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arxiv: 2604.25007 · v1 · submitted 2026-04-27 · 🧮 math.OC

A Constraint-Lifting Framework for Safe and Stable Nonlinear Control

Jhon Manuel Portella Delgado , Ankit Goel This is my paper

Pith reviewed 2026-05-08 02:33 UTC · model grok-4.3

classification 🧮 math.OC
keywords nonlinear controlsafety constraintsdiffeomorphic mappingLyapunov stabilitysigmoid functionsforward invarianceexplicit control law
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The pith

Sigmoid mapping lifts safety constraints into unbounded space to yield explicit stabilizing controllers 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 to create explicit controllers for nonlinear systems that drive the state to a desired equilibrium while keeping it inside a user-specified safe region at all times. Standard approaches such as control barrier functions or model predictive control typically require solving optimization problems repeatedly during operation. The new framework instead applies a smooth sigmoid-based change of coordinates that stretches the safe region into an infinite domain, designs the controller there using special integral Lyapunov functions, and maps the result back to the original coordinates. Stability and safety are then proved using the Barbashin-Krasovskii-LaSalle invariance principle, with the approach illustrated on an attitude-control example.

Core claim

The central claim is that a sigmoid-based diffeomorphic mapping transforms the constrained state space into an unbounded domain. In the new coordinates a controller is synthesized with sigmoid integral functions serving as Lyapunov candidates, then mapped back to produce an explicit law that asymptotically stabilizes the equilibrium and renders the original safe set forward invariant. The proof establishes both properties for the considered class of nonlinear systems.

What carries the argument

The sigmoid-based diffeomorphic mapping that converts the constrained safe set into an unbounded domain, paired with sigmoid integral functions as Lyapunov candidates in the transformed coordinates.

If this is right

  • The resulting control law is explicit and requires no online optimization.
  • Asymptotic convergence to the equilibrium occurs while the safe set remains forward invariant.
  • Numerical conditioning near the boundary is handled by the choice of Lyapunov functions.
  • The guarantees hold for the class of systems where the mapping and Lyapunov candidates are valid.
  • The method is demonstrated on a safe attitude-control problem.

Where Pith is reading between the lines

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

  • The same lifting idea could be tested on systems with model uncertainty to check whether safety remains enforced under disturbances.
  • Implementation on embedded hardware might become simpler because no solver is needed at runtime.
  • Extending the mapping to time-varying safe sets would require checking whether the diffeomorphism and invariance proof still hold.

Load-bearing premise

The nonlinear system must admit the sigmoid diffeomorphism as a valid global coordinate change and the sigmoid integral functions must function as valid Lyapunov candidates without introducing new instabilities.

What would settle it

A simulation or experiment in which the closed-loop trajectory under the derived control law exits the prescribed safe set or fails to converge to the equilibrium would disprove the claimed guarantees.

Figures

Figures reproduced from arXiv: 2604.25007 by Ankit Goel, Jhon Manuel Portella Delgado.

