Nonlinear stabilization via Control Contraction Metrics: A pseudospectral approach for computing geodesics

Abstract

Real-time nonlinear stabilization techniques are often limited by inefficient or intractable online and/or offline computations, or a lack guarantee for global stability. In this paper, we explore the use of Control Contraction Metrics (CCM) for nonlinear stabilization because it offers tractable offline computations that give formal guarantees for global stability. We provide a method to solve the associated online computation for a CCM controller -a pseudospectral method to find a geodesic. Through a case study of a stiff nonlinear system, we highlight two key benefits: (i) using CCM for nonlinear stabilization and (ii) rapid online computations amenable to real-time implementation. We compare the performance of a CCM controller with other popular feedback control techniques, namely the Linear Quadratic Regulator (LQR) and Nonlinear Model Predictive Control (NMPC). We show that a CCM controller using a pseudospectral approach for online computations is a middle ground between the simplicity of LQR and stability guarantees for NMPC.

Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design

“…We now prove path-integrability, i.e. that a solution of (17) exists. We will prove this by contradiction.…”

“…Simulations under LQR diverge rapidly after about 2 seconds. Further results on CCMs for poblems in robotics can be found in [37], and a method for computing geodesics can be found in [17]. [9,9,9] (right).…”

“…Second, we note that the only dependence ρ has on u is via a, which by construction is an s-dependent affine function of u. It follows from (18) that if there is a closed interval S ⊂ [0, 1] such that b > 0 for s ∈ S, then ρ is a globally Lipschitz function of u on S. By standard comparison results, e.g., [1, Thm 3.2], a unique solution to (17) exists on this interval, which contradicts finite escape at s =s.…”

“…The feedback controllers we propose will generally involve real-time optimization to find a minimal-length path with respect to the metric (a geodesic) joining the current state to the desired state. This problem is generally simpler than that in nonlinear model predictive control (MPC) since it has lower dimension, lacks dynamic constraints, and minimal geodesics are guaranteed to exist [17]. If a state-independent metric exists, then geodesics are just straight lines and our method is closely related to well-known methods using quadratic Lyapunov functions, e.g.…”

A Comparison of LPV Gain Scheduling and Control Contraction Metrics for Nonlinear Control

“…Remark 10. The computation of γ can be formulated as a simple nonlinear MPC problem which can be efficiently solved by the pseudospectral approach in Leung and Manchester (2017). Although there may exist multiple geodesics between x * and x, it is proved that the above controller is smooth almost everywhere on R n and continuous at x = x * (Manchester and Slotine, 2017).…”