Non-overshooting stabilisation via state and output feedback

Abstract

The concept of ‘strong stability’ of linear time-invariant (LTI) systems has been introduced in a recent paper (Karcanias, N., Halikias, G., and Papageorgiou, A. (2010), ‘Strong Stability of Internal System Descriptions’, International Journal of Control, 83, 182–205). 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 non-overshooting response in the state-space for arbitrary initial conditions. This article reviews the notion of ‘strong stability’ (Karcanias et al. 2010) and introduces the problem of non-overshooting stabilisation. It is shown that non-overshooting stabilisation under dynamic and static output feedback are, in a certain sense, equivalent problems. Thus, we turn our attention to static non-overshooting stabilisation problems under state-feedback, output injection and output feedback. After developing a number of preliminary results, we give a geometric interpretation to 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 parameterisation of the optimal solution set.