Finite-time Sliding Mode Control for Uncertain Neutral Systems With Time Delays

The finite-time stability (FTS) problem of uncertain neutral time-delay systems via a sliding mode control (SMC) approach is discussed in this paper. First, we construct a suitable sliding mode surface and an SMC law, which can guarantee the system states can reach the sliding mode surface in a finite time and maintain the sliding mode. Then, through the Lyapunov stability theory and the inequality techniques, the finite time stability of the closed-loop system during reaching phase and sliding mode phase is studied, a set of sufficient conditions which ensure the system to be finite-time stability is developed. Finally, a numerical simulation example is given to illustrate the effectiveness of the results.


I. INTRODUCTION
SMC is an effective nonlinear robust control method, which has strong robustness to resist parameter uncertainty and external disturbance of dynamical systems [1], [2]. At the same time, it has the advantage of excellent transient response [3], [4]. Due to these characteristics, SMC is widely used in missile guidance systems [5], motor control systems [6] and other industrial automation fields, and it becomes a hot spots which attracted many scholars in interest and importance. For example, for a second-order nonlinear dynamic system, a new nonlinear sliding mode controller is presented to resist parameter uncertainty and external disturbance [7]. The observer-based SMC problem of phasetype semi-Markovian jump systems was discussed in [4]. In [8], the authors investigated the issue of SMC with adaptive neural networks for a class of nonlinear uncertain systems. A novel asynchronous sliding mode control scheme is proposed in [9], which guarantees the desired finite-time boundedness of Markovian jump systems with sensor and actuator faulty signals. In fact, SMC is used on a variety of systems, such as Markovian jump systems [4], [10], time-delay systems [11], [12], and stochastic systems [13], [14].
For a long time, the investigators focused on Lyapunov asymptotic stability(LAS); however, this theory also has its limitations. The most important point is that there is generally no constraint on the time required for the system to reach steady state, which is not allowed in some time-critical systems, such as robot dynamic stability and communication networks. Based on this issue, Dorato proposed the finite time theory in 1961, since then, many scholars have begun to study this theory [15], [16]. FTS means that the weighted state of a system does not exceed a predetermined threshold during a finite time interval when the initial states of the system are norm-bounded [17], [18]. When there are external disturbances in the system, finite-time boundedness(FTB) will replace FTS. FTS focuses on the transient performance of the system, so it has special application scenarios and has attracted remarkable attention of researchers. Recently, two new finite-time convergence criterions are proposed based on the property of the second order differential equation [19]. For a class of discrete-time nonlinear systems, new control procedures are proposed, which can satisfy the finitetime stabilization [20]. In [21], the authors put forward a finite-time SMC law to satisfy FTB and H ∞ performance requirements. In addition, FTS is also used in fractional order systems [22], [23] and neural networks [24], [25].
Different from the common time-delay systems, the neutral system is a special kind of dynamic system, which is often used in the research of aircraft engine control, immune response, electrodynamics processes and so on [26]- [28]. Neutral systems contain delays in its state and in the derivatives of its state [29]. The above characteristics lead to more complex dynamic behavior of this kind of system, and make this kind of system have better universality than the general time-delay systems [30], [31]. In [32], for a class of uncertain neutral delay systems with mismatched uncertainties, the author proposed a sliding mode control law to guarantee the asymptotic stability of closed-loop systems. The problem of exponential H ∞ output tracking control for a class of switched neutral system with time-varying delay was addressed in [33]. The results of robust stability for a class of uncertain neutral system with time-varying delay can be found in [34].
To date, there have been many studies on finite-time sliding mode control, and many scholars have extended this method to various control systems [9], [21], [35]. However, to our best knowledge, there are still few results on FTS/FTB of uncertain neutral systems. Similar uncertain neutral time delay systems with SMC were studied in [32], [36], but these papers focused on the asymptotic stability of the system rather than FTS. FTS for uncertain systems over reaching phase and sliding motion phase were discussed in [21], [38]; nevertheless, the system in these papers is relatively simple. Different from these literatures, the neutral time-delay systems investigated in this paper have better universality, and the results can better reflect the quantitative relationship between the FTS conditions and the system parameters. As mentioned above, neutral systems have a wide range of applications, FTS based on sliding mode control can guarantee the transient performance of such systems. In addition, the results of neutral time-delay systems can be easily extended to standard time-delay systems. Therefore, the study of this problem is of practical significance, which is our motivation for this paper. We summarize the main contributions of this thesis: (a) Considering the effect of time delay and uncertainty, which makes our model have better universality; (b) The design method of sliding mode controller is given, which has good robustness to resist parameter uncertainty.
(c) Some sufficient conditions for FTS of the SMC system in the finite time interval [0, T ] are given.
(d) The obtained results can describe the relationship between FTS and system parameters, and can be easily solved. Notations:The following notations will be used in this paper: R n and R n×m represent the n-dimensional Euclidean space and the set of n × m real matrices, respectively. Superscript 'T' denotes the transpose of matrix. λ min (·) and λ max (·) stand for the minimum and maximum eigenvalue of matrices. Asterisk( * ) means the term of symmetry. ∥ · ∥ denotes the Euclidean norm operator.

