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International Journal of Control

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The structured controllability radius of linear delay systems

Do Duc Thuan ab

a

School of Applied Mathematics and Informatics , Hanoi University of Science and Technology , 1 Dai Co Viet Str., Hanoi , Vietnam

b

Department of Mathematics, Mechanics and Informatics , Vietnam National University ,

334 Nguyen Trai, Hanoi , Vietnam Published online: 09 Jan 2013

To cite this article: Do Duc Thuan (2013) The structured controllability radius of linear delay systems, International Journal

of Control, 86:3, 512-518, DOI: 10.1080/00207179.2012.746473

To link to this article: http://dx.doi.org/10.1080/00207179.2012.746473

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Vol 86, No 3, March 2013, 512Ờ518

The structured controllability radius of linear delay systems

Do Duc Thuanab*

aSchool of Applied Mathematics and Informatics, Hanoi University of Science and Technology,

1 Dai Co Viet Str., Hanoi, Vietnam;bDepartment of Mathematics, Mechanics and Informatics,

Vietnam National University, 334 Nguyen Trai, Hanoi, Vietnam (Received 3 April 2012; final version received 31 October 2012)

In this article, we shall deal with the problem of calculation of the controllability radius of a delay dynamical systems of the form x0(t)Ử A0x(t)ợ A1x(t� h1)ợ � � � ợ Akx(t� hk)ợ Bu(t) By using multi-valued linear operators, we are able to derive computable formulas for the controllability radius of a controllable delay system in the case where the systemỖs coefficient matrices are subjected to structured perturbations Some examples are provided to illustrate the obtained results

Keywords: linear delay systems; multi-valued linear operators; structured perturbations; controllability radius

1 Introduction

In a lot of applications, there is a frequently arising

question, namely, how robust is a characteristic

qualitative property of a system (e.g controllability)

when the system is subject to uncertainty This work

concerns the robust controllability analysis which has

attracted considerable attention of researchers

recently This article is concerned with linear delay

systems of the form

x0đtỡ Ử A0xđtỡ ợ A1xđt � h1ỡ ợ � � � ợ Akxđt � hkỡ ợ Buđtỡ,

where Ai2 Cn�n, i2 0, k, B 2 Cn�m, xđtỡ 2 Cn and

u(t)2 Cm

The so-called controllability radius is defined by the

largest bound r such that the controllability is

preserved for all perturbations D of norm strictly less

than r For the linear control system _xỬ Ax ợ Bu, one

can define the controllability radius rC(A, B) as

rCđA, Bỡ Ử inffkơD1,D2�k : ơD1,D2� 2 Cn�đnợmỡ,

ơA, B� ợ ơD1,D2� is not controllableg: đ1:2ỡ Here, k�k denotes a matrix norm The problem of

estimating (1.2) is of great importance in mathematical

control theory, and there have been several works in

this direction in recent years (Boley and Lu 1986;

Gahinet and Laub 1992; Gu 2000; Burke, Lewis, and

Overton 2004; Gu et al 2006) One of the most

well-known results was due to Eising (1984), who has

proved the formula

rCđA, Bỡ Ử inf

�2C�minđơA � �IB�ỡ, đ1:3ỡ where �min denotes the smallest singular value of a matrix and the matrix norm in (1.2) is the spectral norm or Frobenius norm The proof of (1.3) was based

on the Hautus characterisation of controllability (Hautus 1969):

đA,Bỡ 2 Kn�n�Kn�mcontrollable()rankơA��I,B� Ử n, 8� 2 C:

đ1:4ỡ Motivated by the recent development in the theory of stability radius (see, e.g Hinrichsen and Pritchard 1986; Hinrichsen and Pritchard 1986; and the extensive literature therein), it is natural, and more general, to consider a problem of computing the structured controllability radius when the pair (A, B) is subjected

to structured perturbations:

ơA, B� ? ơ ~A, ~B� Ử ơA, B� ợ DDE, đ1:5ỡ where D2 Kn�l, E2 Kq�(nợm) are given structure matrices This problem has been solved in recent papers (Karow and Kressner 2009; Son and Thuan 2010) where some formulas of the structured controll-ability radius have been derived

In this article, we shall study the measures of robust controllability of linear delay systems (1.1) By using the unified approach which we have developed in the

*Email: ducthuank7@gmail.com

ISSN 0020Ờ7179 print/ISSN 1366Ờ5820 online

 201 Taylor & Francis

http://dx.doi.org/10.1080/00207179.2012.746473

3

International Journal of Control 513

The structured controllability radius of linear delay systems

Do Duc Thuanab*

aSchool of Applied Mathematics and Informatics, Hanoi University of Science and Technology,

1 Dai Co Viet Str., Hanoi, Vietnam;bDepartment of Mathematics, Mechanics and Informatics,

Vietnam National University, 334 Nguyen Trai, Hanoi, Vietnam (Received 3 April 2012; final version received 31 October 2012)

In this article, we shall deal with the problem of calculation of the controllability radius of a delay dynamical

systems of the form x0(t)Ử A0x(t)ợ A1x(t� h1)ợ � � � ợ Akx(t� hk)ợ Bu(t) By using multi-valued linear

operators, we are able to derive computable formulas for the controllability radius of a controllable delay

system in the case where the systemỖs coefficient matrices are subjected to structured perturbations Some

examples are provided to illustrate the obtained results

Keywords: linear delay systems; multi-valued linear operators; structured perturbations; controllability radius

