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Linear projection schemes like Proper Orthogonal Decomposition can efficiently reduce the dimensions of dynamical systems but are naturally limited, e.g., for convection-dominated problems. Nonlinear approaches have shown to outperform linear methods in terms of dimension reduction versus accuracy but, typically, come with a large computational overhead. In this work, we consider a quadratic reduction scheme which induces nonlinear structures that are well accessible to tensorized linear algebra routines. We discuss that nonintrusive approaches can be used to simultaneously reduce the complexity in the equations and propose an operator inference formulation that respects dynamics on nonlinear manifolds.

Algebraic Riccati equations with indefinite quadratic terms play an important role in applications related to robust controller design. While there are many established approaches to solve these in case of small-scale dense coefficients, there is no approach available to compute solutions in the large-scale sparse setting. In this paper, we develop an iterative method to compute low-rank approximations of stabilizing solutions of large-scale sparse continuous-time algebraic Riccati equations with indefinite quadratic terms. We test the developed approach for dense examples in comparison to other established matrix equation solvers, and investigate the applicability and performance in large-scale sparse examples., 19 pages, 2 figures, 5 tables

The optimization of a controlled process in a simulation without access to the model itself is a common scenario and very relevant to many chemical engineering applications. A general approach is to apply a black-box optimization algorithm to a parameterized control scheme. The success then depends on the quality of the parametrization that should be low-dimensional though rich enough to express the salient features. This work proposes using solution snapshots to extract dominant modes of the temporal dynamics of a process and use them for low-dimensional parametrizations of control functions. The approach is called proper orthogonal decomposition in time (time-POD). We provide theoretical reasoning and illustrate the performance for the optimal control of a methanation reactor.

Dynamic mode decomposition (DMD) is a popular data-driven framework to extract linear dynamics from complex high-dimensional systems. In this work, we study the system identification properties of DMD. We first show that DMD is invariant under linear transformations in the image of the data matrix. If, in addition, the data are constructed from a linear time-invariant system, then we prove that DMD can recover the original dynamics under mild conditions. If the linear dynamics are discretized with the Runge–Kutta method, then we further classify the error of the DMD approximation and detail that for one-stage Runge–Kutta methods; even the continuous dynamics can be recovered with DMD. A numerical example illustrates the theoretical findings.

Output-based controllers are known to be fragile with respect to model uncertainties. The standard $\mathcal{H}_{\infty}$-control theory provides a general approach to robust controller design based on the solution of the $\mathcal{H}_{\infty}$-Riccati equations. In view of stabilizing incompressible flows in simulations, two major challenges have to be addressed: the high-dimensional nature of the spatially discretized model and the differential-algebraic structure that comes with the incompressibility constraint. This work demonstrates the synthesis of low-dimensional robust controllers with guaranteed robustness margins for the stabilization of incompressible flow problems. The performance and the robustness of the reduced-order controller with respect to linearization and model reduction errors are investigated and illustrated in numerical examples., Comment: 20 pages, 4 figures, 3 tables

Reduced-order modeling has a long tradition in computational fluid dynamics. The ever-increasing significance of data for the synthesis of low-order models is well reflected in the recent successes of data-driven approaches such as Dynamic Mode Decomposition and Operator Inference. With this work, we suggest a new approach to learning structured low-order models for incompressible flow from data that can be used for engineering studies such as control, optimization, and simulation. To that end, we utilize the intrinsic structure of the Navier-Stokes equations for incompressible flows and show that learning dynamics of the velocity and pressure can be decoupled, thus leading to an efficient operator inference approach for learning the underlying dynamics of incompressible flows. Furthermore, we show the operator inference performance in learning low-order models using two benchmark problems and compare with an intrusive method, namely proper orthogonal decomposition, and other data-driven approaches., 23 pages, 14 figures

In this paper, we present a method to compute the optimal trajectories of a linear quadratic regulator (LQR) problem corresponding to a single input, index-1 constant coefficient DAE system. Further, we also show that using a suitably designed proportional-derivative (PD) state-feedback controller one can force the trajectories of the corresponding DAE system to its optimal trajectories. Unlike the results present in the literature, the results in this paper are applicable for any arbitrary initial condition of the system.

