From Asymptotic to Designer-Assigned-Time Control: A Review of Stability Notions, Design Mechanisms, and Controller Architectures

1. Introduction and Review Scope

• Lane changes, assembly-line manipulation, and power-grid recovery motivate explicit timing requirements, but do not themselves prove exact settling within a deadline.Time-critical design also evaluates transient behavior alongside steady-state accuracy.Source: §1 · p. 1
• Finite-time stability allows settling time to depend on initial state, fixed-time stability supplies a uniform bound, and predefined-time and prescribed-time approaches let designers select a horizon.The review treats predefined- and prescribed-time approaches as branches with different dominant mechanisms, not consecutive levels of a strict hierarchy.Source: §1 · pp. 1–2

1.1. Scope and Contributions

• The review separates temporal specifications, convergence-shaping mechanisms, and controller architectures, then asks whether nominal guarantees survive in the complete implemented loop.Its contributions include a terminology map, a mechanism-oriented account of Lyapunov structures, and an architectural audit of auxiliary dynamics and validation.Source: §1 · p. 3, §1.1 · p. 4
• The review synthesizes representative analytical and design structures rather than exhaustively listing applications or reducing every method to one Lyapunov formula.It compares studies by the property proved, certified variable and accuracy, initial-condition dependence, assumptions, auxiliary dynamics, and implementation constraints.Source: §1.1 · p. 4
• Implementation analysis checks compatibility with available control authority, measured information, and auxiliary dynamics, without assuming stronger nominal guarantees require greater control effort.The review proceeds from specifications and mechanisms to architectures, illustrative comparisons, and open research directions.Source: §1.1 · p. 4

2. Stability and Convergence-Time Notions

• The review compares certificates for the complete system under stated solution concepts and initial-condition domains, and classifies results by the mechanism producing assigned time rather than by source labels.Discontinuous feedback requires an appropriate differential-inclusion interpretation, and reaching the target must be followed by invariance.Source: §2 · p. 5, §2.1 · p. 5

2.2. Asymptotic Stability

• Asymptotic stability guarantees eventual convergence without quantifying a rate or settling time, while exponential stability adds a decay envelope but does not establish finite settling.The system description assumes an equilibrium and appropriate existence and continuation conditions; local Lipschitz continuity at the equilibrium is not a common requirement for finite-time convergence.Source: §2.2 · p. 5, §2.1 · p. 5, §2.3 · p. 6

2.4. Finite-Time Stability

• Finite-time stability certifies arrival in finite time, but its settling-time bound can depend on the initial state and does not imply a uniform global bound.The review reserves “finite-time stability” for the Bhat–Bernstein equilibrium property; a separate short-time notion only bounds trajectories over a window and need not imply convergence.Source: §2.4 · p. 6

2.5. Fixed-Time Stability

• Fixed-time stability adds a uniform, state-independent upper bound, although actual settling times may still vary with initial conditions.A common certificate combines sublinear and superlinear Lyapunov decay terms for near-origin and large-state behavior, respectively.Source: §2.5 · p. 7

2.6. Predefined-Time Stability

• Predefined-time stability treats assignability as a parameterized fixed-time property: a tuning selected for T_c certifies a settling-time upper bound on the stated domain.The weak property requires only the bound; the strong version additionally requires tightness, and admissible horizons, initial conditions, and tuning restrictions must be stated.Source: §2.6 · p. 7

2.7. Prescribed-Time Convergence

• Prescribed-time control constructs the closed loop around a designer-selected terminal horizon, often using time-varying transformations or gains, but a terminal limit is not automatically exact arrival.Maintaining zero after the horizon requires continuation and equilibrium invariance; bounded physical inputs and internal signals require separate admissibility checks.Source: §2.7 · p. 8

2.8. Input-to-State and Practical Stability Perspectives

• With nonvanishing disturbances or implementation errors, exact convergence may be replaced by input-dependent or practical convergence to a residual neighborhood.Practical predefined-time and prescribed-time formulations bound entry into a neighborhood or accuracy region rather than requiring exact convergence.Source: §2.8 · p. 9

2.9. Terminology Used in the Literature

• The terminology map records how each source’s term is interpreted, the temporal property it actually establishes, and the condition that distinguishes or qualifies that property.Stochastic finite-, fixed-, and predefined-time results state guarantees in probability or expectation; the section’s notions specify temporal properties but not their generating mechanisms or architectures.Source: §2.9 · pp. 9, 11

