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

Abstract

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.

Example of Using Particle Swarm Optimization Algorithm with Nelder–Mead Method for Flow Improvement in Axial Last Stage of Gas–Steam Turbine

“…Nelder-Mead Method: This is a simplex-based method that does not require the calculation of derivatives, making it suitable for optimizing non-smooth functions. The method manipulates a simplex (a geometric figure formed by 𝑁 + 1 vertices in N dimensions) to explore the search space [56]. Key operations include the reflection, expansion, The proposed collaborative parallel algorithm is a combination of particle swarm optimization [50] and the Nelder-Mead method [51], called HNMPSO [52][53][54][55].…”

An Interval-Simplex Approach to Determine Technological Parameters from Experimental Data

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