Elsevier

ISA Transactions

Volume 121, February 2022, Pages 171-179
ISA Transactions

Research article
Iterative learning fault diagnosis and fault tolerant control for stochastic repetitive systems with Brownian motion

https://doi.org/10.1016/j.isatra.2021.03.030Get rights and content

Highlights

  • The fault diagnosis method based on iterative learning algorithm is designed to estimate the state and fault accurately by iteration.

  • Based on the state estimation and fault information, the output feedback active fault tolerant control algorithm is designed to guarantee that the post-fault system is stochastic input-to-state stable.

  • Many existing fault estimation algorithms are established when the first-order derivative or the second-order derivative of the fault was bounded. Different from these methods, the fault estimation algorithm proposed in this paper is feasible without these assumptions, which is more general and practical.

Abstract

In this paper, the issue of iterative learning fault diagnosis (ILFD) and fault tolerant control (FTC) is studied for stochastic repetitive systems with Brownian motion. Different from existing fault diagnosis (FD) methods, a state/fault simultaneous estimation observer based on iterative learning method is designed. The convergence condition of the ILFD algorithm is given. By employing the fault estimation information, the FTC algorithm is proposed to compensate for the fault effect on the system and to keep the stochastic input-to-state stability of the control system. Finally, the simulation results of an induction motor system and a single-link robotic flexible manipulator system are given to show that the proposed method is validated.

Introduction

For practical control systems, it is an urgent task to guarantee the reliability and stability of the system. Any faults in the system may disrupt the normal operation of the system and even lead to disaster. FD and FTC provide a feasible programme to improve the safety and reliability of the equipment operation. Therefore, the research on FD [1], [2] and FTC [3], [4], [5] has obtained more and more attentions.

From another point of view, stochastic systems with Brownian parameter disturbance have caused the extensive research because of the accurate description of the actual system model. A relatively complete framework of input-to-state stability theory and control theory has been established for stochastic systems. In [6], [7], the conditions of input-to-state stability for stochastic systems were studied. In order to address the predictive control problem for the linear stochastic control system with Brownian motion, an optimal controller based on the augmented error system was designed [8]. The controller design based on the descriptor observer for stochastic descriptor systems with Brownian motion was studied and the mean-square exponential stability condition was given in [9]. In the meanwhile, the rich achievements of FTC for stochastic systems with Brownian motion have been obtained. FTC generally is classified as passive fault tolerant control (PFTC) [10], [11] and active fault tolerant control (AFTC) [12]. AFTC can make full use of the fault estimation information to realize FTC, so as to ensure the stable performance of the post-fault control system. In [13], an integrated robust FD and FTC scheme founded on the unknown input observer for stochastic Brownian systems was implemented. By using the sliding mode approach, a novel FD and FTC algorithm was put forward for the nonlinear Itoˆ stochastic system with sensor fault, input and output disturbances in [14]. For the stochastic T-S fuzzy system, the unknown input PD observer and the sliding mode FTC method were proposed in [15]. A new FD method by exploiting the reduced-order observer and the FTC strategy with fault compensation [16] were studied for the switched stochastic system with faults. An extended PI observer was utilized to diagnose the fault and the observer-based FTC technique was proposed for switched fuzzy stochastic systems [17].

In practice, many stochastic systems with Brownian motion are repetitive control systems, for example, industrial robots, chemical processes and servo systems. Therefore, the existing FD and FTC methods are not applicable to stochastic repetitive systems with Brownian motion. The iterative learning scheme [18], [19] is to learn from the experience and performance of the previous iteration, which is suitable for the system with repetitive motion. For discrete time linear stochastic systems, an adaptive iterative learning control (ILC) scheme was put forward [20] to track the batch-varying reference trajectories. In [21], the fault detection and estimation method founded on the PD-type iterative learning scheme was applied for the nonlinear system. The problem of the period intermittent fault estimation for nonlinear uncertain systems was considered in [22], an innovative iterative learning fault diagnosis observer was presented for fault reconstruction. However, the corresponding fault-tolerant control strategy was missing. A new robust adaptive iterative learning FTC law was designed for the high speed train system subject to unknown delays, actuator faults and control input saturations in [23]. The FTC method based on the ILC scheme was proposed [24] for repetitive processes with delay and actuator faults, which can also be extended to the uncertain system. As far as we know, FD and FTC for stochastic repetitive systems with Brownian motion has not yet attracted sufficient attentions. The challenge is how to estimate the fault information accurately and make the post-fault system still keep the stochastic input-to-state stability.

Motivated by the above discussion and analysis, the problem of ILFD and FTC is considered for stochastic repetitive systems with Brownian motion in this paper. The summary for the main contents and contributions is given as follows:

(1) For stochastic repetitive systems with Brownian motion, a FD method based on iterative learning algorithm is developed to estimate the state and fault accurately by iteration;

(2) By utilizing the state estimation and fault information obtained by the FD algorithm, the output feedback AFTC algorithm is investigated to guarantee that the post-fault system is stochastic input-to-state stable;

(3) The faults under investigation can be the constant fault and time-varying fault. It should be pointed out that many existing fault estimation algorithms are established when the first-order derivative or the second-order derivative of the fault was bounded [13], [14]. Different from these methods, the fault estimation algorithm proposed in this paper is feasible without these assumptions, which is more general and practical. At the same time, the comparison results with the fault diagnosis algorithm using the unknown input observer proposed in [13] are given.