3.1. From Convergence Specifications to Decay-Law Design

• The decay law determines temporal behavior: exponential linear-in-V decay is asymptotic, while stronger requirements require changing its functional structure; residuals generally yield a residual-set result, not exact convergence.For time-dependent Lyapunov functions, uniform comparison with the state is also needed to infer state convergence.Source: §3.1 · pp. 11–12

3.2. State-Dependent Shaping: Fractional Powers and Homogeneity

• Fractional-power dissipation and negative-degree homogeneity provide finite-time convergence by strengthening relative decay near the origin, but settling time remains initial-state-dependent (Bhat and Bernstein, 2000; Bhat and Bernstein, 2005).With residual error, the result is practical convergence; steep fractional-power slopes can amplify measurement noise, while smooth regularization replaces exact settling with convergence to a neighborhood.Source: §3.2 · pp. 13–14

3.3. Multi-Region Shaping: Mixed Powers and Bi-Limit Homogeneity

• Mixed-power decay combines sublinear near-origin and superlinear large-state behavior to yield a settling-time bound independent of the initial condition, though the bound may be conservative and residuals weaken exact convergence.Increasing gains can reduce the nominal bound but may demand rapidly growing feedback at large states, which matters under actuator limits.Source: §3.3 · pp. 15–16

3.4. Designer-Assigned-Time Design Mechanisms

• Designer-assigned-time design replaces the emergent fixed-time bound with a specified horizon: predefined-time methods normalize a state-dependent settling law, while prescribed-time methods embed the horizon through time-dependent scaling or transformation.Source: §3.4 · p. 17

3.4.1. Predefined-Time Design Through Settling-Time Normalization

• Predefined-time normalization selects a bound Tc by making the complete nonlinear decay function’s state-independent integral compatible with it, but Lyapunov estimates may be conservative and tighter bounds require stronger dissipation.Under uncompensated residuals, the guarantee becomes practical convergence within a designer-related interval; finite control authority may prevent enforcing the nominal inequality for some states or requested bounds.Source: §3.4.1 · pp. 18–19
• Time normalization makes a uniform convergence-time bound an explicit synthesis variable while allowing the construction to remain autonomous.The section distinguishes the mechanism from whether a certificate yields exact or practical convergence.Source: §3.4.1 · p. 19

3.4.2. Prescribed-Time Design Through Temporal Scaling and Transformation

• Prescribed-time designs place the selected horizon into closed-loop evolution through temporal scaling, state scaling, or time transformation (Song et al., 2017b, 2023).A time transformation maps the finite physical-time interval onto an infinite transformed-time interval, compressing asymptotic convergence into the selected horizon.Source: §3.4.2 · p. 20
• A terminal-limit certificate does not by itself establish bounded inputs or internal signals, or define a valid continuation at and after the prescribed horizon.Feedback terms such as (T_p−t)^−1 may be undefined at the horizon; constructions must establish continuation and preserve equilibrium invariance. Increasing scaling need not make physical input diverge, but admissibility must be verified independently.Source: §3.4.2 · p. 21

3.5. From Desired Decay to Feasible Control

• A requested convergence horizon may be infeasible when the plant’s control channel cannot produce the required dissipation within admissibility limits.The plant’s minimum achievable time, a controller family’s minimum feasible horizon, and a Lyapunov construction’s sufficient certified horizon are distinct; feasibility within a controller family does not establish plant minimum-time reachability.Source: §3.5 · p. 23

4. Architectural Realizations, Implementation Constraints, and Design Guidance

• The three realization groups are primary feedback, augmented and information-structured architectures, and constraint- and implementation-aware designs; these roles can overlap.For example, a backstepping design can also include adaptive approximation, prescribed-performance shaping, and event-triggered implementation.Source: §4 · p. 23

4.1. Primary Feedback Architectures

• Primary feedback architectures convert available plant variables into physical input through direct state feedback, sliding-mode control, or backstepping.They provide different structural routes for embedding a temporal decay mechanism before estimation, learning, communication, or performance layers are added.Source: §4.1 · p. 24

4.1.1. Direct State-Feedback Realizations

• Direct state feedback can impose the desired dissipation without auxiliary controller states, but the command must remain admissible and the certificate must concern the regulated plant state.Aggressive designer-assigned horizons and large-state fixed-time feedback can make admissibility restrictive; simplicity alone does not ensure robustness or implementability.Source: §4.1.1 · pp. 25–26

