Physical Significance Variable Control for a Class of Fractional-Order Systems
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Physical Significance Variable Control for a Class of Fractional-Order Systems Mircea Ivanescu1
· Nirvana Popescu2 · Decebal Popescu2
Received: 2 June 2019 / Revised: 13 August 2020 / Accepted: 21 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper studies a class of fractional-order systems (FOSs) and proposes control laws based on physical significance variables. Lyapunov techniques and the methods that derive from Yakubovici–Kalman–Popov Lemma are used, and the frequency criterions that ensure asymptotic stability of the physical significance variable closedloop system are inferred. The asymptotic stability of the observer system is studied for a sector control law where the output is defined by the physical significance variables. Frequency criterions and conditions for asymptotic stability are determined. The control techniques are extended to a class of linear delay fractional-order systems and nonlinear FOS. Numerical simulations of a class of systems described by fractional-order models show the method efficiency. Keywords Delay fractional-order systems · Fractional-order systems · Frequency criterion · Fractional-order observer
1 Introduction Fractional-order models have been extensively used in the control of a class of systems. Examples include damping mechanical systems, polymer elastic structures and biomechanics systems. The mathematical representation of these systems corresponds to the introduction of fractional-order differential operator in the constitutive equa-
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Mircea Ivanescu [email protected] Nirvana Popescu [email protected] Decebal Popescu [email protected]
1
Department of Mechatronics, University of Craiova, 13 Cuza Street, 1100 Craiova, Romania
2
Department of Computer Science, University Polytehnica of Bucharest, 313 Splaiul Independentei, Bucharest, Romania
Circuits, Systems, and Signal Processing
Fig. 1 Haptic Glove System
Fig. 2 Wheelchair Control System for Disability Human Operator
tions. As a consequence, the governing equation of the system involves fractional-order derivatives and the dynamic model is defined as a fractional-order system (FOS). A large class of systems that monitors or controls the human activities are described by FOS. One of the most significant example is the Haptic Glove System (HGS) for the rehabilitation of the patients that have a diagnosis of a cerebrovascular accident (Fig. 1). The control system of the wheelchairs for the persons with disabilities represents another class of models described by FOS (Fig. 2). In this case, the human operator is represented by the persons with hemiparesis/hemiplegia, with motor restrictions (armor leg-emphasized hemiparesis) with serious brain damage. In the modeling process of these systems, the virtual variables, without a physical significance, are introduced beside the physical variables. Control and stability of FOS is a widely investigated issue by techniques that derive from Lyapunov method. To our knowledge, it should be noted
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