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Prescribed-Time Control and Its Latest Developments
Hefu Ye, Yongduan Song, Frank L. Lewis
TL;DR
Prescribed-time control targets finite-time stability with a settling time preset independently of initial conditions, supporting applications requiring bounded transient duration. The paper surveys PT-control developments, including design techniques, system classes, and unresolved challenges, and reports simulations where PT regulation reaches T = 1s with smoother control action than alternatives.
Problem
Engineering applications such as missile guidance, rendezvous, emergency braking, and robotic obstacle avoidance require transient processes to occur within a given time.
Method
The paper reviews PT-control designs using time-varying transformations, Lyapunov inequalities, and control-gain selection for uncertainties and unknown coefficients.
Results
PT regulation reaches T = 1s, while comparator settling times depend on initial conditions or design parameters; simulations also show smoother control action.
Takeaways & Limitations
Propositions 10–12 indicate that PT settling time can be preset independently of initial conditions and other design parameters.
Takeaways & Limitations
PT control for MIMO nonlinear systems remains an open area, with few results when the control-gain matrix is unknown.
Abstract
from arXiv · showhide
Prescribed-time (PT) control, originated from \textit{Song et al.}, has gained increasing attention among control community. The salient feature of PT control lies in its ability to achieve system stability within a finite settling time user-assignable in advance irrespective of initial conditions. It is such a unique feature that has enticed many follow-up studies on this technically important area, motivating numerous research advancements. In this article, we provide a comprehensive survey on the recent developments in PT control. Through a concise introduction to the concept of PT control, and a unique taxonomy covering: 1) from robust PT control to adaptive PT control; 2) from PT control for single-input-single-output (SISO) systems to multi-input-multi-output (MIMO) systems; and 3) from PT control for single systems to multi-agent systems, we present an accessible review of this interesting topic. We highlight key techniques, fundamental assumptions adopted in various developments as well as some new design ideas. We also discuss several possibles future research directions towards PT control.
I. INTRODUCTION
Prescribed-time control extends finite- and fixed-time stability by allowing the exact settling time to be assigned in advance independently of initial conditions. This survey introduces the concept, reviews its development and design techniques, and organizes results across several system classes.
- Finite-time control can achieve convergence in finite time, but its settling-time estimate depends explicitly on initial conditions.
- Fixed-time control removes dependence on initial conditions for the settling-time upper bound, but the bound may be highly overestimated and is not directly tunable.The overestimate may be hundreds or thousands of times larger than the true settling time.
- PT control designs commonly transform the original system through state scaling or time scaling, then establish Lyapunov inequalities and bounded closed-loop signals.The design also addresses uncertainties, unknown control coefficients, and control-input boundedness.
- The article surveys PT control for SISO and MIMO systems and discusses related developments, design ideas, and future research directions.Its organization includes basic propositions, SISO designs, MIMO results, and connections with finite-time control.
- Prescribed-time control allows users to assign the exact settling time independently of initial conditions and other design parameters.The survey characterizes this property as extending the advantages of finite-, fixed-, and predefined-time control.
- Propositions 1–5 tie settling time to design parameters and initial state, whereas Propositions 10–12 permit user-prescribed settling time independent of those quantities.Propositions 6–9 instead provide computable initial-condition-independent upper bounds.
III. PRESCRIBED-TIME CONTROL FOR SISO SYSTEMS
This section reviews fundamental PT-control topics, emphasizing robust and adaptive control based on time-varying feedback and associated boundedness and convergence concerns.
- The section focuses on robust and adaptive PT control based on time-varying feedback, including convergence and boundedness of inputs and parameter estimates.
A. Preliminaries on prescribed-time control
PT control prescribes a finite settling time through a time-varying stability definition and two principal transformation-based design approaches. State scaling preserves bounded transformed states, while time scaling maps the prescribed interval to an infinite-time axis.
- PT global uniform asymptotic stability in time T uses a time-varying function µ that increases to infinity as t approaches T.The definition bounds the state through a class KL function over [0,T).
- PT stability with non-vanishing perturbations generalizes the unperturbed definition by incorporating disturbance-dependent bounds.
- State scaling multiplies the state by an increasing µ(t) that diverges at T, so bounded transformed dynamics imply the original state converges to zero at T.
- Time scaling uses τ=a(t), mapping t∈[0,T) to τ∈[0,∞), so asymptotic convergence in τ yields prescribed-time convergence in t.The mapping satisfies a(0)=0 and diverges as t approaches T.
- Existing PT studies cover controller structures, time-varying functions, convergence, robustness, observers, and output-feedback design, with many early results targeting SISO stabilization.