1 Introduction

In a lot of applications, there is a frequently arising

question, namely, how robust is a characteristic

qualitative property of a system (e.g controllability)

when the system is subject to uncertainty This work

concerns the robust controllability analysis which has

attracted considerable attention of researchers

recently This article is concerned with linear delay

systems of the form

x0đtỡ Ử A0xđtỡ ợ A1xđt � h1ỡ ợ � � � ợ Akxđt � hkỡ ợ Buđtỡ,

where Ai2 Cn�n, i2 0, k, B 2 Cn�m, xđtỡ 2 Cn and

u(t)2 Cm

The so-called controllability radius is defined by the

largest bound r such that the controllability is

preserved for all perturbations D of norm strictly less

than r For the linear control system _xỬ Ax ợ Bu, one

can define the controllability radius rC(A, B) as

rCđA, Bỡ Ử inffkơD1,D2�k : ơD1,D2� 2 Cn�đnợmỡ,

ơA, B� ợ ơD1,D2� is not controllableg: đ1:2ỡ

Here, k�k denotes a matrix norm The problem of

estimating (1.2) is of great importance in mathematical

control theory, and there have been several works in

this direction in recent years (Boley and Lu 1986;

Gahinet and Laub 1992; Gu 2000; Burke, Lewis, and

Overton 2004; Gu et al 2006) One of the most

well-known results was due to Eising (1984), who has

proved the formula

rCđA, Bỡ Ử inf

�2C�minđơA � �IB�ỡ, đ1:3ỡ where �min denotes the smallest singular value of a matrix and the matrix norm in (1.2) is the spectral norm or Frobenius norm The proof of (1.3) was based

on the Hautus characterisation of controllability (Hautus 1969):

đA,Bỡ 2 Kn�n�Kn�mcontrollable()rankơA��I,B� Ử n, 8� 2 C:

đ1:4ỡ Motivated by the recent development in the theory of stability radius (see, e.g Hinrichsen and Pritchard 1986; Hinrichsen and Pritchard 1986; and the extensive literature therein), it is natural, and more general, to consider a problem of computing the structured controllability radius when the pair (A, B) is subjected

to structured perturbations:

ơA, B� ? ơ ~A, ~B� Ử ơA, B� ợ DDE, đ1:5ỡ where D2 Kn�l, E2 Kq�(nợm) are given structure matrices This problem has been solved in recent papers (Karow and Kressner 2009; Son and Thuan 2010) where some formulas of the structured

controll-ability radius have been derived

In this article, we shall study the measures of robust controllability of linear delay systems (1.1) By using the unified approach which we have developed in the

*Email: ducthuank7@gmail.com

ISSN 0020Ờ7179 print/ISSN 1366Ờ5820 online

 2012 Taylor & Francis

http://dx.doi.org/10.1080/00207179.2012.746473

previous work (Son and Thuan 2010), we are able to derive, as the main result of this article, some formulas for computing the structured controllability radius of linear delay systems under the assumption that the tuple

of coefficient matrices (A0, A1, , Ak, B) is subjected to general structured perturbations of the form

ơA0, A1, , Ak, B�

?ơ ~A0, ~A1, , ~Ak, ~B�

Ử ơA0, A1, , Ak, B� ợ DDE, đ1:6ỡ where D2 Cn�l, E2 Cq�(nợknợm) are given matrices defining the structure of perturbations, D 2 Cl �q is unknown disturbance matrix Moreover, avoiding the restrictive assumption on the matrix norm used in the previous works, throughout this article the norm of matrices is assumed to be the operator norm induced

by arbitrary vector norms on corresponding vector spaces In some particular cases, the main result yield new computable formulas of structured controllability radius of linear delay systems

The organisation of this aritcle is as follows In the next section, we shall recall some notations and some known results from the theory of linear multi-valued operators (see, e.g Cross 1998) which will be used in the sequence Section 3 will be devoted to prove the main results of this article establishing formulas for the structured controllability radius under structured perturbations of the form (1.6) and deriving the computable formulas in some special cases In conclu-sion, we summarise the obtained results and give some remarks of further investigation

2 Preliminaries Let n, m, k, l, q be positive integers Throughout this article, Cn �m will stand for the set of all n� mỜ

matrices, A�2 Cm�n denotes the adjoint matrix of

A2 Cn�m, Cn(Ử Cn�1) is the n-dimensional vector space (of columns of n numbers from C) equipped with the vector norm k � k and its dual space can be identified with (Cn)�Ử (Cn�1)�Ử {u�: u2 Cn}, the vector space of rows of n numbers from C, equipped with the dual norm For u�2 (Cn)� we shall write

u�(x)Ử u�x, 8x 2 Cn For a subset M� Cn, we denote

M?Ử {u�2 (Cn)�: u�xỬ 0 for all x 2 M} Let F :

Cm!! Cnbe a multi-valued operator If the graph of

F , defined by

grF Ử�đx, yỡ 2 Cm� Cn:y2 F đxỡ�, đ2:1ỡ

is a linear subspace of Cm� Cn, thenF is called a linear multi-valued operator The domain and the nullspace ofF are denoted, respectively, by dom F Ử

�

x2 Cm:F đxỡ 6Ử ;� and kerF Ử�x2 dom F :

02 F đxỡ� By definition, if F is a multi-valued linear

operator then F (0) is a linear subspace, and for

x2 dom F , we have the following equivalence

y2 F đxỡ () F đxỡ Ử y ợ F đ0ỡ: đ2:2ỡ Let F : Cm

!

! Cn be a multi-valued linear operator, then for given vector norms on Cnand Cm, the norm of

F is defined by

kF k Ử supn inf

It follows from the definition that inf

and therefore, ifF is single-valued,

kF đxỡk � kF kkxk for all x 2 dom F : đ2:4ỡ

If the spaces under consideration are equipped with the Euclidean norms (i.e kxk Ửpffiffiffiffiffiffiffiffix�x

) then from (2.2) it follows obviously that the following implication holds

y2 F đxỡ, y�2 F đ0ỡ?Ử) dđ0, F đxỡỡ :Ử inf

z 2F đxỡkzk Ử k yk:

đ2:5ỡ For a linear multi-valued operator F : Cm!! Cn, its adjoint operator F�: (Cn)�

!

