Compared to adaptive control of lumped chemical processes which has been well studied in the last decades, the adaptive control problem of transport-reaction processes is a more challenging task that needs more advanced mathematical tools . The spatiotemporal dynamic behavior of such systems which can be mathematically modeled by parabolic partial differential equations (PDEs) usually includes unknown terms due to unidentified terms or uncertainty in transport and reaction mechanisms. This is typical in the chemical and advanced material industries with examples ranging from packed and flow reactors, crystallization and polymerization process, chemical vapor deposition to microelectronic fabrication processes and it showcases the importance of system identification and adaptive control methodologies in industries.
The infinite-dimensional representation of parabolic PDEs which describes the aforesaid transport-reaction processes can in principle be partitioned into a finite-dimensional slow (and possibly unstable) and an infinite dimensional fast (and stable) subsystems . The time-scale separation between the subsystems can also be identified by analytical or statistical approaches. Taking advantage of such separation we apply Galerkin projection to derive a low-dimensional reduced order model (ROM) which can be employed as the basis for adaptive controller designs. To implement such strategy we require the basis functions of the system to discretize the governing PDEs to ROMs in the form of low-dimensional ordinary differential equations (ODEs). For systems with known linear spatial differential operators defined over simple domains we can find an analytical basis functions. However we cannot identify the required basis functions analytically when the spatial differential operator is nonlinear or contains the unknown parameters, or when the process operates over irregular domains. For such cases statistical techniques such as proper orthogonal decomposition (POD) can be applied to compute the dominant empirical basis functions based on the available solution profiles of the system. The essential requirement of the POD-like approaches is the a-priori availability of solution profiles of the governing PDEs which present the global spatiotemporal dynamics of the studied system. To overcome such a limiting requirement we employ adaptive proper orthogonal decomposition (APOD) to recursively compute the set of required empirical basis functions during process evolution [3, 4].
In this work we consider the output feedback control problem of parabolic transport-reaction processes in the presence of unknown parameters. We classify the general form of the parabolic distributed chemical processes based on the required basis functions computation methods. Then to generalize the model reduction based adaptive control method we develop a two-layer system adaptation approach. In the external layer we employ APOD to recursively revise the set of empirical basis functions and reconstruct the ROM as needed during process operation. In the internal layer we derive a Lyapunov-based adaptive output feedback controller to deal with the unknown parameters of the recursively updated ROM. A Luenberger-type dynamic observer is also synthesized to estimate the state of the ROM required by the output feedback control structure. The simulation results are presented for adaptive control of thermal spatiotemporal dynamics in a tubular reactor in the presence of unknown diffusivity coefficient, axial velocity and reaction kinetics.
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