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Enhancing the settling time estimation of a class of fixed-time stable systems
R. Aldana-López, D. Gómez-Gutiérrez, E. Jiménez-Rodríguez, J. D. Sánchez-Torres, M. Defoort
TL;DR
The paper addresses conservative settling-time estimates for fixed-time stable systems, where convergence-time guarantees matter in controlled-system applications. It computes the least upper bound, introduces a modified predefined-time algorithm with that bound set a priori, and develops first- and second-order controllers.
Problem
Settling-time bounds for a widely used class of fixed-time stable systems can be overly conservative, despite convergence-time guarantees being important in control applications.
Method
The paper computes the least upper bound of the settling-time function using geometric conditions, then modifies the system so the prescribed bound Tc is directly a system parameter.
Results
The actual settling-time supremum is Tc, while the classical upper estimate can become unbounded as ϱ approaches zero or infinity; new first- and second-order predefined-time controllers are also introduced.
Takeaways & Limitations
The proposed modification yields strong predefined-time stability with the lowest upper estimate set in advance, improving settling-time estimation for the introduced controllers.
Takeaways & Limitations
The considered nonlinear system assumes constant parameters, continuous dynamics, and an equilibrium at the origin; the controller setting also imposes kp < 1 and kq > 1.
Abstract
from arXiv · showhide
This paper deals with the convergence time analysis of a class of fixed-time stable systems with the aim to provide a new non-conservative upper bound for its settling time. Our contribution is fourfold. First, we revisit the well-known class of fixed-time stable systems, given in (Polyakov et al.,2012}, while showing the conservatism of the classical upper estimate of the settling time. Second, we provide the smallest constant that uniformly upper bounds the settling time of any trajectory of the system under consideration. Third, introducing a slight modification of the previous class of fixed-time systems, we propose a new predefined-time convergent algorithm where the least upper bound of the settling time is set a priori as a parameter of the system. At last, predefined-time controllers for first order and second order systems are introduced. Some simulation results highlight the performance of the proposed scheme in terms of settling time estimation compared to existing methods.
1. Introduction
The paper addresses conservative settling-time estimates for widely used fixed-time systems and develops a predefined-time modification whose least upper bound is set in advance. It also derives controllers for first- and second-order systems.
- Motivation: Convergence time matters in applications requiring observers or controllers to stabilize before a subsequent switching event.Examples include missile guidance, formation flying, consensus, differentiators, and state observers.
- Motivation: Fixed-time stability provides a settling-time bound independent of the initial state, relaxing restrictions caused by finite-time bounds that lack uniformity.The fixed-time concept extends global finite-time stability by requiring the settling-time function to be globally uniformly bounded.
- Problem: The classical upper bound can significantly overestimate the least upper bound, leading to over-tuned gains and poorer control magnitude or noise robustness.The paper identifies overestimation as a practical obstacle for prescribed-time observer and controller implementation.
- Contributions: The paper computes the least upper bound of the settling-time function for a widely used fixed-time system class.The class uses positive parameters α, β, p, q, and k subject to kp < 1 and kq > 1.
- Contributions: A modified system introduces a gain γ/Tc so that Tc is set a priori as the least upper bound of convergence time.This defines strong predefined-time stability for the modified system.
- Results: For the modified system, the actual settling-time supremum remains Tc while the classical estimate can diverge as ϱ → +∞ or ϱ → 0.The paper uses this parameter variation to demonstrate conservatism in the conventional estimate.
- Applications: New predefined-time controllers are introduced for first-order and second-order scalar systems with matched bounded perturbations.These controllers are intended to enhance settling-time estimation.
2. Preliminaries and Definitions
This section defines finite-time, fixed-time, and predefined-time stability and states the scalar-system characterization used to analyze settling times. It distinguishes a predefined time from a strong predefined time.
- System framework: The system framework assumes a nonlinear continuous vector field, constant parameters, and an equilibrium at the origin.The state is x ∈ R^n, parameters are ρ, and f(0; ρ) = 0.
- Finite-time stability: Finite-time stability requires Lyapunov stability and finite-time arrival at the origin for every initial state.The settling-time function is the infimum of times after which the solution remains at zero.
- Finite-time stability: For scalar systems, the paper invokes an if-and-only-if characterization of global finite-time stability based on conditions imposed for every nonzero state.The subsequent settling-time expression is obtained using the scalar trajectory as a change of variables.
- Fixed-time stability: Fixed-time stability adds a uniform bound on the settling-time function over all initial states.Multiple upper bounds may exist, but their least upper bound is sup_x0∈R^n T(x0).
- Predefined-time stability: Predefined-time stability selects a positive settling-time bound Tc as a function of system parameters.The strong form requires sup_x0∈R^n T(x0) = Tc, making Tc the least upper bound.
