Robust stability and robust stabilizability for periodically switched linear systems

Applied Mathematics and Computation 361 (2019) 112–130

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Robust stability and robust stabilizability for periodically switched linear systems Do Duc Thuan a,∗, Le Van Ngoc b,c a

School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet Street, Hanoi, Vietnam Laboratory of Applied Mathematics and Computing, Posts and Telecommunications Institute of Technology, km10 Nguyen Trai Street, Hanoi, Vietnam c Faculty of Mathematics, Mechanics and Informatics, VNU University of Sciences, 334 Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam b

a r t i c l e

i n f o

a b s t r a c t

Keywords: Periodically switched linear systems Exponential stability Structured perturbations Robust stability Robust stabilizability Stability radius

In this paper, the problem of robust stability and robust stabilizability for periodically switched linear systems is studied. The stability radius involving structured perturbations acting on both coefficient matrices and switching moments is introduced and investigated. Some lower bounds for stability radii of periodically switched linear systems are provided. After that, we derive the notion of fast and slow stabilizability for these systems. Some characterizations for robust stabilizability under structured perturbations are established. Several examples are given to illustrate the obtained results. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Switched systems occur in diverse real-life applications, for example in control of mechanical systems, the automotive industry, air traffic and aircraft control, switching power converters, and many other fields, see the recent papers [8,9,23,30,34] and books [21,32], and the references cited therein. These systems are described by a family of subsystems and a rule that controls the switching between them and have attracted a good deal of attention from researchers in recent years, (see, e.g., [13,27,28,30,31]). Besides, periodic models are useful to describe periodic natural phenomena and also arise in linearisation of time-invariant dynamical systems around periodic orbits and when treating multi-rate systems where actuator and sensors operate at different (yet commensurable) frequencies. Thus, in practice, a periodically switched system may serve as an analytical model for a series of wide applications, for example, in electrical circuits, such as switched modulators and filters (see, e.g., [5,33]). In the mathematical setting, the linear switched systems with the periodically switching moments can be described in the form



x˙ (t ) = Aσ (t ) x(t ), x(t0 ) = x0 ,

Aσ (t ) ∈ A := {A1 , . . . , Am },

(1.1)

where the switching signal σ is a piecewise constant and periodic function from [t0 , +∞ ) into {1, . . . , m}, defined by σ (t ) = k for t ∈ [tk−1 + lT , tk + lT ), where k = 1, . . . , m, l = 0, 1, . . . . When studying these systems, stability and stabilizability are important properties to consider. It is known that there are many works devoted to stability and stabilizability analysis of periodically switched continuous-time or discrete-time linear systems, (see, e.g., [1,10–12,16,30]). ∗

Corresponding author. E-mail addresses: [email protected] (D.D. Thuan), [email protected] (L.V. Ngoc).

https://doi.org/10.1016/j.amc.2019.05.025 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.

D.D. Thuan and L.V. Ngoc / Applied Mathematics and Computation 361 (2019) 112–130

113

On the other hand, in recent years, the aspect of developing measures of stability robustness for linear uncertain systems with state space description has received significant attention in system and control theory. These measures can be characterized by stability radii. In its simple form, the stability radius of a given asymptotically stable system x˙ (t ) = Ax(t ) is the maximal γ > 0 for which all the perturbed systems of the form

x˙ (t ) = (A + DE )x(t ),  < γ ,

(1.2)

are asymptotically stable. Here,  is an unknown disturbance matrix, D and E are given matrices defining the structure of the perturbations. The problem of evaluating and calculating this stability radius is of great importance, from both theoretical and practical point of view and has attracted a lot of attention from researchers (see, e.g., [14,15,18,20,24,26] and the references given therein). For a systematic introduction to the topic, the interested readers are referred to the earlier work due to Hinrichsen and Pritchard [17] and their more recent monograph [19], which contains, along with rigorous theoretical developments, also an extensive literature review on the subject. It is remarkable that the similar problems have been considered for many other types of linear dynamical systems, including time-varying and time-delay systems, implicit systems, positive systems as well as linear systems in infinite-dimensional spaces, in both continuous-time and discrete-time models (see, e.g., [2,14,15,20,22,24]). On the basis of the above discussion, there arises a natural question whether one can define measures of stability and stabilizability robustness for periodically switched linear systems and, moreover, how to evaluate and calculate these measures. To the best of our knowledge, such kind of questions has not been addressed so far in the literature, although different aspects of robust analysis for stability of switched systems has been studied already by using the method of Lyapunov functions or linear matrix inequalities (LMIs) (see, e.g., [11,21,32,34]). The purpose of the present paper is to fill this gap. Firstly, we will define the notion of stability radius for the class of periodically switched linear systems and will establish some estimates for this radius under perturbations on both coefficient matrices and switching moments. The obtained estimates are shown to be an exact formula of the stability radius for some switched linear systems having special structures. Secondly, we give some evaluations for the measure of stabilizability robustness for periodically switched linear systems subjected to structured perturbations on coefficient matrices. Note that it is very difficult to find the exact formulas of stability radius and stabilizability radius for periodically switched linear systems under structured perturbations because there are no good characterization for stability and stabilizability of these systems. This is similar with linear time-varying singular systems which we have recently obtained some bounds for perturbations preserving stability (see [22]). The paper is organized as follows. In Section 2 we summarize some preliminary results on matrix analysis and optimizations. In Section 3 the notion of stability radius for periodically switched linear systems is introduced and some formula for estimating this radius are derived. In Section 4, we study robust stabilizability for periodically switched linear systems. We close the paper with a conclusion where some comments for the further work will be given. 2. Preliminaries We begin by recalling some notations and concepts which will be frequently used later. For k ∈ N, k := {1, 2, . . . , k} and Rn×m , Rn+×m denote the set of all n × m matrices with elements in R, R+ , respectively. For a matrix A ∈ Rn×n , A∗ is the conjugate transpose of A and A is called Hermitian if A = A∗ . For any matrix A ∈ Rn×n , the spectral radius and the spectral abscissa of A is denoted by ρ (A ) = max{|λ| : λ ∈ σ (A )}, and μ(A ) = max{λ : λ ∈ σ (A )}, respectively, where σ (A ) := {z ∈ C : det(zIn − A ) = 0} is the set of all eigenvalues of A. A matrix A ∈ Rn×n is called Hurwitz (Schur) if and only if μ(A) < 0 (ρ (A) < 1), respectively. A matrix M ∈ Rn×n is called a Metzler matrix if all off-diagonal elements of M are nonnegative. In particular, for a given matrix A = (ai j ) ∈ Rn×n , we associate the Metzler matrix M (A ) := (aˆi j ) ∈ Rn×n , where

aˆi j := |ai j |, i = j, i, j ∈ n;

aˆii := aii , i ∈ n.

The following results can be deduced by some elementary facts in linear algebra. Lemma 2.1. For all A ∈ Rn×n , we have eA  ≤ e

A+A∗ 2

.

∗

eA v, eA v = ( eA )∗ eA v, v = eA eA v, v = eA

∗ +A

v, v = ( e

A∗ +A 2

A+A∗

eA  = e 2 . Indeed, for all v, we have eA v2 = A∗ +A 2 ) v, v = e 2 v2 .

Remark 2.2. If A ∈ Kn×n is a normal matrix, i.e., AA∗ = A∗ A, then

Lemma 2.3. Let A, B be hermitian matrices, then (i) λmax (A + B ) ≤ λmax (A ) + λmax (B ); (ii) −λmax (A ) ≤ λmax (−A ); (iii) eA  = eλmax (A ) . Lemma 2.4. Let α , β , γ be given positive numbers, and

:= {(x, y ) ∈ R2 : 2xy + α x + β y − γ ≥ 0, x ≥ 0, y ≥ 0}.

