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

Ian R. Manchester, Jean-Jacques E. Slotine

arXiv:1503.03144v3eess.SYmath.OC

TL;DR

Nonlinear stabilization needs constructive conditions for finding feedback laws across trajectories. The paper introduces control contraction metrics, obtaining convex geometric criteria and universal exponential stabilization, with extensions to submanifolds and geodesic-based controllers.

  • Problem

    Finding control Lyapunov functions for nonlinear systems is challenging because their feasible set is not necessarily convex or connected.

  • Method

    The paper studies nonlinear systems through differential dynamics and defines a uniformly bounded control contraction metric that contracts unactuated directions.

  • Results

    A control contraction metric is sufficient for universal exponential stabilizability, including open-loop, sampled-data, and almost-everywhere continuous-feedback forms.

  • Takeaways & Limitations

    The criteria provide a convex, geometrically interpretable, coordinate-invariant approach to nonlinear feedback design and extend to submanifold stabilization.

Abstract

from arXiv · show

We introduce the concept of a control contraction metric, extending contraction analysis to constructive nonlinear control design. We derive sufficient conditions for exponential stabilizability of all trajectories of a nonlinear control system. The conditions have a simple geometrical interpretation, can be written as a convex feasibility problem, and are invariant under coordinate changes. We show that these conditions are necessary and sufficient for feedback linearizable systems, and also derive novel convex criteria for exponential stabilization of a nonlinear submanifold of state space. We illustrate the benefits of convexity by constructing a controller for an unstable polynomial system that combines local optimality and global stability, using a metric found via sum-of-squares programming.

I. INTRODUCTION

The paper frames nonlinear stabilization around finding suitable Lyapunov-like objects, then introduces differential contraction metrics as a convex, coordinate-invariant route to stabilizing all forward-complete trajectories.

  • I. INTRODUCTION: Lyapunov functions model energy dissipation, while control Lyapunov functions generalize this measure so control actions can make it decrease.Finding a CLF remains challenging despite the simple stabilizing feedback formulas available once one is found.
  • I. INTRODUCTION: For nonlinear systems, CLF searches are not necessarily convex or connected, while existing dual representations are generally infinite-dimensional.Finite-dimensional approximations have used gridding or sum-of-squares relaxations.
  • I. INTRODUCTION: Studying stabilization through differential dynamics recovers the simple convexification available for linear systems by generalizing contraction metrics.Contraction analysis applies linear time-varying techniques to nonlinear variational dynamics.
  • I. INTRODUCTION: Universal stabilizability means that every forward-complete solution can be globally stabilized, and the paper gives sufficient conditions for universal exponential stabilizability.The framework also extends to stabilization of submanifolds.
  • I. INTRODUCTION: The proposed criteria require natural contraction in directions orthogonal to control inputs, are convex and coordinate-invariant, and are necessary and sufficient for feedback linearizable systems.Orthogonality and contraction are defined by the chosen metric.
  • I. INTRODUCTION: The controllers generally use minimal-length metric paths, while the main feedback controller may be discontinuous at some state-space points.A sampled-data controller is proposed to ensure solution existence at such points.

III. CONTROL CONTRACTION METRICS

The control contraction metric framework studies nonlinear stabilization through an extended system of state and differential dynamics. A uniformly bounded metric satisfying contraction on unactuated directions yields several forms of universal exponential stabilization.

  • III. CONTROL CONTRACTION METRICS: The analysis uses an extended system pairing the nonlinear system with its differential dynamics.This exposes linear time-varying structure along system solutions.
  • III. CONTROL CONTRACTION METRICS: A known feedback that makes the system contracting for arbitrary additive signals satisfies a matrix contraction inequality.The inequality is expressed using the metric, system Jacobian, input matrix, and feedback Jacobian.
  • III. CONTROL CONTRACTION METRICS: The intrinsic condition requires every nonzero differential displacement orthogonal to all actuated directions to contract at rate λ.This condition is independent of the particular feedback law.
  • III. CONTROL CONTRACTION METRICS: If a uniformly bounded metric satisfies the intrinsic condition, it is a control contraction metric and guarantees universal exponential stabilization.The theorem links the metric condition to open-loop controllability and sampled-data or almost-everywhere continuous feedback stabilization.
  • III. CONTROL CONTRACTION METRICS: A CCM produces a path-integrable differential feedback controller whose integration yields control signals along state-space paths.This requirement is weaker than complete integrability into the Jacobian of a feedback controller.

