Fast positive-real balanced truncation via quadratic alternating direction implicit iteration

Balanced truncation (BT), as applied to date in model order reduction (MOR), is known for its superior accuracy and computable error bounds. Positive-real BT (PRBT) is a particular BT procedure that preserves passivity and stability and imposes no structural constraints on the original state space. However, PRBT requires solving two algebraic Riccati equations (AREs), whose computational complexity limits its practical use in large-scale systems. This paper introduces a novel quadratic extension of the alternating direction implicit (ADI) iteration, which is called quadratic ADI (QADI), that efficiently solves an ARE. A Cholesky factor version of QADI, which is called CEQADI, exploits low-rank matrices and further accelerates PRBT. © 2007 IEEE.published_or_final_versio

Efficient positive-real balanced truncation of symmetric systems via cross-riccati equations

We present a highly efficient approach for realizing a positive-real balanced truncation (PRBT) of symmetric systems. The solution of a pair of dual algebraic Riccati equations in conventional PRBT, whose cost constrains practical large-scale deployment, is reduced to the solution of one cross-Riccati equation (XRE). The cross-Riccatian nature of the solution then allows a simple construction of PRBT projection matrices, using a Schur decomposition, without actual balancing. An invariant subspace method and a modified quadratic alternating-direction-implicit iteration scheme are proposed to efficiently solve the XRE. A low-rank variant of the latter is shown to offer a remarkably fast PRBT speed over the conventional implementations. The XRE-based framework can be applied to a large class of linear passive networks, and its effectiveness is demonstrated through numerical examples. © 2008 IEEE.published_or_final_versio

Efficient Numerical Solution of Large Scale Algebraic Matrix Equations in PDE Control and Model Order Reduction

Matrix Lyapunov and Riccati equations are an important tool in mathematical systems theory. They are the key ingredients in balancing based model order reduction techniques and linear quadratic regulator problems. For small and moderately sized problems these equations are solved by techniques with at least cubic complexity which prohibits their usage in large scale applications. Around the year 2000 solvers for large scale problems have been introduced. The basic idea there is to compute a low rank decomposition of the quadratic and dense solution matrix and in turn reduce the memory and computational complexity of the algorithms. In this thesis efficiency enhancing techniques for the low rank alternating directions implicit iteration based solution of large scale matrix equations are introduced and discussed. Also the applicability in the context of real world systems is demonstrated. The thesis is structured in seven central chapters. After the introduction chapter 2 introduces the basic concepts and notations needed as fundamental tools for the remainder of the thesis. The next chapter then introduces a collection of test examples spanning from easily scalable academic test systems to badly conditioned technical applications which are used to demonstrate the features of the solvers. Chapter four and five describe the basic solvers and the modifications taken to make them applicable to an even larger class of problems. The following two chapters treat the application of the solvers in the context of model order reduction and linear quadratic optimal control of PDEs. The final chapter then presents the extensive numerical testing undertaken with the solvers proposed in the prior chapters. Some conclusions and an appendix complete the thesis

Control and Estimation Oriented Model Order Reduction for Linear and Nonlinear Systems

