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On computing the distance to stability for matrices using linear dissipative Hamiltonian systems. (English) Zbl 1375.93110
Summary: In this paper, we consider the problem of computing the nearest stable matrix to an unstable one. We propose new algorithms to solve this problem based on a reformulation using linear dissipative Hamiltonian systems: we show that a matrix $$A$$ is stable if and only if it can be written as $$A=(J-R)Q$$, where $$J=-J^T$$, $$R \succeq 0$$ and $$Q\succ 0$$ (that is, $$R$$ is positive semidefinite and $$Q$$ is positive definite). This reformulation results in an equivalent optimization problem with a simple convex feasible set. We propose three strategies to solve the problem in variables $$(J,R,Q)$$: (i) a block coordinate descent method, (ii) a projected gradient descent method, and (iii) a fast gradient method inspired from smooth convex optimization. These methods require $$\mathcal{O}(n^3)$$ operations per iteration, where $$n$$ is the size of $$A$$. We show the effectiveness of the fast gradient method compared to the other approaches and to several state-of-the-art algorithms.

MSC:
 93D21 Adaptive or robust stabilization 93C15 Control/observation systems governed by ordinary differential equations 93C05 Linear systems in control theory 93B60 Eigenvalue problems
Software:
CVX; HIFOO; SDPT3; SeDuMi
Full Text:
References:
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