Estimation and synthesis of reachable set for switched linear systems

Automatica 63 (2016) 122–132

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Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Estimation and synthesis of reachable set for switched linear systems✩ Yong Chen a,1 , James Lam a , Baoyong Zhang b a

Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong

b

School of Automation, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China

article

info

Article history: Received 23 July 2014 Received in revised form 3 August 2015 Accepted 2 October 2015 Available online 11 November 2015 Keywords: Bounded peak disturbances Discrete-time switched linear systems Genetic Algorithm (GA) Linear matrix inequality (LMI) Reachable set estimation State-feedback control

abstract This paper focuses on the problems of reachable set estimation and state-feedback controller design for discrete-time switched linear systems under bounded peak disturbances. For the reachable set estimation problem, a Lyapunov-based inequality is developed based on the multiple Lyapunov strategy. By choosing appropriate Lyapunov functions, the ellipsoidal reachable set estimation conditions of discrete-time switched linear systems are obtained. In order to make the estimated ellipsoids as small as possible, three optimization approaches are proposed. Specifically, the Genetic Algorithm is used to search for the optimal parameters satisfying the obtained reachable set estimation conditions. In addition, the statefeedback controller design problem for discrete-time switched linear systems is considered. The function of the controller is to manipulate the reachable set of the closed-loop system to lie within a given ellipsoid or make the reachable set small. Finally, the effectiveness of the obtained results is verified through some numerical examples. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Switched systems have been extensively studied in the past few decades. Typically, a switched system consists of a number of subsystems and there is a switching signal determining which subsystem is active. Owing to the multi-system feature, switched systems have strong modeling capacity to practical systems. Moreover, in control synthesis, switching control is regarded as an effective control strategy to improve system performance. To date, many basic problems concerning switched systems have been considered in the literature, see, e.g., Daafouz, Riedinger, and Iung (2002), Liberzon (2003), Liberzon, Hespanha, and Morse (1999), Lin and Antsaklis (2009) and Zhang, Wang, and Chen (2009) for stability analysis and Chen, Zhang, Karimi, and Zhao (2011), Zhang and Shi (2009) and Zhao and Hill (2008) for H∞ controller design. As a fundamental concept in control theory, the reachable set has received many researchers’ attention. The reachable set of a system is defined as the set containing all system states reachable

✩ This work is supported by GRF HKU 7137/13E. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Hideaki Ishii under the direction of Editor Ian R. Petersen. E-mail addresses: [email protected] (Y. Chen), [email protected] (J. Lam), [email protected] (B. Zhang). 1 Tel.: +852 67314900.

http://dx.doi.org/10.1016/j.automatica.2015.10.033 0005-1098/© 2015 Elsevier Ltd. All rights reserved.

from the origin for a prescribed class of system disturbances. For continuous-time switched linear systems, a preliminary analysis of reachable set was given in Sun, Ge, and Lee (2002). In that work, the switching signal and system disturbances were considered to be arbitrary, that is, switching may occur at any time and there is no restriction in the system disturbances. Under such assumptions, the reachable set of continuous-time switched linear systems can be recursively determined by conditions involving the system matrices. A computational method was also developed in Sun et al. (2002) to tackle the problem of reducing computing burdens of the proposed algorithm. For discrete-time switched linear systems, the corresponding results were presented in Ge, Sun, and Lee (2001). In addition, the structure of the reachable set of switched linear systems was studied in Petreczky (2006) by using a differential geometric approach. Besides the reachable set of a system under arbitrary disturbances, research efforts have also been focused on the reachable set of a system under bounded peak disturbances. For this problem, new challenges often arise since it is generally difficult to obtain the exact characterization of a reachable set. Usually, we have to determine a region that is as small as possible such that this region bounds the reachable set. This is referred to as the reachable set estimation problem. For systems with bounded peak disturbances, one of the recent methods for studying the reachable set estimation problem is the so-called ellipsoidal technique, which aims to determine ellipsoids containing the concerned reachable set. Such ellipsoids can be obtained by using the Lyapunov

Y. Chen et al. / Automatica 63 (2016) 122–132

method and linear matrix inequality (LMI) technique (Boyd, ElGhaoui, Feron, & Balakrishnan, 1994). Very recently, the problem of reachable set estimation of linear systems with both bounded peak disturbances and time delays has received growing attention. In Fridman and Shaked (2003), bounded and time-varying delays were considered in continuous-time linear systems, and the Lyapunov–Razumikhin method was applied to determine the bounding ellipsoids. In order to improve the results given in Fridman and Shaked (2003), the Lyapunov–Krasovskii functional method was developed in Kim (2008), where delay-dependent conditions were presented in terms of matrix inequalities involving only one non-convex parameter. In Zuo, Ho, and Wang (2010), a maximal Lyapunov–Krasovskii functional method was developed for the reachable set estimation of polytopic uncertain systems. When the lower bound of the delay is not zero, a delay decomposition strategy was applied to estimate the reachable set in Nam and Pathirana (2011). In addition, the reachable set bounding problem of discrete-time linear systems was studied in Lam, Zhang, Chen, and Xu (2015) and That, Nam, and Ha (2013). In Chen and Lam (2015) and Feng and Lam (2015), the reachable set estimation problem was considered for singular systems and periodic systems. Besides the reachable set, in Nam, Pathirana, and Trinh (2014), the authors investigated the convergence property of time-delay systems with bounded disturbances under non-zero initial conditions. On the other hand, it is necessary to consider the control synthesis problem for meeting the design requirements related to the reachable set. Two problems are worth considering in this aspect. The first one is to design a controller such that the reachable set of the closed-loop system is contained in a given ellipsoid. In practice, the given ellipsoid may be imposed for safety reasons or due to other special requirements. The second problem is to design a controller such that the reachable set of the closed-loop system is contained in an ellipsoid that is as small as possible. In Zhang, Lam, and Xu (2014), the controller synthesis problem for distributed delay systems has been studied. However, to the best of the authors’ knowledge, the reachable set estimation and controller design problems have not been considered for systems with arbitrary switching. This paper aims to solve this open problem by employing multiple Lyapunov functions. The organization of this paper is as follows. Section 2 gives the problem formulation. The main results of the reachable set estimation and controller design for discrete-time switched linear systems are presented in Section 3. In this section, the ellipsoidal reachable set estimation conditions are first presented. In order to make the bounding ellipsoids as small as possible, three optimization approaches are developed. Genetic Algorithm is adopted to obtain the optimal parameters in the reachable set estimation conditions. In addition, the state-feedback controller design problem is also studied in this section. Some numerical examples are given in Section 4 to verify the obtained results. Finally, Section 5 gives the conclusions of this paper. Notations: The superscript ‘‘T ’’ represents the matrix transposition, Rn stands for the n-dimensional Euclidean space. In symmetric block matrices, the notation ‘‘*’’ is used as an ellipsis for terms that are induced by symmetry. The notation P > 0(P ≥ 0) indicates that P is a real symmetric and positive (semi-) definite matrix. The identity matrix and zero matrix are represented by I and 0 respectively. If their dimensions are not stated explicitly, matrices are assumed to be compatible for algebraic operations.

