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A classification of generalised state space reduction methods for linear multivariable systems. (English) Zbl 0859.93024
The aim of this paper is to establish two algorithms which reduce a linear multivariable system $$\Sigma$$, described by a polynomial matrix model of the form: $A(\rho)\beta(t)= B(\rho)u(t), \qquad y(t)= C(\rho)\beta(t)+ D(\rho)u(t), \tag{$$\Sigma$$}$ to an equivalent model in generalized state space form. Here $$\rho=d/dt$$, $$A(\rho)\in\mathbb{R} [\rho]^{r\times r}$$ with $$\text{rank}_\mathbb{R} A(\rho)=r$$, $$B(\rho)\in \mathbb{R}[\rho]^{r\times m}$$, $$C(\rho)\in \mathbb{R}[\rho]^{p\times r}$$, $$D(\rho)\in \mathbb{R}[\rho]^{p\times m}$$, $$\beta(t)$$ is the pseudostate of $$\Sigma$$, $$u(t)$$ the input vector and $$y(t)$$ the output vector.
More precisely, the authors solve the problem of determining a system: $E\dot x(t)= Ax(t)+ Bu(t), \qquad y(t)= Cx(t)+ Du(t) \tag{$${\Sigma_R}$$}$ equivalent to $$\Sigma$$. This equivalence means that they exhibit identical system properties. The first algorithm is based on the realization of $${\mathcal T}(s)$$ defined by: ${\mathcal T}(\rho)=\begin{pmatrix} A(\rho) &B(\rho) &0\\ -C(\rho) &D(\rho) &I_p\\ 0 &-I_m &0\end{pmatrix}\in \mathbb{R}[\rho]^{\bar r\times\bar r}, \qquad \bar r=r+p+m,$ while the second algorithm is based on a realization of $${\mathcal T}(s)^{-1}$$. In fact, all the known reduction algorithms can be classified by these two different theoretical reduction algorithms which are mentioned above.

##### MSC:
 93C05 Linear systems in control theory 93C35 Multivariable systems, multidimensional control systems 93B11 System structure simplification
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##### References:
  B. D. O. Anderson W. A. Coppel, D. J. Cullen: Strong system equivalence (I). J. Austral. Math. Soc. Ser. B 27 (1985), 194-222. · Zbl 0594.93015  O. H. Bosgra, A. J. J. Van Der Weiden: Realisations in generalised state-space form for polynomial system matrices, and the defìnition of poles, zeros and decoupling zeros at infinity. Internat. J. Control 33 (1981), 393-411. · Zbl 0464.93021  G. E. Hayton A. B. Walker, A. C. Pugh: Infinite frequency structure preserving transformations for general polynomial system matrices. Internat. J. Control 52 (1990), 1-14. · Zbl 0702.93021  N. P. Karampetakis, A. I. G. Vardulakis: Matrix fractions and full system equivalence. IMA J. Math. Control Inform. 9 (1992), 147-160. · Zbl 0777.93052  N. P. Karampetakis, A. I. G. Vardulakis: Generalized state-space system matrix equivalents of a Rosenbrock system matrix. IMA J. Math. Control Inform. 10 (1993), 323-344. · Zbl 0807.93009  T. Shaohua, J. Vandewalle: A singular system realisation for arbitrary matrix fraction descriptions. ISCAS’88, pp. 615-618.  A. I. G. Vardulakis: Linear Multivariable Control, Algebraic Analysis and Synthesis Methods. Nelson-Wiley, London 1991. · Zbl 0751.93002  A. I. G. Vardulakis: On the transformation of a polynomial matrix model of a linear multivariable system to generalised state space form. Proceedings of the 30th IEEE Conference on Decision and Control, Brighton 1991, U.K., pp. 11-13.  G. C. Verghese: Infinite Frequency Behavior in Generalized Dynamical Systems. P\?.D. Dissertation, Stanford Univ., Stanford, CA 1978.  W. A. Wolovich: Linear Multivariable Systems. Springer-Verlag, New York 1974. · Zbl 0291.93002
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