Use of subsystems eigenstructures in dynamically interconnected systems for eigenvalue assignment (Q792279)

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.