1) Open-Loop Control:

The open-loop CCM controller propagates a path from the target trajectory to the current state while applying differential feedback along that path. Over time, the path length shrinks exponentially, yielding exponential stabilization.

  • At the initial time, the controller measures x(t_i) and constructs a smooth path c(t_i) joining x⋆(t_i) to x(t_i).
  • The controller applies u(t)=k_p(c(t),u⋆(t),t,1) using the propagated path and target control trajectory.
  • The path is propagated according to the system dynamics while applying the differential feedback control k_p along its points.
  • The curve length shrinks exponentially on an interval [t_i,∞), establishing the main exponential-stabilization claim.
  • The theorem’s bound is achieved when the initial path is a minimal geodesic; other initial paths preserve the rate but may increase overshoot.

2) Sampled-Data Feedback Controller:

Sampled-data feedback recomputes a minimal geodesic at each sampling time and applies the open-loop controller between samples. Strong CCM conditions support smooth differential feedback and exponential stabilization despite geometric singularities.

  • 2) Sampled-Data Feedback Controller:: At each sample time, the controller measures x(t_i), computes a minimal geodesic to x⋆(t_i), and applies the open-loop control until the next sample.
  • 2) Sampled-Data Feedback Controller:: The sampled-data controller is stabilizing for any choice of sample times, including uniform sampling.
  • 3) Smooth Feedback, Uniquely Defined Almost Everywhere and in a Neighbourhood of x⋆:: Taking the sampling interval to zero yields a continuous controller that selects a minimal geodesic and applies k_p at its endpoint.
  • 3) Smooth Feedback, Uniquely Defined Almost Everywhere and in a Neighbourhood of x⋆:: The continuous controller is smooth away from the cut locus and universally exponentially stabilizing when trajectory time spent on the cut locus has zero measure.
  • 3) Smooth Feedback, Uniquely Defined Almost Everywhere and in a Neighbourhood of x⋆:: Conditions C1 and C2 require contraction orthogonal to control directions and prevent input vector fields from expanding differential lengths.
  • 3) Smooth Feedback, Uniquely Defined Almost Everywhere and in a Neighbourhood of x⋆:: Finsler’s theorem converts C1 into a scalar-multiplier condition, enabling construction of a path-integrable differential gain K(x,t).

B. Dual Metrics and Convexity of Synthesis

A dual-metric change of variables makes CCM synthesis convex in the matrix function W, with finite-dimensional approximations available through basis expansions, gridding, or sum-of-squares relaxations.

  • The transformation η=M(x,t)δx and W=M(x,t)^−1 defines a dual CCM and converts the differential metric representation into cotangent-space coordinates.
  • In dual coordinates, the CCM inequality is linear and therefore convex in the unknown matrix function W.
  • The differential gain can be parameterized as K=YW^−1, with path integrability preserved when Y and K are at most affine in u.
  • Finsler’s theorem introduces a scalar function ρ so the stronger CCM condition becomes jointly convex in W and ρ.
  • Finite-dimensional LMI approximations use finite basis expansions and verify inequalities by gridding or sum-of-squares relaxation.
  • Although integrability constraints are linear in K, they are not jointly convex in W,Y or W,ρ because K depends on matrix inversion.

IV. PROPERTIES OF CONTROL CONTRACTION METRICS

The Riemannian energy induced by a CCM can serve as a control Lyapunov function for any target trajectory. The resulting criteria and constructions retain geometric invariance and extend to submanifold stabilization.

  • A. Riemannian Energy as a CLF: The proof constructs a controller that decreases the Riemannian energy E(x,x⋆,t), making that energy a CLF for any target trajectory.
  • A. Riemannian Energy as a CLF: The first variation expresses the energy derivative as an affine function of the control, enabling pointwise controller selection.
  • A. Riemannian Energy as a CLF: The admissible control set remains a convex half-space or all of R^m, and is always non-empty.
  • A. Riemannian Energy as a CLF: Pointwise min-norm control can reduce control magnitude while retaining the CLF-based stability guarantee.
  • CCM criteria are invariant under affine feedback transformations and smooth coordinate changes, with metrics transformed tensorially.
  • Uniform metric bounds are preserved under coordinate changes whose differential has bounded singular values.

C. Necessity for Feedback Linearizable Systems

For feedback linearizable systems, a control contraction metric is guaranteed and certifies universal stabilizability, while CCM existence is broader because it requires stabilizability without differential integrability.

