Non-Overshooting Quasi-Continuous Sliding Mode Control

• The topic introduces a robust sliding mode control strategy that ensures non-overshooting, finite-time convergence of double integrator systems in the presence of bounded disturbances.
• It employs piecewise control laws with region-specific damping and regularization methods, such as max-value and additive schemes, to suppress chattering.
• Analytical and Lyapunov-based analyses confirm uniform global asymptotic stability, supporting applications in uncertain systems and UAV trajectory tracking.

Non-overshooting quasi-continuous sliding mode control refers to a family of robust nonlinear feedback designs that enforce convergence of second-order (typically double integrator-type) systems to the origin in finite time (or nearly finite time under perturbations) without any overshoot of the primary controlled state. These controllers accomplish robust stabilization against bounded matched disturbances, while guaranteeing that the controlled output decreases monotonically towards the target value—precluding sign reversals—and that the control action remains bounded and quasi-continuous across the state space. Theoretical developments, modifications for chattering suppression, and implementation in both canonical and practical uncertain systems have been advanced in recent literature (Ruderman et al., 5 Dec 2025, Ruderman et al., 5 Jun 2025, Wang et al., 2 May 2024).

1. Mathematical Foundation

The fundamental system for non-overshooting quasi-continuous sliding mode control is the double integrator with matched disturbance: $\dot x_1 = x_2,\qquad \dot x_2 = u + d(t)$ where $x \in \mathbb{R}^2$ is the state, $u\in\mathbb{R}$ is the control input, and $d(t)$ is an essentially bounded disturbance, $\|d\|_\infty\leq D<\infty$.

The original control design by Ruderman and Efimov defined the sliding variable as $s=x_1$, with the quasi-continuous robust control law: $u(t) = -\gamma\,\text{sign}(x_1) - \delta(t)$ with design gain $\gamma>D$, and

$$\delta(t) = \begin{cases} \dfrac{|x_2|}{|x_1|}x_2, & x_1x_2<0 \ 0, & x_1x_2\geq 0 \end{cases}$$

This construction ensures non-overshooting monotonic approach of $x_1$ to zero for all initial conditions, with specific treatment of phase-space quadrants and amplitude limitations (Ruderman et al., 5 Dec 2025).

2. Control Law Structure and State-Space Analysis

The closed-loop state-space is partitioned into regions:

• $U = \{x: x_1x_2 \geq 0\}$ (quadrants I and III): $\delta(t)=0$, control reduces to $u = -\gamma\text{sign}(x_1)$.
• $C = \{x: x_1x_2<0\}$ (quadrants II and IV): $\delta(t)\neq 0$, providing additional nonlinear damping.

Within $U$, the system evolves under constant-acceleration dynamics, ensuring $x_1$ does not change sign—the system decelerates to the switching surface $x_2=0$ without overshoot. The exact exit time from $U$ is $T_U=|x_2(0)|/\gamma$. Entry into $C$ is only possible via transversal crossing, preserving sign-invariance of $x_1$.

In the controlled region $C$, under the set $C_a = \{x_1x_2<0,\ x_2^2<2\gamma|x_1|\}$, trajectories remain confined and the control magnitude $|u|\leq\gamma$. Analytic solution of the unperturbed system yields oscillatory convergence to the origin, with explicit solution

$$\begin{cases} x_1(t) = -\dfrac{\gamma}{\omega^2}\cos(\omega t+\phi) + B\ x_2(t) = \dfrac{\gamma}{\omega}\sin(\omega t+\phi) \end{cases}$$

and finite convergence time $T_{C_a}$. There is strict non-overshooting behavior: $x_1(t)$ decays without sign change throughout both regions. The total exact finite settling time is $T_f=T_U+T_{C_a}$ (Ruderman et al., 5 Dec 2025).

3. Robustness and Lyapunov Analysis

Uniform global asymptotic stability (UGAS) to the origin is established by smooth, piecewise Lyapunov functions. Under matched bounded disturbances, a state transformation

$$\xi(x) = \begin{pmatrix}\sqrt{|x_1|}\,\text{sign}(x_1) \ x_2\end{pmatrix}$$

and Lyapunov candidate with coupling term $\epsilon(x)$ are constructed, enabling bounds: $$\dot V(x) \leq \begin{cases} -\alpha_1\sqrt{|x_1|} - \alpha_2\dfrac{|x_2|^3}{|x_1|}, & x_1x_2<0 \ -\eta|x_2|, & x_1x_2\geq 0 \end{cases}$$ Uniform global asymptotic stability is ensured for

$\gamma > 2\sqrt2 D^{1.5} + D + \frac12$

Further, solution-based analysis shows that only $\gamma>D$ is needed for global finite-time convergence, indicating conservatism in Lyapunov-derived bounds (Ruderman et al., 5 Dec 2025).