4.1.2. Sliding-Mode Realizations

• Sliding-mode guarantees must cover both reaching and sliding phases: finite-time arrival at a manifold does not by itself establish finite-time stabilization of the plant state.Horizon budgets must account for both phases, and discontinuous designs also require a solution concept and post-reaching manifold invariance; bounded reaching time alone does not bound the state transient.Source: §4.1.2 · pp. 26–27

4.1.3. Backstepping Realizations

• Backstepping propagates convergence mechanisms through virtual controls, but its guarantee must hold for the complete composite Lyapunov function, not an individual recursive error.Stagewise dissipation conversion can make bounds more conservative as system order grows, while simultaneous recursive errors require a common composite certificate rather than summed time estimates. Filtering, observers, and adaptation enlarge the closed loop; their errors must be certified or quantified as residuals, and shorter horizons demand faster filtering that increases sensitivity to noise and sampling.Source: §4.1.3 · pp. 29–30

4.2. Augmented and Information-Structured Architectures

• Augmentation with estimation, adaptation, approximation, or distributed information must preserve the nominal convergence mechanism through a certificate that identifies what each auxiliary error must satisfy.Not every auxiliary state needs the regulated variable’s horizon, provided its certified property suffices to preserve that variable’s guarantee.Source: §4.2 · p. 30

4.2.1. Observer-Based and Output-Feedback Realizations

• Output feedback uses measured outputs, optionally with internal controller states or reconstructed state estimates, and its certificate must apply to the augmented dynamics.A full-state controller with a disturbance or extended-state observer remains a state-feedback architecture with an augmentation.Source: §4.2.1 · p. 31
• Output-feedback designs need an augmented-state certificate because nonsmooth or time-varying feedback generally defeats separation and observer transients can invalidate nominal regulation guarantees.Audited designs range from common-horizon certificates for the plant, observer error, and input (Tan et al. (2026b)) to prescribed-performance tracking with auxiliary signals uniformly ultimately bounded (Sui et al. (2025)).Source: §4.2.1 · p. 31

4.2.2. Adaptive and Learning-Based Realizations

• Adaptive and learning augmentations do not themselves certify plant convergence time: residuals often leave only practical or residual-set results unless robust terms or excitation remove them.Interval excitation supports exact preassigned-time tracking despite approximation parameters converging only to a neighborhood (Jiang et al. (2025a)); Wang et al. (2023) report exact fixed-time plant-state convergence with a residual bound on the disturbance-observer error.Source: §4.2.2 · pp. 32–33

4.2.3. Distributed and Cooperative Realizations

• Cooperative deadlines depend on graph structure and information, and auxiliary observer or optimizer convergence must be connected by a joint certificate to agent-level errors.Maintaining an assigned horizon as network size or topology varies may require larger gains or control amplitudes; fully distributed adaptive gains remove some global spectral requirements but not bandwidth, saturation, noise, or energy limits.Source: §4.2.3 · p. 33

4.3. Constraint- and Implementation-Aware Realizations

• Plant, actuator, sensing, communication, and digital limits alter admissible trajectories and effective authority, potentially replacing exact convergence with residual-set guarantees; preserving timing needs redesign and an implementation-level certificate.An assigned horizon may demand unattainable magnitude or bandwidth, violate safety, or rely on updates unavailable to sampled or event-triggered control; trajectories must remain admissible before, at, and after the claimed convergence time.Source: §4.3 · pp. 34–35

4.3.1. Performance-Envelope, Safety-, and Optimization-Constrained Realizations

• Safety, optimization, and performance mechanisms constrain the temporal controller rather than supply its decay law, so feasibility and the target property must be certified separately.CLF-CBF programs may prioritize hard safety over decay, while an MPC prediction horizon alone is not a settling-time guarantee; nonzero terminal envelope widths yield practical or envelope results. Audited cases include practical saturated teleoperation and a prescribed-time safe-boundary claim distinct from regulation-error convergence.Source: §4.3.1 · pp. 35–37

4.3.2. Actuator-Constrained and Regularized Realizations

• For the scalar integrator, a fixed input bound precludes global fixed-time stabilization over unbounded initial states; the saturated benchmark law needs u_max ≳1.5563 to meet its tolerance deadline.This threshold applies to the calibrated law, not the full feedback family. Clipping can invalidate the nominal proof; bounded-by-design feedback, anti-windup, performance relaxation, and regularization address different constraints, but may yield regional or practical guarantees.Source: §4.3.2 · pp. 37–38

