Nonlinear Feedback Control Design via NEOC

Quadratic performance indices associated with linear plants offer simplicity and lead to linear feedback control laws, but they may not adequately capture the complexity and flexibility required to address various practical control problems. One notable example is to improve, by using possibly nonlinear laws, on the trade-off between rise time and overshoot commonly observed in classical regulator problems with linear feedback control laws. To address these issues, non-quadratic terms can be introduced into the performance index, resulting in nonlinear control laws. In this letter, we tackle the challenge of solving optimal control problems with non-quadratic performance indices using the closed-loop neighboring extremal optimal control (NEOC) approach and homotopy method. Building upon the foundation of the Linear Quadratic Regulator (LQR) framework, we introduce a parameter associated with the non-quadratic terms in the cost function, which is continuously adjusted from 0 to 1. We propose an iterative algorithm based on a closed-loop NEOC framework to handle each gradual adjustment. Additionally, we discuss and analyze the classical work of Bass and Webber, whose approach involves including additional non-quadratic terms in the performance index to render the resulting Hamilton-Jacobi equation analytically solvable. Our findings are supported by numerical examples.