Further Results Concerning Delay-Dependent H ∞ Control for Uncertain Discrete-Time Systems with Time-Varying Delay

This paper addresses the problem of H∞ control for uncertain discrete-time systems with time-varying delays. The system under consideration is subject to time-varying norm-bounded parameter uncertainties in both the state and controlled output. Attention is focused on the design of a memoryless state feedback controller, which guarantees that the resulting closed-loop system is asymptotically stable and reduces the effect of the disturbance input on the controlled output to a prescribed level irrespective of all the admissible uncertainties. By introducing some slack matrix variables, new delay-dependent conditions are presented in terms of linear matrix inequalities LMIs . Numerical examples are provided to show the reduced conservatism and lower computational burden than the previous results.


Introduction
During the past decades, considerable attention has been paid to the problems of stability analysis and control synthesis of time-delay systems.Many methodologies have been proposed and a large number of results have been established see, e.g., 1-4 and the references therein .All these results can be generally divided into two categories: delayindependent stability conditions 5, 6 and delay-dependent stability conditions 7-12 .The delay-independent stability condition does not take the delay size into consideration, and thus is often conservative especially for systems with small delays, while the delaydependent stability condition makes fully use of the delay information and is usually less conservative than the delay-independent one.

Notations
Throughout this paper, R n represents the n-dimensional Euclidean space; R m×n is the set of all m × n real matrices.For real symmetric matrices X and Y , the notation X ≥ Y resp., X > Y means that the matrix X − Y is positive semidefinite resp., positive definite .The superscript "T " denotes the transpose.I is an identity matrix with appropriate dimension.Z denotes the set of {0, 1, 2, . ..}. L 2 refers to the space of square summable infinite vector sequences.In symmetric block matrices, we use an asterisk " * " to represent a term that is induced by symmetry.Matrices, if not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

Problem Formulation
Consider the following uncertain discrete-time systems with time-varying delay 20 : where x k ∈ R n , u k ∈ R m , and z k ∈ R p are the state, control input, and controlled output, respectively; ω k ∈ R q is the exogenous disturbance input, which belongs to L 2 .φ k is the initial condition; A 0 , A 1 , B 1 , B 2 , C 0 , C 1 , D 11 , and D 12 are known real constant matrices.The time-varying parameter uncertainties are norm-bounded and meet with where F k is an unknown real time-varying matrix and satisfies the following bound condition:

2.5
Applying this controller to system Σ results in the following closed-loop system: where The robust H ∞ control problem to be addressed in this paper can be formulated as developing a state feedback controller in the form of 2.5 such that 1 the closed-loop system Σ cl is robustly asymptotically stable when ω k 0, for all k ≥ 0; 2 the H ∞ performance z 2 < γ ω 2 is guaranteed for all nonzero ω k ∈ L 2 and a prescribed γ > 0 under the zero-initial condition, for all admissible uncertainties and time-varying delays satisfying 2.2 -2.4 .
At the end of this section, let us introduce some important lemmas which will be used in the sequel.Lemma 2.2 Schur complement 22 .Given constant matrices M, L, Q of appropriate dimensions, where M and Q are symmetric, then Lemma 2.3 see 10 .Let D, E, and F be matrices with appropriate dimensions.Suppose F T F ≤ I, then for any scalar μ > 0, there holds 2.9

Main Results
In this section, some delay-dependent LMI-based conditions will be developed to solve the robust H ∞ control problem formulated in the previous section.First, we will consider the nominal system of system Σ cl with F k 0, for all k > 0, that is, where Theorem 3.1.System Σ ncl is asymptotically stable with a prescribed H ∞ disturbance attenuation level γ > 0, if there exist matrices P > 0, Q > 0, R > 0, W, and Y of appropriate dimensions such that Then, it is easy to see that Now, choose a Lyapunov-Krasovskii functional candidate for the time-delay system Σ ncl as y T Ry .