4. Regularization and Chattering Suppression

The original non-overshooting quasi-continuous SMC exhibits chattering at the origin due to control discontinuity. Two regularization schemes have been introduced (Ruderman et al., 5 Jun 2025):

• Max-value regularization (denominator):

$$u(x) = - \frac{\gamma x_1 + |x_2| x_2}{\max\{\mu,|x_1|\}}$$

• Additive regularization:

$u(x) = - \frac{\gamma x_1 + |x_2| x_2}{|x_1|+\mu}$

where $\mu>0$ is a small parameter defining the boundary layer. Both schemes are globally Lipschitz and coincide with the discontinuous law outside $|x_1|>\mu$. Inside $|x_1|<\mu$, the dynamics are smooth and linear-oscillator-like, yielding exponential decay of the state and suppression of high-frequency control switching.

Residual chattering amplitude under disturbances is $O(\mu)$. For constant disturbance, the steady-state bound is $x_1^* = \frac{\mu}{\gamma}\bar d$; for harmonic resonance, the amplitude is $\mu\sqrt{\tilde d/\gamma}$, all vanishing as $\mu\to 0$ (Ruderman et al., 5 Jun 2025).

5. Extension to General Classes and Aerospace Application

Non-overshooting quasi-continuous sliding mode approaches generalize to uncertain second-order systems and nonzero reference tracking, including time-varying signals (Wang et al., 2 May 2024). Wang & Mao introduced a “2-sliding-mode” (two-stage) architecture in which a first subsystem ensures non-overshooting reachability to a strip in error-space, and a second subsystem guarantees monotonic, non-overshooting convergence under bounded-gain sliding mode action.

For practical chattering suppression, tanh-based quasi-continuous controllers were devised:

• When $|e_1| > e_{1c}$: $$u = k_c\,\tanh[P_c(e_2 + e_{2c}\,\text{sign}(e_1))]$$
• When $|e_1| \leq e_{1c}$: $u = k_2\,\tanh[p(e_2+ k_1 e_1)]$

Rigorous Lyapunov analysis confirms exponential (or finite-time) convergence of the tracking error $e_1$ without overshoot, for all admissible initial errors and above-threshold gains. Application to UAV trajectory tracking demonstrated rms position error below 0.04 m, zero overshoot, and smoothed actuator signals with the controller (Wang et al., 2 May 2024).

6. Implementation, Tuning, and Practical Guidelines

Implementation steps for the regularized non-overshooting quasi-continuous controller include real-time state measurement, denominator computation, control synthesis according to the chosen regularization, and standard explicit or Runge–Kutta time integration:

• Set gain $\gamma$ based on disturbance bound $D$ and robustness margin.
• Choose $\mu$ as small as actuator bandwidth allows, balancing residual error and local oscillation.
• For tanh-based SMC: select boundary-layer widths and table-driven gains according to initial condition partitions; ensure bounded control magnitude.

Numerical experiments confirm convergence properties, monotonic non-overshooting behavior, insensitivity to bounded disturbances, and chattering suppression aligned with theoretical bounds (Ruderman et al., 5 Jun 2025, Wang et al., 2 May 2024, Ruderman et al., 5 Dec 2025).

7. Significance and Comparison with Prior Approaches

Compared to earlier non-overshooting sliding mode designs, the modified quasi-continuous law is globally defined (including $x_1=0$), achieves uniform control amplitude limitation in large portions of the state space, and provides analytic solution formulas for both trajectory and convergence time. The design avoids the unboundedness and ill-definition issues observed in previous designs near the switching boundary. Piecewise-Lyapunov and solution-based analyses provide both strong theoretical foundations and practical verification of robustness and performance (Ruderman et al., 5 Dec 2025, Ruderman et al., 5 Jun 2025).

A summary of key attributes across main variations is given below:

Variant	Overshoot-Free	Finite-Time/Exp	Chattering Level	Bounded Control
Original Quasi-Continuous (Discontinuous)	Yes	Finite-Time	High at Origin	Yes
Max-Value / Additive Regularization	Yes	Finite + Exp	$O(\mu)$	Yes
2-Sliding-Mode, Tanh-Based	Yes	Exponential	None	Yes

All approaches achieve strict non-overshooting monotonic convergence, strong ISS/iISS robustness, and boundedness of the control signal across admissible states. This non-overshooting property is preserved even under disturbance, and practical versions support implementation on physical hardware including UAVs with empirical demonstration of performance (Ruderman et al., 5 Dec 2025, Ruderman et al., 5 Jun 2025, Wang et al., 2 May 2024).