114

D.D. Thuan and L.V. Ngoc / Applied Mathematics and Computation 361 (2019) 112–130

Then,

⎧ α+β ⎪ ⎪ ⎨ αβ + 2γ − 2 γ min {x + y} = β (x,y )∈ ⎪ ⎪ ⎩γ α

if

αβ + 2γ > ω2 ,

if

αβ + 2γ ≤ ω2 and ω = β > α ,

if

αβ + 2γ ≤ ω2 and ω = α > β ,

(2.1)

where ω = max{α , β}. The proofs of Lemma 2.1, Lemma 2.3 and Lemma 2.4 are given in Appendix. We summarize some properties of Metzler matrices which will be used in what follows. Theorem 2.5 [29]. Suppose that M ∈ Rn×n is a Metzler matrix. Then (i) (ii) (iii) (iv)

(Perron-Frobenius) μ(M) is an eigenvalue of M and there exists a nonnegative eigenvector x = 0 such that Mx = μ(M )x. Given α ∈ R, there exists a nonzero vector x ≥ 0 such that Mx ≥ α x if and only if μ(M) ≥ α . (tIn − M )−1 exists and is nonnegative if and only if t > μ(M). Given B ∈ Rn+×n , C ∈ Cn×n . Then

|C | ≤ B ⇒ μ(M + C ) ≤ μ(M + B ). Now, the periodically switched linear system (1.1) can be rewritten in the form

x˙ (t ) =

⎧ A1 x(t ), t0 + lT ≤ t < t1 + lT , ⎪ ⎪ ⎨A2 x(t ), t1 + lT ≤ t < t2 + lT ,

................................................

⎪ ⎪ ⎩Am x(t ), tm−1 + lT ≤ t < tm + lT , x(t0 ) = x0 ; l = 0, 1, . . . ; t ≥ t0 ,

(2.2)

where x(t ) ∈ Rn is unknown vector, A1 , A2 , . . . , Am ∈ Rn×n are given matrices (unnecessarily different), t0 < t1 < · · · < tm := t0 + T , are given switching moments, and T is a given period. Denote by tk = tk − tk−1 for all k = 1, . . . , m the so-called  activation durations. In what follows, we always assume that tk > 0 for all k = 1, . . . , m and m k=1 Ak  > 0. With a fixed n number x0 ∈ R and a periodic T, system (2.2) has an unique solution denoted by x(t; t0 , x0 ; T). Definition 2.6. System (2.2) is called exponentially stable if there exists constants α , K > 0 such that

x(t; t0 , x0 ; T ) ≤ Ke−α (t−t0 )

(2.3)

holds for all t ≥ t0 . Based on the Floquet theory, the author in [16] obtained the following stability criteria for periodically switched linear system (2.2). Theorem 2.7. System (2.2) is exponentially stable if and only if the matrix

R=

m

eAk tk = eAm tm eAm−1 tm−1 . . . eA1 t1

k=1

is Schur. Equivalently, system (2.2) is exponentially stable if and only if the matrix

Q=

 m 1 ln eAk tk T k=1

is Hurwitz.

3. Robust stability for periodically switched linear systems 3.1. Lower bounds for stability radii of periodically switched linear systems Now let us consider the periodically switched system (2.2) with structured perturbations of the form

D.D. Thuan and L.V. Ngoc / Applied Mathematics and Computation 361 (2019) 112–130

x˙ (t ) =

115

⎧ (A1 + D1 1 E1 )x(t ), t0 + lT ≤ t < t1 + lT ⎪ ⎪ ⎨(A2 + D2 2 E2 )x(t ), t1 + lT ≤ t < t2 + lT ..................................................................

(3.1)

⎪ ⎪ ⎩(Am + Dm m Em )x(t ), tm−1 + lT ≤ t < tm + lT x(t0 ) = x0 ; l = 0, 1, . . . ; t ≥ t0 ,

where Dk ∈ Kn×l , Ek ∈ Kq×n are given matrices determining the structure of the perturbations and  = diag(1 , . . . , m ); k ∈ Kl×q are unknown perturbations. These perturbations are endowed with the norm m

∗ :=

m

k=1

k :=

tk k .

k=1

For given matrices Bk ∈ Kn×n , k = 1, . . . , m, we consider the block diagonal matrix B= m k=1 Bk and define the following set of matrices:



UR := B =

m 

Bk : ρ ( R ) ≥ 1; R =

k=1

m



eBk tk .

k=1

According to Theorem 2.7, system (2.2) with the given data A = m / UR . Thus, k=1 Ak is exponentially stable if and only if A ∈ m the set UR consists of all matrices A = k=1 Ak , such that the corresponding system (2.2) is not exponentially stable. Clearly, n×n . Now we want to measure the structured UR is a closed subset in the space of all diagonal matrices m k=1 Ak , where Ak ∈ R distance to instability. Definition 3.1. Assume that (2.2) is exponentially stable. Then stability radius of (2.2) subjected to structured perturbations of the form (3.1) is defined by

rCD,E (A ) := inf{∗ : A + DE ∈ UR },

(3.2)

m

where DE =

k=1 Dk k Ek .

The corresponding matrix R for the perturbed system (3.1) becomes

 := R

m

e(Ak +Dk k Ek )tk .

k=1

Theorem 3.2. If system (2.2) with the data A = lowing inequality:

1 tk 2 max D j E j  m

rCD,E (A ) ≥ −

m

k=1

Ak is exponentially stable, then the stability radius (3.2) satisfies the fol-

λmax (Ak + A∗k ),

k=1

j∈m

or equivalently, if m

tk k  < −

k=1

1 tk 2 max D j .E j  m

λmax (Ak + A∗k )

k=1

j∈m

then the perturbed system (3.1) is still exponentially stable. m (A +D  E )t  k k k k k . We have Proof. Consider a perturbation DE = m k=1 Dk k Ek such that A + DE ∈ UR and let R := k=1 e

 m

) ≤ R  ≤  1 ≤ ρ (R 

 

e(Ak +Dk k Ek )tk  ≤

k=1

m

e(Ak +Dk k Ek )tk 

k=1

Using Lemmas 2.1 and 2.3, item (iii), we get

1≤

m  

e

A k + A ∗ + D k k E k + E ∗ ∗ D ∗ k k k k 2

 m tk ∗ ∗ ∗ ∗  = e 2 λmax (Ak +Ak +Dk k Ek +Ek k Dk ).

tk 

k=1

k=1

Further, Lemma 2.3, item (i) yields

1≤

m

e

tk 2

[λmax (Ak +A∗k )+λmax (Dk k Ek +Ek∗ ∗k D∗k )]

k=1

≤

m

e

tk 2

λmax (Ak +A∗k ) e

tk 2

k=1

From the last relation we find

0≤

m m

tk tk λmax (Ak + A∗k ) + λmax (Dk k Ek + Ek∗ ∗k D∗k ), 2 2 k=1

k=1

λmax (Dk k Ek +Ek∗ ∗k D∗k ) .

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D.D. Thuan and L.V. Ngoc / Applied Mathematics and Computation 361 (2019) 112–130

or equivalently,

0≤

m m

tk tk λmax (Ak + A∗k ) + λmax (Dk k Ek + Ek∗ ∗k D∗k ) 2 2 k=1

k=1

k=1

k=1

k=1

k=1

k=1

k=1

k=1

k=1

 m m

tk tk

≤ λmax (Ak + A∗k ) + λmax (Dk k Ek ) + λmax (Ek∗ ∗k D∗k ) 2 2  m m

tk tk

≤ λmax (Ak + A∗k ) + Dk k Ek + Ek∗ ∗k D∗k  2 2  m m

tk tk

≤ λmax (Ak + A∗k ) + Dk k Ek  + Ek∗ ∗k D∗k  2 2  m m

tk tk

≤ λmax (Ak + A∗k ) + Dk .k .Ek  + Ek∗ .∗k .D∗k  . 2 2 We have

0≤

m m

tk λmax (Ak + A∗k ) + tk Dk k Ek  2 k=1

k=1

k=1

k=1

  m m

tk ≤ λmax (Ak + A∗k ) + tk k  max Dk Ek  . 2 j=1...m Therefore, m

tk k  ≥ −

k=1

1 tk λmax (Ak + A∗k ). 2 max D j E j  m

(3.3)

k=1

j∈m

Combining (3.2) with (3.3), we come to the estimate

1 tk 2 max D j .E j  m

rCD,E (A ) ≥ −

k=1

j∈m

The proof is complete.

