This paper gives a concise parametrization of all identity interactors of a discrete-time multivariable square system. This is performed by means of a state-space description computed from a given particular interactor of the system. The paper then proposes a selection of the parameter which leads to an all-pass closed-loop transfer matrix. This closed-loop system turns out to be equivalent to a certain LQ (linear quadratic) optimal feedback system. A numerical example is given to illustrate the results</p

Inoue, Akira

Liu, Yi

Sugimoto, Kenji

Okayama University Scientific Achievement Repository

EngineeringIndustrial & Management Engineering fieldsOkayama University Year 1995Parametrization of identity interactorsand the discrete-time all-pass propertyKenji Sugimoto Yi LiuOkayama University Okayama UniversityAkira InoueOkayama UniversityThis paper is posted at eScholarship@OUDIR : Okayama University Digital InformationRepository.http://escholarship.lib.okayama-u.ac.jp/industrial engineering/102FP18 - 350 Parametrization of Identity Interactors and the Discrete-Time All-Pass Property Kenji Sugimoto, Yi Liu, and Akira Inoue Department of Information Technology Faculty of Engineering, Okayama University Tsushima-nab 3- 1-1, Okayama, 700 JAPAN Fax: +8 1-86-255-91 36 sugimoto0suri.it.okayama-u.ac.jp Abstract This paper gives a concise parametrization of all iden- tity interactors of a discrete-time multivariable square system. This is performed by means of a state-space description computed from a given particular inter- actor of the system. The paper then proposes a se- lect8ion of the parameter which leads to an all-pass closed-loop transfer matrix. This closed-loop system turns out to be equivalent to a certain LQ (linear quadratic) optimal feedback system. A numerical ex- ample is given to illustrate the results. 1. Introduction A ripecial polynomial matrix called an interactor plays an essential role in various fields of multivari- able control; see [2], (61, [9], just to name a few. This matrix was originally defined by [9] as follows: Let T ( z )  be a given m x m transfer matrix. A polyno- mial matrix is called an interactor iff for some constant nonsingular matrix MT, where and 1 w4 = ( hij[zl '.. 1) (3) with h;j[z] being appropriate polynomials of certain degrees. The interactor is introduced in order to generalize the (concept of the relative degree of scalar systems into the multivariable case. In this sense, however, it is enough to pose Condition (1); the lower triangu- lar structure in (2) and (3) is required only because of algorithmic simplicity as well as guaranteeing unique- ness. Various modifications of this concept have thus been attempted in literature [3], [4], [5].  A natural definition in this context has been given by [3]; namely, a polynomial matrix L T [ ~ ]  is called an identity interactor iff lim L ~ [ z l T ( z )  =Im 2-00 (4) with I,,, denoting the identity matrix. This appears to be more useful than the original definition (1) - (3) in the following sense. In the design of EMM (Ex- act Model Matching) or MRACS (Model Reference Adaptive Control Systems), the feedback system is often designed so that the closed-loop transfer matrix coincides with L~[z] - l ,  see for example [2]. Hence it is less advantageous to require additional conditions as in (2) and (3), which may reduce the degree of freedom in the design. An important feature of the identity interactor is non- uniqueness. In this sense, it should be more desir- able to  parametrize the set of all possible LT[L] which are identity interactors for a given T ( t ) ,  rather than studying a specific one of them. It is also desirable if we can give some guidelines aa to which identity in- teractor should be selected when designing a control system. The objective of this paper is as follows. For a discrete-time invertible (not necessarily known) plant T(z) ,  suppose that we are given a polynomial matrix &[z] and a constant matrix MT in (l), (2), and (3). In what follows, &r[z ]  is called a particular interactor, in order to distinguish it from the identity interactor. Then, we i) parametrize the