- A common time scaling is t=T(1−e^-τ), which gives a(t)=ln T−ln(T−t) and α(τ)=e^τ/T.
- Representative SISO results often assume known control coefficients and omit non-vanishing perturbations, while adaptive designs must additionally ensure bounded parameter estimates.
B. Robust prescribed-time control
Robust prescribed-time control uses state scaling and time scaling to stabilize uncertain scalar systems within a user-prescribed time. The surveyed design addresses unknown disturbances and control coefficients while maintaining bounded closed-loop behavior.
- B. Robust prescribed-time control: State scaling is used to design prescribed-time control for scalar systems with unknown control coefficients and non-vanishing perturbations.The approach transforms the state and establishes boundedness through a Lyapunov argument.
- B. Robust prescribed-time control: Under Assumptions 1–2, the closed-loop system is prescribed-time stable and all internal signals remain bounded over [0, T).The assumptions bound the disturbance through a known function and require the control coefficient to stay away from zero.
- B. Robust prescribed-time control: Table 2 summarizes technical differences across prescribed-time control literature.The table is accompanied by a criterion marking whether each method overcomes the listed limitation.
- B. Robust prescribed-time control: The proposed state-scaling method yields bounded control input and all signals, with the state converging as t approaches T.The result follows from bounded transformed-state dynamics and the scaling properties used in the Lyapunov analysis.
- B. Robust prescribed-time control: The robust design does not require a priori information on the magnitude lower bound of the unknown control coefficient.This control algorithm can be extended to higher-order systems in normal form.
C. Adaptive prescribed-time control
Adaptive prescribed-time control develops scalar-system designs for unknown parameters, including time-invariant and time-varying uncertainties and unknown control directions. Lyapunov analyses establish prescribed-time stability, bounded signals, and convergence under the stated assumptions.
- C. Adaptive prescribed-time control: Adaptive prescribed-time control is organized into designs for time-invariant parameters, time-varying parameters, and adaptive Nussbaum gains.The survey presents these as three basic frameworks for scalar systems.
- 1) Design for systems with time-invariant parameters:: For time-invariant unknown parameters and an unknown nonzero control coefficient with known sign, the adaptive controller achieves prescribed-time stability with bounded internal signals.The design uses time scaling, parameter adaptation, and a gain update law.
- 1) Design for systems with time-invariant parameters:: The time-invariant adaptive design yields x(t) → 0 as t → T while parameter estimates converge to limits and the control input remains bounded.The convergence argument uses a Lyapunov inequality, Barbalat’s Lemma, and bounded adaptive signals.
2) Design for systems with time-varying parameters:
For systems with time-varying parameters, the survey considers a factorized nonlinearity with an unknown bounded parameter and proves prescribed-time stability using adaptive control.
- 2) Design for systems with time-varying parameters:: The nonlinearity is modeled as f(x,t)=θ(t)ψ(x), where ψ(x) is known and θ(t) lies in an unknown compact set.The model permits time-varying parameters without requiring the bound on θ(t) to be known.
- 2) Design for systems with time-varying parameters:: Under the stated assumptions, the adaptive closed-loop system is prescribed-time stable and all internal signals are bounded over [0,T).The result follows from the Lyapunov derivative inequality and the associated transformed-system analysis.
3) Adaptive Nussbaum gain design:
Adaptive Nussbaum-gain design addresses systems with unknown control coefficients and time-varying uncertainty. The resulting controller establishes bounded signals and convergence to a compact set within a preset time.
- 3) Adaptive Nussbaum gain design:: The control coefficient is assumed to stay away from zero while its magnitude and sign remain unknown.An enhanced type B-L Nussbaum function is used to accommodate the unknown control direction.
- 3) Adaptive Nussbaum gain design:: The Nussbaum-function lemma guarantees boundedness of the transformed variable and Lyapunov function under the relevant coefficient bounds.This boundedness supports the subsequent closed-loop stability argument.
- 3) Adaptive Nussbaum gain design:: Under Assumptions 5–6, all internal signals are bounded over [0,T) and the state converges to a compact set within preset time T.The design combines adaptive updates for uncertainty estimates with a Nussbaum-gain mechanism.
- 3) Adaptive Nussbaum gain design:: The adaptive design targets fast or abruptly changing unknown parameters in both feedback and input channels.The survey presents this setting as challenging because the parameters are not restricted to slow variation.
- 3) Adaptive Nussbaum gain design:: Compensating neural-network reconstruction error while preserving prescribed-time stability remains an open research direction.The paper also identifies extension of the first-order design to more general systems as worthwhile.