  • Feedback-linearizable systems: Every feedback linearizable system admits a CCM constructed from a coordinate transformation and a positive definite matrix satisfying an LMI.The metric is W(x,t) = Φ(x,t)PΦ(x,t)′, with P chosen so the projected matrix inequality is negative.
  • Feedback-linearizable systems: The required positive definite matrix P exists when the transformed pair (G,H) is stabilizable.The paper states this as a consequence of standard linear systems theory.
  • Beyond feedback linearization: CCM existence does not require the complete integrability needed to convert a differential coordinate change into an explicit state coordinate transformation.Thus the converse from CCM existence to feedback linearizability does not hold.
  • Submanifold stabilization: For submanifold stabilization, the paper introduces a virtual control system by augmenting the actual input matrix with null-space directions of the manifold definition.A CCM for this virtual system yields exponential stabilization of time-varying level-set submanifolds.
  • Submanifold stabilization: Under the stated CCM conditions, time-varying submanifolds can be exponentially stabilized open-loop, with sampled-data control, or continuously almost everywhere.The construction uses a shadow state on the submanifold and drives the actual state toward it.
  • Submanifold stabilization: For uncontrolled systems, a uniformly bounded dual metric invariant on manifold level sets guarantees exponential convergence to the set Z(t).The corollary requires no actual computation of a control signal and generalizes partial-contraction results.

VI. ILLUSTRATIVE EXAMPLE

The example combines local LQR behavior with global CCM stabilization for an unstable polynomial system that is not feedback linearizable. The CCM controller matches LQR near the origin but remains stabilizing for larger initial conditions.

  • Implication: Convex CCM criteria make it straightforward to combine locally optimal and globally stabilizing control objectives in one controller.The paper contrasts this with the non-convexity of control Lyapunov functions for the same uniting problem.
  • System and design: The example system is not feedback linearizable because the matrix [B, adfB, ad2fB] loses rank at the origin.The construction therefore illustrates CCM design beyond feedback-linearizable systems.
  • System and design: The authors first solve the origin-linearized LQR problem, then constrain W(0) = P^-1 and ρ(0) = 2r^-1 so the CCM and LQR controllers coincide locally.Quadratic state dependence is allowed in W to satisfy the CCM conditions, and the metric is found through the resulting convex search.
  • Simulation results: For small initial conditions, the CCM and LQR responses are virtually identical because the minimal geodesic is close to a straight line and ρ and W barely change.The CCM law therefore approximates simple linear feedback near the target.
  • Simulation results: For larger initial conditions, LQR is not stabilizing and simulations diverge rapidly after about 2 seconds, whereas the CCM controller is stabilizing.Figure 2 compares initial states [0.5, 0.5, 0.5]′ and [9, 9, 9]′ to illustrate the local-optimal and global-stabilizing behaviors.

APPENDIX Proof of Proposition 1:

The appendix proves exponential convergence by constructing differential feedback along metric geodesics and showing that the induced Riemannian distance decreases exponentially under the proposed controllers.

  • Geodesic existence: A quadratic upper bound on the metric’s largest eigenvalue ensures minimal geodesics exist between every pair of points in Rn.The proof uses the Hopf–Rinow theorem and excludes finite escape of geodesics.
  • Proposition and controller construction: A differential feedback controller is constructed from the metric gradient so that the differential storage function satisfies a strict dissipativity inequality.The controller uses kδ(x,δx,u,t) = −ρB′∂V/∂δx, and substitution establishes closed-loop dissipativity.
  • Proposition and controller construction: Path-integrability is established by ruling out finite escape: any blow-up of ρ would force a region where ρ is globally Lipschitz in the control input.Standard comparison results then contradict finite escape along the path.
  • Exponential distance decrease: Along a controlled path, metric energy decreases as dE/dt < −2λE, yielding L(c(t),t) ≤ e^-λ(t−ti)L(c(ti),ti).Taking the initial path to be a minimal geodesic gives the corresponding Riemannian-distance bound.
  • Exponential distance decrease: Uniform boundedness of M converts exponential Riemannian-distance decay into exponential Euclidean convergence.The metric bounds relate Euclidean distance and Riemannian distance through α1 and α2.
  • Controller implementations: The same contraction argument supports open-loop, sampled-data, and continuous feedback constructions, with geodesics recomputed at sampling times for sampled-data control.The continuous-feedback proof uses smooth dependence of geodesics and the feedback law on the state and target.
  • Invariance: Under a feedback transformation, the CCM condition is preserved because the annihilator removes the transformed input directions from the dual inequality.Under coordinate changes, the metric transforms as Mξ = Ψ′MΨ while preserving the contraction inequality.
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