oitacjp 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 MoorePenrose pseudoinverse By the pres ID: 25086 Download Pdf

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oitacjp 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 MoorePenrose pseudoinverse By the pres

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A Simple Derivation of Right Interactor for Tall Transfer Function Matrices and its Application to Inner-Outer Factorization — Continuous-Time Case WATARU KASE Osaka Institute of Technology Department of Electrical and Electronic Systems Engineering 5-16-1 0miya, Asahi-ku, Osaka 535-8585 JAPAN kase@ee.oit.ac.jp 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–Words: interactor matrix, inner-outer factorization, strictly proper plant, continuous-time systems. 1 Introduction The inner-outer factorization for stable transfer func- tion matrices is a useful tool for robust controller anal- ysis and synthesis. It is known as a reasonable method for computer calculation based on the state space rep- resentation [1]. However, the method is only useful for

proper transfer function. In this paper, we will pro- pose to use the interactor [2] to compensate a strictly proper plant. In the several control problems, an interactor ma- trix plays some important roles. Although some pa- pers 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 cal- culation methods were presented ([6]-[9]). However, they were not easy. On the other hand, it is shown that the coefﬁ- cient matrices of the interactor can be obtained by solving a linear matrix equation of

certain type [10]. 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 where nonsquare transfer func- tion 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 interac- tor. The paper is organized as follows. In the next sec- tion, the derivation of an interactor will be presented. For this purpose, Moore-Penrose pseudoinverse will be employed. In Section 3, an application to the inner- outer 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 ) strictly proper

transfer function matrix , there exists a polynomial matrix , which satisﬁes the following equation: lim ) = fullcolumnrank (1) Such an is called a right interactor matrix of Let A, B, C denote a minimal realization of . Setting the coefﬁcient of by ) = sL ··· sS (2) Although the deﬁnition in [2] is restricted the structure of (lower triangular), we do not consider such a restriction since it is not essential in this paper. Recent Researches in Automatic Control and Electronics ISBN: 978-1-61804-080-0 37

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then the the following equation holds as in [10]:

(3) where 1) ∈< = 1 , ..., w CBCAB ··· CA CB ··· CA 0 0 ··· CB ) = sI ··· The integer will be determined later. In [11], the special solution of eqn.(3) is given by using the pseudoinverse of , where and . However, it can not be assumed the special structure of since . At this time, considering the structure of , set (: 1 : K, (4) where (: 1 : denote the submatrix consti- tuted of the ﬁrst -th columns of . Substituting the above equation to eqn.(3), (: 1 : (5) Deﬁne by CBCAB ··· CA (: 1 : (6) the ﬁrst -th rows of eqn.(5) can be written by K. (7) That is, if eqn.(3)

is solvable, its special solution is given by eqn.(4) and must satisfy eqn.(7). Eqn.(7) means that is the set of eigenvectors of which correspond to the eigenvalues at = 1 Since the deﬁnition of is a real symmetric matrix, and thus is also a real symmet- ric. Therefore, the geometric multiplicity of the eigen- value one in equals to the algebraic multiplicity and thus can be found. Therefore, is the least integer when has the eigenvalue at = 1 with multiplicity is constituted of corresponding eigenvectors. Note that from the singular value decomposition of and , every eigenvalue of is

one or zero . In the case where the eigenvalues of is determined, it should be used the eigenvalues and corresponding eigenvectors of Lemma 1 For the interactor derived in the above, the following equation holds: ) = (8) where ) = (9) ··· (Proof.) Since is the pseudoinverse of , the following relation must hold = ( (10) Partitioning by (: 1 : ∈< pw 1) (11) and substituting the above into eqn.(10), we have (: 1 : CBCAB ··· CA 1) (12) Left multiplying to the both side of the above equation, we have (: 1 : CBCAB ··· CA 1) (: 1 : )) (13) From the deﬁnition of an interactor matrix, it

follows that ··· ··· 0 0 ··· = 0 (14) Thus from eqn.(13), we have ··· ··· 0 0 ··· Recent Researches in Automatic Control and Electronics ISBN: 978-1-61804-080-0 38