The Riccati-based approach to robust control is completely understood in theory since long but seldom used for large-scale systems in practice. Only recently, we have transferred iterative algorithms that allow the computation of solutions to LQR Riccati equations in the large-scale setting to the indefinite Riccati equations that appear in robust control [2]. For descriptor systems, the relevant Riccati equations are nonsymmetric and, for large-scale systems, there is no established algorithm that can handle these even in the LQR case. In this paper, we show how the general theory for descriptor systems with index-1 pencil coincides with the theory for standard linear time invariant case with feedthrough terms. This provides an algorithm to characterize and to compute the solution to the generalized indefinite Riccati equations via standard Riccati equations. In view of feasibility for large-scale descriptor systems, we illustrate how to arrive at the equivalent standard Riccati-equation without resorting to the canonical form used in the derivation of the theoretical results.

The optimal control of moving boundary problems receives growing attention in science and technology. We consider the so called two-phase Stefan problem that models a solid and a liquid phase separated by a moving interface. The Stefan problem is coupled with incompressible Navier{Stokes equations. We take a sharp interface model approach and define a quadratic tracking-type cost functional that penalizes the deviation of the interface from the desired state and the control costs. With the formal Lagrange approach and an adjoint system we derive the gradient of the cost functional. The derived formulations can be used to achieve a desired interface position. Among others, we address how to handle the weak discontinuity of the temperature along the interface with mesh movement methods in a finite element framework.

The differential Riccati equation appears in different fields of applied mathematics like control and system theory. Recently, Galerkin methods based on Krylov subspaces were developed for the autonomous differential Riccati equation. These methods overcome the prohibitively large storage requirements and computational costs of the numerical solution. Known solution formulas are reviewed and extended. Because of memory-efficient approximations, invariant subspaces for a possibly low-dimensional solution representation are identified. A Galerkin projection onto a trial space related to a low-rank approximation of the solution of the algebraic Riccati equation is proposed. The modified Davison-Maki method is used for time discretization. Known stability issues of the Davison-Maki method are discussed. Numerical experiments for large-scale autonomous differential Riccati equations and a comparison with high-order splitting schemes are presented.

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.

Riccati-based feedback is commonly applied for the stabilization of flows in theory and in simulations. Nonetheless, there are few attempts to show the convergence of numerically computed feedback gains to the feedback defined by the actual model. In this work, we investigate how standard finite-dimensional formulations approximate the system dynamics and provide sufficient conditions for the convergence of the numerical approximations. The sufficient conditions are partially established for the model problem of the cylinder wake.

We discuss a Krylov subspace projection method for model reduction of a special class of quadratic-bilinear descriptor systems. The goal is to extend the two-sided moment-matching method for quadratic-bilinear ODEs to descriptor systems in an efficient and reliable way. Recent results have shown that the direct application of interpolation based model reduction techniques to linear descriptor systems, without any modifications, may lead to poor reduced-order systems. Therefore, for the analysis, we transform the quadratic-bilinear descriptor system into an equivalent quadratic-bilinear ODE system for which the moment-matching is performed. In view of implementation, we provide algorithms that identify the required Krylov subspaces without explicitly computing the projectors used in the analysis. The benefits of our approach are illustrated for the quadratic-bilinear descriptor systems corresponding to semi-discretized Navier–Stokes equations.

We propose a fluid-rigid body interaction benchmark problem, consisting of a solid spherical obstacle in a Newtonian fluid, whose centre of mass is fixed but is free to rotate. A number of different problems are defined for both two and three spatial dimensions. The geometry is chosen specifically, such that the fluid-solid partition does not change over time and classical fluid solvers are able to solve the fluid-structure interaction problem. We summarise the different approaches used to handle the fluid-solid coupling and numerical methods used to solve the arising problems. The results obtained by the described methods are presented and we give reference intervals for the relevant quantities of interest.

This paper presents a new frequency-selective filter based frequency separated feedback control strategy for disturbance attenuation over multiple frequency ranges. The measured output/state of the plant is firstly separted/divided into auxiliary frequency-limited outputs/state by introducing a group of pre-specified frequency-selective filters. Furthermore, a group of frequency-selective controllers is designed to render the frequency-selective component of the closed-loop system. With the aid of generalized KYP lemma, the design of control gains can be recasted into solutions to a set of linear matrix inequalities(LMIs). Finally, the effectiveness of the proposed method is illustrated via numerical examples.

