Fault Estimation for a Class of Markov Jump Piecewise-Affine Systems: Current Feedback Based Iterative Learning Approach

Yanzheng Zhu ,,, Nuo Xu , Fen Wu ,,,Xinkai Chen ,,, and Donghua Zhou ,,

Abstract—In this paper, the issues of stochastic stability analysis and fault estimation are investigated for a class of continuoustime Markov jump piecewise-affine (PWA) systems against actuator and sensor faults.Firstly, a novel mode-dependent PWA iterative learning observer with current feedback is designed to estimate the system states and faults, simultaneously, which contains both the previous iteration information and the current feedback mechanism.The auxiliary feedback channel optimizes the response speed of the observer, therefore the estimation error would converge to zero rapidly.Then, sufficient conditions for stochastic stability with guaranteed H∞ performance are demonstrated for the estimation error system, and the equivalence relations between the system information and the estimated information can be established via iterative accumulating representation.Finally, two illustrative examples containing a class of tunnel diode circuit systems are presented to fully demonstrate the effectiveness and superiority of the proposed iterative learning observer with current feedback.

I.INTRODUCTION

WITH the rapid development of industrialization and high-precision equipment, faults, as a hidden danger factor, exists in various systems of modern industry.Its main dangers include damage of system components caused by the sudden change of external environment and the system collapse caused by the short circuit inside the module.In order to identify faults and design corresponding modules to reduce risk, a variety of fault estimation strategies have been developed in different situations [1]-[9].For example, using the additive actuator, the time-varying actuator and sensor faults,[1] studies the issues of two types of fault observer design for a class of Markov jump systems; In [6], the issues of distributed fault estimation and fault-tolerant control are investigated for a class of interconnected systems, where both the system and observer can be formulated as two graphical games.As a reliable method to extract fault information, fault estimation is widely applied in various industrial systems, i.e.,high-speed trains [10], wind turbines [11], and wheeled mobile manipulators [12], etc.

It is worth noting that in existing fault estimation strategies,a majority of design methods are based on fault differentiability, as differentiability would ensure that the fault estimation errors can be updated promptly via the fault estimation law[13]-[15].However, such desired properties cannot be attained in complex industrial systems all the time.Inspired by iterative learning control strategies [16], [17], an iterative learning observer is developed to estimate non-differentiable faults.The iterative learning mechanism is required to determine the output information of the system and the result of the previous iteration.This process of repeatedly correcting previous results is operated ceaselessly, and eventually system faults can be estimated accurately in finite time.Therefore,with the iterative learning observer based fault estimation strategy, only a series of repetitive operations is invoked to obtain the desired estimation results.In addition, the fault estimation error can be updated by the iterative process without the involvement of differentiable faults, which is more appropriate in the case of non-differentiable faults.Thus, both the design and the application of iterative learning observer have received significant attention [18]-[20].In [21], the issue of fault estimation is investigated for a class of Markov jump systems for nondifferentiable actuator faults and sensor faults,where a reduced-order fault estimation observer and an iterative learning observer are designed for the augmented system,respectively.In practice, however, the number of iterations cannot be infinite.Determining how to improve iteration law to estimate faults rapidly with the minimum number of iterations is one of the main motivations of this paper.

A Markov jump nonlinear system, as a special kind of nonlinear system, contains both the stochastic switching state subject to specific Markov chain and nonlinear dynamic characteristics, which can be employed to model certain dynamic systems with structural or parameter mutations effectively.Consequently, theories related to Markov jump nonlinear systems have been studied extensively [22]-[25].Furthermore, as a powerful technique to approximate the dynamic characteristics of nonlinear systems, the piecewise-affine (PWA) strategy has been applied to various complex nonlinear systems[26]-[28].The approximation principle of PWA technology is to divide the state space of nonlinear systems into numerous PWA regions, where each state region is approximated by a corresponding affine function respectively, all of which constitute a complete approximation state space.Therefore, various theoretical achievements and application scenarios have been investigated and developed for Markov jump PWA system [29]-[31].For instance, in [29], based on both the modedependent and region-dependent fuzzy affine estimator, the problem of state estimation is studied for a class of discretetime Takagi-Sugeno fuzzy Markov jump affine systems; In[30], the asynchronous region issue between the plant state and the estimator state is investigated for a class of discretetime Markov jump PWA systems with a novel switching path algorithm.However, after reviewing a large number of the existing results, there are fewer reports about the issues of fault estimation for continuous-time Markov jump PWA system against actuator and sensor faults, not to mention using the iterative learning approach, which is another main motivation of this paper.

