STABILIZATION ON INPUT TIME-VARYING DELAY FOR LINEAR SWITCHED SYSTEMS WITH TRUNCATED PREDICTOR CONTROL

. This study is concerned with the stabilization problem for input time-varying delay switched system under the truncated predictor con- trol scheme. The delay in the prediction feedback, is subjected by predicting the future trajectory of the states by system equations and initial condition- s, which is known as truncated prediction feedback (TPF). The TPF is used to construct the state feedback law for stabilizing the linear switched system. By constructing Lyapunov-Krasovskii functions and, the stability condition is derived to ensure the globally asymptotically stable of the state feedback stabi- lization at the origin. When switching system is unstable, truncated predictor control method and Hurwitz convex combination makes the system stable. Fi- nally, a numerical example and their simulation results are given to show the eﬀectiveness of the proposed approach.


1.
Introduction. In the past decades, switching system have been widely studied by many researchers, which are special class of the hybrid system and it is mixture of discrete and continuous system and due to their applications in various fields such as switching and tuning paradigm from adaptive control [21], switched mode power supplies [23] and also in the control of mechanical systems [22], the automotive system [6], aircraft control design [4], switching power converters see [17] and the references therein.
On the other hand, many researchers have been focused the stabilization issues of the switched system with delay under the suitable control. Also, several results are investigated such as, the stochastic optimal control problem studied in [25] for a delayed Markov regime-switching jump-diffusion model and applications to finance. The optimal switching control studied for horizontal (3-D) well's path problem in [10] and in [31] obtained the required gradient of the cost function by using optimum control for time delay switched systems. Further, [14] investigated the stability analysis for uncertain switched time delay systems by using the sliding modes, in [34,33] proved the robust stability and stabilization for uncertain impulsive switched systems by employing the delayed control and guaranteed cost control. In [32], the stabilization of the two dimensional switched system has been formulated by Fornasini-Marchesini for the local state-space model. In this stabilizing condition has been found using switched quadratic Lyapunov function of arbitrary switching signal. Then it has been developed to extend the average dwell time technique using the piecewise Lyapunov function for stabilizing two dimensional switching system except for some switching cases. In [37], they derived the stabilization condition for a single long time varying input delay linear system. The parametric Lyapunov equation method is used to stabilize the open loop system and also by truncating the predictor based control. Also the condition for closed loop system has been found. It tells about the semi-global stability condition by making system to magnitude saturation or energy constraints. In [26], stabilization is obtained by Hurwitz convex combination for the linear switching system with time varying delay. In this unknown constant acts as a delay bound of derivative. The stability of the delay bound system is obtained as a linear matrix inequality. In order to stabilize the switched system, the Hurwitz convex combination plays a major role not only in stability but also a smooth control Lyapunov function. The Hurwitz convex combination of the system matrices is characterized for the dynamic system which switch among a space model of family of positive state. For example, Bialas et.al., in [1] derived the sufficient and necessary condition for stability condition which is based on a Hurwitz convex combination in [3]. The conditions are given the simpler term by Stanislaw Bialas.
In delay differential equation, the time derivatives at the current time depend on the solution and possibly its derivatives at previous time. Time-delays appear in many engineering systems like water quality process [29], aircraft control [9] chemical control systems [28] and laser models [8] and references therein. Recently, the stabilization issue of time-delay switched system has been investigated based on Lyapunov Krasovskii functionals and LMI approach. Stability of time-delay has been traced back to the 18th century. In 1940s research on delay system has been begun in which frequency domain method was used. For explaining transfer function of the real world system was so difficult, so the time domain analysis has become famous. Recently so many results have been found in the time domain which was based on Lyapunov-Krasovskii stability theory, Razumikhin stability theorem, Halanay differential inequality, Barbatla lemma, comparison principle and so on. In [16] Lyapunov Krasovskii functionals was introduced, which is singular type complete quadratic Lyapunov Krasovskii functions with polynomial parameters. In [12,13] quadratic Lyapunov Krasovskii functionals and simple Lyapunov Krasovskii functionals has been developed for finding stability condition for the time-delay system. One of the efficient method for the delay system to be stabilizes is the Lyapunov method. Krasovskii method of Lyapunov functionals [2,20] and the Razumikhin method of Lyapunov functions [19] are the two important methods for the system involving time delay. In this we are using the Lyapunov Krasovskii method to stabilize the system, which cast the condition into linear matrix inequality (LMI). Asymptotic stability of a time-delay are guaranteed by the conditions on a Lyapunov-Krasovskii functional. Prediction feedback method has been paying more attention to the stabilization problem of input delay switched system. For practical implementation the predictor method brings difficulties [24,18,7,27]. To eliminate the infinite dimension feedback laws, a finite dimensional predictor method was introduced which is TPF [15,38,37]. For open loop system TPF has been developed but it is not exponentially unstable [38,15,39]. Later in [35], TPF for general open loop system including exponentially unstable has been proposed. In [30], they have consider the open loop system which has pure imaginary poles on the imaginary axis in which the general TPF for stabilization is not achieved for an arbitrarily large delay. But in closed-loop system asymptotic stability has been achieved. In [36], parametric Lyapunov equation has been used for stabilizing linear system with multiple delay in input. In which TPF law delay and TPF independent law were designed for the systems with open loop poles in the closed left half plane and system with open left half plane or at the origin. In [15,38] low gain feedback design has been used. In [38] by considering low gain feedback design used for the parametric Lyapunov equation and this is termed as TPF.
The motivations by the above discussion, there are two kinds of feedback controllers are studied for time delay systems, that is, finite dimensional controller (allow finite delay) and infinite dimensional controller (allow arbitrarily large delay). The TPF is designed either finite or infinite dimensional controller. In particular, the infinite dimensional TPF is very hard to apply the physical phenomenons. Therefore, the physical implementation necessity, we design the finite dimensional TPF for a linear switching system with time-varying input delay. The controller equation for the feedback gain K I simplifies to [39]. We find the less sufficient condition for asymptotic stabilization of linear switching system under the TPF can be established with the aid of Lyapunov-Krasovskii stability method and Hurwitz convex combination. Finally, numerical example shows that the effectiveness of the derived results. This paper is organized as: In section-II, it speaks about the preliminaries required for the stability analysis. In section-III we derive the stabilization results and Section-IV derived the numerical stimulation to validate the derived theorem.

