On real structured controllability/stabilizability/stability radius: Complexity and unified rank-relaxation based methods

This paper is concerned with the real structured controllability, stabilizability, and stability radii (RSCR, RSSZR, and RSSR, respectively) of linear time invariant systems, i.e., the problems of determining the distance (in terms of matrix norms) between a system to its nearest uncontrollable, unstabilizable, and unstable systems with a prescribed affine structure. The main contributions of this paper lie in two aspects. First, assuming that the involved real perturbations have a general affine parameterization, we prove that determining the RSCR and RSSZR are NP-hard by showing that even checking their feasibilities are NP-hard. This means the hardness results are irrespective of what matrix norm is utilized. The NP-hardness of the 2-norm based RSSR can be obtained from an existing result in robust stability analysis. Second, we develop unified rank-relaxation based algorithms towards these problems, which share two notable features: they are valid both for the Frobenius norm and the 2-norm based problems, and they share the same framework for the RSCR, RSSZR, and RSSR problems. These algorithms exploit the low-rank structure of the original problems and relax the corresponding rank constraints with a regularized truncated nuclear norm term. This leads to problems of difference of convex programs (DC), which can be solved via the sequential convex relaxations with guaranteed convergence to stationary points of the relaxed problems. Moreover, under suitable conditions, a modified version of those algorithms is given to find local optima with performance specification on the corresponding perturbations. Finally, simulations suggest that, our proposed methods, though in a simple framework, may find local optima as good as several existing methods.