To sum things up, motivated by the above discussions, this paper develops stochastic stability analysis withH∞performance and fault estimation issues for a class of continuoustime Markov jump PWA system with sensor faults and nondifferentiable actuator faults.Firstly, the Markov jump PWA system is transformed to the corresponding augmented system with both the system state and the sensor fault.Secondly,a novel mode-dependent PWA iterative learning observer with current feedback is constructed to resolve the non-differentiable fault estimation problem.Thirdly, the issue of stochastic stability with guaranteedH∞performance for the estimation error system and the equivalence relations for the estimated information are established, respectively.The main contributions are summarized as follows:

1) The presented Markov jump PWA system contains both the Markov jump linear system and the ordinary PWA system as two special cases.The mode-dependent PWA iterative learning observer is developed to estimate both the augmented state and the non-differentiable actuator fault.With the increase of iteration numbers, the estimation error can be reduced gradually, and finally both the system state and fault information can be estimated accurately.

2) To ameliorate the iteration law to estimate faults rapidly with the minimum number of iterations, the current feedback mechanism is introduced into the iterative process, so that the designed iterative law includes both the results of the previous iteration based on the measurement output and the feedback information of the current iteration, where the effect of fault estimation errors on the current iteration can be reduced by the feedback mechanism effectively.

3) The mode-dependent and region-dependent Lyapunov function is constructed to adapt stochastic stability with guaranteedH∞performance.The investigated estimation error system illustrates variations of the switching model, the PWA regions, and the iteration process under the current feedback.Furthermore, the iterative relations between the plant information and the estimated information can be obtained.ThesymbolHe{ν}denotesthe relationHe{ν}=ν+νT.The

Notations:Rnstandsforthen-dimensionalEuclideanspace.symbol ∗ represents the transposed term of a symmetric matrix.l2[0,∞) denotes the space of square-integrable vector functions.|·| s tands for the Euclidean vector norm.col{·} represents the column vector.The symbolsTand -1 are the transpose and the inverse of a matrix, respectively.The symbol † denotes the Penrose-Moore inverse.Iand 0 represent the identify matrix and the zero matrix with suitable dimensions,respectively.E{·} is the mathematical expectation.

II.PROBLEM FORMULATION AND PRELIMINARIES

A. System Description

Consider a class of continuous-time Markov jump PWA systems

Fig.1.The block diagram for the iterative learning observer with current feedback from iteration k to iteration k +1.

B. Mode-Dependent PWA Iterative Learning Observer With Current Feedback

To estimate the states of augmented Markov jump PWA system (2), a mode-dependent PWA iterative learning observer with current feedback is designed as follows:

Remark 3: The differences between this paper and [21] are as follows: Firstly, in [21], the design issue of fault estimation is investigated for the normal Markov jump system, while the above issue is extended to the more complex Markov jump PWA system in this paper, which includes both switching modes and PWA regions.The switching process of the fault estimator and estimation error system designed in this paper is more complex.Secondly, in [21], the standard iterative learning law based on output errors is designed to estimate non-differentiable faults.In this paper, a new current feedback mechanism is added to the iterative learning law, which can adjust both the previous iteration error and the current error.In addition, the effectiveness of the current feedback mechanism can be verified in the subsequent simulation.

C. Estimation Error System

D. Preliminaries

In the following, the definitions of stochastic stability and theH∞performance index are restated for the general Markov jump systems and the estimation error system (6), respectively.

Definition 1[32]: Consider a class of continuous-time Markov jump systemsx˙(t)=Fr(t)(x(t)), wherer(t)∈S.The systemissaid tobestochasticallystable for anyinitial conditionx(0)∈Rnx,r(0)∈S,if the following expression holds:

Definition 2: Given a scalar γ ＞0, the estimation error system (6) is said to be stochastically stable and has anH∞performance indexγ, if it is stochastically stable and under the zero initial condition, the following relationship holds for all nonzero ω (t)∈l2[0,∞):

To sum things up, the main objectives of this paper are stated as follows.

