Non-monotonic Lyapunov Functions for Analysis of Nonlinear Systems

Lyapunov’s stability theorem and its many variants comprise a central core of control theory and have a wide array of applications ranging from proving stability of nonlinear systems or designing stabilizing controllers to robustness analysis or proving convergence of combinatorial algorithms. The main challenge in the application of Lyapunov’s theorem is always to find a suitable Lyapunov function, a scalar function of the state that decreases monotonically along trajectories of a dynamical system.

At LIDS, MIT, Amir Ali Ahmadi and Professor Pablo Parrilo are working on finding efficient and systematic ways of searching for Lyapunov functions using techniques from convex optimization, as well as, making theoretical contributions to Lyapunov’s theorem itself. In a recent work, they have shown that by employing higher order derivatives of Lyapunov functions, the monotonicity requirement of Lyapunov’s theorem can be relaxed, therefore enlarging the class of functions that can certify stability. The results are applicable to a variety of dynamical systems including nonlinear, hybrid, uncertain, or time-varying systems. Furthermore, they have shown that efficient numerical algorithms from convex optimization can be used to search for non-monotonic Lyapunov functions.

(a) A typical trajectory of a planar linear time-varying system. A standard time-independent Lyapunov function would have an extremely complicated structure.

(b) Stability can be proven using a time-independent non-monotonic Lyapunov function, which can simply be taken to be the square of the Euclidian norm of the trajectory.

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