Cross-Gramian-based dominant subspaces

We address the problem of model order reduction (MOR) of parametrized dynamical systems. Motivated by reduced basis (RB) methods for partial differential equations, we show that some characteristic components can be transferred to model reduction of parametrized linear dynamical systems. We assume an affine parameter dependence of the system components, which allows an offline/online decomposition and is the basis for efficient reduced simulation. Additionally, error control is possible by a posteriori error estimators for the state vector and output vector, based on residual analysis and primal-dual techniques. Experiments demonstrate the applicability of the reduced parametrized systems, the reliability of the error estimators and the runtime gain by the reduction technique. The a posteriori error estimation technique can straightforwardly be applied to all traditional projection-based reduction techniques of non-parametric and parametric linear systems, such as model reduction, balanced truncation, moment matching, proper orthogonal decomposition (POD) and so on.

The problem of model reduction of linear systems with certain interconnection structure is considered in this paper. To preserve the interconnection structure between subsystems in the reduction, special care needs to be taken. This problem is important and timely because of the recent focus on complex networked systems in control engineering. Two different model reduction methods are introduced and compared in this paper. Both methods are extensions to the well‐known balanced truncation method. Compared with earlier work in the area these methods use a more general linear fractional transformation framework, and utilize linear matrix inequalities. Furthermore, new approximation error bounds that reduce to classical bounds in special cases are derived. The so‐called structured Hankel singular values are used in the methods, and indicate how important states in the subsystems are with respect to a chosen input–output map for the entire interconnected system. It is shown how these structured Hankel singular values can be used to select an approximation order. Finally, the two methods are applied to a model of a mechanical device. Copyright © 2008 John Wiley & Sons, Ltd.

We discuss algorithms for balanced truncation (BT) based model reduction of linear systems. BT is known to have good global approximation properties and to preserve important system properties. A computable error bound allows to choose the order of the reduced-order model adaptively. We will emphasize those aspects that makes the application of BT to models arising in circuit simulation a non-straightforward task. In recent years, these issues have been addressed by several authors. We will survey some of these developments and demonstrate that BT is now suitable for linear descriptor systems encountered in circuit simulation.

Modern power systems have experienced a significant increase in its complexity, and extremely large-scale system models have to be addressed in the study of stability analysis and control. Model reduction is a technique for developing an approximate system model with lower dimensions that shares similar properties to the original system. This work proposes a computationally efficient approach for linear system model reduction in large-scale power systems that is based on the balanced truncation method using an extended Krylov subspace technique. Key algorithm improvements, including sparsity handling of the linearized power system models and efficient computational techniques for solving dual Lyapunov equations, are discussed in detail. In addition, application of the proposed model reduction is extended to unstable systems, and a power system stabilizer (PSS) parameter optimization method based on reduced system models is used to validate the effectiveness of the proposed method. Case studies show that the resulting reduced models preserve the unique characteristics of the original high-dimensional models, including time and frequency domain responses as well as eigenvalues. Numerical results based on test systems with up to 12 $\thinspace$ 685 buses demonstrate the computational efficiency and validity of the proposed approach.

In this paper, we develop a method for solving the problem of minimizing the $H^2$ error norm between the transfer functions of original and reduced systems on the set of stable matrices and two Euclidean spaces. That is, we develop a method for identifying the optimal reduced system from all stable linear systems. However, it is difficult to develop an algorithm for solving this problem, because the set of stable matrices is highly non-convex. To overcome this issue, we show that the problem can be transformed into a tractable Riemannian optimization on the product manifold of the set of skew-symmetric matrices, the manifold of the symmetric positive-definite matrices, and two Euclidean spaces. The stability of the reduced systems constructed using the optimal solutions to our problem is preserved. To solve the reduced problem, the Riemannian gradient and Hessian are derived and a Riemannian trust-region method is developed. The initial point in the proposed approach is selected using the output from the balanced truncation (BT) method. Numerical experiments demonstrate that our method considerably improves the results given by BT in the sense of the $H^2$ norm, and also provides reduced systems that are globally near-optimal solutions to the problem of minimizing the $H^{\infty}$ error norm. Moreover, we show that our method provides a better reduced model than BT from the viewpoint of the frequency response.

Model order reduction of a dynamical linear time-invariant system appears in many applications from science and engineering. Numerically reliable SVD-based methods for this task require in general $\mathcal{O}(n^3)$ floating-point arithmetic operations, with n being in the range 103−105 for many practical applications. In this paper we investigate the use of graphics processors (GPUs) to accelerate model reduction of large-scale linear systems by off-loading the computationally intensive tasks to this device. Experiments on a hybrid platform consisting of state-of-the-art general-purpose multi-core processors and a GPU illustrate the potential of this approach.