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(: 1 : CBCAB ··· CA 1) i ··· ··· 0 0 ··· (: 1 : 1) 1) (15) Since the above equation can be rewritten by ··· = 0 ··· = 0 the following equation also holds: 0 0 ··· ··· ··· ··· = 0 (16) Therefore, using eqns.(15) and (16), can be described by ··· ··· ··· ··· 0 0 ··· ··· 1) 1) (17) If eqn.(8) holds, is having all-pass property in the discrete-time. Therefore, all zeros of the inter- actor 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 := ABA ··· Let ( A, B, denote a minimal representation of HK C, B, K and denote an observer gain which makes the eigenvalues of := be desired ones. Then, there exists a matrix such that W,w (18) W,w := ··· ··· 0 0 ··· Moreover, deﬁning ) := (19) the zeros of det are equivalent to the eigenvalues of (Proof.) Since is the maximum uncontrollable observer gain for the system A, B, K , there are no ﬁnite zeros in B, . Therefore, the

inverse system of B, can be presented by polyno- mial matrix, say . That is, the zeros of det are equivalent to the eigenvalues of Since a right interactor for A, B,C is also the one of B, K [12], can be represented the polynomial matrix with -th degrees as shown in eqn.(19). Thus, the following relation holds: Recent Researches in Automatic Control and Electronics ISBN: 978-1-61804-080-0 39

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Expanding the transfer function matrix using Markov parameters, it can be obtained the following equation: ··· 0 0 ··· 0 0 0 ··· (20) ··· = 0 Since C, is observable, ··· = 0 and combining

the above equations, it can be obtained eqn.(19). Example 1 Consider the following transfer func- tion matrix: ) = + 1 + 2 + 3 + 4 + 5 + 6 Then, a state space realization A, B, C is given by 2 0 0 0 0 3 0 0 0 0 0 0 0 12 0 0 0 0 1 7 0 0 0 0 0 0 0 30 0 0 0 0 1 11 , B 2 1 1 1 4 3 1 1 6 5 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 In this example, = 3 is the ﬁrst integer when 9 0 2 0 6 0 1 0 2 0 has the multiple eigenvalue at = 1 . Then is given by 1 0 0 1 1 2 Thus, the right interactor can be calculated by ) = sI sS 1 0 6 0 5 0 where det ) = 0 Next, design another interactor which has the

desired zeros at . For the realization, let denote an observer gain which makes the closed-loop’s poles be and . Setting := is given by ) = 0 0 0 0 0 sI 8192 4024 0288 2 4801 0357 7731 6024 4 2518 0667 9787 5667 2 4787 5000 5000 5000 0 5000 3 Application to Inner-Outer Factor- ization For a given ) proper and stable trans- fer function matrix which has no invariant zeros on the imaginary axis, it is known the inner-outer fac- torization using state space representation if eqn.(1) holds with ) = . In this section, it will be proposed the method applicable for the case where , which can be

summarized by the follow- ing procedures: step 1 Calculate a right interactor for a given step 2 Calculate a spectral factorization for , i.e., calculate a stable rational Recent Researches in Automatic Control and Electronics ISBN: 978-1-61804-080-0 40

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function matrix satisfying ) = (21) is stable, ) := step 3 Set inner matrix and outer matrix as follows: ) := ) := (22) It is clear that ) = and since ) = (23) hold, 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 where is strictly proper, same idea can be used for the proper case. Let A, B, C denote a minimal state space re- alization of . Then, the state space representation of is given by A, B, C, K , where := BAB ··· := BL , K (24) if the coefﬁcient matrix is given by eqn.(2). Con- sider the following Riccati equation: XA (25) XB ) = 0 where := K. (26) Let and = 0 denote an eigenvalue and corre- sponding eigenvector of . Then, = 0 , since A, B is controllable. On the other hand, λL ··· BL Since all zeros of det lie in the left half

plane, det ) = 0 only if Re . Thus, the pair A, is stabilizable and there exists a positive semi deﬁnite solution for Riccati equation (25) . Moreover, set- ting := (27) it is known that makes BF be stable. Deﬁne and by ) := ( sI B, N ) := FM (28) Then from Riccati equation, the following relation holds: C C CK #" (29) Now, deﬁne by ) = (30) ) = BF , A := BF. Then from eqn.(22), we can obtain ) = BF KF (31) As shown in [14], is also a right interactor for sI if det is Hurwitz. So, is the observer gain of the inverted interactorizing for A, B, F (see [13] for detail), i.e.,