4.3.3. Sampled-Data, Event-Triggered, and Hybrid Implementations

• Digital guarantees require admissible sampling intervals or event-error budgets; event-triggered designs can retain a selected horizon only when their error is covered by the certificate.A shrinking trigger threshold can cause event accumulation near the horizon, so a positive minimum inter-event time is also needed. Sampled-data and event-triggered results range from practical to exact, depending on the design and certified variable.Source: §4.3.3 · pp. 38–39

4.4. Architectural Synthesis and Design Guidance

• The audit codes each study by its strongest theorem for the regulated variable, distinguishing exact from practical guarantees and recording auxiliary dynamics and validation.The examined set is representative, not a population estimate; hardware and hardware-in-the-loop validation cluster in fast-sampled electromechanical platforms and are reported less often for distributed and implementation-aware designs.Source: §4.4 · p. 40

U AW, BLF, PPF N

• Choose a realization conditionally: first specify the temporal requirement and certified target, then test whether the requested horizon is feasible under the plant and implementation constraints.When a nominal guarantee cannot be proved for the complete augmented system, weaken it explicitly rather than infer it from an ideal continuous-time model.Source: U AW, BLF, PPF N · p. 41

5.1. Benchmark System and Common Evaluation Conditions

• The benchmark uses one scalar integrator and two calibrations—matching certified bounds or matching peak input—to distinguish temporal performance from control effort.The fixed- and predefined-time laws share a two-power family, and prescribed-time control is continued after Tc; its bounded-gain variant avoids a zero-order-hold contraction failure at the cost of practical terminal convergence.Source: §5 · p. 42, §5.1 · pp. 43–44

5.2. Temporal Guarantees and Observed Convergence

• 0.393 (FxT) and 0.500 (PdT) are the nominal arrival times, versus exactly 1 for the finite- and prescribed-time laws under certificate-matched calibration.The two-power laws arrive before the shared certificate but require peak inputs of 8 and 6.28, compared with 2 and 3 for the finite- and prescribed-time laws.Source: Figure 7 · p. 47, §5.2 · p. 47
• Finite-time settling grows as 2|x0|^1/2/kf, fixed-time settling approaches 0.785 under a conservative 1.000 certificate, and predefined-time settling approaches but does not reach Tc on bounded initial sets.Prescribed-time arrival is Tc for every initial condition; asymptotic band-entry time grows logarithmically and depends on ε.Source: §5.2 · pp. 47–48, Figure 8 · p. 48
• At equal peak input 4, FxT and PdT are the same law with certificates 2.000 and 1.571; at xref, FT is fastest at 0.500 and AS slowest at 1.727.At |x0| = 10^3, FT settling reaches 15.8, while the two-power law approaches π/2 and prescribed-time remains at 1, with much larger peak inputs for the latter two.Source: §5.2 · p. 48, Table 12 · p. 49

