State estimation for coupled reaction-diffusion PDE systems using modulating functions

Affiliation
Departamento de Ingeniería, Pontificia Universidad Católica del Perú, Avenida Universitaria 1801, Lima 15088, Peru;
Pumaricra Rojas, David;
ORCID
0000-0001-5959-5131
Affiliation
Control Engineering Group, Technische Universität Ilmenau, 98693 Ilmenau, Germany;(M.N.);(J.R.)
Noack, Matti;
Affiliation
Control Engineering Group, Technische Universität Ilmenau, 98693 Ilmenau, Germany;(M.N.);(J.R.)
Reger, Johann;
ORCID
0000-0001-5946-1395
Affiliation
Departamento de Ingeniería, Pontificia Universidad Católica del Perú, Avenida Universitaria 1801, Lima 15088, Peru;
Pérez-Zúñiga, Gustavo

Many systems with distributed dynamics are described by partial differential equations (PDEs). Coupled reaction-diffusion equations are a particular type of these systems. The measurement of the state over the entire spatial domain is usually required for their control. However, it is often impossible to obtain full state information with physical sensors only. For this problem, observers are developed to estimate the state based on boundary measurements. The method presented applies the so-called modulating function method, relying on an orthonormal function basis representation. Auxiliary systems are generated from the original system by applying modulating functions and formulating annihilation conditions. It is extended by a decoupling matrix step. The calculated kernels are utilized for modulating the input and output signals over a receding time window to obtain the coefficients for the basis expansion for the desired state estimation. The developed algorithm and its real-time functionality are verified via simulation of an example system related to the dynamics of chemical tubular reactors and compared to the conventional backstepping observer. The method achieves a successful state reconstruction of the system while mitigating white noise induced by the sensor. Ultimately, the modulating function approach represents a solution for the distributed state estimation problem without solving a PDE online.

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