Consider the following discrete-time switched linear system: (1)

where xk ∈ R is the state vector, switching signal σ (k) is a piecewise constant function of the time and it takes its value in nx

the set I = {1, . . . , N }, N ≥ 1 is the number of subsystems. Ai , and Bω,i , i ∈ I are constant system matrices with appropriate dimensions. ωk ∈ Rnω is the bounded peak disturbance vector satisfying

ωkT ωk ≤ ω2 ,

∀k ≥ 0,

(2)

ω > 0 is a scalar. Remark 1. It should be pointed out that ωk is the exogenous disturbance, the peak of ωk is normally independent of switching signal σ (k) in practice. However, even if in the case that the peak of ωk is dependent on the switching signal σ (k), that is; ωσT (k),k ωσ (k),k ≤ ω2σ (k) , ∀k ≥ 0, where ωi > 0, i ∈ I are scalars. We can make the exogenous disturbance independent of the switching signal through the following transformation. Let wk = ωσ (k),k ω/ωσ (k) and Bω,σ (k) = Bω,σ (k) ωσ (k) /ω, then system (1) can be rewritten as xk+1 = Aσ (k) xk + Bω,σ (k) wk , where wkT wk ≤ ω2 , ∀k ≥ 0. Thus without loss of generality, we assume that the peak of the exogenous disturbance ωk is independent of the switching signal σ (k). The aim of the reachable set estimation problem is to determine a region as small as possible to bound the reachable set of system (1) under bounded peak disturbances satisfying (2). The concerned reachable set is defined as

ℜx , {xk | x0 = 0, xk , ωk satisfy (1), (2), k ≥ 0} .

(3)

It is well known that in the system analysis and synthesis of switched systems, common quadratic Lyapunov function method will lead to a certain level of conservatism. In order to reduce the conservatism, multiple Lyapunov function method, which is a type of non-quadratic Lyapunov function method, is widely used in the system analysis and synthesis of switched systems. In our work, the multiple Lyapunov function method is adopted to solve the reachable set estimation and controller design problems for switched systems. The reachable set is bounded within some ellipsoids. We make use of the bounding ellipsoid in the following form: E (P ) , x ∈ Rnx | xT Px ≤ 1, P > 0 .





(4)

Lemma 1 is a basic reachable set estimation tool for discretetime switched systems and it will be used in later development. Lemma 1. Consider system (1) under bounded peak disturbance (2). Let {Vi (xk ), i ∈ I} be a set of Lyapunov functions satisfying Vi (0) = 0 and Vi (xk ) > 0, ∀xk ̸= 0, i ∈ I. If there exist scalars 0 < αj,i < 1 such that ∀(i, j) ∈ I × I, Vj (xk+1 ) − αj,i Vi (xk ) −

1 − αj,i

ω2

ωkT ωk ≤ 0,

(5)

then system (1) is globally uniformly asymptotically stable (GUAS) and we have Vi (xk ) ≤ 1, ∀i ∈ I for all x0 satisfying Vi (x0 ) ≤ 1, ∀i ∈ I. Proof. On one hand, under non-zero initial conditions, condition (5) guarantees the global uniform asymptotic stability of system (1). Specifically, when ωk = 0, condition (5) can be rewritten as Vj (xk+1 ) − αj,i Vi (xk ) ≤ 0,

2. Problem formulation

xk+1 = Aσ (k) xk + Bω,σ (k) ωk ,

123

∀(i, j) ∈ I × I.

(6)

Noting that 0 < αj,i < 1, then we have Vj (xk+1 ) − Vi (xk ) ≤ − 1 − αj,i Vi (xk ) < 0,





∀(i, j) ∈ I × I, (7)

for xk ̸= 0, which implies that system (1) is GUAS (Daafouz et al., 2002).

124

Y. Chen et al. / Automatica 63 (2016) 122–132

On the other hand, from (2), (5) and 0 < αj,i < 1, we can easily get Vj (xk+1 ) − αj,i Vi (xk ) ≤

1 − αj,i



ωkT ωk

ω2 ≤ 1 − αj,i ,

then system (1) is GUAS and the reachable set ℜx can be bounded by the intersection of a set of ellipsoids given by

∀(i, j) ∈ I × I,

Proof. Choose a quadratic Lyapunov function as

Vj (xk+1 ) − 1 ≤ αj,i (Vi (xk ) − 1) ,

∀(i, j) ∈ I × I.

(9)

Vσ (k) (xk ) = xTk Pσ (k) xk .

Vσ (k) (xk ) − 1

  ≤ ασ (k),σ (k−1) Vσ (k−1) (xk−1 ) − 1   ≤ ασ (k),σ (k−1) ασ (k−1),σ (k−2) Vσ (k−2) (xk−2 ) − 1   ≤ ασ (k),σ (k−1) · · · ασ (1),σ (0) Vσ (0) (x0 ) − 1 .

H = Vj (xk+1 ) − αj,i Vi (xk ) −

Vσ (k) (xk ) − 1 ≤ 0.

