Time-Varying Delayed H ∞ Control Problem for Nonlinear Systems: A Finite Time Study Using Quadratic Convex Approach

: In this manuscript, we consider the ﬁnite-time H ∞ control for nonlinear systems with time-varying delay. With the assistance of a novel Lyapunov-Krasovskii functional which includes some integral terms, a matrix-based on quadratic convex approach, combined with Wirtinger inequalities and some useful integral inequalities, a sufﬁcient condition of ﬁnite-time boundedness is established. A novel feature presents in this paper is that the restriction which is necessary for the upper bound derivative is not restricted to less than 1. Further a H ∞ controller is designed via memoryless state feedback control and a new sufﬁcient conditions for the existence of ﬁnite-time H ∞ state feedback for the system are given in terms of linear matrix inequalities (LMIs). At the end, some numerical examples with simulations are given to illustrate the effectiveness of the obtained result.


Introduction
The occurrence of time delays is an important fact in many of the networking and processing control systems. Such delays can have the capacity to destabilize the control systems and also make some crucial disintegration in the performance of the closed-loop systems, see the references cited therein [1][2][3][4][5][6][7]. While modeling a real control system, the existence of time delays is always taken to be a time-varying one that satisfies the condition d 1 ≤ d(t) ≤ d 2 and d 1 which is not necessarily restricted to be 0. In recent years, the study on finite-time stability (FTS) has increased the research interest from various researches around the world due to the wider applications in mathematical control theory, which has been studied by different approaches in various kinds of systems, see for instance [8][9][10][11][12][13][14][15]. To this extent, the author Dorato in [8], explained the fundamental concepts of stability theory of dynamical systems in finite-time sense. Generally, the given system leads to be finite-time stable if the considered state of the system should be within the bounded limit for a fixed interval of time. From this, one can observe that the concept of finite-time stability mainly attracts the boundedness of a system during a fixed interval of time period. Some of the exciting results for finite-time stability and stabilization with the existence of time-delay have been obtained in [8][9][10][11][12]. Moreover, in some practical I. We consider some new Lyapunov-Krasovskii functional which has not been considered yet in stability analysis of finite-time H ∞ control. The new Lyapunov-Krasovskii functional includes some integral terms of the form t t−h (h − t − s) jẋT (s)R jẋ (s)ds (j = 1, 2) which the integrands are polynomial multiplied byẋ T (s)R jẋ (s)ds (j = 1, 2) and one may estimate an upper bound of the integral by employing some techniques from [22,25], the matrix based quadratic convex approach, the use of a tighter bounding technique and useful integral inequality such as Wirtinger inequality. II. Lyapunov-Krasovskii with the matrix based quadratic convex approach is introduced to formulate finite-time stability criteria and H ∞ performance level where the time-varying delay satisfies 0 ≤ d 1 ≤ d(t) ≤ d 2 , µ 1 ≤ḋ(t) ≤ µ 2 . Moreover, the restriction of upper bound derivative is not necessary restricted less than 1 compared with [20] III. Two numerical examples are given to demonstrate the effectiveness of theoretical result.

Problem statement
In this section, we consider a system with time-varying delay and control input aṡ where x(t) ∈ R n is the state; u(t) ∈ R m is the control input, w(t) ∈ L 2 ([0, ∞], R r ) is a disturbance input and z(t) ∈ R s is the observation output. The delay d(t) is time-varying continuous function which satisfies In this paper, we consider the nonlinear functions satisfying → R n are nonlinear function satisfying the Lipschitz conditions; namely, there exist positive constants β 1 , β 2 , ∀x, y ∈ R n , such that We assume the following restrictions on the nonlinear perturbations The initial condition, φ(. Under the above assumptions on d(.), f (.), g(.) and the initial function φ(t), the system (2) has a unique solution x(t, φ) on [0, T]. For a prescribed scalar γ > 0, we define the performance index as The objective of this paper is to design a memoryless state feedback controller u(t) = Kx(t).

