Nonlinear observers in reflexive Banach spaces

Abstract

Abstract.On an arbitrary reflexive Banach space, we build asymptotic observers for an abstract class of nonlinear control systems with possible compact outputs. An important part of this paper is devoted to various examples, where we discuss the existence of persistent inputs which make the system observable. These results make a wide generalization to a nonlinear framework of previous works on the observation problem in infinite dimension (see [11,18,22,26,27,38,40] and other references therein).Mathematics Subject Classification. 47H020, 47H06, 93B07, 93C20, 93C25.

Recovering the observable part of the initial data of an infinite-dimensional linear system with skew-adjoint generator

“…In the case of exactly observable systems, we call the systems (1.4)-(1.5) forward and backward observers as it is a generalization to infinite-dimensional systems of the so-called Luenberger's observers [29], well-known in control theory. Observers for infinite-dimensional systems are an active topic of research, for both linear or nonlinear systems, and among the large literature, we can cite for instance: Chapelle et al [9], Krstic et al [25], Moireau et al [30], Smyshlyaev and Krstic [35], and Couchouron and Ligarius [10]. For pioneering work, we refer to Baras and Bensoussan [4] and Bensoussan [6].…”

Online parameter identification without Ricatti-type equations in a class of time-dependent partial differential equations: an extended state approach with potential to partial observations

“…Under the common assumption that the parameter q does not depend explicitly on the time variable there are in principle two approaches-motivated by techniques for finite-dimensional problems-for solving the online parameter identification problem. One idea is to consider the dynamical system q t (t) = 0 ( 5 ) with system output z(t) = GS t (q), (6) where S t : Q → V maps q onto u(t), i.e. the solution of (1) evaluated at time t. Then, the goal is to estimate the (parameter) state q(t) ≡ q from observations of the output z(t) described by the operator GS t (q) which is only implicitly defined via (1).…”

“…In the first class, to which the extended Kalman filter and the extended Luenberger observer belong, state estimators are realized as control systems where the available data are fed into the input of the estimator and the output yields the estimated state. For certain infinitedimensional dissipative systems, nonlinear observers belonging to this class are discussed in [5]. However, the assumptions on the structure of (9) made in [5] are neither compatible with online parameter identification in (1) in the sense of ( 5), (6) nor of (7), (8).…”

“…For certain infinitedimensional dissipative systems, nonlinear observers belonging to this class are discussed in [5]. However, the assumptions on the structure of (9) made in [5] are neither compatible with online parameter identification in (1) in the sense of ( 5), (6) nor of (7), (8). The second class of nonlinear observers consists of optimization-based state estimators including the moving (or receding) horizon state estimator which typically is used in the context of nonlinear model predictive control [1,10,12].…”