LPV system analysis via quadratic separator for uncertain implicit systems

Prof. Tetsuya Iwasaki, Department of Control Systems Engineering, Tokyo Institute of Technology

We consider the stability analysis problem for linear parameter varying (LPV) systems described by a nominal linear time-invariant system depending upon time-varying parameters in a linear fractional manner. We assume that the parameters vary within known intervals subject to known bounds on the rate of variations. In order to take the rate bounds into account, we construct an augmented system by taking the time-derivatives of the signals related to the parameters. This simple idea leads to an implicit model for the augmented LPV system. Thus analysis tools for implicit systems may be used to examine stability of LPV systems.

To this end, we first consider a class of linear implicit systems with unknown but constant parameters, and propose a method for assessing the stability using the quadratic separator as a basic tool for the analysis. The idea is to view the uncertain implicit system as an interconnection of the ``nominal'' part and the uncertain component, and to note that the system is stable iff the range spaces of the two components are topologically separated. We show that necessary and sufficient conditions for robust stability can be given in terms of the existence of frequency-dependent quadratic forms that separate the two spaces.

We then apply the result to give a sufficient condition for stability of LPV systems in terms of a constant quadratic separator. Moreover, we show that the stability condition, thus obtained through the quadratic separator for implicit systems, is equivalent to the existence of a Lyapunov function that depends on the parameters in a linear fractional manner. Two extreme cases --- the parameters with arbitrarily slow/fast variations --- are discussed in connection with existing results. Finally, the computational aspects of the proposed stability conditions are addressed in terms of linear matrix inequalities which can be solved efficiently via interior point methods.