Predefined-Time Convergent SMC

• Predefined-time convergent sliding mode control is a robust method that guarantees system states reach a target manifold within a designer-specified time regardless of disturbances.
• It utilizes explicit time-varying terms in sliding surfaces and control laws to overcome singularities and achieve exact convergence in both second-order and high-order systems.
• The approach is underpinned by rigorous Lyapunov stability analysis, adaptive disturbance rejection, and has validated applications in aerospace and robotics.

Predefined-time convergent sliding mode control (PT-SMC) refers to SMC methodologies that guarantee the system state reaches a desired manifold or trajectory in a designer-specified, strictly bounded time regardless of initial conditions or the upper-bound of matched disturbances. Such controllers utilize explicit time-varying terms in both their sliding manifolds and control laws to enforce exact settling at a prescribed convergence time, while avoiding singularities that typically occur near the terminal time in conventional finite-/fixed-time SMC formulations. PT-SMC synthesis spans second-order to general n-th order nonlinear systems, including MIMO cases and systems with disturbances or uncertainties, and underpins advanced applications in aerospace attitude control, cooperative multi-agent systems, and robotic manipulators.

1. Mathematical Foundations and Sliding Surface Design

The core architectural principle in PT-SMC is the embedding of time-explicit terms into the sliding manifold to produce predefined-time convergence dynamics. For a representative second-order system with matched disturbance

$\dot x_1 = x_2,\quad\dot x_2 = f(x)+g(x)u+d(t)$

the time-varying sliding surface for $0\le t<t_f$ is constructed as $s(t) = (t_f-t)x_2(t) + n x_1(t)$, and for $t\ge t_f$ reverts to $s(t) = x_1(t) + x_2(t)$, as detailed in (Chen et al., 2020). This framework generalizes to n-th order systems via

$$S(t) = \sum_{i=0}^{n-1} c_i (t_f-t)^{n-i} x_1^{(i)}(t)$$

where $c_i$ ensures high-order term vanishing as $t\to t_f$. The advantages of these designs include exact time-scale shaping and continuity of the manifold, ensuring that state trajectory contracts to the origin precisely at $t_f$.

2. Prescribed-time SMC Laws and Singularity Avoidance

PT-SMC control laws derive from two-phase Lyapunov analyses, enforcing reaching within a prescribed time regardless of initial conditions. Representative second-order PT-SMC adopts

$$u(t) = g(x)^{-1}\left\{-f(x) - K_1 \mathrm{sign}(s) - K_I s - (t_f-t)^{-2}(n-1)x_2\right\}$$

for $0\le t<t_f$, with singular terms vanishing post $t\ge t_f$ (Chen et al., 2020). The challenge of infinite gain near $t_f$—a central difficulty in classical prescribed-time SMC—is addressed analytically; if parameters satisfy $n>2$, the Lyapunov analysis shows $s=O((t_f-t)^n)$, so despite $(t_f-t)^{-2}\to\infty$, the overall control input $u(t)$ remains bounded.

Nonsingular PT-SMC formulations further enhance regularity by using exponential weighting of the sliding variables: for systems with $$\dot x = -(1/T_c)\|\partial W/\partial x\|^{-2}[\partial W/\partial x]^T$$, Lyapunov quantification yields exact convergence in $T_c$ (Yan, 2020, Xiao et al., 2024). Constraints $0<m_k<1/(n-k)$ for recursive exponents in n-th order systems guarantee bounded control input even at $x\to0$, resolving the classical terminal sliding singularity.

3. Stability Analysis and Lyapunov Theorems

Unified Lyapunov conditions for PT-SMC formalize the settling-time bounds and provide equivalence with prior finite-/fixed-time stability results. The general theorem (Xiao et al., 2024) stipulates that for a radially unbounded Lyapunov $V(x)$ and strictly monotone $\psi(V)$, if

$$\dot V(x) \le -\frac{b-a}{\frac{d\psi}{d(V^p)}\,V^{1-p}} \cdot \frac{1}{pT_c}$$

with $\psi(V)$ bounded on $[a,b)$ and $p\in(0,1)$, then the closed-loop is predefined-time stable with $T_{\text{settle}}\le T_c$. This result facilitates explicit construction of sliding surfaces and control gains whose convergence and input-boundedness can be analytically certified.

Two-phase stability analysis is standard for PT-SMC: in the reaching phase, Lyapunov inequalities integrating time-varying coefficients guarantee the surface is reached in prescribed time. Subsequent invariance in the sliding phase, enforced typically by conventional SMC with quadratic Lyapunov, ensures the state remains at the origin for all $t\ge t_f$ (Chen et al., 2020, Liang et al., 2020).

