Security SMC for networked fuzzy singular systems with semi-Markov switching parameters

In this paper, the issue of security sliding mode control (SMC) is addressed for networked nonlinear singular systems with semi-Markov switching parameters via the Takagi-Sugeno (T-S) fuzzy strategy. The goal is to guarantee a good steady performance of dynamical systems under the framework of deception attacks. First, the common sliding surface is proposed to avoid the potential instability caused by repeated jumps of sliding surface. Then, based on the partly unknown transition rate, sufficient conditions are constructed to realize the stochastic admissibility for the corresponding system. Moreover, a fuzzy security SMC law is designed to drive the state trajectory onto the specified sliding region. Finally, a practical example is shown to validate the proposed algorithm.


I. INTRODUCTION
W ITH the rapid development of modern control technology, communication technology, and computer technology, the traditional point to point control approach cannot satisfy the requirement of actual industrial systems. Networked control systems (NCSs) connect the multiple distributed components through a shared communication network to form the closed-loop control system [1]. Different from the traditional control systems, the shared communication network brings many advantages, such as low cost and easy maintenance. Under the technology wave of industry 4.0, NCSs provide a key technical support for the stable and optimal operation of real-time system, and have been widely applied to automobile manufacturing, robot control, smart home, smart grid, mobile sensor network, and automatic vehicle formation [2]. Although the shared communication network brings a great deal of convenience, the malicious attacks make that NCSs face a severe security challenge. Therefore, it is of great significance to minimize the impact of cyber attacks on NCSs and realize system security.
It is noted that the typical cyber-attacks mainly cover denial-of-service (DoS) attacks [3], deception attacks [4], and replay attacks [5]. For the deception attacks, the attacker injects false data into the communication packet by tampering with the sensor transmission data or control commands, so as to degrade the system performance. In fact, the false data injection attacks are recognized as a typical representative of deception attacks. Recently, a wonderful amount of significant research works have been developed for the corresponding security control system with deception attacks [6][7][8][9].
It is well-known that Markov jump systems (MJSs), as a special kind of dynamical systems [10][11][12][13][14], are well suitable to model physical systems subject to sudden environment disturbances, random failures, and abrupt variation of the operating point, which have extensive applications in the fields of power systems, financial systems, and aerospace [15][16][17][18][19]. For MJSs, the transition rate (TR) between system modes is sojourn-time-independent, in which the sojourn time (ST) obeys an exponential distribution. However, it is difficult for many industrial processes to satisfy this strict condition. More generally, the ST may follow Gaussian distribution, Weibull distribution, and phase distribution. Due to the extended probability distribution, the TR of semi-Markov jump systems (S-MJSs) is ST-dependent. Obviously, S-MJSs are less conservative in modeling practical systems than MJSs. Very recently, lots of remarkable results have been proposed for S-MJSs, such as stability and stabilization [20][21][22][23][24][25], SMC [26][27][28][29][30][31][32][33][34], event-triggered scheme [35,36], time delay [37], and disturbance rejection [38]. As a description form of differential algebraic (difference algebraic) equations, singular systems can better maintain the physical property of dynamical systems [39]. Along with the successful application of singular systems in practical problems, singular systems have become a hot research topic in the control field [40,41]. Combining stochastic switching systems, many significant results have been proposed for singular MJSs and singular S-MJSs [31,34,42,43].
Meantime, many practical systems always exhibit the high nonlinear property, such as economics, systems engineering, medicine, psychology, and welding process, so it is difficult to model and control these systems directly. Takagi-Sugeno (T-S) approach provides an effective tool to characterize complex nonlinear systems [44], in which the main characteristic is that the nonlinear dynamic is represented by a set of local linear systems through the nonlinear fuzzy membership functions. Also, the fuzzy systems can approximate the nonlinear systems with arbitrary accuracy, which is convenient for system analysis. Recently, combining with stochastic switching systems, the research of T-S fuzzy systems has attracted great interest in the control community [34,28,34,[45][46][47][48].
Notations: P > 0 means the positive definite matrix. Ξ{·} is the mathematical expectation. He(P) represents P + P T .

II. PRELIMINARY
Consider the following networked fuzzy singular S-MJSs: where W αβ (α = 1, 2, ..., , β = 1, 2, ..., p) are the fuzzy sets with ς 1 (t), ς 2 (t), ..., ς p (t) being the premise variables. y(t) and u A (t) represent the state and the input. The singular matrix E is considered as is the system mode index and {ı l } l∈N ≥1 is the ST of l−1 between the (l − 1)th transition and lth transition with the probability transitions: where λ τ (ı) ≥ 0 denotes the TR for = τ , and In this paper, the TR is partly unknown in If ∅ = ∅, it is further given by where k m means the mth known TR of ∅ k in the th row of Λ(ı).
Remark 1. In practical systems, there exists the difficulty to obtain precise information of TR owing to some complicated factors. For example, the packet loss and delay always exist in NCSs characterized by the SMP under the ideal framework of of completely known TR. Either the delay or the packet loss is vague owing to stochastic complexity of the network environment, that is, to get an expected TR matrix is seldom possible. Compared with completely unknown TR and bounded TR [20][21][22][23][25][26][27][28][29][30][31][32][33][34][35][36][37][38], partly unknown TR is adopted to model stochastic semi-Markov switching process to reduce some conservatism.

