Transport processes are essential in many chemical plants, which are usually described by partial differential equations , . When diffusive transport is negligible and convective transport is dominant, processes can be described by first-order hyperbolic PDEs. The class of such processes includes tubular reactors, heat exchangers and chromatographic columns, in which the knowledge of the state of the systems is of paramount importance for the operating monitoring and/or subsequent control law synthesis.
Significant research efforts have been focused on the development of observer design methods for PDE systems. In order to respect the distributed nature of these systems, many approaches have been developed, where an important class of observers are back-stepping type observers .
In this work, a general observer design method for the first-order hyperbolic nonlinear partial differential equation systems is presented. In order to deal with the nonlinearity of the system, the equilibrium profile is calculated firstly. Then, the system is linearized around the equilibrium profile and is represented as an infinite-dimensional Hilbert state-space system with infinite-dimensional (distributed) input and output. Based on the infinite-dimensional state-space representation, the formulation of the observer design problem is realized, where our target is to find a state transformation such that the observer to be designed possesses a specific form. Motivated by the Luenberger observer design ideas, the proposed approach in this work is constructed based on the so-called Lyapunov's auxiliary theorem  and single-step approach that achieves desired observer convergence. Taking advantage of the state transformation, the target observer is obtained.
Finally, an easily implementable proposed observer is derived and realized, via simulation, in a fixed-bed chemical reactor, where one elementary reaction takes place.
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