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

Authors: Leipo Liu; Xiangyang Cao; Zhumu Fu; Shuzhong Song
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

The problem of 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.


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## Body

## 1. 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 [1–5] 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 [11–14]. 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 state-dependent 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 [18–23]. 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 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 Section2, 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 aij>0(≥0, <0, ≤0), which is applicable to a vector. aij is in the ith row and jth column of A. A≻B (A⪰B) means that A-B≻0(A-B⪰0). The symbols R, Rn, and Rn×n denote the set of real numbers, the space of the vectors of n-tuples of real numbers, and the space of n×n matrices with real numbers, respectively. R+n is the n-dimensional nonnegative (positive) vector space. Γ· denotes the Gamma function. AT denotes the transpose of matrix A. Matrices are assumed to have compatible dimensions for calculating if their dimensions are not explicitly stated.


## 2. 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. Given0<α<1, the uniform formula of a fractional integral is defined as(1)Dt0t-αft=1Γα∫t0tfτt-τ1-αdτ.For 0<α<1, Riemann-Liouville (RL) definition of fractional derivative is given as(2)Dt0RLtαft=1Γ1-αddt∫t0tfτt-ταdτ,and Caputo definition of fractional derivative is given as(3)Dt0Ctαft=1Γ1-α∫t0tf′τt-ταdτ,where f(t) is an arbitrary integrable function, Dt0t-α is the fractional integral of order α on [t0,t], and Γα denotes the Gamma function with noninteger arguments, Γα=∫0∞e-ttα-1dt. Dt0RLtα and Dt0Ctα represent Riemann-Liouville and Caputo fractional derivatives of order α of f(t) on [t0,t], respectively. These two fractional-order operators are used in this paper. From the above two definitions, we can obtain the following relations between them:(4)Dt0RLtαft=Dt0Ctαft+t-αΓ1-αft0.Lemma 1 (see [23]).
Letα∈(0,1); if f(0)≥0, then Dt0RLtαf(t)≤Dt0Ctαf(t).


### 2.2. Fractional-Order Positive Switched Systems

Consider the following FOPSS:(5)Dt0Ctαxt=Aσtxt+Bσtut,yt=Cσtxt,where x(t)∈Rn, u(t)∈Rm, and y(t)∈Rz represent the system state, control input, and control output, respectively. Dt0Ctα denotes Caputo fractional-order derivative. σ(t):[0,∞)→N_={1,2,…,N} is the switching signal, where N is the number of subsystems; ∀p∈N_, Ap, Bp, and Cp are constant matrices with appropriate dimensions, p denotes the pth systems and tq denotes the qth switching instant.Next, we will present some definitions, lemmas, and inequalities for the FOPSS (5) for our further study.Definition 2 (see [23]).
System (5) is said to be positive if, for any switching signals σ(t) and any initial conditions x(t0)⪰0, the corresponding trajectory satisfies x(t)⪰0 and y(t)⪰0 for all t⪰0.Definition 3 (see [1]).
A matrixA is called a Metzler matrix if the off-diagonal entries of matrix A are nonnegative.Lemma 4 (see [3]).
A matrix is a Metzler matrix if and only if there exists a positive constantς such that A+ςIn⪰0.Lemma 5 (see [23]).
System (5) is positive if and only if Ap, ∀p∈N_, are Metzler matrices and ∀p∈N_, Bp⪰0 and Cp⪰0.Definition 6 (see [21]).
For any switching signalsσ(t) and any T2≥T1≥0, let Nσ(T1,T2) denote the switching numbers of σ(t) over the interval [T1,T2). If there exist N0≥0 and Tα>0 such that(6)NσT1,T2≤N0+T2-T1Tα,then Tα and N0 are called ADT and chattering bound, respectively. Generally speaking, we choose N0=0 in this paper.Definition 7 (FTS).
For given time constantTf and vectors δ≻ε≻0, system (5) is said to be FTS with respect to (δ, ε, Tf, σ(t)), if(7)xTt0δ≤1⟹xTtε≤1,∀t∈0,Tf.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:(8)J=Dt0t-αxTsR1+uTsR2=1Γα∫0tt-sα-1xTsR1+uTsR2ds,where R1≻0 and R2≻0 are two given vectors.Remark 10.
It should be noted that the proposed cost function is different from the nonpositive systems [24–26]. Particularly, if α=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 constantTf and vectors δ≻ε≻0, consider system (5); if there exists a feedback control law u(t) and a positive scalar J∗ such that the closed-loop system is finite-time stable with the respect to (δ,ε,Tf,σ(t)) and the cost function satisfies J≤J∗, then the closed-loop FOPSS is called GCFTS, where J∗ is a guaranteed cost value and u(t) is a guaranteed cost finite-time controller.Lemma 12 (Gronwall-Bellman inequality).
Leta(t), b(t), and g(t) be continuous real-valued functions. If a(t) is nonnegative and g(t) satisfies the integral inequality (9)gt≤at+∫0tbsgsds,then(10)gt≤at+∫0tasbsexp⁡∫stbrdrds.In addition, if a(t) is a constant, then (11)gt≤atexp⁡∫0tbsds.Lemma 13 (Young’s inequality).
Ifa and b are nonnegative real numbers and p and q are positive real numbers such that 1/p+1/q=1, then (12)ab≤app+bqq.Lemma 14 (Cp inequality).
For0<a<1 and any positive real numbers x1,x2,…,xk, (13)∑k=1nxka≤n1-a∑k=1nxka.The aim of this paper is to design a state feedback controller u(t)=K1σ(t)x(t) and a static output feedback controller u(t)=K2σ(t)y(t) and find a class of switching signals σ(t) for FOPSS (5) such that the corresponding closed-loop systems are GCFTS.


