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arxiv: 2604.16991 · v1 · submitted 2026-04-18 · 🧮 math.OC · cs.SY· eess.SY

Semi-definite programs for online control of nonlinear systems with stability guarantees

Xiaoyan Dai This is my paper

Pith reviewed 2026-05-10 06:48 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords semidefinite programmingnonlinear controlonline feedbackLyapunov stabilityrecursive feasibilitystate-dependent representationquadratic performance
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The pith

Time-varying semidefinite programs can be solved sequentially to generate stabilizing feedback controllers and Lyapunov certificates for nonlinear systems.

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

This paper develops a method for online control of nonlinear systems by formulating sequences of time-varying semidefinite programs. Using a state-dependent representation, the solutions to these programs at each step provide both a feedback controller that stabilizes the system and a Lyapunov function that certifies stability while meeting quadratic performance criteria. Compact conditions are established to certify that the sequence of programs remains recursively feasible, along with estimates for the region of attraction. The approach is demonstrated on representative nonlinear systems to show its practical flexibility.

Core claim

For nonlinear systems that admit a state-dependent representation, sequences of time-varying semidefinite programs can be solved online such that their optimal solutions jointly yield a stabilizing feedback controller and a Lyapunov certificate that satisfy stability conditions and quadratic performance specifications, with compact conditions that certify recursive feasibility of the SDP sequence and estimates of the region of attraction.

What carries the argument

Sequences of time-varying semidefinite programs formulated from a state-dependent representation of the nonlinear system, whose optimal solutions provide the controller gains and the Lyapunov matrix.

If this is right

  • The resulting feedback controller stabilizes the closed-loop nonlinear system.
  • Recursive feasibility of the SDP sequence is guaranteed by the derived compact conditions.
  • Quadratic performance specifications are satisfied in addition to stability.
  • Estimates of the region of attraction are obtained from the Lyapunov certificate.

Where Pith is reading between the lines

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

  • If the state-dependent representation can be obtained from data or identification, the method could apply to black-box nonlinear systems.
  • The framework might combine with receding-horizon methods to handle input constraints more explicitly.
  • Region of attraction estimates could guide initial condition selection or controller switching in practice.

Load-bearing premise

A state-dependent representation of the nonlinear dynamics exists such that the time-varying SDPs stay feasible and the extracted controller stabilizes the actual system.

What would settle it

Observing that the closed-loop trajectory diverges or violates the Lyapunov decrease condition despite the SDPs being solved and certified feasible at each step.

Figures

Figures reproduced from arXiv: 2604.16991 by Xiaoyan Dai.

Figure 1
Figure 1. Figure 1: The illustration of the monotonically decreasing sequence. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The evolution of the state variables with the initial condition [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Experimental results with x0 = col(0.1596, 1.2489), γ¯ := λM(P (x0)) λm(P (x0)) γ(x0). formance objectives. The core contributions lie in the formula￾tion of SDPs with explicitly imposed time-varying Lyapunov stability conditions and performance specifications, and in the derivation of compact conditions certifying the recursive feasibility of the proposed SDP sequence. The method is generalizable to setti… view at source ↗
Figure 3
Figure 3. Figure 3: Experimental results with x0 = (0.9727, 0.9341, 0.0945)⊤. C. Inverted pendulum Finally, we consider the Euler discretization of an inverted pendulum represented in (10) with B(x) = col(0, T cos x(1) mℓ ), A(x) = h 1 T T g sin x(1) ℓx(1) 1− T µ mℓ2 i , x(1) ̸= 0 where the left-bottom is replaced by T g ℓ if x(1) = 0. Next, let T = 0.1, g = 9.8, m = 1, ℓ = 1, µ = 0.01. The initial condition is taken as rando… view at source ↗
read the original abstract

This paper develops a semidefinite-programming-based method for online feedback control of nonlinear systems using a state-dependent representation. We formulate sequences of time-varying SDPs whose optimal solutions jointly yield a stabilizing feedback controller and a Lyapunov certificate satisfying stability conditions and quadratic performance specifications. We further establish compact conditions certifying recursive feasibility of the resulting SDP sequences and derive estimates of the region of attraction. Numerical examples on representative nonlinear systems illustrate the flexibility and effectiveness of the proposed method.

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 develops an SDP-based online feedback control method for nonlinear systems via a state-dependent representation of the dynamics. Sequences of time-varying SDPs are formulated whose solutions jointly yield a stabilizing state-feedback controller and a Lyapunov certificate satisfying stability and quadratic performance conditions; compact conditions are derived to certify recursive feasibility of the SDP sequence, together with region-of-attraction estimates. Numerical examples on representative nonlinear systems are used to illustrate the approach.