H = xTk+1 Pj xk+1 − αj,i xTk Pi xk −

(12) 

Remark 2. Lemma 1 provides a basic tool for estimating the reachable set of discrete-time switched systems under bounded peak disturbances. This lemma is applicable for both linear and nonlinear system cases because the system structure is not specified when establishing the proof. In addition, if the Lyapunov functions are chosen to be common, that is, Vi (xk ) = V (xk ), ∀i ∈ I, the condition in Lemma 1 reduces to those developed recently in Lam et al. (2015) and That et al. (2013) for discrete-time non-switched linear systems (except the unnecessary zero initial conditions have been removed). Thus, Lemma 1 can be regarded as a generalization of reachable set estimation method from nonswitched systems to switched systems. It is worth mentioning that the methods in Lam et al. (2015) and That et al. (2013) developed based on common Lyapunov function are also applicable for estimating the reachable set of discrete-time switched systems, but those methods are considerably conservative because, as it is now well known, multiple Lyapunov function methods always lead to less conservative results for switched systems than the common ones under the same framework. Considering this, our study is focused on the use of multiple Lyapunov function method and attempts to give relaxed results. 3. Main results 3.1. Reachable set estimation conditions The reachable set estimation problem for system (1) is studied in this subsection. On the basis of Lemma 1 presented in the previous section, the following result is obtained. Theorem 1. Consider system (1) under bounded peak disturbance (2). If there exist matrices Pi > 0, i ∈ I and scalars 0 < αj,i < 1 such that ∀(i, j) ∈ I × I,



∗



ATi Pj Bω,i 1 − αj,i  ≤ 0, BTω,i Pj Bω,i − I 2

ω

ωkT ωk .

(16)

(13)

1 − αj,i

ωkT ωk ω2  T   = Ai xk + Bω,i ωk Pj Ai xk + Bω,i ωk 1 − αj,i T − αj,i xTk Pi xk − ωk ωk ω2 = xTk ATi Pj Ai xk + ωkT BTω,i Pj Ai xk + xTk ATi Pj Bω,i ωk 1 − αj,i T ωk ωk + ωkT BTω,i Pj Bω,i ωk − αj,i xTk Pi xk − ω2  T   x x = k J k , ωk ωk

As σ (k) can be chosen as an arbitrary subsystem number, (11) means

ATi Pj Ai − αj,i Pi

ω2

Then along the system trajectory of (1), we have

(11)

∀i ∈ I .

1 − αj,i

(10)

For x0 satisfying Vi (x0 ) ≤ 1, ∀i ∈ I, (10) yields

This completes the proof.

(15)

Suppose that σ (k + 1) = j, σ (k) = i, (i, j ∈ I). For simplicity, we define

For any time of k, we have



(14)

(8)

which implies that

Vi (xk ) ≤ 1,

E (Pi ).

i∈I

(17)

where



ATi Pj Ai − αj,i Pi

J =

∗



ATi Pj Bω,i 1 − αj,i  . I BTω,i Pj Bω,i − 2

(18)

ω

Condition (13) implies H ≤ 0, ∀(i, j) ∈ I × I. According to Lemma 1, system (1) is GUAS, and we have Vi (xk ) = xTk Pi xk ≤ 1, ∀i ∈ I for x0 = 0. Thus the reachable set of system (1) can be bounded by the intersection of a set of ellipsoids given in (14).  Remark 3. Note that the conditions obtained in Theorem 1 are not expressed in LMIs, because they contain products of decision variables. However, if the scalars αj,i are fixed, the conditions in (13) are linear with respect to the matrix variables Pi , and thus they can be easily checked by applying standard software. As mentioned previously, the common quadratic Lyapunov function method is also applicable for estimating the reachable set of the switched linear system (1), but the results obtained in this way may be more conservative than the multiple Lyapunov method. In order to show this point clearly, we present a corollary of Theorem 1, whose proof can be established by restricting the variables Pi and αj,i in Theorem 1 to be P and α respectively. Corollary 1. Consider system (1) under bounded peak disturbance (2). If there exist matrix P > 0 and scalar 0 < α < 1 such that ∀i ∈ I,



ATi PAi − α P



∗



ATi PBω,i 1 − α  ≤ 0, BTω,i PBω,i − I 2

(19)

ω

then system (1) is GUAS and the reachable set ℜx can be bounded by the ellipsoid E (P ).

Y. Chen et al. / Automatica 63 (2016) 122–132

Fig. 1. Illustration of Method 1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Remark 4. Both Theorem 1 and Corollary 1 provide sufficient conditions for estimating the reachable set of discrete-time switched linear systems with bounded peak disturbances. Theorem 1 is based on multiple Lyapunov function method, while Corollary 1 is induced by common quadratic Lyapunov function method. Although the number of decision variables involved in Corollary 1 is less than that in Theorem 1, the condition in Corollary 1 is considerably conservative because common variables P and α are required to cater for different system matrices. Such conservatism is obviously reduced in Theorem 1. The conservatism reduction is further shown numerically in the example section. 3.2. Optimization methods for the bounding ellipsoids Note that any Pi > 0 and P > 0 obtained in Theorem 1 and Corollary 1, respectively, can be used to define the bounding ellipsoids containing the reachable set of system (1). Usually, those ellipsoids are expected to be as small as possible. Thus we have to develop some optimization methods to minimize those ellipsoids. Following are possible ways to do that.

125

Fig. 2. Illustration of Method 2. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

of the projection of the bounding ellipsoid E (Pi ) along direction vl , namely, the blue line segment in Fig. 2, is minimized. Here, by choosing different directions vl , we can obtain different optimal bounding ellipsoids. Then the reachable set can be bounded by the intersection of those obtained ellipsoids, namely  l,opt ), where Pil,opt represent the optimal Pi obtained 1≤l≤ND E (Pi in direction vl . As a special case, vl can be chosen to be the standard bases of the Euclidean space. In this case, the length of the projection of the bounding ellipsoid along different axes is minimized. Remark 5. In Nam, Pathirana, and Trinh (2013), the smallest upper bound of the absolute value of any given linear functional of the state vector was determined. The smallest upper bound can also be obtained by Method 2 with a small modification. Let vl = U, where U ∈ Rnx ×1 is a given non-zero constant vector and is not necessarily a unit vector. Then from (22) and the conditions in Theorem 1, we can obtain 1 ≥ xT Pi x ≥ δi,l xT UU T x, thus the , where δi,l,max is smallest upper bound of |z | = U T x is √ 1 δi,l,max

the maximum allowable value obtained from (22).

