arXiv subjects

Leonid Fridman

Publications and source records attributed to Leonid Fridman.

At least 19 recordsLinked to original sources

arXiv2026

Multi-layer barrier adaptation of the discrete-time super-twisting controller

In digital sliding mode control implementations, discretization-induced chattering and inter-sample blindness can severely degrade the closed-loop performance, especially in case of fast perturbations. This paper addresses these challenges for a discrete-time implementation of the super-twisting sliding mode controller. Building upon recent results on barrier-function-modulated super-twisting algorithms, a nested architecture employing multiple barriers is discretized using an eigenvalue-based exact matching approach. The resulting discrete-time controller preserves the adaptive and robustness properties established in continuous time, while ensuring consistent stability behavior at the sampling level. The proposed framework is validated through numerical simulations. The results highlight the effectiveness of multi-layer barrier adaptation for discrete-time sliding mode control applications.

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arXiv2025

Robustness of Delayed Higher Order Sliding Mode Control

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.

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arXiv2025

Multi-layer barrier function-based adaptive super-twisting controller

This article presents an adaptive Super-Twisting Sliding Mode Control framework for uncertain first-order systems, with rate-bounded perturbations, where the bound is constant but unknown. Positive definite barrier functions, when used in self-tuning super-twisting controllers may introduce some conservatism in relation to initial estimations of the perturbation rate bound. Moreover, discrete time implementation of the algorithm does not necessarily guarantee the boundedness of the closed-loop trajectories when sudden changes in the perturbation occur in between two time samples. The salient features of the proposed methodology pertain to extending the use of positive semidefinite barrier functions to Super-Twisting controller adaptation and the employment of a "nested barriers" scheme that ensures boundedness of the solutions even for "unfavourable" perturbations-to-sampling time ratios. The stability of the closed-loop system is assessed via Lyapunov analysis and simulations demonstrate the efficacy of the proposed framework.

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arXiv2025

Discrete implementations of sliding-mode controllers with barrier-function adaptations require a revised framework

Challenges in the discrete implementation of sliding-mode controllers (SMC) with barrier-function-based adaptations are analyzed, revealing fundamental limitations in conventional design frameworks. It is shown that under uniform sampling, the original continuous-time problem motivating these controllers becomes theoretically unsolvable under standard assumptions. To address this incompatibility, a revised control framework is proposed, explicitly incorporating actuator capacity constraints and sampled-data dynamics. Within this structure, the behavior of barrier function-based adaptive controllers (BFASMC) is rigorously examined, explaining their empirical success in digital implementations. A key theoretical result establishes an explicit relation between the actuator capacity, the sampling rate, and the width of the barrier function, providing a principled means to tune these controllers for different application requirements. This relation enables the resolution of various design problems with direct practical implications. A modified BFASMC is then introduced, systematically leveraging sampling effects to ensure finite-time convergence to a positively invariant predefined set, a key advancement for guaranteeing predictable safety margins.

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arXiv2024

Perturbed Integrators Chain Control via Barrier Function Adaptation and Lyapunov Redesign

Lyapunov redesign is a classical technique that uses a nominal control and its corresponding nominal Lyapunov function to design a discontinuous control, such that it compensates the uncertainties and disturbances. In this paper, the idea of Lyapunov redesign is used to propose an adaptive time-varying gain controller to stabilize a class of perturbed chain of integrators with an unknown control coefficient. It is assumed that the upper bound of the perturbation exists but is unknown. A proportional navigation feedback type gain is used to drive the system's trajectories into a prescribed vicinity of the origin in a predefined time, measured using a quadratic Lyapunov function. Once this neighborhood is reached, a barrier function-based gain is used, ensuring that the system's trajectories never leave this neighborhood despite uncertainties and perturbations. Experimental validation of the proposed controller in Furuta's pendulum is presented.

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arXiv2023

Super-twisting based sliding mode control of hydraulic actuator without velocity state

This paper provides a novel surface design and experimental evaluation of a super-twisting algorithm (STA) based control for hydraulic cylinder actuators. The proposed integral sliding surface allows to track a sufficiently smooth reference without using the velocity state which is hardly accessible in the noisy hydraulic systems. A design methodology based on LMI's is given, and the STA gains are designed to be adjusted by only one free parameter. The feasibility and effectiveness of the proposed control method are shown on a standard hydraulic test bench with one linear degree of freedom and passive load, where a typical motion profile is tracked with a bounded average error below 1 % from the total drive effective output.

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arXiv2023

An arbitrary-order predefined-time exact differentiator for signals with exponential growth bound

There is a growing interest in differentiation algorithms that converge in fixed time with a predefined Upper Bound on the Settling Time (UBST). However, existing differentiation algorithms are limited to signals having an $n$-th order Lipschitz derivative. Here, we introduce a general methodology based on time-varying gains to circumvent this limitation, allowing us to design $n$-th order differentiators with a predefined UBST for the broader class of signals whose $(n+1)$-th derivative is bounded by a function with bounded logarithmic derivative. Unlike existing methods whose time-varying gain tends to infinity, our approach yields a time-varying gain that remains bounded at convergence time. We show how this last property maintains exact convergence using bounded gains when considering a compact set of initial conditions and improves the algorithm's performance to measurement noise.

