Fully Automated Stability Verification for Piecewise Affine Systems
One of the most desired properties of a closed-loop control system is stability, as a stable loop is inherently resistant to outside disturbances. Of particular interest is the notion of asymptotic stability. An asymptotically stable system will always converge towards an equilibrium state, once the disturbances have ceased. For hybrid systems, however, there is no known method for proving asymptotic stability directly from the system model. Instead, a promising approach is the use of Lyapunov functions, which can be utilized to show stability indirectly. A Lyapunov function is a formalization of an abstract “energy function” of the system. If the “energy” of the system monotonically decreases over time, converging towards zero in the designated equilibrium state, then a system is asymptotically stable. The existence of such a Lyapunov function proves asymptotic stability, but finding such a function for a hybrid system is not a simple task.
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