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Tham khảo tài liệu 'sabatier agrawal machado advances in fractional calculus episode 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | PSEUDO-RATIONAL APPROXIMATIONS TO FO-LTI SYSTEMS 69 Fig. 3. Step response comparison of the optimum FOLPD and the original model. controller for the optimally reduced model Gr s and let us see if the designed controller still works for the original system. The integer order PID controller to be designed is in the following form Í 1 Ts GcM Kp 1 1 id Nr i 12 The optimum ITAE criterion-based PID tuning formula 43 can be used 0.7303 0.5307T L T 0.5L Kp ---------K t l 13 Ti T 0.5L Td 14 1 0.5L Based on this tuning algorithm a PID controller can be designed for Gr s as follows L 0.63 T 3.5014 K 0.9951 N 10 Ti T 0.5 L Kp 0.7303 0.5307 T L Ti K T L Td 0.5 L T T 0.5 L Kp Ti Td Gc Kp 1 1 Ti s Td s Td N s 1 The parameters of the PID controller are then Kp 3.4160 Tj 3.8164 Td 0.2890 and the PID controller can be written as 1.086s2 3.442s 0.8951 Gc s ----- ----------- ck 0.0289s2 s 70 Xue and Chen Finally the step response of the original FO-LTI with the above-designed PID controller is shown in Fig. 4. A satisfactory performance can be clearly observed. Therefore we believe the method presented in this paper can be used for integer-order controller design for general FO-LTI systems. Time sec Fig. 4. Step response of fractional-order plant model under the PID controller. 5 Concluding Remarks In this paper we presented a procedure to achieve pseudo-rational approximation to arbitrary FO-LTI systems with suboptimum H-norm. Relevant MATLAB codes useful for practical applications are also given in the appendix. Through illustrations we show that the pseudo-rational approximation is simple and effective. It is also demonstrated that this suboptimum approximation method is effective in designing integer order controllers for FO-LTI systems in general form. Finally we would like to remark that the so-called pseudo-rational approximation is essentially by cascading irrational transfer function a time delay and a rational transfer function. Since a delay element is also infinite .
