Guaranteed Cost Finite-Time Control of Fractional-Order Positive Switched Systems

Theproblemof guaranteed cost finite-time control of fractional-order positive switched systems (FOPSS) is considered in this paper. Firstly, a new cost function is defined.Then, by constructing linear copositive Lyapunov functions and using the average dwell time (ADT) approach, a state feedback controller and a static output feedback controller are constructed, respectively, and sufficient conditions are derived to guarantee that the corresponding closed-loop systems are guaranteed cost finite-time stable (GCFTS). Such conditions can be easily solved by linear programming. Finally, two examples are given to illustrate the effectiveness of the proposed method.


Introduction
Positive switched systems are a class of hybrid systems consisting of a family of positive subsystems and a switching law that specifies which subsystem will be activated along the system trajectory at each instant of time.Many remarkable results related to positive switched systems have been presented; see [1][2][3][4][5] and references therein.These results mentioned above refer to the positive switched systems with integer order derivative.However, in many physical, chemical, and economic processes, such as fractional PID control [6,7], fractional electrical networks [8], robotics [9], and electrical machines [10], fractional calculus is more feasible than integer calculations to model the behavior of such systems.Therefore, some researchers have investigated the fractional-order positive systems [11][12][13][14].Particularly, very recently, only a few results about fractional-order positive switched systems have been presented [15,16].Reference [15] considered the controllability of FOPSS for fixed switching sequence.Reference [16] considered the problem of statedependent switching control of FOPSS.But the above results are involved in asymptotic stability, which reflects the asymptotic behavior of the system in an infinite time interval.
By contrast, in many practical applications, one is more interested in what happens over a finite-time interval.Peter [17] has firstly defined finite-time stability (FTS) for linear deterministic systems, which means that, given a bound on the initial condition, the system state does not exceed a certain threshold during a specified time interval.There have been some meaningful results about FTS of positive switched systems [18][19][20][21][22][23].Among these results, it should be pointed out that only one paper has considered the problem of FTS of FOPSS in [23].Moreover, when controlling a real plant, it is also desirable to design a control system which is not only finite-time stable but also guarantees an adequate level of performance.One approach to this problem is the guaranteed cost finite-time control.It has the advantage of providing an upper bound on a given system performance index and thus the system performance degradation incurred by the uncertainties or time delays is guaranteed to be less than this bound [24,25].So it is necessary to study the design problem of guaranteed cost finite-time controller [26,27].In [26], robust finite-time guaranteed cost control for impulsive switched systems with time-varying delay was considered.In [27], guaranteed cost finite-time control was extended to positive switched linear systems with time-varying delays and a cost function for positive systems (or positive switched systems) was also proposed.As we know, fractional-order systems have some different features compared with ordinary systems, like the stability criterion.Then the research of fractional-order 2 Advances in Mathematical Physics switched systems is more challenging than ordinary switched systems.Particularly, for FOPSS, the problem of guarantee cost finite-time control has been rarely reported.
In this paper, guarantee cost finite-time control of FOPSS is considered.A new cost function is firstly proposed, which can utilize the characteristics of nonnegative states of FOPSS.Then the definition of guaranteed cost finite-time stability is also given.By using the ADT approach and copositive Lyapunov functional method, two types of feedback controllers (state feedback controller and static output feedback controller, resp.) and a list of switching signals are designed.Some sufficient conditions are obtained to guarantee that the closed-loop systems are GCFTS.The rest of the paper is organized as follows.In Section 2, problem statements and necessary lemmas are given.In Section 3, we obtained some sufficient conditions of GCFTS of FOPSS based on Lyapunov functional and ADT approach.In Section 4, two examples are given to illustrate the effective of two kinds of controllers, respectively.Section 5 is the conclusion.
Notations.Throughout this paper, A≻0 (⪰0, ≺0, ⪯0) means that   > 0 (≥0, <0, ≤0), which is applicable to a vector.  is in the th row and th column of . ≻  ( ⪰ ) means that − ≻ 0 (− ⪰ 0).The symbols ,   , and  × denote the set of real numbers, the space of the vectors of -tuples of real numbers, and the space of  ×  matrices with real numbers, respectively.  + is the -dimensional nonnegative (positive) vector space.Γ(⋅) denotes the Gamma function.  denotes the transpose of matrix .Matrices are assumed to have compatible dimensions for calculating if their dimensions are not explicitly stated.

