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Universal Adaptive Control of Nonlinear Systems
Brett T. Lopez, Jean-Jacques E. Slotine
TL;DR
Direct adaptive control with certainty equivalence has been difficult to extend to nonlinear systems with unmatched uncertainties. The paper adjusts the adaptation rate online to address parameter-estimation transients, and reports stability results and simulations demonstrating the approach.
Problem
Extending direct certainty-equivalence adaptive control to general nonlinear systems with unmatched uncertainties has remained difficult because estimation transients complicate closed-loop stability.
Source: §I · p. 1Method
The framework combines model-parameterized Lyapunov functions or contraction metrics with online adjustment of the adaptation rate.
Source: §I · p. 1, A. Overview · p. 2Results
Simulations show convergence in each reported strict-feedback trial, 11% of trials without adaptation failed, and the adaptive reference model outperformed a static one in tracking.
Source: A. Example 1: Strict-Feedback System · p. 4, B. Example 2: Contracting System · p. 5Takeaways & Limitations
Abstract
from arXiv · showhide
This work develops a new direct adaptive control framework that extends the certainty equivalence principle to general nonlinear systems with unmatched model uncertainties. The approach adjusts the rate of adaptation online to eliminate the effects of parameter estimation transients on closed-loop stability. The method can be immediately combined with a previously designed or learned feedback policy if a corresponding model-parameterized Lyapunov function or contraction metric is known. Simulation results of various nonlinear systems with unmatched uncertainties demonstrates the approach.
I. INTRODUCTION
The paper addresses the difficulty of extending direct certainty-equivalence adaptive control to nonlinear systems with unmatched uncertainties. It proposes online adaptation-rate adjustment and requires linearly parameterized uncertainty, without restricting the dynamics’ type or structure.
I. INTRODUCTION
- Unmatched uncertainties complicate direct certainty-equivalence design because parameter-estimation transients must be anticipated or made robust to.The paper presents these issues as the main obstacle to a comprehensive direct adaptive-control framework.Source: §I · p. 1
II. PROBLEM FORMULATION
- The uncertainty must be linearly parameterized, but the approach imposes no restriction on the type or structure of the dynamics.The paper calls the approach universal adaptive control.Source: §I · p. 1
I. INTRODUCTION
- Earlier approaches addressed unmatched uncertainties through adaptive backstepping, restricted structures, adaptive CLFs, ISS-based indirect control, or random-basis approximation.The cited approaches have limitations including structural restrictions, difficult adaptive-CLF computation, strict ISS conditions, or extensive tuning.Source: §I · p. 1
- The proposed framework extends certainty-equivalence design to general nonlinear systems with unmatched uncertainties by adjusting the adaptation rate online.Its two core ideas are a model-parameterized family of CLFs and adaptation-rate adjustment to eliminate estimation-transient effects on closed-loop stability.Source: §I · p. 1
A. Overview
The technical results introduce a new CLF for adaptive stabilization with unmatched uncertainties, then extend the approach to contracting systems using a new control contraction metric.
A. Overview
- The section first defines a new CLF and establishes its equivalence to stabilizability.It then uses that CLF for certainty-equivalence adaptive stabilization with online adaptation-rate adjustment.Source: A. Overview · p. 2
- The adaptive stabilization result combines certainty equivalence with online adjustment of the adaptation rate.Source: A. Overview · p. 2
- The approach is extended to contracting systems through a new control contraction metric.Source: A. Overview · p. 2
B. Unmatched Control Lyapunov Functions
An unmatched CLF is a model-indexed family of CLFs, and its existence is equivalent to stabilizability for each parameter value. Given such a function, the unmatched adaptation law yields global asymptotic stability; the framework also extends to contraction metrics.
B. Unmatched Control Lyapunov Functions
- An unmatched CLF exists if and only if the system is stabilizable for each parameter value.Unlike an adaptive CLF, its existence is equivalent to pointwise stabilizability; the cited passage notes adaptive-CLF existence is equivalent to adaptive stabilizability.Source: B. Unmatched Control Lyapunov Functions · p. 2
- Given an unmatched CLF, Theorem 1 guarantees global asymptotic stability under the unmatched adaptation law for any strictly increasing, uniformly positive rate function.The proof uses a Lyapunov-like function; the rate can remain constant when adaptation transients are stabilizing.Source: B. Unmatched Control Lyapunov Functions · pp. 2–3
- The result also applies when the CLF condition supplies an input to a virtual contracting system with the state and desired state as particular solutions.The cited remark attributes this extension to the hierarchical combination property of contracting systems, which generalizes a sliding variable.Source: B. Unmatched Control Lyapunov Functions · p. 3
- The unmatched adaptation law adjusts in response to whether parameter-estimation transients stabilize or destabilize the closed loop.This rate adjustment is the mechanism the theorem uses to extend certainty-equivalence design to unmatched uncertainties.Source: B. Unmatched Control Lyapunov Functions · p. 3
C. Unmatched Control Contraction Metrics
The section extends adaptive certainty-equivalence stabilization to contracting systems using differential contraction metrics rather than an explicit state transformation.
