Composite feedback control of linear singularly perturbed systems: a bond graph approach
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Composite feedback control of linear singularly perturbed systems: a bond graph approach Gilberto Gonzalez-A1
· Noe Barrera-G2 · J. Aaron Padilla3 · Gerardo Ayala4
Received: 17 September 2019 / Revised: 2 September 2020 / Accepted: 12 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract A procedure to obtain a composite control of linear systems with singular perturbations modelled by bond graphs is presented. The state feedback gains of the composite control based on the slow and fast bond graph models are separately designed. The composite control system is formed by: (1) The original Bond Graph in an Integral causality assignment (BGI) with a state feedback of the fast and slow dynamics. (2) An additional bond graph denoted by Singular Perturbed Bond Graph (SPBG) with a state feedback whose storage elements have integral and derivative causality for the slow and fast dynamics, respectively. The advantages of this approach are: (1) From the BGI, the reduced fast models for open and closed loop systems in a direct way are obtained. (2) From the SPBG, the reduced slow models are determined where the change of the causality of the storage elements for the fast dynamics produces inverse matrices required in the traditional approach. (3) The mathematical models are not required. A junction structure of the bond graph with a composite control to determine the mathematical model of the closed loop system is proposed. Finally, the modelling and control of two illustrative examples applying the proposed methodology are described. Keywords Bond graph · Singular perturbations · Composite control
1 Introduction The mathematical model of systems represents a fundamental task in control theory. Frequently, these systems determine high-order state equations. However, many systems can contain different time scales due to parasitic parameters. These
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Gilberto Gonzalez-A [email protected] Noe Barrera-G [email protected] J. Aaron Padilla [email protected] Gerardo Ayala [email protected]
1
Faculty of Electrical Engineering, Graduate Studies Division of the Faculty of Mechanical Engineering, University of Michoacan, Morelia, Mexico
2
Techonological Institute of Morelia, Morelia, Mexico
3
Faculty of Electrical Engineering, University of Michoacan, Morelia, Mexico
4
School of Sciences of Engineering and Technology, Autonomous University of Baja California, Tijuana, Mexico
systems are characterized by having slow and fast dynamics and applying the appropriate procedures, these dynamics can be separated. The corresponding sub-systems of these dynamics permit to have reduced models whose analysis and control design are direct and simpler [1,2]. Composite control has been applied to a variety of systems, different research areas, schemes and for different purposes, for example, a composite control to get the stabilization of discrete time systems is proposed in [3]. The nonlinear and robust composite control for a DC motor is presented in [4]. In [5] it is proposed a composite
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