Preliminaries
The following definition and lemma are necessary in the proof of the main results: Definition 1. [9] The nonlinear system (2) where w(t) is a perturbation satisfying (4). The system (2) is said to be finite-time bounded with respect to (c 1 , c 2 , T, R, d) with 0 < c 1 < c 2 , and R > 0, if Definition 2. [9] The nonlinear system (2) is said to be finite-time H ∞ bounded with respect to (c 1 , c 2 , T, R, d, γ) with 0 < c 1 < c 2 , d ≥ 0, γ > 0, R > 0 and a memoryless state feedback controller u(t) = Kx(t), following conditions should be satisfied: (i) The zero solution of the closed-loop system, where w(t) = 0, is finite-time bounded.
We introduce the following technical lemmas, which will be used in the proof of our results. Proposition 1. [11] Let P ∈ M n×n , R ∈ M n×n be symmetric positive definite matrices. We have Lemma 1. [22] For a given matrix R > 0, the following inequality holds for any continuously differentiable a ω(u)du. Remark 1. : From the above inequality, it can be observed that the inequality in Lemma 1 gives a firm lower bound for b aω T (u)Rω(u)du than Jensen's inequality since 3Γ T 2 RΓ 2 > 0 for Γ 2 = 0. Hence it shows that the inequality (9) is improved than the Jensen's inequality.
Before we introduce some useful integral inequalities, we denote [25] For a given scalar d 1 ≥ 0 and any n × n real matrices Y 1 > 0 and Y 2 > 0 and a vectoṙ y : [−d 1 , 0] → R n such that the integration concerned below is well defined, the following inequality holds for any vector-valued function π 1 (t) : [0, ∞) → R k and matrices M 1 ∈ R k×k and N 1 ∈ R k×n satisfying , where ν 3 (t) is defined in (10).
For any n × n real matrix R 2 > 0 and a vectorẏ : [−d 2 , 0] → R n such that the integration concerned below is well defined, the following inequality holds for any φ i1 ∈ R q and real matrices Z i ∈ R q×q , F i ∈ R q×n satisfying (10).

Lemma 4. [25] Let d(t) be a continuous function satisfying
For any n × n real matrix R 1 > 0 and a vectorẏ : [−d 2 , 0] → R n such that the integration concerned below is well defined, the following inequality holds for any 2n × 2n real matrix S 1 satisfying whereR 1 := diag{R 1 , 3R 1 }; and

Remark 2.
If we substitute d 1 = 0, in Lemma 4, then the inequality can be reduced and it is similar to that of the one in [22]. Also, the dimensions of the slack matrix variables of S 1 is 2n × 2n compared to the dimension 2n × 5n introduced in [23].

Main Results
In this section, we firstly design a memoryless H ∞ feedback control for the addressed system (2) with the inclusion of time-varying delays and then obtain the finite-time stabilizability analysis conditions. Here we derive a novel finite-time stability for the system (2) by using the matrix-based quadratic convex approach with some integral inequalities in [25]. To achieve this status, we choose the following Lyapunov-Krasovskii functional: and P are real matrices to be determined. Before introducing the main result, the following notations of several matrix variables are defined for simplicity: with e i ∈ R n×15n (i = 1, 2, . . . , 15) denoting the i-th row-block vector of the 14n × 14n identity matrix W 1 = diag{W 1 , 3W 1 };and and satisfies z(t) 2 < γ w(t) 2 for all nonzero w ∈ L 2 [0, ∞) if there exist positive definite matrices P, Q j > 0, (j = 0, 1, 2, 3), W 1 , W 2 , R 1 , R 2 , S 1 , Z 1 , Z 2 , Z 3 , N 1 , N 2 , N 3 and Y such that the following linear matrix inequalities (LMIs) hold and For this problem, the feedback control is taken to be of Proof. By finding the time-derivative of V for the considered system (2), we obtaiṅ From (2) and Cauchy inequality, we get the following equality: we obtain the following From (17) and (20), we haveV 1 iṡ (P)e 9 + 2e T 9 (AP + 4BY)e 1 + 2e T 9 DPe 2 + 2e T 9 CPe 12 + 2e 9 Pe 10 + 2e 9 Pe 11 .

Numerical Examples
In this section, we provide two numerical examples with their simulations to demonstrate the effectiveness of our results. Example 1. Consider the nonlinear system with interval time-varying delays which was considered in [7] x We have used theorem 1 to evaluated the value of minimum γ for H ∞ control condition. Where And the condition (15) is satisfied with α = 0.6, T = 10, c 1 = 1, c 1 = 50γ = 4. By using LMI Toolbox in Matlab, it can be shown that the constructed LMI in Theorem (1) is feasible. Further the H ∞ controller feedback gain matrix is obtained as: 1846 . Table 1, shows the value of minimum γ with µ 1 = −0.1 and µ 2 = 0.1 by using Theorem (1). In Table 2, we show the value of minimum γ with µ 1 = 0.05 and µ 2 = 0.1 by using Theorem (1)      x T (t)x(t) Figure 1. Trajectories of x T (t)x(t) in Example 2, the unit of T which is second.

Conclusions
In this paper, finite-time H ∞ control for nonlinear systems with time-varying delay is studied. By using a set of improved Lyapunov-Krasovskii functional including with some integral terms, a matrix-based on quadratic convex, combined with Wirtinger inequalities and some useful integral inequalities were proposed which illustrate the effectiveness of the obtained result in the numerical part. However, the improved method for the restriction on the upper bound of the delay derivative should be considered which means that a fast time-varying delay is allowed without any requirement on the derivative. New sufficient conditions of finite-time boundedness for above-mentioned class of system were given in term of linear matrix inequalities (LMIs).