pith. sign in

arxiv: 2604.24984 · v1 · submitted 2026-04-27 · 🧮 math.OC

Adaptive Constraint-Lifting Control with Stability and Invariance Guarantees

Jhon Manuel Portella Delgado , Ankit Goel This is my paper

Pith reviewed 2026-05-08 02:23 UTC · model grok-4.3

classification 🧮 math.OC
keywords invarianceparameterssystemadaptiveclosed-loopcontrolguaranteessafe
0
0 comments X

The pith

An adaptive constraint-lifting framework achieves asymptotic tracking, closed-loop stability, and forward invariance of safe sets for uncertain strict-feedback nonlinear systems using Lyapunov analysis.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Many real-world machines like motors or robots have limits on how far their states can go to avoid damage or unsafe behavior. When the exact parameters of the system are unknown, designing a controller that respects those limits while still making the machine follow a desired path is difficult. This work uses a strict-feedback structure, where the system dynamics build up in layers, and introduces a lifting transformation that moves the problem into new coordinates. In these lifted coordinates, the original state constraints become automatically satisfied as long as the new variables stay bounded. An adaptive law estimates the unknown parameters in real time. The authors then apply standard Lyapunov stability tools plus an invariance principle to prove that all signals remain bounded, the tracking error goes to zero, and the safe set is never left. They illustrate the idea on a DC motor example with uncertain parameters.

Core claim

A Lyapunov-based stability analysis, combined with the Barbashin-Krasovskii-LaSalle invariance principle, establishes boundedness of all closed-loop signals, asymptotic convergence of the system states to the desired equilibrium, and forward invariance of the safe set under uncertainty.

Load-bearing premise

The system admits a strict-feedback structure with unknown parameters entering both drift and input channels, and that a suitable constraint-lifting transformation exists that converts the constrained problem into an equivalent unconstrained one without introducing new instabilities.

Figures

Figures reproduced from arXiv: 2604.24984 by Ankit Goel, Jhon Manuel Portella Delgado.

Figure 1
Figure 1. Figure 1: State transformations with the constraint-lifting view at source ↗
Figure 2
Figure 2. Figure 2: shows the closed-loop response of (3)–(4) under the control law (32) and the adaptation laws (30) and (31). The results demonstrate successful tracking of x1d while maintaining the state within the prescribed safe set, indicated by the shaded yellow region in the first plot. Furthermore, the state x2 remains within its corresponding safe bounds, as shown by the shaded pink region in the second plot view at source ↗
Figure 3
Figure 3. Figure 3: Estimation errors |Θˆ 1 −Θ1| and |pˆ2 −p2| shown on a logarithmic scale. Consistent with the stability analysis, the parameter estimates do not converge to the true values Θ1 and p2, but remain bounded. APPENDIX I DYNAMICS IN z COORDINATE This appendix derives the system dynamics in the lifted z-coordinates used for controller synthesis. It follows from (8) that z˙1 = x1∂χ1 φ(χ1) ˙χ1 = x1∂χ1 φ(χ1)x −1 1 x˙… view at source ↗
read the original abstract

This paper develops an adaptive tracking controller for a class of nonlinear systems with parametric uncertainty subject to state constraints. The system is characterized by a strict-feedback structure with unknown parameters entering both the drift and input channels. The objective is to design a control law, without knowledge of the unknown parameters, that guarantees closed-loop stability, achieves desired tracking performance, and ensures forward invariance of a prescribed safe set. An adaptive constraint-lifting framework is developed that transforms the constrained control problem into an equivalent unconstrained representation, enabling recursive controller synthesis in lifted coordinates. The proposed design integrates parameter estimation with constraint enforcement without requiring online optimization. A Lyapunov-based stability analysis, combined with the Barbashin-Krasovskii-LaSalle invariance principle, establishes boundedness of all closed-loop signals, asymptotic convergence of the system states to the desired equilibrium, and forward invariance of the safe set under uncertainty. In particular, the analysis characterizes the largest invariant set of the closed-loop system and guarantees convergence despite unknown parameters. The effectiveness of the proposed approach is demonstrated on a DC motor system with uncertain parameters, illustrating accurate tracking performance and safe operation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops an adaptive tracking controller for strict-feedback nonlinear systems with parametric uncertainties affecting both drift and input channels, subject to state constraints. It introduces an adaptive constraint-lifting transformation that recasts the constrained problem as an equivalent unconstrained one, enabling recursive backstepping-style synthesis of the adaptive law without online optimization. A Lyapunov function combined with the Barbashin-Krasovskii-LaSalle invariance principle is used to prove boundedness of all signals, asymptotic convergence of states to the desired equilibrium, and forward invariance of the prescribed safe set. The method is illustrated on a DC-motor example with uncertain parameters.

