Multiple intermittent fault estimation and tolerant control for switched T-S fuzzy stochastic systems with multiple time-varying delays

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Abstract

This paper focuses on the observer-based fault estimation (FE) and active dynamic output feedback controller design for switched Takagi–Sugeno (T-S) fuzzy Itô stochastic systems against multiple time-varying delays as well as intermittent actuator faults and sensor faults. External disturbances are also considered. First, a novel descriptor sliding mode observer (SMO) is investigated to establish the error dynamics. Compared with the existing results, the proposed observer could be utilized to more cases and applied to a wider scope. Next, in light of the online FE information, a fault tolerant controller is suggested to make the closed-loop system mean-square exponentially stable. Furthermore, a novel piecewise fuzzy Lyapunov function is depicted and then the delay dependent sufficient conditions can be got by linear matrix inequalities (LMIs). The designed SMO has less conservatism than the existing ones. The reachability of the sliding mode surface in the estimation error space is assured under the adopted method. And the finite-time boundedness (FTB) issue is discussed. At last, a simulation example is acquired to validate the novelty and validity of the method proposed in the paper.

Introduction

Faults are omnipresent in practical systems and have kinds of characteristics [1], [2], [3], [4], [5], [6]. Constant faults and slow time-varying faults are the most common and simplest faults and general time-varying faults are also very common. At present, there have been a lot of related studies, such as [7], [8], [9]. However, the minor faults and intermittent faults (IFs) are inevitable but the related researches are relatively insufficient, which are worth further studying [10]. Faults may make systems unstable. Hence, fault estimation (FE) as well as fault tolerant control (FTC) is essential [11]. Different from FE and FTC of permanent faults which are researched widely [12], [13], FE and FTC of IFs are urgent to research. In general, the current methods of FE are relatively single, mainly by designing different types of filters and observers [14], [15]. It should be noted that the sliding mode observer (SMO) is one of the most popular observers and is strongly robust against the uncertainties and nonlinearities in the system, which is more stable and effective than some observers such as Luenberger observer [16], [17]. Specially, it is worth pointing out that a SMO can perform FE, fault reconstruction and FTC well and quickly. In addition, because IFs have the characteristics of randomness, intermittence and repeatability, the SMO is very suitable to estimate IFs. The SMO is further studied in the paper subject to FE as well as FTC especially subject to IFs. Whilst, the restrictive assumption that the local input matrices are identical is removed in this paper, which is required in many studies such as [18], [19].

Establishing Takagi–Sugeno (T-S) fuzzy models is one of the most effective approaches to solve kinds of complex nonlinear system problems [20], [21], [22], [23], [24], [25]. Zadeh proposed the opinion that the language description of logical rules could be converted into relative control law [26]. T-S fuzzy control uses fuzzy linguistic variables and logical reasoning as tools, and uses human experience and knowledge to incorporate intuitionistic reasoning into decision making. To the best of the authors’ knowledge, a great number of literatures have been focused on FE as well as FTC by T-S fuzzy models [27], [28], [29], [30], [31], [32]. In [31], the problem of robust FE and FTC for T-S fuzzy systems with time-varying delay subject to actuator faults was discussed. However, the sensor faults were not considered. [32] focused on the issue of robust FE as well as FTC for a series of switched T-S fuzzy stochastic systems against both sensor faults and actuator faults. However, the IF situation was not considered. In conclusion, T-S fuzzy models are worth further studying, so they are adopted in this paper.

On the other hand, when modelling a great deal of practical process, switched systems are also common because environments often abrupt change [33], [34]. Switched T-S fuzzy systems have not been fully investigated and more considerable attention has been devoted to the relevant research at present. [35] dealt with fault detection (FD) problem for switched T-S fuzzy systems subject to unknown input, but stochastic phenomenon that exists commonly in realistic industrial systems was ignored. The control strategies of stochastic systems with Brownian motions, also named Wiener process, relate to Itô differential equations [36], [37], [38]. In [39], the problem of FE and FTC was researched for T-S fuzzy stochastic systems against sensor faults, however, finite-time stability issues were not considered. At the same time, delay even multiple delays always occur in production [40], [41], [42]. [43] focused on the guaranteed cost control problem for uncertain T-S fuzzy stochastic systems with multiple time delays. However, all states were required to be known. Switching behavior, Brownian motions and time delats all can result in system instability. A variety of Lyapunov functions have been correspondingly proposed to make system stable, such as [44], [45], [46], [47], [48], [49], [50]. Besides, another point worth mentioning is that the piecewise fuzzy Lyapunov function is less conservative and shows stronger ability than many ones such as common Lyapunov function, piecewise or fuzzy Lyapunov functions [51]. To mention a few, few results of fuzzy systems have been concentrated on switched stochastic systems against multiple time-varying delays. Moreover, there are also few related studies on piecewise fuzzy Lyapunov function. The preceding discussion content inspires us to carry out this work.