λmax (Ak + A∗k ).



Now, we derive the following corollary which is more easy to check the exponential stability of (2.2).  ∗ Corollary 3.3. Assume that m k=1 tk λmax (Ak + Ak ) < 0. Then system (2.2) is exponentially stable. m 1 t λmax (Ak + A∗k ). By the above proof, it implies that 2 max j∈m D j E j  k=1 k ρ (R ) = ρ (R) < 1. Thus, system (2.2) is exponentially stable.  Proof. We choose the perturbation  = 0 < −

Now, we provide a simple example to illustrate Theorem 3.2. Example 3.4. Consider a periodically switched system (2.2) with m = 2, t1 = t2 = 1, and

⎛

−3 ⎜1 A1 = ⎝ 1 1

1.01 −4 −1.01 0.001

1 1 −2 1

⎞

It is easy to see that

⎛

−6 ⎜2.01 ∗ A1 + A1 = ⎝ 2 1.1

2.01 −8 −0.01 0201

⎛

0.1 −4 0.2⎟ ⎜0 . 3 , A = −1 ⎠ 2 ⎝ 1 −3 1 2 −0.01 −4 0

1.02 −2 0.02 0.1

1 1 −3 1

⎞

⎞

0.1 1 ⎟ −1 ⎠ −2

⎛

1.1 −8 0.201⎟ ⎜1.32 ∗ , A + A2 = ⎝ 0 ⎠ 2 2 −6 1.1

1.32 −4 1.02 1.1

2 1.02 −6 0

⎞

1.1 1.1⎟ , 0 ⎠ −4

and hence λmax (A1 + A∗1 ) = −2.4019, λmax (A2 + A∗2 ) = −1.9047. By Corollary 3.3, system (2.2) is exponentially stable. Now assume that system (2.2) is subjected to structured perturbations of the form

⎛

−3 + δ1 ⎜ 1  A1 = ⎝ 1 + δ1 1

1.01 −4 −1.01 0.001

1 + δ1 1 −2 + δ1 1

⎞

⎛

0.1 − δ1 −4 + δ2 0.2 ⎟  ⎜0.3 − δ2 , A = −1 − δ1 ⎠ 2 ⎝ 1 −3 1 + δ2

1.02 + δ2 −2 − δ2 0.02 0.1 + δ2

1 − δ2 1 + δ2 −3 1 − δ2

⎞

0.1 1 ⎟ −1 ⎠ −2

D.D. Thuan and L.V. Ngoc / Applied Mathematics and Computation 361 (2019) 112–130

117

Fig. 1. A switching thermal systems.

 = A + D  E ,A  = A + D  E with It is easy to see in this model that A 1 1 1 1 1 2 2 2 2 2

⎡ ⎤

⎡

1 ⎢0 ⎥ D1 = ⎣ ⎦, E1 = 1 1 0

0

⎤

1 ! ⎢−1⎥ −1 , D2 = ⎣ ⎦, E2 = 1 0 1

1

1

−1

0

!

and 1 = δ1 , 2 = δ2 . By Theorem 3.2, we get

1 tk 2 max D j .E j  2

rCD,E (A ) ≥ −

λmax (Ak + A∗k ) = 0.7178,

k=1

j∈2

or equivalently, if |δ1 | + |δ2 | < 0.7178 then the perturbed system (3.1) is exponentially stable. Next, we consider a periodically switched system with both stable and unstable subsystems, while the activation durations tk are different. Example 3.5. Let m = 2, t1 = 2; t2 = 1, and

⎛

−3 ⎜0.04 A1 = ⎝ 0.11 −1

1.02 −2 0.3 0.02

−4 ⎜ 0.02 A2 = ⎝ −0.1 1

1.01 −1.01 1 −1

⎛

We have

⎛

−6 ⎜ 1.06 ∗ A1 + A1 = ⎝ 0.26 −0.7

⎞

0.15 0.12 −2 −0.5 0.03 −1 2.05 0.01

1.06 −4 0.42 1.02

⎛

⎞

0.3 1 " 1 ⎟ 1 ⎜0⎟ , D = , E = −1.02⎠ 1 ⎝−1⎠ 1 0 −4 0

⎞

#

0 0

1 . 0

⎛ ⎞

0.1 0 " 0.02⎟ −1 ⎜1⎟ , D = , E = 1 ⎠ 2 ⎝0 ⎠ 2 0 −3 1

⎞

0.26 0.42 −4 −1.52

0 −1

⎛

−0.7 −8 1.02 ⎟ ⎜ 1.03 ∗ , A + A2 = ⎝ −1.52⎠ 2 −0.07 −8 1.1

0 1

0 1

#

1 . 0

1.03 −2.02 0 −0.98

−0.07 0 4.1 1.01

⎞

1.1 −0.98⎟ . 1.01 ⎠ −6

Since,

λmax (A1 + A∗1 ) = −3.1943, λmax (A2 + A∗2 ) = 4.2016, by Theorem 3.2 we get

rCD,E (A ) ≥ −

2

k=1

tk λmax (Ak + A∗k ) = 0.5468. 2 max D j .E j  j=1...2

Example 3.6. Consider a example of periodically switched system are thermal systems for instance the system in Fig. 1, having three rooms: the first of them is heated, while the other two are not. Different thermal transmission coefficients have to be considered depending on whether the doors are closed or open. If we assume that the temperatures in the three rooms are x1 , x2 and x3 , respectively, we can describe the various cases corresponding to the status (periodically open/closed) of the two doors by means of a linear model of the form (1.1), with

$

Ai =

−ai ai 0

ai − ( ai + bi ) bi

%

0 bi − ( bi + θ )

118

D.D. Thuan and L.V. Ngoc / Applied Mathematics and Computation 361 (2019) 112–130

for i ∈ {1, 2, 3, 4}, corresponding to the four combinations open-closed of the two doors (see [4]). Without restrictions, the external temperature is assumed to be T0 = 0. All the coefficients are positive. The coefficients ai and bi may have two values ai ∈ {amin , amax } and bi ∈ {bmin , bmax } with amin < amax and bmin < bmax . So there are four possibilities for (ai , bi ). Precisely, i = 1 : (amin , bmin ); i = 2 : (amax , bmin ); i = 3 : (amin , bmax ); i = 4 : (amax , bmax ). Assume that m = 4, amin = 1, amax = 3 and bmin = 1, bmax = 2. We choose the parameters a1 = b1 = 1, a2 = 3, b2 = 1, a3 = 1, b3 = 2, a4 = 3, b4 = 2, θ = 1,t1 = t3 = 0.5, t2 = t4 = 1. Then, in this model,

$

A1 =

−1 1 0

%

1 −2 1

0 1 , A2 = −2

$

−3 3 0

%

3 −4 1

0 1 , A3 = −2

$

−1 1 0

%

1 −3 2

0 2 , A4 = −3

$

−3 3 0

3 −5 2

%

0 2 . −3

It is easy to have λmax (A1 + A∗1 ) = −0.3961, λmax (A2 + A∗2 ) = −0.4245, λmax (A3 + A∗3 ) = −0.4769, λmax (A4 + A∗4 ) = −0.5167.  By Corollary 3.3, we have 4k=1 tk λmax (Ak + A∗k ) = −1.3777 < 0, system (2.2) is exponentially stable. Assume that system (2.2) is subjected to structured perturbations of the form

$

1 = A

$ 3 = A

%

$

%

$

−1 + δ1 1 + δ1 0

1 + δ1 −2 + δ1 1

0 2 = 1 , A −2

−1 + δ3 1 + δ3 0

1 + δ3 −3 + δ3 2

0 4 = 2 , A −3

%

−3 + δ2 3 + δ2 0

3 + δ2 −4 + δ2 1

0 1 , −2

−3 + δ4 3 + δ4 0

3 + δ4 −5 + δ4 2

0 2 −3

%

 = A + D  E ,A  = A + D  E ,A  = A + D  E ,A  = A + D  E with We have A 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4

& '

1 1 , Ei = 1 0

Di =

1

!