## 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. Given0<α<1, the uniform formula of a fractional integral is defined as(1)Dt0t-αft=1Γα∫t0tfτt-τ1-αdτ.For 0<α<1, Riemann-Liouville (RL) definition of fractional derivative is given as(2)Dt0RLtαft=1Γ1-αddt∫t0tfτt-ταdτ,and Caputo definition of fractional derivative is given as(3)Dt0Ctαft=1Γ1-α∫t0tf′τt-ταdτ,where f(t) is an arbitrary integrable function, Dt0t-α is the fractional integral of order α on [t0,t], and Γα denotes the Gamma function with noninteger arguments, Γα=∫0∞e-ttα-1dt. Dt0RLtα and Dt0Ctα represent Riemann-Liouville and Caputo fractional derivatives of order α of f(t) on [t0,t], respectively. These two fractional-order operators are used in this paper. From the above two definitions, we can obtain the following relations between them:(4)Dt0RLtαft=Dt0Ctαft+t-αΓ1-αft0.Lemma 1 (see [23]).
Letα∈(0,1); if f(0)≥0, then Dt0RLtαf(t)≤Dt0Ctαf(t).


## 2.2. Fractional-Order Positive Switched Systems

Consider the following FOPSS:(5)Dt0Ctαxt=Aσtxt+Bσtut,yt=Cσtxt,where x(t)∈Rn, u(t)∈Rm, and y(t)∈Rz represent the system state, control input, and control output, respectively. Dt0Ctα denotes Caputo fractional-order derivative. σ(t):[0,∞)→N_={1,2,…,N} is the switching signal, where N is the number of subsystems; ∀p∈N_, Ap, Bp, and Cp are constant matrices with appropriate dimensions, p denotes the pth systems and tq denotes the qth switching instant.Next, we will present some definitions, lemmas, and inequalities for the FOPSS (5) for our further study.Definition 2 (see [23]).
System (5) is said to be positive if, for any switching signals σ(t) and any initial conditions x(t0)⪰0, the corresponding trajectory satisfies x(t)⪰0 and y(t)⪰0 for all t⪰0.Definition 3 (see [1]).
A matrixA is called a Metzler matrix if the off-diagonal entries of matrix A are nonnegative.Lemma 4 (see [3]).
A matrix is a Metzler matrix if and only if there exists a positive constantς such that A+ςIn⪰0.Lemma 5 (see [23]).
System (5) is positive if and only if Ap, ∀p∈N_, are Metzler matrices and ∀p∈N_, Bp⪰0 and Cp⪰0.Definition 6 (see [21]).
For any switching signalsσ(t) and any T2≥T1≥0, let Nσ(T1,T2) denote the switching numbers of σ(t) over the interval [T1,T2). If there exist N0≥0 and Tα>0 such that(6)NσT1,T2≤N0+T2-T1Tα,then Tα and N0 are called ADT and chattering bound, respectively. Generally speaking, we choose N0=0 in this paper.Definition 7 (FTS).
For given time constantTf and vectors δ≻ε≻0, system (5) is said to be FTS with respect to (δ, ε, Tf, σ(t)), if(7)xTt0δ≤1⟹xTtε≤1,∀t∈0,Tf.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:(8)J=Dt0t-αxTsR1+uTsR2=1Γα∫0tt-sα-1xTsR1+uTsR2ds,where R1≻0 and R2≻0 are two given vectors.Remark 10.
It should be noted that the proposed cost function is different from the nonpositive systems [24–26]. Particularly, if α=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 constantTf and vectors δ≻ε≻0, consider system (5); if there exists a feedback control law u(t) and a positive scalar J∗ such that the closed-loop system is finite-time stable with the respect to (δ,ε,Tf,σ(t)) and the cost function satisfies J≤J∗, then the closed-loop FOPSS is called GCFTS, where J∗ is a guaranteed cost value and u(t) is a guaranteed cost finite-time controller.Lemma 12 (Gronwall-Bellman inequality).
Leta(t), b(t), and g(t) be continuous real-valued functions. If a(t) is nonnegative and g(t) satisfies the integral inequality (9)gt≤at+∫0tbsgsds,then(10)gt≤at+∫0tasbsexp⁡∫stbrdrds.In addition, if a(t) is a constant, then (11)gt≤atexp⁡∫0tbsds.Lemma 13 (Young’s inequality).
Ifa and b are nonnegative real numbers and p and q are positive real numbers such that 1/p+1/q=1, then (12)ab≤app+bqq.Lemma 14 (Cp inequality).
For0<a<1 and any positive real numbers x1,x2,…,xk, (13)∑k=1nxka≤n1-a∑k=1nxka.The aim of this paper is to design a state feedback controller u(t)=K1σ(t)x(t) and a static output feedback controller u(t)=K2σ(t)y(t) and find a class of switching signals σ(t) for FOPSS (5) such that the corresponding closed-loop systems are GCFTS.


