Optimization based on quasi-Monte Carlo sampling to design state estimators for non-linear systems

State estimation for a class of non-linear, continuous-time dynamic systems affected by disturbances is investigated. The estimator is assigned a given structure that depends on an innovation function taking on the form of a ridge computational model, with some parameters to be optimized. The behaviour of the estimation error is analysed by using input-to-state stability. The design of the estimator is reduced to the determination of the parameters in such a way as to guarantee the regional exponential stability of the estimation error in a disturbance-free setting and to minimize a cost function that measures the effectiveness of the estimation when the system is affected by disturbances. Stability is achieved by constraining the derivative of a Lyapunov function to be negative definite on a grid of points, via the penalization of the constraints that are not satisfied. Low-discrepancy sampling techniques, typical of quasi-Monte Carlo methods, are exploited in order to reduce the computational burden in finding the optimal parameters of the innovation function. Simulation results are presented to investigate the performance of the estimator in comparison with the extended Kalman filter and in dependence of the complexity of the computational model and the sampling coarseness.