2012/06/29 by Majid Zamani, Zamani, Majid, Nathan van de Wouw +3 · 1 citation
Computer Science · Engineering · #Formal Methods in Verification #Control and Stability of Dynamical Systems #Petri Nets in System Modeling
paper · pdf · doi:10.48550/arxiv.1207.0030
Incremental stability is a property of dynamical and control systems,\nrequiring the uniform asymptotic stability of every trajectory, rather than\nthat of an equilibrium point or a particular time-varying trajectory. Similarly\nto stability, Lyapunov functions and contraction metrics play important roles\nin the study of incremental stability. In this paper, we provide\ncharacterizations and descriptions of incremental stability in terms of\nexistence of coordinate-invariant notions of incremental Lyapunov functions and\ncontraction metrics, respectively. Most design techniques providing controllers\nrendering control systems incrementally stable have two main drawbacks: they\ncan only be applied to control systems in either parametric-strict-feedback or\nstrict-feedback form, and they require these control systems to be smooth. In\nthis paper, we propose a design technique that is applicable to larger classes\nof (not necessarily smooth) control systems. Moreover, we propose a recursive\nway of constructing contraction metrics (for smooth control systems) and\nincremental Lyapunov functions which have been identified as a key tool\nenabling the construction of finite abstractions of nonlinear control systems,\nthe approximation of stochastic hybrid systems, source-code model checking for\nnonlinear dynamical systems and so on. The effectiveness of the proposed\nresults in this paper is illustrated by synthesizing a controller rendering a\nnon-smooth control system incrementally stable as well as constructing its\nfinite abstraction, using the computed incremental Lyapunov function.\n