## 3. Main Results

### 3.1. Guaranteed Cost Finite-Time Stability Analysis

In this subsection, we will focus on the problem of GCFTS for FOPSS (5) with u(t)≡0.Theorem 15.
Consider system (5) with u(t)≡0. Given positive constants Tf and λ(λ>1) and vectors δ≻ε≻0 and R1≻0, if there exist positive constants ξ1, ξ2, and μ(μ≥1) and positive vectors vp and p∈N_, such that the following inequalities hold:(14)ApTvp+R1⪯λvp,(15)ξ1ε≺vp≺ξ2δ,(16)vp⪯μvq,(17)vp≺R1,(18)1-α+αTfλ-1Γα+1<ln⁡ξ1ξ2then under the following ADT scheme(19)Tα>Tα∗=Tfln⁡μ+1-αλ-1/Γα+1ln⁡ξ1/ξ2-1-α+αTfλ-1/Γα+1the FOPSS (5) is GCFTS with respect to (δ,ε,Tf,σ(t)) and the guaranteed cost value of system (5) with u(t)=0 is given by(20)J=1Γα∫0tt-sα-1xTsR1+uTsR2ds≤J∗=ξ2μTf/Tα+λξ2μTf/Tα1Γαexp⁡TfTαln⁡μ+1-αλ-1Γα+1+λ-11-α+αTfΓα+11αTfα.Proof.
Construct the multiple linear type Lyapunov-Krasovskii functional for system (5) as follows:(21)Vσtt,xt=xTtvσt,where vp∈R+n, ∀p∈N_.
Denotet0,t1,…,tk as the switching instants over the interval [0,Tf]. Along the trajectory of system (5) with u(t)≡0, we have(22)Dtαt0CVσtt,xt=xTtAσtTvσt.From (14), we obtain(23)Dtαt0CVσtt,xt+xTtR1=xTtAσtTvσt+R1≤λVσtt,xt.Taking the fractional integral Dt-αt0C to both sides of (23) during the period [tk,t) for t∈[tk,tk+1) leads to(24)Vσtt,xt≤Vσtktk,xtk+λΓα∫tktt-sα-1Vσts,xsds-1Γα∫tktt-sα-1xTsR1ds.From (17) and (21), (24) can be rewritten as(25)Vσtt,xt≤Vσtktk,xtk+λΓα∫tktt-sα-1Vσts,xsds-1Γα∫tktt-sα-1xTsvσtds=Vσtktk,xtk+λ-1Γα∫tktt-sα-1Vσts,xsds.According to Lemma 12 and the properties of Gamma function Γα, for t∈[tk,tk+1), we have(26)Vσtt,xt≤Vσtktk,xtkexp⁡λ-1Γα∫tktt-sα-1ds=Vσtktk,xtkexp⁡λ-1αΓαt-tkα=Vσtktk,xtkexp⁡λ-1Γα+1t-tkα.For t∈[tk,tk+1), Vσ(tk)≤μσ(tk)Vσ(tk-)(tk-,x(tk-)) is easily obtained from (16) and (21). Noting that exp⁡(λ-1/Γα+1)t-tkα>0, we have(27)Vσtt,xt≤μVσtk-tk-,xtk-exp⁡λ-1Γα+1t-tkα.Then, for t∈[t0,Tf), we can obtain from (26) and (27) that(28)Vσtt,xt≤Vσtktk,xtkexp⁡λ-1Γα+1t-tkα≤μVσtk-tk-,xtk-exp⁡λ-1Γα+1t-tkα≤μVσtk-1tk-1,xtk-1exp⁡λ-1Γα+1t-tkα+tk-tk-1α≤μ2Vσtk-1-tk-1-,xtk-1-exp⁡λ-1Γα+1t-tkα+tk-tk-1α≤⋯≤μNσt0,TfVσt0t0,xt0exp⁡λ-1Γα+1t-tkα+tk-tk-1α+⋯+t1-t0α.By Lemma 14 and μ>1, we have(29)Vσtt,xt≤Vσt0t0,xt0exp⁡Tf-t0Tαln⁡μ+Nσt0,Tf+11-αTf-t0αλ-1Γα+1.According to Lemma 13, (29) can be rewritten as(30)Vσtt,xt≤Vσt0t0,xt0exp⁡Tf-t0Tαln⁡μ+λ-11-αNσt0,Tf+1+αTf-t0Γα+1=Vσt0t0,xt0exp⁡Tf-t0Tαln⁡μ+1-αλ-1Γα+1Tf-t0Tα+λ-11-α+αTf-t0Γα+1.From (15) and (21), for t∈[0,Tf), we have(31)Vσtt,xt≥ξ1xTtε,(32)Vσtt,xt≤ξ2xTt0δexp⁡TfTαln⁡μ+1-αλ-1Γα+1TfTα+λ-11-α+αTfΓα+1.From (7), (31), and (32), we obtain(33)xTtε≤ξ2ξ1xTt0δexp⁡TfTαln⁡μ+1-αλ-1Γα+1TfTα+λ-11-α+αTfΓα+1≤ξ2ξ1exp⁡TfTαln⁡μ+1-αλ-1Γα+1TfTα+λ-11-α+αTfΓα+1.Substituting (19) into (33), one has(34)xTtε<1.From Definition 11, we conclude that system (5) is GCFTS with respect to (δ,ε,Tf,σ(t)).
Next, we will give the fractional-order guaranteed cost value of system (5).
According to (24) and Vσ(tk)≤μσ(tk)Vσ(tk-)(tk-,x(tk-)), for t∈[tk,tk+1), we can obtain(35)Vσtt,xt≤Vσtktk,xtk+λΓα∫tktt-sα-1Vσts,xsds-1Γα∫tktt-sα-1xTsR1ds≤μVσtk-tk-,xtk-+λΓα∫tktt-sα-1Vσts,xsds-1Γα∫tktt-sα-1xTsR1ds≤μVσtk-1tk-1,xtk-1+λμΓα∫tk-1tkt-sα-1Vσts,xsds-μΓα∫tk-1tkt-sα-1xTsR1ds+λΓα∫tktt-sα-1Vσts,xsds-1Γα∫tktt-sα-1xTsR1ds≤⋯≤μNσt0,tVσt0t0,xt0+λμNσt0,tΓα∫t0t1t-sα-1Vσts,xsds-μNσt0,tΓα∫t0t1t-sα-1xTsR1ds+λμNσt0,tkΓα∫t1t2t-sα-1Vσts,xsds-μNσt0,tkΓα∫t1t2t-sα-1xTsR1ds+⋯+λΓα∫tktt-sα-1Vσts,xsds-1Γα∫tktt-sα-1xTsR1ds.Let t=Tf; then (35) can be turned into(36)Vσtt,xt≤μNσt0,TfVσt0t0,xt0+λμNσt0,TfΓα∫t0Tft-sα-1Vσts,xsds-1Γα∫t0Tft-sα-1xTsR1ds.By Vσ(t)(t,x(t))≥0, we can get(37)1Γα∫t0Tft-sα-1xTsR1ds≤μNσt0,TfVσt0t0,xt0+λμNσt0,TfΓα∫t0Tft-sα-1Vσts,xsds.From (32) and (37), we have(38)1Γα∫t0Tft-sα-1xTsR1ds≤μNσt0,Tfξ2xTt0δ+λμNσt0,TfΓα∫t0Tft-sα-1ξ2xTt0δexp⁡TfTαln⁡μ+1-αλ-1Γα+1TfTα+λ-11-α+αTfΓα+1ds≤ξ2μTf/Tα+λξ2μTf/TαΓαexp⁡TfTαln⁡μ+1-αλ-1Γα+1+λ-11-α+αTfΓα+1∫t0Tft-sα-1ds≤ξ2μTf/Tα+λξ2μTf/TαΓαexp⁡TfTαln⁡μ+1-αλ-1Γα+1+λ-11-α+αTfΓα+11αTfα.Then, the guaranteed cost value of system (5) is given by(39)J=1Γα∫0tt-sα-1xTsR1ds≤J∗=ξ2μTf/Tα+λξ2μTf/Tα1Γαexp⁡TfTαln⁡μ+1-αλ-1Γα+1+λ-11-α+αTfΓα+11αTfα.According to Definition 11, we can conclude that system (5) is GCFTS with respect to (δ,ε,Tf,σ(t)). Thus, the proof is completed.Corollary 16.
ReplaceDt0Ctαx(t) by Dt0RLtαx(t) in Theorem 15. If the conditions in Theorem 15 are held, then the FOPSS (5) is GCFTS with respect to (δ,ε,Tf,σ(t)), and the guaranteed cost value is (20).Proof.
According to (4) and Lemma 1, we can obtain(40)Dtαt0CVσtt,xt+xTtR1≤Dt0RLtαVσtt,xt+xTtR1≤xTtAσtTvσt+R1≤λVσtt,xt.Similar to the proof process of Theorem 15, we can obtain the same results and the proved process is omitted.