0 , i = 1, 2, 3, 4.

and 1 = δ1 , 2 = δ2 , 3 = δ3 , 4 = δ4 . By Theorem 3.2, we get

1 tk 2 max D j .E j  4

rCD,E (A ) ≥ −

λmax (Ak + A∗k ) = 0.3444,

k=1

j∈4

or equivalently, if |δ1 | + |δ2 | + |δ3 | + |δ4 | < 0.3444 then the perturbed system (3.1) is exponentially stable. Finally, we consider a special case of (3.1), namely

x˙ (t ) =

⎧ (A1 + 1 )x(t ), t0 + lT ≤ t < t1 + lT ⎪ ⎪ ⎨(A2 + 2 )x(t ), t1 + lT ≤ t < t2 + lT

............................................................

⎪ ⎪ ⎩(Am + m )x(t ), tm−1 + lT ≤ t < tm + lT x(t0 ) = x0 ; l = 0, 1, . . . ; t ≥ t0 ,

(3.4)

The corresponding matrix R for perturbed system (3.4) becomes

 := R

m

e(Ak +k )tk ,

k=1

where k ∈ Kn×n , k = 1, . . . , m, are unknown perturbations. Using Theorem 3.2 with Dk = Ek = I for all k = 1, . . . , m, we come to the following corollary. Corollary 3.7. Assume that system (2.2) with the given data A = m k=1 Ak is exponentially stable then the stability radius (3.2) satisfies m

tk Ak  ≥ rC (A ) ≥ −

k=1

m

tk λmax (Ak + A∗k ). 2

(3.5)

k=1

Moreover, if Ak is Hermitian and Hurwitz matrix for all k ∈ m and

(

ker(Ak − λmax (Ak )I ) = {0}

k∈m

then we have

rC ( A ) = −

m

tk λmax (Ak + A∗k ). 2 k=1

(3.6)

D.D. Thuan and L.V. Ngoc / Applied Mathematics and Computation 361 (2019) 112–130

119

Proof. Choosing  = −A, we find that A +  = O ∈ UR , hence we get the upper bound for the stability radius in (3.5). If (3.6) holds then there exists a vector v = 0 such that (Ak − λmax (Ak )I )v = 0 for all k ∈ m. This implies that eAk −λmax (Ak )I v = v for all k ∈ m. Now, we choose k = −λmax (Ak )I for all k ∈ m. Then, we have

&

v = R

m

'

e(Ak +k )tk

&

v=

k=1

m

'

e(Ak −λmax (Ak )I )tk

v = v.

k=1

) ≥ 1 and the perturbed system (3.1) is not exponentially stable. Since A is Hermitian and Hurwitz This implies that ρ (R k matrix for all k ∈ m, by definition we get

−

m m m

tk λmax (Ak + A∗k ) = − tk λmax (Ak ) = tk k  ≥ rC (A ). 2 k=1

k=1

k=1

By inequality (3.5), we obtain

rC ( A ) = −

m

tk λmax (Ak + A∗k ). 2 k=1

The proof is complete.



3.2. Periodically switched linear systems with perturbed coefficient matrices and perturbed switching moments Now, we consider the periodically switched linear systems under perturbations on both coefficient matrices and switching moments. Assume that system (2.2) is subjected to structured perturbations of the form

x˙ (t ) =

⎧ (A1 + D1 1 E1 )x(t ), t0 + lT ≤ t < t1 + δt1 + lT ⎪ ⎪ ⎨(A2 + D2 2 E2 )x(t ), t1 + δt1 + lT ≤ t < t2 + δt2 + lT

......................................................................................

(3.7)

⎪ ⎪ ⎩(Am + Dm m Em )x(t ), tm−1 + δtm−1 + lT ≤ t < tm + lT x(t0 ) = x0 ; l = 0, 1, . . . ; t ≥ t0 ,

Here the perturbations of switching moments tk ∈ R are δtk , k = 1, . . . , m − 1. Further, Dk ∈ Kn×l , Ek ∈ Kq×n are given matrices determining the structure of the perturbations, and k ∈ Kl×q are unknown perturbations. Let tk = tk − tk−1 , δt k = δtk − δtk−1 , for all k ∈ m. For the sake of convenience we put δt0 = δtm = 0. The corresponding matrix R for system (3.7) with these structured perturbations becomes

 := R

m

)

e(Ak +Dk k Ek )(tk +δtk ) .

k=1

We are interested in the following set of block matrices



VR := H := diag(H1 , . . . , Hm , δt1 I, δt2 I, . . . , δtm−1 I ) : ρ (R ) ≥ 1; R =

m

)



e(tk +δtk )Hk ,

k=1

where I is n × n-identity matrix. According to Theorem 2.7, system (3.7) with the data A = diag(A1 , . . . , Am , O1 , O2 , . . . , Om−1 ),  := where Ok , k = 1 . . . , m − 1, are n × n- zero matrices, is exponentially stable if and only if A ∈ / VR . Let the perturbation  diag(1 , . . . , m , δt1 I, . . . , δtm−1 I ) be endowed with the norm:

 ∗ := 

m

tk k  + max max j∈m

k=1

 |δ t | |δ t |  j j−1 , , t j t j

(3.8)

where as above δt0 = δtm = 0. We define the stability radius for system (2.2) as

 ∗ : A + D  E ∈ V }, rCD,E (A ) := inf{ R

(3.9)

 E := diag(D  E , . . . , D  E , δt I, . . . , δt where D I ). m m m 1 1 1 1 m−1  ∗ − m 2 m k=1 tk λmax (Ak + Ak ) k=1 tk .Ak  Let a = , c = max{1, a}, b = . Then we obtain the following result which give max j∈m D j E j  2 max j∈m D j .E j  a estimate for stability radius under perturbations on both coefficient matrices and switching moments. Theorem 3.8. Assume that system (2.2) with the data A = diag(A1 , . . . , Am , O1 , O2 , . . . , Om−1 ) is exponentially stable. Then the stability radius (3.9) satisfies the following inequality:

rCD,E

(A ) ≥

⎧ a+1 ⎪ ⎪ if ⎨ 2 (a + b ) − b

⎪ ⎪ ⎩a b

2

a + 2b > c 2 ,

if

a + 2b ≤ a2 and a > 1,

if

a + 2b ≤ 1 and a < 1.

