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Smooth Converse Lyapunov-Barrier Theorems for Asymptotic Stability with Safety Constraints and Reach-Avoid-Stay Specifications

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Document pages: 11 pages

Abstract: Stability and safety are two important aspects in safety-critical control ofdynamical systems. It has been a well established fact in control theory thatstability properties can be characterized by Lyapunov functions. Reachabilityproperties can also be naturally captured by Lyapunov functions for finite-timestability. Motivated by safety-critical control applications, such as inautonomous systems and robotics, there has been a recent surge of interests incharacterizing safety properties using barrier functions. Lyapunov and barrierfunctions conditions, however, are sometimes viewed as competing objectives. Inthis paper, we provide a unified theoretical treatment of Lyapunov and barrierfunctions in terms of converse theorems for stability properties with safetyguarantees and reach-avoid-stay type specifications. We show that if a system(modeled as a perturbed dynamical system) possesses a stability with safetyproperty, then there exists a smooth Lyapunov function to certify such aproperty. This Lyapunov function is shown to be defined on the entire set ofinitial conditions from which solutions satisfy this property. A similar butslightly weaker statement is made for reach-avoid-stay specifications. We showby a simple example that the latter statement cannot be strengthened withoutadditional assumptions.

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