Automatica 44 (2008) 1753–1765 www.elsevier.com/locate/automatica Moving-horizon state estimation for nonlinear discrete-time systems: New stability results and approximation schemes ✩ Angelo Alessandri a , Marco Baglietto b,∗ , Giorgio Battistelli c a Department of Production Engineering, Thermoenergetics, and Mathematical Models, DIPTEM-University of Genoa, P.le Kennedy Pad. D, 16129 Genova, Italy b Department of Communications, Computer and System Sciences, DIST-University of Genoa, Via Opera Pia 13, 16145 Genova, Italy c Dipartimento di Sistemi e Informatica, DSI-Universit` a di Firenze, Via di S. Marta 3, 50139 Firenze, Italy Received 27 March 2006; received in revised form 8 October 2007; accepted 13 November 2007 Available online 21 March 2008 Abstract A moving-horizon state estimation problem is addressed for a class of nonlinear discrete-time systems with bounded noises acting on the system and measurement equations. As the statistics of such disturbances and of the initial state are assumed to be unknown, we use a generalized least-squares approach that consists in minimizing a quadratic estimation cost function defined on a recent batch of inputs and outputs according to a sliding-window strategy. For the resulting estimator, the existence of bounding sequences on the estimation error is proved. In the absence of noises, exponential convergence to zero is obtained. Moreover, suboptimal solutions are sought for which a certain error is admitted with respect to the optimal cost value. The approximate solution can be determined either on-line by directly minimizing the cost function or off-line by using a nonlinear parameterized function. Simulation results are presented to show the effectiveness of the proposed approach in comparison with the extended Kalman filter. c  2008 Elsevier Ltd. All rights reserved. Keywords: State estimation; Moving horizon; Discrete-time nonlinear systems; Approximate solution 1. Introduction Moving-horizon (MH) estimation is a powerful approach that has obtained an increasing success in connection with the rapid diffusion of the so-called model predictive control (see, for an introduction, Mayne, Rawlings, Rao, and Scokaert (2000)). A predictive controller generates a control action by solving an open-loop optimal control problem in which the current state of the plant is used as the initial state. Predictive controllers are somehow dual of MH estimators, which provide estimates of the state variables by using a limited amount of most recent information and propagate the last estimate to the next time instant, where the estimation procedure is repeated. For an insight into the duality between MH estimation and ✩ This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Ben M. Chen under the direction of Editor Ian Petersen. ∗ Corresponding author. Tel.: +39 010 3536548; fax: +39 010 3532154. E-mail addresses: alessandri@diptem.unige.it (A. Alessandri), mbaglietto@dist.unige.it (M. Baglietto), battistelli@dsi.unifi.it (G. Battistelli). control the interested reader is referred to Goodwin, De Don´ a, Seron, and Zhuo (2005). The interest on MH state estimation was originally motivated by its intrinsic robustness, which makes the approach well-suited in the presence of modelling uncertainties and/or numerical errors (Jazwinski, 1968). Recently, researches have focused on the application of such techniques to linear systems (Alessandri, Baglietto, & Battistelli, 2003, 2004, 2005; Rao, Rawlings, & Lee, 2001) and hybrid systems (Alessandri, Baglietto, & Battistelli, 2005; Ferrari-Trecate, Mignone, & Morari, 2002). The literature on nonlinear MH estimation is more restricted. In Moraal and Grizzle (1995) an asymptotic state observer is described that results from the numerical solution of a sequence of nonlinear algebraic equations via Newton’s method. Similar optimization-based solution techniques are employed in Alamir (1999) and Zimmer (1994) to construct stable estimators for continuous-time dynamic systems. In Michalska and Mayne (1995), a MH observer for nonlinear continuous-time systems was proposed that performs estimation at discrete-time instants by approximately 0005-1098/$ - see front matter c  2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2007.11.020