TY - JOUR

T1 - State Estimation for Coupled Reaction-Diffusion PDE Systems Using Modulating Functions

AU - Rojas, David Pumaricra

AU - Noack, Matti

AU - Reger, Johann

AU - Pérez-Zúñiga, Gustavo

N1 - Publisher Copyright:
© 2022 by the authors. Licensee MDPI, Basel, Switzerland.

PY - 2022/7/1

Y1 - 2022/7/1

N2 - 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.

AB - 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.

KW - boundary observer

KW - coupled partial differential equations

KW - modulating function method

KW - reaction-diffusion partial differential equations

KW - state estimation

UR - http://www.scopus.com/inward/record.url?scp=85133135610&partnerID=8YFLogxK

U2 - 10.3390/s22135008

DO - 10.3390/s22135008

M3 - Article

C2 - 35808503

AN - SCOPUS:85133135610

SN - 1424-8220

VL - 22

JO - Sensors (Switzerland)

JF - Sensors (Switzerland)

IS - 13

M1 - 5008

ER -