! (Cm)� and its inverse operator F�1: ImF !! Cm are defined, correspondingly, by

F�đv�ỡ Ử�u�2 đCnỡ�:u�xỬ v�y for allđx, yỡ 2 grF�,

đ2:6ỡ

F�1đ yỡ Ử�x2 Cm:y2 F đxỡ�: đ2:7ỡ Clearly F� and F�1 are also linear multi-valued operators and we have

đF�ỡ�1Ử đF�1ỡ�, kF k Ử kF�k: đ2:8ỡ

It can be proved thatF is surjective (i.e F (Cm)Ử Cn) if and only if F� is injective (i.e F� �1(0)Ử {0}), or, equivalently, F� �1is single-valued Let F : Cm!! Cn, G: Cn!! Cl are the linear multi-valued operators, then the operatorGF : Cm!! Cl, defined by (GF )(x) Ử G(F (x)) for all x 2 dom F , is a linear multi-valued operator and if ImF � dom G or Im G�� dom F� then đGF ỡ�Ử F�G� and

kđGF ỡ�k Ử kF�G�k � kF�k kG�k Ử kF k kGk: đ2:9ỡ

If F is the linear single-valued operator defined by

F (x) Ử FG(x)Ử Gx, where G 2 Cn�mand x2 Cm, then, clearly, the norm of FG defined by (2.3) is just the operator norm of matrix G:

kFGk Ử kGk:

In the sequence, when dealing with this operator in the context of the theory of multi-valued linear operators,

we shall use the notationFG(x)Ử G(x) It is easily seen that the adjoint operator (FG)�: (Cn)�! (Cm)� is also linear single-valued operator which is given by (FG)�(v�)Ử v�G,8v�2 (Cn)� For the sake of simplicity,

we shall identify (FG)� with G�, that reads

đFGỡ�đv�ỡ Ử G�đv�ỡ Ử v�G, 8v�2 đCnỡ�: đ2:10ỡ Remark that the notation G�v is understood, as usual, the product of matrix G�2 Cm�n and column vector

v2 Cnand we have (G�v)�Ử G�(v�)

3 Main results

We consider the linear delay systems with constant delays 05 h15 � � � 5 hk,

x0đtỡ Ử A0xđtỡ ợ A1xđt � h1ỡ

ợ � � � ợ Akxđt � hkỡ ợ Buđtỡ,

xđ0ỡ Ử x0, xđtỡ Ử gđtỡ, 8t 2 ơ�hk, 0ỡ,

8

<

where Ai2 Cn�n, iỬ 0, 1, , k, B 2 Cn�m, and g(t) : [�hk, 0)! Rnis a continuous function System (3.1) is called controllable if for any given initial conditions x0, g(t) and desired final state x1, there exists a time t1,

05 t15 1, and a measurable control function u(t) for

t2 [0, t1] such that x�

t1;x0, gđtỡ, uđtỡ�Ử x1 Define

Pđ�ỡ Ử A0ợ e�h 1 �A1ợ � � � ợ e�h k �Ak� �In: đ3:2ỡ

It is well known that, (Bhat and Koivo 1976; Rocha and Willems 1997),

systemđ3:1ỡ is controllable ()rankơPđ�ỡ, B� Ử n for all � 2 C: đ3:3ỡ Assume that system (3.1) is subjected to structured perturbations of the form

x0đtỡ Ử eA0xđtỡ ợ eA1xđt � h1ỡ ợ ��� ợ eAkxđt � hkỡ ợ eBuđtỡ,

đ3:4ỡ with

ơA0, A1 ., Ak, B�

?ơeA0, eA1, , eAk, eB�

Ử ơA0, A1 ., Ak, B� ợ DDE: đ3:5ỡ Here, D2 Cn�l, E2 Cq�(n(kợ1)ợm) are the given matrices and D 2 Cl �q is the perturbation matrix

The structure matrices D, E determine the structure

of the perturbations DDE We use the notation

AỬ [A0, A1, , Ak]

Definition 3.1: Let system (3.1) be controllable

Given a norm k � k on Cl�q, the controllability radius

of system (3.1) with respect to structured perturbations

of the form (3.5) is defined by

rCđA, B; D, Eỡ Ử inf�kDk : D 2 Cl�q

s.t.ơA, B� ợ DDE not controllable�:

đ3:6ỡ

If [A, B]ợ DDE is controllable for all D 2 Cl�qthen we set rC(A, B; D, E)Ử ợ1

We define

Wđ�ỡ Ử ơPđ�ỡ, B�, Hđ�ỡ Ử

e�h 1 �In 0

e�h k �In 0

2 6 6 6 4

3 7 7 7 5 ,

Eđ�ỡ Ử EHđ�ỡ,

đ3:7ỡ

and the multi-valued operators E(�)W(�)�1D:

Cl!! Cqby setting đEđ�ỡWđ�ỡ�1Dỡđuỡ Ử Eđ�ỡđWđ�ỡ�1đDuỡỡ, 8u 2 Cl, where W(�)�1: Cn!! Cnợm is the (multi-valued) inverse operators of W(�)

Theorem 3.2: Assume that system (3.1) is controllable and subjected to structured perturbations of the form (3.5) Then the controllability radius of (3.1) is given by the formula

rCđA, B; D, Eỡ Ử 1

sup�2CkEđ�ỡWđ�ỡ�1Dk: đ3:8ỡ Proof: Suppose that ơeA, eB� Ử ơA, B� ợ DDE is not controllable for D 2 Cl �q It means, by (3.3), the operator Weđ�ỡ Ử ơePđ�ỡ, eB� is not surjective for some �02 C, where Peđ�ỡ Ử eA0ợ e�h 1 �Ae1ợ � � � ợ

e�h k �Aek� �In: By definitions (3.2) and (3.7), we can deduce

e

Wđ�0ỡ Ử ơePđ�0ỡ, eB� Ử ơeA0, eA1, , eAk, eB�Hđ�0ỡ � �ơIn, 0�

Ử đơA0, A1, , Ak, B� ợ DDEỡHđ�0ỡ � �ơIn, 0�

Ử ơA0, A1, , Ak, B�Hđ�0ỡ � �ơIn, 0� ợ DDEHđ�0ỡ

Ử ơPđ�0ỡ, B� ợ DDEđ�0ỡ Ử Wđ�0ỡ ợ DDEđ�0ỡ:

