Use of subsystems eigenstructures in dynamically interconnected systems for eigenvalue assignment
An eigenvalue assignment problem is considered for a class of interconnected, discrete, linear dynamic systems. The problem reduces to one finding matrices R and N such that the matrix \[ \Gamma = \left[ \begin{matrix} A+BR & BN \\ LC & M \end{matrix} \right] \] has a given set of eigenvalues. A, B and C are block diagonal matrices describing the S subsystem: \(x_ r(k+1)=A_ rx_ r(k)+B_ ru(k)\), \(_ r(k)=C_ r(k),\quad r=1,...S,\) while N and M are block diagonal matrices describing the interconnections: \[ Z_ r(k+1)=M_ rZ_ r(k)+\sum^{s}_{q=1}L_{rq}y_ q(k),\quad U_ r(k)=N_ rZ_ r(k)+\sum^{s}_{q=1}R_{rq}x_ q(k) \] The main result: if, for \(r=1,...S\), (i) \((A_ r,B_ r)\) is controllable, (ii) \((L_ r,M_{rr})\) is controllable, (iii) the eigenvalues of A and of M are disjoint from those of the desired eigenvalues, then there exist matrices N and R which achieve the desired eigenvalues. The eigenstructures of the subsystem interconnections are exploited to achieve the desired eigenvalues, and an iterative scheme is proposed to solve the global problem.
interconnected, discrete, linear dynamic systems
scientific article; zbMATH DE number 3852894
scientific article