3.2.1. Method 1 The first method proposed here has been widely used in the reachable set estimation for non-switched systems. Specifically, as in Boyd et al. (1994), Fridman and Shaked (2003), Kim (2008), Nam and Pathirana (2011) and Zuo et al. (2010), Pi is regarded as a decision variable with respect to (13) and

3.2.3. Method 3 For Method 3, a set of directions represented by unit vectors {vl ∈ Rnx | l = 1, 2, . . . , ND } is also needed. Then the bounding ellipsoid E (Pi ) can be minimized by maximizing scalars δi,l > 0 subject to

Pi ≥ δi I ,

vlT Pi vl ≥ δi,l .

(20)

where δi > 0 is a scalar to be maximized. For this method, we have 1 ≥ xT Pi x ≥ δi xT x,

∀x ∈ E (Pi ),

(24)

For this method, we have 1 (21)

ellipsoid E (Pi ) is contained in the ball B(δi ) = thus the x ∈ Rnx | δi xT x ≤ 1 . The major axis of the bounding ellipsoid E (Pi ), namely, the blue line segment in Fig. 1, is minimized.

δi,l

vlT Pi vl ≥ 1,

this indicates that the point √

(25) 1

δi,l,max

vl , where δi,l,max is the

(22)

maximum allowable value obtained from (24), is in the boundary of the bounding ellipsoid E (Pi ). This can be illustrated in Fig. 3 for second order systems. Similar to Method 2, different optimal bounding ellipsoids can be obtained by choosing different directions. Then the reachable set can be bounded by the intersection of those obtained ellipsoids,  l,opt l,opt namely, 1≤l≤ND E (Pi ), where Pi represents the optimal Pi obtained in direction vl .

 2 1 ≥ x Pi x ≥ δi,l x v v = δi,l vlT x , ∀x ∈ E (Pi ). (23)   Note that vlT x is a scalar and vlT x = |vl | |x| |cos β|, where β is the angle between vl and vector x, represent the length of the projection of vector x along direction vl . In this method, the length

Remark 6. Note that Method 2 is different from Method 3. In Method 2, the length of the projection of the bounding ellipsoid E (Pi ), namely, the blue line segment in Fig. 2, is minimized. While in Method 3, the distance between the intersection points of the bounding ellipsoid E (Pi ) and the straight line represented by λvl (λ is a scalar), namely, the blue line segment in Fig. 3, is minimized.

3.2.2. Method 2 Suppose a set of directions represented by unit vectors {vl ∈ Rnx | l = 1, 2, . . . , ND } is given. Then the bounding ellipsoid E (Pi ) can be minimized by maximizing scalars δi,l > 0 subject to Pi ≥ δi,l vl vlT . For this method, we have T

T

T l l x

126

Y. Chen et al. / Automatica 63 (2016) 122–132

Fig. 3. Illustration of Method 3. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Remark 7. The above three optimization approaches illustrate the process of minimizing one bounding ellipsoid, namely E (Pi ). For multiple Lyapunov function method, the number of bounding ellipsoids is more than one. Then for each ellipsoid, we can apply these three optimization approaches to minimize the ellipsoid, and the reachable set can be bounded by all the obtained ellipsoids.

Fig. 4. Procedure of GA.

3.3. Genetic algorithm

3.4. State-feedback controller design

Theorem 1 provides the reachable set bounding condition for switched systems under bounded disturbances. The bounding ellipsoids can be optimized through those optimization approaches. However, this process is based on given αj,i in Theorem 1. In order to obtain less conservative bounding ellipsoids for the reachable set, we have to determine the optimal αj,i . This purpose can be achieved by using Genetic Algorithm (GA). The GA is a stochastic search method motivated by biological evolution. For more details about GA, see Gen and Cheng (2000) and Winter, Periaux, Galan, and Cuesta (1996). The procedure of GA used in our work is shown in Fig. 4 and can be stated as follows.

Based on the obtained reachable set estimation results, this subsection will consider the state-feedback controller design for discrete-time switched linear systems under bounded peak disturbances. Consider system (1) with control input:

1 Randomly generate m sets of αj,i , (i, j) ∈ I × I, each αj,i set is an individual of the GA population. 2 For each individual, by applying Theorem 1 and one of the optimization approaches, we can obtain the optimal δi or δi,l . Let δ = max{δi , i ∈ I} or δ = max{δi,l , i ∈ I, l ∈ {1, 2, . . . , ND }}, the obtained δ can be regarded as the fitness value of this individual. 3 Set a maximum generation number. If this number is reached, then GA is terminated, otherwise, continue the GA process. 4 Select the individuals for the next generation with a probability according to the fitness value of each αj,i set. 5 Perform crossover and mutation operations to the new generation according to some given probabilities. 6 Go to Step 2. Remark 8. In our work, the population size is constant for all generations. In Step 1, the initial population is randomly generated subject to uniform distribution. In Step 2, if an αj,i set is infeasible for Theorem 1, then a small fitness value is assigned for this individual (say, a fraction of the best fitness value in the population). In Step 4, the elitism preservation strategy and the roulette selection operator are adopted, that means the best individual is retained in the next generation. In Step 5, the crossover and mutation operations are subject to given probabilities, in addition, we adopt single-point crossover operator and single-point mutation operator to reduce the computational cost of GA. The gene index for the crossover and mutation operations is assumed to be a uniformly distributed random variable. In practice, if the system model is complex, we may use some more efficient genetic operators, e.g., multiple-point crossover and mutation operators (Gen & Cheng, 2000).

xk+1 = Aσ (k) xk + Bω,σ (k) ωk + Bu,σ (k) uk ,

(26)

where uk ∈ R is control input vector and Bu,i , i ∈ I are known constant real matrices. For system (26), we will design a statefeedback controller in the following form: nu

uk = Kσ (k) xk ,

(27)

where Kσ (k) is a switched controller gain to be determined. Using this controller, the closed-loop system is obtained as xk+1 = Aσ (k) + Bu,σ (k) Kσ (k) xk + Bω,σ (k) ωk .





