Guidance of a moving sensor for the control of diffusion PDEs based on on-line approximations of feedback kernels

The use of mobile or scanning sensing and actuating devices in the control of distributed parameter systems has been established as an effective means of improving controller performance and enhancing robustness against spatiotemporally varying disturbances. In earlier attempts, the computational requirements for implementing control laws with mobile actuators and sensors was circumvented by algorithmic simplifications of the controller design. To further examine the algorithmic simplifications for implementing a controller with moving actuator and sensor, this work considers the use of feedback kernels for providing the location of the sensor and value of the corresponding static output feedback gain. For a prescribed actuator location, the corresponding full state feedback kernel is computed and instead of feeding it into a state estimator, the proposed scheme replaces the full state feedback by static output feedback with time varying gain and time varying sensor location. The sensor location is found as the location of the maximum of the feedback kernel and the value of the feedback gain is given by the maximum value of the feedback kernel. The resulting controller utilizes a moving sensor with a time varying state output feedback gain to replace a full state feedback controller with state estimator. Two different algorithms are considered; one where the motion of the actuating devices is a priori defined and in another one in which the location of the actuator is changed at discrete time intervals. Numerical studies of a 1D diffusion equation are included to illustrate both algorithms.