Figure 1
Figure 1. Figure 1: Enforcing forward invariance via the constraint lifting framework. has a relative degree greater than one with respect to the control input. In this case, existing approaches typically enforce safety through auxiliary constructions that impose additional constraints on higher-order states, yielding a forward invariant set that is a subset of the original safe set. This observation motivates the following q… view at source ↗
Figure 2
Figure 2. Figure 2: Forward invariant set for the double integrator under a higher-order barrier construction. The admissible set is C0 ∩ C1, which may be a strict subset of C0. forward invariant set is defined by the intersection of mul￾tiple constraint sets, each corresponding to a higher-order derivative condition. This intersection imposes additional restrictions on higher-order states, such as velocities, even when the o… view at source ↗
Figure 3
Figure 3. Figure 3: Various sigmoid functions, their invereses, and the corre￾sponding sigmoid integral function. Note that the legend includes only the ϕ(x) functions, while the associated ψ(z) and V(ζ) functions are shown using the same color scheme to maintain visual consistency. Remark 3: For the controller synthesis in Section III and the example in Section V, the same constraint-lifting function is used for both state v… view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of the constraint-lifting concept, where the constrained dynamics in the original state space (x1, x2) are trans￾formed into equivalent unconstrained dynamics in the lifted state space (z1, z2). x1d. Finally, an additional transformation from (z1, z2) to (ζ1, ζ2) is introduced to simplify the notation used in mapping z back to x as well as the stability analysis. The state transformations from… view at source ↗
Figure 5
Figure 5. Figure 5: State transformations with the constraint-lifting function and its inverse used in the control synthesis. Define χ1 △ = DI(x1)x1, (13) χ2 △ = DI(x2)x2. (14) Note that the state variables χ1 and χ2 are defined to map the constraint boundaries to ±1, that is, when the constraints are satisfied, then, χ1, χ2 ∈ C. Next, define z1 △ = D(x1)ϕ(χ1), (15) z2 △ = D(x2)ϕ(χ2), (16) where ϕ: C → R n is a constraint-lif… view at source ↗
Figure 6
Figure 6. Figure 6: The function cosh2(z2). Note the exponential growth in cosh2(z2). For a value of z2 ≈ 10, cosh2(z2) ≈ 107. We emphasize that both controllers theoretically guar￾antee asymptotic stability, as established by the Lyapunov analysis. However, the corresponding closed-loop dynam￾ics exhibit numerical instability, which imposes practical limitations on implementation. This behavior has been consistently observed… view at source ↗
Figure 8
Figure 8. Figure 8: presents a geometric illustration of the closed￾loop angular velocity trajectories under multiple initial conditions. The yellow-shaded region represents the pre￾scribed safe set, and all trajectories remain confined within this set, demonstrating forward invariance. The initial angular velocities are marked by triangles, and the desired equilibrium (zero angular velocity) is marked at the origin. As expec… view at source ↗
Figure 7
Figure 7. Figure 7: presents a geometric illustration of the closed￾loop Euler angle trajectories under multiple initial condi￾tions and reference commands. The pink-shaded region represents the prescribed safe set, and all trajectories remain confined within this set, demonstrating forward invariance. For each trajectory, the initial attitude is marked by a triangle, and the corresponding desired value is indicated by a squa… view at source ↗
Figure 9
Figure 9. Figure 9: Closed-loop responses of the attitude dynamics: angles (ϕ, θ, ψ) and angular velocities (ω1, ω2, ω3) over time for ran￾domly initialized and commanded trajectories, and control inputs (τ1, τ2, τ3) generated by the control law (38). Note that the angle and angular responses are always contained within the desired safe set S. constructed using strictly increasing sigmoid functions. A class of Lyapunov candid… view at source ↗
read the original abstract

This paper presents a constraint-lifting control framework for designing stabilizing controllers that guarantee the forward invariance of a prescribed safe set. State-of-the-art safety-enforcing methods, such as control barrier functions (CBFs) and model predictive control (MPC), typically rely on solving constrained optimization problems in real time and therefore may not yield an explicit control law that guarantees constraint satisfaction under all conditions. In contrast, the proposed approach develops an explicit control law for a class of nonlinear systems that ensures both asymptotic stabilization of a desired equilibrium and safety preservation of a user-defined set. The central idea is to lift the constrained state space into an unbounded domain using a sigmoid-based diffeomorphic mapping, synthesize the controller in the transformed coordinates, and then map it back to the original coordinates. To address numerical conditioning near constraint boundaries, a special class of Lyapunov candidate functions, called sigmoid integral functions, is introduced. A rigorous stability analysis, based on the Barbashi-Krasovskii-LaSalle invariance principle, establishes asymptotic convergence and safety guarantees. The efficacy of the proposed controller is demonstrated through a safe attitude-control problem.

