Observers for Lipschitz nonlinear systems

An LMI approach to robust fault estimation for a class of nonlinear systems

“…[42,51]) becomes a usual Lipschitz condition [1,27,37,36] with γ being a Lipschitz constant. This appealing property makes the employed strategy more general than those presented in the literature [1,27,37,36].…”

“…[42,51]) becomes a usual Lipschitz condition [1,27,37,36] with γ being a Lipschitz constant. This appealing property makes the employed strategy more general than those presented in the literature [1,27,37,36]. Moreover, a significant progress was recently obtained in the observer design for non-linear systems by introducing the so-called one-sided Lipschitz condition [53], which means that a wider spectrum of systems can be tackled with the new approach.…”

“…It should also be pointed out that the proposed description of non-linearity constitutes an alternative to other approaches presented in the literature [1,27,37,36,33].…”

Observers for interconnected nonlinear and linear systems

“…These include stochastic techniques such as the extended and unscented Kalman filter (see, e.g., Brown and Hwang, 1997;Julier and Uhlmann, 2004) and the particle filter (e.g., Djurić et al, 2003); the use of nonlinear state transformations to achieve linear error dynamics (Krener and Isidori, 1983;Marino and Tomei, 1995); the use of linear observer dynamics in combination with a nonlinear transformation (Kazantis and Kravaris, 1998); design of observer gains to achieve robustness against Lipschitz continuous nonlinearities using, for example, LMIs or Riccati equations (see Thau, 1973;Rajamani, 1998;Zemouche, Boutayeb, and Bara, 2008;Phanomchoeng and Rajamani, 2010); the application of high gain to suppress Lipschitz continuous nonlinearities, both for left-invertible systems (Esfandiari and Khalil, 1987;Saberi and Sannuti, 1990), and non-left-invertible systems (e.g., Gauthier, Hammouri, and Othman, 1992;Bornard and Hammouri, 2002;Grip and Saberi, 2010); the exploitation of monotonic nonlinearities (Arcak and Kokotović, 2001;Fan and Arcak, 2003), or more general nonlinearities satisfying incremental quadratic…”

“…Specifically, many continuoustime designs enable the construction of a Lyapunov function V (t,x) with the properties that α 1 x 2 ≤ V (t,x) ≤ α 2 x 2 , V (t,x) ≤ −α 3 x 2 , and [∂V /∂x](t,x) ≤ α 4 x , wherex is the observation error variable (e.g., Krener and Isidori, 1983;Marino and Tomei, 1995;Rajamani, 1998;Zemouche et al, 2008;Phanomchoeng and Rajamani, 2010;Esfandiari and Khalil, 1987;Saberi and Sannuti, 1990;Gauthier et al, 1992;Bornard and Hammouri, 2002;Grip and Saberi, 2010;Arcak and Kokotović, 2001;Fan and Arcak, 2003;Açıkmeşe and Corless, 2011). This is not surprising, given that many designs are based at least in part on linear theory, which yields quadratic-type Lyapunov functions.…”