120

D.D. Thuan and L.V. Ngoc / Applied Mathematics and Computation 361 (2019) 112–130

l×q ; δ t ∈ R are unknown perturba := diag( , . . . ,  , δt I, . . . , δt Proof. Consider a perturbation  m 1 1 m−1 I ), where k ∈ K k  E ∈ V . We have tions, such that A + D R

 m

 ) ≤ R  =  1 ≤ ρ (R 



 

e(Ak +Dk k Ek )(tk +δt k )  ≤

k=1

m



e(Ak +Dk k Ek )(tk +δt k ) ,

k=1

where δt k = δtk − δtk−1 . Using Lemmas 2.1 and 2.3, item (iii), we get

1≤

 m   m   (Ak +Dk k Ek )(tk +δt k )   Ak +A∗k +Dk 2k Ek +Ek∗ ∗k D∗k (tk +δt k )  e  ≤ e 

k=1

k=1

=

m

e

tk +δt k 2

λmax (Ak +A∗k +Dk k Ek +Ek∗ ∗k D∗k ) .

k=1

Further, Lemma 2.3, item (i) yields

1≤

m

e

tk +δt k 2

[λmax (Ak +A∗k )+λmax (Dk k Ek +Ek∗ ∗k D∗k )]

k=1

≤

m

e

tk +δt k 2

λmax (Ak +A∗k ) e

tk +δt k 2

λmax (Dk k Ek +Ek∗ ∗k D∗k ) .

k=1

From the last relation we find

0≤

m m

tk + δt k tk + δt k λmax (Ak + A∗k ) + λmax (Dk k Ek + Ek∗ ∗k D∗k ). 2 2 k=1

k=1

This implies that

0≤

m m  m

tk δt k tk λmax (Ak + A∗k ) + λmax (Ak + A∗k ) + λmax (Dk k Ek + Ek∗ ∗k D∗k ) 2 2 2 k=1

k=1

k=1

m 

δt k + λmax (Dk k Ek + Ek∗ ∗k D∗k ) 2 k=1

m m m m



tk ≤ λmax (Ak + A∗k ) + δt k Ak  + tk Dk k Ek  + δt k Dk k Ek  2 k=1

k=1

k=1

k=1

m m m m



tk ≤ λmax (Ak + A∗k ) + |δt k |Ak  + tk Dk k Ek  + |δt k |Dk k Ek  2 k=1

k=1

k=1

k=1

m m m

tk = λmax (Ak + A∗k ) + |δtk − δtk−1 |Ak  + tk Dk k Ek  2 k=1

+

m

k=1

k=1

|δtk − δtk−1 |Dk k Ek .

k=1

Thus

0≤

m m m

tk λmax (Ak + A∗k ) + (|δtk | + |δtk−1 | )Ak  + tk Dk k Ek  2 k=1

k=1

+

m

k=1

(|δtk | + |δtk−1 | )Dk k Ek .

k=1

This is equivalent to

0≤

m m m

tk (|δtk | + |δtk−1 | ) λmax (Ak + A∗k ) + tk Ak  + tk Dk k Ek + 2 tk k=1

k=1

m

(|δtk | + |δtk−1 | ) + tk Dk k Ek . tk k=1

k=1

D.D. Thuan and L.V. Ngoc / Applied Mathematics and Computation 361 (2019) 112–130

0≤

m

k=1

121

 |δt | |δt |  2 m t A  tk j j−1 k k k=1 λmax (Ak + A∗k ) + max max , 2 max D j E j  t j t j max D j E j  j=1∈m 

j∈m

j∈m

 |δ t | |δ t |  m j j−1 + tk k  + 2 max max , tk k . t j t j j=∈m k=1 k=1 m

Setting m

tk k  = x, max max j∈m

k=1

 |δ t | |δ t |  j j−1 , = y. t j t j

If b ≤ 0 then the domain is the first quadrant and min {x + y} = 0, implies that rC (A ) ≥ 0. (x,y )∈

If b > 0 then we have



2xy + x + ay − b ≥ 0 x, y ≥ 0, a, b > 0

Applying Lemma 2.4 with α = 1, β = a, γ = b and ω = c = max{1, a}, we get

min{x + y} =

⎧ a+1 ⎪ ⎪ if ⎨ 2 (a + b ) − 2

b

⎪ ⎪ ⎩a b

a + 2b > c 2 ,

if

a + 2b ≤ a2 and a > 1

if

a + 2b ≤ 1 and a < 1.

(3.10)

Combining (3.9) with (3.10), we obtain

⎧ a+1 ⎪ ⎪ if ⎨ 2 (a + b ) − 2 rC ( A ) ≥ b if ⎪a ⎪ ⎩ b

a + 2b > c 2 , a + 2b ≤ a2 and a > 1 a + 2b ≤ 1 and a < 1.

if



The proof is complete.

Remark 3.9. If system (3.7) has no perturbed switching moments, i.e., δtk = 0 for all k ∈ m then we arrive at Theorem 3.2, i.e.,

rCD,E (A ) = rCD,E (A ) ≥ −

m

k=1

tk λmax (Ak + A∗k ). 2 max D j .E j  j=1...m

Remark 3.10. If all the subsystems Ak , k = 1, . . . , m remain unperturbed, while the switching moments ti , i = 1, . . . , m − 1, are contaminated by errors, then from Theorem 3.8 we have

rC0,0

1 (A ) ≥ − 4

m

tk λmax (Ak + A∗k ) , m k=1 tk Ak 

k=1

or equivalently, if

m  |δ t | |δ t |  1 k=1 tk λmax (Ak + A∗k ) j j−1 max max , <− m t j t j 4 j∈m k=1 tk Ak 

then the perturbed system (3.7) is exponentially stable. We end this section by giving a simple example to illustrate Theorem 3.8. Example 3.11. Consider a periodically switched system (2.2) with m = 3, T = 3, t0 = 0, t1 = 1, t2 = 2, t3 = 3, t1 = t2 = t3 = 1, and the perturbed system (3.7) with

⎛

−3 ⎜ 0.1 A1 = ⎝ 0.01 1

0.1 −2 1.01 0

0 1 −4 −1

−2 ⎜1.02 A2 = ⎝ 1.03 1.1

0.1 −2 0.03 0.1

1.01 1.1 −2 −1

⎛

⎞

⎛

⎞

1 1 1 ⎟ ⎜0 ,D = 1 ⎠ 1 ⎝0 −2.01 1

0 " 1⎟ 1 ,E = 1⎠ 1 1 −1

0 1

1 0

−1 ; 0

2 1 0.1⎟ ⎜1 , D = 0.2⎠ 2 ⎝1 −2 0

0 " 0⎟ 1 , E2 = ⎠ 0 −1 1

0 1

1 1

−1 ; 0

⎞

⎛

⎞

#

#

122

D.D. Thuan and L.V. Ngoc / Applied Mathematics and Computation 361 (2019) 112–130

⎛

−3 ⎜0.3 A3 = ⎝ 0.2 −1

0.4 −2 1.02 1

⎞

1.01 −1.2 −1 0.03

⎛

0.12 0 1 ⎟ ⎜1 ,D = −1.02⎠ 3 ⎝1 −2 0

⎞

1 " 0⎟ −1 , E = 1⎠ 3 0 0

0 1

1 0

#

1 . 1

It is easy to compute that λmax (A1 + A∗1 ) = −2.0464, λmax (A2 + A∗2 ) = −0.1548, λmax (A3 + A∗3 ) = −0.8944. By Corollary 3.3, system (2.2) is exponentially stable. Moreover, we can calculate a = 6.8907, b = 5.60 0 0, c = 6.8907, such that a + 2b < a2 and a > 1. By Theorem 3.8, we obtain

rCD,E (A ) ≥

b = 0.8127. a

4. Robust stabilizability for periodically switched linear systems In this section we will study robust stabilizability for periodically switched linear systems subjected to structured perturbations on coefficient matrices. Here, we consider stabilizing these systems by determining suitable activation durations. We first derive the following notion. Definition 4.1. System (2.2) is called fastly stabilizable if there exist a sufficiently small T > 0 and activation durations t1 , t2 , . . . , tm such that system (2.2) is exponentially stable. From now, let the following assumption holds. (H1) there exist η1 > 0, η2 > 0, . . . , ηm > 0 with

m

k=1

ηk = 1 such that A1 η1 + A2 η2 + · · · + Am ηm := C is Hurwitz.

