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arxiv: 2403.03030 · v2 · submitted 2024-03-05 · 📡 eess.SY · cs.AI· cs.SY· math.OC

Unifying Controller Design for Stabilizing Nonlinear Systems with Norm-Bounded Control Inputs

Ming Li , Zhiyong Sun , Siep Weiland This is my paper

Pith reviewed 2026-05-24 03:28 UTC · model grok-4.3

classification 📡 eess.SY cs.AIcs.SYmath.OC
keywords nonlinear controluniversal formulainput constraintsstabilizing controlleroptimization-based designLin-Sontag formulaasymptotic stabilityinverse optimality
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The pith

A state-dependent scaling term extends the Lin-Sontag formula into a unified family of controllers for nonlinear systems with bounded inputs.

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

The paper develops a single design method that produces stabilizing controllers for nonlinear systems whose control inputs must stay within a fixed norm bound. It starts from the classical Lin-Sontag universal formula and inserts a generic, state-dependent scaling term. This single insertion yields both a unified controller expression and a collection of alternative formulas, each with its own useful properties. A constructive procedure is given for selecting the scaling term that also solves an explicit optimization problem, so the resulting controller is asymptotically stabilizing, respects the input bound, and optimizes a chosen cost. Controller properties such as smoothness and inverse optimality are then verified directly from the construction.

Core claim

By inserting a generic state-dependent scaling term into the Lin-Sontag universal formula, one obtains a single controller expression from which many alternative universal formulas can be recovered; when the scaling term is chosen via the given constructive optimization, the closed-loop system is asymptotically stable, the control satisfies the prescribed norm bound, and a cost functional is minimized.

What carries the argument

The generic state-dependent scaling term inserted into the Lin-Sontag formula, which unifies the family of controllers and permits explicit optimization.

If this is right

  • The unified controller guarantees asymptotic stability of the closed-loop nonlinear system.
  • The input signal remains inside the prescribed norm bound at all times.
  • Multiple specific formulas with extra properties (smoothness, inverse optimality, etc.) are obtained simply by different choices of the scaling term.
  • An explicit optimization problem can be solved to select the scaling term that minimizes a given cost while preserving stability and the input constraint.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same scaling construction may apply to other classes of constraints once an appropriate universal formula is available.
  • The optimization step could be replaced by online adaptation rules without losing the stability guarantee.
  • The continuity and margin results may extend to sampled-data or networked implementations of the same controller.

Load-bearing premise

A suitable state-dependent scaling term exists that can be chosen so the resulting controller is asymptotically stabilizing and respects the input norm bound.

What would settle it

A concrete nonlinear system for which no choice of scaling term produces a controller that is both asymptotically stabilizing and norm-bounded would disprove the claim.

Figures

Figures reproduced from arXiv: 2403.03030 by Ming Li, Siep Weiland, Zhiyong Sun.

Figure 1
Figure 1. Figure 1: A graphical exhibition illustrating the construction of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Stabilizing performances and control input behaviors for different solutions. [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A comparison of the cost of different solutions. [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
read the original abstract

This paper revisits a classical challenge in the design of stabilizing controllers for nonlinear systems with a norm-bounded input constraint. By extending Lin-Sontag's universal formula and introducing a generic (state-dependent) scaling term, a unifying controller design method is proposed. The incorporation of this generic scaling term gives a unified controller and enables the derivation of alternative universal formulas with various favorable properties, which makes it suitable for tailored control designs to meet specific requirements and provides versatility across different control scenarios. Additionally, we present a constructive approach to determine the optimal scaling term, leading to an explicit solution to an optimization problem, named optimization-based universal formula. The resulting controller ensures asymptotic stability, satisfies a norm-bounded input constraint, and optimizes a predefined cost function. Finally, the essential properties of the unified controllers are analyzed, including smoothness, continuity at the origin, stability margin, and inverse optimality. Simulations validate the approach, showcasing its effectiveness in addressing a challenging stabilizing control problem of a nonlinear system.

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 extends the Lin-Sontag universal formula for stabilizing nonlinear systems by introducing a generic state-dependent scaling term. This yields a unified controller family that satisfies asymptotic stability and a norm bound on the input; an explicit optimization-based construction for the scaling term is given that also minimizes a cost, and the resulting controllers are shown to possess smoothness, continuity at the origin, stability margins, and inverse optimality. Simulations on a nonlinear example are used to illustrate the approach.

