On the feedback control of the wave equation

This paper addresses the problem of design of collocated and noncollocated controllers for a uniform bar without structural damping. The bar whose dynamics are described by the wave equation is required to perform a rest-to-rest maneuver. A time delay controller whose gains are determined using the root-locus technique is used to control the noncollocated system. The effect of sensor locations on the stability of the system is investigated when the actuator is located at one end of the bar. The critical gains which correspond to a pair of poles entering the right-half of the s-plane and the optimal gains which corresponding to locating the closed-loop poles at the left extreme of the root-locus for each vibration modes are determined. The gain which minimizes a quadratic cost, in the range of the critical gains, is selected as the optimum gain.