III. STOCHASTIC ADMISSIBILITY ANALYSIS
As stated in [31], the repeated switching of sliding surface is inevitable. Meantime, the switching number of sliding surface will be increased by the jumps of semi-Markov switching mode. For some details, please see Ref. [31]. In order to reduce the frequent switching, a common sliding surface is designed as where G  =1 α B T . In fact, B ω does not need to satisfy full column rank.
Design the security SMC law as: where the nonsingular matrix P and the positive constant χ will be given later. Therefore, based on the security SMC law (8), one has Theorem 1. System (9) is stochastically admissible, if there exist Ψ > 0, nonsingular matrix P ω > 0 and scalar φ > 0, such that for ∈ ∅, where Proof. First, we prove that the system (9) is regular and impulse free. Since rank(E) = r < n, there are two nonsingular matrices D and F satisfying the following conditions

VOLUME 4, 2016
According to (10), one has P 2i = 0. Pre-and postmultiplying (11) by F T and F gives * * * Ã T 4α, where * will not be used in the following development. Then, it is got thatÃ T 4α, P 4 + P T 4 Ã 4α, < 0, which means thatÃ 4α, is nonsingular. System (9) is therefore regular and impulse-free. Next, prove the stochastic stability. Choosing Lyapunov function Γ(y(t), ) = y T (t)E T P y(t) results in where q τ is the probability intensity from to τ and (ı) is the cumulative distribution functions of ST.
According to and Taylor-series formula where λ τ = Ξ{λ τ (ı)} = ∞ 0 λ τ (ı) (ı)dı with the probability density function (ı) of ST. Based on the condition (11), it is got that − 2y T (t)P T B κq ψ (y(t), t)sgn(φ (ΨGB ) T s(t)) It follows from (15)- (17) that which means that ℘Γ(y(t), ) < 0, under the condition (12). Therefore, system (9) is stochastically admissible. Remark 2. In comparison with ST subject to the traditional exponential distribution in SMC for MSSs [18,42,43,61], it is required to recompute the weak infinitesimal operator in line with SMP constraint. Under the framework of Taylor-series formula, lim the weak infinitesimal operator can be got. Sufficient conditions are constructed such that the corresponding system is stochastically admissible under the framework of partly unknown TR. However, these conditions (10)- (12) are not solvable directly owing to the existences of equalities in (10), (11), nonlinearity and partly unknown TR in (12). Next, the following Theorem 2 in terms of strict linear matrix inequalities is given to determine P and Ψ. Theorem 2. System (9) is stochastically admissible, if there exist symmetric matrix U ω > 0, matrices Ψ > 0, Q ω ,P and scalars ε > 0, δ > 0, φ > 0, such that the following conditions are satisfied for all ∈ ∅, where with W ∈ R n×(n−r) being with full row rank and satisfying E T W = 0.
Proof. Based on P = U E + WQ and E T W = 0, the condition (10) holds.

IV. REACHABILITY OF SLIDING SURFACE
An appropriate security SMC mechanism is designed to guarantee finite-time reachability of the sliding surface. Theorem 3. Consider the solvable parameters P and Ψ in Theorem 2. The states of system (1) will be driven onto the sliding region under the condition of security SMC law (8) with χ satisfying where > 0.
Proof. Choose Lyapunov function Γ(s(t)) = 1 2 s T (t)Ψs(t). By the expressions (7) and (9), one has ℘Γ(s(t)) where a =max α=1,2,..., { ΨGA α, + ΨGL α, H α, For the following domain it is got that which means ℘Γ(s(t)) ≤ − s(t) < 0 for s(t) = 0. Therefore, the system states can be driven onto the sliding region and kept there in subsequent time. Remark 4. The sliding region is shown for the corresponding proof of reachability analysis. Under the unavoidable chattering effect, the ideal sliding motion will be broken on the sliding surface. In this instance, the sliding motion is set to be a local neighborhood around the predefined sliding surface. Remark 5. For the difficulty Q, a common sliding surface is chosen to reduce the frequent switching of sliding dynamics. Next, the designed SMC law (8) is related to the corresponding parameters κ, q , and ψ (y(t), t) of deception attacks, the specified matrix P , and the positive constant χ . Moreover, a reasonable sliding region is adopted to characterize the reachability analysis. Therefore, under the deception attacks and the partly unknown TR, one has better dynamic performance by the designed SMC mechanism.

Fig. 4 Sliding surface
Remark 6. For the SMC law design, the parameters selection plays an important role in characterizing system performance as well as its impacts on the state signals. First, for given single-link robot arm model (29), the parameters M( t ), L, J ( t ), and D(t) are chosen with appropriate values. Secondly, according to the statistical characteristics of SMP, the TR matrix is considered to be partly unknown in (30). Furthermore, to describe the dynamical behavior of singlelink robot arm model, such as uncertainty and deception attacks, we choose the appropriate parameters L α, , H α, , κ, q , and ψ (y(t), t), for α = 1, 2, = 1, 2, 3. Finally, for given constants φ > 0 and > 0, one has the corresponding parameter P by finding feasible solutions of positive-definite symmetric matrix U ω and real matrix Q ω in Theorem 2. In fact, the parameters selection lies in the appropriate values that are not too big or too small. It is not good for the feasible solutions that these parameters are chosen to be too small or too big. Therefore, it is of importance to choose a compromise value among them to get the feasible solutions at a reasonable computational cost.

VI. CONCLUSION
In this paper, a common sliding surface has been proposed for networked fuzzy singular S-MJSs with false date injection attacks. Based on the partly unknown TR, the feasible conditions have been derived to ensure the sliding dynamics stochastically admissible. Furthermore, by the synthesized security SMC law, the system states can be driven onto the predefined sliding region in a finite time. With the development of modern network communication technology, the corresponding cyber attacks also show a diversified trend. Malicious attackers may launch a variety of cyber attacks at the same time, resulting in the instability of control system performance. In the future, the security SMC will be extended to networked S-MJSs with hybrid cyber attacks.