Design of Optimal Controllers for Unknown Dynamic Systems through the Nelder–Mead Simplex Method

• Published in: Mathematics (Basel) Vol. 9; no. 16; p. 2013
• Main Authors: Tsai, Hsun-Heng, Fuh, Chyun-Chau, Ho, Jeng-Rong, Lin, Chih-Kuang
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.08.2021
• Subjects: Controllers; Experiments; inverted pendulum; Mathematics; Noise measurement; Nonlinear systems; N–M simplex method; optimal control; Optimization; Parameter estimation; Penalty function; performance index; Performance indices; Simplex method; Simulation
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math9162013

This paper presents an efficient method for designing optimal controllers. First, we established a performance index according to the system characteristics. In order to ensure that this performance index is applicable even when the state/output of the system is not within the allowable range, we added a penalty function. When we use a certain controller, if the state/output of the system remains within the allowable range within the preset time interval, the penalty function value is zero. Conversely, if the system state/output is not within the allowable range before the preset termination time, the experiment/simulation is terminated immediately, and the penalty function value is proportional to the time difference between the preset termination time and the time at which the experiment was terminated. Then, we used the Nelder–Mead simplex method to search for the optimal controller parameters. The proposed method has the following advantages: (1) the dynamic equation of the system need not be known; (2) the method can be used regardless of the stability of the open-loop system; (3) this method can be used in nonlinear systems; (4) this method can be used in systems with measurement noise; and (5) the method can improve design efficiency.