Guaranteed Cost Finite-Time Control of Discrete-Time Positive Impulsive Switched Systems

This paper considers the guaranteed cost finite-time boundedness of discrete-time positive impulsive switched systems. Firstly, the definition of guaranteed cost finite-time boundedness is introduced. By using the multiple linear copositive Lyapunov function (MLCLF) and average dwell time (ADT) approach, a state feedback controller is designed and sufficient conditions are obtained to guarantee that the corresponding closed-loop system is guaranteed cost finite-time boundedness (GCFTB). Such conditions can be solved by linear programming. Finally, a numerical example is provided to show the effectiveness of the proposed method.

However, most results mentioned above focus on the classical Lyapunov stability, which guarantees the stability in an infinite-time interval.Different from the Lyapunov stability concept, the finite-time stability requires that the states do not exceed a certain bound during a fixed finitetime interval.The paper [15] firstly defined the definition of finite-time stability (FTS) for linear deterministic systems.Recently, [16] firstly extended the concept of FTS to positive switched systems and gave some FTS conditions of positive switched systems.So far, there have been a few meaningful results about FTS of positive switched systems; see [17][18][19][20].In these results, to make the best of the nature of positivity, the MLCLF approach has been widely used and became a powerful tool for the analysis and synthesis of positive switched systems.Due to the wide application of digital controllers, some researches have been done on the FTS of discrete-time positive switched systems.The paper [21] investigated the problem of robust finite-time stability and stabilization of a class of discrete-time positive switched systems.The paper [22] studied the problem of finite-time control of a class of discrete impulsive switched positive timedelay systems under asynchronous switching, but the effect of disturbance was ignored.
Moreover, in most of practical applications, the researchers are more interested in designing the control system which is not only finite-time stable but also guarantees an adequate level of performance.One method to this problem is the so-called guaranteed cost finite-time control.Some remarkable results have been presented; see [23][24][25][26][27].These results mainly focus on nonpositive systems.Very recently, in [28], guaranteed cost finite-time control was extended to fractional-order positive switched systems and a cost function for fractional-order positive systems (or fractional-order positive switched systems) was proposed.In [29], the problem of guaranteed cost finite-time control for positive switched linear systems with time-varying delays was considered and a cost function of positive systems (or positive switched systems) was also presented.Based on [29], [30] extended guaranteed cost finite-time control to positive switched nonlinear systems with -perturbation.It is worth noting that [28][29][30]  In this paper, we consider the problem of GCFTB of discrete-time positive impulsive switched systems by constructing the MLCLF with average dwell time (ADT) technique.Firstly, the concept of guaranteed cost finite-time boundedness is extended to discrete-time positive impulsive switched systems.Secondly, a state feedback controller is designed and sufficient conditions are obtained to guarantee that the closed-loop system is GCFTB.Some sufficient conditions are obtained by linear programming.
The rest of the paper is organized as follows.Section 2 gives some necessary preliminaries and problem statements.In Section 3, the main results are given.In Section 4, a numerical example is provided.Section 5 concludes the paper.

Preliminaries and Problem Statements
Consider the following discrete-time positive impulsive switched systems: where  ∈ , () ∈   is the system state, and () ∈   represents the control input.() represents switching signal of system and takes values in a finite set  = 1, 2, . . ., ,  ∈  + .In general,   ,   ,   , and   are the th subsystem if () =  ∈ . 0 = 0 is the initial time.  ( ∈  + ) denotes the th impulsive switching instant.Moreover, () =  ∈  means that the th subsystem is active.( − 1) =  and () =  ( ̸ = ) indicate that  is a switching instant at which the system is switched from the th subsystem to the th subsystem.At switching instants, there exist impulsive jumps described by (1).  ,   ,   , and   are constant matrices with suitable dimensions, () ∈   is the exogenous disturbance and defined as with a known scalar  > 0 and a given finite-time threshold value   .
Now we give some new definitions for our further study.Definition 6. Define the cost function of discrete-time positive impulsive switched system (1) as follows: where  1 ≻ 0 and  2 ≻ 0 are two given vectors.
Remark 7. It should be noted that the proposed cost function is different from the general one, such as [26][27][28]; this definition provides a more useful description, because it takes full advantage of the characteristics of nonnegative states of discrete-time positive impulsive switched systems.
Definition 8 (GCFTB).For a given time constant   and two vectors  ≻  ≻ 0, consider discrete-time positive impulsive switched system (1) and cost function (6); if there exist a control law () and a positive scalar  ⋆ such that the closedloop system is FTB with respect to (, ,   , , ()) and the cost function satisfies  ≤  ⋆ , then the closed-loop system is called GCFTB, where  ⋆ is a guaranteed cost value and () is a guaranteed cost finite-time controller.

Main Results
3.1.Guaranteed Cost Finite-Time Boundedness Analysis.In this subsection, we will focus on the problem of GCFTB for discrete-time positive impulsive switched system (1) with () = 0.The following theorem gives sufficient conditions of GCFTB for system (1) with () = 0.
Next, we will give the guaranteed cost value of system (1) with () = 0.

Guaranteed Cost Finite-Time Controller Design.
In this subsection, we are concerned with the guaranteed cost finitetime controller design of discrete-time positive impulsive switched system (1).Under the controller () =  () (), the corresponding closed-loop system is given by By Lemma 2, to guarantee the positivity of system (32),   +     ⪰ 0 should be satisfied, ∀ ∈ .The following Theorem 10 gives some sufficient conditions to guarantee that the closedloop system (32) is GCFTB.
Next, an algorithm is presented to obtain the feedback gain matrices   ,  ∈ .
Step 3. Substituting ]  and   into   =       ]  , f can be obtained.If f −   ⪯ 0, then   are admissible.Otherwise, return to Step 1.

Numerical Example
Consider the discrete-time positive impulsive switched system (1) with the parameters as follows: (42) It is easy to confirm that f −   ≤ 0 and (33) is satisfied; then   are admissible.According to (12), we get  *  = 1.7.The simulation results are shown in Figures 1-3, where the initial conditions of system (1) are (0) = [1, 2]  , which meet the condition   () < 1.The state trajectory of the closedloop system is shown in Figure 1.The switching signal () is depicted in Figure 2. Figure 3 plots the evolution of (), which implies that the corresponding closed-loop system is GCFTB with respect to (, ,   , , ()), and the cost value  * = 3.72, which can be obtained by (35).

Conclusions
In this paper, we have considered the issue of guaranteed cost finite-time control for discrete-time positive impulsive switched systems.Based on the ADT approach, a guaranteed cost finite-time controller is constructed to guarantee that the closed-loop system is GCFTB.Finally, a numerical example is given to illustrate the effectiveness of the proposed method.
are involved in continuous-time positive switched systems.However, the problem of guaranteed Complexity cost finite-time control for discrete-time positive impulsive switched systems is still open, which inspires us for this study.