Optimum Design Strategy for $H_{\infty }$ Control of Time-Delayed Direct Velocity Feedback Systems

In this paper, an optimal $H_{\infty }$ control algorithm using direct velocity feedback is used for the design of control systems in reducing structural seismic responses. An optimum design strategy was developed for deciding two control parameters, $\gamma$ and $\alpha$, which are introduced in the control algorithm and play important roles in system stability and control performance. $\gamma$ is a positive attenuation constant, which denotes a measure of control performance, and $\alpha$ is a control weighting factor, indicating the relative significance between control force requirement and response reduction. Analytical results show that decreases in $\gamma$ or increases in $\alpha$ yield better control performance but require larger control forces. The selection range of $\gamma$ and $\alpha$ for a controlled system yielding overdamped or unstable responses is found. To assure system stability and better performance than linear quadratic regulator (LQR) control, analytical expressions of the upper and lower bounds of $\gamma$ and $\alpha$ are derived for direct velocity feedback control. Therefore, the seismic responses can be effectively reduced with an appropriate selection of $\gamma$ and $\alpha$. In addition, when a time delay in control force execution exists, explicit formulas to calculate the maximum allowable delay time and critical control parameters of $\gamma$ and $\alpha$ are also derived for the design of a stable time-delayed control system. An optimal design flowchart is also provided. The desired control performance can be guaranteed, even with time delay.