1) A novel iterative learning law with current feedback is proposed to estimate the augmented states and the fault in (2)together.By updating the estimated information via the previous iteration and the current feedback channel, the estimation errors converge rapidly.

2) The sufficient conditions for stochastic stability withH∞performance are established for the estimation error system(6), while a mode-dependent PWA observer is implemented for the augmented Markov jump PWA system (2) under the iterative learning law with current feedback.

3) The non-convex conditions for the design process of the PWA observer are improved via the fault reconstruction approach.The equivalence relations between the original information and the estimated information are listed via the iterative representation.

III.MAIN RESULTS

In this section, the stochastic stability withH∞performance of the estimation error system (6) is investigated, then the non-convex design conditions are obtained through the fault reconstruction method.Among them, the estimations of the augmented states and the augmented fault can be realized by two iterative equations, respectively.

Theorem 1: Consider the augmented Markov jump PWA system (2) and the mode-dependent PWA iterative learning observer (3) with current feedback.Then the estimation error system (6) is stochastically stable with anH∞performance indexγif there exist positive definite symmetric matricesPm,Im, matrices Λm,Im,ψm,Im, φm,Im,m∈S, Im∈Im, such that

where

Proof: Consider the Lyapunov function candidate

where

By defining κΘ=minm∈S,Im∈Im{κ(-Θm,Im)}.Then, we get

According to [33], it yields that

where we find that

To sum things up, according to Definition 1, it can be concluded that the estimation error system (6) is stochastically stable.

Next, theH∞performance of the estimation error system (6)i s studied.As(t)≠0, similar to (8), it is easy to determine

It follows from the second equation in (6) that:

Based on (9) and (10), under the zero initial condition, theH∞performance index for the estimation error system (6) is established as follows:

Then the following iterative relations can be obtained:

Proof: According to the iterative learning law for(t) in(4) with current feedback, it yields that

where

By iterating (16) from 1 to ϱ+1

If the following equation holds:

then (17) can be obtained as

The condition (13) can be calculated by (18) and (7) via the Schur complement, according to Theorem 1, the estimation error system (6) is stochastically stable with a guaranteedH∞performance indexγ.Therefore, the estimation erroris bounded and the following equalities hold:

After (20) is substituted by (19), then it gets

Thus, the iterative relation equation (14) can be obtained from (21).

Next, from the first equation in (6) and (21), we get

Combining (22) with (23), we find that

Finally, it is easy to show that the iterative relation equation(15) holds under (24).■

are established, respectively, the estimation error system (6) is stochastically stable with anH∞performance indexγ, where

Proof: In accordance with the technique in [15], (12) can be transformed as

Then, the following inequality can be obtained:

where θm,Imdenotes a small positive scalar.By applying the Schur complement, the inequality (26) can be converted to the minimization issue (25).■

IV.ILLUSTRATIVE EXAMPLES

In this section, first, the estimation performance of the iterative learning observer with current feedback is verified.Secondly, the estimation errors of actuator fault and sensor fault are contrasted at different iterations, respectively.Third, to illustrate the importance and validity of the current feedback method to the iterative process, a comparative simulation is conducted with [21].Finally, a class of tunnel diode circuit systems is employed to show the effectiveness of the proposed iterative learning observer with current feedback.

Example 1: Two modes are considered in the continuoustime Markov jump PWA system (1), where the system matrices are represented in the following:

wherem∈S, Im∈Im, S={1,2} , I1={1,2,3} and I2={1,2,3,4}.The parameters of the transition rate matrix Π=[πmn]are given as

The actuator fault, the sensor fault and the disturbance input are given as follows, respectively:

For simulation purposes, the control input can be set asu(t)=Km,Im x(t), where

By using the above information, under the initial conditionx(0)=col{0.2 -0.2}, the Markov switching process, the state trajectories of the augmented system (2), i.e., the states of the Markov jump PWA system, the sensor fault, the actuator fault,the disturbance input, and their estimations based on the developed iterative learning observer with current feedback can be depicted in Figs.2 and 3(a)-3(c), respectively.It can be observed that all the system states, the actuator fault, the sensor fault and the disturbance input can be estimated by the iterative learning strategy rapidly designed in this paper.As the number of iterations increases, the tracking error can be decreased and the estimation performance can be improved effectively.Furthermore, theH∞performance indexγcan be calculated as γ=8.7826 , and the learning gainsψm,Imand φm,Imcan be calculated as follows:

Fig.2.Fault estimation performance of the iterative learning observer with current feedback (k =0,...,5).