Optimization based controls are advantageous in meeting stringent performance requirements and accommodating constraints. Although computers are becoming more powerful, solving optimization problems in real-time remains an obstacle because of associated computational complexity. Research efforts to address real-time optimization with limited computational power have intensified over the last decade, and one direction that has shown some success is model order reduction. This dissertation contains a collection of results relating to open- and closed-loop reduction techniques for large scale unconstrained linear descriptor systems, constrained linear systems, and nonlinear systems. For unconstrained linear descriptor systems, this dissertation develops novel gramian and Riccati solution approximation techniques. The gramian approximation is used for an open-loop reduction technique following that of balanced truncation proposed by (Moore, 1981) for ordinary linear systems and (Stykel, 2004) for linear descriptor systems. The Riccati solution is used to generalize the Linear Quadratic Gaussian balanced truncation (LQGBT) of (Verriest, 1981) and (Jonckheere and Silverman, 1983). These are applied to an electric machine model to reduce the number of states from $>$100000 to 8 while improving accuracy over the state-of-the-art modal truncation of (Zhou, 2015) for the purpose of condition monitoring. Furthermore, a link between unconstrained model predictive control (MPC) with a terminal penalty and LQG of a linear system is noted, suggesting an LQGBT reduced model as a natural model for reduced MPC design. The efficacy of such a reduced controller is demonstrated by the real-time control of a diesel airpath. Model reduction generally introduces modeling errors, and controlling a constrained plant subject to modeling errors falls squarely into robust control. A standard assumption of robust control is that inputs/states/outputs are constrained by convex sets, and these sets are ``tightened'' for robust constraint satisfaction. However, robust control is often overly conservative, and resulting control strategies cannot take advantage of the true admissible sets. A new reduction problem is proposed that considers the reduced order model accuracy and constraint conservativeness. A constant tube methodology for reduced order constrained MPC is presented, and the proposed reduced order model is found to decrease the constraint conservativeness of the reduced order MPC law compared to reduced order models obtained by gramian and LQG reductions. For nonlinear systems, a reformulation of the empirical gramians of (Lall et al., 1999) and (Hahn et al., 2003) into simpler, yet more general forms is provided. The modified definitions are used in the balanced truncation of a nonlinear diesel airpath model, and the reduced order model is used to design a reduced MPC law for tracking control. Further exploiting the link between the gramian and Riccati solution for linear systems, the new empirical gramian formulation is extended to obtain empirical Riccati covariance matrices used for closed-loop model order reduction of a nonlinear system. Balanced truncation using the empirical Riccati covariance matrices is demonstrated to result in a closer-to-optimal nonlinear compensator than the previous balanced truncation techniques discussed in the dissertation.PHDNaval Architecture & Marine EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/140839/1/riboch_1.pd

Effizientes Lösen von großskaligen Riccati-Gleichungen und ein ODE-Framework für lineare Matrixgleichungen

This work considers the iterative solution of large-scale matrix equations. Due to the size of the system matrices in large-scale Riccati equations the solution can not be calculated directly but is approximated by a low rank matrix ZYZ^*. Herein Z is a basis of a low-dimensional rational Krylov subspace. The inner matrix Y is a small square matrix. Two ways to choose this inner matrix are examined: By imposing a rank condition on the Riccati residual and by projecting the Riccati residual onto the Krylov subspace generated by Z. The rank condition is motivated by the well-known ADI iteration. The ADI solutions span a rational Krylov subspace and yield a rank-p residual. It is proven that the rank-p condition guarantees existence and uniqueness of such an approximate solution. Known projection methods are generalized to oblique projections and a new formulation of the Riccati residual is derived, which allows for an efficient evaluation of the residual norm. Further a truncated approximate solution is characterized as the solution of a Riccati equation, which is projected to a subspace of the Krylov subspace generated by Z. For the approximate solution of Lyapunov equations a system of ordinary differential equations (ODEs) is solved via Runge-Kutta methods. It is shown that the space spanned by the approximate solution is a rational Krylov subspace with poles determined by the time step sizes and the eigenvalues of the matrices of the Butcher tableau of the used Runge-Kutta method. The method is applied to a model order reduction problem. The analytical solution of the system of ODEs satisfies an algebraic invariant. Those Runge-Kutta methods which preserve this algebraic invariant are characterized by a simple condition on the corresponding Butcher tableau. It is proven that these methods are equivalent to the ADI iteration. The invariance approach is transferred to Sylvester equations.Diese Arbeit befasst sich mit der numerischen Lösung hochdimensionaler Matrixgleichungen mittels iterativer Verfahren. Aufgrund der Größe der Systemmatrizen in großskaligen algebraischen Riccati-Gleichung kann die Lösung nicht direkt bestimmt werden, sondern wird durch eine approximative Lösung ZYZ^* von geringem Rang angenähert. Hierbei wird Z als Basis eines rationalen Krylovraums gewählt und enthält nur wenige Spalten. Die innere Matrix Y ist klein und quadratisch. Es werden zwei Wege untersucht, die Matrix Y zu wählen: Durch eine Rang-Bedingung an das Riccati-Residuum und durch Projektion des Riccati-Residuums auf den von Z erzeugten Krylovraum. Die Rang-Bedingung wird durch die wohlbekannten ADI-Verfahren motiviert. Die approximativen ADI-Lösungen spannen einen Krylovraum auf und führen zu einem Riccati-Residuum vom Rang p. Es wird bewiesen, dass die Rang-p-Bedingung Existenz und Eindeutigkeit einer solchen approximativen Lösung impliziert. Aus diesem Ergebnis werden effiziente iterative Verfahren abgeleitet, die eine solche approximative Lösung erzeugen. Bisher bekannte Projektionsverfahren werden auf schiefe Projektionen erweitert und es wird eine neue Formulierung des Riccati-Residuums hergeleitet, die eine effiziente Berechnung der Norm erlaubt. Weiter wird eine abgeschnittene approximative Lösung als Lösung einer Riccati-Gleichung charakterisiert, die auf einen Unterraum des von Z erzeugten Krylovraums projiziert wird. Um die Lösung der Lyapunov-Gleichung zu approximieren wird ein System gewöhnlicher Differentialgleichungen mittels Runge-Kutta-Verfahren numerisch gelöst. Es wird gezeigt, dass der von der approximativen Lösung aufgespannte Raum ein rationaler Krylovraum ist, dessen Pole von den Zeitschrittweiten der Integration und den Eigenwerten der Koeffizientenmatrix aus dem Butcher-Tableau des verwendeten Runge-Kutta-Verfahrens abhängen. Das Verfahren wird auf ein Problem der Modellreduktion angewendet. Die analytische Lösung des Differentialgleichungssystems erfüllt eine algebraische Invariante. Diejenigen Runge-Kutta-Verfahren, die diese Invariante erhalten, werden durch eine Bedingung an die zugehörigen Butcher-Tableaus charakterisiert. Es wird gezeigt, dass diese speziellen Verfahren äquivalent zur ADI-Iteration sind. Der Invarianten-Ansatz wird auf Sylvester-Gleichungen übertragen