5.3. Control Effort and Sensitivity to Implementation Constraints

• Under certificate matching, peak input times observed settling time is approximately 1.94 (FT), 2.70 (PT, η = 3), 3.02 (FxT and PdT), and 6.91 (AS).For AS, the constant ln(|xref|/ε) increases as accuracy is tightened; prescribed-time η = 1 reaches the plant minimum-time limit, while larger η smooths the input at higher peak effort.Source: §5.3 · p. 49
• Below umaxTc = |x0| no law meets the deadline; at the boundary, prescribed-time succeeds only as a zero-margin minimum-time case, while the fixed-calibration AS law is late through umax = 5.In the joint map, autonomous power laws meet deadlines only below their stated sampling reference scales, while the prescribed-time law also becomes late or practical at large periods.Source: §5.3 · pp. 49–50, Figure 9 · p. 50
• At h = 6.7 × 10^-2, sampled FT, FxT, and PdT are practical; at x0 = 100, FxT and PdT diverge at that period, while AS meets its ε-entry deadline throughout the sweep.Capping prescribed-time gain changes the xref, h = 0.107 outcome from practical with a 5.0 × 10^-3 miss to deadline-satisfied with a 3.0 × 10^-5 miss, but leaves exponential rather than exact terminal convergence.Source: §5.3 · p. 50, Figure 10 · p. 51
• At h = 6.7 × 10^-2, AS (Tε = 0.804) and PT (Tε = 0.871) are D, while FT, FxT, and PdT have Tε > 2 and are P.These are observed classes for the finite simulation window, not analytical deadline certificates.Source: Table 13 · p. 52, §5.3 · p. 53
• With d(t) = 0.05 sin 5t, FT, FxT, and PdT approach residual-set radii of 6.3 × 10^-4, 1.6 × 10^-4, and 2.5 × 10^-4, respectively; observed times change by about 3–5%, but exact settling is lost.Finite-time entry is certified only for the larger σ-margin sets; the asymptotic run is practical because its residual radius, 7.2 × 10^-3, exceeds ε.Source: §5.3 · p. 53
• PT rejects the bounded disturbance at the terminal time (ET = 2.4 × 10^-8 against a 2.5 × 10^-8 bound), but its post-Tc continuation retains an oscillation of about 9 × 10^-3 amplitude.Arrival and retention are distinct: the post-Tc residual is set by the continuation gain, not the prescribed-time mechanism.Source: §5.3 · p. 53
• All four class laws meet the same deadline only where control authority and sampling rate permit; when constraints bind, the temporal class determines whether convergence is late, practical, or lost.The reported feasibility region reflects both authority and implementation rate; within it, the class laws differ near the region’s edges.Source: §5.3 · pp. 51, 53, 54, Figure 11 · p. 52
• Observed settling-time superiority does not imply superiority in effort, robustness, or implementability, and the reported feasibility classes are observations over Tsim rather than analytical deadline certificates.Effort indices use the applied input; sampling rows use zero-order hold without saturation.Source: §5.3 · pp. 52, 54

6.1. Feasible Horizons: Limits, Certificate Tightness, and Robust Accuracy

• Develop tightness measures and converse theorems that distinguish a failed certificate search from an unattainable temporal property.Certificates should report arrival or practical-entry margins over the certified initial set and initialization interval; a zero margin leaves no reserve for implementation or disturbances.Source: §6.1 · pp. 55, 57
• Compute feasible horizon regions for a common initial set and admissible inputs, distinguishing plant limits, controller-family limits, and certificate bounds.The analysis must account for magnitude and rate constraints, safety, uncertainty, and internal dynamics; the goal is to identify when no admissible controller can meet a deadline.Source: §6.1 · pp. 55–56
• Compare horizon, accuracy, safety, and control-effort trade-offs under disturbances, faults, actuator limits, and competing optimal or predictive controllers.Open targets include horizon-dependent noise-to-terminal-error bounds, recovery limits after abrupt changes, joint safety and envelope feasibility, and lower bounds on effort or peaking near the physical minimum time.Source: §6.1 · pp. 57–58

6.2. Composition: Auxiliary Dynamics Inside the Certificate

• Establish a nonlinear substitute for separation that certifies the augmented loop when observers, filters, adaptation, reaching phases, or actuator dynamics interact.Temporal guarantees for individual subsystems do not automatically carry over to their interconnection; temporal ISS gains, small-gain conditions, residual bounds, and relative horizons are candidate tools.Source: §6.2 · p. 58
• Make temporal budgets explicit for hidden transients, identification, learned models, and moving references.Allocate deadlines across interacting subsystems while bounding peaking and input demand; include excitation and estimation time, verifiable approximation error over the reachable region, and reference feasibility under actuator limits.Source: §6.2 · pp. 58–59

6.3. Implementation: Terminal Behavior and Finite-Rate Realizations

• Provide one certificate on [0, ∞) covering convergence, terminal transition, and post-arrival invariance under a stated solution concept.It should treat a terminal miss as the continuation's initial condition and distinguish reaching by the deadline, remaining at the target, and recovering after later disturbances.Source: §6.3 · p. 59
• Derive class-preserving discrete-time certificates and computable bounds for regularization and digital implementation.The bounds should relate horizon to sampling, computation delay, quantization, and actuator update rate, including recursive designs and observer coupling, to establish when exact continuous-time guarantees become practical settling claims.Source: §6.3 · pp. 59–60
• Develop computational synthesis and verification for temporal guarantees under sampled, quantized, and saturated implementation.Set-based verification should cover every point in the certified initial set; event-triggered designs should expose minimum inter-event time and its control or accuracy cost, while networked designs should account for heterogeneous resources and connectivity.Source: §6.3 · p. 60

