Predefined-Time Convergent Sliding Mode Control

• PT-SMC is a robust control strategy that explicitly defines convergence time, decoupling settling time from initial conditions and disturbance bounds.
• It integrates terminal sliding mode functions and fixed/predefined-time stability theory to achieve predictable performance with chattering mitigation.
• Practical implementations in aerospace and robotics show PT-SMC’s capacity for tight convergence guarantees and resilience under uncertainty.

Predefined-Time Convergent Sliding Mode Control (PT-SMC) is a robust control strategy that enforces convergence of system trajectories to a desired manifold within an arbitrarily specified, user-defined finite time, independent of initial conditions or bounded matched disturbances. PT-SMC synthesizes techniques from terminal sliding mode control, fixed/predefined-time stability theory, and singularity-avoidance mechanisms, thereby achieving strict performance predictability and chattering mitigation in the presence of system uncertainties and external perturbations.

1. Fundamentals and Definitions

PT-SMC extends classical sliding mode control by embedding a prescribed finite convergence time $T_c$ as an explicit design parameter, decoupling the settling time from initial conditions and disturbance bounds. Three hierarchical notions formalize this:

• Finite-time stability: Trajectories reach equilibrium in a finite time $T(x_0)$, dependent on initial state.
• Fixed-time stability: There exists a uniform upper bound $T^*\,{<}\infty$, such that $T(x_0)\leq T^*$ for all $x_0$.
• Predefined-time stability: $T^*$ can be arbitrarily set a priori by the designer as $T_c$, and the closed-loop controller ensures $x(t)\to x_e$ within $T_c$ for all $x_0$ without conservatism (Yan, 2020).

This is typically achieved by leveraging Lyapunov functions $V(x)$ with time-derivative inequalities of the form

$\dot V(x) \leq -a V(x)^\alpha - b V(x)^\beta,$

where $0 < \alpha < 1 < \beta$, enabling explicit computation of the convergence bound in terms of controller parameters (Xiao et al., 2024).

2. Design Principles of PT-SMC

The PT-SMC methodology involves two principal phases:

1. Predefined-Time Reaching Phase: The system state is driven onto the sliding manifold within $T_c$ by carefully engineered switching or adaptive gains that grow with a singular kernel as $t \rightarrow T_c^{-}$ (e.g., $\kappa(t) = 1/\left[ \alpha (T_c - t) \right]$ for $0 < \alpha < 1$), overriding uncertainties and initial distance (Cruz-Ancona et al., 2021).
2. Sliding Phase (Constrained Tracking): Either a fixed-time or additional predefined-time sliding law ensures trajectories remain on or within a desired neighborhood of the manifold, often employing barrier functions or terminal sliding surfaces for smoothness and robustness (Cruz-Ancona et al., 2021, Liang et al., 2020).

The overall control law unifies these phases, often involving an explicit time-varying structure or parameter switching, with Lyapunov-based analysis guaranteeing global predefined-time convergence.

3. Core Methodologies and Theoretical Guarantees

A. Adaptive Reaching Phase In uncertain MIMO systems,

$$\dot \sigma = G(t, \sigma)[I + \Delta g(t, \sigma)]\,u + f(t, \sigma),$$

the ARPS law

$$u(t) = G^{-1}(t, \sigma) \left[ -\left( \hat \beta(t) + \frac{\|\sigma\|}{\alpha (T_c - t)} \right) \frac{\sigma}{\|\sigma\|} \right], \qquad \dot{\hat \beta}(t) = \|\sigma(t)\|$$

ensures $\sigma$ enters a neighborhood $\|\sigma\| \leq \varepsilon/2$ before $T_c$, regardless of $\|\sigma_0\|$ or the unknown disturbance supremum $d$ (Cruz-Ancona et al., 2021).

B. Prescribed-Time Sliding Mode Surface For $n$th order systems,

$$s(x, t) = \sum_{i=0}^{n-1} c_i (t_f - t)^{n-i} x_1^{(i)},\quad c_i > 0,$$

with a control law containing a $(t_f - t)^{-1}$ term, enables strict convergence to the sliding mode at $t_f$, while design ensures that $s(x, t)$ and its derivatives vanish precisely fast enough to regularize any would-be singularity at $t = t_f$ (Chen et al., 2020).

C. Terminal and Barrier Approaches Terminal sliding mode and barrier-function augmented laws guarantee that, after reaching, tracking errors are confined within an arbitrarily small ball, with the invariance ensured by choosing barrier growth rates and minimum gains according to disturbance estimates (Cruz-Ancona et al., 2021, Liang et al., 2020).

D. Parameterization and Singularity Avoidance The crucial requirement is that exponents in recursive PT surfaces $s_k$ satisfy $0 < m_k < 1$, to ensure $x = 0$ is equilibrium and the convergence is not lost due to repelling dynamics. At the instant sliding occurs (i.e., the next recursion’s argument vanishes), the exponent is switched to $m_k = 1$, yielding bounded control and avoiding $u \to \infty$ (Yan, 2020). Chattering is mitigated by switching exponents only when the calculated $u$ exceeds a designer-specified threshold.

