The problem of estimating the state of a discrete-time linear system can be addressed by minimizing an estimation cost function dependent on a batch of recent measure and input vectors. This problem has been solved by introducing a receding-horizon objective function that includes also a weighted penalty term related to the prediction of the state. For such an estimator, convergence results and unbiasedness properties have been proved. The issues concerning the design of this filter are discussed in terms of the choice of the free parameters in the cost function. The performance of the proposed receding-horizon filter is evaluated and compared with other techniques by means of a numerical example.

An approach to estimation for continuous-time and discrete-time linear systems is proposed that is based on the idea of using switching observers. Convergence conditions have been found to ensure the stability of the error dynamics; in addition, they guarantee the existence of an upper bound to a quadratic cost function of the estimation error. The observer gains may be selected by solving a set of linear matrix inequalities (LMIs). A method is described to design a switching observer that aims to minimize the upper bound to the estimation cost function. Moreover, such a design may be efficiently accomplished by using an LMI algorithm.

Receding-horizon state estimation is addressed for a class of discrete-time systems that may switch among different modes taken from a finite set. The system and measurement equations for each mode are assumed to be linear and perfectly known, but the current mode of the system is unknown, the state variables are not perfectly measurable and are affected by disturbances. The system mode is regarded as an unknown discrete state to be estimated together with the continuous state vector. Observability conditions are found to distinguish the system mode in the presence of bounded system and measurement noises. These results allow one to construct an estimator that relies on the combination of the identification of the discrete state with the estimation of the state variables by minimizing a receding-horizon quadratic cost function. The convergence properties of such an estimator are studied, and simulation results are reported to show the effectiveness of the proposed approach.