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

Abstract

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. Lowdiscrepancy 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.

Stubborn and Dead-Zone Redesign for Nonlinear Observers and Filters

“…where x ∈ R n is the state, u ∈ R p is a known input, y ∈ R m is the measured output, w ∈ R n is some external disturbance, and v ∈ R m represents the sensor measurement noise. For system (3) we suppose to know an observer providing an asymptotic estimatê x of state x. A fairly general expression including, among others, Luenberger observers, Kalman filters, observers for input-affine systems, observers for Lipschitz systems, observer based on the circle criterion, high-gain observers, and low-power high-gain observers, [4], [8], [11], [14], [14], [17]- [19], [27]- [29], [33], [34], [37], [38], [40], [45] (see the details in Section VI), corresponds tȯ…”

“…We suppose that the estimator has already been designed and it satisfies some mild, possibly local, input-to-state stability (ISS) properties [39], [40]. For instance, any of the techniques proposed in [3], [4], [7], [8], [10], [11], [13]- [15], [17]- [19], [27]- [29], [33], [34], [36], [37], [39], [43] enjoy these properties. Then, we propose two different methodologies to redesign the output injection term, both of them preserving ISS.…”

Lyapunov Functions for State Observers of Dynamic Systems Using Hamilton–Jacobi Inequalities

“…ISS sampled-data observers are analyzed in [12]. The design of ISS estimators is addressed in [13][14][15][16].…”

On Hamilton-Jacobi Approaches to State Reconstruction for Dynamic Systems

“…Instead, in this paper, the stability analysis of the error dynamics is conducted by using inputto-state stability [1], where the disturbances are regarded as an input and the role of the state is played by the estimation error incurred by the observer. An estimator (observer or filter) will be said to be input-to-state stable (ISS) if the norm of the estimation is bounded by the sum of two terms, with the first one depending on the magnitude of the initial error and decreasing with time, while the second one increases with the norm of the disturbances [16][17][18][19][20]. Input-to-state stability holds if and only if there exist some suitable Lyapunov functions, which are called ISS Lyapunov functions [21].…”