Solving complex optimal control problems with nonlinear controls using trigonometric functions

Abstract

This study investigates the use of trigonometric functions to resolve two major issues encountered when solving practical optimal control problems (OCPs) that are characterized by nonlinear controls. First, OCPs with constraints on nonlinear controls require the solution to a multipoint boundary value problem, which poses additional computational difficulties. Second, in certain unconstrained OCPs with nonlinear controls, the extremum found from the necessary conditions can be opposite than expected (e.g., a maximum instead of a minimum) due to the absence of control options. The aforementioned issues and their effective resolution by using trigonometric functions are explained through two examples including a second‐order system problem and an aerocapture problem of a slender entry vehicle in the Martian atmosphere.

Controlling Chaos in Van Der Pol Dynamics Using Signal-Encoded Deep Learning

“…Admittedly, reducing PID to PI (proportional-integral) or P (proportional) might either increase the speed or system stability, but neither can learn the features of nonlinear systematic data, which allows more advanced self-adjust control behavior. Another common approach is to model the chaotic behaviors as periodic ones and implement trigonometric commands in hopes of producing predictable periodic system behavior [18]. Model predictive controllers incorporate statistics seeking good results for fuzzy systems [19].…”

Physics-Informed Deep Operator Control: Controlling chaos in van der Pol oscillating circuits

“…Traditionally, proportional-integral-derivative controllers (PID) were widely adopted for controlling nonlinear systems [15,16], which employ closed-loop feedback errors that can tune feedforward command as close to the target output as possible [17], are still common in nowadays's industry. Particularly, a common practice in control theory is to implement trigonometric command as for predictable periodic system behavior [18]. Notably, with such wide applications, PID control experiences sluggish and systematic postpone due to the integral term, and order increasing might lead to system instability [19].…”

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“…Solving optimization problems efficiently is a topic of clear practical interest, as demonstrated by recent research effort on the topic. [14][15][16] Furthermore, the possibility of solving relatively large problems quickly could boost the application of this type of modeling to larger production planning problems, such as the one presented in Reference 17. This article elaborates on the work presented in Reference 13 and presents a reformulation of the constraints introduced by the FCM models that ease the complexity of the optimization problem and allows the use of simpler methods for its solution. This way, results of the application of convex-concave and separable programming to the reformulated problem are presented, showing that separable programming constitutes a very good alternative, both in terms of solution time and reliability in finding the optimum.…”

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