3.6
Taking the forward difference, we have

3.7
For any two matrices of appropriate dimensions Y and W, there holds

3.8
Substituting 3.4 and the previous equality into 3.7 gives Similar to 17 , we have

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After some manipulations, we obtain

3.12
This together with 3.11 gives Then, from 3.7 -3.13 , we have

3.16
In the next, we will prove the conclusion from two aspects.First, we establish the asymptotic stability of system Σ ncl with ω k 0 if 3.2 is satisfied.For this situation, 3.14 becomes where

3.18
By Lemma 2.2, it can be verified that ΔV k < 0 if 3.2 is true.Therefore, system Σ ncl with ω k 0 is asymptotically stable according to the Lyapunov-Krasovskii stability theorem.

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Second, we show that subject to the zero initial condition, the discrete time-delay system Σ ncl has a prescribed H ∞ disturbance attenuation level γ > 0, that is, z 2 < γ ω 2 for all nonzero ω ∈ L 2 .To this end, we introduce the following performance index: where the scalar N ∈ N. Noting the zero initial condition and 3.14 , one can verify that where

3.22
Now, by Lemma 2.2, it follows from 3.2 that Ψ 3 d k < 0, which together with 3.20 ensures that J N < 0. This further implies that z 2 < γ ω 2 holds under the zero initial condition.This complements the proof.
Remark 3.2.In Theorem 3.1, two slack variables Y and W are introduced to reduce some conservatism in the existing delay-dependent conditions for the H ∞ control problem, while no bounding techniques for cross terms are involved.By doing so, we have provided a more flexible condition in 3.2 .The advantage of these introduced variables can be seen from the numerical example later.
Remark 3.3.In 19 , based on a descriptor system transformation method, a delay-dependent condition on the H ∞ control issue for system Σ ncl was proposed.However, there is an additional constraint on the matrix A 1 , that is, A 1 should be nonsingular.While, Theorem 3.1 in this paper gets rid of this constraint.
Very recently, for discrete time-delay system Σ ncl , a less conservative delaydependent H ∞ condition was proposed in 20 .The rationale behind the method lies in providing a finite sum inequality as follows.

3.25
By Theorem 3.1, we can obtain the following delay-dependent H ∞ disturbance attenuation condition, which has been reported in 20 recently.
Corollary 3.5 see 20, Proposition 1 .For a given γ > 0, system Σ ncl is asymptotically stable with a prescribed H ∞ disturbance attenuation level γ for any time-varying delay satisfying 2.4 if there exist matrices 1, 2, 3 with appropriate dimensions such that 3.24 and the following inequality hold: where

3.27
In the following, we will show that the result in 20 can be deduced from Theorem 3.1.
Proof.It is easy to see that 3.26 is equivalent to By 3.24 and using the Schur complement formula, we have This together with 3.28 implies where
Remark 3.6.From Corollary 3.5, it is noted that Theorem 3.1 in this paper is less conservative than Corollary 1 which was reported in 20 .It should be pointed out that neither model transformation e.g., 23 nor bounding technique e.g., 19 is employed here.Although it is proved that the finite sum inequality approach in 20 is better than other reported ones when dealing with delay-dependent stability analysis problem for discrete time-delay systems, it still gives relatively conservative results.
Remark 3.7.Compared with the delay-dependent H ∞ disturbance attenuation condition in 20 , it is worth noting that one of the advantages in our paper is that the inequality in 3.2 involves significantly fewer variables than those in 20 .Specifically, in the case when x k ∈ R n , the number of the variables to be solved in 3.2 is n 7n 3 /2, while in 20 the number of variables is 13n 2 2q 2 6nq 3n /2.When q n, that is, ω k ∈ R n , the number of variables in 20 becomes 21n n 3 /2, which is around 3 times more than those in Theorem 3.1.Therefore, from mathematical and practical points of view, our condition is more desirable than that in 20 .Now, we are in a position to solve the controller gain K from 3.2 .
Define Π 1 diag{I, I, I, I, I, R −1 , I}. Multiplying 3.2 by Π T 1 and Π 1 on the left-hand side and the right-hand side, respectively, yields