2.
Preliminaries. Consider switching systems in the presence of a time-varying input delay, where x ∈ R n , y ∈ R n , is the state u ∈ R n is the input, σ(t) : [0, ∞) → M = 1, 2, .....m is the switching signal which is depending on time t or state x(t); ψ(t) : R → R is a continuously differentiable function with delay, A i ∈ R n×n , B i ∈ R n×p , C i ∈ R q×n are constant matrices. The function ψ(t) is given by where d(t) ≥ 0 is the time-varying delay. From (1), we understand that the predictor design is given by ψ(t).
Hurwitz convex combination condition: We should assume that there exists a Hurwitz linear convex combination F of Since F is Hurwitz there exists a matrix P > 0 such that F T P + P F = −Q, for a given positive definite matrix Q. To make particular quadratic form negative from the sub-region, which is obtain by dividing R n into N sub-regions. This act as one of the sufficient condition for Hurwitz stability of the convex combination of two matrices.
The switching region and law for the given P and Q is given by, for each i ∈ M where M = 1, 2, 3 · · · m.
The switching region is constructed by: Thus m j=1τ i = R n andτ i τ j = ∅. The switching law is given as σ(t) = i when x(t) ∈τ i . Asymptotical stability of the system x (t) = (A i + B i K i )x(t) is guarantee by the presence of Hurwitz convex combination under the switching law. Switching rule: The minimum rule is used to determine the next mode at each switching Remark 1. When x(t) ∈τ i from σ(t) = i, then the i th subsystem is active.
Assumption 1. The continous differentiable function ψ(t) which is also invertible such that for a finite number h ≥ 0 such that ∀t ≥ 0, The analytical solution of (1) can be computed as e Ai(t−s) B i u(ψ(s))ds.
[5]Let P be a positive definite matrix, then the identity holds where ω i ≥ 0 is a scalar and R = −A T i P − P A i + ω i P. Also, R is a positive definite then e A T i t P e Ait ≤ e ωit P.
3. Stabilization Results. For σ(t) = i, the control structure for the system (1) with Assumption 1 is given as where K i is a control gain matrix. Considering solution (8) and also by TPF control law (11) can be derived as where Theorem 3.1. The TPF control law globally asymptotically stabilizes the system at the origin by considering the switching system which satisfies the assumption 1, there exist constants ρ > 0 and ω i ≥ 0, P > 0, Q > 0 such that where γ = 2ρ 2 h 2 e 2ωih β −1 . Then the linear switching system is globally asymptotically stable.
Proof. Let the Lyapunov function is given by Along the trajectories, the time derivative for V 0 (x(t)) is given by by using lemma 2.2 and λ(t), we have, where the condition B i B T i ≤ ρP −1 and K = −B T i P has been used x T (ψ(s))e ωi(t−s) e ωi(s−ψ(s)) x(ψ(s))ds x T (ψ(s))x(ψ(s))ds.
Remark 2. The LMI conditions of Theorem 3.1 is give the guarantee to the globally asymptotic stability of switched system (12). Especially, the Theorem 3.1 is stabilizing the switched system (12), even if the sub-systems are stable, unstable or combination of both.

4.
Example. Consider the class of switching system with unstable subsystems where The linear system x (t) = A 1 x(t) and x (t) = A 2 x(t) are unstable see figure 1 and figure 2. K 1 and K 2 has been chosen as The convex combination F , which is designed using the switching law The switching region (4) for the mode1 and mode 2 with the given P and Q and R 2 =τ 1 τ 2 . Hence globally asymptotically stability of the linear switched system has been established. Further, we choose the initial condition x(t) = [0.3 0.1] T then we get the phase plot and stability behavior of system (16). The phase plot for the mode:1 and mode:    Remark 3. The main contribution of this work is given below: 1. We designed the finite dimensional TPF for a linear switching system with timevarying input delay. 2. We found less sufficient condition for asymptotic stabilization of linear switching system under the TPF can be established with the aid of Lyapunov-Krasovskii stability method and Hurwitz convex combination.

5.
Conclusion. In this paper, we investigated the truncated predictor control for linear switched systems with input time-varying delay and we obtained some sufficient condition for globally asymptotic stability in the method of LMI by using the Truncated Predictor control and Hurwitz convex combination. Finally, we showed   the numerical example for unstable sub-systems are globally asymptotically stabilized, it gave the clear idea to develop the truncated predictor control in linear switched systems with input time-varying delays. Moreover, the input time varying delay allow the uncertainty, the emerging real-world necessity we will develop the TPF for uncertain linear and nonlinear switched system in future.