12 research outputs found
A Contraction Analysis of the Convergence of Risk-Sensitive Filters
A contraction analysis of risk-sensitive Riccati equations is proposed. When
the state-space model is reachable and observable, a block-update
implementation of the risk-sensitive filter is used to show that the N-fold
composition of the Riccati map is strictly contractive with respect to the
Riemannian metric of positive definite matrices, when N is larger than the
number of states. The range of values of the risk-sensitivity parameter for
which the map remains contractive can be estimated a priori. It is also found
that a second condition must be imposed on the risk-sensitivity parameter and
on the initial error variance to ensure that the solution of the risk-sensitive
Riccati equation remains positive definite at all times. The two conditions
obtained can be viewed as extending to the multivariable case an earlier
analysis of Whittle for the scalar case.Comment: 22 pages, 6 figure
Model Predictive Control meets robust Kalman filtering
Model Predictive Control (MPC) is the principal control technique used in
industrial applications. Although it offers distinguishable qualities that make
it ideal for industrial applications, it can be questioned its robustness
regarding model uncertainties and external noises. In this paper we propose a
robust MPC controller that merges the simplicity in the design of MPC with
added robustness. In particular, our control system stems from the idea of
adding robustness in the prediction phase of the algorithm through a specific
robust Kalman filter recently introduced. Notably, the overall result is an
algorithm very similar to classic MPC but that also provides the user with the
possibility to tune the robustness of the control. To test the ability of the
controller to deal with errors in modeling, we consider a servomechanism system
characterized by nonlinear dynamics
Robust Kalman Filtering: Asymptotic Analysis of the Least Favorable Model
We consider a robust filtering problem where the robust filter is designed
according to the least favorable model belonging to a ball about the nominal
model. In this approach, the ball radius specifies the modeling error tolerance
and the least favorable model is computed by performing a Riccati-like backward
recursion. We show that this recursion converges provided that the tolerance is
sufficiently small
Robust Kalman Filtering under Model Perturbations
We consider a family of divergence-based minimax approaches to perform robust
filtering. The mismodeling budget, or tolerance, is specified at each time
increment of the model. More precisely, all possible model increments belong to
a ball which is formed by placing a bound on the Tau-divergence family between
the actual and the nominal model increment. Then, the robust filter is obtained
by minimizing the mean square error according to the least favorable model in
that ball. It turns out that the solution is a family of Kalman like filters.
Their gain matrix is updated according to a risk sensitive like iteration where
the risk sensitivity parameter is now time varying. As a consequence, we also
extend the risk sensitive filter to a family of risk sensitive like filters
according to the Tau-divergence family
Convergence analysis of a family of robust Kalman filters based on the contraction principle
In this paper we analyze the convergence of a family of robust Kalman
filters. For each filter of this family the model uncertainty is tuned
according to the so called tolerance parameter. Assuming that the corresponding
state-space model is reachable and observable, we show that the corresponding
Riccati-like mapping is strictly contractive provided that the tolerance is
sufficiently small, accordingly the filter converges
On the Robustness of the Bayes and Wiener Estimators under Model Uncertainty
This paper deals with the robust estimation problem of a signal given noisy
observations. We assume that the actual statistics of the signal and
observations belong to a ball about the nominal statistics. This ball is formed
by placing a bound on the Tau-divergence family between the actual and the
nominal statistics. Then, the robust estimator is obtained by minimizing the
mean square error according to the least favorable statistics in that ball.
Therefore, we obtain a divergence family-based minimax approach to robust
estimation. We show in the case that the signal and observations have no
dynamics, the Bayes estimator is the optimal solution. Moreover, in the dynamic
case, the optimal offline estimator is the noncausal Wiener filter