Euler’s Method Applied to the Control of Switched Systems

Abstract

Hybrid systems are a powerful formalism for modeling and reasoning about cyber-physical systems. They mix the continuous and discrete natures of the evolution of computerized systems. Switched systems are a special kind of hybrid systems, with restricted discrete behaviours: those systems only have finitely many different modes of (continuous) evolution, with isolated switches between modes. Such systems provide a good balance between expressiveness and controllability, and are thus in widespread use in large branches of industry such as power electronics and automotive control. The control law for a switched system defines the way of selecting the modes during the run of the system. Controllability is the problem of (automatically) synthezing a control law in order to satisfy a desired property, such as safety (maintaining the variables within a given zone) or stabilisation (confinement of the variables in a close neighborhood around an objective point). In order to compute the control of a switched system, we need to compute the solutions of the differential equations governing the modes. Euler's method is the most basic technique for approximating such solutions. We present here an estimation of the Euler's method local error, using the notion of "one-sided Lispchitz constant" for modes. This yields a general control synthesis approach which can encompass several features such as bounded disturbance and compositionality.

Guaranteed phase synchronization of hybrid oscillators using symbolic Euler's method: The Brusselator and biped examples

“…For regions where λ ă 0, the value of rptq is considered to be constant; the value of rptq increases only when xptq occupies a region where λ ą 0 (which corresponds in Fig.2in case x 1 ptq or x 2 ptq is located in the red part of its orbit). See[Fri17].…”

Guaranteed phase synchronization of hybrid oscillators using symbolic Euler's method (verification challenge)

“…x " f pxq with initial condition xp0q " pc 0 1 , c 0 2 q given by Euler's explicit method, and rptq « r 0 e λt is the expanded radius using the one-sided Lipschitz constant λ (also called "logarithmic norm" or "matrix norm") [Söd06; AS12]) associated to f (see [Fri17] for details). 4 It is shown in [Le +17] that B euler ptq contains all the solutions xptq that start at B 0 :…”