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Controller synthesis for incremental stability: Application to symbolic controller synthesis

Incremental stability is a property of dynamical and control systems, requiring the stability and convergence of trajectories with respect to each other, rather than with respect to an equilibrium point or a particular trajectory. Most design techniques providing controllers rendering control systems incrementally stable have two main drawbacks: they can only be applied to control systems in parametric-strict-feedback or strict-feedback form, and they require the control systems to be smooth. In this paper, we propose a controller design technique that is applicable to larger classes of control systems, including a class of non-smooth control systems. Moreover, we propose a recursive way of constructing incremental Lyapunov functions which have been identified as a key tool enabling the construction of finite abstractions of nonlinear control systems. The effectiveness of the proposed results in this paper is illustrated by synthesizing a controller rendering a non-smooth control system incrementally stable as well as constructing its finite abstraction, using the computed incremental Lyapunov function. Finally, using the constructed finite abstraction, we synthesize another controller for the incrementally stable closed-loop system enforcing the satisfaction of logic specifications, difficult (or even impossible) to enforce using conventional techniques.