linear multivariable systems

Abstract

is analytic in C+ ; note, however, that it has a pole at every zero of x(-). Furthermore, N,, is analytic in C, because it is a polynomial in sihence, it is an entire function. Consequently, by (9), for all z E Z [ P ] with ZEC,, rank[ f i y , (z) ] < n o (12) because rank[N',(z)] < 4 ] c R [ N p r (~) l , thus emphasizing that zeros of transmissions have "position" (i n the transfer function matrix) as well as "value" (in the complex plane) [I 11. 4) Since the proof required only that transfer functions be factorizable and that the notion of characteristic polynomial (function) be defined in terms of these factors, it is obvious that this theorem applies to the dbrrihred case (see, e.g., [I31 and [I41 and use the methods of [3D. 5) A reviewer pointed out that the theorem above is a consequence of the general synthesis conditions [15, Theorem 6.21 of Pernebo. Our derivation is more direct and easily extended to the distributed case. REFERENCES and their dynamical interpretation," IEEE Tmnr. Circuits %st., vol. CAS-21, pp. C. A. Dsm and I. D. Schulman, "Zeros and poles of matrix pansfer fundions 3-8. Jan. 1974. C A Desoer and Y. T. Wan& "Linear timeinvariant robust servomechaaism problem: A seli-contained exposition," in Adumced Control and &mt Syrtemr, vol. 16, C. T. h n d c s , Ed. New Academic, to be published-F. M. Callier, V. H. L. Cheng, and C k Desoer, "Dynamic interpretation of poles and hansmission zeros for distributed multivariable systemr5" submitted to the 1980 European Confemce on Circuit Theory and Design, Warsaw, Poland. H. H. Roxnbrock, State-Space and Mdtieruirrble 7kory. New York: Wiley, 1970. A. G. J. MacFarlane and N. Karcantas systems: A survey of the algebraic, geometric and complex-variable theory," Inr. 3.. "Poles and zeros of linear multivariable "Roperlies and calculation of transmission mos of A. C. Pugh, Trammission and system zeros," Inr. 1. Contr. vol. 26, pp. 315-324, T. Kailath, Linear %stems. Englewood Cliffs, NJ: PRnti-Hall, 1980. P. G. Ferreira and S. P. Bhamcharyya, '. O n blocking zeros." IEEE nam. Auromal. C. A. Desm and M. J. Chen, "Desii of multivariable feedback systems with Conv., VOL AGZZ, pp. 258-259, Apr. 1977. stable plant," Univ. California, Berkeley, ERL Memo M80/13. 1980. E. H. Bristol, "A new prooss interaction concept: Pinned zeros." prcprinL R A. Poluiktov. "Constrains due to the plant in synthesis problems of multidimen-sional closed-loop s y s t e m s , " Automat. Remote Conrr. (Iransl. from A t d i F. M. callier and C A. Desoer, "An algebra of W e r functions for distniuted In this paper we decompose the feedback design of the n-dimensional system into two reduced4rder subsystems designs, In (1) the control u is an m-vector and Aii are (n i X n i) matrices; i=1,2; n , + n 2 = n. In most applications a desired effect of feedback is to move the eigenvalues farther left in the complex plane, which also increases their magnitudes. In the proposed two-stage design procedure the nI eigenvalues, which were larger in the open-loop system (l), also become the n1 larger eigenvalues of the designed closed-loop system. In other words, the requirement ~ A , ~ > ~ A i ~ , i = l , ~-,nl; j = n l + l ,. .-, n (2) is satisfied at each design stage. Similar two-stage designs can be based on any block-diagonalizing transformation. The main tool of the design proposed here is an explicitly invertible transformation. Its properties are reviewed first.Then the design procedure is presented. Finally, we show that earlier proposed decompositions such as [I]-[3], applicable to singularly perturbed systems, are special cases of this procedure when eigen-value separation (2) is sufficiently large. AN EXPLICITLY INVF.RTIBLE TRANSFORMATION It can easily be verified that the transformation T= [ L-" I I2 where Ii is the ni Xn, identity, i= 1,2; Mis n1 X n 2 ; and L i s n2 X n l has the following properties. 1) For any L and M the inverse of T is T-' = [ fl, I ,-L M