Passivity, Feedback Equivalence, and the Global Stabilization of Minimum Phase Nonlinear Systems

Passivity, Feedback Equivalence, and the Global Stabilization of Minimum Phase Nonlinear Systems

November 1991 | Christopher I. Byrnes, Fellow, IEEE, Alberto Isidori, Fellow, IEEE, and Jan C. Willems, Fellow, IEEE
This paper investigates the conditions under which a nonlinear system can be rendered passive via smooth state feedback and shows that this is possible if and only if the system has relative degree 1 and is weakly minimum phase. It also proves that weakly minimum phase nonlinear systems with relative degree 1 can be globally asymptotically stabilized by smooth state feedback under suitable controllability-like rank conditions. The paper extends and incorporates recent stabilization schemes for nonlinear systems. The paper discusses dissipative and passive systems, defining dissipativity and passivity in terms of supply rates and storage functions. It introduces the concept of the KYP property, which is a nonlinear extension of the Kalman-Yacubovitch-Popov lemma. The paper then focuses on stabilization by output feedback, showing that passive systems with positive definite storage functions can be stabilized using output feedback. It derives conditions for detectability and observability, and shows that certain stabilization laws can be derived from basic stabilizability properties of passive systems. The paper then discusses feedback equivalence to a passive system, showing that systems with relative degree 1 and weakly minimum phase properties can be feedback equivalent to passive systems. It provides conditions for feedback equivalence and shows that these conditions are necessary and sufficient for feedback equivalence to a passive system. The paper also discusses the global feedback equivalence of systems to passive systems, showing that systems with global weakly minimum phase properties can be globally feedback equivalent to passive systems. The paper concludes by analyzing the feedback equivalence of systems to passive systems in the context of cascade-interconnected configurations.This paper investigates the conditions under which a nonlinear system can be rendered passive via smooth state feedback and shows that this is possible if and only if the system has relative degree 1 and is weakly minimum phase. It also proves that weakly minimum phase nonlinear systems with relative degree 1 can be globally asymptotically stabilized by smooth state feedback under suitable controllability-like rank conditions. The paper extends and incorporates recent stabilization schemes for nonlinear systems. The paper discusses dissipative and passive systems, defining dissipativity and passivity in terms of supply rates and storage functions. It introduces the concept of the KYP property, which is a nonlinear extension of the Kalman-Yacubovitch-Popov lemma. The paper then focuses on stabilization by output feedback, showing that passive systems with positive definite storage functions can be stabilized using output feedback. It derives conditions for detectability and observability, and shows that certain stabilization laws can be derived from basic stabilizability properties of passive systems. The paper then discusses feedback equivalence to a passive system, showing that systems with relative degree 1 and weakly minimum phase properties can be feedback equivalent to passive systems. It provides conditions for feedback equivalence and shows that these conditions are necessary and sufficient for feedback equivalence to a passive system. The paper also discusses the global feedback equivalence of systems to passive systems, showing that systems with global weakly minimum phase properties can be globally feedback equivalent to passive systems. The paper concludes by analyzing the feedback equivalence of systems to passive systems in the context of cascade-interconnected configurations.
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