đ3:9ỡ This implies that there exists y�

0 2 đCnỡ�, y�

such that đWđ�0ỡ ợ DDEđ�0ỡỡ�đ y�0ỡ

Ử Wđ�0ỡ�đ y�0ỡ ợ đEđ�0ỡ�D�D�ỡđ y�0ỡ Ử 0: Since system (3.1) is controllable, by (3.3), W(�0) is surjective, or equivalently W(�0)� �1 is single-valued

D.D Thuan

International Journal of Control 515

In the sequence, when dealing with this operator in the

context of the theory of multi-valued linear operators,

we shall use the notationFG(x)Ử G(x) It is easily seen

that the adjoint operator (FG)�: (Cn)�! (Cm)� is also

linear single-valued operator which is given by

(FG)�(v�)Ử v�G,8v�2 (Cn)� For the sake of simplicity,

we shall identify (FG)� with G�, that reads

đFGỡ�đv�ỡ Ử G�đv�ỡ Ử v�G, 8v� 2 đCnỡ�: đ2:10ỡ

Remark that the notation G�v is understood, as usual,

the product of matrix G�2 Cm�n and column vector

v2 Cnand we have (G�v)�Ử G�(v�)

3 Main results

We consider the linear delay systems with constant

delays 05 h15 � � � 5 hk,

x0đtỡ Ử A0xđtỡ ợ A1xđt � h1ỡ

ợ � � � ợ Akxđt � hkỡ ợ Buđtỡ,

xđ0ỡ Ử x0, xđtỡ Ử gđtỡ, 8t 2 ơ�hk, 0ỡ,

8

<

where Ai2 Cn�n, iỬ 0, 1, , k, B 2 Cn�m, and g(t) :

[�hk, 0)! Rnis a continuous function System (3.1) is

called controllable if for any given initial conditions x0,

g(t) and desired final state x1, there exists a time t1,

05 t15 1, and a measurable control function u(t) for

t2 [0, t1] such that x�

t1;x0, gđtỡ, uđtỡ�Ử x1 Define

Pđ�ỡ Ử A0ợ e�h 1 �A1ợ � � � ợ e�h k �Ak� �In: đ3:2ỡ

It is well known that, (Bhat and Koivo 1976; Rocha

and Willems 1997),

systemđ3:1ỡ is controllable

()rankơPđ�ỡ, B� Ử n for all � 2 C: đ3:3ỡ

Assume that system (3.1) is subjected to structured

perturbations of the form

x0đtỡ Ử eA0xđtỡ ợ eA1xđt � h1ỡ ợ ��� ợ eAkxđt � hkỡ ợ eBuđtỡ,

đ3:4ỡ with

ơA0, A1 ., Ak, B�

?ơeA0, eA1, , eAk, eB�

Ử ơA0, A1 ., Ak, B� ợ DDE: đ3:5ỡ

Here, D2 Cn�l, E2 Cq�(n(kợ1)ợm) are the given

matrices and D 2 Cl �q is the perturbation matrix

The structure matrices D, E determine the structure

of the perturbations DDE We use the notation

AỬ [A0, A1, , Ak]

Definition 3.1: Let system (3.1) be controllable

Given a norm k � k on Cl�q, the controllability radius

of system (3.1) with respect to structured perturbations

of the form (3.5) is defined by

rCđA, B; D, Eỡ Ử inf�kDk : D 2 Cl�q

s.t.ơA, B� ợ DDE not controllable�:

đ3:6ỡ

If [A, B]ợ DDE is controllable for all D 2 Cl�qthen we set rC(A, B; D, E)Ử ợ1

We define

Wđ�ỡ Ử ơPđ�ỡ, B�, Hđ�ỡ Ử

e�h 1 �In 0

e�h k �In 0

2 6 6 6 4

3 7 7 7 5 ,

Eđ�ỡ Ử EHđ�ỡ,

đ3:7ỡ

and the multi-valued operators E(�)W(�)�1D:

Cl!! Cqby setting đEđ�ỡWđ�ỡ�1Dỡđuỡ Ử Eđ�ỡđWđ�ỡ�1đDuỡỡ, 8u 2 Cl,

where W(�)�1: Cn!! Cnợm is the (multi-valued) inverse operators of W(�)

Theorem 3.2: Assume that system (3.1) is controllable and subjected to structured perturbations of the form (3.5) Then the controllability radius of (3.1) is

given by the formula

rCđA, B; D, Eỡ Ử 1

sup�2CkEđ�ỡWđ�ỡ�1Dk: đ3:8ỡ Proof: Suppose that ơeA, eB� Ử ơA, B� ợ DDE is not controllable for D 2 Cl �q It means, by (3.3), the operator Weđ�ỡ Ử ơePđ�ỡ, eB� is not surjective for some �02 C, where Peđ�ỡ Ử eA0ợ e�h 1 �Ae1ợ � � � ợ

e�h k �Aek� �In: By definitions (3.2) and (3.7), we can deduce

e

Wđ�0ỡ Ử ơePđ�0ỡ, eB� Ử ơeA0, eA1, , eAk, eB�Hđ�0ỡ � �ơIn, 0�

Ử đơA0, A1, , Ak, B� ợ DDEỡHđ�0ỡ � �ơIn, 0�

Ử ơA0, A1, , Ak, B�Hđ�0ỡ � �ơIn, 0� ợ DDEHđ�0ỡ

Ử ơPđ�0ỡ, B� ợ DDEđ�0ỡ Ử Wđ�0ỡ ợ DDEđ�0ỡ:

đ3:9ỡ This implies that there exists y�

02 đCnỡ�, y�

06Ử 0 such that

đWđ�0ỡ ợ DDEđ�0ỡỡ�đ y�0ỡ

Ử Wđ�0ỡ�đ y�0ỡ ợ đEđ�0ỡ�D�D�ỡđ y�0ỡ Ử 0:

Since system (3.1) is controllable, by (3.3), W(�0) is surjective, or equivalently W(�0)� �1 is single-valued

Therefore, we have

y�

0Ử �đWđ�0ỡ��1Eđ�0ỡ�D�ỡđD�đ y�

0ỡỡ đ3:10ỡ and, hence, D�đ y�

0ỡ 6Ử 0: By applying D� to the left of the both sides of (3.10), we obtain

D�đ y�0ỡ Ử �đD�Wđ�0ỡ��1Eđ�0ỡ�D�ỡđD�đ y�0ỡỡ:

Therefore, by (2.4),

05 kD�đ y�

0ỡk � kD�Wđ�0ỡ��1Eđ�0ỡ�kkD�đD�đ y�

0ỡỡk

� kD�Wđ�0ỡ��1Eđ�0ỡ�kkD�kkD�đ y�

0ỡk:

Since Im W(�0)�1� dom E(�0)Ử Cnợm, we have, by using (2.9), (E(�0)W(�0)�1)�Ử W(�0)�1 �

E(�0)�Ử W(�0)� �1E(�0)� Further, since W(�0) is surjective, Im

D� dom(E(�0)W(�0)�1)Ử Cnwe have again by (2.9),

đEđ�0ỡWđ�0ỡ�1Dỡ�

Ử D�đEđ�0ỡWđ�0ỡ�1ỡ�

Ử D�Wđ�0ỡ��1Eđ�0ỡ�:

By (2.8), we get

kD�k Ử kDk

kD�Wđ�0ỡ��1Eđ�0ỡ�k

kEđ�0ỡWđ�0ỡ�1Dk

sup�2CkEđ�ỡWđ�ỡ�1Dk: Since the above inequality holds for any disturbance matrixD 2 Cl �qsuch that DDE destroys controllability

of (3.1), we obtain by definition,

rCđA, B; D, Eỡ � 1

sup�2CkEđ�ỡWđ�ỡ�1Dk: đ3:11ỡ

To prove the converse inequality, for any small �4 0 such that sup�2C kE(�)W(�)�1Dk � � 4 0 there exists ��2 C such that kD�W(��)� �1E(��)�k Ử kE(��)W(��)�1Dk � sup�2C kE(�)W(�)�1Dk � � We note further that D�W(��)� �1E(��)� is single-valued, therefore its norm is the operator norm and

� 2 đCqỡ� :kv�

�k Ử 1,

v�

� 2 dom đD�Wđ��ỡ��1Eđ��ỡ�ỡ such that kEđ��ỡWđ��ỡ�1Dk Ử kD�Wđ��ỡ��1Eđ��ỡ�k

Ử kđD�Wđ��ỡ��1Eđ��ỡ�ỡđv��ỡk:

Denoting u�

� Ử �Wđ��ỡ��1đEđ��ỡ�đv�

�ỡỡ 6Ử 0, we have

Wđ��ỡ�đu��ỡ Ử �Eđ��ỡ�đv��ỡ and D�đu��ỡ

Ử �đD�Wđ��ỡ��1Eđ��ỡ�ỡđv��ỡ 6Ử 0:

By HahnỜBanach Theorem, applying with

VỬ fsD�đu�

�ỡ : s 2 Cg � đClỡ�, there exists h�2 Clsuch that kh�k Ử 1, đD�đu�

�ỡỡh� Ử kD�đu�

�ỡk Thus, we can define a rank-one perturbationD�2 Cl�qby setting

D� Ử 1

kD�đu�

�ỡkh�v

�

�: Then, it is obvious thatkD�k � kD�đu�

�ỡk�1: Moreover,

we have D�đu�

�ỡD�Ử v�

� This implies that

kD�k � kD�đu�

�ỡk�1: Thus, we obtain

kD�k Ử kD�đu��ỡk�1Ử kđD�Wđ��ỡ��1Eđ��ỡ�ỡđv��ỡk�1

kEđ��ỡWđ��ỡ�1Dk: Using (2.10),

đD��D�ỡđu��ỡ Ử D��đD�đu��ỡỡ Ử D�đu��ỡD�Ử v��: Hence,đEđ��ỡ�D�

�D�ỡđu�

�ỡ Ử Eđ��ỡ�đv�

�ỡ and, therefore,

Wđ��ỡ�đu��ỡ ợ đEđ��ỡ�D�

�D�ỡđu��ỡ Ử 0, with u�6Ử 0, which implies that the perturbed matrix

ơePđ��ỡ, eB� Ử eWđ��ỡ Ử Wđ��ỡ ợ DD�Eđ��ỡ is non-surjec-tive or, equivalently, by (3.3), system (3.1) is not controllable Thus, by definition,

rCđA, B; D, Eỡ � kD�k

kEđ��ỡWđ��ỡ�1Dk

sup�2CkEđ�ỡWđ�ỡ�1Dk � �: Letting �! 0, we get the required converse inequality Thus, we obtain

rCđA, B; D, Eỡ Ử 1

sup�2CkEđ�ỡWđ�ỡ�1Dk:

The above theorem have been proved similarly with one result for higher order descriptor systems in Son and Thuan (2012), for the case when the norms of matrices under consideration are operator norms induced by arbitrary vector norms in corresponding vector spaces