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 a constraint-lifting framework for nonlinear control that maps a user-defined safe set to an unbounded domain via a sigmoid-based diffeomorphism, synthesizes an explicit stabilizing controller in the lifted coordinates, and pulls the law back to the original coordinates. It introduces sigmoid integral functions as Lyapunov candidates to mitigate numerical ill-conditioning near boundaries and invokes the Barbashin-Krasovskii-LaSalle invariance principle to prove asymptotic stability to a desired equilibrium while guaranteeing forward invariance of the safe set. The approach is illustrated on a safe attitude-control example.

Significance. If the technical conditions on the diffeomorphism and Lyapunov functions hold for the stated class, the framework supplies an explicit control law that simultaneously enforces safety and asymptotic stability without online optimization, offering a computationally lightweight alternative to CBF or MPC methods. The sigmoid-integral construction addresses a practical numerical issue that arises in many coordinate-lifting schemes.

major comments (2)
  1. [Introduction and the section defining the sigmoid diffeomorphism] The abstract and introduction refer to results holding for 'a class of nonlinear systems,' yet the precise membership conditions (e.g., the safe set being star-shaped or diffeomorphic to an open ball so that the sigmoid map is a global C^1-diffeomorphism onto R^n) are not stated. Without these conditions or an explicit verification that the mapping and its inverse are C^1 and that no additional invariant sets are created by the closed-loop dynamics in lifted coordinates, the applicability of the Barbashin-Krasovskii-LaSalle principle and the safety guarantee cannot be confirmed. This is load-bearing for the central claim.
  2. [The section on sigmoid integral functions and stability analysis] The claim that the introduced sigmoid integral functions are positive-definite, proper Lyapunov candidates whose derivatives are non-positive with zero set containing only the target equilibrium rests on the validity of the coordinate change and the closed-loop vector field. The abstract invokes the invariance principle but does not provide the explicit computation of the derivative or the characterization of the largest invariant set; if these steps are omitted or rely on unstated assumptions about the safe-set geometry, the stability proof is incomplete.
minor comments (2)
  1. [Abstract] The abstract contains a typographical error: 'Barbashi-Krasovskii-LaSalle' should read 'Barbashin-Krasovskii-LaSalle'.
  2. [Preliminaries] Notation for the sigmoid mapping and the integral functions should be introduced with explicit domain and range statements to avoid ambiguity when the safe set is not the unit ball.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive review. The comments correctly identify areas where additional clarity on assumptions and explicit proof details would strengthen the manuscript. We address each major comment below and will incorporate the indicated revisions.

read point-by-point responses

  1. Referee: [Introduction and the section defining the sigmoid diffeomorphism] The abstract and introduction refer to results holding for 'a class of nonlinear systems,' yet the precise membership conditions (e.g., the safe set being star-shaped or diffeomorphic to an open ball so that the sigmoid map is a global C^1-diffeomorphism onto R^n) are not stated. Without these conditions or an explicit verification that the mapping and its inverse are C^1 and that no additional invariant sets are created by the closed-loop dynamics in lifted coordinates, the applicability of the Barbashin-Krasovskii-LaSalle principle and the safety guarantee cannot be confirmed. This is load-bearing for the central claim.

    Authors: We agree that the precise membership conditions for the class of systems and safe sets must be stated explicitly to support the central claims. In the revised manuscript we will insert a new paragraph immediately after the problem formulation in the introduction and expand the sigmoid-diffeomorphism section to specify that the safe set is a bounded, open, star-shaped domain (with respect to the origin) that is C^1-diffeomorphic to the open unit ball. Under this geometry the proposed sigmoid map and its inverse are globally C^1, and we will add a short lemma verifying that the closed-loop vector field in lifted coordinates introduces no additional invariant sets beyond the origin. These additions will directly justify the invocation of the Barbashin-Krasovskii-LaSalle principle and the forward-invariance guarantee. revision: yes

  2. Referee: [The section on sigmoid integral functions and stability analysis] The claim that the introduced sigmoid integral functions are positive-definite, proper Lyapunov candidates whose derivatives are non-positive with zero set containing only the target equilibrium rests on the validity of the coordinate change and the closed-loop vector field. The abstract invokes the invariance principle but does not provide the explicit computation of the derivative or the characterization of the largest invariant set; if these steps are omitted or rely on unstated assumptions about the safe-set geometry, the stability proof is incomplete.

