Abstract
This paper introduces a notion of distance between nonlinear dynamical systems which is suitable for a quantitative description of the robustness of stability in a feedback interconnection. This notion is one of several possible generalizations of the gap metric, and applies to dynamical systems which possess a differential graph. It is shown that any system which is stabilizable by output feedback, in the sense that the closed-loop system is input-output incrementally stable and possesses a linearization about any operating trajectory (i.e., about any admissible input-output pair), has a differential graph. A system which possesses a differentiable graph is globally differentiably stabilizable if the linearized model about any admissible input-output trajectory is stabilizable. It follows that if a nonlinear dynamical system is globally incrementally stabilizable, then it is (globally incrementally) stabilizable by a linear (possibly time-varying) controller. A suitable notion of a minimal opening between nonlinear differential manifolds is introduced and sufficient conditions guaranteeing robustness of stability are provided.
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This work was supported in part by the NSF under Grant NSF/ECS-9016050 and the AFOSR under Grant AF/F49620-92-J-0241.
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Georgiou, T.T. Differential stability and robust control of nonlinear systems. Math. Control Signal Systems 6, 289–306 (1993). https://doi.org/10.1007/BF01211498
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DOI: https://doi.org/10.1007/BF01211498