When balanced truncation is used for model order reduction, one has to solve a pair of Lyapunov equations for two Gramians and uses them to construct a reduced-order model. Although advances in solving such equations have been made, it is still the most expensive step of this reduction method. Parametric model order reduction aims to determine reduced-order models for parameter-dependent systems. Popular techniques for parametric model order reduction rely on interpolation. Nevertheless, the interpolation of Gramians is rarely mentioned, most probably due to the fact that Gramians are symmetric positive semidefinite matrices, a property that should be preserved by the interpolation method. In this contribution, we propose and compare two approaches for Gramian interpolation. In the first approach, the interpolated Gramian is computed as a linear combination of the data Gramians with positive coefficients. Even though positive semidefiniteness is guaranteed in this method, the rank of the interpolated Gramian can be significantly larger than that of the data Gramians. The second approach aims to tackle this issue by performing the interpolation on the manifold of fixed-rank positive semidefinite matrices. The results of the interpolation step are then used to construct parametric reduced-order models, which are compared numerically on two benchmark problems.
Model order reduction is a technique that is used to construct low-order approximations of large-scale dynamical systems. In this paper, we investigate a balancing based model order reduction method for dynamical systems with a linear dynamical equation and a quadratic output function. To this aim, we propose a new algebraic observability Gramian for the system based on Hilbert space adjoint theory. We then show the proposed Gramians satisfy a particular type of generalized Lyapunov equations and we investigate their connections to energy functionals, namely, the controllability and observabil…
In this work, the empirical-Gramian-based model reduction methods: Empirical poor man's truncated balanced realization, empirical approximate balancing, empirical dominant subspaces, empirical balanced truncation, and empirical balanced gains are compared in a non-parametric and two parametric variants, via ten error measures: Approximate Lebesgue $L_0$, $L_1$, $L_2$, $L_\infty$, Hardy $H_2$, $H_\infty$, Hankel, Hilbert-Schmidt-Hankel, modified induced primal, and modified induced dual norms, for variants of the thermal block model reduction benchmark. This comparison is conducted via a new me…
We analyze a family of Runge-Kutta based quadrature algorithms for the approximation of the gramians of linear time invariant dynamical systems. The approximated gramians are used to obtain an approximate balancing transformation similar to the approach used in balanced POD. It is shown that hereby rational interpolation is performed, as the approximants span certain Krylov subspaces. The expansion points are mainly determined by the time step sizes and the eigenvalues of the matrices given by the Butcher tableaus.
Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）；Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL), UMR CNRS 9189, CNRS, Ecole Centrale de Lille（里尔信息、信号与自动化研究中心 (CRIStAL)，“CNRS联合研究单位9189”，法国国家科学研究中心，里尔中央理工学院）；Department of Electrical and Computer Engineering, Rice University（莱斯大学电气与计算机工程系）；CNRS CRIStAL UMR 9189, Université de Lille 1, Sciences et Technologies（法国国家科学研究中心 CRIStAL “联合研究单位9189”，里尔第一大学科学与技术学院）
Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）；Faculty of Mathematics, Otto von Guericke University Magdeburg（马格德堡奥托·冯·格里克大学数学学院）
We consider the Bayesian approach to the linear Gaussian inference problem of inferring the initial condition of a linear dynamical system from noisy output measurements taken after the initial time. In practical applications, the large dimension of the dynamical system state poses a computational obstacle to computing the exact posterior distribution. Model reduction offers a variety of computational tools that seek to reduce this computational burden. In particular, balanced truncation is a system-theoretic approach to model reduction which obtains an efficient reduced-dimension dynamical sy…
We investigate model order reduction (MOR) for linear dynamical systems, where a quadratic output is defined as a quantity of interest. The system can be transformed into a linear dynamical system with many linear outputs. MOR is feasible by the method of balanced truncation, but suffers from the large number of outputs in approximate methods. To ameliorate this shortcoming we derive an equivalent quadratic-bilinear system with a single linear output and analyze the properties of this system. We examine MOR for this system via the technique of balanced truncation, which requires a stabilizatio…
We consider linear magneto-quasistatic field equations which arise in simulation of low-frequency electromagnetic devices coupled to electrical circuits. A finite element discretization of such equations on 3D domains leads to a singular system of differential-algebraic equations. First, we study the structural properties of such a system and present a new regularization approach based on projecting out the singular state components. Furthermore, we present a Lyapunov-based balanced truncation model reduction method which preserves stability and passivity. By making use of the underlying struc…
