Time-optimal trajectories of generic control-affine systems have at worst iterated Fuller singularities

We consider in this paper the regularity problem for time-optimal trajectories of a single-input control-affine system on a n-dimensional manifold. We prove that, under generic conditions on the drift and the controlled vector field, any control u associated with an optimal trajectory is smooth out of a countable set of times. More precisely, there exists an integer K, only depending on the dimension n, such that the non-smoothness set of u is made of isolated points, accumulations of isolated points, and so on up to K-th order iterated accumulations.     

High order sliding-mode control for uncertain nonlinear systems with relative degree three

A novel high order sliding-mode control is proposed based on finite state machine by combination of relay algorithm and an improved second order algorithm to solve uncertain nonlinear system stabilization with relative degree three in finite time. This approach drives finite state machines to switch according to sliding variable and first order derivative of it, forces sliding variable, first order derivative and second order derivative of it to zero, without the knowledge of second order derivative of sliding variable, and stabilizes the system in finite time.     

Chattering in dynamic mathematical two-phase flow models

This paper presents a chattering problem which arises in a dynamic mathematical two-phase flow model. The real system under study is also introduced, the DISS test facility, a parabolic-trough solar thermal power plant. The heat transfer fluid in the DISS facility is the steam-water mixture. A dynamic model of this plant, using Modelica as the modeling language, was previously developed in order to study its behavior. Chattering arises in the pipe model reducing the computational performance and hence limiting the applicability of the model. The problem source is studied and analysed together with an approach to the problem which is based on the smooth interpolation of some thermodynamic properties.     

Dynamics of windscreen wiper blades: Squeal noise,reversal noise and chattering

The motion of a windshield wiper blade is modelled by a mass-spring-damper system on a moving frictional surface. The system dynamics is time-varying, since three different regimes of motion, characterized by different degrees of freedom, are possible. Indeed the system, which schematizes a blade cross-section, can experience stick and slip motions when it is in contact with the glass surface, and free-flight motion when it is detached. The contact between the system and the surface is governed by Stribeck׳s friction law and Poisson׳s impact law, which make the dynamics non-smooth. The model is numerically implemented in an event-driven code, and simulations are performed which reproduce the three basic classes of undesired oscillations observed in the motion of real windscreen wipers, i.e., squeal, reversal and chattering noises. Attention is focused on the causes of these vibrations, and remedies for reducing or avoiding them are proposed.     

A chattering-free robust adaptive sliding mode controller for synchronization of two different chaotic systems with unknown uncertainties and external disturbances

In this paper, a robust adaptive sliding mode controller (RASMC) is introduced to synchronize two different chaotic systems in the presence of unknown bounded uncertainties and external disturbances. The structure of the master and slave chaotic systems has no restrictive assumption. Appropriate adaptation laws are derived to tackle the uncertainties and external disturbances. Based on the adaptation laws and Lyapunov stability theory, an adaptive sliding control law is designed to ensure the occurrence of the sliding motion even when both master and slave systems are perturbed with unknown uncertainties and external disturbances. Since the conventional sliding mode controllers contain the sign function, the undesirable chattering is occurred. We propose a new simple adaptive scheme to eliminate the chattering. Finally, numerical simulations are presented to verify the usefulness and applicability of the proposed control strategy.     

Optimal control for a Timoshenko beamContrôle optimal d'une barre de Timoshenko

We consider a linear model of a rotating Timoshenko beam, which is clamped at one end to a disk the other being free. The motion of the beam is controlled by the angular acceleration of the disk. We study the minimization problem of mean square deviation of the Timoshenko beam from a given position. For the minimization problem of the first mode we prove that optimal control is the chattering control, i.e., it has an infinite number of switches in a finite time interval. We construct a suboptimal control with a finite number of switches. To cite this article: M.I. Zelikin, L.A. Manita, C. R. Mecanique 334 (2006).     

Chattering limit for a model of harvesting in a rapidly changing environment

An example of harvesting in an environment that changes rapidly and frequently is examined. The changes are unpredictable, yet their statistical distribution is known. The parameters, however, can be measured on line, and used in the control policy. We show that then chattering systems adequately serve as nominal systems, optimal solutions of which provide near optimal solutions for the system with rapid changes.This research was supported by a grant from the Basic Research Fund, the Israel Academy of Science and Humanities. Zvi Artstein is the incumbent of the Hettie H. Heineman Professorial Chair in Mathematics.     

Non-fragile observer-based passive control for uncertain time delay systems subjected to input nonlinearity

The problem of non-fragile observer-based passive control for uncertain time delay systems subjected to input nonlinearity is investigated by using sliding mode control. A novel control law is established such that the sliding surface in the state-estimation space can be reached in a finite time and chattering reduction is obtained. A sufficient condition for passivity and asymptotic stability of the combined system is derived via linear matrix inequality (LMI). Finally, a simulation example is presented to show the validity and advantages of the proposed method.     

Dynamic soft variable structure control of singular systems

The dynamic soft variable structure control (VSC) of singular systems is discussed in this paper. The definition of soft VSC and the design of its controller modes are given. The stability of singular systems with the dynamic soft VSC is proposed. The dynamic soft variable structure controller is designed, and the concrete algorithm on the dynamic soft VSC is given. The dynamic soft VSC of singular systems which was developed for the purpose of intentionally precluding chattering, achieving high regulation rates and shortening settling times enhanced the dynamic quality of the systems. It is illustrated the feasibility and validity of the proposed strategy by a simulation example, and an outlook on its auspicious further development is presented.     

Continuous output integral sliding mode control for switched linear systems

A continuous output integral sliding mode robustification methodology for switched uncertain linear time invariant systems with state-dependent location transitions and dwell time is presented. The robustifying methodology is based on the adjustment of the super-twisting algorithm gains to assure the convergence time and the attenuation of the chattering. The use of the adjusted STA allows to reconstruct the states theoretically exactly before half of the dwell time without the usage of filters, via a continuous cascade observer. Moreover, it allows to generate a continuous control signal that is turn on after the observer has converged, guaranteeing theoretically exact compensation of the matched uncertainties/perturbations before the dwell time.