### 3.2. Guaranteed Cost Finite-Time Controller Design

In this section, we focus on the problem of guaranteed cost finite-time 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.


#### 3.2.1. State Feedback Controller Design

Consider system (5); under the controller u(t)=K1σ(t)x(t), the corresponding closed-loop system is given by(41)Dtαt0Cxt=Aσt+BσtK1σtxt,yt=Cσtxt,0<α<1.According to Lemma 4, to guarantee the positivity of system (41), Ap+BpK1p should be Metzler matrices, ∀p∈N_. Theorem 17 gives some sufficient conditions to guarantee that the closed-loop system (41) is GCFTS.Theorem 17.
Consider the FOPSS (41). For given constants Tf and λ(λ>1) and vectors δ≻ε≻0, R1≻0, and R2≻0, if there exist constants ξ1, ξ2, and μ(μ≥1) and positive vectors vp and p∈N_, such that (15), (16), (18), and the following conditions hold:(42)Ap+BpK1pareMetzlermatrices,ApTvp+R1+fp+K1pTR2⪯λvp,vp≺R1+K1pTR2,where fp=K1pTBpTvp, then, under the ADT scheme (19), the resulting closed-loop system (41) is GCFTS with respect to (δ,ε,Tf,σ(t)), and the cost value is given by(43)J=1Γα∫0tt-sα-1xTsR1+xTsK1pTsR2ds≤J∗=ξ2μTf/Tα+λξ2μTf/Tα1Γαexp⁡TfTαln⁡μ+1-αλ-1Γα+1+λ-11-α+αTfΓα+11αTfα.Proof.
By Lemma4, we get that Ap+GpK1p are Metzler matrices for each p∈N_. According to Lemma 5, system (41) is positive if ∀p∈N_, Bp are all nonnegative. Replacing Ap in (14) with Ap+BpK1p and letting fp=K1pTBpTvp, μ(μ≥1) satisfy (16), then, under the ADT scheme (19), we easily obtain that the resulting closed-loop system (41) is GCFTS with respect to (δ,ε,Tf,σ(t)) and the guaranteed cost value is given by (43).
The proof is completed.


#### 3.2.2. Static Output Feedback Controller Design

Consider system (5), under the controller u(t)=K2σ(t)y(t), the corresponding closed-loop system is given by(44)Dt0Ctαxt=Aσt+BσtK2σtCσtxt,yt=Cσtxt,0<α<1.According to Lemma 4, to guarantee the positivity of system (44), Ap+BpK2pCp should be Metzler matrices, ∀p∈N_. Theorem 18 gives some sufficient conditions to guarantee that the closed-loop system (44) is GCFTS.Theorem 18.
Consider the FOPSS (44). For given constants Tf and λ(λ>1) and vectors δ≻ε≻0, R1≻0, and R2≻0, if there exist constants ξ1, ξ2, and μ(μ≥1) and positive vectors vp and p∈N_, such that (15), (16), (18), and the following conditions hold:(45)Ap+BpK2pCpareMetzlermatrices,ApTvp+R1+fp+CpTK2pTR2⪯λvp,vp≺R1+CpTK2pTR2,where fp=CpTK2pTBpTvp, then, under the ADT scheme (19), the resulting closed-loop system (44) is GCFTS with respect to (δ,ε,Tf,σ(t)), and the cost value is given by(46)J=1Γα∫0tt-sα-1xTsR1+xTsCpTK2σTR2ds≤J∗=ξ2μTf/Tα+λξ2μTf/Tα1Γαexp⁡TfTαln⁡μ+1-αλ-1Γα+1+λ-11-α+αTfΓα+11αTfα.Proof.
By Lemma4, we get that Ap+GpK2pCp are Metzler matrices for each p∈N_. According to Lemma 5, system (44) is positive if ∀p∈N_, Bp and Cp are all nonnegative. Replacing Ap in (14) with Ap+BpK2pCp and letting fp=CpTK2pTBpTvp, μ(μ≥1) satisfy (16), then, under the ADT scheme (19), we easily obtain that the resulting closed-loop system (44) is GCFTS with respect to (δ,ε,Tf,σ(t)) and the guaranteed cost value is given by (46).
The proof is completed.Next, an algorithm is presented to obtain the feedback gain matricesK1p (or K2p) and p∈N_.Algorithm 19.Step
1. Input the matrices Ap, Bp, Cp, R1, and R2.Step
2. By adjusting the parameter λ and solving the inequalities in Theorem 17 (or Theorem 18) via linear programming, we can get the solutions vp, K1p (or K2p), and fp.Step
3. Then, f~p=K1pTBpTvp (or f~p=CpTK2pTBpTvp) is obtained. If fp-f~p⪰0 and Ap+BpK1p (or Ap+BpK2pCp) are Metzler matrices, then K1p (or K2p) are admissible. Otherwise, return to Step2.


