Jacobian-Based Nonlinear State Estimation

Abstract

Nonlinearity is one of the most challenging issues in controllers and observers design. Most of the practical systems are inherently nonlinear (Slotine et al. (1991); Khalil (2002)). On the other hand, most control schemes and observer/estimater design methods used widely are for linear systems and are therefore not applicable in practice because of the existence of nonlinearities. This chapter deals with the state estimation methods for nonlinear systems stemmed from, however, linear estimator design ideas. Namely, the extensions of Kalman filter, information filter and an $\mathcal{H}_{\infty }$ filter to nonlinear systems will be discussed in this chapter. These extensions are based on the first-order Taylor’s series expansion of the nonlinear process and/or measurement dynamics. The idea is, in some key steps of estimation, to replace the nonlinear process and/or measurement models with their corresponding Jacobians. The three filters to be discussed in this chapter are therefore the extended Kalman filter (EKF), extended information filter (EIF) and extended $\mathcal{H}_{\infty }$ filter (E$\mathcal{H}_{\infty }$F).