VERSAL DEFORMATIONS OF BILINEAR SYSTEMS UNDER OUTPUT-INJECTION EQUIVALENCE

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1 PHYSCON 2013 San Luis Potosí México August 26 August VERSAL DEFORMATIONS OF BILINEAR SYSTEMS UNDER OUTPUT-INJECTION EQUIVALENCE M Isabel García-Planas Departamento de Matemàtica Aplicada I Universitat Politècnica de Catalunya Spain mariaisabelgarcia@upcedu Abstract Bilinear systems under output injection equivalence are considered The aim of this paper is to study what happens when a slight perturbation affects the coefficients of the matrices defining a bilinear equation in C n For this goal we will use the Arnold s techniques that is to say we will construct a versal deformation of a differentiable family of bilinear systems which are the orthogonal linear varieties to the orbits of output injection equivalent bilinear systems Versal deformations provide a special parametrization of bilinear systems space which can be applied to perturbation analysis and investigation of complicated objects like singularities and bifurcations in multi-parameter bilinear systems Key words Bilinear systems versal deformations equivalence relation output injection 1 Introduction A control bilinear system is a control system which is described by linear differential equations in such a way that the control inputs appear as coefficients ẋ = A + unx + bu y = cx 1 with the state vector x C n and input vector u C that we will write as a 4-tuple of matrices A N b c M n C M n C M n 1 C M 1 n C we will denote by M = {A N b c} the space of bilinear systems Notice that the set of linear systems constitute the subclass of bilinear systems for which N = 0 The bilinear control systems have been studied with growing interest in the last years because of the arising problems which challenging practical interest Its methods and applications cross interdisciplinary on the borderland between physics and control proving useful in areas as diverse as spin control in quantum physics and the study of Lie semigroups [3] The stabilization problem analyze the existence of a constant control which renders the resulting linear system to have at least one eigenvalue in the open left half of the complex plan Remember that λu C is an eigenvalue of A N b c under output injection if A+uN λui c < n The eigenstructure of A N b c is quite sensitive to perturbations in the matrices defining the system and one wants therefore to accurately describe how that structure can change when small variations are applied to these coefficients A manner to approach this problem can be by means so called versal deformations of the eigenstructure of the bilinear systems Versal deformation of a differentiable family of square matrices under similarity [1] was constructed by VI Arnold and has been generalized by several authors to matrix pencils under the strict equivalence [4] [9] pairs or triples of matrices under the action of the general linear group [11] pairs of matrices under the feedback similarity [8] Versal deformations provide a special parametrization of matrix spaces which can be effectively applied to local perturbation analysis and investigation of complicated objects like singularities and bifurcations in dynamical systems [1] [2] [4] [8] The theory of miniversal deformations is used to determine which classes of bilinear systems A N b c under a previously defined equivalence relation can be found in all small neighbourhood of a given system 2 Equivalence relation Let us consider the space of bilineal systems M = {A N b c M n C M n C M n 1 C M 1 n C} Many of the works in bilinear systems has concentrated on bilinear systems up to output injection see [10] for example For that we consider the following equivalence relation that we will call output injection equivalence

2 Definition 21 Two 4-tuples A 1 N 1 b 1 c 1 and A 2 N 2 c 2 in M are output injection equivalent if and only if A 2 = P A 1 P 1 + V c 1 P 1 N 2 = P N 1 P 1 + W c 1 P 1 2 = P b 1 r c 2 = sc 1 P 1 for some nonsingular square matrix P Gln; C V W M n 1 C r s C 0 Remark 21 For V = 0 and W = 0 the equivalence relation coincides with the similarity equivalence From this definition it results easily the following proposition Proposition 21 Eigenvalues of the bilinear systems A N b c under output injection are invariant under this equivalence relation Proof A1 + un 1 λui = c 2 P V + uw A1 + un 1 λui 0 s P 1 = c 1 P A1P 1 +up N 1P 1 λui+v c 1P 1 +uw c 1P 1 scp 1 A2 + un 2 λui c 2 It is easy to prove that in this case and for n = 2 the systems can be reduced in the following form Proposition 22 Let A N b c be a 4-tuple Then a If c 0 a1 a2 a β1 b β 2 0 α b1 1 0 b α 2 0 β 2 In all cases if b 1 0 then b 1 = 1 and if b 1 = 0 but 0 then = 1 b If c = 0 = b1 If matrices A and N have a unique common one dimensional invariant subspace then the system is equivalent to λ1 α µ1 β b1 0 λ 2 0 µ 2 and i if λ 1 λ 2 then α = 0 and β 0 ii if λ 1 = λ 2 and A λ 1 I then α = 1 and β 0 iii if λ 1 = λ 2 and A = λ 1 I then α = 0 and β 0 b2 If matrices A and N have no common invariant subspaces then the system is equivalent to λ1 α µ1 0 b1 0 λ 2 β µ 2 with α β 0 b3 If matrices A and N have two common complementary one dimensional invariant subspaces then the system is equivalent to λ1 0 µ1 0 b1 0 λ 2 0 µ 2 and as in the case a if b 1 0 then b 1 = 1 and if b 1 = 0 but 0 then = 1 If b 1 0 then b 1 = 1 and if b 1 = 0 but 0 then = 1 The case c = 0 the possible reduced forms are an extension to 4-tuples A N b c from a classification theorem for couple of matrices A N given but not explicit in [6] The paper is focused on the case c 0 the input injection is allowed For c 0 we have the following result Proposition 23 [7] Let A N b c be a 4-tuple with c 0 if A c and N c are observable pairs and c ca N = 1 = 2 c ca ca 2 N 2 c cạ ca n 2 ca n 1 N n 1 = n 1 3 Then the 4-tuple A N b c is equivalent under equivalence relation considered to the following 4-tuple

3 A r N r b r c r A r = N r = b r = α 11 α 12 α 1n α 22 α 2n b 1 b n c r = Tangent and normal spaces Equivalence relation defined in 2 may be seen as induced by the action of the Lie group G = Gln; C M n 1 C C 0 C 0 in the following manner: Proposition 31 Let x 0 = A N b c M dα x0 g = [P A] + V c [P N] + W c P b br sc cp M g = P V W r s T e G 7 Proof It suffices to compute the first approximation of α x0 I + εp εv εw 1 + εr 1 + εs: α x0 e + ε x 0 + ε[p A] + V c [P N] + W c P b br sc cp e + ε = I + εp εv εw 1 + εr 1 + εs The mapping dα x0 provides a simple description of the tangent space T x0 Ox 0 and as a corollary its normal complement T x0 Ox 0 for any inner scalar product defined on M αg x = x 1 x 1 = A 1 N 1 b 1 c 1 A 1 = P AP 1 + V cp 1 N 1 = P NP 1 + W cp 1 b 1 = P br c 1 = scp 1 M g = P V W r s G x = A N b c M 4 Proposition 32 The tangent space to the orbit of the 4-tuple x 0 is given by T x0 Ox 0 = {[P A] + V c [P N] + W c P b br sc cp P V W r s T e G} 8 The Euclidean scalar product in the spaces M considered in this paper is defined as follows Let us fix a 4-tuple x 0 = A 0 N 0 b 0 c 0 M and define the mapping α x0 P V W r s = αp V W r s x 0 M 5 The equivalence class of the 4-tuple x 0 with respect to the action of G is called the orbit of x 0 and denoted by Ox 0 = {α x0 P V W r s P V W r s G} 6 The mapping α x0 is differentiable and Ox 0 is a smooth submanifold of M Let T e G be the tangent space to the manifold G at the unit element e = I G Since G is an open subset of M n C M n 1 C C C we have T e G = M n C M n 1 C C C and since M is a linear space T x0 M = M Let dα x0 : T e G M be the differential of α x0 at the unit elements e Using 5 we find the following proposition x 1 x 2 = tra 1 A 2+trN 1 N 2 +trb 1 b 2+trc 1 c 2 9 with x i = A i N i b i c i M i = 1 2 and A denotes the conjugate and transpose of a matrix A and tr the trace of corresponding matrices Having defined a scalar product we can compute the orthogonal space to the tangent space Theorem 31 Let x 0 = A N b c M be a 4- tuple then X Y z t T x0 Ox 0 if and only if [A X ] + [N Y ] + bz t c = 0 cx = 0 cy = 0 z b = 0 ct = 0 10 Proof Calling α = [P A] + V c [P N] + W c P b br sc cp and β = X Y z t then < α β >= tr [P A]X + V cx + tr [P N]Y + W cy + tr P bz brz + tr sct cp t = tr [A X ] + [N Y ] + bz t cp + tr cx + tr cy tr z b + tr ct = tr MQ

4 M = [AX ]+[NY ]+bz t c cx cy Q = P V W r s z b ct Obviously the condition of the theorem is sufficient For the necessity remark that in fact we have M 1 = tr MQ = tr M 1 Q 1 [AX ]+[NY ]+bz t c cx cy Q 1 = P M 1 M 2 M 3 M 4 M 5 V M 6 M 7 M 8 M 9 M 10 W M 11 M 12 M 13 M 14 M 15 r M 16 M 17 M 18 M 19 M 20 s z b ct because matrices M i i = 1 20 do not appear in the computation Therefore the first matrix should be the null matrix So the proof is concluded Example 31 Let us consider a bilinear system x 0 = A 0 N 0 b 0 c 0 = According to Theorem 31 the elements X Y z t T x0 Ox 0 can be found by solving the linear system 10 As a result we obtain a general element of T x0 Ox 0 in the form X Y z t with 0 x3 0 x3 X = Y = 0 x 4 0 x 4 z1 z = t = 12 0 x z 1 x 3 x 4 z 1 C are arbitrary parameters; so dimt x0 O 2 x 0 = 3 4 Versal deformation Let M be a differential manifold with the equivalence relation defined by the action of a Lie group G The G- action is described by the mapping g x αg x x αg x M and g G Let U 0 be a neighborhood of the origin of C l A deformation xλ of x 0 is a smooth mapping x : U 0 M such that x0 = x 0 The vector λ = λ 1 λ l U 0 is called the parameter vector The deformation xλ is also called the family of 4-tuples of matrices The deformation xγ of x 0 is called versal if any deformation yµ of x 0 µ = µ 1 µ k U 0 C k is the parameter vector can be represented in some neighborhood of the origin in the following form yµ = αgµ xφµ µ U 0 U 0 13 φ : U 0 C l and g : U 0 G are differentiable mappings such that φ0 = 0 and g0 = e Expression 13 means that any deformation zξ of x 0 can be obtained from the versal deformation xγ of x 0 by an appropriate smooth change of parameters γ = φξ and equivalence transformation gξ smoothly dependent on parameters The versal deformation with minimal possible number of parameters l is called miniversal The following result proved by Arnold [1] for Gln; C acting on M n n C and generalized by Tannenbaum [11] for a Lie group acting on a complex manifold provides the relation between the versal deformation of x 0 and the local structure of the orbit of x 0 Theorem 41 1 A deformation xλ of x 0 is versal if and only if it is transversal to the orbit Ox 0 at x 0 2 Minimal number of parameters of a versal deformation is equal to the codimension of the orbit of x 0 in M l = codim Ox 0 In our particular setup Let us denote by {c 1 c l } a basis of an arbitrary complementary subspace T x0 Ox 0 c to T x0 Ox 0 and by {n 1 n l } a basis of T x0 Ox 0 ; Corollary 41 The deformation xλ = x 0 + is a miniversal deformation l c j λ j 14 j=1 If we take c j = n j j = 1 l in 14 then the corresponding miniversal deformation is called orthogonal Example 41 We explicit a basis {c 1 c l } for the case A simplest miniversal deformation is A N b c + X Y z t with X Y z t = x 7 0 x 5 x 8 0 y 4 y 7 0 y 5 y 8 z y 9 0

5 The basis is obtained placing a 1 in a single parameter and 0 otherwise for each one of them Given a bilinear system x 0 = A N b c the homogeneity of the orbits allow us to consider canonical reduced forms to write down explicitly the bases {c 1 c l } and {n 1 n l } 5 Structural stability In a similar way to [5] from the miniversal deformation in Section 35 we can deduce conditions for a 4- tuple of matrices to be structurally stable according to the usual definition Definition 51 A 4-tuple of matrices A N b c M is called structurally stable if and only if it has a neighborhood formed by 4-tuples equivalent to it-that is to say if A N b c is an interior point of its orbit Because of homogeneity of the orbits we have: Proposition 51 A 4-tuple of matrices A N b c M is structurally stable if and only if so are all other 4-tuples in its orbit Structural stability is equivalent to the nonexistence of deformations in the following sense: Proposition 52 A 4-tuple of matrices x 0 = A N b c M is structurally stable if and only if dim T x0 Ox 0 = 0 Proof Analogous to [5] proposition 43 From Sections 3 and 4 it is immediate to see how Theorem 31 can be used to characterize the structural stability of a 4-tuple of matrices because the above dimension is zero if and only if the only solution of the system 10 in Section 3 is the zero one In our particular setup all reduced forms presented have continuous invariants so all miniversal deformation are not zero We can consider strata defined as the infinite union of orbits of the 4-tuples having the same type of reduced form varying only on the values of continuous invariants It is obvious that the strata are invariant under equivalence defined and we have the following proposition Proposition 53 For n = 2 i The stratum consisting of 4-tuples of the type a1 in 22 Sx 0 = { β } 0 β 2 b2 with β 1 0 is stable under the equivalence relation considered ii The remaining strata consisting of 4-tuples of the type a1 with β 1 = 0 a2 a3 b1 b2 b3 in 32 are not stable under the equivalence relation considered Proof i It suffices to compute a miniversal deformation of the 4-tuple obtaining β1 0 β 2 0 β1 + y 1 0 β 2 + y z 2 that obviously all 4-tuples on the deformation are not in the orbit but in the same stratum ii It is easy to observe that any small perturbation of the 4-tuples in these strata contain 4-tuples belonging in the stratum of the type a1 References V I Arnold On matrices depending on parameters Russian Math Surveys 26: pp J V Burke and M L Overton Stable perturbations of nonsymmetric matrices Linear Algebra Appl pp E David Bilinear Control Systems Applied Mathematical Sciences Vol A Edelman E Elmroth and B Kågström A geometric approach to perturbation theory of matrices and matrix pencils Part I: Versal deformations SIAM J Matrix Anal Appl pp J Ferrer M ā I García-Planas Structural Stability of Quadruples of Matrices Linear Algebra and its Applications 241/ pp Sh Friedland Similarity of families of matrices An article in CRC Handbook of Linear Algebra 2005 M ā I García-Planas Similarity of pairs of linear maps defined modulo a subspace Linear and Multilinear Algebra vol M ā I García-Planas A Mailybaev Reduction to versal deformations of matrix pencils and matrix pairs with application to control theory SIAM Journal on Matrix Analysis vol 24 4 pp M I Garcia-Planas and V V Sergeichuk Simplest miniversal deformations of matrices matrix pencils and contragredient matrix pencils Linear Algebra Appl pp P M Pardalos V Yatsenko Optimization and control of bilinear systems: theory algorithms and applications Ed Springer 2008 A Tannenbaum Invariance and System Theory: Algabraic and geometric Aspects Lecture Notes in Math 845 Springer-Verlag 1981