24. Optimal Nonlinear Control Using a Non-quadratic Cost Function for Scalar Systems

Various types of controllers have been studied and implemented to mitigate the effects of excitations from natural hazards. Linear control laws are most often applied, for active devices as well as part of the controller for semiactive devices, primarily due to their simple design and use. However, researchers have investigated nonlinear control of structures, showing performance improvements relative to that with linear controllers. The optimal linear control law that minimizes a quadratic cost function can be obtained using typical solvers like Matlab’s lqr command. In contrast, the optimal nonlinear controller generally requires minimizing a non-quadratic cost function, which is difficult and analytical solutions may only exist when the cost function is in particular forms. This paper presents a comparison of some of the analytical and numerical methods for finding optimal nonlinear controllers, particularly for cost functions that are even order powers of the states and quadratic in the control. A scalar model (i.e., a scalar state-space equation) is used for the comparisons since analytical solutions exist for some of the methods. Even though the methods would all result in the same linear control law, it is demonstrated that the methods’ differing assumptions give rise to different optimal nonlinear control laws, with different performance in different excitations.