Adaptive Techniques for the Online Estimation of Spatial Fields With Mobile Sensors

The regression-based reconstruction of spatial fields by static sensors requires the number of measurements be at least equal to the number of unknown parameters in order to provide a unique solution. Improvements include the use of mobile sensor networks to arrive at higher accuracy estimates. An alternate approach utilizes adaptive techniques to update the parameter estimates online. Such adaptive techniques need persistence of excitation to ensure convergence of the parameter and function estimates. It is shown that the motion of just a single sensor is a necessary condition for persistence of excitation and, subsequently, of function convergence. Having a prescribed sensor guidance is also a sufficient condition for function convergence. First, a Lyapunov-based adaptation and guidance is presented and it is shown that persistence of excitation necessitates the time dependence of the outer product of the regressor vector with itself and which in turn requires the motion of the sensor within the spatial domain. Subsequently, a gradient-based adaptive scheme relaxes the Lyapunov-based sensor guidance to any user-defined guidance as long as the outer product of the associated rank-one regressor vector with itself has a uniformly positive definite integral. The latter adaptation is conducive to the inclusion of platform dynamics resulting in a vehicle control design needed to attain parameter convergence. Extension of the centralized solution to distributed estimation using a network of mobile sensors is also examined. These results are demonstrated with simulations in one and two dimensions.