II. PROBLEM FORMULATION AND PRELIMINARIES
Consider the uncertain neutral system with time-delay as follows: where x(t) ∈ R n is the system state, u(t) ∈ R m is the control input, G ∈ R n×n , A 1 ∈ R n×n , A 2 ∈ R n×n , B ∈ R n×m are know constant matrices. h > 0 and d > 0 are time delay constants. τ takes the maximum value of h and d, f (x(t)) denotes non-linear known function, which satisfies the Lipchitz constraint ∥f (x(t))∥ ≤ β∥x(t)∥, here β is a positive scalar. ∆A 1 and ∆A 2 are uncertain matrices that satisfy the following condition: where, M, N 1 , N 2 are constant matrices of appropriate dimensions, F (t) is an unknown time-varying matrix that satisfies F (t) T F (t) ≤ I, ϕ(θ) ∈ R n×n is a continue initial function vector.
Firstly we give a necessary definition and some lemmas. Definition 1(FTS) For given c 1 , c 2 , T and R > 0, where Lemma 1 [37] For some real matrices of appropriate dimensions S 1 , S 2 , E(t), here, E T (t)E(t) ≤ I and a scalar υ , the following inequality holds: [38] For given c 1 , c 2 , T and R > 0, the system (1) is FTS about (c 1 , c 2 , T, R), if and only if there exists an auxiliary scalar satisfying c 1 < c * < c 2 such that the system is FTS about (c 1 , c * , τ * , R) during reaching phase and FTS about (c * , c 2 , T, R) during sliding motion phase, here, τ * is the time that the system state reaches the sliding surface.

III. MAIN RESULTS
We design the following integral sliding variable where H is chosen such that HB is nonsingular, which can be attained by H = B T L with L > 0 . K 1 , K 2 are controller gains to be designed later. The derivative of s(t) giveṡ

A. REACHABILITY ANALYSIS
In this subsection, an appropriate SMC law is designed, which can ensure the system state reaches the sliding surface in a finte time τ * and maintain sliding motion for all subsequent time.
Theorem 1 The state trajectories of uncertain neutral system (1) can reach the sliding surface in finite time [0, τ * ] with the SMC as . Here, ρ > 0, sign(·) stands for sign function.
Proof. Take the Lyapunov function as We get Thus .

C. FTS OVER SLIDING MOTION PHASE WITHIN [τ * , T ]
In this subsection, we will analyze FTB for the system (1) during the sliding motion phase. When the system state move along the sliding surface, the equivalent control u eq can be solved byṡ(t) = 0 .

IV. NUMERICAL SIMULATION
Using the method in Remark 3 to solve the LMIs in Theorem 3 and Theorem 5, we can obtain We note that the control law (6) contains a sign function sign(s(t)), which can cause severe chattering. High-frequency chattering will affect the actual control accuracy and may cause system oscillation to destroy controller components. Therefore, we replace the traditional sign function with a smooth approximate sigmoid function (35). Simulation results with the initial condition x(0) = 0.42 −0.7 0.48 T are shown in Fig.1 -Fig.6 . Fig.1 and  satisfies finite-time stability about (c1, c2, T, R) by the SMC law we designed. Fig.4 shows the curve of sliding variable s(t). Fig.5 depicts the evolution of control input u(t) with the switching function sign(s(t)), we can see that the control input begins to jitter at high frequency when the system state reaches the sliding surface and then remains in this state, which we do not want to see. Fig.6 depicts the evolution of control input u(t) with the switching function sigm(s(t)), compared to Fig.5, we can find that the addition of sigmoid function can effectively reduce the chattering phenomenon.

V. CONCLUSION
Based on sliding mode control, we have investigated the finitetime stability problem of a class of uncertain neutral systems. A suitable integral-type sliding surface and an SMC law are designed, which can ensure the system satisfies FTS in time interval given. The obtained conditions are given in the form of LMIs for easy solution, a numerical simulation shows the practicability of the proposed method. The neutral time-delay systems studied in this paper have better universality than general time-delay systems, such as wireless transmission system, standard time-delay system and so on, can be converted into this kind of neutral system, FTS can ensure the transient performance requirements of such systems. It is noteworthy that the results we obtained contain a large number of variables and parameters, which increases the computational complexity, so how to simplify the results and reduce the conservativeness, deserves further research. In the future, we will use adaptive sliding mode control to study uncertain neutral systems with mixed time delays. In addition, we shall extend the proposed method to Markovian jump systems, and investigate the asynchronous sliding mode control problem.