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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:
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