Preliminaries and Problem Statements
2.1.Fractional-Order Calculus.Fractional-order integrodifferential operator is the generalization of integer order integrodifferential operator.There are different definitions of the fractional-order integral or derivative.Given 0 <  < 1, the uniform formula of a fractional integral is defined as For 0 <  < 1, Riemann-Liouville (RL) definition of fractional derivative is given as and Caputo definition of fractional derivative is given as where () is an arbitrary integrable function,  0  −  is the fractional integral of order  on [ 0 , ], and Γ() denotes the Gamma function with noninteger arguments, Γ() = ∫ ∞ 0  −  −1 .respectively.These two fractional-order operators are used in this paper.From the above two definitions, we can obtain the following relations between them: Lemma 1 (see [23]).Let  ∈ (0, 1); if (0) ≥ 0, then Next, we will present some definitions, lemmas, and inequalities for the FOPSS (5) for our further study.
Lemma 4 (see [3]).A matrix is a Metzler matrix if and only if there exists a positive constant  such that  +   ⪰ 0.
Definition 7 (FTS).For given time constant   and vectors  ≻  ≻ 0, system ( 5) is said to be FTS with respect to (, ,   , ()), if Advances in Mathematical Physics 3 Remark 8.This description of FTS is different from the one in [23], but they are the same definitions in nature.
Definition 9. Define the cost function of system (5) as follows: where  1 ≻ 0 and  2 ≻ 0 are two given vectors.
Remark 10.It should be noted that the proposed cost function is different from the nonpositive systems [24][25][26].Particularly, if  = 1, this definition is turned into the definition of cost function in positive switched system [27].This proposed definition provides a more useful description, because it takes full advantage of the characteristics of nonnegative states of FOPSS.
Now we give the definition of GCFTS for the FOPSS (5).
Definition 11 (GCFTS).For given time constant   and vectors  ≻  ≻ 0, consider system (5); if there exists a feedback control law () and a positive scalar  * such that the closed-loop system is finite-time stable with the respect to (, ,   , ()) and the cost function satisfies  ≤  * , then the closed-loop FOPSS is called GCFTS, where  * is a guaranteed cost value and () is a guaranteed cost finite-time controller.
In addition, if () is a constant, then Lemma 13 (Young's inequality).If  and  are nonnegative real numbers and  and  are positive real numbers such that Lemma 14 (  inequality).For 0 <  < 1 and any positive real numbers  1 ,  2 , . . .,   , The aim of this paper is to design a state feedback controller () =  1() () and a static output feedback controller () =  2() () and find a class of switching signals () for FOPSS (5) such that the corresponding closed-loop systems are GCFTS.

Guaranteed Cost Finite-Time Stability Analysis.
In this subsection, we will focus on the problem of GCFTS for FOPSS (5) with () ≡ 0.
Next, we will give the fractional-order guaranteed cost value of system (5).

Corollary
Proof.According to (4) and Lemma 1, we can obtain Similar to the proof process of Theorem 15, we can obtain the same results and the proved process is omitted.

Guaranteed Cost Finite-Time Controller Design.
In this section, we focus on the problem of guaranteed cost finitetime controller design of system (5).The state feedback controller and static output feedback controller will be designed to ensure the corresponding close-loop system is GCFTS, respectively.

State Feedback Controller Design.
Consider system (5); under the controller () =  1() (), the corresponding closed-loop system is given by According to Lemma 4, to guarantee the positivity of system (41),   +    1 should be Metzler matrices, ∀ ∈ .Theorem 17 gives some sufficient conditions to guarantee that the closed-loop system (41) is GCFTS.
The proof is completed.
Next, an algorithm is presented to obtain the feedback gain matrices  1 (or  2 ) and  ∈ .
Step 2. By adjusting the parameter  and solving the inequalities in Theorem 17 (or Theorem 18) via linear programming, we can get the solutions V  ,  1 (or  2 ), and   .

Numerical Examples
In this section, two examples are given to illustrate the effectiveness of the two controllers proposed above.
Example 2. A fractional-order positive electrical circuit system was modeled in [28].Accordingly, a type positive fractional-order electrical circuit system could be modeled by (5).Under the static output feedback controller (49) Let  = 0.9,  = 1.08, and  = 1.1.Solving the inequalities in Theorem 18 by linear programming, we have  4-6, and the initial conditions of system (5) are
} is the switching signal, where  is the number of subsystems; ∀ ∈ ,   ,   , and   are constant matrices with appropriate dimensions,  denotes the th systems and   denotes the th switching instant.
(19)s easy to verify that   +      2 are Metzler matrices, ∀ ∈ .Then, according to(19), we can obtain  *  = 1.3591 s.Choose   = 1.4 s >  *  .Under the static output feedback controller, the simulation results are shown in Figures