The following lemma has been proved by using Lyapunov theory or Theorem 2.7 in previous section (see [7,16,25]). Lemma 4.2. Assume that (H1) holds. Then, system (2.2) is fastly stabilizable. We now derive a bound for perturbations preserving fast stabilizability. Theorem 4.3. Assume that (H1) holds and system (2.2) is subjected to structured perturbations in the form (3.1). If m

 k  <

k=1

1 max ηk k∈m

max

i, j∈m; s≥0

then the perturbed system (3.1) is still fastly stabilizable. + *  Proof. We first claim that μ C + m k=1 ηk Dk k Ek < 0 if (4.1) holds. Indeed, + * m μ C + k=1 ηk Dk k Ek ≥ 0. Then, there exists s ∈ C : s ≥ 0, x0 ∈ Cn , x0 = 0 such that

$

C+

m

(4.1)

Ei (sI − C )−1 D j 

assume

on

the

contrary

that

%

ηk Dk k Ek x0 = sIx0 .

k=1

Since μ(C) < 0, (sI − C ) is invertible and hence

*

sI − C

m +−1

η k D k k E k x 0 = x 0 .

(4.2)

k=1

Let i0 be an index such that Ei0 x0  = maxk∈m Ek x0 . It follows from (4.2) that Ei0 x0  = 0. Multiplying both sides of (4.2) from the left by Ei0 , we get m

*

Ei0 sI − C

+−1

η k D k k E k x 0 = E i 0 x 0 .

k=1

It follows that m

* + Ei0 sI − C −1 Dk ηk k Ek x0  ≥ Ei0 x0 .

k=1

This gives

max ηk max k∈m

i, j∈m; s≥0

*

Ei sI − C

+−1

$ D j

m

k=1

%  k   E i 0 x 0  ≥  E i 0 x 0  ,

D.D. Thuan and L.V. Ngoc / Applied Mathematics and Computation 361 (2019) 112–130

or equivalently,

*

max ηk max

i, j∈m; s≥0

k∈m

Ei sI − C

+−1

$ D j

m

123

%  k  ≥ 1 .

k=1

It implies that m

 k  ≥

k=1

1 max ηk max

i, j∈m; s≥0

k∈m

This conflicts with (4.1). Thus,

$

μ

m

* + . Ei sI − C −1 D j  $

%

η k ( A k + D k k E k ) = μ C +

k=1

m

% η k D k k E k < 0 .

k=1

By Lemma 4.2, the perturbed system (3.1) is fastly stabilizable. The proof is complete.



We assume addition that (H2) there exists a Hurwitz matrix A0 and η1 > 0, η2 > 0, . . . , ηm > 0, A m ηm ) ≤ A 0 ; p×n l×p ∈ Rn+×l , Ek+ ∈ R+ , + ∈ R+ such that (H3) the exist D+ k k

m

k=1

ηk = 1 such that M (A1 η1 + A2 η2 + · · · +

|Dk | ≤ D+k , |Ek | ≤ Ek+ , |k | ≤ +k , ∀k = 1, . . . , m. By comparing with positive matrices, we obtain the following result for robust stabilizability of the perturbed system (3.1). Theorem 4.4. Assume that (H2)-(H3) hold and system (2.2) is subjected to structured perturbations in the form (3.1). If m

+k  <

k=1

1

(4.3)

max ηk max Ei+ A−1 D+j  0 k∈m

i, j∈m

then the perturbed system (3.1) is still fastly stabilizable. + *  + + + Proof. We first claim that μ A0 + m k=1 ηk Dk k Ek < 0 if (4.3) holds. Indeed, since A0 is a Metzler matrix and  + + + D+ , E + , + are nonnegative for any k ∈ m, A0 + m k=1 ηk Dk k Ek is a Metzler matrix. Assume on the contrary that μ0 := k* k k + m + + + μ A0 + k=1 ηk Dk k Ek ≥ 0. By the Perron-Frobenius Theorem 2.5 (i), there exists x0 ∈ Rn+ , x0 = 0 such that

$

A0 +

m

%

ηk D+k +k Ek+ x0 = μ0 x0 .

k=1

By assumption, μ(A0 ) < 0. Therefore (μ0 In − A0 ) is invertible and this implies that

*

μ0 In − A0

m +−1

ηk D+k +k Ek+ x0 = x0 .

(4.4)

k=1

Let i0 be an index such that Ei+ x0  = maxk∈m Ek+ x0 . It follows from (4.4) that Ei+ x0  = 0. Multiplying both sides of (4.4) from the left by Ei+ , we get

0

0

0

m

Ei+0

*

μ0 In − A0

+−1

ηk D+k +k Ek+ x0 = Ei+0 x0 .

k=1

It follows that m

* + Ei+0 μ0 In − A0 −1 D+k ηk +k Ek+ x0  ≥ Ei+0 x0 .

k=1

This gives

* +

max ηk max Ei k∈m

i, j∈m

μ0 In − A0

+−1

$ D+j 

m

k=1

% +k  Ei+0 x0  ≥ Ei+0 x0 ,

124

D.D. Thuan and L.V. Ngoc / Applied Mathematics and Computation 361 (2019) 112–130

or equivalently,

max ηk max  k∈m

i, j∈m

Ei+

*

μ0 In − A0

+−1

$ D+j



m

%   ≥ 1. + k

(4.5)

k=1

On the other hand, the resolvent identity gives

(0In − A0 )−1 − (μ0 In − A0 )−1 = (μ0 − 0 )(0In − A0 )−1 (μ0 In − A0 )−1 . Since A0 is a Metzler matrix and μ0 (A0 ) < 0, by Theorem 2.5 (iii) (0.In − A0 )−1 ≥ 0 and (μ0 In − A0 )−1 ≥ 0. This yields,

Ei+ (−A0 )−1 D+j ≥ Ei+ (μ0 In − A0 )−1 D+j ≥ 0, for any i, j ∈ m. By monotonicity of an operator norm associated with a given pair of monotonic vector norms, we have + + −1 + Ei+ A−1 0 D j  ≥ Ei (μ0 In − A0 ) D j .

(4.6)

From (4.5) and (4.6), it implies that m

+k  ≥

k=1

1 max ηk max Ei+ A−1 D+j  0 k∈m

.

i, j∈m

*

However, this conflicts with (4.3) and hence μ A0 +

m

k=1

+ ηk D+k +k Ek+ < 0. Now, there exists a γ > 0 such that

, , $ % , , m m

, , 0≤, ηk A k + γ I , = M ηk Ak + γ I ≤ A0 + γ I. , k=1 , k=1

This implies that

, , , , m m

, , ηk D+k +k Ek+ + γ I , η k ( A k + D k k E k ) + γ I , ≤ A 0 + , k=1 , k=1

By Theorem 2.5 (iv),

$

μ

m

$

%

η k ( A k + D k k E k ) + γ I = μ

k=1

$

m

% η k ( A k + D k k E k ) + γ I

k=1

≤ μ A0 +

$ = μ A0 +

m

% η

+ k Dk



+ + E k k

k=1 m

+γI

% η

+ k Dk



+ + E k k

+ γ I.

k=1

+ + *m * m + + + Thus μ k=1 ηk (Ak + Dk k Ek ) ≤ μ A0 + k=1 ηk Dk k Ek < 0 and hence by Lemma 4.2 the perturbed system (3.1) is fastly stabilizable. The proof is complete.  In particular, we consider the problem of robust stabilizability of the periodically switched linear systems of (2.2) under unstructured perturbations, i.e, Di = Ei = I for all i ∈ m. We have the following corollary. Corollary 4.5. If (H2) and (H3) hold and Di = Ei = I for all i ∈ m m

k=1

+k  <

1 max ηk A−1  0

,

(4.7)

k∈m

then the perturbed system (3.1) is still fastly stabilizable. Now we derive the notion of slow stabilizability for periodically switched linear systems of (2.2). Definition 4.6. System (2.2) is called slowly stabilizable if there exist a sufficiently large T > 0 and activation durations t1 , t2 , . . . , tm such that system (2.2) is exponentially stable. Consider the assumption: (H4) there exists at least one of A1 , A2 , . . . , Am is Hurwitz.

