A new approach to the synthesis of parameter dependent Lyapunov functions for the estimation of the stability margin in the presence of slowly-varying parameters.

In this paper we consider a MIMO asymptotically stable linear plant. For such a system the classical concepts of {\em quadratic stability margin} and {\em multivariable gain margin} can be defined. These margins have the following interpretation: consider the closed loop system composed of the plant and some real parameters, one inserted in each channel of the loop; then any time-varying (time-invariant) parameters whose amplitudes are smaller than the quadratic stability (multivariable gain) margin result in a stable closed loop system. For time-varying parameters whose magnitudes are between these two stability measures, stability may depend on the {\em rate of variation} of the parameters. Therefore it makes sense to consider the stability margin given by the maximal allowable rate of variation of the parameters which guarantees stability of the closed loop system. As shown in previous papers, an estimate of this margin can be obtained with the aid of parameter dependent Lyapunov functions. In this paper we propose a new approach to the search of parameter dependent Lyapunov functions which leads to a {\em much less conservative} algorithm to estimate such margin at the price of a greater computational burden; this limits the application of the methodology to systems containing up to three parameters. In this context, an example will show that the proposed methodology introduces a strong improvement with respect to existing techniques.