The state observation problem is tackled for a system of

Model-based control and advanced process monitoring of Distributed-Parameter Systems (DPSs), governed by Partial Differential Equations (PDEs), typically require full state information. However, the available measurements of DPS are typically located on the boundary of the spatial domain that motivates the need of the state observer [

For linear infinite-dimensional systems the Luenberger observer theory was established by replacing matrices with linear operators [

Design methods, which are not relying on any discretization or finite-dimensional approximation (thereby preserving the infinite-dimensional representation of the system during the entire design process) and which are yielding the observer gains in the explicit form, have only recently been investigated. In this context, the backstepping method appears to be a particularly effective systematic design approach which can be applied for a broad class of systems governed by PDEs [

For systems governed by parabolic PDEs defined on a one-dimensional (1D) spatial domain, a systematic observer design approach using boundary sensing is introduced in [

More recently, high-dimensional systems of

In this work, the observer design is developed for a class of

Particularly, in the present context, two output injections are needed in the observer dynamics (one distributed along the spatial domain and another one located at the uncontrolled boundary).

The structure of the paper is as follows. After introducing some useful notation in Section

The notation used throughout is fairly standard.

With reference to a generic real-valued symmetric matrix

The following

The following lemma is in order.

The error system (

Employing the Leibnitz differentiation rule, the spatial differentiation of (

In turn, the temporal differentiation of (

By evaluating (

Considering (

The present paper is confined to the case in which all the coupled PDEs (

For later use, the following result is reproduced.

Problem (

By the invertible change of variables

Following [

Setting

Integrating (

Integrating (

An explicit form of

By substituting (

The method of successive approximations is then applied to show that (

Provided that this recursion converges, solution

Let

Since variables

In order to apply the mathematical induction method, suppose that

Then, by employing (

It is readily shown (cf. [

Thus, by mathematical induction, (

The solution to the integral equation (

Converting (

Transformation (

Relation (

Taking advantage of the explicit solution (

The stability features of the target error dynamics (

If the design matrix

Consider the Lyapunov function

Integrating by parts taking into account the BCs (

The next theorem specifies the proposed observer design and summarizes the main result of this paper.

The observer (

In Lemma

The asymptotic stability features of (

From now on, we follow [

To validate the proposed observer, system (

The initial conditions are set to

The unstable behaviour of the plant subject to the open-loop input vector

Spatiotemporal evolution of

Observer (

Spatiotemporal evolution of

Temporal evolution of the norm

To provide a more valuable validation of the proposed scheme, we consider the coupled temperature-concentration dynamics of a Chemical Tubular Reactor (CTR) at low fluid superficial velocities, when convection terms become negligible, dealt with in [

Its linearization around the constant profiles

The open-loop control input

Spatiotemporal evolution of

Temporal evolution of the norm

The backstepping-based anticollocated observer design of a system of

The authors declare that there is no conflict of interests regarding the publication of this paper.

The research leading to these results has received funding from the Research Project “Modeling, Control and Experimentation of Innovative Thermal Storage Systems,” funded by Sardinia Regional Government under Grant Agreement no. CRP-60193, and from the Research Project “RODEO—RObust Decentralised Estimation fOr Large-Scale Systems,” funded by the Italian Ministry for Foreign Affairs under Grant Agreement PGR00152.