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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ayyagari, R.: Control Engineering:A Comprehensive Foundation, 1st edn. Vikas Publication (2003)Google Scholar
  2. 2.
    Blondel, V.D., Tstiskilis, J.N.: Np-hardness of some linear control design problems. SIAM J. Control Optim. 35, 2118–2127 (1997)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brogan, W.L.: Modern Control Theory, 3rd edn. Prentice Hall (1991)Google Scholar
  4. 4.
    Callier, F.M., Desoer, C.A.: Linear System Theory, 1st edn. Springer, NY (1991)CrossRefMATHGoogle Scholar
  5. 5.
    Vidyasagar, M., Blondel, V.D.: Probabilistic solutions to some np-hard matrix problems. Automatica 37, 1397–1405 (2001)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.National Institute of TechnologyTiruchirappalliIndia

Personalised recommendations