We consider the stabilization of incompressible fluid flow using linearized and spatially discretized models. In order to potentially work in applications, the designed controller must stabilize the discrete model with a robustness margin that covers linearization, discretization, and modeling errors. We expand on previous results that a linearization error in the infinite-dimensional model amounts to a coprime factor uncertainty and show that H∞-robust controllers can compensate this in the discrete approximation. In numerical experiments, we quantify the robustness margins and show that the H∞-robust controller, unlike the LQG-controller, is capable of stabilizing nonlinear incompressible Navier-Stokes equations with an inexact linearization.

Linearization based controllers for incompressible flows have been proven to work in theory and in simulations. To realize such a controller numerically, the infinite dimensional system has to be linearized and discretized. The unavoidable consistency errors add a small but critical uncertainty to the controller model which will likely make it fail, especially when an observer is involved. Standard robust controller designs can compensate small uncertainties if they can be qualified as a coprime factor perturbation of the plant. We show that for the linearized Navier-Stokes equations, a linearization error can be expressed as a coprime factor perturbation and that this perturbation smoothly depends on the size of the linearization error. In particular, improving the linearization makes the perturbation smaller so that, eventually, standard robust controller will stabilize the system., Comment: 13 pages

The differential Sylvester equation and its symmetric version, the differential Lyapunov equation, appear in different fields of applied mathematics like control theory, system theory, and model order reduction. The few available straight-forward numerical approaches if applied to large-scale systems come with prohibitively large storage requirements. This shortage motivates us to summarize and explore existing solution formulas for these equations. We develop a unifying approach based on the spectral theorem for normal operators like the Sylvester operator $\mathcal S(X)=AX+XB$ and derive a formula for its norm using an induced operator norm based on the spectrum of $A$ and $B$. In view of numerical approximations, we propose an algorithm that identifies a suitable Krylov subspace using Taylor series and use a projection to approximate the solution. Numerical results for large-scale differential Lyapunov equations are presented in the last sections., 30 pages, 51 figures

In the context of Galerkin discretizations of a partial differential equation (PDE), the modes of the classical method of Proper Orthogonal Decomposition (POD) can be interpreted as the ansatz and trial functions of a low-dimensional Galerkin scheme. If one also considers a Galerkin method for the time integration, one can similarly define a POD reduction of the temporal component. This has been described earlier but not expanded upon -- probably because the reduced time discretization globalizes time which is computationally inefficient. However, in finite-time optimal control systems, time \textit{is} a global variable and there is no disadvantage from using a POD reduced Galerkin scheme in time. In this paper, we provide a newly developed generalized theory for space-time Galerkin POD, prove its optimality in the relevant function spaces, show its application for the optimal control of nonlinear PDEs, and, by means of a numerical example with Burgers' equation, discuss the competitiveness by comparing to standard approaches.

The Navier-Stokes equations are commonly used to model and to simulate flow phenomena. We introduce the basic equations and discuss the standard methods for the spatial and temporal discretisation. We analyse the semi-discrete equations – a semi-explicit nonlinear DAE – in terms of the strangeness index and quantify the numerical difficulties in the fully discrete schemes, that are induced by the strangeness of the system. By analysing the Kronecker index of the difference-algebraic equations, that represent commonly and successfully used time stepping schemes for the Navier-Stokes equations, we show that those time-integration schemes factually remove the strangeness. The theoretical considerations are backed and illustrated by numerical examples.

We consider mechanical systems where the dynamics are partially constrained to prescribed trajectories. An example for such a system is a building crane with a load and the requirement that the load moves on a certain path. Modelling the system using Newton's second law- "The force acting on an object is equal to the mass of that object times its acceleration."- and enforcing the servo constraints directly leads to differential-algebraic equations (DAEs) of arbitrarily high index. Typically,the model equations are of index 5 which already poses high regularity conditions. Also, common approaches for the numerical time-integration will likely fail. If one relaxes the servo constraints and considers the system from an optimal control point of view, the strong regularity conditions vanish and the solution can be obtained by standard techniques. By means of a spring-mass system, we illustrate the theoretical and expected numerical difficulties. We show how the formulation of the problem in an optimal control context works and address the solvability of the optimal control system. We discuss that the problematic DAE behavior is still inherent in the optimal control system and show how its evidences depend on the regularization parameters of the optimization., Oberwolfach Preprints;2015,18

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.