IV. PRESCRIBED-TIME CONTROL FOR MIMO SYSTEMS
PT control for MIMO nonlinear systems remains an important open problem, motivated by applications requiring optimized transient timing and complicated unknown control gains.
- PT control for MIMO nonlinear systems is theoretically and practically important for missile guidance, weather forecasting, flight control, and robotic obstacle avoidance.
- Existing results are sparse, especially for unknown control gain matrices, with limited findings on PT stabilization, regulation, and tracking.
- Reported applications include a 7-DoF robot manipulator, spacecraft rendezvous, Euler–Lagrange regulation, and tracking with unknown control gains.
A. Square system
For square and non-square MIMO systems, PT controllers establish prescribed-time stability under structured uncertainty assumptions, while unknown nonlinear perturbations and control matrices remain central design challenges.
- Square system: The square-system analysis uses uncertainty bounds, skew-symmetry cancellation, and Lyapunov arguments to show bounded signals and X(t) → 0 as t → T.
- Square system: Under a square unknown control matrix whose symmetric part is positive definite, the proposed controller achieves PT stability with all internal signals bounded on [0,T).
- Non-Square system: For non-square systems, the gain matrix is factorized as B(X,t)=A(X,t)M(X,t), with A known full row rank and A(M+M⊤)A⊤ positive definite.
- Non-Square system: Under this factorization, the resulting controller achieves PT stability and bounded internal signals over [0,T).
- Non-Square system: A major challenge is handling unknown nonlinear perturbations and relaxing assumptions on the control matrix for more general PT algorithms.
V. LATEST DEVELOPMENTS IN PRESCRIBED-TIME CONTROL
PT consensus extends prescribed-time control to networked agents through time-varying feedback, overcoming dependencies on initial conditions and design parameters that affect earlier finite-time protocols.
- Time-varying feedback has enabled PT consensus for integrator networks, directed and undirected graphs, containment, leader-following, and time-base-generator designs.
- For single-integrator agents, the protocol uses local neighborhood errors and a time-varying function defined on [0,T).
- Prescribed-time consensus protocol: Earlier consensus settling time depends on gains, initial state, and topology, while reducing settling time can increase control effort.
- Prescribed-time consensus protocol: The PT protocol is designed to circumvent these earlier shortcomings by enabling prescribed-time consensus with connected undirected graphs when c ≥ 1/λ2(L).
- Prescribed-time consensus protocol: The resulting control input remains C1 smooth and bounded over [0,T).
B. Prescribed-time containment protocol
Prescribed-time containment extends consensus to networks with multiple leaders, while comparisons with earlier controllers clarify PT control’s relation to finite-time designs and implementation trade-offs.
- Prescribed-time containment protocol: With a directed spanning tree led by a root node and c ≥ 2λmax(Ṕ)/λ1( Q̃), containment is attained in prescribed time.
- Prescribed-time containment protocol: The containment protocol maintains C1-smooth, bounded control inputs over [0,T).
- PT control is compared with FT, FxT, and PdT controllers through simulations and discussion of numerical implementation issues.
- The PT controller shares the property that its control gain tends to ∞ as t → T and contains the FT controller as a special case.
- Controller structure: PT control input need not become large when the feedback signal decays faster than the high-gain function grows.
B. Discussion on Implementation
PT control implementations use practical modifications to avoid unbounded gains, while simulations compare FT, FxT, PdT, and PT control on a double integrator. PT achieves regulation at a user-set time with smoother control action, but several broader design limitations remain.
- Implementation: Practical PT implementations either schedule T slightly beyond the desired time or cap the scaling function near T, sacrificing some precision to avoid unbounded gains.These approaches preserve bounded control input and promote implementation without an exorbitant control effort.
- Simulation comparison: Double-integrator simulations compare FT, FxT, PdT, and PT controllers under two initial-condition scenarios.The scenarios use x1(0) = 0.2 and 0.4, with x2(0) = −0.2 and 0.
- Simulation comparison: PT achieves finite-time regulation at T = 1s, whereas FT depends on initial conditions and FxT has an overestimated settling-time bound tied to design parameters.The comparison also reports settling-time overestimation for PdT and increased control input when initial conditions become slightly larger.
- Simulation comparison: PT produces smoother control action than the other simulated methods and avoids the chattering phenomenon observed in the FT, FxT, and PdT plots.The cited simulations identify this smoother action as a distinguishing PT characteristic.
- Challenges and future opportunities: Open challenges include adaptive PT control with changing unknown parameters, output-feedback PT control for general systems, PT tracking under extra uncertainties, and higher-order multi-agent dynamics.The survey also identifies lower-conservative algorithms and improved closed-loop performance as future research topics.