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On Controllability and Normalizability for Linear Descriptor Systems

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Abstract

Complete and strong controllability properties for linear descriptor systems are studied. We show that complete and strong controllability for a descriptor system is equivalent to the controllability for some suitably designed normal systems under natural assumptions. The normal systems are obtained by a numerically stable algorithm which is solely based on singular value decomposition of system matrices. The developed technique has also been exploited in designing some appropriate feedbacks such that the closed loop system becomes normal and asymptotically stable in square case and nonsingular in rectangular case. The presented theory is applied to some real-life problems for the illustration.

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Acknowledgments

The research of the second author was supported by the Council for Scientific and Industrial Research (CSIR), New Delhi, under Grant Number 25(0195)11/EMR-II. The third author of the paper gratefully acknowledges the support provided by the University Grants Commission, New Delhi, in the form of Senior Research Fellowship.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Patna, Patna, India

    Vikas Kumar Mishra, Nutan Kumar Tomar & Mahendra Kumar Gupta

Authors
  1. Vikas Kumar Mishra
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  2. Nutan Kumar Tomar
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  3. Mahendra Kumar Gupta
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Corresponding author

Correspondence to Nutan Kumar Tomar.

Appendix

Appendix

Algorithm to find the matrix S used in Theorem 1

In the following steps, SVD means singular value decomposition.

  1. 1.

    \(B=U_1\begin{bmatrix} D_1 \\ 0_{(m-r) \times r}\\ \end{bmatrix} V_1^T\) (SVD of B).

  2. 2.

    \(Q=\begin{bmatrix} V_1D_1^{-1}&0_{r \times (m-r)} \\ 0_{(m-r) \times r}&I_{m-r} \\ \end{bmatrix}U^T_1\).

  3. 3.

    \(E_2=\begin{bmatrix} 0_{(m-r) \times (n+r-m)}&I_{m-r} \\ \end{bmatrix}\)QE.

  4. 4.

    \(E_2=U_2\begin{bmatrix} D_2&0_{(m-r) \times (n+r-m)}\\ \end{bmatrix}V_2^T\) (SVD of \(\tilde{E}\)).

  5. 5.

    \(S_0=V_2\begin{bmatrix} 0_{(m-r) \times (n+r-m)}&D_2^{-1}U_2^T \\ I_{(n+r-m)}&0_{(n+r-m) \times (m-r)} \\ \end{bmatrix}\).

  6. 6.

    \(S=\begin{bmatrix} 0_{n \times (m-n)}&S_0 \\ \end{bmatrix}\)Q.

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Mishra, V.K., Tomar, N.K. & Gupta, M.K. On Controllability and Normalizability for Linear Descriptor Systems. J Control Autom Electr Syst 27, 19–28 (2016). https://doi.org/10.1007/s40313-015-0218-y

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  • DOI: https://doi.org/10.1007/s40313-015-0218-y

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