A Computationally Faster Randomized Algorithm for NP-Hard Controller Design Problem

Abstract

In Finite Dimensional Linear Time-Invariant (FDLTI) control systems the following problem is important from a computational perspective. Given $A \in \Re^{n\times n}$, $B \in \Re^{n\times m}$ such that rank of the composite matrix $\left[B\,:\,AB\,:\,\cdots\,:\, A^{n-1}B\right]\,\in\,\Re^{n\times mn}$ is full, and a n ^th order polynomial χ with constant coefficients, compute a matrix $K\,=\,\left[k_{ij}\right] \in \Re ^{m\times n}$ such that characteristic polynomial of A + BK  =  χ. It is proven that when m  >  1 and when the elements of the matrix K are constrained such that $\underline{k}_{ij}\, \leq\, k_{ij} \,\leq \,\overline{k}_{ij}$, the problem belongs to the class NP-hard. In this paper, we provide a computationally efficient polynomial time algorithm to this problem using randomization. We show that the number of matrices K satisfying the given specification follows an interesting distribution w.r.t the matrix norm ∥ K ∥. We give several examples wherein the algorithm outputs the desired K matrices in polynomial time.