

Method: Model Training

The Baum Welch and the Forward-Backward algorithms are commonly used in HMM which does not allow us to consider the input condition nor the multiple output case. That is why the adapted algorithms are used which able to consider multiple inputs and outputs during the training session. The detail adaptation is discussed in [START_REF] Shahin | Estimating IOHMM parameters to compute remaining useful life of system[END_REF]]. The forward-backward algorithm is given in (1) and (2).

• Forward algorithm: Basis: Recursion:

(1) here , and presents multiple sequences where is the number of outputs of the system.

• Backward algorithm:

Basis:

is the length of the sequence.

Recursion:

(2) here .

The Baum-Welch algorithm takes and , applies the Baum Welch algorithm to update the parameter in an iterative process.

• The Baum Welch algorithm:

The probability of being in state at time given the observed sequences and the parameters of is given below by ( 3):

(3)

The probability of being in state and at time and given the observed sequences of the input operating conditions and the parameters of is given by ( 4):

(4)

Parameters update:

o Initial state probability:

, where , where N is the number of hidden states.

(5) o Transition probabilities:

(6) here o Emission probabilities:

(7) here Now, these steps are repeated iteratively until the changes between two consecutive results fall into a given tolerance and fixed the parameters of the model as , which is used in diagnostic of the system.

Method: Diagnostic

The Viterbi algorithm is used to diagnostic the system using the trained model . It is an interesting algorithm that estimates the maximum path using (8). This algorithm is also adapted to IOHMM [START_REF] Shahin | Estimating IOHMM parameters to compute remaining useful life of system[END_REF]]. It computes the maximum likelihood state sequence considering operating conditions as input.

The Viterbi algorithm: Basis: Recursion:

(8) here .

Based on this set of algorithms the next section explains the confidence interval and the bootstrap method.

BOOTSTRAP-IOHMM

The bootstrap-IOHMM method is a sampling technique used to estimate IOHMM parameters statistic by sampling a dataset with replacement. It can be used to estimate measures of accuracy, such as confidence intervals, the sample mean, standard deviation, variance, etc. Resampling with replacement selects some random dataset from the original sample and put those back into the sample again for another selection. Resampling size should be equal to the sampling size which may have some repeated dataset. This technique maintains data structure but reshuffles values, extrapolating to the data population. This repeated process uses the new sample to generate the sampling distribution of the mean. Some important definitions are needed to understand for understanding the bootstrapping.

1. Synchronize the system with IOHMM and training the model using the available data sets.

a. Use (1) to ( 7) for parameter estimation.

b. Provide a confidence interval for each of the estimate parameters of the model using the bootstrap-IOHMM method (9).

2. Use the trained up IOHMM to estimate the system diagnostic according to a given data set.

a. Use the Viterbi algorithm to compute the system health condition at any time given a new data set.

b. Provide a comparison between the model trained with bootstrap method and without bootstrap method.

The goal is to show how the bootstrap method is useful when a small data-amount can be used to estimate parameters of a big data population.

Data Generating

To validate the procedure, we have generated the data sequences using a given model structure. Later, this (original) model is compared with the estimated model in the result section.

Model architecture:

• Data unit is discrete: continuous signal data could be converted into discrete unit.

• Model type is left-right: a system cannot comeback from the bad health to good health.

• Hidden states are three (could be more).

• Observation symbols are three for each output (could be more).

• Transition matrices are two according to two input modes. The matrices have zero on the lower parameters of the diagonal because of the left-right property. Switching model is illustrated in Fig. 2.

Fig. 2. Model switching by input

• Emission matrices: two for two outputs

• Initial state distribution: (assumed as in good health)

A set of 1000 data is generated using a simulator following this given model architecture. This data sequences are used in IOHMM training to estimate the parameters with 95% confidence intervals. The bootstrap approach is applied where the 30 data sets are randomly selected with data replacement for multiple times. The iteration was about 1000 times.

Results

Training results

Fig. 3 shows the confidence interval for a total of 36 parameters of the transition and emission matrices. The Xaxis of each rectangle represents the probabilities and the Yaxis represents the boot execution number.