The issues of FE and FTC are investigated in the paper for a series of switched T-S fuzzy stochastic systems against multiple time-varying delays, external disturbances, sensor faults as well as intermittent actuator faults. Firstly, a SMO is employed to perform FE. In addition, the observer gain matrices are collected. Secondly, the sliding mode controller is acquired to make the closed-loop system stable. Thirdly, sufficient stability conditions of the overall closed-loop system are suggested by piecewise fuzzy Lyapunov function. Fourthly, the reachability of the sliding mode surface is assured in the estimation error space under the adopted control method. Sixthly, FTB is discussed. Finally, an example is described to validate the availability of the addressed approach. The contributions in the paper could be highlighted as follows:

  • (1)

    For the switched T-S fuzzy stochastic systems subject to multiple time-delays and intermittent actuator faults, the FE and FTC are provided for the first time. The approach in the paper performs FE and FTC against intermittent actuator faults and sensor faults and is able to be applied to a wider scope.

  • (2)

    In the SMO, the introduction of proportional gain, derivative gain as well as sliding mode gain makes the observer design flexible and feasible.

  • (3)

    By a tricky matrix parameter, the influence of Brownian motion is removed from the suggested sliding mode function. Furthermore, the reachability of the sliding mode surface is assured strictly.

  • (4)

    Different from fuzzy Lyapunov function such as [52], which only considered the effect of membership functions on system stability, or piecewise Lyapunov function such as [35], which only considered the effect of switching on system stability, piecewise fuzzy Lyapunov function considers the effect of both membership functions and switching behavior on system stability. By employing the novel piecewise fuzzy Lyapunov function, delay-dependent results are obtained easily, the conservativeness of the finding maximum delay bounds is decreased and the exponential stability can be guaranteed, which converges faster than asymptotical stability.

In the paper, the remainder is organized as below. The problem formulation along with preliminaries is presented in Section 2. A SMO is suggested and observer gain matrices including proportional gain, derivative gain and sliding mode gain are designed in Section 3. In Section 4, the sliding mode for the switched T-S fuzzy stochastic system is shown against multiple state delays. And the stability conditions of the corresponding closed-loop system are provided by piecewise fuzzy Lyapunov function in the form of LMIs. In Section 5, the reachability of the sliding mode surface is assured in the estimation error space under the adopted control method. FTB issue is discussed in Section 6. In Section 7, a simulation examples is given to demonstrate the effectiveness and advantages of the proposed approach. At last, the conclusions are drawn in Section 8.

Notation: sym(U¯)=U¯+U¯T; The symbol * in a matrix means the transposed element in the symmetric position; P{U¯} means the occurrence probability of event U¯; E{U¯} stands for the expectation of event U¯; D(ν¯,ϖ¯) denotes the LMI region. Specifically, if complex numbers c¯+d¯iD(ν¯,ϖ¯), we have ν¯<c¯<0, |c¯+d¯i|<ϖ¯.

Section snippets

Problem statement

In this paper, this switched T-S fuzzy stochastic system with multiple time-varying delays is presented by IF-THEN rules. The ith rule of T-S fuzzy system is provided as below.

Plant Rule i:

IF μ1(t) is πi1j, and,, and μl(t) is πilj,

THEN{dx(t)=[k=0SAkijx(tτk(t))+Buiju(t)+Bajαa(t)fa(t)+Bwijw(t)]dt+k=0SBkijx(tτk(t))dW(t),y(t)=C0jx(t)+Fsfs(t),yz(t)=Czijx(t),x(t)=ψ(t),t[τ¯,0],where x(t) ∈ Rn stands for the state, u(t) ∈ Rm denotes the system input, αa(t)fa(t)Rna means the intermittent

SMO design

In this section, the main results for the switched T-S fuzzy stochastic systems are given against multiple time-varying delays. In the following, a SMO method is presented to estimate the intermittent actuator faults fa(t) as well as the sensor faults fs(t).

