Reachable Set and Robust Mixed Performance of Uncertain Discrete Systems with Interval Time-Varying Delay and Linear Fractional Perturbations

: In this paper, the mixed performance and reachable set of uncertain discrete systems with slow variation interval time-varying delay are considered. The original uncertain discrete systems with interval time-varying delay are ﬁrst transformed into a switched system. Then, the proposed improved results are used to guarantee the stability and reachable set of the uncertain system with slow variation interval time-varying delay. The mixed performance ( H 2 / H ∞ ) can be derived in the same formulation simultaneously. The design scheme of robust switched control is also developed in this paper. The gains of the controller can be designed and switched to achieve stabilization and mixed performance of the system according to the delay value. Some comparisons with published results are made to show the main contribution of the proposed approach. Finally, some numerical examples are illustrated to show the main results.


Introduction
Time delay is confronted in many practical systems, such as chemical engineering systems, hydraulic systems, inferred grinding models, neural networks, population dynamic models, and rolling mills. Instability or bad performance may cause the appearance of time delay in many control systems [1][2][3]. It is also interesting to note that interval time-varying delay is a more suitable type that describes the physical environment for signal transmission [4][5][6][7]. Hence, we consider herein the uncertain discrete system with interval time-varying delay.
In recent years, many performance issues of switched systems have been considered. The dynamics of switched systems comprise a family of subsystems and use a switching signal to handle the switching between the subsystems. Switched systems are encountered in many practical models, such as automotive engine control, chemical processes, constrained robotics, multi-rate control, power systems and power electronics, robot manufacture, stepper motors, and water quality control [8,9]. It is also well known that many complicated nonlinear dynamical system behaviors, such as multiple limit cycles and chaos, may be caused by switching among subsystems [8,9]. Hence, the stability and performance of switched systems have been investigated in recent years [10][11][12][13][14][15]. It is also interesting to note that the stability of a switched system can be guaranteed by selecting an appropriate switching signal, even when each subsystem is unstable [13,14]. Hence, switching control can be designed by using the switching signal.
The reachable set of a control system is used to show the set containing all states of the system reachable from the origin for a bounded peak disturbance [4,6,10,12,[16][17][18]. In [4], the reachable set of systems of a switched system was investigated by a multiple Lyapunov strategy. In [9], the dwell time approach was applied to estimate the reachable set of the switched system under consideration. In [6], the reachable set of a discrete system with interval time-varying delay was estimated by the Lyapunov method. In [12], the switched system approach in [4] was applied to estimate the reachable set of the system under consideration in [6]. The switched system approach in [4] was illustrated to be less conservative by the proposed numerical examples.
The H ∞ performance of systems has been used in recent years to inspect the effect of regulated output with respect to disturbance input and guarantee that the closed-loop system is stable [19][20][21]. The H 2 performance of systems is another requirement that has been applied to minimize a quadratic performance index about the initial state of a system under no disturbance input [22]. Hence, the H 2 /H ∞ mixed performance has been an interesting research topic in recent years [23][24][25]. In this paper, the reachable set and mixed H 2 /H ∞ performance are considered. Minimization of the H ∞ performance with the H 2 norm constraint is developed. Moreover, it is well known that linear fractional perturbation is a more general representation than norm bounded uncertainties of the systems under consideration [13,19,21,[25][26][27]. To the best of the authors' knowledge, there are few results considering the reachable set and mixed performance of an uncertain discrete system with interval time-varying delay and linear fractional perturbations. In this paper, the LMI optimization approach in [28] is used to minimize H ∞ performance under some H 2 norm constraints. The main contributions of this paper can be highlighted as follows: (1) Reachable set estimation and mixed H 2 /H ∞ performance for an uncertain discrete system with interval time-varying delay and linear fractional perturbations are considered in this paper. (2) A new improved analytic result is proposed based on the approach developed in [4].
Less conservative results for an uncertain discrete system with slow variation interval time-varying delay are provided for more accurate estimation of the reachable set. The H 2 /H ∞ performance can also be guaranteed from the design scheme. (3) The LMI optimization approach is used to guarantee the minimization of the reachable set and achievement of mixed performance of the system under consideration. The proposed conditions can be solved easily by the Matlab LMI toolbox.
The remainder of this paper is organized as follows. The problem formulation is given in Section 2. The main results are given in Section 3. Section 4 provides some numerical examples to illustrate the main results. Finally, conclusions are drawn in Section 5.
Notation: For a matrix A, we denote the transpose by A T and symmetric positive (negative) definite by A > 0 (A < 0). A ≤ B means that matrix B − A is symmetric positive semi-definite. Sym(X) = X + X T , and I and 0 denote the identity matrix and zero matrix with appropriate dimensions, respectively; 0 l×m and I n denote the zero matrix with dimensions l × m and identity matrix with dimensions n × n, respectively. Diag [ A 1 · · · A r ] denotes the block diagonal matrix with matrices A 1 · · · A r on its diagonal, E qi = 0 n×(i−1)n I n 0 n×(q−i)n ∈ n×qn , i = 1, 2, · · · , q, q = 2, 3, · · · , 7, , and x denotes the Euclidean norm of vector x.

