Stability and l 1-Gain Control of Positive Switched Systems with Time-Varying Delays via Delta Operator Approach

This paper investigates the problems of stability and l 1 -gain controller design for positive switched systems with time-varying delays via delta operator approach.Thepurpose is to design a switching signal and a state feedback controller such that the resulting closedloop system is exponentially stablewith l 1 -gain performance. Based on the average dwell time approach, a sufficient condition for the existence of an l 1 -gain controller for the considered system is established by constructing an appropriate copositive type LyapunovKrasovskii functional in delta domain. Moreover, the obtained conditions can unify some previously suggested relevant methods in the literature of both continuousand discrete-time systems into the delta operator framework. Finally, a numerical example is presented to explicitly demonstrate the effectiveness and feasibility of the proposed method.


Introduction
Positive systems mean that their states and outputs are nonnegative whenever the initial conditions and inputs are nonnegative [1,2].A positive switched system consists of a family of positive subsystems and a switching signal, coordinating the operation of various subsystems to specify when and how the switching takes place among the subsystems.Recently, due to the broad applications in communication systems [3,4], formation flying [5], viral mutation dynamics under drug treatment [2], and systems theories [6][7][8][9][10], positive systems have been highlighted and investigated by many researchers [11][12][13][14].It has been shown that a linear copositive Lyapunov functional is powerful for the analysis and synthesis of positive systems [15][16][17].
The delta operator, a novel method with good finite word length performance under fast sampling rates, has drawn considerable interest in the past three decades.As we know, the standard shift operator was mostly adopted in the study of control theories for discrete-time 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.Until Goodwin et al. proposed a delta operator method in [18] to take the place of the traditional shift operator, the above problem is avoided.It was shown that delta operator requires smaller word length when implemented in fixedpoint digital control processors than shift operator does [19].The delta operator model can be regarded as a useful approach to deal with discrete-time systems under high sampling rates through the analysis methods of continuoustime systems [20][21][22][23].Based on significant early investigations such as [24][25][26] studying the basic properties and performance of delta operator model, numerical properties and practical applications of delta operator model have been extensively investigated [27][28][29].The delta operator is defined by ( ( + ) −  ())  ,  ̸ = 0, where  is a sampling period.When  → 0, the delta operator model will approach the continuous system before discretization and reflect a quasicontinuous performance.
In addition, exogenous disturbances are commonly unavoidable in practical process, and the output will be inevitably affected by the disturbance in a system.Because of the peculiar nonnegative property of positive systems, the  1gain (or  1 -gain) index [39] can characterize the disturbance rejection property, by means of which we can limit the effect of disturbance in a prescribed level.Some results on  1 -gain (or  1 -gain) analysis and control for positive systems have been reported in the literature [39,40].However, few results on the issue of  1 -gain performance for positive switched systems via delta operator approach are proposed, which motivates the current research.
In this paper, we focus our attention on investigating the stability and  1 -gain controller design for positive switched systems with time-varying delays via delta operator approach.The main contributions of this paper are fourfold.(1) The positive switched systems via delta operator approach are investigated for the first time.(2) By applying the average dwell time approach, sufficient conditions of exponential stability for positive switched delta operator systems are derived.Moreover, the results obtained can be applied to both continuous-time systems and discrete-time systems.
(3)  1 -gain performance analysis of the underlying system is developed.(4) A state feedback controller design scheme is proposed such that the corresponding closed-loop system is exponentially stable with an  1 -gain performance.
The remainder of the paper is as follows.The problem formulation and some necessary lemmas are provided in Section 2. In Section 3, the issues of stability,  1 -gain performance analysis, and control of the underlying system are developed.A numerical example is presented to demonstrate the feasibility of the obtained results in Section 4. In Section 5, concluding remarks are given.

