Boundary observer design for cascaded ODE — Hyperbolic PDE systems: A matrix inequalities approach

Abstract

Boundary observer design for a system of ODEs in cascade with hyperbolic PDEs is studied. An infinite dimensional observer is used to solve the state estimation problem. The interconnection of the observer and the system is written in estimation error coordinates and analyzed as an abstract dynamical system. The design of the observer is performed to achieve global exponential stability of the estimation error with respect to a suitable norm and with a tunable convergence rate. Sufficient conditions in the form matrix inequalities are given for the design of the observer. The effectiveness of the approach is shown in a numerical example.

On a nodal observer for a semilinear model for the flow in gas networks

“…In order to show this, we use an upper bound for the source term in ( 16) that is given by the continuously differentiable function σ e (R e + , R e − ). In fact, for R ∈ B(M ) for all e ∈ E due to (10) we have |σ e (R e + , R e − )| 4 ν e M 2 . Moreover, it has to be shown that the iteration map maps from B(M ) into B(M ).…”

“…For results about the recovery of an unknown initial state using an observer see [20]. In [10], the design of boundary observers for a linear system of ODEs in cascade with hyperbolic PDEs is studied and more references on observer design are given. We want to emphasize that the novelty of our contribution is the construction of an observer for a system that is governed by networked semilinear pdes and uses observations that are located pointwise in space, whereas in the previous contributions distributed observations coming from subdomains in space have been considered.…”

Spectral stabilization of linear transport equations with boundary and in-domain couplings

“…Finally, we have to prove (15). Let R ∈ H −1 and let (R n ) n be a sequence of elements of H converging to R in H −1 .…”

“…As a consequence, to use the proposed method, one needs to measure the state R on all the domain, which is sometimes impossible in industrial applications (see [10] for example). Some works [5,11,15,17,31] solve this difficulty designing a boundary observer of the state R. Here for simplicity, it is assumed that the observation of the state is complete in order to focus only on the effect of the control.…”

Unknown Input Observer design for coupled PDE/ODE linear systems

“…This is due to the presence of the cross term introduced in the Lyapunov functional (17). A specific approach to get an LMI-based design algorithm from (18) has been proposed recently in [23].…”