Abstract
The present study uses linear quadratic regulator (LQR) theory to control a vibratory system modeled by a fractional-order differential equation. First, as an example of such a vibratory system, a viscoelastically damped structure is selected. Second, a fractional-order LQR is designed for a system in which fractional-order differential terms are contained in the equation of motion. An iteration-based method for solving the algebraic Riccati equation is proposed in order to obtain the feedback gains for the fractional-order LQR. Third, a fractional-order state observer is constructed in order to estimate the states originating from the fractional-order derivative term. Fourth, numerical simulations are presented using a numerical calculation method corresponding to a fractional-order state equation. Finally, the numerical simulation results demonstrate that the fractional-order LQR control can suppress vibrations occurring in the vibratory system with viscoelastic damping.
Highlights
Fractional calculus is a form of calculus in which the order of differentiation and integration is expanded or generalized from an integer number to a non-integer number
A viscoelastically damped structure is considered as an example of a fractionalorder vibratory system in order to explain linear quadratic regulator (LQR) control
Using the feedback gains obtained with the iteration-based method and the fractional-order state observer, it is proven that fractional-order LQR control can be carried out for a controlled object that includes fractional-order derivative states in its state equation
Summary
Fractional calculus is a form of calculus in which the order of differentiation and integration is expanded or generalized from an integer number to a non-integer number. Dynamical systems, which cannot be described sufficiently by conventional integer-order calculus, have been modeled with fractional calculus, and controllers that use fractional calculus have been designed. In the present paper, a novel iteration-based method is first proposed to solve the algebraic Riccati equation to derive the fractional-order LQR. The remainder of the present paper is organized as follows
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