On Finite-Time Feedback Control for Switched Discrete-Time Systems under Fast and Slow Switching

The problem of finite-time stabilization for switched discrete-time systems under both fast and slow switching is addressed. In the fast switching case, the designed static state feedback controller combines controllers for each subsystem and resetting controller at switching instant, it is shown that the resetting controller can reduce the conservativeness on controller design.Then the results are extended to output feedback controller design. Under slow switching, both static state feedback and output feedback controller are designed with admissible average dwell time, respectively. Several numerical examples are given to illustrate the proposed results within this paper.


Introduction
A switched system consists of a finite number of subsystems that are switched according to the time variation of the system's mode of operation.Many physical processes possess switched and hybrid nature [1][2][3], and switched systems arise in many engineering applications, for example, in motor engine control [4], constrained robotics [5], and networked control systems [6].Due to their ubiquity in modern engineering problems, switched systems are receiving increasing interest and attention as the recent books [1,2] and survey articles [3,7,8] indicate.Stability and stabilization are the main concerns in the field of switched systems.Many Lyapunov function techniques are effective tools dealing with switched systems [9][10][11].Dwell time and average dwell time approaches were employed to study the stability and stabilization of time-dependent switched systems [12][13][14].
On the other hand, the finite-time stability is a concept dealing with the boundness of system during a fixed finitetime interval, which mainly focuses on admitting the state does not exceed a certain bound during a fixed finite-time interval, for instance, to avoid saturations or the excitation of nonlinear dynamics.A distinction should be made between classical Lyapunov stability and finite-time stability (or shorttime stability).The concept of Lyapunov asymptotic stability is largely known to the control community; conversely a system is said to be finite-time stable if, once we fix a time interval, its state does not exceed some bounds during this time interval.Often asymptotic stability is enough for practical applications, but there are some cases where large values of the state are not acceptable, for instance, in the presence of saturations.In these cases, we need to check that these unacceptable values are not attained by the state; for these purposes finite-time stability could be used.Some early results about finite-time stability can be found in [15][16][17].More recently, [18] investigated the output feedback finitetime stabilization for continuous linear system.Finite-time stability and stabilization for discrete linear system were investigated in [19].Moreover, the robust finite-time control for linear switched discrete-time system with norm-bounded disturbance was considered in [20], and continuous-time case is considered in [21].
Similar as indicated for asymptotic stability, the finitetime stability is supposed to be affected significantly by switching among several subsystems [20,21].Thus, besides Lyapunov asymptotic stability, another important system property we are interested in is the boundness of the state during the short time interval in which the switching occurs, and it is explicit that the boundness of state during the short time interval could be influenced significantly by the switching.
To avoid the state reaching the unacceptable large values caused by switching during the short time interval, the boundness property of state, that is, the finite-time stability, needs to be considered when we design the controller and switching law.However, most of the existing stability results were about Lyapunov stability; few results have been reported in literature about the finite-time stability of switched systems so far.This motivates the present study.
In this paper, solutions for finding state and output feedback controller guaranteeing the finite-time stabilization of switched discrete-time system are given in both fast switching and slow switching case.In fast switching case, a natural idea is to find a single common positive matrix satisfying finite-time stabilization condition for each subsystem to ensure the finite-time stability of closed-loop switched system which is similar to the common Lyapunov function approach dealing with asymptotic stability.However, it often yields overly conservativeness.To reduce the conservativeness our approach is based on searching for a set of positive matrices; furthermore it is worth noting that the controller combines the controller for each subsystem and resetting controller at switching instant which generates a proper state impulsive jumping which could further reduce the conservativeness in finite-time controller design.In slow switching case, the controller design is proposed including the controllers for subsystems and the admissible average dwell time.The results in fast and slow switching case are both extended to output feedback case.Several numerical simulations are proposed to show the effectiveness of our approach.
The rest of this paper is organized as follows.In Section 2 the problem formulation and some preliminaries are introduced.The main results, finite-time stabilization under fast switching and slow switching, are given in Sections 3 and 4, respectively; several numerical simulations are given to illustrate our proposed results.Conclusions are given in Section 5.
Notations.The notations used in this paper are fairly standard.The superscript "" stands for matrix transposition, R  denotes the -dimensional Euclidean space and Z + represents the set of nonnegative integers, and the notation ‖ ⋅ ‖ refers to the Euclidean norm.In addition, in symmetric block matrices, we use * as an ellipsis for the terms that are introduced by symmetry and diag{⋅ ⋅ ⋅ } stands for a blockdiagonal matrix. min () and  max () stand for the smallest and the largest eigenvalue of matrix .The notation  > 0 ( ≥ 0) means  is real symmetric and positive definite (semipositive definite).