    Authors: The stability-analysis section already derives the Lie derivative of the sigmoid-integral Lyapunov function along the closed-loop dynamics and characterizes the largest invariant set on which the derivative vanishes as containing only the target equilibrium. Nevertheless, we acknowledge that these steps can be presented with greater explicitness. In the revision we will (i) write out the full expression for the derivative, (ii) detail the application of the invariance principle step by step, and (iii) explicitly connect the star-shaped geometry assumption to the positive-definiteness, properness, and the zero set of the derivative. This will render the proof self-contained without altering its substance. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard tools

full rationale

The paper introduces a sigmoid-based diffeomorphic mapping to lift the safe set and proposes sigmoid integral functions as Lyapunov candidates, then applies the standard Barbashin-Krasovskii-LaSalle invariance principle in the transformed coordinates to conclude asymptotic stability and forward invariance. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or renamed known result; the central claims are conditional on the mapping being a global C1-diffeomorphism and the integral functions being valid Lyapunov candidates for the stated class of systems, which are presented as modeling assumptions rather than derived from the outputs. The framework is self-contained against external mathematical benchmarks (diffeomorphisms, Lyapunov theory, invariance principle) with no visible self-referential loops in the provided derivation outline.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The framework depends on the existence of a global diffeomorphism induced by the sigmoid map and on the validity of the new integral functions as Lyapunov candidates; these are introduced without external verification in the abstract.

axioms (2)
  • domain assumption The sigmoid-based mapping is a diffeomorphism from the constrained state space onto an unbounded domain.
    Invoked to lift the original constrained problem into an unbounded coordinate system where standard stabilization techniques apply.
  • ad hoc to paper Sigmoid integral functions are valid Lyapunov candidates that remain well-conditioned near constraint boundaries.
    New class of functions introduced specifically to handle numerical issues at the boundary of the safe set.
invented entities (1)
  • sigmoid integral functions no independent evidence
    purpose: Lyapunov candidates that address numerical conditioning near constraint boundaries
    Introduced in the abstract to enable the stability proof; no independent evidence of their properties is supplied.

pith-pipeline@v0.9.0 · 5487 in / 1430 out tokens · 46120 ms · 2026-05-08T02:33:54.755272+00:00 · methodology

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Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages

  1. [1]

    The corresponding safe set is C0 △ = {x ∈ R2 : h(x) ≥ 0} = {x ∈ R2 : |x1| ≤ 1}

    Illustrative Double-Integrator Example : To illustrate the conservatism introduced by higher-order barrier con- structions, consider the double integrator ˙x1 = x2, ˙x2 = u, (8) with the state constraint h(x) △ = 1 − x2 1 ≥ 0. The corresponding safe set is C0 △ = {x ∈ R2 : h(x) ≥ 0} = {x ∈ R2 : |x1| ≤ 1}. (9) Since h has relative degree two with respect t...

    work page

  2. [2]

    In particular, we will require the sigmoid function to be strictly increasing in order to ensure the existence of its inverse

    Sigmoid Functions : We revisit the definition and key properties of sigmoid functions, presented in [34]. In particular, we will require the sigmoid function to be strictly increasing in order to ensure the existence of its inverse. Definition 2.1 (Simple sigmoids): A function σ : R → (−1, 1) is a simple sigmoid if it satisfies the following conditions

    work page

  3. [3]

    σ is a smooth function, that is, σ(x) ∈ C ∞

    work page

  4. [4]