6.4. Temporal Guarantees Beyond the Finite-Dimensional ODE Setting

• Temporal definitions and certificates must match the native solution concept across memory, stochastic, distributed-parameter, switched, and hybrid systems.For PDEs, specify the convergence norm and quantify reduced-order control against the distributed state; for switched and hybrid systems, distinguish physical time, flow time, jump count, and dwell time.Source: §6.4 · pp. 60–61
• Fractional and delayed systems require careful target and solution concepts, while certificates should cover both target entry and persistence of the complete state or history.Under standard assumptions, exact finite-time stability is excluded for Caputo-type systems with continuous right-hand sides and the classical equilibrium concept; delay results may certify the current state without settling the history segment.Source: §6.4 · p. 61
• Expected settling-time bounds, probabilistic deadlines, and almost-sure deadlines are distinct, so comparisons need matched probability criteria and explicit noise, initial-set, and control limits.High-probability and almost-sure guarantees under bounded actuation are especially important because gain cannot suppress diffusion without physical cost.Source: §6.4 · p. 61

6.5. Benchmarks, Failure Boundaries, and Experimental Validation

• The audit finds hardware validation concentrated on fast-sampled electromechanical platforms and reports often compare unlike quantities as though equivalent.Certificate- and effort-matched calibration with persistence-based settling measures is proposed; reporting should also identify evidence level and implementation conditions, including sampling, actuation limits, and failed trials.Source: §6.5 · pp. 61–62
• A shared benchmark suite should test varied control architectures and report infeasible-deadline boundaries, not only successful trajectories.Suggested cases include relative-degree-two sliding-mode, strict-feedback backstepping, output-feedback, and constrained nonlinear systems, evaluated over common initial sets, disturbances, sampling grids, and accuracy definitions.Source: §6.5 · p. 62
• Progress requires computable feasibility limits, certificates that include auxiliary dynamics, and guarantees preserved under sampled, triggered, saturated, and regularized implementation.These criteria should establish certified minimum times for a stated initial set, input class, and accuracy, and certify continuation after the deadline rather than treating convergence time as a nominal parameter.Source: §6.5 · p. 62

7. Conclusions

• Open directions include one certificate for arrival, terminal transition, retention, and post-arrival robustness, plus sampled-data guarantees and shared multi-architecture benchmarks.Other proposed problems include model-error bounds, resource-aware deadlines, and computational verification.Source: §7 · p. 63
• A temporal claim must distinguish finite settling for each admissible initial condition, a uniform or selectable bound, a deadline or terminal limit, and practical convergence to a retained set.Exact first arrival and a selectable bound’s tightness are separate properties.Source: §7 · p. 63
• The same decay law can be realized by different architectures, whose auxiliary dynamics may change the certified variable and the scope of the guarantee.The audit records these qualifications for its examined studies, not as evidence of their prevalence in the literature.Source: §7 · pp. 63–64
• The scalar benchmark separates plant reachability, a controller-family limit, fixed-tuning performance, and certified bounds, but does not numerically validate architectural synthesis or rank stability classes.Its comparisons apply only at the mechanism layer.Source: §7 · p. 64
• Saturation, regularization, sampling, and event triggering create new closed loops, so a continuous-time certificate carries over only when the implemented loop is certified.The review identifies class-preserving implementation as a foundational open problem.Source: §7 · p. 64

Funding

• The research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.Source: Funding · p. 64

Data availability

• No new experimental data were generated, and MATLAB scripts reproducing the Section 5 numerical comparison are available from the author upon reasonable request.Source: Data availability · p. 64

A. Architectural Audit of the Examined Studies

• Table A1 records design architecture and augmentations, reported temporal class, certified variable, guarantee type, auxiliary dynamics, and validation for examined and additional representative designs.Guarantee codes distinguish exact, practical, qualified-horizon, and bounded or envelope results; validation codes distinguish numerical, hardware-in-the-loop, and physical experiments.Source: A. Architectural Audit of the Examined… · pp. 64–65, Table A1 · p. 65
• Entries are grouped by the architecture generating physical input or by a distinguishing observer, adaptation, cooperation, or constraint-handling feature; mechanisms and augmentations may overlap.The coding follows the strongest property proved for the certified variable, not the source’s title or conclusion.Source: A. Architectural Audit of the Examined… · p. 64

