A note on computing the minimum distance between Lyapunov functions

Many system stability and robustness problems can be reduced to the solution of particular structures of Lyapunov equations. There exists a positive definite quadratic Lyapunov that establishes absolute stability if and only if a particular set of matrices is simultaneously Lyapunov stable. Design methods which rely on controller scheduling can be characterized in terms of the simultaneous Lyapunov stability problem. Such an approach foresees the design of several controllers for a fixed number of operating points of the plant, and then a switch from one controller to another during the maneuver. By supposing that between two operating points the controlled system can be described by means of the convex combination of the systems in the given points, it is possible to search for an upper bound on the maneuver velocity which guarantees stability. Since this upper bound depends on the distance between the Lyapunov functions in the operating points, an algorithm is presented that allows minimizing such a distance in a finite number of steps