Tilman Utz

Trajectory planning for quasilinear parabolic distributed parameter systems based on finite-difference semi-discretisations

In this contribution, a flatness-based approach is considered for the solution of the trajectory planning problem for quasilinear parabolic distributed parameter systems (DPS) by making use of finite-difference semi-discretisations. It is shown that the method yields solutions which are equivalent to results known from the infinite-dimensional trajectory planning for a certain class of quasi-linear parabolic DPS. Furthermore, the methodology being proposed can also be applied to systems with general analytic nonlinearities. As analytical convergence results are not available in this case, a numerical test criterion for the convergence behaviour is suggested.

Trajectory Planning for a Two-Dimensional Quasi-Linear Parabolic PDE Based on Finite Difference Semi-Discretizations

Abstract In this contribution, trajectory planning is considered for a quasi-linear heat-conduction problem defined on a two-dimensional domain. For this, the governing partial differential equation is discretized using finite differences. It is shown that the resulting finite-dimensional system is differentially flat, which serves as a basis for trajectory planning. Simulation results show the applicability of the proposed approach to a heat-conduction problem with physical parameters depending on the state.

Model Predictive Control and Moving Horizon Estimation of a Large-Scale Chemical Reactor Model

Abstract The presented contribution concerns the model predictive control (MPC) and moving horizon estimation (MHE) of a catalytic fixed-bed reactor model. Rigorous modeling for these systems leads to systems of (transport) partial differential equations. Following a so-called early lumping approach for the model predictive control and estimation yields high-dimensional systems of ordinary differential equations and therefore the need to solve large-scale dynamic optimization problems online. It is shown how a tailored gradient method and efficient numerical integration can be combined to solve the concerned optimization methods in a time-efficient way.

Trajectory planning and receding horizon tracking control of a quasilinear diffusion-convection-reaction system

Abstract In this contribution, the tracking control design for setpoint transitions of a quasi-linear diffusion-convection-reaction system with boundary control is considered. At first, a flatness-based feedforward controller is designed based on a finite-difference spatial semi-discretization of the system. In order to stabilize the system around the desired trajectory, a receding horizon tracking controller based on the nonlinear tracking error dynamics is presented. The resulting two-degrees-of-freedom control scheme allows for stabilization and tracking of desired output trajectories in the nominal case as well as in the presence of disturbances and under input constraints.

Optimal control of induction heating processes using FEM software

This contribution presents the optimal control of an induction heating process. The formal Lagrangian technique is applied to the optimal control problem to derive the first-order optimality conditions in terms of a first optimize then discretize approach. It is shown how a gradient algorithm in combination with FEM software can be used to numerically solve the optimality conditions consisting of partial differential algebraic equations of several coupled physical domains in a straightforward manner.

MOTION PLANNING FOR THE HEAT EQUATION WITH RADIATION BOUNDARY CONDITIONS BASED ON FINITE DIFFERENCE SEMI-DISCRETIZATIONS

Abstract In this contribution, motion planning for the temperature distribution in a 1-dimensional slab with radiation boundary conditions is considered. For this, the infinite-dimensional model of the slab is spatially discretized using finite differences. It is shown that the discretized finite-dimensional model is flat and hence serves as a basis for a feedforward control design. For the heat equation with constant parameters, it is shown that the obtained feedforward control converges to the feedforward control for the infinite-dimensional problem in the limit as the discretization step size tends to zero. For the heat equation with temperature dependent parameters, simulation results illustrate the convergence behavior.

Transformation approach to constraint handling in optimal control of the heat equation

Abstract In this contribution, an approach is presented for the optimal control of the boundary-controlled heat equation, which is subject to state- and input-constraints. Thereby, suitably chosen asymptotic saturation functions are used to reformulate the original infinite-dimensional system in new coordinates. The new unconstrained optimal control problem can then be solved with methods of unconstrained optimization. The method is demonstrated for a heat-up problem where both state and input constraints become active.

Efficient state constraint handling for MPC of the heat equation

This contribution is concerned with model predictive control (MPC) of systems governed by partial differential equations (PDEs) and subject to state and input constraints. In particular, the numerically efficient handling of the underlying optimal control problem (OCP) is considered. It is shown that the state- and input-constrained OCP can be transformed by using saturation functions into an unconstrained OCP. This then can be solved numerically by means of well-established optimization methods. For a numerically efficient implementation of the MPC, a first discretize then optimize-approach is used in conjunction with a tailored gradient method. Both the transformation into an unconstrained OCP and its numerical solution are investigated for a simple heat conduction problem.

Dynamische Optimierung von Multiphysik-Problemen am Beispiel induktiver Heizvorgänge

Zusammenfassung Der Beitrag stellt einen Ansatz zur dynamischen Optimierung von Multiphysik-Problemen am Beispiel induktiver Heizvorgänge vor. Hierfür werden im Rahmen eines first-optimize-then-discretize-Ansatzes zunächst die Optimalitätsbedingungen in Form von partiellen Differentialgleichungen aufgestellt. Dies ermöglicht eine elegante numerische Lösung des Problems durch den Einsatz eines Gradientenverfahrens in Verbindung mit FEM-Software (FEM – Finite-Elemente-Methode). Neben der Auslagerung des numerischen Aufwands liegen die Vorteile des Ansatzes insbesondere darin, dass auch Probleme auf komplexen Geometrien relativ einfach behandelt werden können. Die geeignete Formulierung und numerische Lösung des Optimierungsproblems wird anhand von Simulationsergebnissen für die induktive Aufheizung und Oberflächenhärtung eines Zahnrads präsentiert.

Kinetic modelling of an industrial semi-batch alkoxylation reactor for control studies

Semi-batch processes are gaining further importance in the chemical industry, due to the trend to produce more speciality products with higher margins. Besides a constantly high product quality, a maximum space-time-yield is a principal goal of process control for semi-batch processes. This requires to drive the process along its limits, which often is impossible with standard PID control schemes. More sophisticated control methodologies potentially allow for a maximisation of the space-time-yield while keeping the product quality high. However the scientific development and test of such methodologies is often conducted using very simplified reactor models, which may allow for a demonstration of the basic feasibility but neglect additional effects, which hamper the transfer to industrial processes. Therefore a more detailed model of a semi-batch process, suitable for the development and test of control schemes is presented in this contribution. Using this model, a simulation study elucidates the limitations of classical temperature control concepts when aiming for a maximisation of the space-time-yield.