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

Abstract

We provide an amendment to the first theorem of "Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design" by Manchester & Slotine in the form of an additional technical condition required to show integrability of differential control signals. This technical condition is shown to be satisfied under the original assumptions if the input matrix is constant rank, and also if the strong conditions for a CCM hold. However a simple counterexample shows that if the input matrix drops rank, then the weaker conditions of the original theorem may not imply stabilizability of all trajectories. The remaining claims and illustrative examples of the paper are shown to remain valid with the new condition.

A Nonlinear Model Predictive Control Framework Using Reference Generic Terminal Ingredients

“…Assuming A, B are polynomial, consider matrices X, Y polynomial in r (with a specified order d) and ensure that the matrix in (19) is SOS. A similar approach is suggested in [23] to find a control contraction metric (CCM) for continuous-time systems (which is a strongly related problem). This approach is not pursued here since most systems require a polynomial of high order to approximate the nonlinear dynamics and the computational complexity grows exponentially in n d , thus prohibiting the practical application.…”

“…Solve (20) using Θ according to (22) or Remark 7. Gridding: Select (r i , r + i ) satisfying (23). Solve (19) for all (r i , r + i ).…”

“…For the discrete-time convex approach the polytopic description Θ (22) and the hyperbox description Θ × Ω (Remark 7) are considered. For the gridding, (x 1 , x 3 , u r , u + r ) ∈ R 4 is gridded using 10 4 points, of which approximately 8.000 satisfy the conditions (23) and are considered in the optimization problem (19).…”

Decentralized nonlinear feedback design with separable control contraction metrics

“…A differentiable feedback law for system (5) and a feedback law for system (2) are obtained using the following notion of contraction metric, recalled from [28].…”

“…The following result, recalled from [6, Theorem 1] (see [28] for a proof), formalizes how a feedback law is obtained from a control-contraction metric for system (2). Proposition 1.…”