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arXiv2022

On the Notion of Safe Sliding Mode Control

Within the framework of Sliding Mode Control, safety critical control looks at two problems of theoretical significance: construct a sliding manifold that does not intersect a given unsafe set of the state space, and design a robust controller that safely takes the system trajectory to the manifold during the reaching phase and enforces the sliding motion. In this technical note, we address both problems. We introduce the notion of Safe Sliding Manifold (SSM) of relative degree one, and show that it is possible to construct it from the gradient of a class of Lyapunov-like energy functions, previously reported in the literature, used to asses stability and safety of a nominal system. The constructed SSM allows us providing a robust safe controller for a fair general class of uncertain nonlinear systems.

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arXiv2022

Sensitivity analysis and experimental evaluation of PID-like continuous sliding mode control

Continuous higher order sliding mode (CHOSM) controllers represent an efficient tool for disturbance rejection. For the systems with relative degree r, CHOSM approaches provide theoretically exact compensation of the matched Lipschitz perturbation, ensuring the finite-time convergence to the (r+1)-th sliding-mode set, by using only information on the sliding output and its derivatives up to the order (r-1). In this paper, we investigate the disturbance rejection properties of a PID-like CHOSM controller, as the simplest and intuitively clear example which incorporates nonlinear actions on the output error, its derivative, and integration of its sign. We use the harmonic balance approach and develop an analysis of propagation of the matched Lipschitz perturbation through the control loop in frequency domain. The resulted solution appears in form of the Bode-like loci which depend also on the amplitude of harmonic disturbances. Such amplitude-frequency characteristics allow certain comparability with standard disturbance sensitivity functions of a linear PID-controlled system in frequency domain. Also a simple and straightforward design procedure for the robust linear PID controller targeting the second-order system plants under investigation is provided for benchmarking. Additional (parasitic) actuator dynamics, which can lead to self-induced steady oscillations, i.e. chattering, is ditto respected. A detailed experimental case study, accomplished on an electro-mechanical actuator in the laboratory setting, highlight and make the pros and cons of both PID and CHOSM controllers well comparable for a broadband disturbance rejection.

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arXiv2022

Analysis of relay-based feedback compensation of Coulomb friction

Standard problem of one-degree-of-freedom mechanical systems with Coulomb friction is revised for a relay-based feedback stabilization. It is recalled that such a system with Coulomb friction is asymptotically stabilizable via a relay-based output feedback, as formerly shown in [1]. Assuming an upper bounded Coulomb friction disturbance, a time-optimal gain of the relay-based feedback control is found by minimizing the derivative of the Lyapunov function proposed in [2] for the twisting algorithm. Furthermore, changing from the discontinuous Coulomb friction to a more physical discontinuity-free one, which implies a transient presliding phase at motion reversals, we analyze the residual steady-state oscillations. This is in the sense of stable limit cycles, in addition to chattering caused by the actuator dynamics. The numerical examples and an experimental case study accompany the provided analysis.

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arXiv2022

Barrier Function-Based Adaptive Continuous Higher-Order Sliding Mode Controllers with Unbounded Perturbations

In this paper, two classes of continuous higher order adaptive sliding mode controllers based on barrier functions are developed for a perturbed chain of integrators with unbounded perturbations. Both classes provide finite-time convergence of system states to a predefined domain using a continuous control signal. The first class of adaptive controllers does not require any assumptions about the perturbation; however, it can provide unbounded control gains. To ensure bounded control gains, and in the case of a Lipschitz perturbation, a second class of adaptive controllers, called the adaptive higher order Super-Twisting (HOST) algorithm, is developed.

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arXiv2021

Multivariable Super-Twisting Algorithm for Systems with Uncertain Input Matrix and Perturbations

This paper proposes a Lyapunov approach to the design of a multivariable generalized Super-Twisting algorithm (MGSTA), which is able to control a system with perturbations and uncertain control matrix, both depending on time and the system states. The presented procedure shows that, under reasonable assumptions for the uncertainties, it is always possible to find a set of constant gains for the MGSTA in order to ensure global and robust finite-time stability of the system's outputs. Simulation results on an omnidirectional mobile robot illustrate the performance of the MGSTA.

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arXiv2021

Robust exact differentiators with predefined convergence time

The problem of exactly differentiating a signal with bounded second derivative is considered. A class of differentiators is proposed, which converge to the derivative of such a signal within a fixed, i.e., a finite and uniformly bounded convergence time. A tuning procedure is derived that allows to assign an arbitrary, predefined upper bound for this convergence time. It is furthermore shown that this bound can be made arbitrarily tight by appropriate tuning. The usefulness of the procedure is demonstrated by applying it to the well-known uniform robust exact differentiator, which is included in the considered class of differentiators as a special case.

