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

Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 235)

## 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.

### Keywords

Randomized Algorithms Computational Complexity

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