Controller Design for Uncertain Neutral System with Mixed Time-Varying Delays

and Applied Analysis 3 Lemma 4 (see [17]). For any real positive scalars α, β (where α > β) and a positive definite symmetric matrix S, then the following inequality holds for a vector function ω : [β, α] → R which can let the integrals converge:


Introduction
Dynamical systems with time delays and uncertain parameters have been of considerable interest over the past decades.In fact, time delays are always the important source of system instability and poor performance [1][2][3][4].As a special class of time-delay systems, the neutral type time-delayed system has also received some attention in recent years.This timedelayed system contains time delays both in its state and in the derivative of its states.Moreover, neutral time-delayed systems are frequently encountered in many dynamics, such as automatic control, distributed network system containing lossless transmission line, heat exchangers, and population ecology.Various analysis approaches have been utilized to find stability criteria and control design conditions for asymptotic stability of neutral time delays [5][6][7][8][9][10].
It is now worth pointing out that the control performances mentioned above concern the desired behavior of control dynamics over an infinite-time interval and it always deals with the asymptotic property of system trajectories.For controlling a dynamical system, it can meet the requirements of asymptotic stability, but it will not reflect the transient characteristics.Asymptotic stability is unable to satisfy the transient requirements of industrial production if there exists large amount of overshoot, oscillation change, and nonlinear disturbance within a finite-time interval.To deal with this transient performance of control dynamics, Dorato gave the concept of finite-time stability [11] (or short-time stability) in the early 1960s.Then, the relevant concepts of finite-time bounded (FTB) [12], finite-time stabilization [13], finite-time  ∞ control [14], and finite-time  2 - ∞ [15] control have been revisited in form of linear matrix inequalities (LMIs) techniques.And this transient performance is widely applied to time-delay systems, uncertain systems, nonlinear systems, stochastic systems, and so forth.However, to the best of our knowledge, very few results in the literature consider the related control problems of neutral time-varying delays in the finite-time interval.
On the other hand, the  2 - ∞ performance has attracted considerable attention as an important performance evaluation index when it was first proposed in 1989 [16].In engineering practice, although the study of the impact of noise and delay on the system performance is important, the extremum problem of the controlled output cannot be ignored, because the controlled output should be controlled within a certain range.In control theory and engineering application, the  2 - ∞ control has very important significance that lies in its performance index which can control the output value minimization.Unfortunately, up to now, the theme of  2 - ∞ control design of uncertain neutral systems with timevarying delays has received little attention.
Motivated by the above discussion, this paper focuses on the problem of finite-time  2 - ∞ controller design for a class where x() ∈ R  is the state, u() ∈ R  is the controlled input, y() ∈ R  is the controlled output, and w() ∈ R  is the disturbance input that belongs to  2 [0, +∞) and for a given positive number  and constant time , the following form is satisfied: ℎ() and () are time-varying delays and satisfy where ℎ, , ℎ  , and   are constant scalars.() ∈  2 [− max{ℎ, }, 0] is the continuous initial function.A, A  , C, D, and F ∈ R × are known constant matrices, and ΔA(), ΔA  (), ΔC(), ΔD(), and ΔF() are unknown timevariant matrices representing the norm-bounded parameter uncertainties and satisfy the following form: where M 1 , M In this paper, we consider the state feedback controller as follows: where K is the unknown controller gain and ΔK() is the time-varying controller gain which satisfies Then, we can get the following closed-loop control system: The main purpose of this paper is to design an appropriate resilient state feedback controller (7), such that the closedloop control system Σ is finite-time bounded and satisfies the given performance index constraints.
Before proceeding with the study, we give the relevant definitions and lemmas first.Definition 1.For given positive scalars  1 , , and  and a symmetrical positive determined matrix R, the closed-loop system Σ is robust finite-time bounded (FTB) with respect to ( 1 ,  2 , , R, ), if there exists a positive constant  2 with  2 >  1 , such that, for all the external disturbances w() satisfying condition (2), the following formula is satisfied: Remark 2. If the disturbance input is not present in the closed-loop system, that is, w() = 0, the concept of FTB will reduce into finite-time stability (FTS).It is worth mentioning that Lyapunov stability and finite-time stability are two different concepts.The former is largely known to the control characteristic in infinite-time interval, but the latter concerns the boundedness analysis of the controlled states within a finite-time interval.Obviously, a finite-time stable system may not be Lyapunov stochastically stable and vice versa.
Definition 3. The state feedback controller in the form of ( 7) is considered as a robust finite-time  2 - ∞ controller for the closed-loop system Σ, if the system Σ is FTB with respect to ( 1 ,  2 , , R, ) and under the zero initial condition, there exist two positive scalars  and  for all disturbance which satisfy condition (2), such that where ‖y()‖ Lemma 4 (see [17]).For any real positive scalars ,  (where  > ) and a positive definite symmetric matrix S, then the following inequality holds for a vector function  : [, ] → R  which can let the integrals converge: Lemma 5 (see [17]).For any positive scalar ℎ and positive definite symmetric matrix S, the following inequality is satisfied: Lemma 6 (see [15]).For any given appropriate dimension matrix H and E, if there exists a matrix W() which satisfied W  ()W() ≤ I and a scalar  > 0, then

Main Results
In this section, our main purpose is to solve the design problem of a resilient robust finite-time  2 - ∞ controller for a class of uncertain neutral systems with mixed time-varying delays.
Theorem 7. Given positive scalars  1 , , , and , positive definite symmetric matrix R, and time-delay parameters ℎ > 0, ℎ  > 0,  > 0, and   > 0, the closed-loop system Σ is FTB with respect to ( where Proof.Construct a positive definite Lyapunov function as follows: where We take the time derivative of () along the trajectory of system Σ and it yields the following: For any symmetric positive definite matrices W  ,  = 1, 2, . . ., 6, the following equations are satisfied according to Leibniz-Newton lemma: where According to (20)-( 21), we can obtain Since Q 1 , Q 2 , Q 3 , and Q 4 are positive definite symmetric matrices, we have where   6 , Recalling formula (24) and Lemmas 4 and 5 and using Schur complement, we can get Then, formula (27) can be written as which can be guaranteed by condition (16).This completes the proof.

Conclusions
This paper studied the delay-dependent resilient robust finite-time  2 - ∞ control problem for a class of uncertain neutral time-delayed system with mixed time-varying delays.
A state feedback controller is designed by using LMI technique and free weighting matrices, such that the closed-loop  (t)  x 1 (t) x T (t)Rx(t) controlled system is FTB and satisfies the input-output  2 - ∞ performance matrices.The simulation results verify the effectiveness of the design method.We will consider the finite-time observer for neutral time-delayed system in the future.