## 3.1. Guaranteed Cost Finite-Time Stability Analysis

In this subsection, we will focus on the problem of GCFTS for FOPSS (5) with u(t)≡0.Theorem 15.
Consider system (5) with u(t)≡0. Given positive constants Tf and λ(λ>1) and vectors δ≻ε≻0 and R1≻0, if there exist positive constants ξ1, ξ2, and μ(μ≥1) and positive vectors vp and p∈N_, such that the following inequalities hold:(14)ApTvp+R1⪯λvp,(15)ξ1ε≺vp≺ξ2δ,(16)vp⪯μvq,(17)vp≺R1,(18)1-α+αTfλ-1Γα+1<ln⁡ξ1ξ2then under the following ADT scheme(19)Tα>Tα∗=Tfln⁡μ+1-αλ-1/Γα+1ln⁡ξ1/ξ2-1-α+αTfλ-1/Γα+1the FOPSS (5) is GCFTS with respect to (δ,ε,Tf,σ(t)) and the guaranteed cost value of system (5) with u(t)=0 is given by(20)J=1Γα∫0tt-sα-1xTsR1+uTsR2ds≤J∗=ξ2μTf/Tα+λξ2μTf/Tα1Γαexp⁡TfTαln⁡μ+1-αλ-1Γα+1+λ-11-α+αTfΓα+11αTfα.Proof.
Construct the multiple linear type Lyapunov-Krasovskii functional for system (5) as follows:(21)Vσtt,xt=xTtvσt,where vp∈R+n, ∀p∈N_.
Denotet0,t1,…,tk as the switching instants over the interval [0,Tf]. Along the trajectory of system (5) with u(t)≡0, we have(22)Dtαt0CVσtt,xt=xTtAσtTvσt.From (14), we obtain(23)Dtαt0CVσtt,xt+xTtR1=xTtAσtTvσt+R1≤λVσtt,xt.Taking the fractional integral Dt-αt0C to both sides of (23) during the period [tk,t) for t∈[tk,tk+1) leads to(24)Vσtt,xt≤Vσtktk,xtk+λΓα∫tktt-sα-1Vσts,xsds-1Γα∫tktt-sα-1xTsR1ds.From (17) and (21), (24) can be rewritten as(25)Vσtt,xt≤Vσtktk,xtk+λΓα∫tktt-sα-1Vσts,xsds-1Γα∫tktt-sα-1xTsvσtds=Vσtktk,xtk+λ-1Γα∫tktt-sα-1Vσts,xsds.According to Lemma 12 and the properties of Gamma function Γα, for t∈[tk,tk+1), we have(26)Vσtt,xt≤Vσtktk,xtkexp⁡λ-1Γα∫tktt-sα-1ds=Vσtktk,xtkexp⁡λ-1αΓαt-tkα=Vσtktk,xtkexp⁡λ-1Γα+1t-tkα.For t∈[tk,tk+1), Vσ(tk)≤μσ(tk)Vσ(tk-)(tk-,x(tk-)) is easily obtained from (16) and (21). Noting that exp⁡(λ-1/Γα+1)t-tkα>0, we have(27)Vσtt,xt≤μVσtk-tk-,xtk-exp⁡λ-1Γα+1t-tkα.Then, for t∈[t0,Tf), we can obtain from (26) and (27) that(28)Vσtt,xt≤Vσtktk,xtkexp⁡λ-1Γα+1t-tkα≤μVσtk-tk-,xtk-exp⁡λ-1Γα+1t-tkα≤μVσtk-1tk-1,xtk-1exp⁡λ-1Γα+1t-tkα+tk-tk-1α≤μ2Vσtk-1-tk-1-,xtk-1-exp⁡λ-1Γα+1t-tkα+tk-tk-1α≤⋯≤μNσt0,TfVσt0t0,xt0exp⁡λ-1Γα+1t-tkα+tk-tk-1α+⋯+t1-t0α.By Lemma 14 and μ>1, we have(29)Vσtt,xt≤Vσt0t0,xt0exp⁡Tf-t0Tαln⁡μ+Nσt0,Tf+11-αTf-t0αλ-1Γα+1.According to Lemma 13, (29) can be rewritten as(30)Vσtt,xt≤Vσt0t0,xt0exp⁡Tf-t0Tαln⁡μ+λ-11-αNσt0,Tf+1+αTf-t0Γα+1=Vσt0t0,xt0exp⁡Tf-t0Tαln⁡μ+1-αλ-1Γα+1Tf-t0Tα+λ-11-α+αTf-t0Γα+1.From (15) and (21), for t∈[0,Tf), we have(31)Vσtt,xt≥ξ1xTtε,(32)Vσtt,xt≤ξ2xTt0δexp⁡TfTαln⁡μ+1-αλ-1Γα+1TfTα+λ-11-α+αTfΓα+1.From (7), (31), and (32), we obtain(33)xTtε≤ξ2ξ1xTt0δexp⁡TfTαln⁡μ+1-αλ-1Γα+1TfTα+λ-11-α+αTfΓα+1≤ξ2ξ1exp⁡TfTαln⁡μ+1-αλ-1Γα+1TfTα+λ-11-α+αTfΓα+1.Substituting (19) into (33), one has(34)xTtε<1.From Definition 11, we conclude that system (5) is GCFTS with respect to (δ,ε,Tf,σ(t)).
Next, we will give the fractional-order guaranteed cost value of system (5).
According to (24) and Vσ(tk)≤μσ(tk)Vσ(tk-)(tk-,x(tk-)), for t∈[tk,tk+1), we can obtain(35)Vσtt,xt≤Vσtktk,xtk+λΓα∫tktt-sα-1Vσts,xsds-1Γα∫tktt-sα-1xTsR1ds≤μVσtk-tk-,xtk-+λΓα∫tktt-sα-1Vσts,xsds-1Γα∫tktt-sα-1xTsR1ds≤μVσtk-1tk-1,xtk-1+λμΓα∫tk-1tkt-sα-1Vσts,xsds-μΓα∫tk-1tkt-sα-1xTsR1ds+λΓα∫tktt-sα-1Vσts,xsds-1Γα∫tktt-sα-1xTsR1ds≤⋯≤μNσt0,tVσt0t0,xt0+λμNσt0,tΓα∫t0t1t-sα-1Vσts,xsds-μNσt0,tΓα∫t0t1t-sα-1xTsR1ds+λμNσt0,tkΓα∫t1t2t-sα-1Vσts,xsds-μNσt0,tkΓα∫t1t2t-sα-1xTsR1ds+⋯+λΓα∫tktt-sα-1Vσts,xsds-1Γα∫tktt-sα-1xTsR1ds.Let t=Tf; then (35) can be turned into(36)Vσtt,xt≤μNσt0,TfVσt0t0,xt0+λμNσt0,TfΓα∫t0Tft-sα-1Vσts,xsds-1Γα∫t0Tft-sα-1xTsR1ds.By Vσ(t)(t,x(t))≥0, we can get(37)1Γα∫t0Tft-sα-1xTsR1ds≤μNσt0,TfVσt0t0,xt0+λμNσt0,TfΓα∫t0Tft-sα-1Vσts,xsds.From (32) and (37), we have(38)1Γα∫t0Tft-sα-1xTsR1ds≤μNσt0,Tfξ2xTt0δ+λμNσt0,TfΓα∫t0Tft-sα-1ξ2xTt0δexp⁡TfTαln⁡μ+1-αλ-1Γα+1TfTα+λ-11-α+αTfΓα+1ds≤ξ2μTf/Tα+λξ2μTf/TαΓαexp⁡TfTαln⁡μ+1-αλ-1Γα+1+λ-11-α+αTfΓα+1∫t0Tft-sα-1ds≤ξ2μTf/Tα+λξ2μTf/TαΓαexp⁡TfTαln⁡μ+1-αλ-1Γα+1+λ-11-α+αTfΓα+11αTfα.Then, the guaranteed cost value of system (5) is given by(39)J=1Γα∫0tt-sα-1xTsR1ds≤J∗=ξ2μTf/Tα+λξ2μTf/Tα1Γαexp⁡TfTαln⁡μ+1-αλ-1Γα+1+λ-11-α+αTfΓα+11αTfα.According to Definition 11, we can conclude that system (5) is GCFTS with respect to (δ,ε,Tf,σ(t)). Thus, the proof is completed.Corollary 16.
ReplaceDt0Ctαx(t) by Dt0RLtαx(t) in Theorem 15. If the conditions in Theorem 15 are held, then the FOPSS (5) is GCFTS with respect to (δ,ε,Tf,σ(t)), and the guaranteed cost value is (20).Proof.
According to (4) and Lemma 1, we can obtain(40)Dtαt0CVσtt,xt+xTtR1≤Dt0RLtαVσtt,xt+xTtR1≤xTtAσtTvσt+R1≤λVσtt,xt.Similar to the proof process of Theorem 15, we can obtain the same results and the proved process is omitted.