The purpose of this paper is to develop a model reduction theory for linear quantum stochastic systems that are commonly encountered in quantum optics and related fields, modeling devices such as optical cavities and optical parametric amplifiers, as well as quantum networks composed of such devices. Results are derived on subsystem truncation of such systems and it is shown that this truncation preserves the physical realizability property of linear quantum stochastic systems. It is also shown that the property of complete passivity of linear quantum stochastic systems is preserved under subsystem truncation. A necessary and sufficient condition for the existence of a balanced realization of a linear quantum stochastic system under sympletic transformations is derived. Such a condition turns out to be very restrictive and will not be satisfied by generic linear quantum stochastic systems, thus necessary and sufficient conditions for relaxed notions of simultaneous diagonalization of the controllability and observability Gramians of linear quantum stochastic systems under symplectic transformations are also obtained. The notion of a quasi-balanced realization is introduced and it is shown that all asymptotically stable completely passive linear quantum stochastic systems have a quasi-balanced realization. Moreover, an explicit bound for the subsystem truncation error on a quasi-balanceable linear quantum stochastic system is provided. The results are applied in an example of model reduction in the context of low-pass optical filtering of coherent light using a network of optical cavities.

Balanced truncation is a well-known technique for model-order reduction with a known uniform reduction error bound. However, its practical application to large-scale problems is hampered by its cubic computational complexity. While model-order reduction by projection to approximate dominant subspaces without balancing has produced encouraging experimental results, the approximation error bound has not been fully analyzed. In this paper, a square-integral reduction error bound is derived for unbalanced dominant subspace projection by using a frequency-domain solution of the Lyapunov equation. Such an error bound is valid in both the frequency and time domains. Then, a dominant subspace computation scheme together with three Krylov subspace options is introduced. It is analytically justified that the Krylov subspace for moment matching at low frequencies is able to provide a better dominant subspace approximation than the Krylov subspace at high frequencies, while a rational Krylov subspace with a proper real shift parameter is capable of achieving superior approximation than the Krylov subspace at low frequency. A heuristic method of choosing a real shift parameter is also introduced based on its new connection to the discretization of a continuous-time model. The computation algorithm and theoretical analysis are then examined by several numerical examples to demonstrate the effectiveness. Finally, the dominant subspace computation scheme is applied to the model-order reduction of two large-scale interconnect circuit examples.

Riemannian optimal model reduction of linear port-Hamiltonian systems

Model reduction by walsh function techniques

This letter develops a structure preserving ${H^{2}}$ optimal model reduction method of linear second-order systems. The model reduction problem is formulated as an optimization problem on the product manifold of the two manifolds of symmetric positive definite matrices and two Euclidean manifolds. Reduced systems constructed by the optimal solution preserve the structure of the original second-order system. A Riemannian metric of the manifold is chosen in such a way that the manifold is geodesically complete, i.e., the domain of the exponential map is the whole tangent space for all points on the manifold. The Riemannian gradient of the objective function is derived for solving the problem by using a Riemannian steepest descent method. The geodesic completeness of the manifold guarantees that all points generated by the steepest descent method are on the manifold, and thus, our method naturally preserves the second-order structure. Furthermore, we suggest how to choose an initial point in the proposed algorithm. The initial point is given by solving another optimization problem on the Stiefel manifold. As a result, we can expect that the value of the objective function at the initial point is relatively small. Numerical experiments illustrate that the proposed method can give a reduced system which is sufficiently close to the original system even if the dimension of the reduced system is small.