(28)

In this subsection, two problems will be investigated.

• Problem 1. Given an ellipsoid E (Po ), where Po > 0 is a known constant real matrix, design a state-feedback controller (27) such that the reachable set of closed-loop system (28) is contained in the ellipsoid E (Po ). • Problem 2. Design a state-feedback controller (27) such that the reachable set of closed-loop system (28) is contained in an ellipsoid, and the ellipsoid should be as small as possible. Remark 9. The open-loop system is not necessarily stable in the state-feedback controller design problem. One can design a statefeedback controller for Problem 1 and Problem 2 as long as conditions in Theorems 2 and 3 are satisfied. 3.4.1. Solutions to Problem 1 Theorem 2. Consider system (26) under bounded peak disturbance (2). Given a matrix Po > 0, if there exist matrices Xi > 0, Wi , i ∈ I and scalars 0 < αj,i < 1 such that ∀(i, j) ∈ I × I,



−Xj  ∗  ∗

Ai Xi + Bu,i Wi −αj,i Xi

∗

Bω,i 0



1 − αj,i  ≤ 0, I − 2



(29)

ω

∃m ∈ I such that Xm ≤ Po−1 ,

(30)

then there exists a state-feedback controller in the form of (27) such that closed-loop system (28) is GUAS and the reachable set ℜx can be

Y. Chen et al. / Automatica 63 (2016) 122–132

bounded by the given ellipsoid E (Po ), and the desired controller can be obtained with the controller gains given by Ki = Wi Xi−1 .

(31)

Proof. Applying the change of variable Wi = Ki Xi to (29), we can get

 −X j  ∗  ∗





Ai + Bu,i Ki Xi −αj,i Xi

Bω,i 0



≤ 0, 1 − αj,i  − I 2

∗

∀(i, j) ∈ I × I. (32)



ω

Pre-multiplying and post-multiplying diag Xj , Xi



−1

−1

, I to (32), 

−1

= Pi , we have   Pj Ai + Bu,i Ki Pj Bω,i  −αj,i Pi 0 ≤ 0, 1 − αj,i  ∗ − I 2 ω

and let Xi

 −P j  ∗  ∗



∀(i, j) ∈ I × I. (33)

By using the Schur complement equivalence, inequality (33) is equivalent to



Ai + Bu,i Ki

T

Pj Ai + Bu,i Ki − αj,i Pi





∗ 

Ai + Bu,i Ki

T

Bω,i Pj Bω,i −

T



Pj Bω,i

1 − αj,i

ω2

I

 ≤ 0,

∀(i, j) ∈ I × I.

(34)

Remark 10. Problem 1 is different from the reachable set estimation problem. When estimating the reachable set, we want to make the bounding region as small as possible. While for Problem 1, we just want to obtain a state-feedback controller such that the reachable set of closed-loop system is contained in a given ellipsoid. Essentially, Problem 1 is a feasibility problem and thus GA is inappropriate for it. For this problem, we can search the feasible αj,i with fixed step size from 0 to 1. In practice, one may make some simplifications to reduce the computational cost, for instance, let αj,i = αi and search for the feasible αi . Similar to Corollary 1, one can arrive at the common quadratic Lyapunov method for Problem 1. Corollary 2. Consider system (26) under bounded peak disturbance (2). Given a matrix Po > 0, if there exist matrices X > 0, Wi , i ∈ I and a scalar 0 < α < 1 such that ∀i ∈ I, Ai X + Bu,i Wi −α X

X ≤ Po−1 ,

∗

(35)

ω

(36)

then there exists a state-feedback controller in the form of (27) such that closed-loop system (28) is GUAS and the reachable set ℜx can be bounded by the given ellipsoid E (Po ), and the desired controller can be obtained with the controller gains given by Ki = Wi X −1 .

Theorem 3. Consider system (26) under bounded peak disturbance (2). If there exist matrices Xi > 0, Wi , i ∈ I and scalars 0 < αi,j < 1 such that (29) holds, then there exists a state-feedback controller in the form of (27) such that closed-loop system (28) is GUAS and the reachable  set ℜx can be bounded by the intersection of a set of ellipsoids i∈I E (Xi−1 ), and the desired controller can be obtained with the controller gains given by (31). The proof of Theorem 3 can be established by following a similar line as in the proof of Theorem 2. Next, we have to apply those optimization approaches to minimize the reachable set of the closed-loop system. For Method 1, in order to minimize the ith ellipsoid, we can maximize δi > 0 with respect to (29) and

(37)

(38)

This inequality is equivalent to Xi ≤ δ i I

(39)

where δ i = δ . Therefore, the optimal ellipsoid E (Xi ) can be i obtained by minimizing δ i > 0 with respect to (29) and (39). When minimizing different ellipsoids, we can obtain different δ i and different controllers. Overall, the controller we should adopt is the controller corresponding to the minimum δ i . −1

1

Remark 11. In order to apply Method 2 or Method 3, an inequality of the form Xi−1 ≥ δi,l vl vlT or vlT Xi−1 vl ≥ δi,l should be imposed. However, it is difficult to get Xi from these two inequalities directly, thus Method 2 and Method 3 are inappropriate when solving Problem 2. In addition, similar to the reachable set estimation problem, GA can be used to search for the optimal αj,i in Theorem 3. 4. Illustrative examples 4.1. Example 1 Consider a second order discrete-time switched linear system (1) with the following parameters:



0 A1 = −0.2



Bω,i 0  1 − α  ≤ 0, − 2 I

3.4.2. Solutions to Problem 2 In order to solve Problem 2, we can apply the optimization methods developed earlier. Note that by using the multiple Lyapunov method, the reachable set of switched systems can be bounded within the intersection of several ellipsoids. When designing a state-feedback controller, it is possible to minimize each ellipsoid separately. However, minimizing different ellipsoids will obtain different controllers, thus we should choose one controller among them according to some criteria.