4. Practical Implementation and Tuning

Explicit guidelines for parameter selection and implementation emerge from the literature:

• Prescribed time $T_c$: Directly set by system performance requirements (e.g., transient bounds, mission-critical time constraints).
• Exponent selection ($m_k$): For $n$th order, $m_k \leq 1/(n - k)$, yielding total settling time $n T_c$ for the recursive design (Yan, 2020).
• Gain tuning: Increasing adaptive or fixed gains ($\kappa$, $K^*$) reduces settling time but increases peak effort and potential chattering.
• Barrier function choice: Determines tightness of steady-state tube versus required minimum admissible gain—tighter tubes necessitate greater gains (Cruz-Ancona et al., 2021).
• Offset and smoothing: Sampling-practicality requires replacing $(T_c - t)$ with $(T_c - t - \delta)$, $\delta > 0$, to prevent near-singularities and account for discretization (Chen et al., 2020).

A summary of parameter requirements appears below:

Parameter	Constraint/Role	Typical Value/Guideline
$T_c$	Predefined settling time	Chosen by designer
$m_k$	Recursive exponent	$0 < m_k < 1/(n-k)$
Adaptive gain	Drives reach within $T_c$	Embeds singular kernel as $t\to T_c^{-}$
$\gamma$, $\rho$	TSM shape/fixed-time surface	$0<\gamma<1,\;0<\rho<1$
$K_\mathrm{BF}$	Barrier function minimum gain	$K_\mathrm{BF}(0)=\bar\beta \ge 0$

5. Main Variants and Comparative Analysis

PT-SMC synthesizes and generalizes several previous approaches:

• Predefined-Time Terminal SMC (PTSM): Employs TSM surrogates using non-Lipschitz and bounded functions to ensure continuous attractivity of the origin, rigorously decoupling the convergence time from initial condition and system nonlinearity (Liang et al., 2020).
• Barrier-Function Augmented PT-SMC: Integrates a switching mechanism from adaptive reaching phase to a steady-state tube regulated by barrier functions, crucial for MIMO and high-uncertainty regimes (Cruz-Ancona et al., 2021).
• Non-Singular Recursive Schemes: Establish switching criteria for exponents $m_k$ within the sliding mode recursion so the resulting control law is globally bounded and insusceptible to actuator saturation or chattering (Yan, 2020).

PT-SMC is compared with finite-time and fixed-time SMC:

Scheme	Settling Time Dependence	Control Structure	Robustness/Chattering
Finite-time	Initial state	Homogeneous, non-uniform	Nonuniform, dependent on $x_0$
Fixed-time	Uniform, not a priori chosen	Inflexible, often conservative	Lacks transparent tuning
Predefined-time	Designer-chosen $T_c$	Explicit $T_c$ embedding	Parameterizable/mitigable

Notably, PT-SMC yields explicit, non-conservative, and tight bounds on settling time.

6. Application Domains and Simulation Evidence

Applications span tracking and regulation tasks in MIMO uncertain systems, spacecraft attitude control, and robotic manipulators. For instance:

• In rigid-body attitude tracking (spacecraft), explicit PT-SMC achieves $\|q - q_d\| \approx 10^{-4}$ at the specified $t_f$ with performance independent of initial conditions. Shorter $t_f$ values demand higher torques (Chen et al., 2020).
• In multi-DoF robot manipulators and general EL systems, PT-SMC guarantees convergence within $T_{\mathrm{pre}}$, maintains robustness under bounded perturbations, and provides systematics for saturating control effort or energy minimization (Xiao et al., 2024, Liang et al., 2020).

Extensive Monte Carlo simulation studies reveal that, under various uncertainty realizations, both state and sliding variables reach equilibrium strictly within the prescribed total convergence intervals (Yan, 2020).

7. Challenges, Open Issues, and Outlook

Design of PT-SMC remains sensitive to appropriate choice of exponents and gains; mis-selection (e.g., $m_k = 1$ during the reaching phase) leads to non-attractivity of the origin or delayed convergence (Yan, 2020). Singularities can arise if the sliding variable or its derivatives do not vanish sufficiently rapidly, especially under discretization. Chattering mitigation requires ancillary logic or adaptive smoothing.

A plausible implication is that future PT-SMC research will emphasize:

• Automated gain/exponent selection algorithms via convex or global optimization.
• Extension to time-delay, distributed, or underactuated systems.
• Integration with disturbance/uncertainty observers for real-time gain adaptation without conservatism (Chen et al., 2020).
• Practical deployment studies within high-uncertainty mechatronic platforms, with emphasis on implementation overheads and digital effects.

The PT-SMC paradigm provides a transparent, systematic, and robust controller synthesis framework for modern nonlinear control applications requiring hard, deterministic temporal guarantees (Cruz-Ancona et al., 2021, Chen et al., 2020, Liang et al., 2020, Xiao et al., 2024, Yan, 2020).