Noncollocated point control of nondispersive distributed-parameter systems using time delays

Abstract

This paper deals with the noncollocated, time-delayed active point control of continuous systems. It considers systems of finite spatial extent which can be modeled by the undamped wave equation. The paper presents a new method of control of the system using noncollocated sensors and actuators. By using the physical properties of the mode shapes of vibration of the system it is shown that the modal response at a given location in the system can be reconstructed from the time-delayed modal responses at (at most) three different locations in the system. This result is then used to motivate a closed-loop control design which is capable of stabilizing the system and dampening the vibrations in all its modes, using dislocated sensor and actuator locations. It is found that the cost of dislocating the sensors from the actuator is at most an additional two sensors. For special types of boundary conditions this cost may be reduced to one additional sensor, or even be completely eliminated. Simple finite-dimensional controllers, commonly used in control design, are found to suffice. The results are valid for rather general conditions at the boundaries of the continuum. Explicit conditions are provided to obtain the bounds on the controller gains to ensure stability of the closed-loop control design. These bounds are obtained in terms of the locations of the sensors and the actuators. Simulation results, which validate the control methodology and the theoretical bounds on the gain, are also presented. © 1991.