3.36
Let P −1 X. Defining Π 2 diag{X, X, X, I, X, I, I}, then after performing congruence transformations on 3.36 by Π 2 , we have where

3.39
It is clear that 3.38 is a nonlinear matrix inequality in the matrix variables X, Q, R, Y , W, and K, due to the existence of the nonlinear term −hX R −1 X.In order to solve the desired controller K, we will propose three methods in the sequel.
Let R P, that is, take a particular Lyapunov-Krasovskii functional in 3.5 .Then, the following result holds naturally.Theorem 3.8.System Σ ncl is asymptotically stable with a prescribed H ∞ disturbance attenuation level γ > 0, if there exist matrices X > 0, Q > 0, Y , W, and K of appropriate dimensions such that the following LMI holds: Moreover, a robustly stabilizing state feedback controller is given by 2.5 with K KX −1 .
Remark 3.9.Theorem 3.8 provides a simple method in solving the controller gain K by introducing a special Lyapunov-Krasovskii functional.Although it has some good merits, it may bring some conservatism due to the restriction of R P .
From 3.38 and 3.42 , the following theorem follows immediately.
Theorem 3.10.System Σ ncl is asymptotically stable with a prescribed H ∞ disturbance attenuation level γ > 0, if there exist matrices X > 0, Q > 0, R > 0, Y , W, and K of appropriate dimensions such that the following LMI holds: Moreover, a robustly stabilizing state feedback controller is given by 2.5 with K KX −1 .
Remark 3.11.It is clear that there also exists conservatism because of the replacement In the sequel, we will resort to the cone complementary linearization method 24 to further reduce the conservatism.Introduce a new matrix variable S > 0, which satisfies X R −1 X ≥ S.

3.44
It is easily seen that inequality 3.44 is more general than that in 3.42 .Note that 3.44 is equivalent to , we obtain the following theorem.

3.48
Moreover, a robustly stabilizing state feedback controller is given by 2.5 with K KX −1 .
Remark 3.13.As one can see that the inequality conditions in Theorem 3.12 are not strict LMI conditions due to the equation constraints in 3.48 .However, by resorting to the cone complementary linearization method in 24 and the optimization solver in 25 , the nonconvex feasibility problem formulated by 3.46 , 3.47 , and 3.48 can be transformed into the following nonlinear minimization problem subject to LMIs: minimize Tr SS XX R R subject to 3.46 and 3.47 ,

3.49
According to the cone complementarity problem CCP in 24 , if the solution of the above minimization problem is 6n, we can say from Theorem 3.12 that system Σ ncl is asymptotically stable with a prescribed H ∞ disturbance attenuation level γ > 0 via the controller 2.5 with K KX −1 .Although it is very difficult to always find the global optimal solution, the proposed nonlinear minimizatiion problem is easier to solve than the original nonconvex feasibility problem.Based on the linearization method in 24 , we can solve the above nonlinear minimization problem using an iterative algorithm presented in the following.Algorithm 3.14.We have the following steps.

Set
Step 4. If matrix 3.46 is satisfied and for some sufficient small scalar δ > 0, then decrease γ ini to some extent and set γ s0 γ ini and go to Step 2. If one of the conditions in 3.47 and 3.51 is not satisfied within a specified number of iterations, then exit.Otherwise, set k k 1 and go to Step 3. Now, we are in a position to present the delay-dependent robust conditions concerning H ∞ control of system Σ with uncertainties based on Theorems 3.8, 3.10, and 3.12, respectively.By Lemma 2.3, we can easily have the following results.Theorem 3.15.System Σ is asymptotically stable with a prescribed H ∞ disturbance attenuation level γ > 0, if there exist a scalar ε > 0, matrices X > 0, Q > 0, Y , W, and K of appropriate dimensions such that the following LMI holds: where Ξ 1 is defined in 3.40 , and Moreover, a robustly stabilizing state feedback controller is given by 2.5 with K KX −1 .
Theorem 3.16.System Σ is asymptotically stable with a prescribed H ∞ disturbance attenuation level γ > 0, if there exist a scalar ε > 0, matrices X > 0, Q > 0, R > 0, Y , W, and K of appropriate dimensions such that the following LMI holds: where Ξ 2 , D, E are defined in 3.43 , 3.53 , and 3.54 , respectively.A robustly stabilizing state feedback controller is given by 2.5 with K KX −1 .

