Variable structure control of linear time invariant fractional order systems using a finite number of state feedback law (original) (raw)
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Stabilization of fractional order systems using a finite number of state feedback laws
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In this paper, the stabilization of linear timeinvariant systems with fractional derivatives using a limited number of available state feedback gains, none of which is individually capable of system stabilization, is studied. In order to solve this problem in fractional order systems, the linear matrix inequality (LMI) approach has been used for fractional order systems. A shadow integer order system for each fractional order system is defined, which has a behavior similar to the fractional order system only from the stabilization point of view. This facilitates the use of Lyapunov function and convex analysis in systems with fractional order 1 < q < 2. To this end, an extremumseeking method is used for obtaining Lyapunov function and defining a suitable sliding sector in order to enable switching between available control gains for system stabilization. Consequently, using the LMI approach in fractional order systems, necessary and suf-ficient conditions are provided for stabilization of systems with fractional order 1 < q < 2 using a limited number of available state feedback gains which lead to variable structure control.
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This paper presents the stabilization problem of a linear time invariant fractional order (LTI-FO) switched system with order 1 < q < 2 by a single Lyapunov function whose derivative is negative and bounded by a quadratic function within the activation regions of each subsystem. The switching law is extracted based on the variable structure control with a sliding sector. First, a sufficient condition for the stability of an LTI-FO switched system with order 1 < q < 2 based on the convex analysis and linear matrix inequality (LMI) is presented and proved. Then a single Lyapunov function, whose derivative is negative, is constructed based on the extremum seeking method. A sliding sector is designed for each subsystem of the LTI-FO switched system so that each state in the state space is inside at least one sliding sector with its corresponding subsystem, where the Lyapunov function found by the extremum seeking control is decreasing. Finally, a switching control law is designed to switch the LTI-FO switched system among subsystems to ensure the decrease of the Lyapunov function in the state space. Simulation results are given to show the effectiveness of the proposed VS controller.
International Journal of Robust and Nonlinear Control, 2010
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Bulletin of the Polish Academy of Sciences Technical Sciences, 2014
In this paper, the stabilization problem of a autonomous linear time invariant fractional order (LTI-FO) switched system with different derivative order in subsystems is outlined. First, necessary and sufficient condition for stability of an LTI-FO switched system with different derivative order in subsystems based on the convex analysis and linear matrix inequality (LMI) for two subsystems is presented and proved. Also, sufficient condition for stability of an LTI-FO switched system with different derivative order in subsystems for more than two subsystems is proved. Then a sliding sector is designed for each subsystem of the LTI-FO switched system. Finally, a switching control law is designed to switch the LTI-FO switched system among subsystems to ensure the decrease of the norm of the switched system. Simulation results are given to show the effectiveness of the proposed variable structure controller.