Robust $H_{\infty }$ Control Considering Actuator Saturation. I: Theory

This paper represents the first part of a two-part study devoted to the theory and application of $H_{\infty }$ state feedback optimal control to civil structures in the presence of actuator limitations and time varying parametric uncertainties. The primary goals when applying optimal control theory to civil structures is the maintenance of stability and the achievement of specific performance criteria, including control efficiency, in the face of random disturbances. Two important issues in achieving these goals are the consideration of nonlinear actuator saturation effects and unknown, time-varying, parametric uncertainties. Most importantly both of these issues must be addressed concomitantly within the same control design. Robust $H_{\infty }$ state feedback controllers are developed here that achieve the desired $H_{\infty }$ norm bound while accounting for prespecified bounds on the time-varying parametric uncertainties. Stability of these controllers in the presence of nonlinear actuator saturation can be proven through the construction of a Lyapunov function for the saturated control system using a nonlinear state space model and new mathematical programming techniques. Application of these newly developed algorithms to an actively controlled 33‐story structure in Tokyo is discussed in part 2 of this study.