We present a balanced truncation model reduction approach for a class of nonlinear systems with time-varying and uncertain inputs. First, our approach brings the nonlinear system into quadratic-bilinear~(QB) form via a process called lifting, which introduces transformations via auxiliary variables to achieve the specified model form. Second, we extend a recently developed QB balanced truncation method to be applicable to such lifted QB systems that share the common feature of having a system matrix with zero eigenvalues. We illustrate this framework and the multi-stage lifting transformation …
Considering the use of dynamical systems in practical applications, often only limited regions in the time or frequency domain are of interest. Therefor, it usually pays off to compute local approximations of the used dynamical systems in the frequency and time domain. In this paper, we consider a structure-preserving extension of the frequency- and time-limited balanced truncation methods to second-order dynamical systems. We give a full overview about the first-order limited balanced truncation methods and extend those methods to second-order systems by using the different second-order balan…
Model order reduction (MOR) is often applied to spatially-discretized partial differential equations to reduce their order and hence decrease computational complexity. A reduced system can be obtained, e.g., by time-limited balanced truncation, a method that aims to construct an accurate reduced order model on a given finite time interval $[0, T]$. This particular balancing related MOR technique is studied in this paper. An $L^2_T$-error bound based on the truncated time-limited singular values is proved and is the main result of this paper. The derived error bound converges (as $T\rightarrow …
We introduce a model reduction approach for linear time-invariant second order systems based on positive real balanced truncation. Our method guarantees asymptotic stability and passivity of the reduced order model as well as the positive definiteness of the mass and stiffness matrices. Moreover, we receive an a priori gap metric error bound. Finally, we show that our method based on positive real balanced truncation preserves the structure of overdamped second order systems.
Numerical computation and new output bounds for time-limited balanced truncation of discrete-time systems
Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）；KU Leuven, Electrical Engineering (ESAT), Kulak Kortrijk Campus（荷语鲁汶大学，电气工程（ESAT），库拉克科特赖克校区）
A low-rank balanced truncation approach for large-scale RLCk model order reduction based on extended Krylov subspace and a frequency-aware convergence criterion
In this paper, the problem of full state approximation by model reduction is studied for stochastic and bilinear systems. Our proposed approach relies on identifying the dominant subspaces based on the reachability Gramian of a system. Once the desired subspace is computed, the reduced order model is then obtained by a Galerkin projection. We prove that, in the stochastic case, this approach either preserves mean square asymptotic stability or leads to reduced models whose minimal realization is mean square asymptotically stable. This stability preservation guarantees the existence of the redu…
In standard balanced truncation model order reduction, the initial condition is typically ignored in the reduction procedure and is assumed to be zero instead. However, such a reduced-order model may be a bad approximation to the full-order system, if the initial condition is not zero. In the literature there are several attempts for modified reduction methods at the price of having no error bound or only a posteriori error bounds which are often too expensive to evaluate. In this work we propose a new balancing procedure that is based on a shift transformation on the state. We first derive a …
Equilibrium or stationary solutions usually proceed through the exact balance between hyperbolic transport terms and source terms. Such equilibrium solutions are affected by truncation errors that prevent any classical numerical scheme from capturing the evolution of small amplitude waves of physical significance. In order to overcome this problem, we compare two commonly adopted strategies: going to very high order and reduce drastically the truncation errors on the equilibrium solution, or design a specific scheme that preserves by construction the equilibrium exactly, the so-called well-bal…
The aim in model order reduction is to approximate an input-output map described by a large-scale dynamical system with a low-dimensional and cheaper-to-evaluate reduced order model. While high fidelity can be achieved by a variety of methods, only a few of them allow for rigorous error control. In this paper, we propose a rigorous error bound for the reduction of linear systems with balancing-related reduction methods. More specifically, we consider the simulation over a finite time interval and provide an a posteriori adaption of the standard a priori bound for Balanced Truncation and Balanc…
We analyze a structure-preserving model order reduction technique for delay and stochastic delay equations based on the balanced truncation method and provide a system theoretic interpretation. Transferring error bounds based on Hankel operators to delay systems, we find error estimates for the difference between the dynamics of the full and reduced model. This analysis also yields new error bounds for bilinear systems and stochastic systems with multiplicative noise and non-zero initial states.
