Static Output Feedback Control for Fuzzy Systems With Stochastic Fading Channel and Actuator Faults

This work focuses on the issue of static output feedback control for Takagi-Sugeno (T-S) fuzzy systems in the discrete-time domain. Both the fading channel and actuator faults are determined by a set of stochastic variables. Specifically, an actuator fault is described by a nonhomogeneous Markov chain, and fading channels are characterized by the <inline-formula> <tex-math notation="LaTeX">$l$ </tex-math></inline-formula>-order Rician fading model. Using the Lyapunov-Krasovskii function, sufficient conditions are established, and controller gains are developed. Finally, the practical mass-spring-damping model is expressed to verify the practicability of the theoretical results.


I. INTRODUCTION
In recent decades, comprehensive physical applications have encouraged an increase in the amount of attention given to nonlinear systems. Compared with linear systems, nonlinear systems are generally difficult to explore. To address this, many efficient methodologies have been proposed. Among them, the T-S fuzzy model has been extensively recognized. Because of its influential approximation capacity, the T-S fuzzy model has been used to approximate many complex nonlinear plants [1]- [5]. In fact, in addition to this approach, nonlinear plants can be divided into finite weighted sums of linear subsystems. Note that by adopting the T-S fuzzy framework, the sophisticated theories and approaches for linear systems can be extended to the analysis of nonlinear systems. In addition, by means of parallel distributed compensation (PDC), the controllers/filters can also be solved. In light of the aforementioned discussion, the analysis and synthesis of T-S fuzzy systems have attracted much attention [6]- [10].
In networked control systems (NCSs), control techniques have been widely studied and investigated. As an efficient The associate editor coordinating the review of this manuscript and approving it for publication was Jianquan Lu . tool in NCSs, the state feedback control (SFC) strategy plays a significant role in modeling NCSs. In this strategy, the states are required to be constantly accessible. However, for uncertain/unknown state conditions, SFC seems to be unrealistic. To tackle this situation, a static output feedback control (SOFC) scheme has been developed for NCSs [11]- [18]. Compared with other control laws, the SOFC has been widely applied due to its unique structure. In light of the above observations, applying SOFCs for T-S fuzzy systems is more realistic, which partly inspires this work.
In addition, the signals are transmitted via shared communication channels, which may result in multiple path fading and other unanticipate factors, such as fading channel (FC), data collision, dropout loss, et al. Among them, the FC is more general, where the received signals are modeled by various paths with different probability distributions. In fact, because of its probabilistic sequences, FC models are more general than unideal measurements governed by Bernoulli sequences. In this regard, FCs abate the performance and quality of plants. To eliminate the drawbacks, filtering/control issues for NCSs with FCs have been addressed [19]- [23]. However, much attention has been devoted to normal channels, and the reported results offer few insights into SOFCs for fuzzy systems. Following this trend, this study seeks to solve this problem.
Based on the above observations, we aim to establish an SOFC for T-S fuzzy systems with FCs and actuator faults. The major contribution of this work is as follows. (1) To characterize the property of incomplete measurements, the phenomena of FCs are covered, all of which are obeyed by probabilistic variables. By referring to a Rician fading model, the missing measurement can be settled. (2) The actuator faults are described by a nonhomogeneous Markov chain, where timevarying transition probabilities are polytope structured. (3) To better model reality, randomly occurring uncertainties are revealed. (4) By establishing a Lyapunov-Krasovskii function, a mode-dependent SOFC design is formed.

II. PRELIMINARY
Consider the fuzzy systems depicted by IF-THEN rules: Rule p: IF ζ 1 (k) is M p1 , and · · · and ζ r (k) is M pr , Then where M pq (p = 1, 2, · · · , t, q = 1, 2, · · · , r) are fuzzy sets, r is the number of ''IF-THEN'' rules. ζ (k) = [ζ 1 (k), ζ 2 (k), · · · , ζ r (k)] indicates the premise variable. δ(k) ∈ R n y , y(k) ∈ R n y , z(k) ∈ R n z , u(k) ∈ R n u are, respectively, the state vector, measured output, controlled output signal and control input. ω(k) ∈ R n w means the disturbance input that residing in l 2 [0, ∞). A p , B p , C p , D p , F p and H p are known matrices with proper dimensions. The matrix A p is characterized by where M p and N p are known matrices. p (k) is an unknown matrix with the form of p (k) p (k) ≤ I . The stochastic variable (SV) α(k) is utilized to depict the parameter uncertainties in a probabilistic way. Here, for SV α(k), it is obvious that i.e., A p (k) = π (k). Although an easier SOFC design can be acquired for system (1), the addressed results may be conservative due to neglect of the cross term. In this work, probabilistic parameter uncertainty is applied. In contrast to TVSS, norm-bounded uncertainty achieves less conservative results.
For simplification, one denotes

