Stabilization of Switched Systems with Time-Varying Delays Using Delta Operator Approach

This paper considers the problems of the robust stability and robust H ∞ controller design for time-varying delay switched systems using delta operator approach. Based on the average dwell time approach and delta operator theory, a sufficient condition of the robust exponential stability is presented by choosing an appropriate Lyapunov-Krasovskii functional candidate. Then, a state feedback controller is designed such that the resulting closed-loop system is exponentially stable with a guaranteed H ∞


Introduction
Switched systems are a kind of hybrid systems consisting of a set of discrete event dynamic subsystems or continuous variable dynamic subsystems and a switching rule which defines a particular subsystem working during a certain interval of time.Switched systems have numerous applications in network control systems [1], robot control systems [2], intelligent traffic control systems [3], chemical industry control systems [4], and many other areas [5,6].Many important achievements on stability and stabilization of switched systems have been developed [7][8][9][10].It was shown in the literature that the average dwell time (ADT) method is a powerful tool to deal with the stability of switched systems.
The delta operator which is a novel approach with good finite word length performance under fast sampling rates has been investigated by many researchers due to their extensive applications [11][12][13], for instance, optimal filtering [14], signal processing [15], robust control [16], system identification [17], and so forth.As stated in [15], the standard shift operator was mostly adopted in the study of control theories for discretetime systems.However, the dynamic response of a discrete system does not converge smoothly to its continuous counterpart when the sampling period tends to zero; namely, data are taken at high sampling rates.The delta operator method can solve the above problem.In addition, it was shown in [15] that delta operator requires smaller word length when implemented in fixed-point digital control processors than shift operator does.So far, some useful results on delta operator systems have been formulated in [18][19][20][21].As is well known, time delay phenomena which often cause instability or undesirable performance in control systems are involved in a variety of real systems, such as chaotic systems, and hydraulic pressure systems [22].In the past years, a mass of results on delta operator systems with time delay have appeared [23][24][25][26][27].The delta operator is defined by where  is a sampling period.When  → 0, the delta operator model will approach the continuous system before discretization and reflect a quasicontinuous performance [28].
It should be noted that external disturbances are generally inevitable, and the output will be subsequently affected by disturbances in the system.Some results on  ∞ control were developed by many researchers to restrain the external disturbances [29][30][31][32][33].The  ∞ control problem for a class of discrete systems was solved by using delta operator approach [34].Low order sampled data  ∞ control using the delta operator was reported in the literature [35].Robust  ∞ control for a class of uncertain switched systems using delta operator was investigated [36].However, few results on the issues of robust stability and  ∞ controller design for delta operator switched systems with time-varying delay are presented, which motivates the present investigations.
In this paper, we concentrate our interest on investigating the stability and  ∞ controller design problems for delta operator switched systems with time-varying delay.The main contributions of this paper can be summarized as follows: (1) by constructing a new Lyapunov-Krasovskii functional candidate and using the average dwell time approach, an exponential stability criterion for the considered system is proposed and (2) a state feedback controller design scheme is developed such that the corresponding closed-loop system is exponentially stable with a guaranteed  ∞ performance.
The remainder of the paper is organized as follows.The formulation of the considered systems and some corresponding definitions and lemmas is given in Section 2. In Section 3, the exponential stability analysis and  ∞ control for the underlying system are developed.A numerical example is given to illustrate the feasibility and effectiveness of the proposed method in Section 4. Finally, concluding remarks are presented in Section 5.
Notations.‖ ⋅ ‖ 2 denotes the Euclidean norm. min (⋅) and  max (⋅) denote the minimum and maximum eigenvalues of a matrix, respectively;   means the transpose of matrix ;  denotes the set of all real numbers;   represents the dimensional real vector space;  × is the set of all ( × )dimensional real matrices.The notation  > 0(≥ 0) means that the matrix  is positive (nonnegative) definite; diag{⋅ ⋅ ⋅ } refers to the block-diagonal matrix;  is the identity matrix of appropriate dimension. 2 [ 0 , ∞) stands for the space of square summable functions on [ 0 , ∞).

Problem Formulation
Consider the following delta operator switched system with time-varying delay: where () ∈   is the state vector, () ∈   denotes the controlled output, and where   ,   ,   ,   , and   are known real constant matrices of suitable dimensions and   () is an unknown time-varying matrix which satisfies To obtain the main results, we first give some definitions and lemmas which will be essential in our later development.
Lemma 5 (see [20]).Let , , , and  be real matrices of appropriate dimensions with  satisfying  =   ; then, for all    ≤ , if and only if there exists a scalar  such that Lemma 6 (see [28]).For any time function () and (), the following equation holds: where  is the sampling period.
The objectives of the paper are (1) to find a class of switching signal () such that system (2) is exponentially stable with a guaranteed  ∞ performance and (2) to determine a class of switching signal and design a state feedback controller () =  () () for the following delta operator switched system with time-varying delay: such that the corresponding closed-loop system is exponentially stable with a guaranteed  ∞ performance.

Robust Stability Analysis.
In this section, we will focus on the stability of system (2) with () = 0. Theorem 7.For a given positive constant 0 <  < 1/, if there exist scalars   and positive definite symmetric matrices   and   ,  ∈ , with appropriate dimensions, such that where   =     +      +   + (1 − )( −  + 1)  , then system (2) with () = 0 is exponentially stable for any switching signal () with the following average dwell time scheme: where  ≥ 1 satisfies Proof.Choose the following Lyapunov-Krasovskii functional candidate for the th subsystem where Taking the delta operator manipulations of Lyapunov functional candidate   () along the trajectory of system ( 2 Combining ( 17)-( 19), we have where Applying Lemma 4, we can obtain that Ω  < 0 is equivalent to } to pre-and post-multiply both sides of (22), respectively, we have where and   =  −1  ; then, substituting (3) into (23) and applying Lemmas 4 and 5, we obtain that ( 23) is equivalent to (12).Thus, from (12), we can easily obtain It follows from (24) that Let  1 < ⋅⋅ ⋅ <   denote the switching instants of () over the interval [ 0 , ).Consider the following piecewise Lyapunov functional for system (2): From ( 14), we obtain It can be obtained from ( 24), (27), and Definition 3 that Considering the definition of  () (), it yields that where Combining ( 29) and ( 30), we have Therefore, system (2) with () = 0 is exponentially stable under the average dwell time scheme (13).
The proof is completed.

(33)
Then we have the following corollary.
Proof.Equation ( 12) in Theorem 7 can be directly derived from (36).Thus, system (2) is exponentially stable.We are now in a position to show the  ∞ performance of system (2).
Therefore, one has, for  ∈ [  ,  +1 ), where Following the proof line of (28), we obtain Under the zero initial condition, we get Mathematical Problems in Engineering Namely, Multiplying both sides of (46) by  − () ( 0 ,) leads to From Definition 3 and ( 13), we have Combining ( 47) and (48) leads to Then, summing both sides of (49) from  0 to ∞ leads to According to Definition 2, we can conclude that the theorem is true.
The proof is completed.
3.3. ∞ Controller Design.In this section, a state feedback controller () =  () () will be designed for system (11) such that the corresponding closed-loop system (51) is exponentially stable and satisfies an  ∞ performance.Consider where   ,  ∈ , are the controller gains to be determined.B are uncertain real-valued matrices with appropriate dimensions and have the following form Theorem 11.Consider system (11).For given positive constants  and 0 <  < 1/, if there exist scalars   , positive definite symmetric matrices   and   , and any matrices   ,  ∈ , of appropriate dimensions, such that where 53) is directly obtained.The proof is completed.
We are now in a position to give an algorithm for determining   and  *  .
Step 1. Input the system matrices.
Step 3. By (54), with the obtained   and   , one can compute the gain matrices   .

Numerical Example
Consider system (11) with parameters as follows:  Figure 1 and the state responses of the corresponding closedloop system are given in Figure 2.
From Figures 1 and 2, it is easy to see that the designed controller can guarantee that the resulting closed-loop system is exponentially stable.This demonstrates the effectiveness of the proposed method.

Conclusions
In this paper, the robust stability and  ∞ controller design problems for time-varying delay switched system using delta operator approach have been investigated.By using the average dwell time approach and constructing a Lyapunov-Krasovskii functional candidate, sufficient conditions for the existence of a state feedback  ∞ controller are presented.Finally, a numerical example is given to illustrate the feasibility of the proposed approach.In our future work, we will study the problem of robust  ∞ filtering for delta operator switched systems with uncertainties and time-varying delays.
() ∈   represents the disturbance input belonging to  2 [ 0 , ∞).  means the time  =  and  > 0 is the sampling period;  0 is the initial instant.() : [ 0 ,∞) →  = {1, 2, . . ., } is the switching signal with  being the number of subsystems.() is the time-varying delay satisfying 0 ≤  ≤ () ≤  for known constants  and .() is the discrete vector-valued initial function.  ,   , and   are constant matrices with proper dimensions.Â and Â are uncertain real-valued matrices with appropriate dimensions and have the following form: 0 <  < 1/, if there exist scalars   and positive definite symmetric matrices   , ∀ ∈ , of appropriate dimensions, such that For given positive constants  and 0 <  < 1/, if there exist scalars   and positive definite symmetric matrices   and   ,  ∈ , of appropriate dimensions, such that ) 3.2. ∞ Performance Analysis.The following theorem gives sufficient conditions for the existence of an  ∞ performance level for system (2).Theorem 10.