    σ is an odd function, that is, σ(−x) = −σ(x)

    work page

  5. [5]

    σ has y = ±1 as horizontal asymptotes, that is, limx→∞ σ(x) = 1

    work page

  6. [6]

    The following result, reproduced from [34], establishes that if the simple sigmoid is strictly increasing, then its inverse exists

    σ(x)/x is a completely convex function in (0, 1). The following result, reproduced from [34], establishes that if the simple sigmoid is strictly increasing, then its inverse exists. Proposition 2.1: Let y = σ(x) be a strictly increasing simple sigmoid, that is, σ′(x) > 0 for all x ∈ R, where σ′ denotes the derivative of σ. Then,

    work page

  7. [7]

    η △ = σ−1 : (−1, 1) → R exists

    work page

  8. [8]

    η(y) is a strictly increasing function, analytic in the interval (−1, 1)

    work page

  9. [9]

    η′(y) = 1 /σ′(η(y)), where η′ and σ′ denote the first derivatives of η and σ, respectively

    work page

  10. [10]

    η(y)/y is absolutely monotone in (0, 1)

    work page

  11. [11]

    Constraint-Lifting Function : Using the strictly increas- ing sigmoid functions presented above, we define the constraint-lifting function, which is used to convert a constrained control problem into an equivalent uncon- strained problem. In particular, the constraint-lifting func- tion ϕ : C → Rn is constructed using the inverse of a strictly increasing ...

    work page

  12. [12]

    The sigmoid integral function will be used to construct candidate Lyapunov functions in Section III

    Sigmoid Integral Function : Using the sigmoid functions presented above, we define the sigmoid integral function. The sigmoid integral function will be used to construct candidate Lyapunov functions in Section III. Definition 2.2: Let σ : R → R be a simple sigmoid function. The sigmoid integral function V : R → [0, ∞) is defined as V(ζ) △ = ∫ ζ 0 σ(s)ds. ...

    work page

  13. [13]

    V(0) = 0 , and for all ζ ∈ R \ {0}, V(ζ) > 0

    work page

  14. [14]

    Note that the sigmoid integral functions V(ζ) exhibit quadratic behavior near ζ = 0 and approach linear behavior for |ζ| ≫ 0

    ∂ζV (ζ) = σ(ζ), and thus the sigmoid integral function is continuous. Note that the sigmoid integral functions V(ζ) exhibit quadratic behavior near ζ = 0 and approach linear behavior for |ζ| ≫ 0. Table II lists the sigmoid integral functions V(ζ) corresponding to each sigmoid function in the third column and Figure 3 shows these sigmoid integral functions...

    work page

  15. [15]

    (30) Differentiating ( 30) and using ( 21) yields ˙V1 = eT 1 G1(z1, z2)

    e1 stabilization: Consider the function V1 △ = 1 2 eT 1 e1. (30) Differentiating ( 30) and using ( 21) yields ˙V1 = eT 1 G1(z1, z2). (31) If z2 is chosen such that G1(z1, z2) = −ke1, where k1 > 0, then ˙V1 < 0. However, since z2 is not the control input, G1(z1, z2) cannot be arbitrarily chosen. Instead, we define e2 △ = G1(z1, z2) − (−k1e1) = Φ( ζ1)ψ(ζ2) ...

    work page

  16. [16]

    As shown in Appendix II, note that dt n∑ i=1 V(ζ2i) = ψ(ζ2)TDI(x2) ˙z2

    e2 stabilization: Next, consider the function V △ = V1 + n∑ i=1 V(ζ2i), (34) where V is a sigmoid integral function given by ( 12). As shown in Appendix II, note that dt n∑ i=1 V(ζ2i) = ψ(ζ2)TDI(x2) ˙z2. (35) Thus, ˙V = −k1eT 1 e1 + eT 2 e1 + ψ(ζ2)TDI(x2)( F2(z1, z2) + G2(z1, z2)u ) . (36) Substituting ψ(ζ2) from ( 32) in the equation above yields ˙V = −k...

