Adaptive fault-tolerant control for continuous-time Markovian jump systems with signal quantization
Introduction
Over the past decades, Markovian jump systems (MJSs) have attracted far-ranging attention in theoretical research field and have abroad applications in industrial processes, economic problems and so on. It is mainly because that this kind of system is able to describe random and abrupt switching phenomenon associated with plants structure. The stability analysis for various classes of MJSs have been widely investigated such as [8], [12], [17], [25], [30], [31], [32] and the references therein. The corresponding combination of different issues with MJSs have also been extensively studied, for example, network-based time-delay communication problems [18], H∞ control problems [37], sliding mode control problems [19], [20], etc.
It is worth noticing that in modern industrial engineering, the physical constraints can always influence the system’s structure and unexpected faults of actuators may occur inevitably, which can cause severe performance of deterioration of system. For example, an actuator of the vehicle system is stuck and failed to deflect the certain control state, it may result in catastrophic accident. Therefore fault-tolerant control (FTC) has become a relatively active researching field recently [7], [9], [16], [28], [29], [38]. The relevant research content includes its applications on linear systems, nonlinear systems and MJSs, etc. In particularly, the FTC system which selects a precomputed adaptive control law can maintain its stability and the acceptable performance fairly well. In recent years, compensation of unknown actuator failures has been realized by novel adaptive fault-tolerant controller design approaches, which estimate the efficiency factor online and some elementary results have been achieved. It is worth noting that a disturbance or disturbance-free scenario is considered in most previous fault-tolerant schemes, the system parameter uncertainty which will make the situation more complex has always been neglected.
On the other hand, the controller design is usually implemented via digital computer in modern industrial control systems [26], so the signal quantization is always encountered in this situation [3], [10], [11], [24], [39], [42]. Great efforts have been spent on quantized control and this issue has become another research forefront in academic field. In previous existing literatures, coder and decoder sides are supposed to have the same quantization sensitivity parameters. It should be pointed that this assumption is impractical due to the hardware flaws. The mismatched initializations at both sides of quantized control systems have been in-depth studied and the mismatched relation can be depicted by a time-invariant ratio. The quantizers can be dynamical or static type [4], [5], [41]. However, under particular circumstances the state vector has to be quantized in the sensor side before transmitting to controller and only quantized signals are accessible in the controller side. This situation makes the stability analysis of systems more difficult. Little attention has been focused on the synthesis of MJSs with mismatch between the quantization levels at the coder and decoder sides. At the same time, there is few work focused on simultaneous quantization of state and input, that is, we consider digital communication links of sensor–controller and controller–actuator together.
As a significant challenge, adaptive fault-tolerant compensation controller design for quantized MJSs needs to attract more attention [2], [6], [15], [23], [34], [35], [36]. Actually, this design is of great importance since the proposed fault-tolerant controller is capable of properly adjusting corresponding control parameters by using appropriate adaptive laws [1], [14], [21], [22], [27], [40], [43], then the actuator fault can be automatically compensated. In this design issue, sometimes only quantized data rather than exact state measurements can be used to synthesize certain terms of controller, the asynchronous occurrence of quantization parameters mismatch should also be dealt with correctly. Hence, an effective adaptive control method needs to be studied to finally make sure the states of system converge to zero asymptotically and the solutions of the quantized adaptive closed-loop system are uniformly bounded and asymptotically stable almost surely, which motivates the study of this paper. The main contributions in this work are summarized as follows: (i) This paper investigates the design of robust adaptive fault-tolerant compensation controller for MJSs with time-varying parameter uncertainty, external disturbance and actuator faults. The proposed controller can online adjust controller parameters to estimate the unknown lower or upper bounds of corresponding terms and automatically compensate the parameter uncertainty, external disturbance and actuator faults; (ii) a novel mode-dependent non-linear control law for MJSs with input quantization mismatch is constructed and it can be proved that the solutions of the overall closed-loop system are uniformly bounded, which is asymptotically stable almost surely; (iii) both sensor–controller and controller–actuator are connected via digital communication channels, dynamical quantizers are updated online and the quantization effects can be completely compensated via injecting quantizer parameters into the controller gains.
The remaining part of this paper is organized as follows. In Section 2, the problem formulation and preliminary results are presented. In Section 3, the quantized adaptive fault-tolerant controller design approach for closed-loop system is proposed. Another novel design approach, which can deal with the more complicated situation is analyzed in Section 4. In Section 5 a simulation example is provided to prove the effectiveness of the methods and the paper is concluded in Section 6.
Section snippets
Problem statement
Consider the following continuous-time MJS with N modes on a fixed given probability space (Ω, Z, P), Ω is the sample space. Z is the algebra of events and P is the probability measure defined by Z. where x(t) ∈ Rn is the state space vector, is the fault control term, stands for the output signal from the lth actuator that has failed in the hth faulty mode, ul(t) represents the input signal of the lth
Fault-tolerant control with state quantization
In this section, to achieve the desired control objectives, the adaptive control law is constructed as where is the estimate of unknown matrix K1i(t) in Eq. (11) and is updated by the following adaptive laws where Γil is any positive constant, and bil is the lth column of Bi, Pi is a positive definite matrix satisfying inequalities (10) and (11), is finite.
Fault-tolerant control with both state and input quantization
In this section, the structure of the control system we consider is shown in Fig. 2, where both sensor–controller and controller–actuator are connected by digital communication channels. In corresponding quantizers the τc1(t) is updated online, τc2(t) is a specified constant. From the schematic diagram, it can be seen that in the controller side only quantized signal of x(t) is available, while x(t) is unknown. The control input signal received in the actuator side is and v(t) is
Numerical example
In this section, the example mentioned in [29] is borrowed to illustrate the validity of the controller design method proposed. Considering the problem of input quantization mismatch mentioned in Section 3, the time-varying ratio parameter model g(t) is described as Table 1.
The system (1) with switching operating modes is depicted by
The
Conclusion
This paper has investigated the adaptive fault-tolerant control problem for a class of continuous-time MJSs with signal quantization, actuator faults, parameter uncertainty and disturbance. In order to compensate the effects caused by these factors, a dynamical uniform quantization strategy and an effective adaptive controller are proposed to illustrate the solutions of the overall closed-loop system are uniformly bounded, which are asymptotically stable almost surely. More network-induced
Acknowledgement
This work was supported by the National Natural Science Foundation of China (61473096, 61690212, 61333003, 61673133) and the New Century Excellent Talents Program of the Ministry of Education of the Peoples Republic of China under Grant NCET-13-0170.
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