D.D. Thuan and L.V. Ngoc / Applied Mathematics and Computation 361 (2019) 112–130

125

Lemma 4.7 [7,16]. Assume that (H4) holds. Then, system (2.2) is slowly stabilizable. Now, we define the functions

γk (Ak , Dk , Ek ) =

⎧ ⎨0

1 ⎩ sup Ek (sI − Ak )−1 Dk 

if

μ ( Ak ) ≥ 0;

if

μ ( Ak ) < 0.

(4.8)

s≥0

Theorem 4.8. Assume that (H4) holds and system (2.2) is subjected to structured perturbations in the form (3.1). If m

k  < max γk (Ak , Dk , Ek ),

(4.9)

k∈m

k=1

then the perturbed system (3.1) is still slowly stabilizable.  Proof. Assume that m k=1 k  < γk0 (Ak0 , Dk0 , Ek0 ) = maxk∈m γk (Ak , Dk , Ek ). We claim that μ (Ak0 + Dk0 k0 Ek0 ) < 0. Assume on the contrary that μ(Ak0 + Dk0 k0 Ek0 ) ≥ 0. Then there exist s ∈ C : s ≥ 0, x0 ∈ Cn , x0 = 0 such that

-

.

Ak0 + Dk0 k0 Ek0 x0 = sIx0 .

Since 0 < γk0 (Ak0 , Dk0 , Ek0 ), μ(Ak ) < 0. Thus (sI − Ak0 ) is invertible and hence

*

sI − Ak0

+−1

D k 0 k 0 E k 0 x 0 = x 0 .

(4.10)

It follows from (4.10) that Ek0 x0  = 0. Multiplying both sides of (4.10) from the left by Ek , we get

*

Ek0 sI − Ak0

+−1

D k 0 k 0 E k 0 x 0 = E k 0 x 0 .

It follows that

* + Ek0 sI − Ak0 −1 Dk0 k0 Ek0 x0  ≥ Ek0 x0 ,

and hence m  k=1

 k  ≥  k 0 

≥

1

*

Ek0 sI − Ak0

+

−1 D

k0



≥

*

1

sup Ek0 sI − Ak0

s≥0

+

−1 D

k0



(4.11)

= γk0 (Ak0 , Dk0 , Ek0 ) = max γk (Ak , Dk , Ek ). k∈m

This conflicts with (4.9). Thus μ(Ak0 + Dk0 k0 Ek0 ) < 0 and hence by Lemma 4.7, the perturbed system (3.1) is slowly stabilizable. The proof is complete.  Finally, assume that (H5) there exists a Hurwitz stable matrix B0 such that

M (Ak ) ≤ B0 , k ∈ S ⊂ m; (H5) (H6) the exist D+ ∈ Rn+×l , Ek+ ∈ R+ , + ∈ R+ k k p×n

|Dk | ≤

|Ek | ≤

D+ , k

| k | ≤ 

Ek+ ,

l×p

+ , k

such that

k ∈ S.

Similarly with the proof of Theorem 4.4, we get the following result for robust slow stabilizability of periodically switched linear systems. Theorem 4.9. Assume that (H5)-(H6) hold and system (2.2) is subjected to structured perturbations in the form (3.1). If m

+k  < max k∈S

k=1

1

Ek+ B−1 D+  0 k

,

(4.12)

then the perturbed system (3.1) is still slowly stabilizable. Corollary 4.10. Assume that (H5)-(H6) hold and Dk = Ek = I for all k ∈ S. If m

k=1

+k  <

1

B−1  0

,

then the perturbed system (3.1) is still slowly stabilizable.

(4.13)

126

D.D. Thuan and L.V. Ngoc / Applied Mathematics and Computation 361 (2019) 112–130

9 8 7

abs(eig(R)

6 5 4 3 2 1 0

0

0.5

1

1.5

2 T

2.5

3

3.5

4

Fig. 2. Magnitudes of the eigenvalues of R.

We illustrate Theorem 4.8 by the following example. Example 4.11. Consider system (2.2) with m = 2 and

⎛

−2 ⎜ 0 A1 = ⎝ 0.01 1

⎡ ⎤

0 −2 1.01 0

1 ⎢0 ⎥ D1 = ⎣ ⎦, E1 = 1 0 1

⎞

0.12 1 −2 1

0

⎛

1 −4 1⎟ ⎜ 1 , A = 1⎠ 2 ⎝ 1 −3 0.03

⎡ ⎤

0 −2.02 1.02 0.11

0

!

⎢1⎥ 0 , D2 = ⎣ ⎦, E2 = 1 0

1

0

⎞

0.01 0 −1 0.11

1

0.02 0.15⎟ , 0.01⎠ −3

!

0 .

1

Clearly, A1 , A2 are Hurwitz stable Metzler matrices and D1 , E1 , D2 , E2 are non-negative matrices we can calculate

γ1 (A1 , D1 , E1 ) =

1

1

= 0.4637, γ2 (A2 , D2 , E2 ) = = 1.8460, E1 A−1 D1  E2 A−1 D2  1 2 . max γk (Ak , Dk , Ek ) = max γ1 (A1 , D1 , E1 ), γ2 (A2 , D2 , E2 ) = 1.8460. k=1,2

The system (2.2) is subjected to structured perturbations of the form

⎛

−2 + δ1 ⎜ 0  A1 = ⎝ 0.01 1 + δ1

0.12 + δ1 1 −2 1 + δ1

0 −2 1.01 0

⎞

⎛

1 −4 1⎟  ⎜ 1 + δ2 , A = 1⎠ 2 ⎝ 1 −3 0.03 + δ2

0 −2.02 1.02 0.11

0.01

δ2

−1 0.11 + δ2

⎞

0.02 0.15⎟ 0.01⎠ −3

 = A + D  E ,A  = A + D  E with and  = δ ,  = δ . By Theorem 4.8, if It is easy to see in this model that A 1 1 1 1 1 2 2 2 2 2 1 1 2 2 |δ1 | + |δ2 | < 1.8460 then the perturbed system (3.1) is still slowly stabilizable. We end this section by giving a simple example to illustrate Theorem 4.4. Example 4.12. Consider system (2.2) with m = 2 and

"

A1 =

−3 1

#

2 , D1 = 0

"

#

−1 , E1 = 0

"

−1 1

#

0 ; 0

D.D. Thuan and L.V. Ngoc / Applied Mathematics and Computation 361 (2019) 112–130

127

Switching 100 $x (t)$ 1

$x (t)$ 2

50

0

−50

−100

−150

0

2

4

6

8

10

Time(sec) Fig. 3. The exponential stability of periodically switched linear systems under structurbed perturbations with T = 2.

Switching 14000 $x (t)$ 1

$x (t)$

12000

2

10000 8000 6000 4000 2000 0 −2000

0

2

4

6

8

10

Time(sec) Fig. 4. The unstability of periodically switched linear systems under structurbed perturbations with T = 3.

128

D.D. Thuan and L.V. Ngoc / Applied Mathematics and Computation 361 (2019) 112–130

" A2 =

2 1

with

*  1 = δ1

#

−2 , D2 = −1

"

+ * δ2 ,  2 = δ3

#

0 , E2 = −1

"

#

0 0

1 , −1

+ δ4 , where δ1 , δ2 , δ3 , δ4 ∈ R are unknown perturbations.

√ 3 T 17 . The eigenvalues of A1 are − ± and the eigenvalues of A2 are {0, 1}. Note that al2 2 2 though neither A1 nor A2 is Hurwitz, A1 + A2 is Hurwitz. Thus, it follows from Lemma 4.2 that for sufficiently small T, the system is fastly stabilizable, i.e., it is exponentially stable when T = 2, whereas it is unstable when T = 3, or equivalently, according to Lemma 4.2, the magnitude of the eigenvalues of R as functions of T are plotted by the dashed and dashdot curves on Fig. 2. We check the assumptions of Theorem 4.4 in the following Assume that t1 = t2 =

(H2) there exists a Hurwitz stable matrix A0 =

-

−0.5 1

.