Formula (3.8) gives us a unified framework for computation of controllability radii, however, it is not easy to be used because this formula involves calcula-tion of the norm of the multi-valued linear operator E(�)W(�)�1D which do not have an explicit represen-tation We now derive from this result more compu-table formulas for the particular case, where the norm

of the matrices under consideration is the spectral norm (i.e the operator norm induced by Euclidean

vector norms of the formkxk ¼pffiffiffiffiffiffiffiffix�x

) To this end, we need the following lemmas

Lemma 3.3: Assume that G2 Cn�phas full row rank:

rank G¼ n and Cn, Cp are equipped with Euclidean

norms Then, for the linear operator FG(z)¼ Gz,

we have

dð0, F�1

G ð yÞÞ ¼ kGyyk, kF�1

G k ¼ kGyk, ð3:12Þ where Gydenotes the Moore-Penrose pseudoinverse of G

Proof: See Lemma 3.3 in Son and Thuan (2010) œ

Lemma 3.4: Assume that �2 Cn�(nþm) has full row

rank, M2 Cq�(nþm) has full column rank and the

operator norms are induced by Euclidean vector norms

Then, we have

kM��1Dk ¼ kð�ðM�MÞ�1=2ÞyDk, ð3:13Þ where y denotes the Moore-Penrose pseudoinverse,

D2 Cn�land ��1is the (multi-valued) inverse operator

of �

Proof: See Corollary 3.7 in Son and Thuan (2010) or

Lemma 4.2 in Son and Thuan (2012) œ

We note that if system (3.1) is controllable, then

W(�) have full row rank, and if E has full column rank,

then E(�) have full column rank for all �2 C By

Lemma 3.4 and Theorem 3.2, we obtain

Theorem 3.5: Assume that E has full column rank and

the operator norms are induced by Euclidean vector

norms Then we have

sup�2C�

��Wð�Þ½Eð�Þ�Eð�Þ��1=2�yD�:

ð3:14Þ The above theorem covers many existing results as

particular cases Indeed, for k¼ 0, we obtain the main

result in Karow and Kressner (2009) Further, it is easy

to see that if k¼ 0 and D, E are the identity matrices

in Cn�n and C(n(kþ1)þm)�(n(kþ1)þm), respectively, then

Theorem 3.5 is reduced to the formula of Eising (1984)

as a particular case

It is worth to mention that if coefficient matrices

Ai, B are subjected to separate structured

perturba-tions, then it may not be possible to cover this case by

the model (3.5) with the full blockD Next, we consider

a particular case of separate structured perturbations,

which can be covered by the model (3.5) and thus the

above result are applicable Assume that system (3.1) is

subjected to separate perturbations of the form

B ? eB¼ B þ DBDBEB,

Ai? eAi¼ Aiþ DA iDA iEA i, for all i2 0, k, ð3:15Þ

where DA i¼ DB2 Cn�l, EA i2 CqAi �n, EB2 CqB �m, for all i2 0, k, are given matrices and DB2 Cl�qB,

DA i 2 Cl�qAi, for all i2 0, k, are the perturbation matrices It is easy to see that the perturbation model (3.15) can be rewritten in the form

½A0, A1, , Ak, B�

?½eA0, eA1, , eAk, eB�

¼ ½A0, A1, , Ak, B� þ bDDbE, where bD¼ DB, bE¼ diagðEA 0, EA 1, , EA k, EBÞ and the perturbation

D ¼ ½DA 0,DA 1, ,DA k,DB�:

In this situation, we define

b

Eð�Þ ¼ bEHð�Þ ¼

EA 0 0

e�h 1 �EA 1 0

e�h k �EA k 0

2 6 6 6 6

3 7 7 7 7 : ð3:16Þ

Theorem 3.6: Assume that system (3.1) is subjected to separate structured perturbations of the form (3.15) Then, if system (3.1) is controllable then

rCðA,B;DB, EB, EA i, i2 0,kÞ ¼ 1

sup�2CkbEð�ÞWð�Þ�1Dbk:

ð3:17Þ Let us consider system (3.1) subjected to perturba-tions of the form

B ? eB¼ B þ DB,

Ai? eAi¼ Aiþ �iDA i, for all i2 0, k, ð3:18Þ where �i2 C, i 2 0, k are given scalar parameters, not all zero, and DA i 2 Cn�n, i2 0, k, DB2 Cn�m are unknown matrices Then, we can apply Theorem 3.6

to calculate the controllability radii of system (3.1) under structured perturbations (3.18) Define

�ð�Þ ¼ j�0jpþX

k

i ¼1

j�ijpje�h i �p

j: ð3:19Þ

Now, we will derive the formula of the controllability radius for linear delay systems under affine perturba-tions (3.18), which is nearly similar with the one for higher order descriptor systems in Son and Thuan (2012)

Corollary 3.7: Assume that the controllable system (3.1) is subjected to perturbations of the form (3.18) and

International Journal of Control 517

vector norms of the formkxk ¼pffiffiffiffiffiffiffiffix�x

) To this end, we need the following lemmas

Lemma 3.3: Assume that G2 Cn�p has full row rank:

rank G¼ n and Cn, Cp are equipped with Euclidean

norms Then, for the linear operator FG(z)¼ Gz,

we have

dð0, F�1

G ð yÞÞ ¼ kGyyk, kF�1

G k ¼ kGyk, ð3:12Þ where Gydenotes the Moore-Penrose pseudoinverse of G

Proof: See Lemma 3.3 in Son and Thuan (2010) œ

Lemma 3.4: Assume that �2 Cn�(nþm) has full row

rank, M2 Cq�(nþm) has full column rank and the

operator norms are induced by Euclidean vector norms

Then, we have

kM��1Dk ¼ kð�ðM�MÞ�1=2ÞyDk, ð3:13Þ

where y denotes the Moore-Penrose pseudoinverse,

D2 Cn�land ��1 is the (multi-valued) inverse operator

of �

Proof: See Corollary 3.7 in Son and Thuan (2010) or

Lemma 4.2 in Son and Thuan (2012) œ

We note that if system (3.1) is controllable, then

W(�) have full row rank, and if E has full column rank,

then E(�) have full column rank for all �2 C By

Lemma 3.4 and Theorem 3.2, we obtain

Theorem 3.5: Assume that E has full column rank and

the operator norms are induced by Euclidean vector

norms Then we have

sup�2C�

��Wð�Þ½Eð�Þ�Eð�Þ��1=2�yD�:

