Strong stabilization via state and output feedback

The concept of “strong stability” of LTI systems has been introduced in a recent paper [KHP2]. This is a stronger notion of stability compared to alternative definitions (e.g. stability in the sense of Lyapunov, asymptotic stability), which allows the analysis and design of control systems with no overshooting response in the phase-space for arbitrary initial conditions. Here, after a brief review of “strong stability” and its relation with other stability notions, we introduce the problem of “strong stabilisation”. It is shown that strong stabilisation under dynamic and static output feedback are, in a certain sense, equivalent problems. Thus, we turn our attention to static strong stabilization problems under state-feedback, output injection and output feedback. After developing a number of preliminary results, we give a geometric interpretation of the problem in terms of the intersection of an affine hyperplane and the interior of an open convex cone. A solution to the problem is finally obtained via Linear Matrix Inequalities, along with the complete parametrisation of the optimal solution set of strongly stabilising matrices. The paper concludes by establishing simple (sufficient) strong stabilisation conditions using a linear programming (LP) methodology and by extending our results to robust strong stabilisation problems under two separate model uncertainty descriptions.