) = (32) Substituting the above equation and eqn.(30) to eqn.(22), we can obtain ) = (33) Example 2 Same plant is considered as shown in Example 1. Using the solution of Riccati equation (25), the outer matrix is given by ) = 2 0 0 0 0 2 1 3 0 0 0 0 1 1 0 0 0 12 0 0 4 3 0 0 1 7 0 0 1 1 0 0 0 0 0 30 6 5 0 0 0 0 1 11 1 1 012 942 024 116 012 348 0 0 009 209 018 582 009 789 0 0 Recent Researches in Automatic Control and Electronics ISBN: 978-1-61804-080-0 41

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whereas the inner matrix is given by ) = 843 968 0 0 0 0 968 843 0 0 0 0 0 0 800 0 0 0 0 0 0 200 0 0 0 0 0 0 400 0 0 0 0 0 0

600 038 005 000 000 000 000 141 004 000 000 000 000 114 003 000 000 000 000 11 711 491 396 263 640 301 193 46 845 936 12 227 462 27 638 881 237 237 527 408 817 Note that all zeros of det are included as poles of . However, zeros of the interactor are canceled and yields to ) = 84309 96740 01332 5 36645 96740 84309 64847 93587 00570 05150 88165 0 23670 00236 19346 23670 52660 00686 15569 40825 81650 i.e., 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 sys- tems can be found in [14]. References: [1] B. A. Francis, “A course in control theory”, Lecture Notes in Control and Information Science vol. 88 , Springer-Verlag, NY, 1987. [2] W. A.

Wolovich and P. L. Falb, “Invariants and canon- ical form under dynamic compensations”, SIAM Journal of Control and Optimization , vol. 14 , pp.996- 1008, 1976. [3] X. Xin and T. Mita, “Inner-outer factorization for non-square proper functions with inﬁnite and ﬁ- nite j -axis zeros”, International Journal of Control vol. 71 , pp.145-161, 1998. [4] A. Varga, “Computations of inner-outer factorizations of rational matrices”, IEEE Transactions on Auto- matic Control , vol. 43 , pp.684-688, 1998. [5] C. Oara and A. Varga, “Computations of general inner-outer and spectral

factorizations”, IEEE Trans- actions Automatic Control , vol. 45 , pp.2307-2325, 2000. [6] R. W. Rogozinski, A. P. Paplinski and M. J. Gibbard, “An algorithm for the calculation of a nilpotent inter- actor matrix for linear multivariable systems”, IEEE Transactions on Automatic Control , vol. 32 , pp.234- 237, 1987. [7] S. Bittani, P. Colaneri and M. F. Mongiovi, “Singular ﬁltering via spectral interactor matrix”, IEEE Trans- actions on Automatic Control , vol. 40 , pp.1492-1497, 1995. [8] B. Huang, S. L. Shah and H. Fujii, “The unitary in- teractor matrix and its estimation using

closed-loop data”, Journal of Process Control , vol.—bf 7, pp.195- 207, 1997. [9] T. Mita, T. K. Nam and X. Xin, “Sliding mode con- trol for invertible systems based on a direct design of interactors”, Asian Journal of Control , pp.242-250, 2002. [10] Y. Mutoh and R. Ortega, “Interactor structure estima- tion for adaptive control of discrete-time multivari- able nondecouplable systems”, Automatica , vol. 29 pp.635-647, 1993. [11] W. Kase and Y. Mutoh, “A simple derivation of inter- actor matrix and its applications”, International Jour- nal of Systems Science , vol. 40 , pp.1197-1205, 2009.

[12] Y. Mutoh, “A note on the invariant properties of in- teractors”, International Journal of Control , vol. 62 pp.1247-1252, 1995. [13] Y. Mutoh and P. N. Nikiforuk, “Inversed interactoriz- ing and triangularizing with an arbitrary pole assign- ment using the state feedback”, IEEE Transactions on Automatic Control , vol. 37 , pp.630-633, 1992. [14] W. 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 Automa- tion , pp.1516-1521, 2011. Recent Researches

in Automatic Control and Electronics ISBN: 978-1-61804-080-0 42

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