C. Unmatched Control Contraction Metrics
- Contraction analysis derives stabilizing feedback from differential dynamics without constructing an explicit state transformation.The nominal differential dynamics are given by ˙δx = A(x, u, t)δx + B(x, t)δu.Source: C. Unmatched Control Contraction Metrics · p. 3
- The Riemannian energy bound ˙E(x, t) ≤ −2λE(x, t) yields exponential convergence to the desired state at rate λ.The energy is defined along a path from the desired state to the current state.Source: C. Unmatched Control Contraction Metrics · p. 3
- Certainty equivalence removes the parameter-adaptation term from ˙Wθ, making computation of an unmatched control contraction metric tractable.For matched uncertainty, the metric need not depend on unknown parameters.Source: C. Unmatched Control Contraction Metrics · p. 4
- If an unmatched control contraction metric exists, Theorem 2 guarantees global asymptotic stability for any strictly increasing, uniformly positive scaling function υ(ρ).The result uses the unmatched adaptation law.Source: C. Unmatched Control Contraction Metrics · p. 4
A. Example 1: Strict-Feedback System
In a strict-feedback system with unmatched uncertainties, simulations show convergence under the proposed adaptive controller and rate adjustment in most trials.
A. Example 1: Strict-Feedback System
- The example is a strict-feedback system with unmatched uncertainties, a commonly used benchmark in nonlinear adaptive control.The paper states that the approach applies to a broader class of systems.Source: A. Example 1: Strict-Feedback System · p. 4
- The objective was to stabilize the origin from initial states and true parameters sampled uniformly from bounded sets.The simulation time step was dt = 0.005.Source: A. Example 1: Strict-Feedback System · p. 4
- Across 100 simulation experiments, the state-vector norm converged to the origin in every trial; without adaptation, 11% of trials failed, while the adaptation rate adjusted in the majority of trials.Source: A. Example 1: Strict-Feedback System · pp. 4–5
B. Example 2: Contracting System
In a contracting-system tracking example, the proposed controller outperforms a same-metric controller with a fixed reference model and maintains desired behavior as the reference changes.
B. Example 2: Contracting System
- The system is neither feedback linearizable nor in strict-feedback form, and parameters θ1 and θ2 are unmatched.The example tracks a desired trajectory.Source: B. Example 2: Contracting System · p. 5
- The contraction metric is non-flat, so a quadratic control Lyapunov function does not exist; geodesics and a pointwise min-norm controller are computed each step.The metric was found using YALMIP and SOS.Source: B. Example 2: Contracting System · p. 5
- The proposed approach has superior tracking performance to the same-metric controller with a fixed reference model, as assessed by Riemannian energy.The energy indicates closed-loop tracking error.Source: B. Example 2: Contracting System · p. 5
- The system retains the desired behavior even as its reference model changes.The blue curve in Fig. 2b shows this behavior.Source: B. Example 2: Contracting System · p. 5
V. CONCLUDING REMARKS
The framework uses online adaptation-rate adjustment to eliminate parameter-estimation transient effects on stability and apply certainty equivalence to unmatched-uncertainty systems. The conclusion also points to extensions in learned control and adaptive safety.
V. CONCLUDING REMARKS
- Online adaptation-rate adjustment eliminates parameter-estimation transient effects on stability and lets the method use certainty equivalence for unmatched-uncertainty systems.The framework synthesizes stabilizing controllers through Lyapunov or contraction theory.Source: §V · p. 5
- Fig. 2 reports lower Riemannian energy than a static-reference controller while the closed-loop system retains desired behavior as the reference model adapts.The paper describes Riemannian energy as an indicator of closed-loop tracking error.Source: Figure 2 · p. 5
- Proposed extensions include data-driven control, adaptive safety through barrier functions, and output feedback with adaptive nonlinear observers.For dynamical-system learning, the authors propose learning a family of optimal policies for possible model realizations and combining them with online adaptation.Source: §V · p. 5
- The connection between adaptive stabilizability and stabilizability for each θ remains an area for further investigation.Source: §V · p. 5
VI. APPENDIX
The appendix defines adaptive control Lyapunov functions and presents uccm-related arguments, including a convergence condition for the state.
A. Adaptive Control Lyapunov Functions
- An aclf is a smooth positive-definite function that is radially unbounded in x.The supplied definition begins with Va over R^n × R^p × R.Source: A. Adaptive Control Lyapunov Functions · p. 5
B. Universal Adaptive Control with uccm
- For a geodesic under a uccm, the energy satisfies ˙Eθ(x, t) ≤ −2λEθ(x, t).Source: B. Universal Adaptive Control with uccm · p. 6
- If υ(ρ) is uniformly positive and Eˆθ(x, t) = 0 if and only if x = xd, then x converges to xd as t approaches infinity.Source: B. Universal Adaptive Control with uccm · p. 6
C. Example 1: Backstepping Controller Derivation
The example’s controller derivation invokes the standard backstepping technique.
C. Example 1: Backstepping Controller Derivation
- The example’s derivation uses the standard backstepping technique.Source: C. Example 1: Backstepping Controller… · p. 6