E DO, BLF Hw

• Examples in the disturbance-observer and barrier-related group include prescribed-time integral sliding-mode control with an ESO and performance transformation, and adaptive-barrier reaching with a fixed-time disturbance observer.The listed applications include a manipulator with faults and a PMSM, respectively.Source: E DO, BLF Hw · p. 66
• The group also lists prescribed-time performance-function designs for load-frequency control and fixed-time ESO designs for VTOL UAVs.Source: E DO, BLF Hw · p. 66

E PPF N

• Shao et al. (2026a) combine a TTR-based performance tube with asymmetric-barrier sliding-mode control for second-order platoon systems.Source: E PPF N · p. 66

P PPF N

• Periodic-delayed-feedback super-twisting, FxT reaching laws, and prescribed-time sliding phases appear in high-order, RBFNN-based, and cooperative-spacecraft designs.Source: P PPF N · p. 67

P DO, Flt

• Backstepping studies combine disturbance observers, differentiators, command filters, fuzzy or neural elements, and barrier or performance functions; reported targets include plant, attitude, and composite errors.Examples include FxT sliding-mode disturbance observation for a rectifier, extended observers for a DC microgrid, and prescribed-time composite-error claims.Source: P DO, Flt · p. 67

U AW, BLF, PPF N

• Backstepping and cascade designs address mobile robots, fixed-wing UAVs, electro-hydraulic systems, and slung loads, with FxT position and swing errors reported for a three-loop load controller.Source: U AW, BLF, PPF N · p. 67

P Est N

• Observer and output-feedback examples cover humanoid walking, motor drives, mobile robots, piezo systems, marine vehicles, and UAVs, with FxT walking-position and speed-error claims among the reported results.Source: P Est N · p. 68

E Obs, FC Hw

• Wang et al. (2026g) pair a fixed-time ESO with synchronized ADRC for load-frequency control in heterogeneous units, targeting ACE and per-unit power consensus.Source: E Obs, FC Hw · p. 68

E ESO, Sw Hw

• Observer, neural, and adaptive designs span switched USV control, PMSM speed control, and stand-alone inverter voltage and current regulation.Other examples include a fuzzy observer with an ESO and prescribed-time filter for a constrained manipulator, and a bounded-gain prescribed-time ESO for USV formation.Source: E Obs, FC Hw · p. 68, E ESO, Sw Hw · pp. 68–69

• Adaptive and learning-based examples span USV control, spacecraft formation, robotic arms, and cooperative vehicle spacing, with prescribed-time regulated variables or spacing errors reported.Source: P Ad HIL · p. 69
• Adaptive designs combine neural or fuzzy approximation with observers, command filters, backstepping, or fault compensation across UAV, continuum-robot, and underwater-vehicle applications.Source: P Ad HIL · pp. 69–70
• Actor-critic and critic-learning designs address servo control, spacecraft pursuit-evasion, and multi-drive systems, with fixed-time claims for relative states, weights, or bounded-set convergence.Source: P Ad HIL · p. 70
• Distributed designs address consensus, formation, output regulation, and resource allocation, including prescribed-time state claims at each node and optimizer-error claims.Source: P Ad HIL · p. 70
• Cooperative-control examples report fixed- or predefined-time formation, dispatch, allocation, and bipartite errors under switching, quantized inputs, faults, or signed graphs.Source: P Ad HIL · pp. 70–71
• Further networked examples combine disturbance or neural observers with event triggering, attack resilience, saturation handling, or intermittent secondary control.Applications include containment, microgrid frequency and power sharing, and consensus tracking.Source: P Ad HIL · p. 71

• Tang et al. (2026) combine optimized backstepping, an identifier-actor-critic, and saturation handling; the reported target is FxT containment error.Source: P Trg N · p. 71
• Feng et al. (2026) combine a predefined-time disturbance observer and sliding-mode control with quantitative prescribed-performance functions and dynamic event-triggering for UAV–UGV formation.Source: P Trg N · p. 71

P DO, PPF, Trg

• Tian et al. (2026b) combine topology switching, a PdT observer, RL, saturation, and faults for USVs, with PdT formation errors reported.The result passage also lists P Obs, Cr, AW, and Trg.Source: P DO, PPF, Trg · p. 71

P Obs N

• Ma and Zhu (2026) use a hierarchical estimator and algebraic disturbance reconstruction for EL agents, reporting synchronized PdT formation error.Source: P Obs N · p. 71

