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arxiv: 2509.05191 · v2 · submitted 2025-09-05 · 📡 eess.SY · cs.SY

Feedback Linearisation with State Constraints

Songlin Jin , Yuanbo Nie , Morgan Jones This is my paper

Pith reviewed 2026-05-18 18:41 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords feedback linearisationstate constraintsrelative degreeswitching controllernonlinear controlsystem augmentation
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The pith

Augmenting dynamics before feedback linearisation keeps state constraints simple in nonlinear control.

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

The paper introduces a method to augment the dynamics of nonlinear control systems to embed state constraints before applying feedback linearisation. This prevents the state constraints from transforming into complicated nonlinear constraints in the linearised system. The augmentation creates ill-defined relative degrees at the constraint boundaries, but these are addressed using a switching feedback linearisation controller. Experiments demonstrate that this approach allows effective constraint handling while retaining the benefits of linearisation for controller design.

Core claim

By augmenting the system dynamics to capture state constraints prior to feedback linearisation, the method ensures that constraints stay simple in the original nonlinear system, with ill-defined relative degrees at boundaries overcome by a switching controller.

What carries the argument

Dynamics augmentation to embed state constraints, combined with a switching FBL controller to handle ill-defined relative degrees at boundaries.

If this is right

  • State constraints remain straightforward rather than becoming complex in the linearised system.
  • The switching controller maintains stability and performance when relative degrees change at boundaries.
  • The approach extends the applicability of feedback linearisation to systems with state constraints.

Where Pith is reading between the lines

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

  • This method might apply to other linearisation techniques facing similar constraint issues.
  • Implementation on physical systems could reveal practical challenges like switching transients.
  • Further analysis could determine conditions for smooth switching without performance loss.

Load-bearing premise

That the augmentation successfully incorporates the state constraints and that a switching controller can be designed to handle the resulting ill-defined relative degrees while preserving system stability.

What would settle it

An experiment showing that the augmented system violates the intended state constraints or that the switching controller leads to instability at the boundaries.

Figures

Figures reproduced from arXiv: 2509.05191 by Morgan Jones, Songlin Jin, Yuanbo Nie.

Figure 1
Figure 1. Figure 1: Flowchart for closed-form controller synthesis a [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Figures associated with Examples 1-2 showing the p [PITH_FULL_IMAGE:figures/full_fig_p029_2.png] view at source ↗
read the original abstract

Feedback Linearisation (FBL) is a widely used technique that applies feedback laws to transform input-affine nonlinear control systems into linear control systems, allowing for the use of linear controller design methods such as pole placement. However, for problems with state constraints, controlling the linear system induced by FBL can be more challenging than controlling the original system. This is because simple state constraints in the original nonlinear system become complex nonlinear constraints in the FBL induced linearised system, thereby diminishing the advantages of linearisation. To avoid increasing the complexity of state constraints under FBL, this paper introduces a method to first augment system dynamics to capture state constraints before applying FBL. We show that our proposed augmentation method leads to ill-defined relative degrees at state constraint boundaries. However, we show that ill-defined relative degrees can be overcome by using a switching FBL controller. Numerical experiments illustrate the capabilities of this method for handling state constraints within the FBL framework.

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 manuscript proposes augmenting the dynamics of input-affine nonlinear systems to embed state constraints before feedback linearisation (FBL). This produces ill-defined relative degrees exactly at the constraint boundaries. The authors assert that a switching FBL controller resolves the issue, with numerical experiments illustrating performance on constrained systems.

Significance. If the switched closed-loop system can be shown to remain stable and constraint-compliant, the approach would preserve the advantages of FBL while avoiding the conversion of simple state constraints into complex nonlinear ones in the linearised coordinates. The structural augmentation idea is a clear contribution, and the identification of the relative-degree issue at boundaries is correctly diagnosed. Numerical experiments provide preliminary evidence of feasibility.

major comments (2)
  1. [Switching controller section] Switching controller section: no common Lyapunov function, average dwell-time condition, or vector-field matching condition across the switching surface is derived or referenced. This is load-bearing because, without such analysis, trajectories may exit the feasible set or the linearising feedback may become undefined when the augmented state reaches the boundary with nonzero velocity.
  2. [Augmentation and FBL application] Augmentation step prior to FBL: the explicit form of the augmented vector fields and the precise calculation showing ill-defined relative degree at the boundary are not supplied. This step is central to the claim that augmentation successfully embeds constraints while leaving only the relative-degree issue to be solved by switching.
minor comments (2)
  1. [Numerical experiments] The description of how the switching surface is detected and the control law is blended (or not) during transitions lacks implementation-level detail.
  2. [Method] Notation for the augmented state vector should be introduced with an explicit equation to avoid confusion with the original state.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and for recognizing the structural augmentation as a clear contribution. We address each major comment below, clarifying the manuscript content where possible and committing to revisions that strengthen the presentation without overstating the current results.

read point-by-point responses

  1. Referee: [Switching controller section] Switching controller section: no common Lyapunov function, average dwell-time condition, or vector-field matching condition across the switching surface is derived or referenced. This is load-bearing because, without such analysis, trajectories may exit the feasible set or the linearising feedback may become undefined when the augmented state reaches the boundary with nonzero velocity.

    Authors: We agree that the absence of an explicit stability analysis for the switched system is a limitation. The manuscript currently relies on numerical experiments to illustrate that the switching controller keeps trajectories inside the feasible set and avoids undefined feedback. In the revision we will add a dedicated subsection that references standard results on switched nonlinear systems, discusses the applicability of average dwell-time conditions, and derives a vector-field matching condition at the switching surface under the assumption that the augmented state approaches the boundary with bounded velocity. This will directly address the concern that trajectories could exit the feasible set. revision: yes

  2. Referee: [Augmentation and FBL application] Augmentation step prior to FBL: the explicit form of the augmented vector fields and the precise calculation showing ill-defined relative degree at the boundary are not supplied. This step is central to the claim that augmentation successfully embeds constraints while leaving only the relative-degree issue to be solved by switching.

    Authors: The augmentation augments the original input-affine dynamics with auxiliary states whose vector fields are constructed so that the constraint boundaries become invariant manifolds. In the revised manuscript we will insert the explicit expressions for the augmented drift vector field f_aug(x, z) and input vector field g_aug(x, z), together with the step-by-step Lie-derivative calculation that shows the relative degree becomes undefined precisely when the augmented state lies on the boundary (the coefficient of the input in the highest-order derivative vanishes). This will make the central claim fully transparent. revision: yes

Circularity Check

0 steps flagged

No circularity: augmentation and switching construction is forward and self-contained.

full rationale

The paper defines an augmentation of the input-affine dynamics to embed state constraints, applies standard FBL to the augmented system, identifies the resulting ill-defined relative degree at the constraint boundary as a direct consequence of the augmentation, and proposes a switching FBL controller to address it. This sequence is a structural construction whose outputs are not algebraically or definitionally equivalent to its inputs; the claims rest on the explicit vector-field modification and the switching logic rather than on any fitted parameter renamed as prediction, self-referential definition, or load-bearing self-citation. Numerical experiments are presented as external validation. No equation or central claim reduces to its own premise by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard assumptions of input-affine nonlinear control theory and the existence of feedback linearization under suitable relative-degree conditions; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Input-affine nonlinear systems admit feedback linearization when relative degree conditions are satisfied.
    Implicitly invoked when FBL is applied after augmentation.

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