Significance. If the central stability and invariance claims are rigorously established, the work would provide a useful non-optimization-based route to safe adaptive control for uncertain strict-feedback systems. The constraint-lifting idea, when paired with standard adaptive backstepping, could reduce computational burden relative to CBF or MPC approaches while retaining explicit guarantees; the DC-motor simulation offers a concrete, reproducible test case.

major comments (2)
  1. [Stability analysis section] Stability analysis (the section invoking the Barbashin-Krasovskii-LaSalle principle): the proof applies the invariance principle to conclude asymptotic convergence to the largest invariant set (and hence to the equilibrium). However, the abstract simultaneously describes a 'tracking controller' and convergence 'to the desired equilibrium.' If the reference signal is time-varying, the closed-loop vector field is explicitly time-dependent, violating the autonomy assumption required by the standard LaSalle theorem. The manuscript must either (i) restrict to constant references (making the system autonomous) or (ii) invoke an appropriate extension such as the LaSalle-Yoshizawa theorem or uniform continuity arguments. This point is load-bearing for the asymptotic-convergence claim.
  2. [Constraint-lifting transformation section] Constraint-lifting transformation (the section defining the lifted coordinates and the associated error dynamics): the abstract states that the transformation converts the constrained problem into an 'equivalent unconstrained representation' without introducing new instabilities. No explicit error bounds, Lipschitz constants, or invariance-preserving properties of the lifting map are supplied in the provided abstract or summary. Because the entire recursive design and the subsequent Lyapunov analysis rest on this equivalence, the transformation must be shown to be a diffeomorphism that preserves the safe-set invariance property under the unknown-parameter dynamics.
minor comments (2)
  1. [Abstract] The abstract claims both 'tracking performance' and convergence 'to the desired equilibrium.' Clarify whether the reference is a constant set-point or a time-varying trajectory; this affects the interpretation of the invariance result.
  2. [Notation and preliminaries] Notation for the lifted state and the adaptive parameter vector should be introduced once and used consistently; several symbols appear without prior definition in the abstract-level description.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on the stability analysis and constraint-lifting transformation. We address each major comment below, indicating the revisions that will be incorporated to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Stability analysis section] Stability analysis (the section invoking the Barbashin-Krasovskii-LaSalle principle): the proof applies the invariance principle to conclude asymptotic convergence to the largest invariant set (and hence to the equilibrium). However, the abstract simultaneously describes a 'tracking controller' and convergence 'to the desired equilibrium.' If the reference signal is time-varying, the closed-loop vector field is explicitly time-dependent, violating the autonomy assumption required by the standard LaSalle theorem. The manuscript must either (i) restrict to constant references (making the system autonomous) or (ii) invoke an appropriate extension such as the LaSalle-Yoshizawa theorem or uniform continuity arguments. This point is load-bearing for the asymptotic-convergence claim.

    Authors: We agree that the autonomy assumption is essential for the standard Barbashin-Krasovskii-LaSalle invariance principle. In the manuscript the reference signal is constant, so the closed-loop vector field is autonomous and the invariance principle applies directly to establish convergence to the largest invariant set, which is the desired equilibrium. The phrase 'tracking controller' is used loosely to indicate regulation to this constant setpoint. We will revise the abstract, introduction, and stability section to state explicitly that the reference is constant, remove any ambiguity about time-varying signals, and add a brief remark confirming autonomy of the closed-loop dynamics. revision: yes

  2. Referee: [Constraint-lifting transformation section] Constraint-lifting transformation (the section defining the lifted coordinates and the associated error dynamics): the abstract states that the transformation converts the constrained problem into an 'equivalent unconstrained representation' without introducing new instabilities. No explicit error bounds, Lipschitz constants, or invariance-preserving properties of the lifting map are supplied in the provided abstract or summary. Because the entire recursive design and the subsequent Lyapunov analysis rest on this equivalence, the transformation must be shown to be a diffeomorphism that preserves the safe-set invariance property under the unknown-parameter dynamics.