E Est N

• Zhu et al. (2026b) pair an event-triggered fuzzy-parameter ZNN with multi-robot formation and report PdT voltage and current-sharing errors.Source: E Est N · p. 71
• Cui et al. (2026b) use a PdT gradient/Hessian estimator with fuzzy backstepping for optimal consensus, reporting PdT output to a time-varying optimum.Source: E Est N · p. 72
• Ding et al. (2026c) combine arc-length consensus with a PdT state estimator for UAV elliptical formation, reporting PdT arc-length error.Source: E Est N · p. 72
• Other studies report PdT errors in envelopes, formation residuals, spacing, and synchronization, alongside PT consensus and formation errors.The reported contexts include UAV-USV formation-containment, vehicle platoons, MAS, and signed-graph formation.Source: E Est N · pp. 71–72

U Obs, NN, Sw, PPF

• Zhou et al. (2026c) combine frozen time-varying scaling, a two-power law, and distance PPF for rigid formation, reporting PT distance errors.Source: U Obs, NN, Sw, PPF · p. 72
• Li et al. (2026b) combine a PT cluster estimator and NN PT synchronization with faults and saturation for USV-UAV systems, reporting PT cluster consensus error.The passage also identifies FDI.Source: U Obs, NN, Sw, PPF · p. 72

P Est, NN, FC, Sw

• An et al. (2026) use a variable-gain ESO, tracking differentiator, and PT performance function for a CAV platoon, reporting PT spacing error in an envelope.Source: P Est, NN, FC, Sw · p. 72
• Garg et al. (2022) apply a CLF-CBF QP to ACC and multi-robot settings, reporting FxT safe-set and target-set results.Source: P Est, NN, FC, Sw · p. 72

U ESO, Dif, PPF N

• The CLF-CBF QP study reports FxT safe-set and target-set results, while a separate result reports PT distance to a safe set.The passages identify ACC and multi-robot settings for the QP and cite Alavi Nasab et al. (2025) for the PT safe-set result.Source: U ESO, Dif, PPF N · pp. 72–73
• Other listed applications include current-constrained DC-DC voltage regulation, constrained USV cluster formation, and output-constrained mobile-robot control.Source: U ESO, Dif, PPF N · p. 73
• The related-work inventory includes tracking and state constraints, saturation-aware control, and PT control with CBF safety filtering for WMRs.Source: U ESO, Dif, PPF N · p. 73

P Sat N

• The inventory spans event-triggered and adaptive designs, terminal constraints, quantized control, and finite-time control barrier functions across varied systems.Examples include stochastic systems, AUVs, memristive neural networks, and manipulators.Source: P Sat N · p. 73
• The remaining studies list sampled-data output-feedback, event- or self-triggered consensus, and FxT PPF designs with observers or critic-based ADP.The identified settings include leader-follower consensus, spacecraft attitude, and cable-robot wrench control.Source: P Sat N · p. 73
• Listed finite-time results include quasi finite-time synchronization error, a state bound over a window, and FxT position and velocity errors.Source: P Sat N · pp. 73–74

E Obs, QP

• Lin et al. (2026) pair FxT PPF with BLF and saturation compensation for a macro-micro stage, while Hua et al. (2026a) use FxT PPF in nonsingular SMC with dynamic ETC for a quadrotor.The listed transformed error is mapped to a set.Source: E Obs, QP · p. 74

P PPF, Trg

• The listed PPF and triggering studies include NN backstepping with Nussbaum and memory ETC under stochastic delay, plus intermittent self-triggered control with bounded gain.Source: P PPF, Trg · p. 74

P Ad, Flt, PPF

• The applications also include PdT torque estimation with sEMG admittance and ETC for teleoperation, and RISE disturbance observation with APF and time-scale transformation for spacecraft proximity.The spacecraft setting lists collision and visibility.Source: P PPF, Trg · p. 74, P Ad, Flt, PPF · p. 74

P PPF, Trg

• Golestani et al. (2026) combine adaptive DSC, bounded performance-rate PPF, and vanishing σ-modification.Source: P PPF, Trg · p. 74

P Ad, Flt, PPF

• Kim and Yoo (2026) combine dual-channel dynamic ETC, an impulsive PT tracker, and saturation compensation for a UUV.Source: P Ad, Flt, PPF · p. 74