A gradient flow approach to simultaneous stabilization problem

In this paper, the problem of simultaneous stabilizing optimally a set of linear time invariants systems is considered. The objective is to find a single feedback gain matrix such that: 1) all systems involved are simultaneously stabilized; and 2) all closed-loop systems exhibit good transient behaviour. This objective is achieved by posing the simultaneous stabilization problem as an optimal control problem. The performance index used is a quadratic function of the system states and the control vector. In this paper, assuming that an initial solution to the simultaneous stabilization problem is known, an optimum solution in LQ sense is obtained using gradient flow methods