    Authors: The constraint-lifting map is constructed in the full manuscript as a C^1 diffeomorphism between the safe set and the lifted domain, with a smooth inverse. We show that solutions of the lifted error dynamics remain in one-to-one correspondence with solutions of the original system inside the safe set, thereby preserving forward invariance. To make these properties fully explicit, we will add in the revised transformation section: (i) the explicit Lipschitz constants of the map and its inverse, (ii) uniform bounds on the state-error between original and lifted coordinates, and (iii) a short lemma confirming that the diffeomorphism maps the safe set onto itself and that invariance is preserved under the parameter-adaptive dynamics. revision: yes

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides insufficient detail to enumerate free parameters, axioms, or invented entities; no explicit fitting or new postulated objects are named.

pith-pipeline@v0.9.0 · 5490 in / 1069 out tokens · 67396 ms · 2026-05-08T02:23:37.713493+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

  1. [1]

    Set invariance in control,

    F. Blanchini, “Set invariance in control,” Automatica, vol. 35, no. 11, pp. 1747–1767, 1999

    work page 1999

  2. [2]

    Discrete- time reference governors and the nonlinear control of systems with state and control constraints,

    E. G. Gilbert, I. Kolmanovsky, and K. T. Tan, “Discrete- time reference governors and the nonlinear control of systems with state and control constraints,” Interna- tional Journal of Robust and Nonlinear Control , vol. 5, no. 5, pp. 487–504, 1995

    work page 1995

  3. [3]

    Ref- erence and command governors for systems with con- straints: A survey on theory and applications,

    E. Garone, S. D. Cairano, and I. Kolmanovsky, “Ref- erence and command governors for systems with con- straints: A survey on theory and applications,” Auto- matica, vol. 75, pp. 306–328, 2017

    work page 2017

  4. [4]

    Control barrier function based quadratic programs for safety critical systems,

    A. D. Ames, X. Xu, J. W . Grizzle, and P . Tabuada, “Control barrier function based quadratic programs for safety critical systems,” IEEE Transactions on Auto- matic Control , vol. 62, no. 8, pp. 3861–3876, 2016

    work page 2016

  5. [5]

    Adaptive control of pure feedback systems without inversion of parame- ter estimates,

    J. M. Portella Delgado and A. Goel, “Adaptive control of pure feedback systems without inversion of parame- ter estimates,” International Journal of Control, vol. 99, no. 2, pp. 358–366, 2026

    work page 2026

  6. [6]

    Barrier lyapunov functions for the control of output-constrained nonlin- ear systems,

    K. P . Tee, S. S. Ge, and E. H. Tay, “Barrier lyapunov functions for the control of output-constrained nonlin- ear systems,” Automatica, vol. 45, no. 4, pp. 918–927, 2009

    work page 2009

  7. [7]

    Backstepping control for output- constrained nonlinear systems based on nonlinear map- ping,

    T. Guo and X. Wu, “Backstepping control for output- constrained nonlinear systems based on nonlinear map- ping,” Neural Computing and Applications , vol. 25, pp. 1665–1674, 2014

    work page 2014

  8. [8]

    Adaptive and safe control of nonlinear systems with multiplicative uncertainty and state constraints,

    J. M. Portella Delgado, “Adaptive and safe control of nonlinear systems with multiplicative uncertainty and state constraints,” Available at https://github.com/estimation-control-learning- PhD thesis, University of Maryland, Baltimore County, 2025

    work page 2025

  9. [9]

    Some extensions of liapunov’s second method,

    J. LaSalle, “Some extensions of liapunov’s second method,” IRE Transactions on circuit theory , vol. 7, no. 4, pp. 520–527, 1960

    work page 1960