Control Lyapunov Value Functions
Standard reachability solves for the minimum time to a goal set. However, (1) the system is not guaranteed to stay inside the goal, and (2) the goal set must be predefined. We seek to modify reachability analysis to make a control Lyapunov-like function. This “control Lyapunov value function,” or CLVF, identifies the smallest control invariant set around the origin and provides both the region from which the system can stabilize to this set at a desired rate, as well as the control policy to achieve a stable trajectory.
Abstract:
In this paper, we seek to build connections between control Lyapunov functions (CLFs) and Hamilton-Jacobi (HJ) reachability analysis. CLFs have been used extensively in the control community for synthesizing stabilizing feedback controllers. However, there is no systematic way to construct CLFs for general nonlinear systems and the problem can become more complex with input constraints. HJ reachability is a formal method that can be used to guarantee safety or reachability for general nonlinear systems with input constraints. The main drawback is the well-known ``curse of dimensionality.'' In this paper we modify HJ reachability to construct what we call a control Lyapunov-Value Function (CLVF) which can be used to find and stabilize to the smallest control invariant set around a point of interest. We prove that the CLVF is the viscosity solution to a modified HJ variational inequality (VI), and can be computed numerically, during which the input constraints and exponential decay rate are incorporated. This process identifies the region of exponential stability to the smallest possible control invariant set given the desired input bounds and desired rate. Finally, a feasibility-guaranteed quadratic program (QP) is proposed for online implementation.