Research articleIterative learning fault diagnosis and fault tolerant control for stochastic repetitive systems with Brownian motion☆
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 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 can be expressed as , represents the identity matrix with proper dimension, and is the expectation of the stochastic variable. The trace of matrix is defined to be the sum of eigenvalues of , which is recorded as . denotes the set of functions and is continuously once differentiable in and twice in .
Section snippets
Preliminaries and system model description
The stochastic linear Brownian system is considered as follows where is the state of the control system (1), and represent the input and the output of the control system (1), respectively. denotes the fault, is the Brownian motion and and . and are known parameter matrices. It is supposed that and the condition that 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 where denotes the state estimation vector, represents the output estimation vector and denotes the fault estimation vector. is the gain matrix of the FD observer to be determined. 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
The following reliable output of the system is obtained to eliminate the effect caused by the sensor fault
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
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 where and represent the -axis component of the motor stator current and the stator voltage. and represent the -axis component of the motor stator current and the stator voltage. , and
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.
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