Gradient flow approach to LQ cost improvement for simultaneous stabilization problem

In this paper we consider LQ cost optimization for the simultaneous stabilization problem. The objective is to find a single simultaneously stabilizing feedback gain matrix such that all closed-loop systems exhibit good transient behaviour. The cost function used is a quadratic function of the system states and the control vector, This paper proposes to seek an optimization solution by solving an ordinary differential equation which is a gradient flow associated with the cost function, Two examples are presented to illustrate the effectiveness of the proposed procedure.