ð3:14Þ The above theorem covers many existing results as

particular cases Indeed, for k¼ 0, we obtain the main

result in Karow and Kressner (2009) Further, it is easy

to see that if k¼ 0 and D, E are the identity matrices

in Cn�n and C(n(kþ1)þm)�(n(kþ1)þm), respectively, then

Theorem 3.5 is reduced to the formula of Eising (1984)

as a particular case

It is worth to mention that if coefficient matrices

Ai, B are subjected to separate structured

perturba-tions, then it may not be possible to cover this case by

the model (3.5) with the full blockD Next, we consider

a particular case of separate structured perturbations,

which can be covered by the model (3.5) and thus the

above result are applicable Assume that system (3.1) is

subjected to separate perturbations of the form

B ? eB¼ B þ DBDBEB,

Ai? eAi¼ Aiþ DA iDA iEA i, for all i2 0, k, ð3:15Þ

where DA i ¼ DB2 Cn�l, EA i 2 CqAi �n, EB2 CqB �m, for all i2 0, k, are given matrices and DB2 Cl�qB,

DA i 2 Cl�qAi, for all i2 0, k, are the perturbation matrices It is easy to see that the perturbation model

(3.15) can be rewritten in the form

½A0, A1, , Ak, B�

?½eA0, eA1, , eAk, eB�

¼ ½A0, A1, , Ak, B� þ bDDbE, where bD¼ DB, bE¼ diagðEA 0, EA 1, , EA k, EBÞ and the

perturbation

D ¼ ½DA 0,DA 1, ,DA k,DB�:

In this situation, we define

b

Eð�Þ ¼ bEHð�Þ ¼

EA 0 0

e�h 1 �EA 1 0

e�h k �EA k 0

2 6 6 6 6

3 7 7 7 7

Theorem 3.6: Assume that system (3.1) is subjected to separate structured perturbations of the form (3.15)

Then, if system (3.1) is controllable then

rCðA,B;DB, EB, EA i, i2 0,kÞ ¼ 1

sup�2CkbEð�ÞWð�Þ�1Dbk:

ð3:17Þ Let us consider system (3.1) subjected to

perturba-tions of the form

B ? eB¼ B þ DB,

Ai? eAi¼ Aiþ �iDA i, for all i2 0, k, ð3:18Þ where �i2 C, i 2 0, k are given scalar parameters, not all zero, and DA i 2 Cn�n, i2 0, k, DB2 Cn�m are unknown matrices Then, we can apply Theorem 3.6

to calculate the controllability radii of system (3.1) under structured perturbations (3.18) Define

�ð�Þ ¼ j�0jpþX

k

i ¼1

j�ijpje�h i �p

j: ð3:19Þ

Now, we will derive the formula of the controllability radius for linear delay systems under affine perturba-tions (3.18), which is nearly similar with the one for higher order descriptor systems in Son and Thuan

(2012)

Corollary 3.7: Assume that the controllable system (3.1) is subjected to perturbations of the form (3.18) and

the vector spaces are endowed with the p-norm with

05 p 5 1 Then,

rCðA, B; �i, i2 0, kÞ ¼ 1

sup�2C

�

�� P ð�Þ

�ð�Þ 1=p, B

��1�

� , ð3:20Þ

and if p¼ 2,

rCðA, B; �i, i2 0, kÞ

sup�2C

�

�� Pð�Þffiffiffiffiffiffiffi

�ð�Þ

p , B

�y�

�

¼ inf

�2C�min

� P ð�Þ ffiffiffiffiffiffiffiffiffiffi

�ð�Þ

� :

ð3:21Þ Proof: We see in model (3.18) that

DA i ¼ DB¼ In, EA i ¼ �iIn, EB¼ Im, for all i2 0, k:

b

D¼ In, bE¼ diagð�0In, �1In, , �kIn, ImÞ, and by (3.16)

b

Eð�Þ ¼

�0In 0

�1e�h 1 �In 0

�ke�h k �In 0

2 6 6 4

3 7 7

5:

Therefore,

ðbEð�ÞWð�Þ�1DbÞðvÞ

¼

�0y1

�1e�h 1 �y1

�ke�h k �y1

u1

0 B B B B

1 C C C C :y12 Cn, u12 Cms.t Pð�Þy1þ Bu1¼ v

8

>

>

>

>

>

>

9

>

>

>

>

>

>

:

Let w1:¼ �(�)1/py1 with �(�) defined by (3.19)

We have

�

� Z1

u1

� ����p

¼

� j�0jpþX

k

i ¼1

j�ijpje�hi �p

j

�

k y1kpþ ku1kp

¼

�

� w1

u1

� ����p

, for each Z1

u 1

� �

2 ðbEð�ÞWð�Þ�1DbÞðvÞ: This implies that

dð0, bEð�ÞWð�Þ�1DbðvÞÞ

¼ inf���� w1

u1

� ����

� :

�

Pð�Þ

�ð�Þ1=p, B

�

w1

u1

� �

¼ v

�

¼ d

� 0,

�

Pð�Þ

�ð�Þ1=p, B

��1 ðvÞ

� :

Therefore, by Theorem 3.6, we obtain formula (3.20)

Note that for the matrix spectral norm and G2 Cn�m,

1

kG y k¼ �minðGÞ, the smallest singular value of G Thus,

by Lemma 3.3, we obtain formula (3.21) œ Example 3.8: Let us consider the second-order time-delay system

x0ðtÞ ¼ A0xðtÞ þ A1xðt � 1Þ þ BuðtÞ, ð3:22Þ where

A1¼ 1 1

0 0

, A0¼ 0 0

1 1

, B¼ 0

1

� � :

