A NEW RECURSIVE FORMULATION FOR FAST COMPUTING OF UNKNOWN INPUT ESTIMATION A tracking application Vincent Sircoulomb, Ghaleb Hoblos, Houcine Chafouk IRSEEM, Technopôle du Madrillet, avenue Galilée, BP 10024, 76 801 Saint-Étienne-du-Rouvray, France sircoulomb@esigelec.fr, hoblos@esigelec.fr, chafouk@esigelec.fr José Ragot CRAN, 2 avenue de Haye, 54 516 Vandoeuvre-lès-Nancy, France jose.ragot@ensem.inpl-nancy.fr Keywords: Unknown input estimation, Kalman filtering, recursive algorithm, computation, tracking Abstract: This paper introduces a new form for the Recursive Input Estimation problem. The proposed algorithm needs the inversion at each sample time of one n y × n y and one n u × n u matrices, whereas the classical formulation requires the inversion of two n y × n y matrices. Consequently, a reduction of computation time occurs in the case where n y > n u , n y being the number of measurements and n u the number of inputs of the system under consideration. The new formulation obtained is tested on a tracking application, and its results compared with those issued from the classical formulation. 1 INTRODUCTION The problem of both estimating the state and input of a dynamical system from noisy measurements has been leading for a long time to intensive re- search. The solution to such a problem can help in applications like fault tolerant control (through actuator bias detection and estimation) (Hou and Patton, 1998a), manoeuvering targets tracking (Aley and Delinger, 1973), evaluation of reaction rates in chemical reactors (Mhamdi and Marquardt, 1999) or estimation of accelerometers and gyroscopes errors in inertial navigation (Grewal et al., 2001). In order to solve this kind of problem, different approach can be found, such as model inversion (Sil- verman, 1969), Unknown Input Observers (Darouach et al., 1994) and Filters (Hou and Patton, 1998b), In- teracting Multiple Model (IMM) (Mazor et al., 1998) or inclusion of the unknown inputs into the state vec- tor (Bar-Shalom and Birmiwal, 1982) (leading to an augmented state vector). Another approach, called In- put Estimation, has been proposed by (Chan et al., 1979), recursively developed in (Wang and Varshney, 1993) and adapted to a class of time-varying inputs in (Lee and Tark, 1999), (Jilkov and Li, 2002). Its advantage compared to the others approach is unde- niable: Indeed, its applicability conditions are far less restrictive than the model inversion or Unknown In- put Observers/Filters ones; Its computational burden is lower than the IMM one (Mazor et al., 1998); its structure allows decoupling of state and input estima- tion, contrary to the augmented state vector approach. However, this algorithm presents a drawback : It needs the inversion of two squares matrices of di- mension n y × n y , where n y is the number of measure- ments done on the system under consideration, while a Kalman filter (Kalman, 1960) using an augmented state vector needs only one inversion. The purpose of this article is to propose a new formulation of least squares estimator, by using the information form (An- derson and Moore, 1979). The two matrices to inverse are then of dimension n y × n y and n u × n u , where n u is the number of inputs. Such a formulation reduces the computation time in the case where n y is greater than n u , and moreover, provides a simpler algorithm. In this paper, the Recursive Input Estimation (RIE) is firstly presented. Then, the new formulation of RIE using information matrices is proposed in sec- tion 2. Finally, the proposed method is applied on a tracking application in section 3. hal-00292238, version 1 - 30 Jun 2008 Author manuscript, published in "International Conference on Informatics in Control, Automation and Robotics, ICINCO 2008, Madeira : Portugal (2008)"