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A Lyapunov-like Characterization of Predefined-Time Stability
Esteban Jiménez-Rodríguez, Aldo Jonathan Muñoz-Vázquez, Juan Diego Sánchez-Torres, Michael Defoort, Alexander G. Loukianov
TL;DR
The paper addresses how to certify and design systems whose settling-time bound can be assigned in advance, including under uncertainty. It develops a Lyapunov-like framework, proves equivalence with previous autonomous-system theorems, extends the analysis to predefined-time ultimate boundedness, and applies the results to robust controllers. The framework supports continuous and discontinuous controllers for desired manifolds, with the control scheme validated in simulation.
Problem
Fixed-time stability bounds settling time but does not generally allow its upper bound to be arbitrarily selected through tunable parameters.
Method
The paper develops Lyapunov-like sufficient conditions, extends them to predefined-time ultimate boundedness, and uses the framework to design robust controllers for affine systems.
Results
The framework establishes equivalence among previous Lyapunov-like theorems for predefined-time stability of autonomous systems.
Takeaways & Limitations
Continuous controllers provide predefined-time ultimate boundedness, while discontinuous controllers provide predefined-time stability relative to a desired manifold.
Abstract
from arXiv · showhide
This technical note studies Lyapunov-like conditions to ensure a class of dynamical systems to exhibit predefined-time stability. The origin of a dynamical system is predefined-time stable if it is fixed-time stable and an upper bound of the settling-time function can be arbitrarily chosen a priori through a suitable selection of the system parameters. We show that the studied Lyapunov-like conditions allow to demonstrate equivalence between previous Lyapunov theorems for predefined-time stability for autonomous systems. Moreover, the obtained Lyapunov-like theorem is extended for analyzing the property of predefined-time ultimate boundedness with predefined bound, which is useful when analyzing uncertain dynamical systems. Therefore, the proposed results constitute a general framework for analyzing predefined-time stability, and they also unify a broad class of systems which present the predefined-time stability property. On the other hand, the proposed framework is used to design robust controllers for affine control systems, which induce predefined-time stability (predefined-time ultimate boundedness of the solutions) w.r.t. to some desired manifold. A simulation example is presented to show the behavior of a developed controller, especially regarding the settling time estimation.
I. INTRODUCTION
Finite-time methods may have settling times that depend unboundedly on initial conditions, while fixed-time stability bounds settling time without allowing its bound to be freely assigned. Predefined-time stability addresses this limitation through tunable parameters and Lyapunov-like analysis.
- Motivation: Finite-time algorithms can have settling times that are unbounded functions of the initial conditions.This motivates stronger stability notions for applications with time-response constraints.
- Motivation: Fixed-time stability bounds the settling-time function, improving on finite-time stability.
- Motivation: Fixed-time stability does not generally permit arbitrary selection of the convergence-time bound through system parameters.
- Predefined-time stability: Predefined-time stability allows the settling-time upper bound to be chosen arbitrarily through suitable parameter selection.
- Paper contributions: The paper develops Lyapunov-like sufficient conditions, relates them to previous theorems, extends them to ultimate boundedness, and applies them to controller design and simulation.
A. Notation
The notation section defines the paper’s real-number domains, derivative notation, ball notation, signed powers, and Beta and Gamma function conventions.
- Number sets: R, R+, R≥0, and ¯R+ denote real, positive, nonnegative, and extended nonnegative real numbers, respectively.
- Geometric notation: Br(x) denotes the open radius-r ball centered at x in Rn.
- Signed powers: The notation ⌊x⌉h represents the signed power |x|^h sign(x), with corresponding vector notation defined componentwise through x.
- Derivatives: A dot over x denotes its first derivative with respect to time, while θ′(z) denotes differentiation with respect to z.
- Special functions: The Beta, incomplete Beta, incomplete Gamma, and regularized incomplete Gamma functions are introduced with their stated domains.
B. On predefined-time stability
The paper defines predefined-time stability as fixed-time stability with an arbitrarily assignable settling-time bound, then develops related boundedness concepts and system interpretations.
- B. On predefined-time stability: Parameter-dependent systems can be represented as controlled systems whose feedback law contains the tunable parameters.
- B. On predefined-time stability: Predefined-time stability requires fixed-time stability together with parameter choices that realize any prescribed positive settling-time bound.
- B. On predefined-time stability: A fixed-time stable system may still have a convergence time that cannot be reduced arbitrarily by tuning its parameters.
- B. On predefined-time stability: If a system has no tunable parameters, its origin cannot be predefined-time stable.
- B. On predefined-time stability: Predefined-time ultimate boundedness with predefined bound requires selecting parameters so all solutions enter a chosen radius by a chosen time.
C. Class K1 functions
Class K1 functions provide the scalar transformation framework used in the paper’s Lyapunov analysis, with monotonicity, bounded range, differentiability, and invertibility properties.
- C. Class K1 functions: A class K1 function is continuous, strictly increasing, starts at zero, and approaches one as its argument tends to infinity.
- C. Class K1 functions: Differentiable class K1 functions have a positive continuous derivative and admit an integral representation normalized to one.
