This article was downloaded by: [York University Libraries] On: 18 October 2013, At: 04:54 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK International Journal of Control Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tcon20 The structured controllability radius of linear delay systems Do Duc Thuan a b a School of Applied Mathematics and Informatics , Hanoi University of Science and Technology , 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 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content This article may be used for research, teaching, and private study purposes Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions International Journal of Control Vol 86, No 3, March 2013, 512–518 The structured controllability radius of linear delay systems Do Duc Thuanab* a School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Dai Co Viet Str., Hanoi, Vietnam; bDepartment of Mathematics, Mechanics and Informatics, Vietnam National University, 334 Nguyen Trai, Hanoi, Vietnam Downloaded by [York University Libraries] at 04:54 18 October 2013 (Received 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) ẳ A0x(t) ỵ A1x(t h1) ỵ ỵ Akx(t hk) þ 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 system’s 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 proved the formula 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 x tị ẳ A0 xtị ỵ A1 xt h1 ị þ � � � þ Ak xðt � hk Þ þ BuðtÞ, t � 0, ð1:1Þ n�n n�m n where Ai C , i 0, k, B C , xðtÞ C and u(t) 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 x_ ẳ Ax ỵ Bu, one can define the controllability radius rC(A, B) as rC A, Bị ẳ inffkẵD1 , D2 k : ẵD1 , D2 Cnnỵmị , ẵA, B ỵ ẵD1 , D2 is not controllableg: ð1:2Þ 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 wellknown results was due to Eising (1984), who has *Email: ducthuank7@gmail.com ISSN 0020–7179 print/ISSN 1366–5820 online ß 2013 Taylor & Francis http://dx.doi.org/10.1080/00207179.2012.746473 http://www.tandfonline.com rC A, Bị ẳ inf ẵA IBị, 2C 1:3ị where � 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): ðA,BÞ Knn Knm controllable()rankẵA I,B ẳ n, C: ð1:4Þ 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: ~ B ~ ẳ ẵA, B ỵ DDE, ẵA, B ? ẵA, 1:5ị where D Knl, E Kq(nỵm) 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 controllability 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 International Journal of Control 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 ½A0 , A1 , , Ak , B� ~ ? ½A~ , A~ , , A~ k , B� Downloaded by [York University Libraries] at 04:54 18 October 2013 ¼ ½A0 , A1 , , Ak , B ỵ DDE, 1:6ị where D Cnl, E Cq(nỵknỵm) are given matrices defining the structure of perturbations, D 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 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 conclusion, we summarise the obtained results and give some remarks of further investigation Preliminaries Let n, m, k, l, q be positive integers Throughout this article, Cn�m will stand for the set of all n � m– matrices, A� Cm�n denotes the adjoint matrix of A Cn�m, Cn(¼ 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)� ¼ (Cn�1)� ¼ {u� : u Cn}, the vector space of rows of n numbers from C, equipped with the dual norm For u� (Cn)� we shall write u� (x) ¼ u� x, 8x Cn For a subset M � Cn, we denote M? ¼ {u� (Cn)� : u� x ¼ for all x M} Let F : ! Cn be a multi-valued operator If the graph of Cm ! F , defined by gr F ẳ x, yị Cm Cn : y F ðxÞ , ð2:1Þ is a linear subspace of Cm � Cn, then F is called a linear multi-valued operator The domain and the of F are �denoted, respectively, �by dom F ¼ �nullspace x Cm : F xị 6ẳ ; and ker F ẳ x dom F : F ðxÞ By definition, if F is a multi-valued linear 513 operator then F (0) is a linear subspace, and for x dom F , we have the following equivalence y F xị () F xị ẳ y ỵ F ð0Þ: ð2:2Þ ! C be a multi-valued linear operator, Let F : C ! then for given vector norms on Cn and Cm, the norm of F is defined by n � kF k ¼ sup inf k yk : x dom F , kxk ẳ : 2:3ị m n y2F ðxÞ It follows from the definition that inf k yk � kF kkxk y2F ðxÞ for all x dom F , and therefore, if F is single-valued, kF ðxÞk � kF kkxk for all x dom F : ð2:4Þ If the spaces under consideration are ffi equipped with the pffiffiffiffiffiffiffi Euclidean norms (i.e kxk ¼ x� x) then from (2.2) it follows obviously that the following implication holds y F ðxÞ, y� F 0ị? ẳ) d0, F xịị :ẳ inf kzk ẳ k yk: z2F ðxÞ ð2:5Þ ! Cn, For a linear multi-valued operator F : C ! n � ! m � � its adjoint operator F : (C ) ! (C ) and its ! Cm are defined, inverse operator F �1: Im F ! correspondingly, by � � F � v ị ẳ u Cn ị : u x ¼ v� y for all ðx, yÞ grF , m 2:6ị F yị ẳ x Cm : y F ðxÞ : ð2:7Þ Clearly F � and F �1 are also linear multi-valued operators and we have F ị1 ẳ F ị , kF k ẳ kF k: 2:8ị It can be proved that F is surjective (i.e F (Cm) ¼ Cn) if and only if F � is injective (i.e F � �1(0) ¼ {0}), or, ! Cn, equivalently, F � �1 is single-valued Let F : Cm ! n ! l G: C ! C are the linear multi-valued operators, ! Cl, defined by (GF )(x) ¼ then the operator GF : Cm ! G(F (x)) for all x dom F , is a linear multi-valued operator and if Im F � dom G or Im G� � dom F then GF ị ẳ F G and kGF ị k ẳ kF G k � kF � k kG� k ¼ kF k kGk: ð2:9Þ If F is the linear single-valued operator defined by F (x) ¼ F G(x) ¼ Gx, where G Cn�m and x Cm, then, clearly, the norm of F G defined by (2.3) is just the operator norm of matrix G: kF G k ¼ kGk: D.D Thuan 514 In the sequence, when dealing with this operator in the context of the theory of multi-valued linear operators, we shall use the notation F G(x) ¼ G(x) It is easily seen that the adjoint operator (F G)� : (Cn)� ! (Cm)� is also linear single-valued operator which is given by (F G)� (v� ) ¼ v� G, 8v� (Cn)� For the sake of simplicity, we shall identify (F G)� with G� , that reads ðF G Þ� ðv� Þ ¼ G� ðv� Þ ¼ v� G, 8v� ðCn Þ� : ð2:10Þ Remark that the notation G� v is understood, as usual, the product of matrix G� Cm�n and column vector v Cn and we have (G� v)� ¼ G� (v� ) Downloaded by [York University Libraries] at 04:54 18 October 2013 Main results We consider the linear delay systems with constant delays h1 � � � hk, < x tị ẳ A0 xtị ỵ A1 xt h1 ị 3:1ị ỵ ỵ Ak xt hk ị ỵ Butị, : x0ị ẳ x0 , xtị ẳ gtị, 8t ẵhk , 0ị, where Ai Cn�n, i ¼ 0, 1, , k, B Cn�m, and g(t) : [�hk, 0) ! Rn is 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, control t1 1, and a measurable � � function u(t) for t [0, t1] such that x t1 ; x0 , gtị, utị ẳ x1 Define Pị ẳ A0 ỵ eh1 A1 ỵ ỵ ehk Ak In : 3:2ị It is well known that, (Bhat and Koivo 1976; Rocha and Willems 1997), system 3:1ị is controllable ()rankẵPị, B ẳ n for all � C: ð3:3Þ Assume that system (3.1) is subjected to structured perturbations of the form e0 xðtÞ þ A e1 xðt � h1 Þ þ � � ỵ A ek xt hk ị ỵ Butị, e x0 tị ẳ A 3:4ị with ẵA0 , A1 , Ak , B� e0 , A e1 , , A ek , B� e ? ẵA ẳ ẵA0 , A1 , Ak , B ỵ DDE: of system (3.1) with respect to structured perturbations of the form (3.5) is defined by rC A, B; D, Eị ẳ inf kDk : D Clq s.t ẵA, B ỵ DDE not controllable : lq If [A, B] ỵ DDE is controllable for all D C set rC(A, B; D, E) ẳ ỵ1 3:6ị then we We define In eh1 In 6 Wị ẳ ẵPị, B, Hị ẳ 6 h e k In 0 Im Eị ẳ EHị, and the multi-valued ! Cq by setting Cl ! operators 7 7, 7 3:7ị E()W()1D: EịWị1 Dịuị ẳ EịWị1 Duịị, 8u Cl , ! Cnỵm is the (multi-valued) where W()1: Cn ! 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 rC A, B; D, Eị ẳ : sup2C kEịWị1 Dk 3:8ị e B e ẳ ẵA, B ỵ DDE is not Proof: Suppose that ẵA, controllable for D Cl�q It means, by (3.3), the e ẳ ẵPị, e B e is not surjective for operator Wị e0 ỵ eh1 A e1 ỵ ỵ e ẳA some C, where Pị �hk � e Ak � �In : By definitions (3.2) and (3.7), we can e deduce e0 , A e1 , , A ek , B�Hð� e ị ẳ ẵP e ị, B e ẳ ẵA e W ị ẵIn , ẳ ð½A0 , A1 , , Ak , B ỵ DDEịH0 ị ẵIn , ẳ ẵA0 , A1 , , Ak , B�Hð�0 ị ẵIn , ỵ DDEH0 ị 3:5ị ẳ ẵP0 ị, B ỵ DDE0 ị ẳ W0 ị ỵ DDE0 ị: 3:9ị Here, D Cnl, E Cq(n(kỵ1)ỵm) are the given matrices and D Cl�q is the perturbation matrix The structure matrices D, E determine the structure of the perturbations DDE We use the notation A ¼ [A0, A1, , Ak] This implies that there exists y0 Cn ị , y0 6ẳ such that Definition 3.1: Let system (3.1) be controllable Given a norm k � k on Cl�q, the controllability radius Since system (3.1) is controllable, by (3.3), W(�0) is surjective, or equivalently W(0) is single-valued W0 ị ỵ DDE0 ÞÞ� ð y�0 Þ ¼ Wð�0 Þ� ð y�0 Þ ỵ E0 ị D D ị y0 ị ẳ 0: International Journal of Control Therefore, we have y�0 ¼ �ðWð�0 Þ��1 Eð�0 Þ� D� ÞðD� ð y�0 ÞÞ ð3:10Þ and, hence, D y0 ị 6ẳ 0: By applying D� to the left of the both sides of (3.10), we obtain D y0 ị ẳ D Wð�0 Þ ��1 � � � Eð�0 Þ D ÞðD ð y�0 ÞÞ: Therefore, by (2.4), kD� ð y�0 Þk � kD� Wð�0 Þ��1 Eð�0 Þ� kkD� ðD� ð y�0 ÞÞk Downloaded by [York University Libraries] at 04:54 18 October 2013 � kD� Wð�0 Þ��1 Eð�0 Þ� kkD� kkD� ð y�0 Þk: Since Im W(�0)�1 � dom E(�0) ẳ Cnỵm, we have, by using (2.9), (E(0)W(0)1) ẳ W(�0)�1 E(�0)� ¼ W(�0)� �1E(�0)� Further, since W(�0) is surjective, Im D � dom(E(�0)W(�0)�1) ¼ Cn we have again by (2.9), ðEð�0 ÞWð�0 Þ�1 DÞ� kD� Wð�0 Þ��1 E0 ị k kD k ẳ kD u ịk1 ẳ kD W ị1 E ị ịv ịk1 ẳ : kEð�� ÞWð�� Þ�1 Dk Using (2.10), ðD�� D� Þðu�� Þ ¼ D�� ðD� ðu�� ÞÞ ¼ D� ðu�� ÞD� ¼ v�� : kEð�� ÞWð�� Þ�1 Dk : � sup2C kEịWị1 Dk ẳ Since the above inequality holds for any disturbance matrix D Cl�q such that DDE destroys controllability of (3.1), we obtain by definition, ð3:11Þ To prove the converse inequality, for any small � such that sup�2C kE(�)W(�)�1Dk � � there exists �� C such that kD� W(��)� �1E(��)� k ¼ 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 hence there exists v�� ðCq Þ� : kv�� k ¼ 1, ��1 � � � v� dom ðD Wð�� Þ Eð�� Þ Þ such that kE ịW ị1 Dk ẳ kD W ị1 E Þ� k ¼ kðD� Wð�� Þ��1 Eð�� Þ� Þðv�� Þk: Denoting u ẳ W ị1 E ị v ịị 6ẳ 0, we have W ị u ị ẳ E ị ðv�� Þ and D� ðu�� Þ Then, it is obvious that kD� k � kD� ðu�� Þk�1 : Moreover, we have D u ịD ẳ v This implies that kD� k � kD� ðu�� Þk�1 : Thus, we obtain rC A, B; D, Eị kD k ẳ kEð�0 ÞWð�0 Þ�1 Dk � : sup�2C kEð�ÞWð�Þ�1 Dk : sup�2C kEð�ÞWð�Þ�1 Dk h� v� : kD� u ịk with u 6ẳ 0, which implies that the perturbed matrix e ị, B e ẳ W e ị ẳ W ị ỵ DD E ị is non-surjec½Pð� tive or, equivalently, by (3.3), system (3.1) is not controllable Thus, by definition, By (2.8), we get rC A, B; D, Eị D ẳ W ị u ị ỵ E ị D D ịu ị ẳ 0, ¼ D� Wð�0 Þ��1 Eð�0 Þ� : � By Hahn–Banach Theorem, applying with V ẳ fsD u ị : s Cg � ðCl Þ� , there exists h� Cl such that kh k ẳ 1, D u ịịh ¼ kD� ðu�� Þk Thus, we can define a rank-one perturbation D� Cl�q by setting Hence, ðEð�� Þ� D�� D ịu ị ẳ E ị v ị and, therefore, ¼ D� ðEð�0 ÞWð�0 Þ�1 Þ� kD� k ¼ kDk 515 ẳ D W ị1 E ị ịv ị 6ẳ 0: Letting � ! 0, we get the required converse inequality Thus, we obtain rC A, B; D, Eị ẳ The proof is complete : sup�2C kEð�ÞWð�Þ�1 Dk œ 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 calculation of the norm of the multi-valued linear operator E(�)W(�)�1D which not have an explicit representation We now derive from this result more computable 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 D.D Thuan 516 vector norms of the form kxk ¼ need the following lemmas pffiffiffiffiffiffiffiffi x� x) To this end, we Lemma 3.3: Assume that G Cn�p has full row rank: rank G ¼ n and Cn, Cp are equipped with Euclidean norms Then, for the linear operator F G(z) ¼ Gz, we have y d 0, F G yịị ẳ kG yk, y kF G k ẳ kG k, 3:12ị where Gy denotes the Moore-Penrose pseudoinverse of G Proof: See Lemma 3.3 in Son and Thuan (2010) œ Downloaded by [York University Libraries] at 04:54 18 October 2013 n(nỵm) Lemma 3.4: Assume that � C has full row rank, M Cq(nỵm) has full column rank and the operator norms are induced by Euclidean vector norms Then, we have kM1 Dk ẳ kM Mị1=2 ịy Dk, 3:13ị where y denotes the Moore-Penrose pseudoinverse, D Cn�l and ��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 � 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 rC ðA, B; D, Eị ẳ y : sup2C WịẵEị Eð�Þ��1=2 D� ð3:14Þ The above theorem covers many existing results as particular cases Indeed, for k ¼ 0, we obtain the main result in Karow and Kressner (2009) Further, it is easy to see that if k ¼ and D, E are the identity matrices in Cnn and C(n(kỵ1)ỵm)(n(kỵ1)ỵm), 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 perturbations, then it may not be possible to cover this case by the model (3.5) with the full block D 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 e ẳ B ỵ DB DB EB , B?B ei ẳ Ai ỵ DA DA EA , Ai ? A i i i for all i 0, k, ð3:15Þ where DAi ¼ DB Cn�l , EAi CqAi �n , EB CqB �m , for all i 0, k, are given matrices and DB Cl�qB , DAi Cl�qAi , for all i 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� e0 , A e1 , , A ek , B� e ? ½A b E, b ¼ ½A0 , A1 , , Ak , B ỵ DD b ẳ DB , E b ¼ diagðEA0 , EA1 , , EAk , EB ị and the where D perturbation D ẳ ½DA0 , DA1 , , DAk , DB �: In this situation, we define EA0 e�h1 � E A1 6 b ¼ EHị b Eị ẳ6 6 hk EAk 4e 0 EB 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 rC ðA, B;DB , EB , EAi , i 0, kị ẳ : b b Dk sup2C kEịWị ð3:17Þ Let us consider system (3.1) subjected to perturbations of the form e ẳ B ỵ DB , B?B ei ẳ Ai ỵ i DA , Ai ? A i for all i 0, k, ð3:18Þ where �i C, i 0, k are given scalar parameters, not all zero, and DAi Cn�n , i 0, k, DB 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 ị ẳ j0 jp ỵ k X iẳ1 ji jp jehi p j: 3:19ị Now, we will derive the formula of the controllability radius for linear delay systems under affine perturbations (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 the vector spaces are endowed with the p-norm with p Then, rC ðA, B; �i , i 0, kị ẳ and if p ẳ 2, � , ð3:20Þ � Pð�Þ � � sup�2C � �ð�Þ1=p , B � � rC ðA, B; �i , i 0, kị ẳ Downloaded by [York University Libraries] at 04:54 18 October 2013 Proof: �� �y � ¼ inf �min � Pð�Þ � �2C � sup�2C � pffiffiffiffiffiffiffi , B � � �ð�Þ � � Pð�Þ pffiffiffiffiffiffiffiffiffiffi , B : ị kGy k ẳ Gị, 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 timedelay system x0 tị ẳ A0 xtị ỵ A1 xt 1ị ỵ Butị, A1 ẳ Therefore, �1 b b DÞðvÞ ðEð�ÞWð�Þ 80 � > > y1 > > > > >B > C �h1 � > > � e y > > B C 1 > > > B C > > > > B C �hk � > > y1 A > > @ �k e > > > > : ; u1 � , � � 0 A0 ¼ , 1 � � Bẳ : We see that Wị ẳ ẵPị, B� ¼ We see in model (3.18) that Therefore, we get b ¼ In , E b ¼ diagð�0 In , �1 In , , �k In , Im Þ, and by (3.16) D �0 In �h1 � In �1 e b Eị ẳ 7: �hk � In �k e Im 3:22ị where 3:21ị DAi ẳ DB ẳ In , EAi ¼ �i In , EB ¼ Im , for all i 0, k: 517 � 1�� e�� � : 1 �� e �� It follows that rank W(�) ¼ for all � 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 � � 1 0 0 1 ỵ 1 ỵ 2 ? , 1 ỵ ỵ ỵ where �i C, i 1, are disturbance parameters The above-perturbed model can be represented in the form ½A0 , A1 , B� ? ½A0 , A1 , B� þ DDE with � � � 1 D¼ ,E ¼ 1 0 0 1 � Let w1 :¼ �(�)1/py1 with �(�) defined by (3.19) We have �� ��p � � k X � Z1 � p �hi �p � � ¼ j�0 jp þ j� j je j k y1 kp þ ku1 kp i � u � i¼1 �� ��p � w1 � � ¼� � u �, � � �1 b b DÞðvÞ: This implies that for each Zu11 EịWị and D ẳ [1 2] It implies that 1 Eị ẳ e e Therefore, by Theorem 3.6, we obtain formula (3.20) Note that for the matrix spectral norm and G Cn�m, Thus, for each v C, the problem of computing d(0, E(�)W(�)�1D(v)) is reduced to the calculation of the �1 b b DðvÞÞ d ð0, Eð�ÞWð�Þ �� � � ��� �� � w1 Pị w1 : , B ẳ v ¼ inf � � u � u1 �ð�Þ1=p ��1 Pị , B vị : ẳ d 0, �ð�Þ1=p � : We have, for v C, EịWị1 Dvị v ẳ EịWị1 v p B C ẳ Eị@ q A : ịp ỵ q ẳ e p ỵ e ịq ỵ r ẳ v r v ỵ p ẳ :p2C : ỵ 1ịv ỵ 1ịp D.D Thuan 518 distance from the origin to the straight line in C2 whose equation can be rewritten in the form x2 � (� � 1)x1 ẳ 2v with x1 ẳ v ỵ p, x2 ẳ ỵ 1ịv ỵ 1ị p: Note �that � if � ¼ then this line is reduced to the point vv : Assume that � 6¼ and let C2 be endowed with the vector norms k � k1, then we can deduce, 2jvj � j� 1jjx1 j ỵ jx2 j j 1j þ 1Þ maxfjx1 j, jx2 jg �� �� � x1 ẳ j 1j ỵ 1ị x � : This implies Downloaded by [York University Libraries] at 04:54 18 October 2013 Acknowledgements This work was supported financially by NAFOSTED (Vietnam National Foundation for Science and Technology Development) �� �� � x1 � 2jvj � � x j 1j ỵ , 2v which yields the equality if x2 ¼ j1jỵ1 and x1 ẳ eix2, where is chosen such that (1 � �)ei’ ¼ j� � 1j Therefore, � kEịWị1 Dk1 ẳ sup d 0, EịWị1 Dvị jvjẳ1 < if � 6¼ 0, ¼ j� � 1j ỵ : if ẳ 0: Thus, by Theorem rC ðA0 , A1 , B; D, EÞ ¼ 12 : 3.2, we obtain 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 Ax ¼ b, x K � Cm, where K is a closed convex cone, as well as for controllability radius of convex processes x_ F ðxÞ, t � 0: These problems are the topics of our further study 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 ... 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. .. 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...International Journal of Control Vol 86, No 3, March 2013, 512–518 The structured controllability radius of linear delay systems Do Duc Thuanab* a School of Applied Mathematics and Informatics,