set of all identity interactors LT[.z] 4403 for fixed T(z) ;  ii) select, in this set, a parameter such that N ( z )  := LT[Z]-' satisfies NT(z- ' )N(r)  = 0 vz E Q: (5)  for some constant a; and iii) show that the feedback achieving N ( z )  as a closed- loop transfer matrix is LQ optimal for some weight- ings. The item i) enables us to design &[z] by adjusting this parameter: for example, we can allocate closed- loop poles at specified points other than the origin. In this paper, however, we use this degree of freedom for achieving the item ii). This condition corresponds to what is called the all-pass property in the continuous- time case, and guarantees that llN(z)11 is constant at all frequencies. In the design of EMM, we usually apply a feedback so as to  cancel invariant zeros inside the unit disk, and put a pre-compensator C ( z )  so that N ( z ) C ( z )  has a desirable frequency property (see Fig. 1). In such a case, we may want N ( z )  to  have a Bode gain-plot as flat as possible, although a phase lag is inevitable because of time delay. Figure 1: A block diagram of EMM In fact, for diagonalizable systems (i.e., the case where HT[z]  = Im), it is natural to adopt a diago- nal interactor with zf1, . . ., z fm in diagonal entries. In this case, Condition (5) is automatically satisfied. In view of this fact, the item ii) gives a natural gen- eralization of the diagonal interactor into the non- diagonalizable case. All-pass interactor is also found in [4 ] ,  by the name of unitary interactor, in the con- text of explicit formula of the LQ optimal control. The item iii) is given in order to clarify the control- theoretic meaning of the obtained interactor. Throughout the paper we use the bracket [ .]  as in A[%] for polynomial matrices, and the parenthesis (.) as in A( z )  for rational function matrices, in order to  avoid confusion of these two different categories. 2. Parametrization of Interactors Consider the discrete-time system Z E R " ,  uEIRm, y E I R m .  Throughout the paper, we assume that the (square) transfer matrix " ( 2 )  = C(zln - A)-'B is nonsingu- lar on the field of all rational functions, and that this system has no unstable invariant zeros. We will not use the explicit values of ( A , B , C )  but will use only the knowledge of a corresponding particular interac- tor <T[z]  together with M T ,  so that the result may be applied to adaptive control design in future work. Lemma 1 Consider E T [ z ]  and MT in (l), (2), and (3) for a system (6), (7). Then, any polynomial matrix LT[L]  is an identity interactor iff 2'00 lim ~ T [ r ] t T [ z ] - '  = MT-'. (8) Proof (Necessity) If LT[z]  is an identity interactor, then we have lim ~ T [ z ] t T [ z ] - '  Z'OO = (lim L T [ z ] T ( z ) ) (  lim ET[Z]T(Z)) - '  Z'CO Z'OO = MT-' by definition. Hence (8) holds. Sufficiency can also be shown similarly. 0 Lemma 2 Consider &[z] and MT in (l), (2), and (3). Since &[z]-' is strictly proper, there exists a minimal realization, denoted by (A, B, e) : &.[z]-l = C(Z1, - A)-1B, s[t] := (Zld - A)-'Bt&] (9) where d is the McMillan degree of &[z]- l .  Then (10) is a polynomial matrix. Furthermore, any identity interactor &[z]  is parametrized as LT[4  = MT-I(<T[Z]  + RS[Z]) (11) in terms of a constant parameter matrix R 4404 Proof First, consider the coprime factorization (dd -A)-lB = ~ ? [ ~ ] e [ ~ ] - ~  by polynomial matrices. In view of (9), it follows that E[%] = (T[%]U[%], er[%] = U[%] for some unimodular polynomial matrix U[%]. Hence S [ 4  = r [ z ] ~ [ z ] - l ,  which is a polynomial matrix. The latter half is a direct consequence of the condi- tBon (8 )  and an elementary property of polynomial matrices [8]. 