# Switched Observer Design for a Class of Non-Linear Systems
## 1 Introduction  
State observer design for continuous-time dynamical systems is still a topic of significant research interest in control engineering. In several applications, a reliable state estimation of the unmeasurable state variables is required, particularly, when they are employed for either monitoring or model-based control design purposes. The state observer design problem is substantially more challenging when dealing with non-linear dynamical systems, which have attracted a lot of interest in control and systems literature for several decades. As a result, numerous results addressing the design of state observers and filters for non-linear systems were proposed, having different levels of conservatism; see, e.g., [1; 2; 3; 4; 5] and the references therein. Throughout recent decades, significant attention arose for the class of Lipschitz non-linear control systems that have been extensively studied in the literature. As we know, most physical systems models satisfy the Lipschitz condition, at least locally. Specifically, in the context of state estimation, the so-called Lipschitz non-linear observer has been thoroughly investigated in the literature (see, for instance, [6; 7; 8; 9; 10]). The limitation of all these approaches is that the synthesis conditions usually become unfeasible for systems with large Lipschitz constants. Some relatively recent results offer a proper reformulation of non-linear dynamical systems by the application of mathematical principles that, for the general class of Lipschitz systems, directly lead to the linear parameter varying (LPV) approach and, hence, to less restrictive synthesis conditions [11; 12]. On the other hand, many stabilization techniques on the field of dynamical systems use the concept of switching structure to compensate for system variations and fluctuations caused by uncertainties and perturbations. Switching stabilization methods have been proven to be successful against parameter uncertainties [13; 14; 15; 16; 17]. The general approach consists of constructing a state-dependent switching strategy and adaptive stabilization control law consisting of a set of feedback gains, each of which is selected at certain time instants according to the switching strategy, which is derived from the Lyapunov stability theory. Most of the previously published LMI-based observer design methods for non-linear systems consider a single observer gain matrix, whereas a set of observer gain matrices in a switched scheme of implementation are also needed in order to achieve and/or increase the feasibility range of the design parameters as well as to improve the state estimation performance. A recent result of importance develops a hybrid switched-gain observer for non-monotonic non-linear systems [18] and whose application has been depicted in [19; 20; 21; 22]. Furthermore, the hybrid switched-gain observer design has been extended for systems with disturbances in both the state dynamics and in the output measurements, and its \(H_{\infty}\) performance has been investigated [23]. In regard to non-linear systems with asynchronous switching, some recent results involve observer-based control for time-delayed non-linear systems [24] and robust state and sensor fault estimation [25; 26; 27]. Although the results from the literature above show a considerable advancement in the development of workable switched observer design methods, most of them consider asynchronous switching schemes of implementation for systems that incorporate switching in their dynamics, which limits their applicability to practical non-linear systems where switching may be included to take advantage of the previously mentioned benefits of this scheme. In particular, this study considers an LPV representation of the non-linear state estimation error dynamics, which is inferred directly from the general Lipschitz condition and the application of the differential mean value theorem (DMVT), requiring no extra assumptions on system non-linearities.