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Feedback realization of open loop diagonalizers. (English) Zbl 0811.93027
For a given full row rank transfer matrix (rational matrix function) \(Z(s)\) and a given \(p\times p\) strictly proper diagonal matrix \(\Lambda(s)\), the set of open loop \(\Lambda\)-diagonalizers, \(\text{OLD} (Z,\Lambda)\) is defined as \[ \text{OLD} (Z,\Lambda):= \{\text{proper } M(s)\mid Z(s) M(s)= \Lambda\}. \] Assuming the \(Z(s)= C(sI- A)^{-1} B\) is realized as the transfer function of a linear system \[ \begin{aligned} \dot x(t) &= Ax(t)+ Bu(t), \qquad x(0)=0,\\ y(t) &= Cx(t),\\ y(s) &= Z(z) u(s)\end{aligned} \] (where \(u(s)\) is the Laplace transform of \(u(t)\)), the author considers the problem of realizing elements of \(\text{OLD} (Z,\Lambda)\) via implementation of a feedback control law of one of the following types:
i) Dynamic state feedback: \(u(s)=- F(s) x(s)+ Gv(s)\), ii) Constant state feedback: \(u(s)=- Fx(s)+ Gv(s)\), iii) Dynamic output feedback: \(u(s)=- Z_ c(s) y(s)+ Gv(s)\), iv) Constant output feedback: \(u(s)=- Z_ c y)s_ + Gv(s)\). In the above, \(F(s)\) and \(Z_ c(s)\) are proper and \(F\) and \(Z_ c\) are constant compensators in the feedback path, while \(G\) is a full column rank constant precompensator. The first part of the paper discusses properties of open loop diagonalizers (old’s) which admit a feedback realization of one of the above types. Then dynamic (constant) output feedback decoupling problems are formulated as determining an open loop diagonalizer which admits the desired feedback realization. Finally, solutions to these problems are obtained by determining the conditions of existence for such open loop diagonalizers.

93B52 Feedback control
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