# Fractional-Order LQR and State Observer for a Fractional-Order Vibratory System
## 1 Introduction  
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. However, fractional calculus is not novel or strange and has a long history that traces back to [PERSON], who was one of the founders of calculus [1]. During these past 30 years, fractional calculus has been applied in engineering fields, especially in control engineering. In control engineering, 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 other words, a revolution in the construction methods of control system has occurred. Fractional calculus has been introduced into proportional-integral-differential (PID) control systems as a typical classical control method. Several examples can be cited: PI\({}^{\lambda}\)D\({}^{u}\) control [2], robust fractional-order PI control using the auto-tuning method [3], the application of this fractional-order PI control method to a manipulator robot [4], fractional-order PD control of a smart beam [5, 6], a tuning rule for a fractional-order PD controller for a second-order system [7], fractional-order PID control for a magnetically levitated system [8], fractional-order PD control to suppress vibrations that appear in a three-story structure [9], a graphical design method for tuning a fractional-order PI or PD controller [10], a fractional-order PID controller to suppress vibration that occurs in a cantilever beam using electromagnetic actuators [11], the meaning of the parameters used in fractional-order PID controllers [12], fractional-order PID control for a pumped storage unit [13], fractional-order PID load-frequency control for a power system [14], tuning rules for fractional-order PID controllers [15], PI\({}^{\lambda}\)D\({}^{u}\) control for trajectory tracking by a 3-DOF parallel manipulator [16], fractional-order PID control for an automatic voltage regulatingsystem [17], and fractional-order PID control for a hydraulic turbine regulating system [18]. Thus, research examples are too numerous to mention. Fractional calculus has also been used for linear quadratic regulator (LQR) control as a typical modern control theory. In the sense that the controller is applicable to controlled systems with fractional-order dynamics, we use the terminology \"fractional-order LQR\" in this paper. Similarly, some examples can be cited: a state-feedback LQR controller for fractional-order systems [19], an LQR-based PID controller design using a fractional-order integral performance index [20], a fractional-order LQR [21], a fractional-order LQR control to suppress vibrations appearing in a three-story building [22], and fractional-order PID control for time-delay systems [23]. Again, research examples are too numerous to count. However, it is known that the procedure to derive a fractional-order LQR, as described in a previous report [21], is too complicated to be practical. Therefore, in the present paper, a novel iteration-based method is first proposed to solve the algebraic Riccati equation to derive the fractional-order LQR. Since all states of the controlled system are fed back in the LQR control scheme, all of the states are assumed to be detected. However, in reality, it is very difficult to measure the states originating from the fractional-order derivative terms. A state observer is then established in order to estimate all of the states, including the fractional-order derivative states. The fractional-order LQR control system is then constructed using the estimated states. Next, by way of example, a viscoelastic damper system modeled with a fractional-order differential equation is considered, and fractional-order LQR control is conducted for the viscoelastic damper system. The effectiveness of the proposed design method is then confirmed.