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.

Flow control, input/output maps, discretization, reduced order modeling.The construction of reduced order models for flow control via a direct discretization of the input/output behavior of thesystem is discussed. The spatially discretized equations are linearized such that an explicit formula for the correspondinginput/output map can be used to generate a matrix representation of the input/output map. Estimates for the approximationerror are derived and the applicability is illustrated via a numerical example for the control of a driven cavity flow.

In a recent work [Baumann et al. 'Discrete Input/Output Maps and their Relation to Proper Orthogonal Decomposition' (2015)], a generalized version of Proper Orthogonal Decomposition (POD) has been formulated based on generalized measurements replacing the snapshot matrix of the standard POD. We extend this approach in two directions. Firstly, we add a parameter dependence as a third dimension of the measurement matrix. Secondly, we use a higher-order SVD to obtain optimized low-dimensional bases for both the space and the time discretization. Then, applying a space-time Galerkin scheme, a time-dependent PDE can be transformed into a small system of algebraic equations -- the reduced model. We illustrate the properties and benefits of this approach for an example of nonlinear Burgers equation with varying diffusion parameters.

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.

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.

In the simulation of flows, the correct treatment of the pressure variable is the key to stable time-integration schemes. This paper contributes a new approach based on the theory of differential- algebraic equations. Motivated by the index reduction technique of minimal extension, a remodelling of the flow equations is proposed. It is shown how this reformulation can be realized for standard finite elements via a decomposition of the discrete spaces and that it ensures stable and accurate approxi- mations. The presented decomposition preserves sparsity and does not call on variable transformations which might change the meaning of the variables. Since the method is eventually an index reduction, high index effects leading to instabilities are eliminated.

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.

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.

The concept of turnpike connects the solution of long but finite time horizon optimal control problems with steady state optimal controls. A key ingredient of the analysis of the turnpike is the linear quadratic regulator problem and the convergence of the solution of the associated differential Riccati equation as the terminal time approaches infinity. This convergence has been investigated in linear systems theory in the 1980s. We extend classical system theoretic results for the investigation of turnpike properties of standard state space systems and descriptor systems. We present conditions for turnpike in the nondetectable case and for impulse controllable descriptor systems. For the latter, in line with the theory for standard linear systems, we establish existence and convergence of solutions to a generalized differential Riccati equation., Comment: 31 pages, 1 figure

In this article, we investigate exponential lag synchronization results for the Cohen-Grossberg neural networks (C-GNNs) with discrete and distributed delays on an arbitrary time domain by applying feedback control. We formulate the problem by using the time scales theory so that the results can be applied to any uniform or non-uniform time domains. Also, we provide a comparison of results that shows that obtained results are unified and generalize the existing results. Mainly, we use the unified matrix-measure theory and Halanay inequality to establish these results. In the last section, we provide two simulated examples for different time domains to show the effectiveness and generality of the obtained analytical results., Comment: 20 pages, 18 figures

This repository includes the most fundamental implementations used for my thesis and all included figures. Abstract: While CFD simulations constantly become more complex, data availability also increases. By exploiting this abundance, modal decomposition and model reduction techniques exhibit an immense potential to improve hemodynamic research. On this behalf, this work derives the dynamic mode decomposition (DMD) method and examines its capabilities based on a straight vessel-like geometry. Furthermore, as patient-specific treatment demands fast adaptation and investigation of flow param- eters, a Navier Stokes equation-based modeling approach is introduced regarding the vessel’s mass-inflow as a system variable. While calculations show that the basic DMD algorithm provides quantitatively accurate results, the applied method of the dynamic mode decomposition with control (DMDc) lacks functional outcomes. Furthermore, different characteristics and variants of the method are demonstrated, providing a basis for various future assessments and applications. &nbsp