System- and Data-Driven Methods and Algorithms

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This first volume focuses on real-time control theory, data assimilation, real-time visualization, high-dimensional state spaces and interaction of different reduction techniques

Grid generation for the solution of partial differential equations

A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given

Dynamic iteration and model order reduction for magneto-quasistatic systems

Our world today is becoming increasingly complex, and technical devices are getting ever smaller and more powerful. The high density of electronic components together with high clock frequencies leads to unwanted side-effects like crosstalk, delayed signals and substrate noise, which are no longer negligible in chip design and can only insufficiently be represented by simple lumped circuit models. As a result, different physical phenomena have to be taken into consideration since they have an increasing influence on the signal propagation in integrated circuits. Computer-based simulation methods play thereby a key role. The modelling and analysis of complex multi-physics problems typically leads to coupled systems of partial differential equations and differential-algebraic equations (DAEs). Dynamic iteration and model order reduction are two numerical tools for efficient and fast simulation of coupled systems. Formodelling of low frequency electromagnetic field, we use magneto-quasistatic (MQS) systems which can be considered as an approximation to Maxwells equations. A spatial discretization by using the finite element method leads to a DAE system. We analyze the structural and physical properties of this system and develop passivity-preserving model reduction methods. A special block structure of the MQS model is exploited to to improve the performance of the model reduction algorithms

Glosarium Matematika

273 p.; 24 cm

Structure-Preserving Model Reduction of Physical Network Systems

This paper considers physical network systems where the energy storage is naturally associated to the nodes of the graph, while the edges of the graph correspond to static couplings. The first sections deal with the linear case, covering examples such as mass-damper and hydraulic systems, which have a structure that is similar to symmetric consensus dynamics. The last section is concerned with a specific class of nonlinear physical network systems; namely detailed-balanced chemical reaction networks governed by mass action kinetics. In both cases, linear and nonlinear, the structure of the dynamics is similar, and is based on a weighted Laplacian matrix, together with an energy function capturing the energy storage at the nodes. We discuss two methods for structure-preserving model reduction. The first one is clustering; aggregating the nodes of the underlying graph to obtain a reduced graph. The second approach is based on neglecting the energy storage at some of the nodes, and subsequently eliminating those nodes (called Kron reduction).</p