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

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

Ensembles of Hyperbolic PDEs: Stabilization by Backstepping

For the quite extensively developed PDE backstepping methodology for coupled linear hyperbolic PDEs, we provide a generalization from finite collections of such PDEs, whose states at each location in space are vector-valued, to previously unstudied infinite (continuum) ensembles of such hyperbolic PDEs, whose states are function-valued. The motivation for studying such systems comes from traffic applications (where driver and vehicle characteristics are continuously parametrized), fluid and structural applications, and future applications in population dynamics, including epidemiology. Our design is of an exponentially stabilizing scalar-valued control law for a PDE system in two independent dimensions, one spatial dimension and one ensemble dimension. In the process of generalizing PDE backstepping from finite to infinite collections of PDE systems, we generalize the results for PDE backstepping kernels to the continuously parametrized Goursat-form PDEs that govern such continuously parametrized kernels. The theory is illustrated with a simulation example, which is selected so that the kernels are explicitly solvable, to lend clarity and interpretability to the simulation results.Comment: 16 pages, 4 figures, to be publishe

Observer design for multivariable transport-reaction systems based on spatially distributed measurements

This paper is concerned with the design of observers for a class of one-dimensional multi-state transport-reaction systems considering distributed in-domain measurements over the spatial domain. A design based on the Lyapunov method is proposed for the stabilization of the estimation error dynamics. The approach uses positive definite matrices to parameterize a class of Lyapunov functionals that are positive in the Lebesgue space of integrable square functions. Thus, the stability conditions can be expressed as a set of LMI constraints which can be solved numerically using sum of squares (SOS) and standard semi-definite programming (SDP) tools. In order to evaluate the proposed methodology, a state observer is designed to estimate the variables of a nonisothermal tubular reactor model. Numerical simulations are presented to demonstrate the potentials of the proposed observer.Campus Arequip

State estimation for coupled PDE systems using Modulation Functions

This master thesis is devoted to the state estimation of a particular form of PDE systems, coupled parabolic PDEs with spatial dependent coefficients. This form of PDEs represent some dynamic systems such as Tubular Reactors, Diffusion in lithium-ion cells and Diffusive Gradient in Thin Films sensor. Other methods for this problem use "Backstepping" observers, in which the estimation error system is transformed into another system that is stable, reducing the problem to calculate the Kernel functions making the transformation possible. In some cases this calculation is not simple, also the simulation in real time of the observer system, that is also a PDE, can be difficult. The method presented in this thesis uses the properties of the so-called Modulating Functions in order to estimate the states. The procedure con- sists of generating an orthonormal basis of functions that can represent the state as a combination of them. Then auxiliary systems are formed from the original systems with boundary conditions that help in the simplification of the problem. Resolving these auxiliary systems, result in the calculation of the Modulating kernels. All of these steps can be made offline and do not have to be repeated. The functions are used together with the orthonormal basis in the online part, that consists of an inte- gration of a combination of the kernel functions, inputs and outputs of the system in a time window. Finally, with a matrix multiplication the coefficients for the ba- sis expansion of the state can be obtained, resulting in the desired state estimation. The present method is tested in systems that resemble the forms of the dynamics of Tubular Reactors and the performance is compared to other methods.Diese Masterarbeit widmet sich der Zustandsschätzung einer bestimmten Art von Systemen, gekoppelten partiellen Differenzialgleichungen mit raumabhängigen Ko- effizienten. Diese besondere Form von PDEs repräsentiert einige dynamische Sys- teme wie Röhrenreaktoren, Diffusion in Lithium-Ionen-Batterien und Gradienten in dünnen Schichten. Andere Methoden für dieses Problem benutzen "Backstep- ping" Beobachter, bei denen das Schätzfehlersystem in ein anderes stabiles System transformiert wird, wodurch das Problem reduziert wird, um die Kernfunktionen zu berechnen, die die Transformation ermöglichen. In manchen Fällen ist diese Berech- nung nicht einfach. Auch die Simulation in Echtzeit des Beobachters System, das auch eine PDE ist, kann sehr schwierig sein. Die in dieser Arbeit vorgestellte Meth- ode verwendet die Eigenschaften der sogenannten Modulationsfunktionen, um die Zustände zu schätzen. Das Verfahren besteht darin, eine Orthonormalbasis von Funktionen zu erzeugen können, die den Zustand als Kombination von ihnen repräsen- tieren, dann werden Hilfssysteme gebildet von dem ursprünglichen Systemen mit Randbedingungen, die bei der Vereinfachung helfen, von dem Problem. Das Au- flösen dieser Hilfssysteme ergibt die Berechnung der modulierende Kerne. Alle diese Schritte können offline durchgeführt und müssen nicht wiederholt werden. Die Funktionen werden zusammen mit der Orthonormalbasis im Online-Teil ver- wendet. Dieser Teil besteht aus einer Integration einer Kombination der Kernfunk- tionen, Eingaben und Ausgaben des Systems in einem Zeitfenster. Schließlich kön- nen die Koeffizienten zur Basiserweiterung mit einer Matrixmultiplikation berech- net werden, was zu der gewünschte Zustandsschätzung führt. Das Verfahren wird am Beispiel der Dynamik eines Rohreaktors getestet und die Ergebnisse werden mit anderen Methoden verglichen

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal