A Simple Derivation of Right Interactor for Tall Transfer Function Matrices and its Application to Inner-Outer Factorization Continuous-Time Case

A Simple Derivation of Right Interactor for Tall Transfer Function Matrices and its Application to Inner-Outer Factorization Continuous-Time Case ATARU KASE Osaka Institute of Technology Department of Electrical and Electronic Systems Engineering 5-16-1 miya, Asahi-ku, Osaka 535-8585 JAPAN kase@eeoitacjp Abstract: An interactor matrix plays several important roles in the control systems theory In this paper, we present a simple method to derive the right interactor for tall transfer function matrices using Moore-Penrose pseudoinverse By the presented method, all zeros of the interactor lie at the origin The method will be applied to the inner-outer factorization It will be shown that the stability of the interactor is necessary to calculate the factorization Key ords: interactor matrix, inner-outer factorization, strictly proper plant, continuous-time systems 1 Introduction The inner-outer factorization for stable transfer function matrices is a useful tool for robust controller analysis and synthesis It is known as a reasonable method for computer calculation based on the state space representation 1 However, the method is only useful for proper transfer function In this paper, we will propose to use the interactor 2 to compensate a strictly proper plant In the several control problems, an interactor matrix plays some important roles Although some papers were shown along with the above idea, those method were very complex since the derivations of the interactor were also complex 3-5 Some new calculation methods were presented (6-9) However, they were not easy On the other hand, it is shown that the coefficient matrices of the interactor can be obtained by solving a linear matrix equation of certain type 1 The authors presented a method to solve the equation by using Moore-Penrose pseudoinverse 11 But the method can be only used for square transfer function matrices In this paper, the method presented in 11 will be extended to the case nonsquare transfer function matrices For our application to the inner-outer factorization, we will consider the derivation of the right interactor, unlike our previous work 11 It will be shown that the proposed interactor has all-pass property in the discrete-time, so all of its zeros lie at the origin However, it can not be used since the origin is not stable for the continuous-time systems Thus, we will also show the zeros assignment of the interactor The paper is organized as follows In the next section, the derivation of an interactor will be presented For this purpose, Moore-Penrose pseudoinverse will be employed In Section 3, an application to the innerouter factorization will be shown The result is very simple but some properties should be proved, which will be shown in Appendix Concluding remarks will be given in Section 4 2 Simple Derivation of Interactor Matrix For a given m p (m p) strictly proper transfer function matrix G(s), there exists a p p polynomial matrix L(s), which satisfies the following equation: lim G(s)L(s) = K (full column rank) (1) z Such an L(s) is called a right interactor matrix of G(s) 1 Let (A, B, C) denote a minimal realization of G(s) Setting the coefficient of L(s) by L(s) = sl 1 + + s w L w = ss w 1 (s)l, (2) 1 Although the definition in 2 is restricted the structure of L(s) (lower triangular), we do not consider such a restriction since it is not essential in this paper ISBN: 978-1-6184-8- 37

