Robustness of Delayed Higher Order Sliding Mode Control

Abstract: In this paper, the feasibility of recently developed higher order delayed sliding mode controllers is addressed. With this aim the robustness against the measurement noise and mismatched perturbations for the systems governed by such controllers is established using ISS implicit Lyapunov-Razumikhin function approach. To illustrate proposed results, a simulation example validating the efficiency of the method is provided.

• The paper establishes input-to-state stability (ISS) of delayed HOSMC using an implicit Lyapunov-Razumikhin approach under measurement noise and mismatched perturbations.
• The methodology employs LMI-based design parameters to achieve hyperexponential convergence while significantly reducing chattering compared to conventional methods.
• Numerical validations confirm that the delayed HOSMC delivers continuous control action and effective disturbance rejection, making it viable for practical control applications.

Robustness Analysis of Delayed Higher Order Sliding Mode Control

Introduction

The paper "Robustness of Delayed Higher Order Sliding Mode Control" (2512.18018) rigorously investigates the robustness properties of delayed higher order sliding mode controllers (HOSMCs). The main focus is to establish input-to-state stability (ISS) of these controllers in the presence of both measurement noise and mismatched perturbations for linear systems with arbitrary relative degree. The approach leverages the implicit Lyapunov-Razumikhin (ILR) method, significantly advancing prior work that was limited to noise-free settings.

Background and Problem Formulation

Higher order sliding mode control (HOSMC) methods are well-known for their robustness against matched uncertainties and finite-time convergence properties. Conventional HOSMC, however, suffers from two principal drawbacks: excessive chattering due to discontinuous control action, and implementation limitations, especially for orders higher than two. Recent paradigms introduce time-delay in the feedback, yielding delayed SMC and delayed HOSMC algorithms that generate continuous control signals, theoretically eliminating chattering and offering hyperexponential convergence.

The system under analysis is: $$\dot{x}(t) = Ax(t) + b(u(t) + d(t)) + \delta(t), \quad y(t) = x(t) + w(t)$$ where $d(t)$ is a bounded matched disturbance, $w(t)$ is measurement noise, and $\delta(t)$ is a mismatched disturbance orthogonal to $b$. The goal is to design a controller achieving uniform hyperexponential stabilization of the origin for all admissible $d$, and ISS in the presence of bounded $w$ and $\delta$. The controller is allowed to depend on the history of output values, thus being functionally delayed.

Delayed HOSMC Synthesis via Implicit Lyapunov Functions

The authors employ the implicit Lyapunov function (ILF) technique for controller synthesis, where the Lyapunov function $V_y(y)$ is implicitly defined through a polynomial in the output: $Q(V_y(y), y) = 0$ By extending this to a history-dependent functional $\Psi(y_t)$, the delayed HOSMC law is then: $$u(y_t) = YX^{-1} D_{\mathbf{r}}(\Psi^{-1}(y_t)) y(t)$$ with $\Psi(y_t)$ constructed to guarantee that the Lyapunov-Razumikhin conditions for hyperexponential stability are satisfied. The main tuning parameters, $X, Y$, and associated Lyapunov attributes, are computed via tractable LMIs, retaining compatibility with standard HOSMC synthesis. The free parameters $\chi > 1$ and delay $\eta>0$ offer adjustment of the convergence rate and delay margin.

Robustness Results: ISS under Measurement Noise and Mismatched Perturbations

The ISS analysis is the core contribution, extending robustness guarantees to bounded measurement noise and mismatched perturbations under mild structural constraints. Through careful utilization of the ILR technique and leveraging recent extensions to time-delay systems, the paper proves:

• The closed-loop system is ISS with respect to $w$ and $\delta$ (with $b^\top \delta = 0$), uniformly for all matched disturbances $d$ within the admissible set, when the parameter $\chi$ is chosen arbitrarily close to $1$.
• The LMI-based design ensures that all requisite robustness margins can be enforced via explicit relationships between Lyapunov function levels and disturbance magnitudes, leading to quantitative asymptotic gain estimates with respect to noise/perturbation amplitudes.

A technical obstacle surmounted in the analysis is the non-homogeneity of the system (owing to $\chi>1$ and delay), preventing direct recourse to classical homogeneity-based ISS proofs; the paper instead successfully adapts the Lyapunov-Razumikhin apparatus for this purpose.

Numerical Validation

Simulation results illustrate the effectiveness and practical implementation aspects of the delayed HOSMC scheme compared to classical (nondelayed) HOSMC. The main observed features are:

• The delayed HOSMC provides continuous control action with considerably attenuated chattering, even under nontrivial, rapidly-varying matched disturbances, bounded measurement noise, and mismatched perturbation vectors.
• The convergence to the origin in state space is hyperexponential, and, post-transient, the control input identifies and compensates the matched disturbance up to the noise/perturbation floor.
• In comparison, the reference HOSMC exhibits faster convergence but with pronounced chattering, underscoring the advantage of the delay-based modification in actuator-friendly scenarios.

Implications and Future Directions

From a theoretical perspective, this work cements the delayed HOSMC approach as robust not only to matched uncertainties (as classical SMC/HOSMC) but also to bounded measurement noise and a class of mismatched perturbations—reaching ISS, not merely bounded-input bounded-state properties. The constructive LMI formulation enables straightforward gain and rate tuning, and suggests extensions to both observer-based output feedback and dynamic output feedback frameworks.

Practically, the findings make the delayed HOSMC a compelling alternative for applications where continuous control and robustness to broadband disturbance spectra are required, especially in sampled-data and networked control systems subject to delays, noise, and modeling uncertainty.

Future research lines may focus on:

• Observer integration for output-feedback under partial/noisy state measurements, exploiting the proven ISS characteristics for cascade robustness.
• Extension to nonlinear and MIMO plant structures, potentially mediated by coordinate/homogenization transformations.
• Discretization and sampled-data implementation under asynchronous delays and quantization effects, which are crucial for embedded and cyber-physical system realization.

Conclusion

The paper establishes, via implicit Lyapunov-Razumikhin analysis, uniform ISS of delayed higher order sliding mode controllers with respect to bounded measurement noise and mismatched perturbations. The approach provides explicit design formulas and quantitative gain estimates, validated through numerical simulations which confirm dramatically reduced chattering without sacrificing robustness or accelerated convergence. These results further advance the state-of-the-art in robust continuous-time sliding mode control, with significant implications for both theory and application.