    work page

  17. [17]

    Handling constraints in optimal control with saturation functions and system extension,

    K. Graichen, A. Kugi, N. Petit, and F. Chaplais, “Handling constraints in optimal control with saturation functions and system extension,” Systems & Control Letters, vol. 59, no. 11, pp. 671–679, 2010

    work page 2010

  18. [18]

    Reference and command governors for systems with constraints: A survey on theory and applications,

    E. Garone, S. Di Cairano, and I. Kolmanovsky, “Reference and command governors for systems with constraints: A survey on theory and applications,” Automatica, vol. 75, pp. 306–328, 2017

    work page 2017

  19. [19]

    Balanced control between performance and saturation for constrained nonlinear systems,

    P. Wang, H. Wang, X. Zhang, and S. S. Ge, “Balanced control between performance and saturation for constrained nonlinear systems,” International Journal of Robust and Nonlinear Con- trol, vol. 33, no. 10, pp. 5675–5690, 2023

    work page 2023

  20. [20]

    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,” International Journal of robust and nonlinear control, vol. 5, no. 5, pp. 487–504, 1995

    work page 1995

  21. [21]

    Fast reference governors for systems with state and control constraints and disturbance inputs,

    E. G. Gilbert and I. Kolmanovsky, “Fast reference governors for systems with state and control constraints and disturbance inputs,” International Journal of Robust and Nonlinear Control: IF AC-Affiliated Journal, vol. 9, no. 15, pp. 1117–1141, 1999

    work page 1999

  22. [22]

    Backstepping con- trol integrated with lyapunov-based model predictive control,

    Y. Kim, T. H. Oh, T. Park, and J. M. Lee, “Backstepping con- trol integrated with lyapunov-based model predictive control,” Journal of Process Control, vol. 73, pp. 137–146, 2019

    work page 2019

  23. [23]

    Integration of model predictive control and backstepping approach and its stability analysis,

    Y. Kim, T. Park, and J. M. Lee, “Integration of model predictive control and backstepping approach and its stability analysis,” IF AC-PapersOnLine, vol. 51, no. 18, pp. 405–410, 2018

    work page 2018

  24. [24]

    Schlagenhauf, P

    J. Schlagenhauf, P. Hofmeier, T. Bronnenmeyer, R. Paelinck, and M. Diehl, “Cascaded nonlinear mpc for realtime quadrotor position tracking￿￿this research has been funded by the company kiteswarms in the kite project (zvk2017022302) with the university of freiburg. ” IF AC-PapersOnLine, vol. 53, no. 2, pp. 7026–7032, 2020, 21st IF AC World Congress. [Onlin...

    work page 2020

  25. [25]

    Model predictive path following control of a quadrotor in constrained environments,

    D. Wang, C. Zhao, J. Hu, and Q. Pan, “Model predictive path following control of a quadrotor in constrained environments,” in 2020 IEEE 16th International Conference on Control and Automation (ICCA), 2020, pp. 719–724

    work page 2020

  26. [26]

    Adaptive backstepping control of quadrotor uavs with output constraints and input saturation,

    J. Li, L. Wan, J. Li, and K. Hou, “Adaptive backstepping control of quadrotor uavs with output constraints and input saturation,” Applied Sciences, vol. 13, no. 15, p. 8710, 2023

    work page 2023

  27. [27]

    Barrier lyapunov func- tions for the control of output-constrained nonlinear systems,

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

    work page 2009

  28. [28]

    Tuning functions based adaptive backstepping con- trol for uncertain strict-feedback nonlinear systems using bar- rier lyapunov functions with full state constraints,

