NP-Hardness of Some Linear Control Design Problems

We show that some basic linear control design problems are NP-hard, implying that, unless P=NP, they cannot be solved by polynomial time algorithms. The problems that we consider include simultaneous stabilization by output feedback, stabilization by state or output feedback in the presence of bounds on the elements of the gain matrix, and decentralized control. These results are obtained by first showing that checking the existence of a stable matrix in an interval family of matrices is NP-hard.

[1]  W. Wonham.  Linear Multivariable Control: A Geometric Approach , 1974 .

[2]  B. Anderson,et al.  Output feedback stabilization and related problems-solution via decision methods , 1975 .

[3]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[4]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[5]  Mathukumalli Vidyasagar,et al.  Control System Synthesis , 1985 .

[6]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[7]  D. Bernstein.  Some open problems in matrix theory arising in linear systems and control , 1992 .

[8]  Svatopluk Poljak,et al.  Checking robust nonsingularity is NP-hard , 1993, Math. Control. Signals Syst..

[9]  A. Nemirovskii.  Several NP-hard problems arising in robust stability analysis , 1993 .

[10]  G. Coxson,et al.  Computational complexity of robust stability and regularity in families of linear systems , 1993 .

[11]  Vincent D. Blondel,et al.  Simultaneous Stabilization Of Linear Systems , 1993 .

[12]  Christopher L. DeMarco,et al.  The computational complexity of approximating the minimal perturbation scaling to achieve instability in an interval matrix , 1994, Math. Control. Signals Syst..

[13]  Vincent D. Blondel,et al.  Survey on the State of Systems and Control , 1995, Eur. J. Control.

[14]  Marie-Françoise Roy,et al.  On the combinatorial and algebraic complexity of quantifier elimination , 1996, JACM.