Xi−1 ≥ δi I .

According to Theorem 1, system (28) is GUAS and the reachable set ℜx is contained in the intersection of the ellipsoids i∈I E (Pi ),  namely i∈I E (Xi−1 ). Furthermore, if there exists m ∈ I such that Xm ≤ Po−1 , then we have xT Po x ≤ xT Xm−1 x ≤ 1. This means that ellipsoid E (Xm−1 ) is contained in the ellipsoid E (Po ). Thus the reachable set ℜx can be bounded by the given ellipsoid E (Po ). 

 −X  ∗  ∗

127

0.7 , −0.6



0.2 Bω,1 = , −0.4





Bω,2

 −0.6 A2 = −0.7   −0.6 = , 0.4

0.4 , 0.2



(40)

the disturbance of system is bounded by (2) with ω = 1. In the following, Theorem 1 and Corollary 1 will be used to estimate the reachable set of this system. Method 1 and Method 2 will be applied to minimize the bounding ellipsoids. For Theorem 1, GA is used to search for the optimal αj,i , and the population size is 20. For Corollary 1, the optimal α is searched with a constant step size of 0.0001.

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Theorem 1:

α1,1 = 0.7986, α1,2 = 0.8289, α2,1 = 0.8319, α2,2 = 0.7195, δ1 = 0.1056, δ2 = 0.2010,   0.1056 −0.0005 P1 = , −0.0005 0.2291   0.2212 −0.0421 P2 = . −0.0421 0.0879

(43)

Corollary 1:

α = 0.9589, δ = 0.0138,  0.0139 P = −0.0011

Fig. 5. Fitness value of each generation.

 −0.0011 . 0.0162

(44)

2nd step (v2 = [01]T ): Theorem 1:

α1,1 = 0.8265, α1,2 = 0.7942, α2,1 = 0.8663, α2,2 = 0.6529, δ1 = 0.2756, δ2 = 0.0925,   0.0578 0.0005 P1 = , 0.0005 0.2756   0.1757 −0.0275 P2 = . −0.0275 0.0970

(45)

Corollary 1:

α = 0.9585, δ = 0.0166,  0.0135 P = −0.0014

Fig. 6. Bounding ellipsoids optimized by Method 1.

4.2. Example 2 (41)

Corollary 1:

α = 0.9593, δ = 0.0134,  0.0138 P = −0.0011

Consider a third order discrete-time switched linear system (1) with the following parameters: 0.15 0.22 0.24

 A1 =

 −0.0011 . 0.0162

(46)

The best and average fitness values of each generation in the computation process of GA are shown in Fig. 7. The bounding ellipsoids obtained by Theorem 1 and Corollary 1 are shown in Fig. 8. The state of system (40) is bounded by the intersection of the obtained ellipsoids. As can be seen from Figs. 6 and 8, the ellipsoids obtained by Theorem 1 are smaller than that by Corollary 1. Thus Theorem 1 provides less conservative results than Corollary 1.

Applying Method 1, we can obtain the following results: Theorem 1:

α1,1 = 0.8021, α1,2 = 0.9116, α2,1 = 0.8482, α2,2 = 0.7047, δ1 = 0.1288, δ2 = 0.0866,   0.1292 0.0056 P1 = , 0.0056 0.2185   0.1493 −0.0206 P2 = . −0.0206 0.0933

 −0.0014 . 0.0167

(42)

The best and average fitness values of each generation in the computation process of GA are shown in Fig. 5. The bounding ellipsoids obtained by Theorem 1 and Corollary 1 are shown in Fig. 6. As can be seen from Fig. 6, the state of system (40) is bounded by the intersection of the obtained ellipsoids. The switching signal is arbitrary and the system disturbance ωk is assumed to be uniformly distributed over [−1, 1]. When applying Method 2, vl is chosen to be the standard bases of the Euclidean space, we can obtain the following results: 1st step (v1 = [10]T ):

0.01 0.10 0.11

 −0.14 0 0.22

,

  −0.03 0.26 0.13 −0.21 0.12 , A2 = 0.01 0.07 0.14 0.31     0.32 0.21 Bω,1 = −0.15 , Bω,2 = 0.04 , 0.04 −0.13

(47)

the disturbance of system is bounded by (2) with ω = 1. In the following, Theorem 1 will be used to estimate the reachable set of this system. Method 3 will be applied to minimize the bounding ellipsoids. In order to reduce the computational cost, we let αj,i = αi in this example, and search for the optimal αi with constant step size 0.0001 from 0 to 1.

Y. Chen et al. / Automatica 63 (2016) 122–132

(a) 1st step.

129

(b) 2nd step. Fig. 7. Fitness value of each generation.

ellipsoids for the reachable set. The intersection of those ellipsoids is shown in Fig. 10. Note that the bounding region here is the intersection of the obtained ellipsoids. In order to illustrate the bounding region clearly, those ellipsoids are omitted in the figure. As can be seen from Fig. 10, the state of the system can be bounded by the obtained bounding region. The switching signal is arbitrary and the system disturbance ωk is assumed to be uniformly distributed over [−1, 1]. 4.3. Example 3 Consider a second order discrete-time switched linear system (26) with the following parameters:

  −0.3 1.3 , −1.1 1.2     0.2 −0.5 Bω,1 = , Bω,2 = , −0.4 0.4     0.2 0.1 Bu,1 = , Bu,2 = , 0.3 0.7 

A1 =

Fig. 8. Bounding ellipsoids optimized by Method 2.