The remainder of this paper is arranged as below. Section 2 gives the system model description of a class of stochastic linear Brownian systems. The design of the FD algorithm is given in Section 3. The design of the FTC algorithm is carried out in Section 4. Section 5 gives the simulation results of two practical examples. Then, the corresponding conclusions is given in Section 6.

Notation: The following standard notations will be used in this paper, the pseudo inverse of matrix X can be expressed as X+, I represents the identity matrix with proper dimension, and E is the expectation of the stochastic variable. The trace of matrix XRn×n is defined to be the sum of eigenvalues of X, which is recorded as traceX. C2 denotes the set of functions V(t,x) and V(t,x) is continuously once differentiable in t and twice in x.

Section snippets

Preliminaries and system model description

The stochastic linear Brownian system is considered as follows dx(t)=(Ax(t)+Bu(t)+HF(t))dt+Dx(t)dv(t)y(t)=Cx(t)+ WF(t)where x(t)Rn is the state of the control system (1), u(t)Rm and y(t)Rp represent the input and the output of the control system (1), respectively. F(t)Rr denotes the fault, v(t) is the Brownian motion and Ev(t) = 0 and Ev2(t) = t. A,B,H,D,C and W are known parameter matrices. It is supposed that rank[B,H]=rankB and the condition that rp should be satisfied.

Consider the

Fault diagnosis

In order to obtain the state and fault estimation information, the following state observer and iterative learning diagnosis algorithm are constructed dxˆk(t)=(Axˆk(t)+Bsat(uck(t))+HFˆk(t)+K1(yk(t)yˆk(t)))dtyˆk(t)=Cxˆk(t)+WFˆk(t) Fˆk+1(t)=Fˆk(t)+L(yk(t)yˆk(t))where xˆk(t) denotes the state estimation vector, yˆk(t) represents the output estimation vector and Fˆk(t) denotes the fault estimation vector. K1 is the gain matrix of the FD observer to be determined. L is the learning operator to be

Fault tolerant control

According to the state and fault estimation information, the controller is reconstructed to compensate validly the effect generated from the fault uck(t)=K2xˆk(t)B+HFˆk(t)

The following reliable output of the system is obtained to eliminate the effect caused by the sensor fault yck(t)=yk(t)WFˆk(t)=Cxk(t)+WF̃k(t)

Putting the fault tolerant controller (25) into the system (3) and replacing the actual measurement output with the reliable output, it can be obtained the following closed-loop system d

Simulation examples

An induction motor system and a single-link robotic flexible manipulator system are considered in this section, which are used to verify the validity of the FD and FTC algorithm.

Example 1

Consider the following induction motor system did=(RsLsid+ωsiq+1Lsud)dt+0.001iddvdiq=(ωsidRsLsiq+1Lsuq)dt+0.01iqdvwhere id and ud represent the d-axis component of the motor stator current and the stator voltage. iq and uq represent the q-axis component of the motor stator current and the stator voltage. Rs, ωs and L

Conclusions

For the stochastic repetitive system with the system fault and Brownian parameter perturbations, an iterative learning algorithm for fault estimation has been investigated. By employing the iterative learning fault diagnosis method, the system fault can be accurately estimated by iteration. Then, a FTC algorithm has been designed by utilizing the fault estimation information to make the post-fault closed-loop system still remain stochastic input-to-state stable. Simulation results have been

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (29)

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    Meanwhile, as science and technology development and equipment becomes more sophisticated and automated, some new techniques have been applied for fault diagnosis. For example: condition-based maintenance (CBM) was proposed (Ayo-Imoru and Cilliers, 2018; Ignat, 2013), which recommends maintenance decisions based on information collected during condition monitoring (Caruso et al., 2021; Jakab et al., 2021); Samir Khan and Piyush Agarwal applied artificial intelligence and deep learning to system diagnosis (Khan and Yairi, 2018; Agarwal et al., 2021); Han studied the fault diagnosis (FD) and fault-tolerant tracking control (FTTC) problems as a kind of discrete-time systems with faults and delays in actuators and measurements, and illustrated the effectiveness of their proposed fault diagnosis and optimal fault-tolerant tracking controller (OFTTC) with real design examples (Han et al., 2017); and Li studied the iterative learning fault diagnosis (ILFD) and fault-tolerant control (FTC) problems of a stochastic repetitive system with Brownian motion (Li et al., 2021). However, the abovementioned models are complicated in structure and require a large amount of mathematical calculations, which undoubtedly prohibits them from being used and promoted.

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This work was supported by the NSFC, China grants (No. 61973278, 61374128) and Basic Research Projects of Key Scientific Research Projects of Colleges and Universities in Henan, China 21zx007.

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