Abstract: Feedback control determines how systems respond to measured states to achieve desired objectives. The linear quadratic regulator (LQR), developed in the 1960s, became one of the most widely applied control methods due to its elegant solution. The robust control extension for bounded disturbances, the disturbance attenuation regulator (DAR), formulated by American and Soviet researchers in the 1960s, has remained unsolved for six decades. 

 

This dissertation provides a complete state feedback solution to the DAR for linear systems, addressing signal bound disturbances where cumulative magnitude is constrained over the horizon, and stage bound disturbances where magnitudes are independently constrained at each time step. We also provide a complete feedback solution to the constrained LQR with signal and stage bound control constraints. The optimal controls to DAR and constrained LQR are nonlinear in the state and require solving a tractable convex optimization; the control is then explicit. To obtain these results, we develop fundamental results for minmax optimization. 

 

We extend these contributions to receding horizon control, also known as model predictive control (MPC), and establish closed-loop stability for the stage bound DAR within this framework for nonlinear systems.