A Lyapunov-like Characterization of Predefined-Time Stability

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.Source: §I · p. 1
• Fixed-time stability bounds the settling-time function, improving on finite-time stability.Source: §I · p. 1
• Fixed-time stability does not generally permit arbitrary selection of the convergence-time bound through system parameters.Source: §I · p. 1

Predefined-time stability

• Predefined-time stability allows the settling-time upper bound to be chosen arbitrarily through suitable parameter selection.Source: §I · p. 1

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.Source: §I · p. 1

Number sets

• R, R+, R≥0, and ¯R+ denote real, positive, nonnegative, and extended nonnegative real numbers, respectively.Source: A. Notation · p. 1

Geometric notation

• Br(x) denotes the open radius-r ball centered at x in Rn.Source: A. Notation · p. 1

Signed powers

• The notation ⌊x⌉h represents the signed power |x|^h sign(x), with corresponding vector notation defined componentwise through x.Source: A. Notation · p. 1

Derivatives

• A dot over x denotes its first derivative with respect to time, while θ′(z) denotes differentiation with respect to z.Source: A. Notation · p. 1

Special functions

• The Beta, incomplete Beta, incomplete Gamma, and regularized incomplete Gamma functions are introduced with their stated domains.Source: A. Notation · p. 1

B. On predefined-time stability

• Parameter-dependent systems can be represented as controlled systems whose feedback law contains the tunable parameters.Source: B. On predefined-time stability · p. 2
• Predefined-time stability requires fixed-time stability together with parameter choices that realize any prescribed positive settling-time bound.Source: B. On predefined-time stability · p. 2
• A fixed-time stable system may still have a convergence time that cannot be reduced arbitrarily by tuning its parameters.Source: B. On predefined-time stability · p. 2
• If a system has no tunable parameters, its origin cannot be predefined-time stable.Source: B. On predefined-time stability · p. 2
• Predefined-time ultimate boundedness with predefined bound requires selecting parameters so all solutions enter a chosen radius by a chosen time.Source: B. On predefined-time stability · p. 2

C. Class K1 functions

• A class K1 function is continuous, strictly increasing, starts at zero, and approaches one as its argument tends to infinity.Source: C. Class K1 functions · p. 2
• Differentiable class K1 functions have a positive continuous derivative and admit an integral representation normalized to one.Source: C. Class K1 functions · p. 2
• The derivative and function can be viewed as a probability density and cumulative distribution function of a positive random variable.Source: C. Class K1 functions · p. 2
• Every class K1 function is bijective, so its inverse exists.Source: C. Class K1 functions · p. 2
• Class K1 functions remain within the class under composition with class K∞ functions, while inverse compositions can belong to class K∞.Source: C. Class K1 functions · p. 3
• Examples include exponential, rational, and regularized incomplete-Beta constructions.Source: C. Class K1 functions · p. 3

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.Source: §III · p. 3
• 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.Source: §III · p. 3

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].Source: §III · p. 3

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.Source: §III · pp. 3–4
• 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.Source: §III · p. 4

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.Source: A. Problem statement · p. 4
• 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.Source: A. Problem statement · p. 4
• 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.Source: A. Problem statement · p. 4

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.Source: A. Problem statement · p. 4

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.Source: A. Problem statement · p. 4

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.Source: B. Proposed solution · p. 4
• Setting ρ4 = 0 gives µ = 0, so the origin s = 0 becomes predefined-time stable.Source: B. Proposed solution · pp. 4–5
• The predefined time Tc = ρ1 and predefined bound b = δρ4 can be selected independently because they depend on different parameters.Source: B. Proposed solution · p. 5

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.Source: B. Proposed solution · pp. 4–5
• 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.Source: B. Proposed solution · p. 5

• The framework designs robust controllers for uncertain affine control systems.Source: §V · p. 5
• The simulation includes trajectories of x1, x2, and ||e|| over time for the discontinuous controller.Source: Figure 3 · p. 6
• Continuous controllers provide predefined-time ultimate boundedness, whereas discontinuous controllers provide predefined-time stability to a desired manifold.Source: §V · p. 6
• The theoretical findings were validated through a numerical simulation demonstrating the effectiveness of the proposed control scheme.Source: §V · p. 6
• Further research is required to exploit the Lyapunov-like conditions for controller design in particular classes of nonlinear systems.Source: §V · p. 6