Problem Formulation and Mixed Performance Analysis
Consider the following uncertain discrete system with interval time-varying delay: where x(k) ∈ n is the state; w(k) ∈ m is the disturbance input; z(k) ∈ l is the regulated output; φ is the initial function; τ(k) is the interval time-varying delay satisfying τ m ≤ τ(k) ≤ τ M ; τ m and τ M are two given positive integers; perturbed matri- where M x ∈ n×n , M z ∈ l×l , Ξ j , j = 1, 2, N l , l = 0, 1, 3, 4, and N kw , k = 2, 5 are some given constant matrices with appropriate dimensions. Γ j (k), j = 1, 2, are unknown matrices representing the parameter perturbations which satisfy Now, we consider the interval time-varying delay τ(k) satisfying the following condition: where λ is a non-negative integer with 1 < τ m ≤ τ(k) ≤ τ M . If λ = 1, we have the following three possible conditions: τ(k + 1) = τ(k), τ(k + 1) = τ(k) + 1, and τ(k + 1) = τ(k) − 1. By using the switching approach, the original system (1) can be rewritten as where switching signal With respect to τ(k) = τ m and τ(k) = τ M , we have σ(k) = 1 and σ(k) = N, respectively. With the constraint in (3), we have We define We then have With the consideration for the reachable set, we may assume bounded peak disturbance satisfying where − w > 0 is a given constant. We give the following definition.
If the parameter 2 is selected as ∞, the disturbance input w should be constrained in L 2 (0, ∞). Then we say that the system (1) with (2) and (7) is asymptotically stable with H 2 measure α and H ∞ performance γ. If the disturbance is constrained in (7) for a given w > 0, we can find a set R where x(k) ∈ R, ∀k > 0. This set R ⊆ n is called the reachable set of the system.
The following lemmas will be used to derive the main results in this paper. (1) S < 0;

Lemma 2 ([26,27]).
Suppose that ∆ 1 (t) is defined in (2a) and satisfies (2c); then, for real matrices U, W, and X with X = X T , the following statements are equivalent: (I) The following inequality is satisfied: where the matrix Ξ 1 is as defined in (2c).
then the system is asymptotically stable and we have Proof. With w(k) = 0, we have the following result from (8a): This implies that the system is asymptotically stable. With the bounded peak disturbance in (7) and V i (X(k)) ≤ 1, we have (6) with bounded peak disturbance in (7). If there

Lemma 4 ([12]). Consider a switched system in
then the system is asymptotically stable and the reachable set is bounded by a hyper-sphere S(r) For the above proposed condition in (8b), it is difficult to solve and decompose the system with uncertainties; we developed the following results to find the feasible solution.
Lemma 5. Consider a switched system in (6) with bounded peak disturbance in (7). If there exist then the system is asymptotically stable and the reachable set is bounded by a hyper-sphere S(r) with r = min Proof. Define the following Lyapunov function: We have the following result from (9): By the Schur complement in Lemma 1, the conditions in (9) imply ∆(k) < 0 in (10b). This proof can be completed in view of Lemmas 3 and 4. Remark 1. Lemmas 4 and 5 can be applied to guarantee the asymptotic stability and reachable set of system (1) with (2) and (7). However, the proposed condition in (9) can be used to solve the system under consideration with perturbations and obtain LMI conditions that are easy to solve without uncertain elements by the LMI toolbox of Matlab.