## 3.2. Guaranteed Cost Finite-Time Controller Design

In this section, we focus on the problem of guaranteed cost finite-time 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.


### 3.2.1. State Feedback Controller Design

Consider system (5); under the controller u(t)=K1σ(t)x(t), the corresponding closed-loop system is given by(41)Dtαt0Cxt=Aσt+BσtK1σtxt,yt=Cσtxt,0<α<1.According to Lemma 4, to guarantee the positivity of system (41), Ap+BpK1p should be Metzler matrices, ∀p∈N_. Theorem 17 gives some sufficient conditions to guarantee that the closed-loop system (41) is GCFTS.Theorem 17.
Consider the FOPSS (41). For given constants Tf and λ(λ>1) and vectors δ≻ε≻0, R1≻0, and R2≻0, if there exist constants ξ1, ξ2, and μ(μ≥1) and positive vectors vp and p∈N_, such that (15), (16), (18), and the following conditions hold:(42)Ap+BpK1pareMetzlermatrices,ApTvp+R1+fp+K1pTR2⪯λvp,vp≺R1+K1pTR2,where fp=K1pTBpTvp, then, under the ADT scheme (19), the resulting closed-loop system (41) is GCFTS with respect to (δ,ε,Tf,σ(t)), and the cost value is given by(43)J=1Γα∫0tt-sα-1xTsR1+xTsK1pTsR2ds≤J∗=ξ2μTf/Tα+λξ2μTf/Tα1Γαexp⁡TfTαln⁡μ+1-αλ-1Γα+1+λ-11-α+αTfΓα+11αTfα.Proof.
By Lemma4, we get that Ap+GpK1p are Metzler matrices for each p∈N_. According to Lemma 5, system (41) is positive if ∀p∈N_, Bp are all nonnegative. Replacing Ap in (14) with Ap+BpK1p and letting fp=K1pTBpTvp, μ(μ≥1) satisfy (16), then, under the ADT scheme (19), we easily obtain that the resulting closed-loop system (41) is GCFTS with respect to (δ,ε,Tf,σ(t)) and the guaranteed cost value is given by (43).
The proof is completed.