then the the following equation holds as in 1: T w 1 L = J w 1 (3) L 1 L 2 K L =, J w 1 =, m(w 1) p L i R p p (i = 1, 2,, w) L w CB CAB CA w 1 B CB CA w 2 B T w 1 = CB SI w p (s) = s s w The integer w will be determined later In 11, the special solution ξ of eqn(3) is given by ξ = T w 1 J w 1 using the pseudoinverse T w 1 of T w 1, p = m and K = However, it can not be assumed the special structure of K since p m At this time, considering the structure of J w 1, set L = T w 1 J w 1 = T w 1 (:, 1 : m)k, (4) T w 1 (:, 1 : m) denote the submatrix constituted of the first m-th columns of T w 1 Substituting the above equation to eqn(3), T w 1 T w 1 (:, 1 : m)k = J w 1 (5) Define Λ by Λ = CB CAB CA w 1 B T w 1 (:, 1 : m), (6) the first m-th rows of eqn(5) can be written by ΛK = K (7) That is, if eqn(3) is solvable, its special solution is given by eqn(4) and K must satisfy eqn(7) Eqn(7) means that K is the set of eigenvectors of Λ which correspond to the eigenvalues at λ = 1 Since the definition of T w 1, T w 1T w 1 is a real symmetric matrix, and thus Λ is also a real symmetric Therefore, the geometric multiplicity of the eigenvalue one in Λ equals to the algebraic multiplicity and thus K can be found Therefore, w is the least integer when Λ has the eigenvalue at λ = 1 with multiplicity p K is constituted of corresponding eigenvectors Note that from the singular value decomposition of T w 1 and T w 1, every eigenvalue of T w 1T w 1 is one or zero In the case the eigenvalues of Λ is determined, it should be used the eigenvalues and corresponding eigenvectors of T w 1 T w 1 Lemma 1 For the interactor L(s) derived in the above, the following equation holds: L (s)l(s) = L (8) L (s) = (s 1 ) (9) = s 1 1 + s 2 2 + + s w w (Proof) Since T w 1 is the pseudoinverse of T w 1, the following relation must hold Partitioning T w 1 by T w 1 T w 1 = (T w 1 T w 1) T (1) T w 1 = T w 1 (:, 1 : m) P, P R pw m(w 1) (11) and substituting the above into eqn(1), we have T w 1 CB (:, 1 : m) CAB CA w 1 B +P m(w 1) p T w 2 = Tw 1(T T w 1 )T (12) Left multiplying to the both side of the above equation, we have T w 1 CB (:, 1 : m) CAB CA w 1 B + P m(w 1) p T w 2 = Jw 1(T T w 1 )T = K T (T w 1 (:, 1 : m))t = (13) From the definition of an interactor matrix, it follows that L w L w 1 L 2 L w L 3 T w 2 = (14) L w Thus from eqn(13), we have L w L w 1 L 1 L w L 2 L w ISBN: 978-1-6184-8- 38

= ( T w 1 (:, 1 : m) CB CAB CA w 1 B L w L w 1 L 1 ) + L w L 2 P m(w 1) p T w 2 L w = T w 1 (:, 1 : m) m p(w 1) K = p p(w 1) L (15) Since the above equation can be rewritten by L w L w 1 L w = LT 1 L w = = 1 L w 1 + 2 L w 1 L = = 1 LT 2 L =, the following equation also holds: L 1 L 2 L 1 L w 1 L w 2 L 1 = (16) Therefore, using eqns(15) and (16), L (s)l(s) can be described by L (s)l(s) s 1 L(s) = s 2 L(s) s w L(s) s w 1 s w 2 L w L w 1 L 1 = L w L 2 L 1 I p L w L w 1 L s 1 1 s 1 w s w 1 = p p(w 1) L p p(w 1) s 1 w = L (17) If eqn(8) holds, L(s) is having all-pass property in the discrete-time Therefore, all zeros of the interactor lie at the origin But the origin is not stable for the continuous-time systems Therefore, the zeros of interactor should be assigned to the stable region Theorem 2 Define H := AB A 2 B A w B L Let (Ã, B, C) denote a minimal representation of (A HK C, B, K C) and denote an observer gain which makes the eigenvalues of à := à C be desired ones Then, there exists a matrix L such that T,w L = n p J w 1 B à B à w B C B T,w := CÃw 1 B C B Moreover, defining (18) ˆL(s) := S w (s)l, (19) the zeros of detˆl(s) are equivalent to the eigenvalues of à (Proof) Since H is the maximum uncontrollable observer gain for the system (A, B, K C), there are no finite zeros in (Ã, B, C) Therefore, the inverse system of (Ã, B, C) can be presented by polynomial matrix, say ˆL(s) That is, the zeros of det ˆL(s) are equivalent to the eigenvalues of à Since a right interactor for (A, B, C) is also the one of (Ã, B, K C) 12, ˆL(s) can be represented the polynomial matrix with w-th degrees as shown in eqn(19) Thus, the following relation holds: à B C S w (s)l = ISBN: 978-1-6184-8- 39