Problem Formulation
Consider the following switched delta operator system with time-varying delays: Using the traditional shift operator approach to discretize the system, the following discrete form in -domain can be obtained ( = 0, 1, 2, . ..): where   =   ,   = (∫  0   ),   = , and   = .When  → 0, lim  → 0   =  and lim  → 0   = 0.The movement of the system poles towards stable boundary makes the system defective with the increase in the sampling rates.However, by utilizing the delta operator approach, we can obtain the following system expressed in delta domain: where   = (  − )/,   =   /,   = , and   = .
When  → 0, lim  → 0   =  and lim  → 0   = .It can be seen that the system matrices are the same as those of the original continuous system, alleviating the problems encountered with fast sampling.
Remark 2. Since a delta operator system can be regarded as a quasicontinuous system when  → 0, the term () can be utilized like ẋ () in normal continuous-time systems.
Remark 4. Definition 3 follows the general positivity definition of a positive system, which means that the state and output are nonnegative whenever the initial condition and input are nonnegative [1,2].
Proof.From the definition of delta operator , the discrete form of system (2) can be obtained as follows: Combining Lemma 2 in [41] and Lemma 1 in [42], one can obtain the remaining proof easily.
Remark 7. In the light of Lemma 2.1 of [43], it is clear that the th subsystem in system (2) is positive if and only if ( +   ) ⪰ 0, Α  ⪰ 0,   ⪰ 0,   ⪰ 0, and   ⪰ 0, for all  ∈ .Thus we can have an equivalent expression of Lemma 5: system (2) is positive under any switching signals if and only if it consists of a family of positive subsystems.Definition 8 (see [44]).System (2) with () = 0 is said to be exponentially stable under () if, for constants  > 0 and  > 0, the solution () satisfies where Definition 9 (see [45]).For any switching signal () and any  2 >  1 ≥ 0, let   ( 1 ,  2 ) denote the number of switches of () over the interval [ 1 ,  2 ).For given   > 0 and  0 ≥ 0, if the inequality holds, then the positive constant   is called an average dwell time and  0 is called a chattering bound.Without loss of generality, one chooses  0 = 0 in this paper.

Main Results
This section will focus on the problems of stability analysis and  1 -gain controller design for positive switched delta operator systems with time-varying delays.

Stability Analysis.
First, we consider the following switched positive delta operator system: where  +   ⪰ 0,   ⪰ 0 for  ∈ , and   is defined the same as system (2).
Sufficient conditions of exponential stability of system (12) are provided in the following theorem.
Theorem 12.Given a positive constant 0 <  < 1/, if there exist ]  ,   ,   ∈   + , such that, for all  ∈ , where where  ≥ 1 satisfies Furthermore, the state decay of system (12) is given by where Proof.Choose the following piecewise copositive type Lyapunov functional for the th subsystem in system (12): where For simplicity,   (, ()) is written as   () (correspondingly, (, ()) is written as ()) in the later section of the paper.
This completes the proof.
Then we have the following corollary.
We can obtain sufficient conditions of exponential stability of system (31) by Theorem 12.

𝑙 1 -Gain Analysis.
The following theorem establishes sufficient conditions of exponential stability with  1 -gain property for system (2).
Remark 18.When  = 1 in Theorem 17, summing both sides of ( 44) from  =  0 to ∞ leads to which gives the standard  1 -gain performance.This completes the proof.
Consider the controller design of the following positive switched delta operator system without time delay: where Ã() =  () +  () .Then we directly have the following corollary.
Corollary 20.Considering system (54), for given positive scalars 0 <  < 1/ and , if there exist ]  ∈   + and   ∈   , such that, for all  ∈ , where c and ẽ have been defined in Theorem 17 and   =       ]  , then the corresponding closed-loop system is positive and exponentially stable with a prescribed  1 -gain performance level  for any switching signals () with average dwell time (14), where  ≥ 1 satisfies (30).
Based on Theorem 19, one is now in a position to present an effective algorithm for constructing the desired controller.Algorithm 21.Consider the following.
Step 3. By the equation   =       ]  with the obtained   and ]  , one can get the gain matrices   .
Step 4. Check condition (53) in Theorem 19.If it holds, go to Step 5; otherwise, adjust the parameter  and return to Step 2.

Numerical Example
Consider positive switched delta operator system (11) consisting of two subsystems described by the following.
Subsystem 1: and the state feedback gain matrices can be obtained as follows: Obviously, condition (53) is satisfied.
According to (15), we have  = 2.0193.Then from ( 14), we get   >  *  = 1.7880.Choosing   = 2, the simulation results are shown in Figures 1 and 2, where the initial conditions are (0) = [0.20.3]  and () = [0 0]  ,  = −2, −1, and the exogenous disturbance input is () = 0.05 −0.5 which belongs to  1 [0, ∞).The switching signal with average dwell time   = 2 is shown in Figure 1 and the state responses of the corresponding closed-loop system are given in Figure 2. From the simulation results, it can been seen that the closedloop system is exponentially stable with a prescribed  1 -gain performance level  = 2.

Conclusions
In this paper, the stability and  1 -gain controller design problems for positive switched systems with time-varying delays via delta operator approach have been investigated.By constructing a copositive type Lyapunov-Krasovskii functional and using the average dwell time approach, we proposed sufficient conditions of exponential stability and  1gain performance for the considered system.The desired state feedback  1 -gain controller was designed such that the corresponding closed-loop system is exponentially stable and satisfies an  1 -gain performance.Finally, a numerical example was presented to demonstrate the feasibility of the obtained results.In our future work, we will study the robust stabilization problem of positive switched systems with uncertainties and time-varying delays via delta operator approach.

Figure 2 :
Figure 2: State responses of the closed-loop system.
,   ,   ,   , and   are constant matrices with appropriate dimensions.