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1

ℎ-Stability of set differential equations

Boukthir Sihem, Ghanmi Boulbaba, Basdouri Imed, Ichalal Dalil, Lerbet Jean

Archives of Control Sciences

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2023

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Vol. 33, No. 4

761--788

EN

In this paper, we introduce the notion of h-stability for set-valued differential equations. Necessary and sufficient conditions are established by using Lyapunov theory. Then, based on the obtained results, we study the ℎ-stability of perturbed and cascaded systems. Finally, an example illustrates the proposed theorems.

2

Design of an adaptive fuzzy variable structure compensator for the nonholonomic mobile robot in trajectory tracking task

Begnini M., Bertol W., Martins N. A.

Control and Cybernetics

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2018

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Vol. 47, no. 3

239--275

EN

A robust adaptive kinematic control strategy, based on the methodology of variable structure control is considered in this paper. Because the dynamics of mobile robots is subject to uncertainties and disturbances, a fuzzy compensator is adopted to estimate them. In order to minimize the tracking errors and to attenuate the chattering phenomenon, an adaptation law for the fuzzy compensator is obtained by Lyapunov stability theory so as to asymptotically stabilize the control system as well as guarantee the convergence of the tracking errors. In terms of comparison with the boundary layer variable structure controller, simulations and experiments verify the feasibility and eﬀectiveness of the proposed kinematic control strategy for the nonholonomic mobile robots under the incidence of uncertainties and disturbances.

3

Trajectory tracking of a wheeled mobile robot with uncertainties and disturbances: proposed adaptive neural control

Martins N. A., Alencar M., Lombardi W. C., Bertol D. W., Pieri E. R., Filho H. F.

Control and Cybernetics

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2015

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Vol. 44, no. 1

47--98

EN

This paper analyses a trajectory tracking control problem for a wheeled mobile robot, Rusing integration of a kinematic neural controller (KNC) and a torque neural controller (TNC), in which both the kinematic and dynamic models contain uncertainties and disturbances. The proposed adaptive neural controller (PANC) is composed of the KNC and the TNC and is designed with use of a modeling technique of Gaussian radial basis function neural networks (RBFNNs). The KNC is a variable structure controller, based on the sliding mode theory and is applied to compensate for the disturbances of the wheeled mobile robot kinematics. The TNC is an inertia-based controller composed of a dynamic neural controller (DNC) and a robust neural compensator (RNC) applied to compensate for the wheeled mobile robot dynamics, bounded unknown disturbances, and neural network modeling errors. To minimize the problems found in practical implementations of the classical variable structure controllers (VSC) and sliding mode controllers (SMC), and to eliminate the chattering phenomenon, the nonlinear and continuous KNC and RNC of the TNC are applied in lieu of the discontinuous components of the control signals that are present in classical forms. Additionally, the PANC neither requires the knowledge of the wheeled mobile robot kinematics and dynamics nor the timeconsuming training process. Stability analysis, convergence of the tracking errors to zero, and the learning algorithms for the weights are guaranteed based on the Lyapunov method. Simulation results are provided to demonstrate the eﬀectiveness of the proposed approach.

4

New fault tolerant control strategies for nonlinear Takagi-Sugeno systems

Ichalal D., Marx B., Ragot J., Maquin D.

International Journal of Applied Mathematics and Computer Science

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2012

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Vol. 22, no. 1

197-210

EN

New methodologies for Fault Tolerant Control (FTC) are proposed in order to compensate actuator faults in nonlinear systems. These approaches are based on the representation of the nonlinear system by a Takagi-Sugeno model. Two control laws are proposed requiring simultaneous estimation of the system states and of the occurring actuator faults. The first approach concerns the stabilization problem in the presence of actuator faults. In the second, the system state is forced to track a reference trajectory even in faulty situation. The control performance depends on the estimation quality; indeed, it is important to accurately and rapidly estimate the states and the faults. This task is then performed with an Adaptive Fast State and Fault Observer (AFSFO) for the first case, and a Proportional-Integral Observer (PIO) in the second. Stability conditions are established with Lyapunov theory and expressed in a Linear Matrix Inequality (LMI) formulation to ease the design of FTC. Furthermore, relaxed stability conditions are given with the use of Polya's theorem. Some simulation examples are given in order to illustrate the proposed approaches.

5

Modelowanie i sterowanie układu kula-belka

Burghardt A., Giergiel J.

Modelowanie Inżynierskie

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2009

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T. 6, nr 37

55-62

PL

Zaprezentowane podejście do procesu modelowania obiektu typu US z wykorzystaniem równań Lagrange'a umożliwiło otrzymanie modelu matematycznego w formie umożliwiającej jego wykorzystanie w procesie syntezy adaptacyjnego algorytmu sterowania. Ponadto otrzymany model matematyczny zastosowano do budowy emulatora obiektu kula-belka, który był wykorzystywany podczas badań symulacyjnych. Zaproponowany algorytm sterowania pozwolił na stabilizację pozycji kuli na belce, co wykazano w badaniach symulacyjnych oraz podczas weryfikacji na obiekcie rzeczywistym. Dalsze prace wykorzystujące zaproponowany model będą prowadzone w kierunku nie tylko stabilizacji pozycji kuli, ale również w kierunku sterowania nadążnego ruchem kuli po belce.

EN

The ball-and-beam problem is a benchmark for testing control algorithms. This paper we presents analysis and implementation adaptive control schemes on ball and beam system. Dynamic model for the system was derived using Lagrange equations. Control algorithm solves a stabilization under-actuated ball and beam system was created by using Lyapunov theory. The controllers designed for the system have been simulated by Matlab/Simulink.

6

Dynamika układów hybrydowych

Czornik A.

Zeszyty Naukowe. Automatyka / Politechnika Śląska

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2008

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z. 151

31-36

PL

Badanie dynamiki układów hybrydowych jest źródłem wielu interesujących i trudnych problemów matematycznych. Celem tej pracy jest zreferowanie postępów w badaniach własności dynamiki układów hybrydowych ze szczególnym uwzględnieniem problemu stabilności. Z problemem stabilności ściśle związane są problemy eksponencjalnego wzrostu i twierdzenia odwrotne do twierdzenia Lapunowa. Oba te zagadnienia zostaną zreferowane. W pracy zostanie zasygnalizowanych również kilka otwartych problemów.

EN

The study of the hybrid systems dynamics gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to review progress made in research of hybrid systems dynamics. We concentrate our attention on stability. Closely related to the concept of stability are the notations of exponential growth and converse Lyapunov theorems, both of which are discussed. We also point out some problems that remain open.