### 3.2.2. Static Output Feedback Controller Design

Consider system (5), under the controller u(t)=K2σ(t)y(t), the corresponding closed-loop system is given by(44)Dt0Ctαxt=Aσt+BσtK2σtCσtxt,yt=Cσtxt,0<α<1.According to Lemma 4, to guarantee the positivity of system (44), Ap+BpK2pCp should be Metzler matrices, ∀p∈N_. Theorem 18 gives some sufficient conditions to guarantee that the closed-loop system (44) is GCFTS.Theorem 18.
Consider the FOPSS (44). For given constants Tf and λ(λ>1) and vectors δ≻ε≻0, R1≻0, and R2≻0, if there exist constants ξ1, ξ2, and μ(μ≥1) and positive vectors vp and p∈N_, such that (15), (16), (18), and the following conditions hold:(45)Ap+BpK2pCpareMetzlermatrices,ApTvp+R1+fp+CpTK2pTR2⪯λvp,vp≺R1+CpTK2pTR2,where fp=CpTK2pTBpTvp, then, under the ADT scheme (19), the resulting closed-loop system (44) is GCFTS with respect to (δ,ε,Tf,σ(t)), and the cost value is given by(46)J=1Γα∫0tt-sα-1xTsR1+xTsCpTK2σTR2ds≤J∗=ξ2μTf/Tα+λξ2μTf/Tα1Γαexp⁡TfTαln⁡μ+1-αλ-1Γα+1+λ-11-α+αTfΓα+11αTfα.Proof.
By Lemma4, we get that Ap+GpK2pCp are Metzler matrices for each p∈N_. According to Lemma 5, system (44) is positive if ∀p∈N_, Bp and Cp are all nonnegative. Replacing Ap in (14) with Ap+BpK2pCp and letting fp=CpTK2pTBpTvp, μ(μ≥1) satisfy (16), then, under the ADT scheme (19), we easily obtain that the resulting closed-loop system (44) is GCFTS with respect to (δ,ε,Tf,σ(t)) and the guaranteed cost value is given by (46).
The proof is completed.Next, an algorithm is presented to obtain the feedback gain matricesK1p (or K2p) and p∈N_.Algorithm 19.Step
1. Input the matrices Ap, Bp, Cp, R1, and R2.Step
2. By adjusting the parameter λ and solving the inequalities in Theorem 17 (or Theorem 18) via linear programming, we can get the solutions vp, K1p (or K2p), and fp.Step
3. Then, f~p=K1pTBpTvp (or f~p=CpTK2pTBpTvp) is obtained. If fp-f~p⪰0 and Ap+BpK1p (or Ap+BpK2pCp) are Metzler matrices, then K1p (or K2p) are admissible. Otherwise, return to Step2.


## 3.2.1. State Feedback Controller Design

Consider system (5); under the controller u(t)=K1σ(t)x(t), the corresponding closed-loop system is given by(41)Dtαt0Cxt=Aσt+BσtK1σtxt,yt=Cσtxt,0<α<1.According to Lemma 4, to guarantee the positivity of system (41), Ap+BpK1p should be Metzler matrices, ∀p∈N_. Theorem 17 gives some sufficient conditions to guarantee that the closed-loop system (41) is GCFTS.Theorem 17.
Consider the FOPSS (41). For given constants Tf and λ(λ>1) and vectors δ≻ε≻0, R1≻0, and R2≻0, if there exist constants ξ1, ξ2, and μ(μ≥1) and positive vectors vp and p∈N_, such that (15), (16), (18), and the following conditions hold:(42)Ap+BpK1pareMetzlermatrices,ApTvp+R1+fp+K1pTR2⪯λvp,vp≺R1+K1pTR2,where fp=K1pTBpTvp, then, under the ADT scheme (19), the resulting closed-loop system (41) is GCFTS with respect to (δ,ε,Tf,σ(t)), and the cost value is given by(43)J=1Γα∫0tt-sα-1xTsR1+xTsK1pTsR2ds≤J∗=ξ2μTf/Tα+λξ2μTf/Tα1Γαexp⁡TfTαln⁡μ+1-αλ-1Γα+1+λ-11-α+αTfΓα+11αTfα.Proof.
By Lemma4, we get that Ap+GpK1p are Metzler matrices for each p∈N_. According to Lemma 5, system (41) is positive if ∀p∈N_, Bp are all nonnegative. Replacing Ap in (14) with Ap+BpK1p and letting fp=K1pTBpTvp, μ(μ≥1) satisfy (16), then, under the ADT scheme (19), we easily obtain that the resulting closed-loop system (41) is GCFTS with respect to (δ,ε,Tf,σ(t)) and the guaranteed cost value is given by (43).
The proof is completed.