1.5 0.8

1.2 , −0.6



A2 =

(49)

the disturbance of system is bounded by (2) with ω = 1. In the following, Theorems 2 and 3 will be used to design the statefeedback controller. Note that A1 is not a Schur stable matrix, thus the reachable set of the open-loop system cannot be bounded by any ellipsoid. For Problem 2, GA is used to search for the optimal αj,i in Theorem 3, and the population size is 20. Problem 1: Given 0.3 0.25

 Po = Fig. 9. Illustration of direction v(θ1 , θ2 ).

v(θ1 , θ2 ) =

cos(θ2 ) cos(θ1 ) cos(θ2 ) sin(θ1 ) , sin(θ2 )



(50)

by using Theorem 2, we can get a feasible result as follows.

In order to apply Method 3, we need a set of directions represented by unit vectors. For third order system (47), the unit vector can be chosen as



0.25 , 0 .7



(48)

where θ1 , θ2 ∈ [0, π ). Note that due to the symmetry of the ellipsoid, it is not necessary to consider θ1 and θ2 in [π , 2π ). In this π example, we choose 180 different θ1 and θ2 with a step size of 180 . The direction v(θ1 , θ2 ) can be illustrated in Fig. 9. By using Method 3 to minimize the ellipsoids subject to the conditions in Theorem 1, we can get 180 × 180 different bounding

α1,1 = α2,1 = 0.0500, α1,2 = α2,2 = 0.5250,   4.3195 −2.2263 X1 = , −2.2263 1.2431   4.5598 −1.9752 X2 = , −1.9752 1.1288   W1 = −16.7336 8.6253 ,   W2 = 12.3105 −5.9891 ,

(51)

and the corresponding state-feedback controller is obtained as K1 = −3.8700



0.0077 ,



K2 = 1.6587



 −2.4034 . (52)

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Y. Chen et al. / Automatica 63 (2016) 122–132

(a) 1st and 2nd viewing angle.

(b) 3rd and 4th viewing angle. Fig. 10. Bounding region obtained for Example2.

1.6853 −0.5857

 X1 =

X2 = 104 ×

 −0.5857 , 0.4062

5.9850 0.7575



W1 = −6.1607





2.3141 ,



W2 = 104 × 8.1064



0.7575 , 0.0959

1.0259 ,



(53)

and the corresponding state-feedback controller is obtained as K1 = −3.3588



0.8541 ,

K2 = −0.7661





6.0517 . (54)



When minimizing E (X2−1 ),

α1,1 = 0.5015, α2,1 = 0.3900, Fig. 11. System state of the closed-loop system for Problem 1.

Under this controller, the system state of the closed-loop system is shown in Fig. 11. The switched signal is arbitrary and the disturbance ωk is assumed to be uniformly distributed over [−1, 1]. As can be seen from the figure, the state of the closedloop system is bounded within the given ellipsoid E (Po ). Problem 2: For this problem, we want to make the reachable set of the closed-loop system as small as possible. This purpose can be achieved by applying Method 1. Using Method 1 to minimize E (X1−1 ) and E (X2−1 ), respectively, we can obtain the following results. When minimizing E (X1−1 ),

α1,1 = 0.4076, α2,1 = 0.4076, δ 1 = 1.9129,

α1,2 = 0.5953, α2,2 = 0.2434,

α1,2 = 0.7928, α2,2 = 0.5455,

δ 2 = 1.8620,   0.8536 −0.5154 X1 = 103 × , −0.5154 0.3116   1.7116 −0.4758 X2 = , −0.4758 0.3538   W1 = 103 × −3.3076 1.9975 ,   W2 = 3.5398 −1.3750 ,

(55)

and the corresponding state-feedback controller is obtained as K1 = −3.3397



0.8861 ,

K2 = 1.5777





 −1.7642 . (56)

The best and average fitness values of each generation in the computation process of GA are shown in Fig. 12. In this example, the fitness value is δi = δ¯1 . As δ 1 > δ 2 , we choose the second i

Y. Chen et al. / Automatica 63 (2016) 122–132

(a) 1st step.

131

(b) 2nd step. Fig. 12. Fitness value of each generation.

Fig. 13. System state of the closed-loop system for Problem 2.

obtained controller. When applying this controller, the system state of the closed-loop system is shown in Fig. 13. As can be seen from the figure, the state of the closed-loop system is bounded within ellipsoid E (X2−1 ). 5. Conclusion In this paper, the problems of reachable set estimation and state-feedback controller design for discrete-time switched linear systems under bounded peak disturbances have been investigated. Multiple Lyapunov function method has been used to determine the bounding ellipsoids for the reachable set of the considered systems. In order to make the bounding ellipsoids as small as possible, three types of optimization approaches have been developed. GA was used to determine the optimal parameters in the reachable set estimation conditions. Moreover, the statefeedback controller design conditions have been obtained for the sake of manipulating the reachable set of the closed-loop system. The effectiveness of the obtained results has been verified through numerical examples. References Boyd, S., El-Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory. Philadelphia: SIAM. Chen, Y., & Lam, J. (2015). Estimation and synthesis of reachable set for discrete-time periodic systems. Optimal Control Applications and Methods, http://dx.doi.org/10.1002/oca.2211. published online.