Main Results
From (6), the condition in (9) is equivalent to By using Lemma 2, the condition in (11a) is equivalent to The following LMI result is proposed to guarantee the mixed performance and reachable set of system (1) with (2) and (7).
have a feasible solution with some positive definite symmetric matrices P, , and positive constants α, γ, δ i ,η xi and η zi , andε ij , i, j ∈ N − , then the system (1) with (2) is asymptotically stable with H 2 asure α and H ∞ performance γ = √ γ, and the reachable set is bounded by a hyper-sphere S(a) with a = min Proof. The first part of the proof about stability and the reachable set can be completed using Lemma 5 and derivations in (11). Now we wish to solve the uncertain system with mixed H 2 /H ∞ performance. We define the following Lyapunov function: where P ∈ n·(τ M +1)×n·(τ M +1) . We define Y(k) = X T (k)w T (k) T , and from the difference of Lyapunov function (13) along the solutions of system (6), we have where We define , and from (14b) and Lemma 2, we can define Ω i in (12c). By using Lemma 2, the condition Summing the equation (8a) from 0 to 2 , we have With zero initial condition ( By the definition of Lyapunov function V(X(k)) in (13), we have From the above derivations, the following condition can be guaranteed: By using (12d) with Lemma 1, we have With w(k) = 0, (12d), and (15), (16), we have This completes the proof. For simplicity, we can obtain the following optimization result by δ = δ i and α 0 = α ij , ∀i, j ∈ N − . Corollary 1. Suppose there exists a constant 0 < α 0 < 1, such that the optimization problem where γ = γ 2 , has a feasible solution with some positive definite symmetric matrices P, P i ∈ n·(τ M +1)×n·(τ M +1) , i ∈ N − , and positive constants α, γ, η xi , η zi , and ε ij , i, j ∈ N − . Then, the system (1) with (2) is asymptotically stable with H 2 measure α and H ∞ performance γ = √ γ, and the reachable set is bounded by a hyper-sphere S(a) with a = min i∈N δ i . (3) and (5), the constant λ ≥ 0 represents the variation in the delay and switching signal with N = τ M − τ m + 1. If the value is small (for example, λ = 1 or λ = 2), we can say that it is a discrete system with slow variation in time-varying delay. When λ = 1, for the LMI condition terms in (8b),

Remark 2. Under the variation constraint in
. Hence, the proposed results for a system under a slow variation constraint on time-varying delay will be less conservative than those without this constraint.

