Polynomial system optimization methods for nonlinear control parameter selection in engineering systems

Abstract

Nonlinear engineering systems involve complex interrelationships between parameters and sensitivities to operating conditions. Finding optimal control parameters for such systems becomes an optimization challenge. Typical tuning methods tend to focus on nominal performance and often lead to controllers which are not robust enough under the influence of uncertainty in parameters, disturbances, and unfavorable operating conditions. This paper introduces the Polynomial Minimax Optimization (PMMO) method as a robust approach to nonlinear control parameter selection. In PMMO, the dynamics of the nonlinear system response are modeled via polynomial basis function with parameters and interactions, and the uncertain parameters are modeled via uncertainty sets. The objective is a combination of several measures like tracking error, settling time, overshoot, and robustness which is defined as a minimax problem in order to minimize the maximum achievable objective under the uncertainty domain. Various engineering constraints, such as limits on the controller parameters, actuators, states and performance are considered in the optimization procedure.  A nonlinear DC motor controlled by PID is used as a case study to analyse the proposed framework for nominal, low, medium, high, and adverse uncertainty cases. Comparative analysis to gradient optimization, Genetic Algorithm, Particle Swarm Optimization, and Differential Evolution reveals that the PMMO framework is capable of generating competitive nominal solutions while minimizing performance degradation. Sensitivity and degree-of polynomial analysis further suggests that there is an optimal tradeoff between accuracy, robustness, and complexity for the PMMO approach.