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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.

93C05 Linear systems in control theory
93C35 Multivariable systems, multidimensional control systems
93B11 System structure simplification
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