818 research outputs found
Control and State Estimation of the One-Phase Stefan Problem via Backstepping Design
This paper develops a control and estimation design for the one-phase Stefan
problem. The Stefan problem represents a liquid-solid phase transition as time
evolution of a temperature profile in a liquid-solid material and its moving
interface. This physical process is mathematically formulated as a diffusion
partial differential equation (PDE) evolving on a time-varying spatial domain
described by an ordinary differential equation (ODE). The state-dependency of
the moving interface makes the coupled PDE-ODE system a nonlinear and
challenging problem. We propose a full-state feedback control law, an observer
design, and the associated output-feedback control law via the backstepping
method. The designed observer allows estimation of the temperature profile
based on the available measurement of solid phase length. The associated
output-feedback controller ensures the global exponential stability of the
estimation errors, the H1- norm of the distributed temperature, and the moving
interface to the desired setpoint under some explicitly given restrictions on
the setpoint and observer gain. The exponential stability results are
established considering Neumann and Dirichlet boundary actuations.Comment: 16 pages, 11 figures, submitted to IEEE Transactions on Automatic
Contro
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
On Polynomial Stability of Coupled Partial Differential Equations in 1D
We study the well-posedness and asymptotic behaviour of selected PDE-PDE and
PDE-ODE systems on one-dimensional spatial domains, namely a boundary coupled
wave-heat system and a wave equation with a dynamic boundary condition. We
prove well-posedness of the models and derive rational decay rates for the
energy using an approach where the coupled systems are formulated as feedback
interconnections of impedance passive regular linear systems.Comment: 12 pages, 1 figure, accepted for publication in the Proceedings of
"Semigroups of Operators: Theory and Applications", Kazimierz Dolny, Poland,
October 201
Semi-definite programming and functional inequalities for Distributed Parameter Systems
We study one-dimensional integral inequalities, with quadratic integrands, on
bounded domains. Conditions for these inequalities to hold are formulated in
terms of function matrix inequalities which must hold in the domain of
integration. For the case of polynomial function matrices, sufficient
conditions for positivity of the matrix inequality and, therefore, for the
integral inequalities are cast as semi-definite programs. The inequalities are
used to study stability of linear partial differential equations.Comment: 8 pages, 5 figure
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