Sensor Choice for Minimum Error Variance Estimation

A Kalman filter is optimal in that the variance of the error is minimized by the estimator. It is shown here, in an infinite-dimensional context, that the solution to an operator Riccati equation minimizes the steady-state error variance. This extends a result previously known for lumped parameter systems to distributed parameter systems. It is shown then that minimizing the trace of the Riccati operator is a reasonable criterion for choosing sensor locations. It is then shown that multiple inaccurate sensors, that is, those with large noise variance, can provide as good an estimate as a single highly accurate (but probably more expensive) sensor. Optimal sensor location is then combined with estimator design. A framework for calculation of the best sensor locations using approximations is established and sensor location as well as choice is investigated with three examples. Simulations indicate that the sensor locations do affect the quality of the estimation and that multiple low-quality sensors can lead to better estimation than a single high-quality sensor.