Non-smooth techniques for stabilizing linear systems

We discuss closed-loop stabilization of linear time-invariant dynamical systems, a problem which frequently arises in controller synthesis, either as a stand-alone task, or to initialize algorithms for Hinfin synthesis or related problems. Classical stabilization methods based on Lyapunov or Riccati equations appear to be inefficient for large systems. Recently, non-smooth optimization methods like gradient sampling [J.V. Burke et al., 2002] have been successfully used to minimize the spectral abscissa of the closed-loop state matrix (the largest real part of its eigenvalues). These methods have to address the non-smooth and even non-Lipschitz character of the spectral abscissa function. In this work, we develop an alternative non-smooth technique for solving similar problems, with the option to incorporate second-order elements to speed-up convergence to local minima. Using several case studies, the proposed technique is compared to more conventional approaches including direct search methods and techniques where minimizing the spectral abscissa is recast as a traditional smooth non-linear mathematical programming problem.