Expanding the transfer function matrix using Markov parameters, it can be obtained the following equation: C B Cà B CÃw 1 B C B Cà w 2 B L = J w 1 (2) C B C Cà CÃ2 B à B à w 1 B L = Since ( C, à ) is observable, B à B à w 1 B L = and combining the above equations, it can be obtained eqn(19) Example 1 Consider the following transfer function matrix: s + 1 s + 2 G(s) = s + 3 s + 4 s + 5 s + 6 Then, a state space realization (A, B, C) is given by 2 2 1 1 3 12 4 3 A =, B = 1 7 3 6 5 1 1 1 1 C = 1 1 In this example, w = 3 is the first integer when 9 2 1 Λ = 2 6 2 1 2 9 has the multiple eigenvalue at λ = 1 Then K is given by 1 K = 1 1 2 Thus, the right interactor L(s) can be calculated by L(s) = si 2 s 2 I 2 s 3 I 2 L 1 L 2 L 3 7 3 7 3 = ssi 2 2 (s), 6 4 5 5 5 5 det L(s) = 5s 4 Next, design another interactor ˆL(s) which has the desired zeros at s = 12, 14, 16, 18 For the realization, let denote an observer gain which makes the closed-loop s poles be 12, 14, 16 and 18 Setting à := à C, ˆL(s) is given by B à B Ã2 B Ã3 B ˆL(s) = SI 3 2 (s) C B Cà B CÃ2 B C B Cà B C B 38192 1424 5288 2481 6357 27731 = I 2 si 2 s 2 I 2 s 3 I 2 7624 42518 3667 19787 35667 24787 5 5 5 5 2 2 I 2 4 2 3 Application to Inner-Outer Factorization For a given m p (m p) proper and stable transfer function matrix G(s) which has no invariant zeros on the imaginary axis, it is known the inner-outer factorization using state space representation if eqn(1) holds with L(s) = In this section, it will be proposed the method applicable for the case L(s), which can be summarized by the following procedures: step 1 Calculate a right interactor L(s) for a given G(s) step 2 Calculate a spectral factorization for G(s)L(s), ie, calculate a stable rational ISBN: 978-1-6184-8- 4

function matrix H(s) satisfying L (s)g (s)g(s)l(s) = H (s)h(s)(21) H 1 (s) is stable, G (s) := G T ( s) step 3 Set inner matrix H i (s) and outer matrix H o (s) as follows: H i (s) := G(s)L(s)H 1 (s), H o (s) := H(s)L 1 (s) (22) It is clear that H i (s)h o (s) = G(s) and since H i (s)h i (s) = {G(s)L(s)H 1 (s)} G(s)L(s)H 1 (s) = {H 1 (s)} H (s)h(s)h 1 (s) = (23) hold, H i (z) is an inner Since it is hard to carry out the above calculations based on transfer function matrix, we will present the method based on the state space representation Although we only consider the case G(s) is strictly proper, same idea can be used for the proper case Let (A, B, C) denote a minimal state space realization of G(s) Then, the state space representation of G(s)L(s) is given by (A, ˆB, C, K), Γ := B AB A w 1 B L ˆB := AΓ + BL, K = CΓ, (24) if the coefficient matrix L(s) is given by eqn(2) Consider the following Riccati equation: A T X + XA + C T C (25) (XB + C T K)R 1 (B T X + K T C) = R := K T K (26) Let λ and x T denote an eigenvalue and corresponding eigenvector of A Then, x T B, since (A, B) is controllable On the other hand, x T ˆB = x T B(L + λl 1 + + λ w L w ) = x T BL(λ) Since all zeros of detl(s) lie in the left half plane, detł(λ) = only if Reλ < Thus, the pair (A, ˆB) is stabilizable and there exists a positive semi definite solution X for Riccati equation (25) Moreover, setting F := R 1 ( ˆB T X + K T C), (27) it is known that X makes A ˆBF be stable Define M(s) and N(s) by M(s) := (si A) 1 ˆB, N(s) := F M(s) (28) Then from Riccati equation, the following relation holds: { + N (s)}r{ + N(s)} = M C (s) I T C C T K M(s) p K T C K T K = {G(s)L(s)} G(s)L(s) (29) Now, define H(s) by H(s) = R 1/2 { + N(s)} = R 1/2 A ˆB (3) F H 1 A ˆBF ˆB (s) = R 1/2, A F := A F ˆBF Then from eqn(22), we can obtain H i (z) = A ˆBF ˆB R 1/2 (31) C KF K As shown in 14, L(s) is also a right interactor for F (si A) 1 B if detl(s) is Hurwitz So, ˆB is the observer gain of the inverted interactorizing for (A, B, F ) (see 13 for detail), ie, L 1 (s) = AF B (32) F Substituting the above equation and eqn(3) to eqn(22), we can obtain H o (s) = R 1/2 A B (33) F Example 2 Same plant is considered as shown in Example 1 Using the solution X of Riccati equation (25), the outer matrix is given by 2 2 1 1 3 12 4 3 H o (s) = 1 7 3 6 5 1 1 1 12 942 24 116 12 348 9 29 18 582 9 789 ISBN: 978-1-6184-8- 41

