In this thesis we explore the problem of finding optimal sensor/actuator locations to achieve the minimum square error/least effort. The solution for the optimal sensor/actuator is often combinational, this means in order to solve for the solution we have to look at many parameters in the system and those parameters change frequently. In this thesis, we propose two methods to achieve this goal: the first one is based on gradient flow differential in which it provides the global optimal solution for the placement, and the second one is based on the evaluation of the Hessian matrix at the critical points. The optimal sensor/actuator location found using the gradient flow or Hessian matrix is usually not sparse. However in practical settings, the optimal sensor/actuator locations are often determined by discrete numbers such as ones and zeroes. We then propose some methods of relating the optimal sensor/actuator locations found using the gradient flow or Hessian matrix to the optimal sensor locations in the practical settings. Next we test the performance of the method we proposed by comparing the square error/effort of the system using the optimal sensor/actuator locations we found and the optimal sensor locations in the practical settings in multiple dimensions and with selected number of sensors/actuators.
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