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The optimal control of chemostat reactors is relevant for the commercial manufacture of products in industrial plants. The microorganisms in the reactor produce different amounts of product depending on their age structure. In order to optimally affect this structure, the concentration of nutrient solution in the reactor can be varied. The modeling of the population of microorganisms leads to a hyperbolic semi-linear PDE of first order with integral boundary conditions. In order to calculate an optimal control, an Early- as well as a Late-Lumping approach is used. For the discretization of the system the finite difference method is applied. Simulations are used to compare the two approaches and verify the optimality of the control. The reference trajectory can be tracked precisely, while the input constraints are satisfied.

We outline an image reconstruction algorithm that is capable of reconstructing an object from the averaged Fourier intensity of finite crystals of that object with arbitrary lattice occupancies. The algorithm is tested via simulations where the 3-dimensional electron density of a protein molecule is successfully reconstructed from such intensity measurements.

Coupled parabolic partial differential equations (PDEs) are encountered in many mechatronic applications, such as the thermal control of elastic deformations. Our contribution considers spatially distributed inputs which actuate the PDEs through a bounded spatial function. The proposed method allows an inversion of the input-output dynamics and can be used for feedforward control and the compensation of disturbances. A crucial prerequisite is the ability to represent the spatial characteristic as a solution to an ordinary differential equation (ODE). This makes it possible to analyze the input-output behavior of the PDE system by virtue of differential geometric arguments. We extract a finite-dimensional subsystem containing all relevant properties of the input-output behavior. For instance, the outputs' relative degrees can be discussed using ODE methods. Furthermore, the convergence of the inverse system can be guaranteed if the internal dynamics are stable; a natural assumption for inversion-based approaches.

This contribution focuses on the optimal design and pneumatic deformation control of thin circular plates with spatially-varying thickness. The work is motivated by pneumatically actuated deformable mirrors, which are used to compensate for wavefront disturbances in adaptive optics. To achieve the required deformation profiles with high precision, the equilibrium deformation is optimized with respect to the plate's thickness. For designing an asymptotic tracking control, the finite relative degree of the distributed parameter system is exploited. Since the nonlinear pressure dynamics actuate the governing partial differential equation (PDE) in a distributed way, finite-order feedback linearization can be applied. In addition, static nonlinear effects in the pneumatic valve and infinite-dimensional internal dynamics have to be taken into account. Finally, we demonstrate the controller's performance by means of simulations.

Scale-up is a major challenge in bioprocess engineering. Because physically achievable mixing times in large-scale bioreactors are limited, gradients of dissolved oxygen are likely to occur. Since the impact on performance of production strains in this environment is poorly understood, we address this question by developing a distributed parameter model. This approach makes it possible to model the adaptive behavior of Escherichia coli during reoccurring switches between oxygen free and oxygen rich environments. We use a set of first-order hyperbolic partial differential equations actuated by spatially varying activation dynamics. The design of the distributed parameter model is based on a scale-down bioreactor, which is used for an experimental validation of the simulation results.

This article analyzes a class of distributed parameter systems, which consist of a boundary-controlled partial differential equation (PDE) and a static boundary-value problem (BVP) generating the measured output. We focus on the steady-state reachability of given output profiles and construct corresponding input signals using modal decomposition techniques. Furthermore, we identify which PDE modes contribute to the output and establish state estimates. An Euler-Bernoulli beam that deforms under thermal load is chosen as an illustrative example.

In this contribution we introduce an efficient simulation scheme for a class of hyperbolic dynamic systems, which model the age-structured evolution of a microorganism population in a bio reactor. Due to the non-local boundary condition and the semilinear transient behavior, standard methods like finite-differences do not lead to satisfactory results, even at high approximation orders. This gives rise to advanced simulation schemes: The presented algorithm utilizes the framework of weighted residuals and preserves crucial properties of the PDE system. For instance, we incorporate the steady state solution of the PDE system and guarantee a vanishing steady state approximation error — independent of the order.