- Predefined-time stability: The paper notes that fixed-time stability may follow from bi-limit homogeneity even when an explicit settling-time bound is unavailable.Predefined-time stability is introduced to distinguish cases where such a bound is set in advance.
3. On the least upper bound for the settling time of a class of fixed-time stable systems
The paper derives the least upper bound of the settling-time function for a fixed-time stable system and uses it to characterize predefined-time stability. The analysis also demonstrates that a classical estimate can substantially overestimate settling time.
- Least upper bound: Theorem 1 establishes that the system is fixed-time stable and that the supremum of its settling-time function equals γ.The result follows from the settling-time expression and the finiteness of γ under the stated parameter conditions.
- Predefined-time characterization: A Lyapunov condition using the parameters α, β, p, q, k and γ characterizes predefined-time stability with predefined time Tc.If the relevant inequality is an equality, Tc is the strong predefined time and equals the least upper bound of settling times.
- Predefined-time characterization: For the modified system (3), the Lyapunov candidate V(x) = |x| verifies predefined-time stability with strong predefined time Tc.The example applies the characterization directly to system (3).
- Numerical illustration: With α = 4, β = 1, Tc = 1, p = 0.5, q = 3, and k = 1.5, simulations give supx0∈R T(x0) = Tc = 1.The trajectories are evaluated for several initial conditions.
- Numerical illustration: For the same parameters, the classical upper estimate is Tmax(4) = 4.4331s, whereas the least upper estimate is Tc = 1s.The comparison illustrates the overestimation of the conventional bound.
- Bound comparison: The classical bound can become unbounded as the parameter ϱ increases or approaches zero, while its best estimate is attained at arg minϱ>0 Tmax(ϱ) = 1.The least upper bound of the settling time remains Tc when the other parameters are fixed.
4. Application to robust predefined-time stabilization for first and second order systems
The paper applies the predefined-time result to robust first- and second-order controllers under bounded perturbations. The designs guarantee convergence within prescribed times, with simulations comparing the proposed estimates against the classical bound.
- 4.1. First-order predefined-time controllers: The first-order objective is predefined-time stabilization despite an unknown but bounded perturbation.The perturbation satisfies |∆(t)| ≤ δ, with δ known.
- 4.1. First-order predefined-time controllers: The first-order controller guarantees predefined-time stability with predefined time Tc when ζ ≥ δ and the system parameters satisfy kp < 1 and kq > 1.The result follows from the predefined-time stability theorem.
- 4.1. First-order predefined-time controllers: For p = 0.5, q = 3, k = 1.5, and α = 1/β = ϱ = 4, the proposed bound is Tc = 1s versus Tmax(ϱ)|ϱ=4 = 4.4331s from.The example uses disturbance ∆(t) = sin(2πt/5).
- 4.2. Second-order predefined-time controllers: The second-order design targets predefined-time stabilization of both state variables despite an unknown but bounded perturbation.The design uses a sliding variable and parameters satisfying the stated positivity and gain constraints.
- 4.2. Second-order predefined-time controllers: The second-order controller guarantees predefined-time stability with Tc = Tc1 + Tc2.The proof first establishes predefined-time stability for the sliding variable, then for the reduced-order dynamics.
- 4.2. Second-order predefined-time controllers: With Tc1 = Tc2 = 0.5s, the second-order example guarantees predefined-time stability with Tc = 1s under the considered perturbation.The example applies the controller to a perturbed second-order system.
5. Conclusion
The paper concludes that the classical settling-time estimate can be conservative and proposes a modified algorithm whose least upper estimate is prescribed in advance.
- 5. Conclusion: The classical upper-bound condition can be too conservative, with the estimate tending to infinity while the actual settling time remains bounded by Tc.The paper uses this discrepancy to motivate a new settling-time estimate.
- 5. Conclusion: The modified fixed-time algorithm is strongly predefined-time with a lowest upper settling-time estimate set in advance as Tc.The Lyapunov inequality is modified so equality makes Tc the lowest upper estimate.
- 5. Conclusion: The resulting framework is applied to predefined-time controllers for first- and second-order systems.The conclusion identifies these controllers as the final application of the proposed result.
Appendix A. Auxiliary Results
The appendix introduces the Beta function and develops an integral expression used in an auxiliary proposition under the system-parameter constraints.
- Appendix A. Auxiliary Results: The Beta function B(a, b) is introduced for positive real numbers a and b.It is used in the auxiliary analysis.
- Appendix A. Auxiliary Results: Proposition 1 considers positive parameters α, β, p, q, and k satisfying pk < 1 and qk > 1.The proposition states an auxiliary result under these constraints.
- Appendix A. Auxiliary Results: The proof rewrites the left-hand side of the auxiliary inequality before applying a variable change.The transformed integral is then expressed using the Beta function definition.