Accordingly, system (3) can be rewritten as
In limited source circumstances, all signals are transformed through a shared communication network. The phenomena of fading channels (FCs) cannot be neglected in many circumstances, which may lead to different transmission rates among channels. To reveal the different capacities of a channel, the lth-order Rice FC model [20]- [23] is expressed as follows: where l k = min{l, k}, ν(k) is an external disturbance, and β s (k) symbolize the channel coefficients that are mutually independent SVs satisfying  VOLUME 8, 2020 In the following, a more general actuator fault (AA) with the property of time-varying is developed, it yields where where µν (k) is the time-varying TP, and µν (k) ≥ 0, ν∈M µν (k) = 1. To characterize the TP matrix (k) form of polytope structure: where ϕ n (k) ≥ 0 and N n=1 ϕ n (k) = 1. For n = 1, 2, · · · , N with N symbolizes the vertices number of (k), n are the vertex matrices. For ϑ k = µ ∈ M, the system AA coefficient matrix of the µ-th mode is symbolized by µ .
Rule p: IF ζ 1 (k) is M p1 , and · · · and ζ r (k) is M pr , Then where K p is the corresponding controller gain to be solved. By resorting to T-S fuzzy strategies [3][4][5][6], (8) implies where Kh = t p=1h p (ζ (k))K p . In terms of (4)- (6) and (9), we obtain the closed-loop fuzzy system as below: where Remark 3: In reality, one may experience bias faults, especially for physical plants with multiple actuators. Recently, mode-independent AA has been fully studied in To remove the mode-independent restriction and reflect the stochastic occurrence of AA, a homogeneous MC is utilized in [21]. In fact, it is unrealistic that the TP of a homogeneous MC remains time invariant. To eliminate this drawback, a nonhomogeneous MC accounting for the characterization of the time-varying TP is adopted.
Before proceeding further, the necessary definition and lemmas are introduced.
Definition 1 [20]: The system (10) with ω(k) = 0 and ν(k) = 0 is said to be stochastic stable (SS), if for any (δ 0 , ϑ 0 ), such that The object of this work is to explore the H ∞ fuzzy control problem for system (10) such that the following conditions satisfied: (i) System (10) is SS in mean square; (ii) Under zero initial condition and a performance level γ , the controlled output z(k) satisfies Lemma 1 [16]: For compatible matrices X = X , Y and Z , U satisfying U U ≤ I , then X + Y U Z + Z U Y < 0, if and only if there exists ε > 0 such that [18]: If there exist a scalar ε and matrices X , Y , Z and U satisfying In what follows, sufficient conditions are attained such that the system (10) is SS in mean square.
To analyze the H ∞ performance for the system (10) under zero initial condition, the following index function is adopted: By utilizing the inequalities (21)- (22), J (T ) can be further improved as below: By resorting to the inequalities (15) and (27) such that Letting T → ∞, (28) results in Therefore, we conclude that the system (10) is SS with H ∞ performance index γ . This completes the proof.
Next, in Theorem 2, the control design scheme is elicited.

V. CONCLUSION
In this study, an SOFC for nonlinear systems with FCs and AAs has been explored using the T-S fuzzy model. Both FCs and AAs are determined by a set of stochastic variables. An AA is described by a nonhomogeneous MC that accounts for the characterization of time-varying TPs. FCs are characterized by the l-order Rician fading model. By applying the Lyapunov-Krasovskii function and PDC technique, sufficient conditions are established, and controller gains are developed. Finally, the practical MSDM is expressed to verify the practicability of the theoretical results. In the near future, our attention will be shifted to sliding mode control scheme for complexity with FC [27], [28].