13 R.emark 1 Lemmas 1 and 2 are nothing but a gen- eralization of the fact that "For a scalar system of relative degree d, any polynomial of degree d can be tamken as an interactor." Remark 2 S[z]  is the numerator polynomial ma- trix that is known in the polynomial matrix approach [8]l. This matrix, however, does not necessarily have the well-known arranged structure with monomials 1, z, z2, - . in block-diagonal entries. This is because ST[%] is not column proper in general nor is (A, B, C) in the reachability canonical form. Remark 3 In practice, it is necessary to choose a Hurwitz polynomial matrix L*[z] (i.e., det L*[t] # 0 if Izl 2 1) among those satisfying ( l l ) ,  so that the closed-loop system LT[%]-* is stable. Taking this into co:nsideration, we may say that "the identity interac- toirs are parametrized by R such that La"[%] is Hur- witz." 3. Selection of An Interactor In this section we show how to select an interac- tor LT[%] such that the closed-loop transfer matrix N ( z )  = LT[%]-' has the all-pass property (5). This is done by making full use of the degree of freedom of the identity interactors revealed in Section 2. Theorem 1 Consider ( ~ [ z ]  and MT in (I), (2), and (3), and the realization (A,B,C) as in Lemma 2. Co:mpute a positive semi-definite solution P to the discrete-time Riccati equation P iiii A T P A - A T ~ B ( B T P B ) - l B T P A ~ ~ T ~ ,  (12) ancl obtain the LQ optimal gain Finally, take an identity interactor LT[z] := MT-~(&[z] + ROS[z]). (14) Then, the closed-loop transfer matrix N ( z )  := LT[%]-' satisfies ( 5 )  for o := M T ~ B ~ P B M T .  Proof It is well known (e.g., [l]) that if zo is given by (12) and (13), then the Kalman equation WT(z-')BTPBW(z) = gT(z-')cTc'Z(z) (15) holds, where W(%> :=I, + EoZ(t), Z( r )  := (%Id - A)-- lB.  Also, note that in this case CZ(%) = ST[%]-' from (9) and the definition of g(z). Kalman equation (15) is reduced to Hence the WT(r-')BTPBW(2) = ( & - [ Z - ' ] - ' ) ~ & [ Z ] - ~ .  (16) On the other hand, by substituting (10) into (14) we have MTLT[%](T[%]-' = 1, + Eo(%Id - A)-'B = w(Z). Hence by pre- and post-multiplying (16) respectively by [ ~ ~ [ z - ' ]  and &[z],  we have t*T[%-l]M*TBTPBMTLT[r] = I, Thus LT[%] satisfies the condition. 0 We briefly mention a method for numerically com- puting LT[z] given above. First, note that HT[z] is unimodular, and hence, H*[z]-' is a polynomial ma- trix. Then we see that &[z]-' = diag(zf', a ,  zf")-'H~[z]-' gives a left coprime factorization by polynomial ma- trices. By using this factorization, a realization for &-[z]-l can easily be derived in the observable canoni- cal form. For example, when the particular interactor is diagonal (i.e., HT[%] = Im), then the tripplet A = block-diag 4405 is such a realization. Here e p  is the unit m- dimensional vector with unity only in the p-th en- try, and the sizes of the diagonal blocks in A are fi,. , fm, respectively. If, on the other hand, the particular interactor is non-diagonal, then it suffices to  construct B by putting the coefficients of H T [ I ' ] - ~  in suitable positions. It is fairly easy to solve the Riccati equation in this case, as will be shown in a later example. We should note here that if a particular interactor is diagonal, then the equation has a trivial solution P = I d ,  xo = 0, and hence ~ T [ z ]  = diag(zfl , . . . ,zfm) is the very LT[z] that we want. In other words, for the diagonalizabIe case, the particular interactor itself gives the all-pass property ( 5 )  (see Section 1). 4. LQ optimality Now we should note that K O  introduced in Section 3 is a mere imaginary gain for (A,l?,c) and not an actual feedback law. The feedback corresponding to the resulting L T [ ~ ] ,  however, turns out to be LQ optimal in reality. This section is devoted to showing this. Theorem 2 Let the system ( 6 )  be reachable. Apply a stabilizing feedback U = -Kx to this system, and define N ( z )  := C(zI,, - A + BK)-'B. Then, if N ( z )  satisfies N T ( z - I ) N ( z )  = @ v z  E Q: (17) for some positive definite constant matrix a, then I< is expressed as K = ( B ~ P B ) - ~ B ~ P A ,  (18) where P is a positive semi-definite solution of the Ric- cati equation P = P A  - P B ( B ~  PB) -l B~ P A  + cT C. (19) Proof This can be regarded as a special case of the inverse problem of optimal control. We can not apply, however, an existing solution technique given in [7], since the control weighting R was assumed to  be positive definite there, while R = 0 in the present problem. Instead, we will show it in the following way. Define A K : = A - B K .  