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)


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.


Randomized Algorithms Computational Complexity 


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.National Institute of TechnologyTiruchirappalliIndia

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