The reduction of the large order models of complex systems into a sufficiently low dimensional order in such a way that important dynamic characteristics of the real system are preserved, is a nontrivial task. The motivations for such a model order reduction are either to reduce computations for analysis and practical control design or to simplify the control system structure. In the present article, major techniques in model order reduction are briefly discussed. The characteristic similarities and differences are highlighted, their advantages and disadvantages are underlined, and further research requirements and trends are commented upon. Introduction and Perspective One of the central issues in the active control of complex systems such as large flexible space structures (LFSS) is the derivation of a correct mathematical model for both the controlled and the uncontrolled dynamical systems. Theoretically, there are infinitely many elastic modes or degrees of freedom (DOF) in the distributed parameter (DP) models of LFSS, usually with very low natural damping. Moreover, the flexible modes contribute to the actual deformation of the structure.' The truly infinite dimensional character of LFSS models has to be approximated by some fidelity finite (but usually very large) dimensional model. The normal approach taken by engineers to achieve this end is via modal models with a large number of modes that provide a reasonable representation of the spacecraft dynamic characteristics. However, a difficult problem still remains to be the development of a model of lowenough dimensional order that it can be utilized by the onboard controller, yet high enough dimensional order that it preserves the dynamic characteristics of the real system represented and controlled. The motivations for such a reduction are either to reduce computations for analysis and practical control design or to simplify the control system structure. Discretization procedures will not be discussed herein and it will be assumed that a large finite dimensional model has been generated somehow and for implementation and other practical considerations, the model needs to be further reduced. *This work was-supported, in part, by the Air Force Wright Aeronautical, Air Force Systems Command, Flight Dynamics Lab, Structures and Dynamics Division, while the author was employed with HR-Textron, Valencia, California. There are various approaches to model order reduction; some are or pseudo-optimal and others are ad-hoc methods based on practical considerations and engineering experience and judgment. These techniques are known by different names such as reduction, condensation, economization, aggregation, optimal projection, and other combinations thereof. It has often been pointed out that such techniques generally constitute application of Rayleigh-Ritz/Galerkin optimization, and matrix transformation methods to the eigenvalue/eigenvector formulation for structural dynamic problems. The reduction or the condensation methods are based on transformations of the coordinates in the equations of motion that essentially maintain the invariance of the quadratic forms of the potential and kinetic energies. An important feature of these methods is that the reduced order model often loses the basic characteristics of the original system. Considerable progress was made by Likins, Ohkami, and wong5 in this respect. However, they failed to develop a general enough criterion that could reduce the system model in an sense without significantly affecting the eigenvalues of the original model. The optimization or the mathematical reduction procedure, on the other hand, is based on the reduction of the eigenvalue problem to a smaller size based on some optimality criterion (usually quadratic). In these techniques, the reductions are carried out without actually resorting to approximations, i .e. , without truncatiag any coordinates or states of the original system. Balanced model reduction of linear timeinvariant dynamical systems is essentially based on the controllability and observability relations of the states of the system. Subsystem models are obtained by deleting those states that contribute the least to the controllability and observability (or the impulse response) of the original system7 and thus are only in this sense. A more recent approach to model reduction is proposed in Skelton's work,* where each state of the system model is assigned a relative to a given basis, via a quadratic criterion, and the states with the least cost are deleted in a systematic manner. The resulting reduced model is a function of the state-space basis, and thus there is no guarantee for optimality for all choices. The latest development in model order reduction techniques is the work by Hyland and erns stein.^ Herein first order necessary conditions for reduced order modeling of linear time-invariant systems are derived via a pair of modified Lyapunov equations coupled by a nonorthogonal projection. This approach reveals the possibility of multiple extrema forsome of the abovementioned methods. Copyright @ 1988 by the American Institute of Aeronautics and Astronautics. Inc. All Rights Reserved.

<p style='text-indent:20px;'>We consider model order reduction of a nonlinear cable-mass system modeled by a 1D wave equation with interior damping and dynamic boundary conditions. The system is driven by a time dependent forcing input to a linear mass-spring system at one boundary. The goal of the model reduction is to produce a low order model that produces an accurate approximation to the displacement and velocity of the mass in the nonlinear mass-spring system at the opposite boundary. We first prove that the linearized and nonlinear unforced systems are well-posed and exponentially stable under certain conditions on the damping parameters, and then consider a balanced truncation method to generate the reduced order model (ROM) of the nonlinear input-output system. Little is known about model reduction of nonlinear input-output systems, and so we present detailed numerical experiments concerning the performance of the nonlinear ROM. We find that the ROM is accurate for many different combinations of model parameters.

This paper addresses the model reduction of high-order linear systems within the framework of the incompressible Navier-Stokes equations. We look for reduced-order models that capture the response of some specific sensor whatever the initial flow condition and in the presence of any time-dependent external forcing. Namely, this work deals with the accurate modeling of the input-output dynamics of a fluid system when considering each degree of freedom of the system as an input, and the given measurement as the output. In the case of complex or realistic flows, the number of inputs is too large to apply the standard balanced truncation procedure. To alleviate this problem, we introduce a method called input projection. Input projection is shown to be analogous to the output projection procedure introduced by Rowley, Int. J. Bifurcation. Chaos Appl. Sci. Eng. 15, 997 (2005). To illustrate the model reduction, we consider the dynamics of a globally stable flow over a rounded backward-facing step. Reduced-order models are obtained by projecting the full original system onto: (i) the basis of the leading balanced modes computed from the input-projected systems and (ii) the most observable modes. The balanced models are observed to accurately capture the transient growths along the separated flow whatever the input while outperforming the models based on the most observable modes.

Model reduction of linear discrete systems by matching the discrete Laguerre spectra of the unit step responses of the original and reduced models is first discussed. In order to preserve the stable property of the original model, combined methods which use conventional stable methods to determine the parameters of denominator of the reduced models and use the discrete Laguerre spectrum matching to determine the parameters of the numerator are discussed