0 , and −0.5

1 2

η1 = η2 = , η1 + η2 = 1 such that M (A1 η1 + A2 η2 ) ≤

A0 ; ∈ R2+×1 , + ∈ R1+×2 , Ek+ ∈ R2+×2 such that (H3) the exist D+ k k

" #

|D1 | ≤ D+1 =

* 1 , |  1 | ≤ + = |δ1 | 1 0

" # |D2 | ≤ D+2 =

* 0 , |  2 | ≤ + 2 = |δ3 | 1

It is easy to see that + + −1 + + −1 + E1+ A−1 0 D1 = E2 A0 D2 = E2 A0 D1 =

"

#

" + 1 |δ2 | , |E1 | ≤ E1+ =

#

1

0 , 0

+ 0 |δ4 | , |E2 | ≤ E2+ =

1 . 1

"

0

−2 + , E1+ A−1 0 D2 = −2

#

" #

0 . 0

Let R2 be endowed with 1-norm. By Theorem 4.4, the perturbed system (3.1) is fastly stabilizable if

max{|δ1 |, |δ2 |} + max{|δ3 |, |δ4 |} < 0.5. Thus, if we choose the parameters δ1 = 0.15, δ2 = 0.1, δ3 = 0.17, δ4 = −0.25 satisfy max{|δ1 |, |δ2 |} + max{|δ3 |, |δ4 |} < 0.5 then the perturbed system (3.1) is fastly stabilizable. Moreover, in this case, the experimental orbit of this system is exponentially stable with T = 2 and unstable with T = 3, shown on Figs. 3 and 4. 5. Conclusion In this paper we have investigated robust stability and robust stabilizability for periodically switched linear systems. The lower bounds for the stability radius, when both coefficient matrices and switching moments are perturbed, are obtained. Some characterizations for robust stabilizability under structured perturbations are established. It may be of interest to study the following problems in the future: • Compute the stability radius of switched linear systems when coefficient matrices are subjected to structured perturbations. • Extend the obtained results to periodically switched linear descriptor systems, see [1,35]. Acknowledgments The authors would like to express their gratitude to Prof. Pham Ky Anh, Prof. Nguyen Khoa Son and Prof. Vu Hoang Linh for several helful discussions. The authors also gratefully thank the reviewers for useful comments that led to the improvements of the paper. This work was supported by NAFOSTED project 101.01–2017.302. Appendix Proof of Lemma 2.1. Let s1 (A ) ≥ s2 (A ) ≥ · · · ≥ sn (A ) and λ1 , λ2 , . . . , λn , where |λ1 | ≥ |λ2 | ≥ · · · ≥ |λn |, be singular values and eigenvalues of a matrix A ∈ Kn×n , respectively. It is proved (see, [3]), that:

|λ1 | ≤ A = s1 (A ) = A∗ A1/2 = AA∗ 1/2 . Further, s1 (Am ) = ||Am || ≤ ||A||m = sm ( A ), 1 [s21 (A )]m = [s1 (A∗ A )]m . Thus, ∗ s1 ( A∗ m Am ) ≤ sm 1 ( A A ).

hence

s21 (Am ) ≤ s21m (A ).

On

other

hand

s1 [(Am )∗ Am ] = s21 (Am ) ≤ s21m (A ) =

D.D. Thuan and L.V. Ngoc / Applied Mathematics and Computation 361 (2019) 112–130

129

A

Now, replacing A with e m , we obtain

-

∗

A∗

A

.

m m s1 ( eA eA ) ≤ sm . 1 e e

Letting m → ∞, and using the Lie product formula A

B

eA+B = lim (e m e m )m , m→∞

we have ∗

s1 ( eA eA ) ≤ s1 ( eA Since

( eA )∗ s21

=

∗ eA ,

∗

+A

we get

).

eA 

(6.1) =

∗ eA ,

( e ) = s1 [ ( e ) e ] ≤ s1 ( e A ∗ A

A

A∗ +A

hence s1

) = s1 [ ( e

( eA )

A∗ +A 2

= s1

) ]=

A∗ +A 2

2

∗ ( eA ).

s21

(e

A∗ +A 2

Using (6.1) we find

).

A+A∗ 2

Thus, s1 (eA ) ≤ s1 (e ), or equivalently eA  ≤ e . For a hermitian matrix D ∈ Cn×n , we denote by λmax (D) the largest eigenvalue of D. It is well known (see, [3]) that λmax (D ) = maxv=1 vT Dv.  Proof of Lemma 2.3. (i) For all v with v = 1, we have

vT (A + B )v = vT Av + vT Bv ≤ λmax (A ) + λmax (B ), hence,

λmax (A + B ) = max vT (A + B )v ≤ λmax (A ) + λmax (B ). v=1

(ii) Since −λmax (A ) ∈ σ (−A ), it follows that −λmax (A ) ≤ λmax (−A ). (iii) Observe that if a matrix A is hermitian, then so is eA . Besides eA is positive define, hence

||eA || = maxA μ = max eλ = eλmax (A) . μ∈ σ ( e )

λ∈σ (A )

 Proof of Lemma 2.4. Clearly, the function ϕ (x, y ) := x + y is continuous and coercive on the closed subset , hence it attends a minimum on . Consider the Lagrange function

L(x, y,

λ ) = x + y − λ(2xy + α x + β y − γ ).

According to KKT conditions (see, [6]), we have

Lx = 1 − 2λy − αλ ≥ 0,

(6.2)

x(1 − 2λy − αλ ) = 0,

(6.3)

Ly = 1 − 2λx − βλ ≥ 0

(6.4)

y(1 − 2λx − βλ ) = 0,

(6.5)

λ(2xy + α x + β y − γ ) = 0, λ ≥ 0

(6.6)

2xy + α x + β y − γ ≥ 0, x ≥ 0, y ≥ 0,

(6.7)

If λ = 0 then by (6.3) and (6.5), we get x = y = 0, hence from (6.7) we find γ ≤ 0, which contradicts γ > 0. Thus λ = 0. The inequality (6.6) implies that λ > 0 and

2xy + α x + β y − γ = 0.

(6.8)

Case 1. If x = y = 0 then γ = 0, which contradicts the assumption γ > 0.

γ . Relation (6.2) implies that αβ + 2γ ≤ β 2 . Besides, αβ < β αβ + 2γ ≤ β 2 , hence α < β . In this case, the only KKT point is (x1 , y1 , λ1 ) := (0, γ /β , 1/β ) and ϕ attains the value Case 2. If x = 0, y > 0 then by (6.5), we have λ =

ϕmin =

γ . β

1

β

and y =

130

D.D. Thuan and L.V. Ngoc / Applied Mathematics and Computation 361 (2019) 112–130

Case 3. If x > 0, y = 0, then by (6.3) λ =

1

α

. Eq. (6.8) implies that x =

γ . From (6.4), we get αβ + 2γ ≤ α 2 and β < α . α

Moreover, the only KKT point is (x2 , y2 , λ2 ) := (γ /α , 0, 1/α ) and the corresponding value

ϕmin =

γ . α

1 − βλ 1 − αλ , y= . Further, inserting the values of x and y 2λ 2λ 1 into (6.8) we come to the equation 1 − (αβ + 2γ )λ2 = 0, which gives a positive solution λ =  and αβ + 2γ Case 4. Let x > 0, y > 0. From (6.3) and (6.5) it follows that x =

⎧  ⎪ ⎨x = αβ + 2γ − β 2  ⎪ ⎩y = αβ + 2γ − α 2

Since x > 0, y > 0, we find αβ + 2γ > β 2 , and αβ + 2γ > α 2 , hence αβ + 2γ > ω2 . Thus, we get

ϕmin =



αβ + 2γ −

α+β 2

.

Combining the above four cases, we come to formula (2.1).



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