Jan Heiland

LQG-Balanced Truncation Low-Order Controller for Stabilization of Laminar Flows

Recent theoretical and simulation results have shown that Riccati based feedback can stabilize flows at moderate Reynolds numbers. We extend this established control setup by the method of LQG-balanced truncation. In view of practical implementation, we introduce a controller that bases only on outputs rather than on the full state of the system. Also, we provide a very low dimensional observer so that the control actuation can be computed in an online fashion.

Best practices for replicability, reproducibility and reusability of computer-based experiments exemplified by model reduction software

Over the recent years the importance of numerical experiments has gradually been more recognized. Nonetheless, sufficient documentation of how computational results have been obtained is often not available. Especially in the scientific computing and applied mathematics domain this is crucial, since numerical experiments are usually employed to verify the proposed hypothesis in a publication. This work aims to propose standards and best practices for the setup and publication of numerical experiments. Naturally, this amounts to a guideline for development, maintenance, and publication of numerical research software. Such a primer will enable the replicability and reproducibility of computer-based experiments and published results and also promote the reusability of the associated software.

A New Discretization Framework for Input/Output Maps and Its Application to Flow Control

We discuss the direct discretization of the input/output map of linear time-invariant systems with distributed inputs and outputs. At first, the input and output signals are discretized in space and time, resulting in a matrix representation of an approximated input/output map. Then the system dynamics is approximated, in order to calculate the matrix representation numerically. The discretization framework, corresponding error estimates, a SVD-based system reduction method and a numerical application in optimal flow control are presented.

Discrete Input/Output Maps and their Relation to Proper Orthogonal Decomposition

Current control design techniques require system models of moderate size to be applicable. The generation of such models is challenging for complex systems which are typically described by partial differential equations (PDEs), and model-order reduction or low-order-modeling techniques have been developed for this purpose. Many of them heavily rely on the state space models and their discretizations. However, in control applications, a sufficient accuracy of the models with respect to their input/output (I/O) behavior is typically more relevant than the accurate representation of the system states. Therefore, a discretization framework has been developed and is discussed here, which heavily focuses on the I/O map of the original PDE system and its direct discretization in the form of an I/O matrix and with error bounds measuring the relevant I/O error. We also discuss an SVD-based dimension reduction for the matrix representation of an I/O map and how it can be interpreted in terms of the Proper Orthogonal Decomposition (POD) method which gives rise to a more general POD approach in time capturing. We present numerical examples for both, reduced I/O map s and generalized POD.

Exponential Stability and Stabilization of Extended Linearizations via Continuous Updates of Riccati Based Feedback

Summary Many recent works on the stabilization of nonlinear systems target the case of locally stabilizing an unstable steady-state solution against small perturbations. In this work, we explicitly address the goal of driving a system into a nonattractive steady state starting from a well-developed state for which the linearization-based local approaches will not work. Considering extended linearizations or state-dependent coefficient representations of nonlinear systems, we develop sufficient conditions for the stability of solution trajectories. We find that if the coefficient matrix is uniformly stable in a sufficiently large neighborhood of the current state, then the state will eventually decay. On the basis of these analytical results, we propose a scheme that is designed to maintain the stabilization property of a Riccati-based feedback constant during a certain period of the state evolution. We illustrate the general applicability of the resulting algorithm for setpoint stabilization of nonlinear autonomous systems and its numerical efficiency in 2 examples.

A Differential-Algebraic Riccati Equation for Applications in Flow Control

We investigate existence and structure of solutions to quadratic control problems with semiexplicit differential-algebraic constraints as they appear in the modeling of flows. We introduce a decoupled representation of the state and the formal adjoint equations to identify the conditions for the existence of solutions. We derive a differential-algebraic Riccati equation formulated in the original coefficients for the efficient numerical computation of the optimal solution.

SYSTEMATIC DISCRETIZATION OF INPUT/OUTPUT MAPS AND CONTROL OF PARTIAL DIFFERENTIAL EQUATIONS

We present a framework for the direct discretization of the input/output map of dynamical systems governed by linear partial differential equations with distributed inputs and outputs. The approximation consists of two steps. First, the input and output signals are discretized in space and time, resulting in finite dimensional spaces for the input and output signals. These are then used to approximate the dynamics of the system. The approximation errors in both steps are balanced and a matrix representation of an approximate input/output map is constructed which can be further reduced using singular value decompositions. We present the discretization framework, corresponding error estimates, and the SVD-based system reduction method. The theoretical results are illustrated with some applications in the optimal control of partial differential equations.