Fig. 3. Distribution of matrices parameters

Both the transition matrices are having some zeros on corresponding parameters. These parameters did not get any transition probability during the training following the nature of the system. The left-right model is used in data simulation as mentioned earlier. This is the reason the transition matrices have zeros on (2,1), (3,1), (3,2) position (highlight in Fig. 4).

The green circle on the position (3,3) presents the absorbent state with a 100% probability. Besides these four, all the other parameters are estimated with a 95% confidence interval (see Table 1). is trained with about 20% data while 80% data are used to train matrix Now, if the matrices organized with the mean value then we find the estimated parameters of IOHMM as following:

• Estimated transition parameters:

is the low stressed (e.g. low speed) model transitions where the mean transition probability from the first state to the last state is .

is the high stressed (e.g. high speed) model transitions where the mean transition probability from the first state to the last state is . • Estimated emission parameters: presents emission probabilities for the first output sequences (e.g. temperature) and is for second output (e.g. vibration).

• Initial state distribution: (estimated as good health) Fig. 5 presents the distance between the estimated parameters and the original parameters used in data simulation. The parameters estimated twice: the first one is with the bootstrap-IOHMM method where the model uses the data with replacement technique from the simulated for training and the second one is the classical method without bootstrap which uses all the data samples. The CI mean is the estimated parameters using the bootstrap-IOHMM method. There is another circle is inside each box which represents the second estimated parameters (without bootstrap). This time the IOHMM trained by all the simulated data (1000 sequences) while the bootstrap-IOHMM uses a selective approach. However, both the estimated parameters are very close to the original parameters. Therefore, we can say that the bootstrap method-IOHMM is very crucial and effective while a small data amount is available to estimate the parameters of a big data population.

The next section diagnostic the system health by using both the estimated parameters and explains the difference between them.

Diagnostic results

A new data set is given to the model to estimate the health state of the system. The diagnostic method uses the Viterbi algorithm and calculates the state distribution from the starting point to the breakdown point of the system. The method shows the estimated time about the system degradation stays in each state (Fig. 6). The first two parts of the figure are the estimated diagnostic using the models with and without bootstrap given two data sequences (in third part). Two diagnostic results show a similar transition from one state to another state, but distribution is not the same. Despite the fact, their distributions are not same they still give the same max path shown in fourth part of the figure. Degradation level of the system health can be found in this max path given by the Viterbi algorithm, which fulfils the objective of this paper. This method could be very handy while the data sequence from health degradation is not easy to get, because degradation is a slow process that needs a long time to produce the data. So, the bootstrap-IOHMM can be used to estimate an effective diagnostic of system health.

CONCLUSIONS

This paper proposed a bootstrap-IOHMM method considering multiple operating conditions. 95% confidence intervals are given for each of the parameters of the IOHMM model. The estimated parameters used in the diagnostic application and show the results with and without applying the bootstrap method. The comparison between the results is explained to show the importance of the proposed method. This contribution could be used in the PHM domain to estimate different RUL considering the operating condition and control the system operation.

Fig. 1 .

 1 Fig. 1. DBN representation of 3-States IOHMM.

2. 1

 1 Model Structure2.1.1 Transition ProbabilityIOHMM evolves in a sequence of the variable which holds one of the hidden states at each time instant is discrete time)

Fig. 4 .

 4 Fig. 4. Parameter distribution for the first transition matrix.

Fig. 5 .

 5 Fig. 5. Parameter distance between learned and original model parameters The figure presents transition and emission parameter distances where only the diagonal probabilities except the absorbent states are compared. The blue box represents the CI bounds, the red line inside the box represents the CI mean and the star symbol represents the original parameters.The CI mean is the estimated parameters using the bootstrap-IOHMM method. There is another circle is inside each box which represents the second estimated parameters (without bootstrap). This time the IOHMM trained by all the simulated data (1000 sequences) while the bootstrap-IOHMM uses a selective approach. However, both the estimated parameters are very close to the original parameters. Therefore, we can say that the bootstrap method-IOHMM is very crucial and effective while a small data amount is available to estimate the parameters of a big data population.