Examples
In this section, two examples are used to demonstrate the effectiveness of the proposed methods.
Example 4.1.Consider the following discrete-time systems with time-varying delay: where and d k is a delay satisfying 2.4 .In the following, different cases of d k are involved.
Case 1. Delay d k is time invariant.First, suppose h h 64.For this situation, we will compare the results in Theorems 3.15, 3.16, and 3.17.For this reason, we calculate the minimum value of γ for which system 4.1 is robustly stabilizable via state feedback 2.5 .The obtained results are listed in Table 1, from which we can see that the conditions in Theorem 3.17 are less conservative than those in Theorems 3.15 and 3.16.Therefore, we will only compare the results in Theorem 3.17 with those in the previous literatures in the sequel.
Second, when h h 64, Zhang and Han 20 also calculated the achieved minimum H ∞ performances γ, the corresponding controller gain K, and the iterations.Here, in order to show much less conservative results or lower computational burden of Theorem 3.17 than 20 , we give Table 2. Noting from this table, we conclude that in order to achieve the same disturbance attenuation level, Theorem 3.17 needs significantly less iterations and smaller gain.From this table, we also have verified Remark 3.9.
Third, in 19, 20 , the authors also calculated the achieved minimum H ∞ performance for h 64, respectively.However, according to Theorem 3.17, much less H ∞ performance is obtained, which is listed in Table 3  Under this case, Fridman and Shaked 19 concluded that system 4.1 can be stabilized for all h ≤ 43.In 20 , it is obtained that system 4.1 is robustly stabilizable for h ≤ 48.However, by Theorem 3.17, we have that system 4.1 is robustly stabilizable for h ≤ 52.The details are shown in Table 5.
In 20 , Zhang and Han also gave the minimum H ∞ performances γ and corresponding controller gain K for a set of h when h 48.Here, we also present a table to demonstrate the lower computational complexity and smaller controller gains than those in 20 , which are listed in Table 6.

4.3
From Table 7, we can see that the condition in our paper can obtain a smaller H ∞ performance γ than 18, 27 for this example.

Conclusions
The problem of H ∞ control for uncertain discrete-time systems with time-varying delay has been studied.By introducing slack matrix variables, delay-dependent LMI based conditions have been developed to design a stable state feedback controller, which ensures the asymptotic stability of the resulting closed-loop system and guarantees a prescribed disturbance attenuation level irrespective of all the admissible uncertainties.Numerical examples have been provided to demonstrate the effectiveness and applicability of the proposed approach.

Example 4 . 2 .
Consider the discrete-time systems 4.1 this example was first presented in 27 and E 1, 2, 3, 4 are known constant matrices of appropriate dimensions describing how the uncertainty F k enters the nominal matrices of system Σ .dkdenotes the time-varying delay satisfyingh ≤ d k ≤ h, ∀k ∈ Z , 2.4where h and h are positive integer numbers.

Table 1 :
The achieved minimum H ∞ performances γ in this paper and corresponding controller gain K for h h 64.

Table 2 :
The iterations and corresponding controller gain K for h h 64 under different cases of γ.

Table 4 .
. From this table, one can see that Theorem 3.17 in this paper provides much less H ∞ performances.Now, we are in a position to calculate the maximum delay bound h, which can guarantee that system 4.1 is robustly stabilizable via state feedback 2.5 .The details are given in Case 2. Delay d k is time varying.

Table 3 :
The achieved minimum H ∞ performances γ and corresponding controller gain K for h h 64.

Table 4 :
The maximum delay bound h for system 4.1 under Case 1.

Table 5 :
The maximum delay bound h for system 4.1 under Case 2.

Table 6 :
The corresponding controller gain K for different h and γ when h 48.

Table 7 :
The achieved minimum H ∞ performances γ and corresponding controller gain K.