Chen, Y., Zhang, L., Karimi, H., & Zhao, X. (2011). Stability analysis and H∞ controller design of a class of switched discrete-time fuzzy systems. In Proceedings of the 50th IEEE conference on decision and control and European control conference, CDC-ECC (pp. 6159–6164), Orlando, FL, USA. Daafouz, J., Riedinger, P., & Iung, C. (2002). Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach. Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 47, 1883–1887. Feng, Z., & Lam, J. (2015). On reachable set estimation of singular systems. Automatica, 52, 146–153. Fridman, E., & Shaked, U. (2003). On reachable sets for linear systems with delay and bounded peak inputs. Automatica, 39, 2005–2010. Ge, S., Sun, Z., & Lee, T. (2001). Reachability and controllability of switched linear discrete-time systems. Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 46, 1437–1441. Gen, M., & Cheng, R. (2000). Genetic algorithms and engineering optimization. New York: John Wiley & Sons. Kim, J. (2008). Improved ellipsoidal bound of reachable sets for time-delayed linear systems with disturbances. Automatica, 44, 2940–2943. Lam, J., Zhang, B., Chen, Y., & Xu, S. (2015). Reachable set estimation for discretetime linear systems with time delays. International Journal of Robust & Nonlinear Control, 25, 269–281. Liberzon, D. (2003). Switching in systems and control. Berlin: Birkhauser. Liberzon, D., Hespanha, J., & Morse, S. (1999). Stability of switched systems: a Liealgebraic condition. Systems & Control Letters, 37, 117–122. Lin, H., & Antsaklis, P. (2009). Stability and stabilizability of switched linear systems: a survey of recent results. Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 54, 308–322. Nam, P., & Pathirana, P. (2011). Further result on reachable set bounding for linear uncertain polytopic systems with interval time-varying delays. Automatica, 47, 1838–1841. Nam, P., Pathirana, P., & Trinh, H. (2013). Linear functional state bounding for perturbed time-delay systems and its application. IMA Journal of Mathematics and Control Information, http://dx.doi.org/10.1093/imamci/dnt039. Nam, P., Pathirana, P., & Trinh, H. (2014). Convergence within a polyhedron: controller design for time-delay systems with bounded disturbances. IET Control Theory & Application, 9, 905–914. Petreczky, M. (2006). Reachability of linear switched systems: differential geometric approach. Systems & Control Letters, 55, 112–118. Sun, Z., Ge, S., & Lee, T. (2002). Controllability and reachability criteria for switched linear systems. Automatica, 38, 775–786. That, N., Nam, P., & Ha, Q. (2013). Reachable set bounding for linear discrete-time systems with delays and bounded disturbances. Journal of Optimization Theory and Applications, 157, 96–107. Winter, G., Periaux, J., Galan, M., & Cuesta, P. (1996). Genetic algorithms in engineering and computer science. New York: John Wiley & Sons. Zhang, B., Lam, J., & Xu, S. (2014). Reachable set estimation and controller design for distributed delay systems with bounded disturbances. Journal of the Franklin Institute, 351, 3068–3088. Zhang, L., & Shi, P. (2009). Stability, l2 -gain and asynchronous control of discretetime switched systems with average dwell time. Institute of Electrical and Electronics Engineers.Transactions on Automatic Control., 54, 2192–2199. Zhang, L., Wang, C., & Chen, L. (2009). Stability and stabilization of a class of multimode linear discrete-time systems with polytopic uncertainties. IEEE Transactions on Industrial Electronics, 56, 3684–3692. Zhao, J., & Hill, D. (2008). On stability, L2 -gain and H∞ control for switched systems. Automatica, 44, 1220–1232. Zuo, Z., Ho, D., & Wang, Y. (2010). Reachable set bounding for delayed systems with polytopic uncertainties: The maximal Lyapunov-Krasovskii functional approach. Automatica, 46, 949–952.

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Yong Chen was born in Changsha, Hunan province, China, in 1988. He received his B.Eng. degree in Automation (ranking No. 10 among the 126 students of the major) from Harbin Institute of Technology (HIT), Harbin, Heilongjiang province, China, in 2010. He is now a Ph.D. candidate in the department of Mechanical Engineering at The University of Hong Kong. He has passed the CFA Level III exam in R the Chartered Financial Analyst⃝ program. His current research interests include switched systems, stochastic systems, periodic systems, fuzzy systems, positive systems in control theory, micro quadrotor in control engineering and empirical study of financial market.

Professor J. Lam received a B.Sc. (1st Hons.) degree in Mechanical Engineering from the University of Manchester, and was awarded the Ashbury Scholarship, the A.H. Gibson Prize, and the H. Wright Baker Prize for his academic performance. He obtained the M.Phil. and Ph.D. degrees from the University of Cambridge. He is a recipient of the Croucher Foundation Scholarship and Fellowship, the Outstanding Researcher Award of the University of Hong Kong, and the Distinguished Visiting Fellowship of the Royal Academy of Engineering. He is a Cheung Kong Chair Professor, Ministry of Education, China. Prior to joining the University of Hong Kong in 1993 where he is now Chair Professor of Control Engineering, Professor Lam held lectureships at the City University of Hong Kong and the University of Melbourne.

Professor Lam is a Chartered Mathematician, Chartered Scientist, Chartered Engineer, Fellow of Institute of Electrical and Electronic Engineers, Fellow of Institution of Engineering and Technology, Fellow of Institute of Mathematics and Its Applications, and Fellow of Institution of Mechanical Engineers. He is Editor-in-Chief of IET Control Theory and Applications and Journal of The Franklin Institute, Subject Editor of Journal of Sound and Vibration, Editor of Asian Journal of Control, Cogent Engineering, Associate Editor of Automatica, International Journal of Systems Science, International Journal of Applied Mathematics and Computer Science, Multidimensional Systems and Signal Processing, and Proc. IMechE Part I: Journal of Systems and Control Engineering. He is a member of the IFAC Technical Committee on Networked Systems. His research interests include model reduction, robust synthesis, delay, singular systems, stochastic systems, multidimensional systems, positive systems, networked control systems and vibration control. He is a Highly Cited Researcher (Thomson Reuters, 2014). Baoyong Zhang received his B.Sc. degree and M.Sc. degree from Qufu Normal University, Qufu, China, in 2003 and 2006, respectively, and Ph.D. degree from Nanjing University of Science and Technology (NJUST), Nanjing, China, in 2011. From March 2008 to June 2008, he was a Research Associate in the Department of Mechanical Engineering, the University of Hong Kong, Hong Kong. From November 2008 to July 2009, he was a Visiting Fellow at the School of Computing and Mathematics, the University of Western Sydney, Penrith NSW, Australia. From December 2011 to December 2012, he was a Postdoctoral Fellow in the Department of Mechanical Engineering, the University of Hong Kong, Hong Kong. He joined the School of Automation at NJUST as Lecturer in November 2010 and was selected as Young Professor of NJUST in August 2014. His research interests include robust control and filtering, time-delay systems, stochastic systems, switched systems, nonlinear systems, LPV systems and complex networks.