    Y. Soukkou, M. Tadjine, A. Soukkou, M. Nibouche, and H. Nouri, “Tuning functions based adaptive backstepping con- trol for uncertain strict-feedback nonlinear systems using bar- rier lyapunov functions with full state constraints,” European Journal of Control, vol. 70, p. 100783, 2023

    work page 2023

  29. [29]

    Integrator backstep- ping using barrier functions for systems with multiple state constraints,

    K. B. Ngo, R. Mahony, and Z.-P. Jiang, “Integrator backstep- ping using barrier functions for systems with multiple state constraints,” in Proceedings of the 44th IEEE Conference on Decision and Control. IEEE, 2005, pp. 8306–8312

    work page 2005

  30. [30]

    Adaptive barrier-lyapunov-functions based control scheme of nonlinear pure-feedback systems with full state constraints and asymp- totic tracking performance,

    B. Niu, X. Wang, X. Wang, X. Wang, and T. Li, “Adaptive barrier-lyapunov-functions based control scheme of nonlinear pure-feedback systems with full state constraints and asymp- totic tracking performance,” Journal of Systems Science and Complexity, vol. 37, no. 3, pp. 965–984, 2024

    work page 2024

  31. [31]

    Consensus-based distributed formation control of multi-quadcopter systems: Bar- rier lyapunov function approach,

    N. Sadeghzadeh-Nokhodberiz and N. Meskin, “Consensus-based distributed formation control of multi-quadcopter systems: Bar- rier lyapunov function approach,” IEEE Access, vol. 11, pp. 142 916–142 930, 2023

    work page 2023

  32. [32]

    Barrier-lyapunov- function-based backstepping control for pmsm servo system with full state constraints,

    Z. Yin, B. Wang, C. Du, and Y. Zhang, “Barrier-lyapunov- function-based backstepping control for pmsm servo system with full state constraints,” in 2019 22nd International Con- ference on Electrical Machines and Systems (ICEMS). IEEE, 2019, pp. 1–5

    work page 2019

  33. [33]

    Barrier lyapunov function-based adaptive asymptotic tracking of nonlinear systems with unknown virtual control coefficients,

    Y.-X. Li, “Barrier lyapunov function-based adaptive asymptotic tracking of nonlinear systems with unknown virtual control coefficients,” Automatica, vol. 121, p. 109181, 2020

    work page 2020

  34. [34]

    Control of nonlinear systems with partial state constraints using a barrier lyapunov function,

    K. P. Tee and S. S. Ge, “Control of nonlinear systems with partial state constraints using a barrier lyapunov function,” International Journal of Control, vol. 84, no. 12, pp. 2008–2023, 2011

    work page 2008

  35. [35]

    Stabilizing dynamic controllers for hybrid systems: A hybrid control lyapunov function approach,

    S. Di Cairano, W. M. H. Heemels, M. Lazar, and A. Bemporad, “Stabilizing dynamic controllers for hybrid systems: A hybrid control lyapunov function approach,” IEEE Transactions on Automatic Control, vol. 59, no. 10, pp. 2629–2643, 2014

    work page 2014

  36. [36]

    Adaptive optimized backstepping tracking control for full-state constrained non- linear strict-feedback systems without using barrier lyapunov function method,

    B. Zhu, N. Xu, G. Zong, and X. Zhao, “Adaptive optimized backstepping tracking control for full-state constrained non- linear strict-feedback systems without using barrier lyapunov function method,” Optimal Control Applications and Methods, vol. 45, no. 5, pp. 2051–2075, 2024

    work page 2051

  37. [37]

    Safe backstepping with control barrier functions,

    A. J. Taylor, P. Ong, T. G. Molnar, and A. D. Ames, “Safe backstepping with control barrier functions,” in 2022 IEEE 61st Conference on Decision and Control (CDC). IEEE, 2022, pp. 5775–5782

    work page 2022

  38. [38]

    Safe control synthesis for multicopter via control barrier function backstepping,

