The Stability of a Class of Fractional Order Switching System with Time-Delay Actuator (original) (raw)
Related papers
ISA Transactions, 2012
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.
On the Stabilization of Linear Time Invariant Fractional Order Commensurate Switched Systems
Asian Journal of Control, 2014
In this paper, the stabilization problem of a linear time invariant fractional order (LTI-FO) switched system is outlined. First, the sufficient condition for stability of a commensurate LTI-FO switched system based on the convex analysis and linear matrix inequality (LMI) is presented. Then, a single Lyapunov function is constructed based on the optimization method. Then, 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 optimization method 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 variable structure controller.
Stability of fractional order switching systems
Computers & Mathematics with Applications, 2013
This paper addresses the stabilization issue for fractional order switching systems. Common Lyapunov method is generalized for fractional order systems and frequency domain stability equivalent to this method is proposed to prove the quadratic stability. Some examples are given to show the applicability and effectiveness of the proposed theory.
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.
Communications in Nonlinear Science and Numerical Simulation, 2011
In this paper, an approach based on the variable structure control is proposed for stabilization of linear time invariant fractional order systems (LTI-FOS) using a finite number of available state feedback controls, none of which is capable of stabilizing the LTI-FOS by itself. First, a system with integer order derivatives is defined and its existence is proved, which has stability equivalent properties with respect to the fractional system. This makes it possible to use Lyapunov function and convex analysis in order to define the sliding sector and develop a variable structure control which enables the switching between available control gains and stabilizing the fractional order system.
Analysis and Control Design for a Class of Fractional Order Time-Delay Systems
Algerian Journal of Signals and Systems, 2020
In this paper, we consider a class of fractional order time-delay systems and propose a fractional order control design for their stabilization. The controller parameter’s adjustment is achieved in two steps: first, the relay approach is used to compute satisfactory classical PID coefficients, namely kp, Ti and Td. Then, the fractional orders ʎ and µ are optimized using performance criteria. Simulation results show the efficiency of the proposed design technique and its ability to enhance the PID control performance.
TURKISH JOURNAL OF ELECTRICAL ENGINEERING & COMPUTER SCIENCES, 2014
This paper presents a design method for a sliding mode controller with the contribution of a fractional order differential operator. The conventional sliding mode controller has been widely studied in different control applications. This paper proposes that the fractional order differential operator enlarges the output span of the classical sliding mode controller to obtain a better-fitting control signal for enhanced control performance. The sliding surface and the equivalent control law are modified with the addition of a fractional differential operator and a conventional one. The proposed sliding mode controller with fractional order differentiation is applied to the unstable time delay systems successfully.
Design of Fractional Order Sliding Mode Controller for a class of nonlinear systems
2018
In this article, a novel nonlinear sliding mode controller is proposed to control a class of nonlinear systems. The proposed control scheme is based on conformable fractional order operators. The stability analysis is performed using Lyapunov direct method. Simulation results show high convergence speed, chattering reduction and small control effort.
2017
Stability and stabilization of delayed linear fractional order systems with input delay using linear matrix inequalities were considered in the present study. At first, the input delay fractional order system was changed into a free delay fractional order system using an alternate variable. Thereafter, using a state feedback, the systems were turned into a closed loop. Later on, the system stabilization was examined by applying linear matrix inequality theorems and, finally, an example was used to demonstrate the efficiency of the proposed method.
Stability Analysis and Fractional Order Controller Design for Control System
2017
In this paper, a new approach to stability for fractional order control system is proposed. Here a dynamic system whose behavior can be modeled by means of differential equations involving fractional derivatives. Applying Laplace transforms to such equations, and assuming zero initial conditions, causes transfer functions with no integer powers of the Laplace transform variable s to appear. In recent time, the application of fractional derivatives has become quite apparent in modeling mechanical and electrical properties of real materials. Fractional integrals and derivatives have found wide application in the control of dynamical systems when the controlled system and the controller are described by a set of fractional order differential equations. In the existing work, a fractional order system has been signified by a higher integer order system. Fractional calculus provides an excellent instrument for the description of memory and hereditary properties of various materials and pr...