Fig.3.Fault estimation performance of the iterative learning observer with current feedback (k =0,...,5).

Second, in order to further illustrate the influence of iteration number on the estimation error, the estimation errors of actuator fault and sensor fault are plotted in Figs.4(a) and 4(b), respectively, where the estimation error at each iteration can be obtained in sequence.It can be clearly seen that the fault estimation error is reduced by increasing the number of iterations effectively, which fully shows that the designed iterative learning observer with current feedback plays a satisfactory effect.

Third, to show the switching trajectories of states at different modes and PWA regions, by letting the artificial bounds|x1|≤1.5, |x2|≤1.5, and using the MPT toolbox in MATLAB[34], the trajectories of both the Markov jump PWA system states and the estimated states are plotted in Fig.5 under the initial conditionx(0)=[-0.7 0.7]T.It can be observed from Fig.5 that the system state can be tracked by the estimation state rapidly, which shows that the designed iterative learning observer with current feedback is satisfactory for tracking operation under the complex multi-mode and multi-region scenes.

Fourth, Fig.6 compares the estimation performance of designed iterative learning observer with current feedback in this paper and an iterative learning observer in [21].Using comparisons with the same number of iterations (k=5), in the case not considering current feedback, all the system states,the actuator fault, the sensor fault and the disturbance input cannot be tracked in a timely manner, and the estimated states fluctuate greatly; In the case considering current feedback, all the mentioned factors above can be tracked instantly.Therefore, it can be concluded that the designed current feedback module plays a significant role in the fault estimation.

Example 2: In order to further show the effectiveness of the proposed iterative learning observer with current feedback, a class of tunnel diode circuit systems [29] is presented as follows:

Fig.4.Fault estimation errors with different iteration numbers.

wherex1(t) andx2(t) stand for the voltage across the capacitor and the current flowing through the inductance, respectively, whereL=1 H, andC=0.04 F.Rr(t)is the load resistance satisfying the Markov switching rule, which varies between two modes, i.e.,R1=20 Ω ifr(t)=1 andR2=30 Ω ifr(t)=2.ω(t) represents the disturbance input.The voltage value is set between -3 V and 3 V to avoid diode breakdown,and we assume that the current changes between -3 A and 3 A.Therefore, the artificial bounds are defined as |x1|≤3 and|x2|≤3.Then, the state space of the tunnel diode circuit system can be divided into a series of PWA regions for each mode via the PWA technique.The system parameter matrices can be obtained in the following:

Fig.5.Trajectories of system state and estimated state at different modes and PWA regions.

Fig.6.Comparisons between the iterative learning observer with current feedback in this paper and iterative learning observer in [21] ( k=5).

The rest of the system parameters are the same as in Example 1.Then, Figs.7(a) and 7(b) plot the estimation errors of actuator fault and sensor fault to obtain the estimation error at each iteration in sequence, respectively.It can be clearly observed that the fault estimation error is reduced by increasing the number of iterations effectively, which fully shows that the designed iterative learning observer with current feedback is effective to resolve the fault estimation of tunnel diode circuit system.

V.CONCLUSIONS

Fig.7.Fault estimation errors in the tunnel diode circuit system.

This paper has addressed the issue of iterative learning based fault estimation with current feedback for a class of continuous-time Markov jump PWA systems.The augmented system was constructed based on the original Markov jump PWA system with sensor and actuator faults firstly, where the actuator fault is non-differentiable.A novel PWA iterative learning observer with current feedback was designed to estimate both the augmented states and the faults.The stochastic stability with guaranteedH∞performance and the constructions of iterative equations for states and faults were studied,respectively.Finally, the effectiveness and the utility of the proposed iterative learning strategy were demonstrated via two numerical examples.