    J. Kim and Y. Kim, “Safe control synthesis for multicopter via control barrier function backstepping,” in 2023 62nd IEEE Conference on Decision and Control (CDC), 2023, pp. 8720– 8725

    work page 2023

  39. [39]

    Constant-sum high- order barrier functions for safety between parallel boundaries,

    K. H. Kim, M. Diagne, and M. Krstić, “Constant-sum high- order barrier functions for safety between parallel boundaries,” IEEE Control Systems Letters, vol. 9, pp. 1447–1452, 2025

    work page 2025

  40. [40]

    Stabilization with guaranteed safety using control lyapunov–barrier function,

    M. Z. Romdlony and B. Jayawardhana, “Stabilization with guaranteed safety using control lyapunov–barrier function,” Automatica, vol. 66, pp. 39–47, 2016

    work page 2016

  41. [41]

    Control barrier functions: The- ory and applications,

    A. D. Ames, S. Coogan, M. Egerstedt, G. Notomista, K. Sreenath, and P. Tabuada, “Control barrier functions: The- ory and applications,” in 2019 18th European control conference (ECC). Ieee, 2019, pp. 3420–3431

    work page 2019

  42. [42]

    Control lyapunov-barrier function-based model predictive control of nonlinear systems,

    Z. Wu, F. Albalawi, Z. Zhang, J. Zhang, H. Durand, and P. D. Christofides, “Control lyapunov-barrier function-based model predictive control of nonlinear systems,” Automatica, vol. 109, p. 108508, 2019

    work page 2019

  43. [43]

    Über die lage der integralkurven gewöhnlicher differentialgleichungen,

    M. Nagumo, “Über die lage der integralkurven gewöhnlicher differentialgleichungen,” Nippon Sugaku-Butsurigakkwai Kizi Dai 3 Ki, 1942

    work page 1942

  44. [44]

    Set invariance in control,

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

    work page 1999

  45. [45]

    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 Automatic Control, vol. 62, no. 8, pp. 3861–3876, 2016

    work page 2016

  46. [46]

    Rectified control barrier functions for high-order safety constraints,

    P. Ong, M. H. Cohen, T. G. Molnar, and A. D. Ames, “Rectified control barrier functions for high-order safety constraints,” IEEE Control Systems Letters, 2024

    work page 2024

  47. [47]

    Backstepping control for output- constrained nonlinear systems based on nonlinear mapping,

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

    work page 2014

  48. [48]

    Output feedback control for constrained pure-feedback systems: A non-recursive and transformational observer based approach,

    X. Huang, Y. Song, and C. Wen, “Output feedback control for constrained pure-feedback systems: A non-recursive and transformational observer based approach,” Automatica, vol. 113, p. 108789, 2020

    work page 2020

  49. [49]

    Nonlinear control of the air spindle testbed with constraints,

    M. Sampei, J. Shen, and N. H. McClamroch, “Nonlinear control of the air spindle testbed with constraints,” in Proceedings of the 2003 American Control Conference, 2003., vol. 1. IEEE, 2003, pp. 483–488

    work page 2003

  50. [50]

    Char- acterization of a class of sigmoid functions with applications to neural networks,

    A. Menon, K. Mehrotra, C. K. Mohan, and S. Ranka, “Char- acterization of a class of sigmoid functions with applications to neural networks,” Neural networks, vol. 9, no. 5, pp. 819–835, 1996

    work page 1996

  51. [51]

    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

  52. [52]

    Perspectives on euler angle singularities, gimbal lock, and the orthogonality of applied forces and applied moments,

    E. G. Hemingway and O. M. O’Reilly, “Perspectives on euler angle singularities, gimbal lock, and the orthogonality of applied forces and applied moments,” Multibody system dynamics, vol. 44, no. 1, pp. 31–56, 2018. Jhon Portella received the B.Sc. degree in Mechanical Engineering from the Pontificia Universidad Católica del Perú (PUCP), Lima, Peru, in 201...

    work page 2018