We analyze structure-preserving model order reduction methods for Ornstein-Uhlenbeck processes and linear S(P)DEs with multiplicative noise based on balanced truncation. For the first time, we include in this study the analysis of non-zero initial conditions. We moreover allow for feedback-controlled dynamics for solving stochastic optimal control problems with reduced-order models and prove novel error bounds for a class of linear quadratic regulator problems. We provide numerical evidence for the bounds and discuss the application of our approach to enhanced sampling methods from non-equilib…
Most model reduction methods reduce the state dimension and then temporally evolve a set of coefficients that encode the state in the reduced representation. In this paper, we instead employ an efficient representation of the entire trajectory of the state over some time interval of interest and then solve for the static coefficients that encode the trajectory on the interval. We use spectral proper orthogonal decomposition (SPOD) modes, which are provably optimal for representing long trajectories and substantially outperform any representation of the trajectory in a purely spatial basis (e.g…
We investigate an ultraweak variational formulation for (parameterized) linear differential-algebraic equations (DAEs) w.r.t. the time variable which yields an optimally stable system. This is used within a Petrov-Galerkin method to derive a certified detailed discretization which provides an approximate solution in an ultraweak setting as well as for model reduction w.r.t. time in the spirit of the Reduced Basis Method (RBM). A computable sharp error bound is derived. Numerical experiments are presented that show that this method yields a significant reduction and can be combined with well-kn…
We present a novel reformulation of balanced truncation, a classical model reduction method. The principal innovation that we introduce comes through the use of system response data that has been either measured or computed, without reference to any prescribed realization of the original model. Data are represented by sampled values of the transfer function {or the impulse response} corresponding to the original model. We discuss parallels that our approach bears with the Loewner framework, another popular data-driven model reduction method. We illustrate our approach numerically in both conti…
Vibration and dissipation in vibro-acoustic systems can be assessed using frequency response analysis. Evaluating a frequency sweep on a full-order model can be very costly, so model order reduction methods are employed to compute cheap-to-evaluate surrogates. This work compares structure-preserving model reduction methods based on rational interpolation and balanced truncation with a specific focus on their applicability to vibro-acoustic systems. Such models typically exhibit a second-order structure and their material properties as well as their excitation may be depending on the driving fr…
We study output tracking and disturbance rejection for an Euler-Bernoulli beam with Kelvin-Voigt damping. The system has distributed control and pointwise observation. As our main result we design a finite-dimensional low-order internal model based controller that is based on a spectral Galerkin method and model reduction by Balanced Truncation. The performance of the designed controller is demonstrated with numerical simulations and compared to the performance of a low-gain internal model based controller.
We study robust output regulation for parabolic partial differential equations and other infinite-dimensional linear systems with analytic semigroups. As our main results we show that robust output tracking and disturbance rejection for our class of systems can be achieved using a finite-dimensional controller and present algorithms for construction of two different internal model based robust controllers. The controller parameters are chosen based on a Galerkin approximation of the original PDE system and employ balanced truncation to reduce the orders of the controllers. In the second part o…
In this paper we consider a geothermal energy storage in which the spatio-temporal temperature distribution is modeled by a heat equation with a convection term. Such storages often are embedded in residential heating systems and control and management require the knowledge of some aggregated characteristics of that temperature distribution in the storage. They describe the input-output behaviour of the storage and the associated energy flows and their response to charging and discharging processes. We aim to derive an efficient approximative description of these characteristics by a low-dimen…
