HJB-based optimal nonlinear control for multiple objectives in different levels of excitations

Linear control laws are the most common control strategies, primarily due to their simplicity in both design and implementation. While there are a multitude of accepted techniques for designing linear control strategies for mitigating the response of structures due to external loads and excitations, there are no elegant and widely-applied nonlinear methods. While the optimal linear control law that minimizes a quadratic cost function can be carried out simply with standard tools (e.g., MATLAB´s lqr command), a nonquadratic cost function leads to a Hamilton-Jacobi-Bellman (HJB) partial differential equation that results in an optimal nonlinear controller. However, finding the exact analytical solution of the HJB equation is very difficult; it may only have a solution when the cost function is cast in a particular form. This paper suggests a particular form of a nonlinear control law for civil structures designed to minimize one objective (a serviceability and performance measure) for small excitation and small response, and a different objective (life safety) for larger motion. An analytical approach is presented to cast the nonquadratic cost function in a form for which the HJB equation can be solved. The resulting optimal nonlinear control law can be written as the sum of a linear term, which is related to an LQR problem and is effective in small excitations, and some nonlinear terms that dominate when the excitation level is large. The performance of the proposed nonlinear control strategy is demonstrated by a numerical simulation for a simple building model subjected to different historical earthquakes and a synthetic ground motion, showing that it can successfully achieve both life safety (drift reduction) and serviceability (acceleration reduction) objectives at different excitation levels.