For convenience of the later description and analysis, we define x¯(t)=[xT(t),faT(t),(Fsfs(t))T]T and f(t)=[ϵfa(t)+f˙a(t)fs(t)]. Then the system (2) can be redescribed as{G¯jdx¯(t)=i=1rϱij(μ(t)){[A¯0ijx¯(t)+k=1SA¯kijx¯(tτk(t))+B¯uiju(t)+B¯

System stability analysis

Recalling from (9), it yields thatS¯jdx¯^(t)=S¯jdx¯c(t)+L¯djdy(t)=i=1rϱij(μ(t)){[(A¯0ijL¯pijC¯0j)x¯^(t)A¯0ij(S¯j)1L¯djy(t)+L¯pijC¯0j(S¯j)1L¯djy(t)W¯y(t)+k=1SA¯kijx¯^(tτk(t))+B¯uiju(t)+L¯sjus(t)]dt+L¯djdy(t)}.It is worth pointing out that C¯0j(S¯j)1L¯dj=Iny, A¯0ij(S¯j)1L¯dj=W¯. Hence, (18) can be written asS¯jdx¯^(t)=i=1rϱij(μ(t)){[(A¯0ijL¯pijC¯0j)x¯^(t)+W¯y(t)+L¯pijy(t)W¯y(t)+s=1SA¯kijx¯^(tτk(t))+B¯uiju(t)+L¯sjus(t)]dt+L¯djdy(t)}.It further follows thatS¯jdx¯^(t)=i=1rϱij(μ(t)){[

The Reachability Control of Sliding Motion

This section is concerned with the reachability of the sliding mode surface s(t). The solution e(t) of (22) can be written ase(t)=0tj=1pσj(θ)i=1rϱij(μ(θ))(S¯j)1{[(A¯0ijL¯pijC¯0j)e(θ)+k=1SA¯kije(θτk(θ))+L¯sjus(θ)B¯fjf(θ)B¯wijw(θ)B¯ajα˜a(θ)fa(θ)]dθk=0SB¯kijx¯(θτk(θ))dW(θ)}.

Substituting (73) into (24), it follows that:s(t)=j=1pσj(θ)i=1rϱij(μ(θ)){0t(B¯fj)T(S¯j)TP2ϱj(S¯j)1[(A¯0ijL¯pijC¯0j)e(θ)+k=1SA¯kije(θτk(θ))+L¯sjus(θ)B¯fjf(θ)B¯wijw(θ)B¯ajα˜a(θ)fa(θ)]dθ0tk=0SCϱjC¯0j(S¯j)

Finite-time stability

Definition 4

For a time interval [t¯,t˜], assume there are scalars c2 > c1 > 0 and a positive weighted matrix R, then the system (2) with u(t)=0 is FTB with respect to (c1,c2,[t¯,t˜],R,W¯[t¯,t˜],), ifξT(t¯)Rξ(t¯)c1ξT(t)Rξ(t)c2,t[t¯,t˜]where w(t)W¯[t¯,t˜], is denoted in Assumption 4.

Lemma 6 (Partitioning Strategy)

[61]. The closed-loop system (29) is FTB with respect to (c1,c2,[0,T],R,W¯[0,T],) denoted in Definition 4, if and only if there is an auxiliary scalar c* with c* ∈ (c1, c2) such that the system (29) is FTB with respect to

Simulation results

An example is got to demonstrate the effectiveness of the addressed method in this section.

The mechanical system dynamics for a single-link direct-drive manipulator actuated by a permanent magnet brush dc motor and the electrical system dynamics for the dc motor are simultaneously studied. They can be regarded as a part of electromechanical system with the following data [32].We can't comment on the formulas. F_S should be changed to F_s.τ1(t)=0.2+0.1sin(t),τ2(t)=0.4+0.2cos(t),A011=[0a00θ11b1a0θ

Conclusions

This paper has focused on the observer-based FE and observer-based controller design for switched T-S fuzzy Itô stochastic systems against multiple time-varying delays as well as intermittent actuator faults and sensor faults. External disturbances have also been considered. First, a novel descriptor SMO has been investigated to construct the error functions. Then, based on the information of online FE, a fault tolerant controller has been suggested to make the closed-loop system mean-square

Acknowledgements

The work was supported by the National Natural Science Foundation of China (61627809, 61433004, 61621004), and Liaoning Revitalization Talents Program (XLYC1801005).

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