Preliminaries
In this paper, a switched discrete-time system we consider is described as follows: where () ∈ R  is the discrete state, () ∈ R  is the control input, and () ∈ R  is the measurement of the system.  ,   , and   are real known constant matrices with appropriate dimensions.() : Z + → I = {1, 2, . . ., } is a piecewisely constant function of discrete-time , called switching law or switching signal, which takes its values in finite set I.  > 0 is the number of subsystems.
For simplicity, at any arbitrary discrete-time  ∈ Z + the switching signal () is denoted by .Given switched system ((1a), (1b)), the switching sequence can be defined as S := { 0 ,  1 ,  2 , . . .,   , . ..},where  0 denotes the initial time and   denotes the th switching instant.In this paper, we assume that the switching signal is available in real time; that is, the activated subsystem is explicitly known at each switching instant and the corresponding controller can be activated immediately.
For switched discrete-time system, the general conception of finite-time stability concerns the boundness of discrete state () over finite discrete-time interval [0, ],  ∈ Z + with respect to given initial condition  0 .This conception can be formulized through following definition.
Definition 1 (finite-time stability).Switched system is said to be finite-time stable with respect to (, , , ), where 0 ≤  < ,  is a positive definite matrix, and Then we recall the following lemma, which will be used in the proof of our main results.

Finite-Time Stabilization under Fast Switching
In this section we consider the finite-time stabilization for switched discrete-time system under fast switching.A set of static state feedback controllers are given as () =   (), ∀ ∈ Z + \S during the subsystems activation time and () =  , (), ∀ ∈ S and  ̸ = , (, ) ∈ I × I at the switching instant when system ((1a), (1b)) switches from subsystem  to .Then the closed-loop switched system becomes Remark 3. From (5a) and (5b) we see that the state impulsive jumping idea is applied since the state feedback controller combines controllers for each subsystem and resetting controller generating a proper state impulsive jumping at switching instant.Before deriving the conditions for finite-time stabilization of switched system ((1a), (1b)), some explicit facts are recalled.For a symmetric positive definite matrix  ∈ R × , it is easy to verify that  can be factorized according to  = ( 1/2 )   1/2 , where  1/2 ∈ R × is a symmetric positive definite matrix.And for any positive definite matrix  ∈ R × , there always exist  −1 ∈ R × which is positive definite.We present our first result on controller design ensuring the closed-loop system ((5a), (5b)) finite-time stable by following theorem.
It is worth mentioning that the class of switching signal is not specified during above discussion; thus Theorem 4 supplies us with a sufficient condition for finite-time stability of switched system ((5a), (5b)) under arbitrarily fast switching.We note that the result in Theorem 4 depends on the parameter .For a fixed , conditions (6a), (6b), and (6c) can be expressed in an LMI form.Obviously, in order to find a suitable , a one parameter search may be necessary; nevertheless this does not represent a hard computational problem.When we let  , =   , ∀(, ) ∈ I × I, the following design results can be derived.Corollary 6.The closed-loop switched system ((5a), (5b)) is finite-time stable with respect to (, , , ), if there exist a set of matrices   ,   ,  ∈ I, and positive scalars  > 0,  > 0, and  ≥ 1 such that the following conditions are satisfied ∀(, ) ∈ I × I:

Remark
Then the set of state feedback controllers is given by   =    −1  .
Furthermore, if we chose  =   , ∀ ∈ I, then we get the following results based on search for a single positive matrix.