Significance. If the central claims hold, the work would provide a computationally attractive, optimization-based route to online nonlinear control with explicit stability and performance guarantees, extending SDP techniques beyond linear or quasi-LPV settings. The compact recursive-feasibility conditions and region-of-attraction estimates are potentially valuable for real-time implementation. The numerical examples demonstrate flexibility, but the overall significance hinges on whether the state-dependent representation is shown to be exact and the Lyapunov decrease is rigorously transferred to the original vector field.

major comments (2)
  1. [§3] §3 (State-dependent representation and SDP formulation): The central stability guarantee rests on the exact equivalence between the original nonlinear dynamics ẋ = f(x,u) and the state-dependent form ẋ = A(x)x + B(x)u. No general algorithm is supplied for constructing A(x) and B(x) from arbitrary f, nor is a proof given that the LMI decrease condition enforced by the SDP implies V̇(x,K(x)x) < 0 along the true f. This equivalence is load-bearing for the claim that SDP solutions certify closed-loop stability.
  2. [§4] §4 (Recursive feasibility and region of attraction): The compact conditions certifying recursive feasibility of the time-varying SDP sequence are derived under the maintained validity of the state-dependent representation along the closed-loop trajectory. No invariant set or explicit bound is provided to guarantee that future states remain inside the domain where the representation holds exactly, which directly affects whether the recursive-feasibility claim transfers to the original nonlinear system.
minor comments (2)
  1. [§2] Notation for the online versus sequence SDP should be introduced more explicitly in the problem statement to avoid ambiguity when referring to the time-varying formulation.
  2. [§5] The numerical examples would be strengthened by overlaying the estimated region of attraction on the state trajectories or phase portraits.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. We address each major comment below, providing clarifications and indicating where revisions will be made to strengthen the presentation.

read point-by-point responses

  1. Referee: [§3] §3 (State-dependent representation and SDP formulation): The central stability guarantee rests on the exact equivalence between the original nonlinear dynamics ẋ = f(x,u) and the state-dependent form ẋ = A(x)x + B(x)u. No general algorithm is supplied for constructing A(x) and B(x) from arbitrary f, nor is a proof given that the LMI decrease condition enforced by the SDP implies V̇(x,K(x)x) < 0 along the true f. This equivalence is load-bearing for the claim that SDP solutions certify closed-loop stability.

    Authors: The state-dependent representation is exact by construction: for the systems under consideration, A(x) and B(x) are chosen such that A(x)x + B(x)u ≡ f(x,u) holds identically in the relevant domain. This is standard in the literature for input-affine nonlinear systems, where A(x) can be obtained by factoring the state-dependent terms (e.g., for polynomial f via algebraic methods). Since the vector fields are identical, the time derivative of the Lyapunov function V along the closed-loop trajectory is the same whether computed with f or with the state-dependent form. The SDP enforces the LMI condition on the state-dependent representation, which therefore guarantees V̇ < 0 for the original dynamics. We acknowledge that a universal algorithmic procedure for arbitrary f is not provided, as the focus is on the control synthesis assuming the representation is available; we will revise §3 to include an explicit statement on the equivalence and a brief note on construction for common classes of systems. revision: partial

  2. Referee: [§4] §4 (Recursive feasibility and region of attraction): The compact conditions certifying recursive feasibility of the time-varying SDP sequence are derived under the maintained validity of the state-dependent representation along the closed-loop trajectory. No invariant set or explicit bound is provided to guarantee that future states remain inside the domain where the representation holds exactly, which directly affects whether the recursive-feasibility claim transfers to the original nonlinear system.

    Authors: The recursive feasibility conditions and the region-of-attraction (RoA) estimate are indeed predicated on the state-dependent representation remaining valid. The RoA estimate is derived as a sublevel set of the Lyapunov function within the domain of validity of the representation. To ensure the claim transfers rigorously, the estimated RoA can be taken as the intersection with the validity domain, which is invariant under the closed-loop dynamics by construction of the Lyapunov decrease. We will add a paragraph in §4 clarifying this intersection and confirming that the recursive feasibility holds within this invariant set for the original system. revision: yes

Circularity Check

0 steps flagged

No circularity: forward SDP construction from given state-dependent representation

full rationale

The paper presents a constructive method that takes an assumed exact state-dependent representation of the nonlinear dynamics as input and builds time-varying SDPs whose LMI constraints are derived from standard Lyapunov decrease conditions and quadratic performance criteria. Recursive feasibility conditions and region-of-attraction estimates are then proved directly from the SDP solutions under that representation. No step reduces a claimed prediction or certificate to a fitted parameter or self-referential definition; the stability guarantee is conditional on the representation being exact and the SDPs remaining feasible, which is stated as an assumption rather than derived tautologically. No self-citation chains, ansatz smuggling, or renaming of known results appear as load-bearing elements in the derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the method implicitly relies on standard convex-optimization assumptions (feasibility of SDPs, existence of quadratic Lyapunov functions) that are not stated or justified here.

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