Fig. 6 .

 6 Fig. 6. Diagnostic over the time from start to end.

Table 1

 1

shows 31 parameters of transition matrices, emission matrices, and the initial state distributions. Each row represents different information (lower bound, upper bound, mean, standard error) about a parameter. Parameters having zero value are ignored in the table.

Table 1 . Bootstrap parameters

 1

	Para-	Lower	Higher	Mean	Standard
	meter	bound	bound	value	Error
			Transition Matrix
		0.9783	0.9791	0.9787	1.96
		0.0209	0.0217	0.0213	1.96
		0.9508	0.9523	0.9515	3.91
		0.0477	0.0492	0.0485	3.91
		1	1	1	0
			Transition Matrix
		0.8428	0.8477	0.8453	0.0012
		0.1523	0.1572	0.1547	0.0012
		0.7861	0.7916	0.7889	0.0014
		0.2084	0.2139	0.2111	0.0014
		1	1	1	0
			Emission Matrix
		0.8970	0.8984	0.8977	3.70
		0.0506	0.0517	0.0512	2.82
		0.0506	0.0517	0.0511	2.80
		0.0524	0.0591	0.0557	17
		0.8929	0.8999	0.8964	18
		0.0470	0.0487	0.0479	4.32
		0.0498	0.0501	0.0500	0.894
		0.0503	0.0507	0.0505	1.07
		0.8993	0.8998	0.8995	1.29

Confidence interval (CI): Confidence interval estimated from observed statistical data, which may contain an unknown population parameter. The CI communicates the accuracy of a probabilistic estimate. It expresses a range in which it is fairly certain that the population parameter is present. The range-width depends on the variation within the population of interest and the sample size. [START_REF] Efron | Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy[END_REF]] Population variation: If all values in a large data population are almost the same, then the sample also has a small variation. It gives a small confidence interval. On the other hand, more varied data will lead to more varied samples, which makes less sure that the sample average is close to the population mean. That means the CI is large in this case. The greater variation of the data leads to a wider CI.

Sample size:

The sample size also affects the width of a confidence interval. Small samples differ more from each other and have less information. There is more variation due to a sampling error. The CI may be larger. On the other hand, larger samples will be more similar. The effect of the sampling error is less, and the information is more. The confidence interval may be smaller in this case. [START_REF] Efron | Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy[END_REF] Calculating confidence intervals: The confidence interval calculation for a mean uses formula (9). ( 9) where is the sample mean, is the t-distribution which depends on the sample size and the chosen level of confidence, is the sample standard deviation and is the sample size.

Sample: A sample is a selection of observations from the population of interest. Different selection criterion is simple, random, convenient, systematic, clustered, layered, etc.

Sampling error: A sample is only a selection of objects from the population. It will never be a perfect representation of the population. Different samples of the same population will yield different results. This is called sampling error or sampling variation. There will always be a sampling error. [START_REF] Efron | Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy[END_REF] The sample means: Defined as the average of observations in the sample of the population. The sample mean is considered as the estimate of the population mean.

Sample standard deviation: It is the average distance of the sample data from the sample mean.

Useful:

• Bootstrapping is useful for modelling non-normal data

• It provides unknown statistic properties (e.g. PCA results)

• It has unknown statistic properties and shows the standard calculation such as a 95% confidence interval.

• Easy to measure result accuracy or compute standard error, sample mean, standard deviation, variance etc.

NUMERICAL APPLICATION

A system has represented in this application which has two outputs to be observed applying an operating condition with two different modes. The operating modes provide two different transition matrices. The degradation of the system assumed to have three hidden states (good, moderate, bad) for easy and simple computation. Each of the states emits two outputs with two probabilities which are represented by two emission matrices. There are three discrete symbols considered in the emitted observation sequences.

The proposed model is designed to use the sensor data as a unit. The system temperature and the system vibration are used as two observations, and the speed of the system is considered as an input where the speed has two modes (high and low). The goal is divided into two steps: