Analysis of behavior of a simple eigenvalue of singular system

Transcription

1 Proceedings of the 213 International onference on Systems, ontrol and Informatics nalysis of behavior of a simple eigenvalue of singular system Maria Isabel García-Planas, Sonia Tarragona Matemàtica plicada I, Universitat Politècnica de atalunya, Minería 1, Esc, 1-3, 838 Barcelona, Spain mariaisabelgarcia@upcedu Departamento de Matemáticas, Universidad de León, ampus de Vegazana s/n, 2471, León, soniatarragona@unileones bstract Small perturbations of simple eigenvalues with a change of parameters is a problem of general interest in applied mathematics The aim of this work is to study the behavior of a simple eigenvalue of singular linear system family Epẋ px + Bpu, y px smoothly dependent on real parameters p p 1,, p n Index Terms Singular linear systems, Eigenvalues, Eigenvectors, Perturbation 1 INTRODUTION Let us consider a finite-dimensional singular linear time-invariant system Eẋt xt + But xt yt xt x, 1 where x is the state vector, u is the input or control vector, E, M n, B M n m, M p n and ẋ dx/dt We will represent the systems as quadruples of matrices E,, B, In the case where E I n the systems are standard and we will denote them, as triples, B, Singular systems are found in engineering systems such as electrical, chemical processing circuit or power systems, aircraft guidance and control, mechanical industrial plants, acoustic noise control, among others, and they have attracted interest in recent years Sometimes it is possible to change the value of some eigenvalues introducing proportional and derivative feedback controls in the system and proportional and derivative output injection The values of the eigenvalues that can not be modified by any feedback proportional or derivative and/or output injection proportional or derivative, correspond to the eigenvalues of the singular pencil se B, that we will simply call eigenvalues of the quadruple E,, B, Perturbation theory of linear systems has been extensively studied over the last years starting from the works of Rayleigh and Schrodinger [6], and more recently different works as [2], [5],[4], can be found This treatment of eigenvalues is a tool for efficiently approximating the influence of small perturbations on different properties of the unperturbed system Small perturbations of simple eigenvalues with a change of parameters is a problem of general interest in applied mathematics and concretely, this study for the kind of systems under consideration have some interest because in the case where m p < n, the most generic types of systems have n m simple eigenvalues The obtained results can be applied to analyze the frequency and damping perturbations in models of flexible structures for example, see [7], [8] In the sequel and without lost of generality, we will consider systems such that matrices B and have full and m p < n 123

2 Proceedings of the 213 International onference on Systems, ontrol and Informatics 2 FEEDBK ND OUTPUT INJETION EQUIVLENE RELTION Definition 21: Two quadruples E,, B, and E,, B, are called equivalent if, and only if, there exist matrices P, Q Gln;, R Glm;, S Glp;, F B, F E B M m n and F, F E M n p such that E QEP + F E P + QBF B E, QP + F P + QBF B, B QBR, SP, 2 ssociated to each eigenvalue there is an eigenvector defined in the following manner: or written in a matrix form Definition 23: i v E B M n 1 is an eigen- vector of this system corresponding to the B eigenvalue λ if and only if, there exist a vector w M m 1 such that Q FE E B P S F B Q F E R λ E + BF B P E B + BF B B v w S F B R, It is easy to check that this relation is an equivalence relation singular system E,, B,, for which there exist matrices FE B and/or F E such that E + BF E B + FE is invertible is called standardizable, and in this case there exist matrices P, Q, FE B, F E such that QEP + QBFE B + F E P I n onsequently the equivalent system is standard Notice that the standardizable character is invariant under the equivalence relation considered In the case where the original system is standard and if we want to preserve this condition under the equivalence relation we restrict the operation to the case where Q P 1, FE B and F E Definition 22: Let E,, B, be a system λ is an eigenvalue of this system if and only if λ E B < λe B We denote by σe,, B, the set of eigenvalues of the quadruple E,, B, and we call it the spectrum of the system The continuous invariants under this equivalence are the eigenvalues of the system that they are defined as follows Proposition 21: Let E,, B, be a system The spectrum of this system is invariant under equivalence relation considered Proof: It suffices to observe that λe B Q λf E F λe B S P λfe B F B R for all FE B, F B ii u M 1 n is a left eigenvector of the system corresponding to the eigenvalue λ if and only if, there exist a vector ω M 1 p such that λ u ω E + FE + F B, for all FE, F Proposition 22: Let λ be an eigenvalue and v an associated eigenvector of the E,, B, Then λ is an eigenvalue and v an associated eigenvector of E + BFE + F E, + BF B + F, B, for all FE B, F E, F B, F Proof: Let w w λ FE B F Bv I λ FE F λ E B I v I λ FE B F B I w I λ FE F λ E B v I λ FE B F Bv + w I λ FE F λ E B v I w I λ FE F I Proposition 23: Let λ be an eigenvalue and u an associated left eigenvector of the E,, B,

3 Proceedings of the 213 International onference on Systems, ontrol and Informatics Then λ is an eigenvalue and u an associated left eigenvector of E + BFE + F E, + BF B + F, B, for all F E B, F E, F B, F Proof: nalogous to the proof of proposition 22, taking ω ω u λ FE F Remark 21: Unlike the case of triples of matrices E,, B see [4] if λ is an eigenvalue of the quadruple E,, B, it is not necessarily a generalized eigenvalue of the pair E, Example 21: Let E,, B, be a system with E I, 1 2, B 3, 1 1 λe B det 3λ + 3 Then, the eigenvalue of the system is λ 1 Observe that v 3, 3 t is an eigenvector associated to λ 1 there exist w 1 But detλe λλ 2, so the eigenvalues of the pair E, are λ 1 and λ 2 2 Definition 24: n eigenvalue λ of the system E,, B, is called simple if and only if verifies the following conditions λ E B i λe B 1, ii and λ E B E λ E B λ E B λe B + Proposition 24: The simple character is invariant under equivalence relation considered Proof: onsidering Q Q λ FE F S FE Q λ FE F S P P λ FE B F B R P FE B λ FE B F B r Therefore, Q λ E B E λ E B λ E B E λ E B P Proposition 25: Let λ be a simple eigenvalue of the standard system, B, Then, there exist an associate eigenvector v and an associate left eigenvector u such that u v 1 Proof: If λ is a simple eigenvalue, the system can be reduced to 1 B, 1, λ 1, λ I with 1 B 1 n 1 1 In this reduced form it is easy to observe that v,,, 1 t is an eigenvector and v,,, 1 is a left eigenvector verifying u v 1 Now, taking into account propositions 22 and 23, we can check easily that P v is an eigenvector of the system, B, and u P 1 is a left eigenvector for some invertible matrix P Remark 22: In general, for singular systems this result fails, as we can see in the following example Example 22: Let E,, B, be a singular system with E,, B and 1 It is easy to observe that λ 3 is a simple eigenvalue of this system and all possible eigenvectors are v α, t with α and all possible left eigenvectors are u, β with β learly u v But, we have the following more general result Proposition 26: Let λ be a simple eigenvalue of the singular system E,, B, with m p 1 λe B and n + 1 Then, there exist an associate eigenvector v and an associate left eigenvector u such that u Ev

4 Proceedings of the 213 International onference on Systems, ontrol and Informatics Proof: If λ is a simple eigenvalue λ E B v w E λ E B v 1 for all v 1 and w 1 So taking v 1 and w 1 we have that Ev Suppose now that u Ev, in this case we have that Ev Ker u ω λ E B Ev Im Then, λ B v1 for some v w 1, w 1 1 v, w because Ev So λ E B v w E λ E B v 1 w 1 w 1 ans λ can not be simple Therefore u Ev 3 NLYSIS OF PERTURBTION OF SIMPLE EIGENVLUES Standard systems We begin studying the case of standard systems in order to make more comprehensive the study So, we consider systems in the form ẋ x + Bu, y x with M n, B M n m and M m n represented as a triple of matrices, B, Let, B, be a linear system and assume that the matrices, B, smoothly depend on the vector p p 1,, p r of real parameters The function p, Bp, p is called a multi-parameter family of linear systems Eigenvalues of linear system functions are continuous functions λp of the vector of parameters In this section, we are going to study the behavior of a simple eigenvalue of the family of linear systems p, Bp, p Let us consider a point p in the parameter space and assume that λp λ is a simple eigenvalue of p, Bp, p, B,, and vp v is an eigenvector, ie there exists w M m 1 such that v B w λ v v Equivalently + B F Bv B w λ v v, F B M m n Now, we are going to review the behavior of a simple eigenvalue λp of the family of standard linear systems The eigenvector vp corresponding to the simple eigenvalue λp determines a one-dimensional nullsubspace of the matrix operator smoothly B dependent on p Hence, the eigenvector vp and corresponding wp can be chosen as a smooth function of the parameters We will try to obtain an approximation by means of their derivatives We write the eigenvalue problem as pvp Bpwp λpvp pvp 3 Taking the derivatives with respect to p i, we have λp p p λpi vp vp + Bp wp Bp wp p vp p vp t the point p, we obtain λp p v + Bp p p i w p λ I vp wp B p p 4 p vp v p p This is a linear equation system for the unknowns λp, vp and wp Lemma 31: Let v and u be an eigenvector and a left eigenvector respectively, corresponding to the 4 126

5 Proceedings of the 213 International onference on Systems, ontrol and Informatics simple eigenvalue λ of the system E,, B, Then, the matrix λ I T B v u + has full Proof: It suffices to consider the system in the reduced form 1 B, 1, λ 1 Proposition 31: The system 4 has a solution if and only if λp p u ω i p Bp v p w 5 where u is a left eigenvector for the simple eigenvalue λ of the system, B, Proof: The system 4 can be rewritten as λp p Bp v p w p vp 6 λ I B wp p We have that 4 has a solution if and only if 6 has a solution Premultiplying both sides of the equation 6, by u, ω u ω λp λp u ω p i I u ω p I p Bp p p v w Bp p We obtain a solution for λp λ ;p λp p u ω p p p Bp p v w v p w v w u v Using the normalization condition, that is to say, taking v such that u v 1, we have: λp p p u ω i Bp v p p w p Knowing λp we can deduce vp p p First of all, we observe that if u v 1, then u vp and we can take vp such that u vp 1 normalization condition, it suffices to take as vp 1 the vector u vp So vp u vp u vp onsequently we can consider the compatible equivalent system: λp p p Bp λ I + v u B p v w vp wp p 7 In our particular case where m p, the system has a unique solution vp wp p T 1 λp p Bp p p v Taking the partial derivative 2 / p j on both sides of both equations in the eigenvalue problem 3, we can obtain a second order approximation for eigenvalues B Singular systems Now we consider singular systems as in 1 written as quadruple of matrices E,, B, In this case the eigenvalue problem is written as pvp Bpwp λpepvp pvp 8 Taking the derivatives with respect to p i, we have λp Ep + λp Ep p p λpep vp t the point p, we obtain vp + Bp wp Bp wp p vp p vp w 5 127

6 Proceedings of the 213 International onference on Systems, ontrol and Informatics λp λp E + λ Ep p p v + Bp p w λ E vp p B wp p p p v vp p 9 This is a linear equation system for the unknowns, vp and wp Suppose now, systems E,, B, with m λe B p 1 and n + 1 Lemma 32: Let v and u be an eigenvector and a left eigenvector respectively, corresponding to the simple eigenvalue λ of the system E,, B, Then, the matrix T λ E B + E v u E has full Proof: First of all we proof that E v u E λ u ω E + E v u E B v w E u ω v u E v w u E v 2, so, E v u E In the other hand v / Ker E v u E, because u E v 2 u E v u E v Suppose now, that λ E + E v u E B v w for some vectors v and w Then λ u ω E + E v u E B v w E u ω v u E v u w E v u E v Taking into account that u E v we have that u E v, so E v u E v, then v is an eigenvector of the system corresponding to the eigenvalue λ linearly independent of v, but λ is simple Proposition 32: The system 9 has a solution if and only if λp E p u ω i E + λ p Bp v w p 1 where u is a left eigenvector for the simple eigenvalue λ of the system E,, B, Proof: nalogously to the proof of proposition 31 we observe that proposition 26 permits to clear the unknown λp from equation 1 On the other hand, taking into account that u E v, we have that u Epvp in vp a neighborhood of the origin So, u E p Lemma 32 permits to obtain vp and wp Example 31: onsider now, the following two-parameter differentiable family of systems Ep, p, Bp, p with Bp Ep I, p p1, p p1 + p 2, p 1 + p p 1, 1 + p 2 1 t p, we have that λ 1 is a simple eigenvalue, v 3 3 t a right eigenvalue with w 1 and u, 1 with ω 1 a left eigenvector Then λ 1 1 p t , λ 1 1 p t ONLUSION In this work families of singular systems in the form Epẋ px + Bpu, y px smoothly dependent on a vector of real parameters p p 1,, p n have been considered study of the behavior of a simple eigenvalue of this family of singular linear system is analyzed and a description of a first approximation of the eigenvalues and corresponding eigenvectors have been obtained p 6 128

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