Robust Stabilization of Uncertain Systems Based on Energy Dissipation Concepts

Abstract Robust stability conditions obtained through generalization of the notion of energydissipation in physical systems are discussed in this report. Linear time-invariant(LTI) systems which dissipate energy corresponding to quadratic power functions arecharacterized in the time-domain and the frequency-domain, in terms of linear matrixinequalities (LMIs) and algebraic Riccati equations (AREs). A novel characterizationof strictly dissipative LTI systems is introduced in this report. Sufficient conditionsin terms of dissipativity and strict dissipativity are presented for (1) stability ofthe feedback interconnection of dissipative LTI systems, (2) stability of dissipativeLTI systems with memoryless feedback nonlinearities, and (3) quadratic stability ofuncertain linear systems. It is demonstrated that the framework of dissipative LTIsystems investigated in this report unifies and extends small gain, passivity and sectorconditions for stability. Techniques for selecting power functions for characterizationof uncertain plants and robust controller synthesis based on these stability results areintroduced. A spring-mass-damper example is used to illustrate the application ofthese methods for robust controller synthesis.Keywords: State space characterization of dissipative LTI systems, stability of inter-connected dissipative systems, Linear Matrix Inequalitiess (LMIs) for robust stability.

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