Remark 3.
In Theorem 1 and Corollary 1, we can take minimization about γ = γ 2 or α with respect to the LMI conditions in (12). The optimal H 2 measure or H ∞ performance of the system can be finished using the Matlab toolbox from our proposed results. Smaller values of γ and α will imply better disturbance attenuation and H 2 measure, respectively. The minimization of δ can be used to give a more exact evaluation of the reachable set. Now we provide a procedure around the results of Corollary 1 to ensure the stability and reachable set for the system under consideration. With the selection of α 0 , the optimization in (17) of Corollary 1 can be formulated as follows: Step 1. Set δ 1 = · · · = δ j−1 = δ j+1 = · · · = δ N = 1000 (or a larger constant) and perform minimization on δ j ; the optimization in (17) has a feasible solution.
Step 2. Minimize the value γ with a feasible solution for optimization in (17).
Step 3. The value α can be minimized for a feasible solution of optimization in (17) at this final step.
Consider the following uncertain discrete system with interval time-varying delay and control input: where x(k) ∈ n is the state; w(k) ∈ m is the disturbance input; z(k) ∈ l is the regulated ouput; u(k) ∈ p is the control input; φ is the initial function ; τ(k) is the interval time-varying delay satisfying τ m ≤ τ(k) ≤ τ M ; τ m and τ M are two positive integers; perturbed matrices and B zu are constant matrices with appropriate dimensions; and ∆A i (k), ∆A zi (k), ∆B xw (k),∆B xu (k), ∆B zw (k), and ∆B zu (k) are some perturbed matrices that satisfy the following linear fractional perturbation conditions: where M x ∈ n×n , M z ∈ l×l , Ξ j , j = 1, 2, N l , l = 0, 1, 3, 4, N 2w , N 2u , N 5w , and N 5u , are some given constant matrices with appropriate dimensions. Γ j (k), j = 1, 2, are unknown matrices representing the parameter perturbations which satisfy By using the switching approach, the original system (1) can be rewritten as: where switching signal σ(k) = τ(k) − (τ m − 1) ∈ N, N = τ M − τ m + 1, and τ σ(k) (k) = τ(k). With respect to τ(k) = τ m and τ(k) = τ M , we have σ(k) = 1 and σ(k) = N, respectively. With the known time delay τ(k) and switching approach, we can choose the control input as follows: where K i ∈ p×n , i ∈ N − , are selected by the developed results. With X(k) as defined in (6), we have From the results in (11a) with (21), we have We define By using Lemma 2, the condition in (22b) is equivalent to The following LMI result is proposed to design the switching control in (20d) to guarantee the asymptotic stability and reachable set of system (18) with (7) and (19).
, matricesǨ i ∈ p×n , and positive constants δ i , ε ij , i, j ∈ N − , then the system (18) with (7) and (19) is asymptotically stabilizable by the switching control in (20d) with , and the reachable set is bounded by a hyper-sphere S(a) with a = min i∈N δ i .
Proof. From (23b) and Lemma 1, we have WithP i = P i −1 , the LMI condition in (12b) can be achieved from (23b) by pre-and postmultipling the matrix P i . The remainder of the proof regarding stabilization and the reachable set follows from Lemma 5 and derivations in (22).
From the switching control gains K i , i ∈ N − , in Theorem 2, we have the following formulation: By Equations (24) and (25), the following results can be obtained from Corollary 1 and Theorem 2.
Step 2. We minimize the value γ; a feasible solution can be found with γ = 0.1501.
We can conclude that the system (18) with (7), (19), and (29) is asymptotically stabilizable by the switching control in (20d) with (30), and the reachable set is bounded by hyper-sphere S(a) with a = min i∈N δ i = 1.5874. By Corollary 2 with the switching control gains in (30), we can conclude that the system (18) with (7), (19), and (29) is asymptotically stabilizable with H 2 measure α =93.6876 and H ∞ performance γ = √ γ = 0.7559 by the switching control in (20d) with K i in (30), and the reachable set is bounded by hyper-sphere S(a) with a = min We can conclude that the system (18) with (7), (19), and (29) is asymptotically stabilizable by the switching control in (20d) with (31), and the reachable set is bounded by hyper-sphere S(a) with a = min i∈N δ i = 1.6362. By Corollary 2 with the switching control gains in (30), we can conclude that the system (18) with (7), (19), and (29) is asymptotically stabilizable with H 2 measure α =85.0073 and H ∞ performance γ = √ γ = 0.7708 by the switching control in (20d) with K i in (31), and the reachable set is bounded by hyper-sphere S(a) with a = min i∈N δ i = 1.6362. Under no perturbations, no initial conditions, and the disturbance inputs w(k) = 1 × (−0.95) k shown in Figure 1, the regulated outputs z(k) ∈ 2 of the discrete system (18), (19), and (29) without control and with switching control (20d) with (30) are shown in Figures 2 and 3, respectively. Figure 3 shows the good disturbance attenuation effect with H ∞ performance γ = 0.7708 for switching control (20d) with (30). The state trajectories x(k) ∈ 2 and reachable set with switching control (20d) with (30) are shown in Figures  4 and 5, respectively. Under zero disturbance and initial state function φ T (k) = [0.5 − 1], k = −4, · · · , 0, the state trajectories x(k) ∈ 2 and regulated outputs of system (18), (19), and (29) with switching control (20d) with (30) are shown in Figures 6 and 7, respectively. The H 2 measure can be observed to be guaranteed in Figure 7. The interval time-varying delay used in this paper is depicted in Figure 8.        With the above conditions (1) and (2), the switching controls in (20d) with (28a) and (28b) can used to improve the reachable set and H ∞ performance γ. The H 2 measure α may produce more conservative results with than without switching control. Some comparisons are made in Table 2   Table 2. Some comparisons regarding the proposed results of system (1) or (18) with (7) and (29).

Interval Time-Varying Delay
Conditions Reachable Set and Performance [6] 1 ≤ τ(k) ≤ 4 No perturbations (β = 0) and no control Fail [4] Fail The proposed results in this paper

Conclusions
In this paper, the mixed H 2 /H ∞ performance control and reachable set of uncertain discrete systems with interval time-varying delay and linear fractional perturbations were investigated. Some LMI-based conditions were proposed to guarantee the mixed performance and reachable set of the system under consideration. When the interval timevarying delay has a slow variation condition, the proposed results will be less conservative. Our numerical simulations showed that our proposed method may yield better and more flexible results than those in past research.