Then AK is stable, and hence the Lyapunov equation P = A K T P A ~  + Q (20) admits a positive definite solution P, where Q := CTC. Using this P ,  we will show (18). To this end, we expand N ( z )  = CBr-l+ C A K B Z - ~  + C A K ~ B Z - ~  + ..- Then we substitute it into (17), and compare the both hand sides with respect to z: constant term B ~ Q B  + B ~ A ~ ~ Q A ~ B  + B ~ A ~ ~ ~ Q A ~ ~ B  + -.. = cp, t - l  term B ~ Q A ~ B  + T A ~ ~ Q A ~ ~ B  + B T A ~ 2 T Q A ~ 3 B  + a . *  = 0, z-2 term B T & A ~ ' B  +B T A ~ T & A ~ 3 B  + B T A ~ 2 T Q A ~ 4 B  + * e *  = 0, ... ... (21) where AKZT = (AK')*. Now observe that, from (20), we have 00 P = C ( A ~ ~ ) ' Q A ~ ' .  i=O Hence we have B ~ P B  = cp (22) \ I  B ~ P A ~ [ B ,  AKB,  A K ~ B , . . . ]  = o from (21). By the first equality BTPB is positive def- inite. By the second one, together with reachability of the pair ( A K ,  B), 0 = BTPAK = BTP(A - B K )  = BTPA - BTPBK holds. Pre-multiplying the both hand-sides in this equality by (BTPB)- ' ,  we obtain (18). Finally, sub- stituting (18) into (20), we see that P is a solution to the Riccati equation (19). 0 Corollary In particular, if we construct EMM by means of the identity interactor given in Theorem 1, 4406 thein the closed-loop transfer matrix coincides with the one by an optimal regulator. Proof Since we have assumed that the plant T(z)  is nionsingular, so is N ( z )  and hence @ cannot be sin- gular. This means that all assumptions in Theorem 2 are satisfied. 0 5. An Example Consider the 2 x 2 transfer matrix of a plant and assume that the relative degrees of t l l  and t21  are iunity, while those of tlz and t 2 2  are two. Then this system does not have a diagonal interactor, but has ;a particular interactor of the form, say ( ~ [ z ]  = ( ) diag(z, z2)>. (23) 2 z + 5  1 Supose that, for simplicity, MT = I2 in this case. We now apply the algorithm given in Sections 2 and 3. f:, begin with, a minimal realization of < ~ [ z ] - ~  is givexi by 0 0 0  0 1 0  ji=( o o o ) ,  B=( -2 :5 0 Y ) ,  (24) Next we solve the discrete-time Riccati equation. This is rather easy to solve since A is a nilpotent matrix. Take Po = I d  and set pk+1 = ATPkk; - ATPkB(BT&B)-lBTfjkA + CTc k=O, l , . ’ . .  (26) recursively. Then the sequence converges to  P in fi- nite times of recursion (in this case, twice). Hence we have 1 0 0  0 0 1  P = =  (0 0.2 o ) ,  go= (; -Oa4 -2 0 ). (27) On the ohter hand, from (10) it follows that 1 0  S[4 = ( ; ; ) , and hence 0.21 -0.4% 2 r 2 + z  2- 2% L T [ . f ]  = is the desired interactor. Acknowledgments: The authors wish to thank Lorant, Co. Ltd, Osaka, Japan, for their finantial support. References (11 B. D. 0. Anderson and J. B. Moore, Linear Optimal Control, 2nd ed., Englewood Cliffs, NJ: Prentice-Hall, 1990. [2] H. Elliott and W. A. Wolovich, “A parame- ter adaptive control structure for linear multivariable systems,” IEEE Trans. Automat. Contr., vol. AC-27, no. 2, pp. 340-352, 1982. [3] H. Elliott and W. A. Wolovich, “Parametriza- tion issues in multivariable adaptive control,” Auto- matica, vol. 20, pp. 533-545, 1984. [4] Y. Peng and M. Kinnaert, “Explicit solution to the singular LQ regulation problem,” IEEE Trans. Automat. Contr., vol. AC-37, no. 5, pp. 633-636, 1992. [5] M. W. Rogozinski, et. al, “An algorithm for the calculation of a nilpotent interactor matrix for lin- ear multivariable systems,” IEEE Trans. Automat. Contr., vol. AC-32, no. 3, pp. 234-237, 1987. [6] S .  L. Shar et. al, “Multivariable adaptive con- trol without a prior knowledge of the delay matrix,” Systems & Control Let., vol. 9, pp. 295-306, 1987. [7] I<. Sugimoto and Y. Yamamoto, “Solution to  the inverse regulator problem for discrete-time sys- tems,” Int. J. of Contr., vol. 48, no. 3, pp. 1285-1300, 1988. [8] W. A. Wolovich, Linear Multivariable Systems, Springer-Verlag, 1974. [9] W. A. Wolovich and P. L. Falb, “Invariants and canonical forms under dynamic compensation,” SIAM J. Contr., & Opt., vol. 14, no. 6, pp. 996-1008, 1976. 4407 

Parametrization of identity interactors and the discrete-time all-pass property

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