as the inner matrix is given by 3843 968 968 3843 18 12 H i (s) = 14 16 38 5 141 4 114 3 11711 5491 8396 7263 64 731 9193 46845 936 12227 5462 27638 881 237 237 527 48 817 Note that all zeros of detˆl(s) are included as poles of H i (s) However, zeros of the interactor are canceled and H i (s) yields to 38439 9674 61332 536645 9674 38439 864847 393587 H i (s) = 57 515 88165 2367 236 19346 2367 5266 686 15569 4825 8165 ie, the parameters of the interactor does not appear 4 Conclusions A simple derivation of a right interactor was given It was shown that the proposed interactor has all-pass property in the discrete-time It was also shown that the zeros assignment method of the interactor Applying the result, it was presented a method of inner-outer factorization based on the state space representation The method was independent on the choice of the interactor This fact is quite natural since the interactor is not an essential but a technical tool for the factorization The result for the discrete-time systems can be found in 14 References: 1 B A Francis, A course in H control theory, Lecture Notes in Control and Information Science, vol88, Springer-Verlag, NY, 1987 2 A olovich and P L Falb, Invariants and canonical form under dynamic compensations, SIAM Journal of Control and Optimization, vol14, pp996-18, 1976 3 X Xin and T Mita, Inner-outer factorization for non-square proper functions with infinite and finite jω-axis zeros, International Journal of Control, vol71, pp145-161, 1998 4 A Varga, Computations of inner-outer factorizations of rational matrices, IEEE Transactions on Automatic Control, vol43, pp684-688, 1998 5 C Oara and A Varga, Computations of general inner-outer and spectral factorizations, IEEE Transactions Automatic Control, vol45, pp237-2325, 2 6 R Rogozinski, A P Paplinski and M J Gibbard, An algorithm for the calculation of a nilpotent interactor matrix for linear multivariable systems, IEEE Transactions on Automatic Control, vol32, pp234-237, 1987 7 S Bittani, P Colaneri and M F Mongiovi, Singular filtering via spectral interactor matrix, IEEE Transactions on Automatic Control, vol4, pp1492-1497, 1995 8 B Huang, S L Shah and H Fujii, The unitary interactor matrix and its estimation using closed-loop data, Journal of Process Control, vol bf 7, pp195-27, 1997 9 T Mita, T K Nam and X Xin, Sliding mode control for invertible systems based on a direct design of interactors, Asian Journal of Control, 5, pp242-25, 22 1 Y Mutoh and R Ortega, Interactor structure estimation for adaptive control of discrete-time multivariable nondecouplable systems, Automatica, vol29, pp635-647, 1993 11 Kase and Y Mutoh, A simple derivation of interactor matrix and its applications, International Journal of Systems Science, vol4, pp1197-125, 29 12 Y Mutoh, A note on the invariant properties of interactors, International Journal of Control, vol62, pp1247-1252, 1995 13 Y Mutoh and P N Nikiforuk, Inversed interactorizing and triangularizing with an arbitrary pole assignment using the state feedback, IEEE Transactions on Automatic Control, vol37, pp63-633, 1992 14 Kase and Y Mutoh, A simple derivation of right interactor for tall plant and its application to inner-outer factorization, Proceedings of the 19th Mediterranean Conference on Control and Automation, pp1516-1521, 211 ISBN: 978-1-6184-8- 42