4. Robustness to Disturbances and Uncertainties

PT-SMC controllers integrate disturbance observers or adaptive estimation strategies to reject bounded matched uncertainties. For example, spacecraft attitude control employs an observer governed by $\dot z = -\Lambda(w)z+\dots$, providing disturbance estimates $\hat d$ such that the control torque

$$T(t) = J(w\times(Jw)-a) - (T(q)J^{-1})^{-1}[K_2\mathrm{sign}(s) + \dots]$$

has its $K_2$ selected to dominate the observer error bound $|e_d(t)|\le |e_d(0)|e^{-\lambda_{\min}t}$, ensuring robust rejection (Chen et al., 2020). In the context of uncertain MIMO systems, adaptive reaching phase strategies employ time-scaled adaptive gain update laws—$\dot{\hat{\beta}}(t)=\|\sigma(t)\|$—and time-singular proportional gains to guarantee reaching within $T_r$, independent of disturbance magnitude (Cruz-Ancona et al., 2021).

Adaptive and nonsingular PT-SMC methods further involve barrier function switching in the post-reaching phase to maintain output in a compact set around the sliding manifold, with sub-level gain adaptation proportional to real-time disturbance norm (Cruz-Ancona et al., 2021, Yan, 2020). These constructs guarantee robustness without excessive gain overshoot and mitigate chattering.

5. Parameter Selection, Design Guidelines, and Practical Issues

The explicit parameterization of PT-SMC settling times via control or manifold coefficients enables designer control over system responsiveness versus input peak magnitude and regularity. For recursive sliding surfaces, exponents $m_k$ must be strictly less than $1/(n-k)$ for nonsingularity, with switching to $m_k\to1$ in an $\epsilon$-neighborhood to avoid discontinuous control and excessive chattering (Yan, 2020). Time constants $(T_1,\,T_2)$ in two-phase designs, as well as barrier gains and adaptive observer rates, are tuned according to actuator limits, disturbance bounds, and desired proximity to equilibrium (Xiao et al., 2024, Cruz-Ancona et al., 2021).

Chattering caused by frequent parameter switching is resolved by only activating proximity thresholds $\epsilon$ when the raw control effort exceeds pre-defined limits (Yan, 2020). For multi-agent systems or systems with input matrix uncertainty, switching to high-gain barrier control in a predefined vicinity ensures output remains within the specified neighborhood while minimizing unnecessary gain escalation (Cruz-Ancona et al., 2021).

6. Representative Applications and Simulation Results

PT-SMC is demonstrated in spacecraft attitude tracking, robotic manipulator path following, cooperative pursuit-evasion, and Markovian jump systems. For spacecraft, PT-SMC achieves quaternion tracking errors below $10^{-4}$ in user-specified times of $30$–$40\,s$, with torque peak proportional to convergence speed and manifold parameter $n$ (Chen et al., 2020). Robotic manipulators implementing PTSM surfaces reach zero error in exactly $\mathcal{T}_f=\mathcal{T}_s+\mathcal{T}_c$, with total error vanishing for all cases tested, demonstrating superiority to conventional finite-/fixed-time SMC (Liang et al., 2020). Cooperative interception autopilots utilize predefined-time sliding mode on acceleration error, obtaining tracking zero within $\le t_a$ for arbitrary input variation and reduced energy consumption benchmarked against established finite-time SMC (Gopikannan et al., 12 Jan 2026).

PT-SMC also enables finite/predefined-time convergence to switching surfaces in Markovian jump systems via LMI-based synthesis of sliding surfaces and robust/adaptive gain selection (Zohrabi et al., 2017). Simulations confirm exact adherence to the prescribed time regardless of initial errors or disturbance structure.

7. Extensions, Limits, and Practical Challenges

While PT-SMC provides rigorous predefined-time guarantees, challenges remain with management of unbounded time-varying gains near $t_f$ and avoidance of actuator saturation—necessitating tailored selection of design constants and proximity switching rules. Precise timing for phase switching and minimization of chattering under frequent gain-switching are essential for implementation robustness (Cruz-Ancona et al., 2021, Yan, 2020).

Extensions to MIMO systems with uncertain control matrices employ barrier functions for adaptive reaching, maintaining finite control effort and guaranteeing convergence to small output sets within a predefined time (Cruz-Ancona et al., 2021). Multi-phase architectures, disturbance observers, and Lyapunov-based nonsingular designs provide systematic pathways for robust PT-SMC synthesis across diverse nonlinear systems (Xiao et al., 2024, Chen et al., 2020).

In summary, predefined-time convergent sliding mode control is a mathematically rigorous and practically validated paradigm for enforcing strict time-bounded convergence in nonlinear, uncertain, and high-order systems, with concrete construction techniques, Lyapunov analysis, and gain selection strategies documented in the primary literature (Chen et al., 2020, Yan, 2020, Liang et al., 2020, Cruz-Ancona et al., 2021, Xiao et al., 2024, Zohrabi et al., 2017, Gopikannan et al., 12 Jan 2026).