Feedback Synthesis for Nonlinear Systems Via Convex Control Lyapunov Functions

This paper introduces computationally efficient methods for synthesizing explicit piecewise affine (PWA) feedback laws for nonlinear discrete-time systems, ensuring robustness and performance guarantees. The approach proceeds by optimizing a configuration-constrained PWA approximation of the value function of an infinite-horizon min-max Hamilton-Jacobi-Bellman equation. Here, robustness and performance are maintained by enforcing the PWA approximation to be a generalized control Lyapunov function for the given nonlinear system. This enables the generation of feedback laws with configurable storage complexity and pre-determined evaluation times, based on a selected configuration template. The framework’s effectiveness is demonstrated through a constrained Van der Pol oscillator case study, where an explicit PWA controller with certified ergodic performance and specified complexity is synthesized over a large operational domain.

💡 Research Summary

This paper presents a novel computational framework for synthesizing explicit, piecewise affine (PWA) feedback controllers for constrained, uncertain nonlinear discrete-time systems. The core challenge addressed is the computational intractability of solving the infinite-horizon min-max Hamilton-Jacobi-Bellman (HJB) equation for such systems. The proposed method circumvents this by constructing a PWA approximation of the optimal value function that is enforced to be a generalized control Lyapunov function (CLF) for the original nonlinear dynamics.

The methodology begins by defining the nonlinear system model and introducing key assumptions regarding the Lipschitz continuity of the dynamics (Assumption 1) and the bounded non-convexity of the stage cost (Assumption 2). A generalized CLF is defined as a convex, radially unbounded function M that satisfies a dissipation inequality (4) with a supply rate (L - d), where d is a drift parameter. Lemma 1 establishes that if such a CLF exists, it provides a certified upper bound (d) on the system’s asymptotic average (ergodic) cost, generalizing the traditional stability-centric role of CLFs to economic performance.

The computational breakthrough comes from restricting the search for CLFs to a specific, tractable class: functions whose epigraphs are configuration-constrained polyhedra P(z), as defined in (8). This polyhedron is parameterized by a vector z and has a pre-specified combinatorial structure (number of facets, vertices, and their adjacency) encoded by a matrix E. The corresponding PWA function M_z (9) has configurable storage complexity and deterministic evaluation time. Theorem 1 provides the crucial sufficient condition: if the dissipation inequality holds at all vertices v_i of P(z) for corresponding control inputs u_i, considering the worst-case disturbance within a set D_i that accounts for local nonlinearity errors, then M_z satisfies the CLF condition for all states. This reduces an infinite-dimensional functional inequality to a finite set of constraints.

Leveraging this result, the paper formulates a convex optimization problem (13). The decision variables are the polyhedron parameters z and the drift bound d. The constraints enforce the vertex-wise CLF conditions from Theorem 1, along with domain constraints. Solving this problem yields an optimal pair (z*, d*), where M_z* serves as an approximate value function and a valid CLF. An explicit PWA feedback law is then constructed by storing the control actions u_i* associated with each vertex/region of M_z*.

The framework’s efficacy is demonstrated through a detailed case study on a constrained Van der Pol oscillator. The authors synthesize an explicit PWA controller with a pre-specified complexity (number of facets) over a large operational domain. Numerical simulations confirm that the controller successfully stabilizes the nonlinear system to the origin while guaranteeing the certified ergodic performance bound d*. This demonstrates the method’s ability to handle complex nonlinear dynamics, provide robust performance guarantees, and deliver controllers with predictable memory and runtime characteristics—all with offline computation and no online optimization. The work effectively bridges the gap between the theoretical rigor of CLFs/HJB theory and the practical needs for implementable, certifiable control laws for nonlinear systems.