Optimal $H_{\infty }$ Output Feedback Control Systems with Time Delay

An $H_{\infty }$ direct output feedback control algorithm through minimizing the entropy, a performance index measuring the tradeoff between $H_{\infty }$ optimality and $H_2$ optimality, is developed in this paper to reduce the earthquake response of structures. To achieve optimal control performance and assure control system stability, the strategy to select both control parameters $\gamma$ and $\alpha$ is extensively investigated considering the control force execution time delay. It is found that a lower bound of $\gamma$ and an upper bound of $\alpha$ exist. The selection beyond these values will cause the control system instability. For a damped structure, analytical expressions of direct output feedback gains, controlled frequencies and damping ratios are derived. It can be proved that the conventional LQR control is a special case of the developed $H_{\infty }$ control. In real active control, control force execution time delay cannot be avoided. This paper gives explicit formulas of maximum allowable delay time and critical control parameters for the design of a stable control system. Some solutions are also proposed to lengthen maximum allowable delay times.