Corollary 7.
The closed-loop switched system ((5a), (5b)) is finite-time stable with respect to (, , , ), if there exist a set of matrices ,   ,  ∈ I, and positive scalars  > 0,  > 0, and  ≥ 1 such that the following conditions are satisfied ∀(, ) ∈ I × I: Then the set of state feedback controllers is given by   =    −1 .
Obviously, the results in both Theorem 4 and Corollary 6 are based on searching for a set of positive matrices, which would be explicitly less conservative than that based on searching for a common positive matrix in Corollary 7.Moreover, the main advantage of Theorem 4 is that the controller for switched system ((1a), (1b)) designs a reset controller  , =  ,  −1  at switching instant, which could reduce the conservativeness compared to Corollary 6 only considering controller for each subsystem.To compare the conservativeness of the above proposed results, a numerical simulation is given in following example.Example 8. Consider a second order switched system with two subsystems given by We chose  = √ 2,  = , and  = 10.Then we assume that  is not ascertained; our aim in this example is to design the controller ensuring the minimum value  min through optimization procedure (23).In order to illustrate the advantages of our approach based on searching for a set of positive matrices and applying resetting controller at switching instant, three approaches are used.
(1) Firstly we design controller based on searching for a single positive matrix by Corollary 7; the controllers are and the minimum value  min = 7.0739.The simulation result under a random generated switching signal is given in Figure 1.Note that "o" denotes the initial state (0) and "+" denotes the discrete state (), ∀ ∈ {1, 2, . . ., 10}.The solid line denotes the bound for initial state (0) and the dotted line is the bound for discrete state (), ∀ ∈ {1, 2, . . ., 10}.
(2) Then we propose the simulation results by Corollary 6 based on searching for a set of positive matrices but without resetting controller.In this case, we only design controller for each subsystem.The controllers are The minimum value  min = 3.8430.The simulation result is given in Figure 2.
(3) Finally we use the proposed approach by Theorem 4 based on searching for a set of positive matrices and with The minimum value  min = 3.1179 which explicitly has the smallest value among the three proposed approaches.The simulation result is given in Figure 3.
Comparing three simulation results in this example, we can see that the conservativeness can be reduced by searching for a set of positive matrices and further considering the resetting controller design.
Similar to state feedback controller design and based on Assumption 9, the following result can be derived.

Finite-Time Stabilization under Slow Switching
In this section, the finite-time stabilization under slow switching is considered.Now we consider a class of average dwell time switching; the definition of average dwell time is given as follows.
The following closed-loop switched system is considered: Thus, the objective here is to design controller () =   (),  ∈ I, ensuring the finite-time stability of closed-loop switched system ((1a), (1b)) with the admissible average dwell time.
Theorem 13.The closed-loop switched system (40) is finitetime stable with respect to (, , , ), if there exist a set of matrices   ,   ,  ∈ I, and positive scalars  > 0,  > 0, and  ≥ 1 such that the following conditions are satisfied Then the set of state feedback controllers is given by   =    −1  ,  ∈ I.
Proof.Substituting   =     ,  ∈ I, into (41a) we get For pre-and postmultiplying (42) by the symmetric matrix [ ], the following equivalent condition of (42) is obtained: Let   = Then let   =  −1/2    −1/2 ,  ∈ I; we get On the other hand ∀ ∈ I we have We chose ( = √ 2,  = 5,  = ,  = 50); by Theorem 13 the state feedback controller is designed as and the admissible average dwell time   > 9.9366; thus we chose a periodical switching that the switching occurs every 10 seconds which is shown in Figure 4.
The simulation result is in Figure 5.
It is explicit that the controller and the switching law with the admissible average dwell time meet requirement of finitetime stabilization.
Then from the same guideline in Theorem 13, (62b) and (62c) guarantee the finite-time stability of closed-loop switched system (61).

2 Figure 1 :
Figure 1: State response and minimum bound by Corollary 7.

Figure 2 :Figure 3 :
Figure 2: State response and minimum bound without reset controller.
5. Once the state bound  is not ascertained, a set of optimized controller gains   =    −1  and  , =  ,  −1  with minimum value  min is of interest.Since from (6c) we have  2 >    2 /, the minimum value  min can be found by solving optimization problem min    2 / subject to (6a) and (6b).Furthermore, when  is fixed and letting  = 1,  =  be derived through optimization procedure according to  and the minimum  min = √   2 , which can be implemented on some numerical optimization software tools such as the optimization toolbox of Matlab to ascertain the optimized value  and finally ascertain the minimum  min = √   2 .And when the admissible initial state bound  is not ascertained, a set of optimized controller gains with the maximum  max with a fixed  is of interest, and the search for  max can be executed by a similar way.