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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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