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arXiv2021

A uniform reaching phase strategy in adaptive sliding mode control

In adaptive sliding mode control methods, an updating gain strategy associated with finite-time convergence to the sliding set is essential to deal with matched bounded perturbations with unknown upper-bound. However, the estimation of the finite time of any adaptive design is a complicated task since it depends not only on the upper-bound of unknown perturbation but also on the size of initial conditions. This brief proposes a uniform adaptive reaching phase strategy (ARPS) within a predefined reaching-time. Moreover, as a case of study, the barrier function approach is extended for perturbed MIMO systems with uncertain control matrix. The usage of proposed ARPS in the MIMO case solves simultaneously two issues: giving a uniform reaching phase with a predefined reaching-time and adapting to the perturbation norm while in a predefined vicinity of the sliding manifold.

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arXiv2021

Super Twisting based Lyapunov Redesign for Uncertain Linear Delay Systems

We present a new continuous Lyapunov Redesign (LR) methodology for the robust stabilization of a class of uncertain time-delay systems that is based on the so-called Super Twisting Algorithm. The main feature of the proposed approach is that allows one to simultaneously adjust the chattering effect and achieve asymptotic stabilization of the uncertain system, which is lost when continuous approximation of the unit control is considered. At the basis of the Super Twisting based LR methodology is a class of Lyapunov-Krasovskii functionals, whose particular form of its time derivative allows one to define a delay-free sliding manifold on which some class of smooth uncertainties are compensated.

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arXiv2020

Chattering analysis of Lipschitz continuous sliding-mode controllers

Lipschitz continuous sliding-mode controllers (LCSMC) are developed as the integral of discontinuous SMC, producing control signals of finite slope. Nevertheless, LCSMC still generate chattering in the presence of fast parasitic dynamics. In this paper, an analysis of chattering in systems driven by LCSMC is performed using the Harmonic Balance (HB) approach. Two kinds of LCSMC are considered: the first one is based on a linear sliding variable (LSV) and the second one on a terminal switching variable (TSV). Predictions of the amplitude and frequency of self-excited oscillations allowed to compute the average power consumed by the controller, in order to maintain the trajectories into the real sliding mode. A comparison of LCSMC with the Super-Twisting controller (STC), which produce a continuous control signal with infinite slope, is performed. Theoretical predictions and simulation results confirm that LCSMC may induce fast-oscillations (chattering) of smaller amplitude and average power than those ones caused by the STC. But, surprisingly, the chattering generated by LSV-LCSMC could be smaller than that one caused by TSV-LCSMC, when the actuators are fast enough. On the other hand, it tuns that if the sliding dynamics of the LSV-LCSMC closed-loop is of similar speed as the actuators dynamics, the system can loose even practical stability.

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arXiv2020

A Lyapunov Approach to Barrier-Function Based Time-Varying Gains Higher Order Sliding Mode Controllers

In this paper, we present Lyapunov-based {\color{black}time varying} controllers for {\color{black}fast} stabilization of a perturbed chain of integrators with bounded uncertainties. We refer to such controllers as {\color{black}time varying} higher order sliding mode controllers since they are designed for nonlinear Single-Input-Single-Output (SISO) systems with bounded uncertainties such that the uncertainty bounds are unknown. %{\color{blue} OLD: Our main result states that, given any neighborhood $\varepsilon$ of the origin, we determine a controller insuring, for every uncertainty bounds, that every trajectory of the corresponding closed loop system enters $\varepsilon$ and eventually remains there. Furthermore, based on the homogeneity property, a new asymptotic accuracy, which depends on the size of $\varepsilon$, is presented.} We provide a time varying control feedback law insuring verifying the following: there exists a family $(D(t))_{t\geq 0}$ of time varying open sets decreasing to the origin as $t$ tends to infinity, such that, for any unknown uncertainty bounds and trajectory $z(\cdot)$ of the corresponding system, there exists a positive positve $t_z$ for which $z(t_z)\in D(t_z)$ and $z(t)\in D(t)$ for $t\geq t_z$. %enters convergence in finite time of all the trajectories to a time varying domain $D(t)$ shrinking to the origin and their maintenance there. Hence, since the function $η(t)$ tends to zero, this leads the asymptotic convergence of all the trajectories to zero. The effectiveness of these controllers is illustrated through simulations.

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arXiv2019

Barrier function-Based Variable Gain Super-Twisting Controller

In this paper, a variable gain super-twisting algorithm based on a barrier function is proposed for a class of first order disturbed systems with uncertain control coefficient and whose disturbances derivatives are bounded but they are unknown. The specific feature of this algorithm is that it can ensure the convergence of the output variable and maintain it in a predefined neighborhood of zero independent of the upper bound of the disturbances derivatives. Moreover, thanks to the structure of the barrier function, it forces the gain to decrease together with the output variable which yields the non-overestimation of the control gain.

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