Instantaneous optimal method for vibration control of linear sampled-data systems with time delay in control

In an actively controlled system, time delay exists inevitably. Neglecting time delay may cause degradation of control performance or even induce instability to the dynamic system. In this paper, instantaneous optimal control method for vibration control of linear sampled-data systems with time delay in control is investigated. By a peculiar integral transformation, the first order state equation with time delay is transformed into the standard first order state equation, which contains no time delay. Then the optimal controller is designed based on the numerical algorithm of the regular fourth order Runge-Kutta method. Since the obtained controller contains integral term, which is not practical for control implementation, the numerical algorithm for this integral term is investigated too. Since the controller is deduced directly from the time-delay differential equation, the control method presented is prone to guarantee system stability. Thus the presented control method can be applicable to the case of large time delay. The performance of the control method is demonstrated by numerical simulation. Simulation results indicate that this control method is feasible and is an attractive strategy for dealing with the time delay in vibration control systems and is effective in suppressing maximum structural responses. Instability in structural responses may occur if the systems with time delay are controlled using the controller designed in the case of no time delay.