- C. Class K1 functions: The derivative and function can be viewed as a probability density and cumulative distribution function of a positive random variable.
- C. Class K1 functions: Every class K1 function is bijective, so its inverse exists.
- C. Class K1 functions: Class K1 functions remain within the class under composition with class K∞ functions, while inverse compositions can belong to class K∞.
- C. Class K1 functions: Examples include exponential, rational, and regularized incomplete-Beta constructions.
III. A LYAPUNOV CHARACTERIZATION OF PREDEFINED-TIME STABILITY
The paper presents a Lyapunov-like theorem for predefined-time stability that unifies previous autonomous-system results and extends the framework to predefined-time ultimate boundedness.
- Lyapunov characterization: Theorem 1 provides sufficient Lyapunov-like conditions under which the origin is predefined-time stable, with a selectable settling-time bound Tc.The conditions use a continuous, positive definite, radially unbounded function V and tunable system parameters.
- Lyapunov characterization: The proof uses a comparison function to bound V along trajectories and establish convergence to the origin within the prescribed time.The settling-time bound satisfies sup T(x0) ≤ Tc, and equality holds when the differential inequality is an equality.
- Unification of prior results: All previous Lyapunov-like theorems for predefined-time stability of autonomous systems are shown to be equivalent particular forms of the proposed theorem.Different choices of κ and theorem parameters recover earlier results, including the cited theorems in [15]–[17].
- Ultimate boundedness: The framework extends Lyapunov analysis to predefined-time ultimate boundedness with a predefined bound, including systems without an equilibrium at the origin.When V(x)=α(||x||), the ultimate bound equals the selected parameter µ.
- Ultimate boundedness: The ultimate-boundedness proof shows that trajectories enter a positively invariant sublevel set within at most Tc time units and remain within the resulting state bound.For general V, the bound is expressed through α1^-1(α2(µ)); when V(x)=α(||x||), the bound is µ.
A. Problem statement
The problem statement considers affine systems with disturbances and seeks feedback laws that drive trajectories to a desired manifold, or its vicinity, within an arbitrarily selected time and keep them there.
- Problem statement: The disturbance vector includes plant-parameter variations and external unknown perturbations, while the input matrix B(x) is continuous with rank m.These assumptions define the uncertain affine control setting.
- Problem statement: The control objective is to make trajectories of the affine system reach the desired manifold, or a vicinity of it, in an arbitrarily selected time Tc and remain there.The manifold is represented by the smooth mapping s(x,t), which expresses equality-constraint error.
- Problem statement: The derivative of the manifold variable combines nominal dynamics, control action, disturbance effects, and explicit time variation.The control design assumes rank[G(x,t)B(x)] = m so the input can be selected through a virtual control u.
- Uncertainty and robustness: The lumped perturbation Δ(x,t) is assumed globally bounded by a known constant δ.No smoothness, Lipschitz-continuity, or continuity conditions are imposed on this perturbation.
- Controller objectives: The design seeks either predefined-time stability with a discontinuous controller or predefined-time ultimate boundedness with a continuous controller.The continuous alternative sacrifices exact convergence to the manifold for convergence to a prescribed vicinity.
B. Proposed solution
The proposed controller framework establishes predefined-time ultimate boundedness with a tunable bound and reduces this to predefined-time stability when the bound parameter is zero. It is applied to trajectory tracking, with simulations illustrating continuous and discontinuous controller cases.
- Proposed controller: The closed-loop trajectories are predefined-time ultimately bounded with predefined bound b = δρ4 for all t ≥ Tc = ρ1.The parameters satisfy κ ∈ K1, ρ1 > 0, 0 ≤ ρ2 < 1, ρ3 > δ, and ρ4 ≥ 0.
- Proposed controller: Setting ρ4 = 0 gives µ = 0, so the origin s = 0 becomes predefined-time stable.
- Proposed controller: The predefined time Tc = ρ1 and predefined bound b = δρ4 can be selected independently because they depend on different parameters.
- Tracking example: For a planar point tracking a reference trajectory with unknown but bounded derivative, the feedback signal is designed using the controller from Corollary 1.The tracking error is s(x,t) = x − r(t), and the reference derivative satisfies sup_t ||ṙ(t)|| ≤ δ.
- Tracking example: The simulations compare continuous control reaching a 0.01-vicinity of the origin with discontinuous control reaching the origin.The figures show state and reference trajectories, tracking-error norms, and planar point trajectories for both cases.
V. CONCLUSION
The framework supports robust controller design for uncertain affine systems, with continuous and discontinuous controllers providing distinct predefined-time properties. Numerical simulation validates the proposed control scheme, while further research is needed for particular nonlinear-system classes.
- The framework designs robust controllers for uncertain affine control systems.
- The simulation includes trajectories of x1, x2, and ||e|| over time for the discontinuous controller.
- Continuous controllers provide predefined-time ultimate boundedness, whereas discontinuous controllers provide predefined-time stability to a desired manifold.
- The theoretical findings were validated through a numerical simulation demonstrating the effectiveness of the proposed control scheme.
- Further research is required to exploit the Lyapunov-like conditions for controller design in particular classes of nonlinear systems.