# Fractional-Order LQR and State Observer for a Fractional-Order Vibratory System
## 8 Conclusions  
The present study investigated whether the LQR control method could be applied to a state equation with fractional-order derivative states. A design method for a fractional-order LQR control system was proposed in order to deal with a controlled object that included fractional-order derivative terms in the equation of motion.  
First, in order to explain the design method, a viscoelastic damper was modeled by an equation of motion including a fractional-derivative term. The equation of motion was able to be transformed into a fractional-order state equation.  
Second, a fractional-order LQR was applied to control the dynamics of the viscoelastically damped structure. A design method for fractional-order LQR control was proposed using the Lyapunov equation and Lagrange's method for undetermined multipliers. A new  
Figure 10: Results for states with fractional-order LQR control (proposed method: blue lines, Sierociuk and Vinagre method (S–V method): red broken lines).  
iteration-based approach to solve the algebraic Riccati equation was established in order to obtain the feedback gains for the fractional-order LQR.  
Third, a fractional-order state observer was constructed in order to estimate the fractional-order derivative states. All states for the viscoelastic damper system were successfully estimated using the fractional-order state observer.  
Fourth, a numerical calculation algorithm corresponding to a differential equation containing fractional derivative terms was introduced. The numerical calculation algorithm was based on the Grunwald-Letnikov definition of fractional differentiation. Numerical calculation results using the introduced method and the exact solution of the viscoelastic damper model were compared in order to investigate its calculation accuracy.  
Finally, using the introduced numerical simulation method, LQR control effect simulations were carried out. The numerical simulations revealed that the iterative method was valid for obtaining the feedback gains for the fractional-order LQR. The fractional-order LQR controller was confirmed to be capable of successfully suppressing vibrations appearing in a vibratory system with viscoelasticity. Moreover, the fractional-order LQR obtained by the Sierociuk and Vinagre method and that derived by the proposed method were compared with respect to their control effects.  
However, the fractional-order state observer in the present paper is not optimal, and therefore needs to be improved. In addition, control effects by a fractional-order LQR to suppress vibrations appearing in a vibratory system with viscoelasticity must be compared with control effects by an integer-order LQR.  
Conceptualization, M.K.; methodology, A.T., T.Y. and N.K.; software, A.T. and N.K.; validation, A.T.; formal analysis, A.T. and T.Y.; investigation, A.T. and M.K.; resources, N.K. and M.K.; data curation, A.T. and T.Y.; writing--original draft preparation, A.T. and M.K.; writing--review and editing, M.K.; visualization, A.T.; supervision, N.K. and M.K.; project administration, M.K. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. Not applicable. Not applicable. Not applicable. Not applicable. Not applicable. Conflicts of Interest: The authors declare no conflict of interest.