Computing Closest Stable Nonnegative Matrix

Abstract

The problem of finding the closest stable matrix for a dynamical system has many applications. It is studied for both continuous and discrete-time systems and the corresponding optimization problems are formulated for various matrix norms. As a rule, nonconvexity of these formulations does not allow finding their global solutions. In this paper, we analyze positive discrete-time systems. They also suffer from nonconvexity of the stability region, and the problem in the Frobenius norm or in the Euclidean norm remains hard for them. However, it turns out that for certain polyhedral norms, the situation is much better. We show that for the distances measured in the max-norm, we can find an exact solution of the corresponding nonconvex projection problems in polynomial time. For the distance measured in the operator $\ell_{\infty}$-norm or $\ell_{1}$-norm, the exact solution is also efficiently found. To this end, we develop a modification of the recently introduced spectral simplex method. On the other hand, for all these three norms, we obtain exact descriptions of the region of stability around a given stable matrix. In the case of the max-norm, this can be seen as an extension onto the class of nonnegative matrices, the Kharitonov theorem, providing a stability criterion for polynomials with interval coefficients [V. L Kharitonov, Differ. Uravn., 14 (1978), pp. 2086--2088; K. Panneerselvam and R. Ayyagari, Internat. J. Control Sci. Engrg., 3 (2013), pp. 81--85]. For practical implementation of our technique, we developed a new method for approximating the maximal eigenvalue of a nonnegative matrix. It combines the local quadratic rate of convergence with polynomial-time global performance guarantees.