We see that

Wð�Þ ¼ ½Pð�Þ, B� ¼ 1� � 1 0

e�� e��� � 1

:

It follows that rank W(�)¼ 2 for all � 2 C Therefore,

by (3.3), the system is controllable Assume that the control matrix [A0, A1, B] is subjected to structured perturbation of the form

? 1þ �1 1þ �1 �2 �2 �2

�1 �1 1þ �2 1þ �2 1þ �2

,

where �i2 C, i 2 1, 2 are disturbance parameters The above-perturbed model can be represented in the form

½A0, A1, B� ? ½A0, A1, B� þ DDE with

D¼ 1 1

� � , E¼ 1 1 0 0 0

andD ¼ [�1�2] It implies that

Eð�Þ ¼ e1�� e1�� 01

:

We have, for v2 C,

Eð�ÞWð�Þ�1DðvÞ

¼ Eð�ÞWð�Þ�1� �vv

¼

�

Eð�Þ

p q r

0 B

1 C

A : ð1��Þpþq ¼ e��pþðe����Þqþr ¼ v

�

¼

þ�p ð�þ1Þvþ�ð��1Þp

:p2 C

� :

Thus, for each v2 C, the problem of computing d(0, E(�)W(�)�1D(v)) is reduced to the calculation of the

distance from the origin to the straight line in C2

whose equation can be rewritten in the form

x2� (� � 1)x1¼ 2v with

x1¼ v þ �p, x2¼ ð� þ 1Þv þ �ð� � 1Þ p:

Note that if �¼ 0 then this line is reduced to the

point � �vv

: Assume that � 6¼ 0 and let C2 be endowed with the vector normsk � k1, then we can deduce,

2jvj � j� � 1jjx1j þ jx2j � ðj� � 1j þ 1Þ maxfjx1j, jx2jg

¼ ðj� � 1j þ 1Þ��� x1

x2

� ����

1

: This implies

�

� x1

x2

� ����

1

� 2jvj j� � 1j þ 1, which yields the equality if x2¼ 2v

j��1jþ1and x1¼ ei’x2, where ’ is chosen such that (1� �)ei’¼ j� � 1j

Therefore,

kEð�ÞWð�Þ�1Dk1¼ sup

jvj¼1

d�

0, Eð�ÞWð�Þ�1DðvÞ�

¼

2 j� � 1j þ 1 if �6¼ 0,

8

<

:

rCðA0, A1, B; D, EÞ ¼1

2:

4 Conclusion

In this article, we developed a unifying approach to the

problem of calculating the controllability radius of

linear delay systems, which is based on the theory of

linear multi-valued operators We obtained some

general formulas of complex controllability radii

under the assumption that the system coefficient

matrices are subjected to structured perturbations

These results unify and extend many existing results to

more general cases Moreover, it has been shown that

from our general results, some easily computable

formulas can be derived Our approach can be

developed further for calculating the distance from

ill-posedness of conic systems of the form Ax¼ b,

x2 K � Cm, where K is a closed convex cone, as well as

for controllability radius of convex processes

_

x2 F ðxÞ, t � 0: These problems are the topics of our

further study

Acknowledgements This work was supported financially by NAFOSTED (Vietnam National Foundation for Science and Technology Development)

References Bhat, K.P.M., and Koivo, H.N (1976), ‘Modal Characterisations of Controllability and Observability in Time Delay Systems’, IEEE Transactions on Automatic Control, 21, 292–293

Boley, D.L., and Lu, W.S (1986), ‘Measuring How Far a Controllable System is from Uncontrollable One’, IEEE Transactions on Automatic Control, 31, 249–251

Burke, J.V., Lewis, A.S., and Overton, M.L (2004),

‘Pseudospectral Components and the Distance to Uncontrollability’, SIAM Journal of Matrix Analysis and Applications, 26, 350–361

Cross, R (1998), Multi-valued Linear Operators, New York: Marcel Dekker

Eising, R (1984), ‘Between Controllable and Uncontrollable’, Systems & Control Letters, 5, 263–264 Gahinet, P., and Laub, A.J (1992), ‘Algebraic Riccati Equations and the Distance to the Nearest Uncontrollable Pair’, SIAM Journal on Control Optimization, 4, 765–786

Gu, M (2000), ‘New Methods for Estimating the Distance to Uncontrollability’, SIAM Journal on Matrix Analysis and Applications, 21, 989–1003

Gu, M., Mengi, E., Overton, M.L., Xia, J., and Zhu, J (2006), ‘Fast Methods for Estimating the Distance to Uncontrollability’, SIAM Journal on Matrix Analysis and Applications, 28, 447–502

Hautus, M.L.J (1969), ‘Controllability and Observability Conditions of Linear Autonomous Systems’, Nederlandse Akademic van Wetenschappen Proceedings, Series A, 72, 443–448

Hinrichsen, D., and Pritchard, A.J (1986), ‘Stability Radii of Linear Systems’, Systems & Control Letters, 7, 1–10 Hinrichsen, D., and Pritchard, A.J (1986), ‘Stability Radius for Structured Perturbations and the Algebraic Riccati Equation’, Systems & Control Letters, 8, 105–113 Karow, M., and Kressner, D (2009), ‘On the Structured Distance to Uncontrollability’, Systems & Control Letters,

58, 128–132

Rocha, P., and Willems, J.C (1997), ‘Behavioral Controllability of Delay Differential Systems’, SIAM Journal on Control Optimization, 35, 254–264

Son, N.K., and Thuan, D.D (2010), ‘Structured Distance to Uncontrollability Under Multi-perturbations: an Approach using Multi-valued Linear Operators’, Systems

& Control Letters, 59, 476–483

Son, N.K., and Thuan, D.D (2012), ‘The Structured Controllability Radii of Higher Order Systems’, Linear Algebra and its Applications, accepted for publication