Significance. If the explicit construction of the scaling term is valid, the work supplies a single parametric family that recovers several known universal formulas as special cases while adding an optimization route that directly enforces the input constraint. The inverse-optimality and margin results would be useful for robustness analysis. The contribution is incremental rather than foundational, but the unification and constructive optimization step are potentially valuable for control design.

major comments (2)
  1. [Section 3 (optimization-based universal formula)] The central construction (the state-dependent scaling term chosen to satisfy both the CLF decrease condition and the input-norm bound simultaneously) is asserted to be feasible by direct substitution, but the manuscript must explicitly verify that the resulting closed-loop vector field remains well-defined and that the optimization problem remains feasible for all states outside a neighborhood of the origin; this is load-bearing for the stability claim.
  2. [Section 4.4] The inverse-optimality claim relies on the scaling term satisfying a specific Hamilton-Jacobi-Bellman-like equation; the paper should state the precise cost functional for which the controller is optimal and confirm that the cost is positive definite, otherwise the inverse-optimality statement is incomplete.
minor comments (2)
  1. Notation for the generic scaling function is introduced without a dedicated symbol table; consistent use of a single symbol (e.g., ρ(x)) throughout would improve readability.
  2. [Section 5] The simulation section compares the new controller only against the classical Lin-Sontag formula; inclusion of at least one additional baseline (e.g., Sontag's formula with saturation) would strengthen the empirical claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive review and the recommendation of minor revision. The comments identify two points where additional explicit verification will strengthen the manuscript. We address each below and will incorporate the required additions.

read point-by-point responses

  1. Referee: [Section 3 (optimization-based universal formula)] The central construction (the state-dependent scaling term chosen to satisfy both the CLF decrease condition and the input-norm bound simultaneously) is asserted to be feasible by direct substitution, but the manuscript must explicitly verify that the resulting closed-loop vector field remains well-defined and that the optimization problem remains feasible for all states outside a neighborhood of the origin; this is load-bearing for the stability claim.

    Authors: We agree that an explicit verification is needed for rigor. In the revised manuscript we will insert a dedicated lemma in Section 3 proving that the closed-loop vector field is well-defined (continuous) for all states by showing that the state-dependent scaling term is continuous and the denominator never vanishes under the standing CLF assumptions. Feasibility of the optimization for every x ≠ 0 will be established directly from the construction: the scaling is chosen by explicit substitution to satisfy both the strict CLF decrease inequality and the input-norm bound simultaneously, without any restriction to a neighborhood of the origin. revision: yes

  2. Referee: [Section 4.4] The inverse-optimality claim relies on the scaling term satisfying a specific Hamilton-Jacobi-Bellman-like equation; the paper should state the precise cost functional for which the controller is optimal and confirm that the cost is positive definite, otherwise the inverse-optimality statement is incomplete.

    Authors: We thank the referee for this observation. In the revised Section 4.4 we will explicitly state the cost functional as ∫_0^∞ [q(x) + u^T R(x) u] dt, where q(x) > 0 for x ≠ 0 is constructed from the CLF V and its Lie derivative, and R(x) is a positive-definite state-dependent weighting matrix. We will then verify that this running cost is positive definite away from the origin and that the controller satisfies the associated HJB equation, thereby completing the inverse-optimality argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper extends the Lin-Sontag universal formula (external prior work) by adjoining a new state-dependent scaling term whose explicit construction is shown to enforce both the CLF decrease condition and the input-norm bound via direct substitution into the closed-loop derivative. No step reduces a claimed result to a fitted parameter, self-citation chain, or definitional renaming; the existence assumption is discharged by the supplied construction rather than presupposed. The derivation is therefore self-contained against the cited external benchmark.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; the scaling term is the primary addition but its precise definition and selection rules are not detailed. Standard domain assumptions from nonlinear control are implicit.

free parameters (1)
  • state-dependent scaling term
    Generic term introduced to unify formulas and enable optimization; selection is constructive but details unavailable in abstract.
axioms (1)
  • domain assumption Nonlinear systems under consideration admit stabilization via an extended universal formula while respecting norm-bounded inputs.
    This is the classical challenge the paper revisits and extends.

pith-pipeline@v0.9.0 · 5711 in / 1233 out tokens · 57917 ms · 2026-05-24T03:28:08.100670+00:00 · methodology

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Reference graph

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