## 3.2.2. Static Output Feedback Controller Design

Consider system (5), under the controller u(t)=K2σ(t)y(t), the corresponding closed-loop system is given by(44)Dt0Ctαxt=Aσt+BσtK2σtCσtxt,yt=Cσtxt,0<α<1.According to Lemma 4, to guarantee the positivity of system (44), Ap+BpK2pCp should be Metzler matrices, ∀p∈N_. Theorem 18 gives some sufficient conditions to guarantee that the closed-loop system (44) is GCFTS.Theorem 18.
Consider the FOPSS (44). For given constants Tf and λ(λ>1) and vectors δ≻ε≻0, R1≻0, and R2≻0, if there exist constants ξ1, ξ2, and μ(μ≥1) and positive vectors vp and p∈N_, such that (15), (16), (18), and the following conditions hold:(45)Ap+BpK2pCpareMetzlermatrices,ApTvp+R1+fp+CpTK2pTR2⪯λvp,vp≺R1+CpTK2pTR2,where fp=CpTK2pTBpTvp, then, under the ADT scheme (19), the resulting closed-loop system (44) is GCFTS with respect to (δ,ε,Tf,σ(t)), and the cost value is given by(46)J=1Γα∫0tt-sα-1xTsR1+xTsCpTK2σTR2ds≤J∗=ξ2μTf/Tα+λξ2μTf/Tα1Γαexp⁡TfTαln⁡μ+1-αλ-1Γα+1+λ-11-α+αTfΓα+11αTfα.Proof.
By Lemma4, we get that Ap+GpK2pCp are Metzler matrices for each p∈N_. According to Lemma 5, system (44) is positive if ∀p∈N_, Bp and Cp are all nonnegative. Replacing Ap in (14) with Ap+BpK2pCp and letting fp=CpTK2pTBpTvp, μ(μ≥1) satisfy (16), then, under the ADT scheme (19), we easily obtain that the resulting closed-loop system (44) is GCFTS with respect to (δ,ε,Tf,σ(t)) and the guaranteed cost value is given by (46).
The proof is completed.Next, an algorithm is presented to obtain the feedback gain matricesK1p (or K2p) and p∈N_.Algorithm 19.Step
1. Input the matrices Ap, Bp, Cp, R1, and R2.Step
2. By adjusting the parameter λ and solving the inequalities in Theorem 17 (or Theorem 18) via linear programming, we can get the solutions vp, K1p (or K2p), and fp.Step
3. Then, f~p=K1pTBpTvp (or f~p=CpTK2pTBpTvp) is obtained. If fp-f~p⪰0 and Ap+BpK1p (or Ap+BpK2pCp) are Metzler matrices, then K1p (or K2p) are admissible. Otherwise, return to Step2.


## 4. Numerical Examples

In this section, two examples are given to illustrate the effectiveness of the two controllers proposed above.Example 1.
Under the state feedback controlleru(t)=K1σ(t)x(t), consider the FOPSS (5) with the parameters as follows: (47)A1=-6212-5112-5,B1=0.20.30.2,C1=2.532T,R1=211,ε=0.40.20.3,A2=-411.51-4221.8-5,B2=0.30.30.2,C2=324T,R2=122,δ=212.Let α=0.9, μ=1.1, and λ=1.12. Solving the inequalities in Theorem 17 by linear programming, we have(48)v1=2.01881.32131.5412,v2=2.08831.26231.5862,ξ1=5.2301,ξ2=1.1471,K11=1.03900.17542.0385,K12=1.94330.24810.9922.It is easy to verify that Ap+BpK1p are Metzler matrices, ∀p∈N_. Then, according to (19), we can obtain Tα∗=2.8233s. Choosing Tα=2.9s>Tα∗. Under the state feedback controller, the simulation results are shown in Figures 1–3, the initial conditions of system (5) are x(0)=[0.20.30.1]T, which satisfy xT(0)δ≤1. The state trajectories of the closed-loop system with ADT are shown in Figure 1. The switching signal σ(t) with ADT is depicted in Figure 2. Figure 3 plots the evolution of xT(t)ε of system (5). The cost value is J∗=62.2654, which can be obtained by (43).Figure 1
State trajectories of closed-loop system (5).Figure 2
Switching signal of system (5) with ADT.Figure 3
The evolution ofxT(t)ε of system (5).Example 2.
A fractional-order positive electrical circuit system was modeled in [28]. Accordingly, a switching type positive fractional-order electrical circuit system could be modeled by (5). Under the static output feedback controller u(t)=K2σ(t)y(t), consider system (5) with the parameters as follows: (49)A1=-200-1,B1=2001,C1=1001,R1=21.6,ε=0.010.02,A2=-100-3,B2=1002,C2=1001,R2=1.21,δ=0.81.Let α=0.9, μ=1.08, and λ=1.1. Solving the inequalities in Theorem 18 by linear programming, we have(50)v1=1.01561.2353,v2=1.07651.2241,ξ1=6.8058,ξ2=1.3769,K21=0.28230.36840.28230.3684,K22=0.20090.31740.20090.3174.It is easy to verify that Ap+CpBpK2p are Metzler matrices, ∀p∈N_. Then, according to (19), we can obtain Tα∗=1.3591s. Choose Tα=1.4s>Tα∗. Under the static output feedback controller, the simulation results are shown in Figures 4–6, and the initial conditions of system (5) are x(0)=[0.50.3]T, which satisfy xT(0)δ≤1. The state trajectories of the closed-loop system (5) with ADT are shown in Figure 4. The switching signal σ(t) with ADT is depicted in Figure 5. Figure 6 plots the evolution of xT(t)ε of system (5). The cost value is J∗=107.2803, which can be obtained by (46).Figure 4
State trajectories of closed-loop system (5).Figure 5
Switching signal of system (5) with ADT.Figure 6
The evolution ofxT(t)ε of system (5).


## 5. Conclusions

This paper has dealt with the problem of guaranteed cost finite-time control for FOPSS. A novel fractional-order cost function is introduced. By using ADT approach and constructing multiple linear copositive Lyapunov functions, two kinds of controllers are designed, and some sufficient conditions are obtained to guarantee that the closed-loop system is GCFTS. Such sufficient conditions can be solved by linear programming. Finally, two examples are provided to show the effectiveness of the proposed method.



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*Source: 2695894-2017-05-24.xml*