Riemannian Optimal Model Reduction of Stable Linear Systems

Riemannian optimal model reduction of linear port-Hamiltonian systems

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.

This paper presents a framework for model order reduction based on Balanced Truncation (BT) method and Artificial Neural Networks (ANN). As social, architectural and technical models are witnessing rapid complexity increase, it has become essential to find solutions to obtain a lower complex version of models while still preserving the model integrity. The main idea of the proposed framework is to reduce uncertainty by cascading two different Model Order Reduction (MOR) approaches, the BT technique and the Jordan Recurrent Neural Network (JRNN). The reduced model obtained by the JRNN network is estimated using the Instrument Variable (IV) estimation approach. Finally, the BT reduced model and the estimated JRNN reduced model are cascaded to obtain the final reduced order version of the original plant. The cascading procedure takes the powerfulness of both techniques and present them in a single system capable to compensate for possible errors and uncertainties. Simulation results prove the efficiency of the proposed framework in terms of the deduction strength and the integrity level.

Studies a class of balanced truncation algorithms applicable to relative/multiplicative model reduction. These algorithms seek to balance the controllability gramian of a given transfer function and the observability gramian of its right inverse. For this reason, the algorithms are referred to as inverse-weighted balanced truncation (IWBT) algorithms. It is shown that by using IWBT algorithms one can derive relative and multiplicative L/sub /spl infin// error bounds that are known to hold for other reduction algorithms. It is also shown that the balanced stochastic truncation (BST) method is actually one special, but an optimal version of the IWBT algorithms. As such, the authors' result also serves to establish the fact that the available error bounds pertaining to BST algorithms actually hold for IWBT algorithms.

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.

This paper studies stability- and symmetry-preserving H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> optimal model reduction problems of linear systems, which include linear gradient systems as a special case. The problem is formulated as a nonlinear optimization problem on the product manifold of the manifold of symmetric positive-definite matrices and two Euclidean spaces. To solve the problem by using the trust-region method, the gradient and Hessian of the objective function are derived. Furthermore, it is shown that if we restrict our systems to gradient systems, the gradient and Hessian can be obtained more efficiently. More concretely, by symmetry, we can reduce linear matrix equations to be solved. In addition, by a simple example, we show that the solutions to our problem and a similar problem in some literature works are not unique, and the solution sets of both problems do not contain each other in general. Also, it is revealed that the attained optimal values do not coincide. Numerical experiments show that the proposed method gives a reduced system with the same structure with the original system although the balanced truncation method does not.

A new mixed method for relative error model order reduction is proposed. In the proposed method the frequency domain balanced stochastic truncation method is improved by applying the generalized singular perturbation method to the frequency domain balanced system in the reduction procedure. The frequency domain balanced stochastic truncation method, which was proposed in [15] and [17] by the author, is based on two recently developed methods, namely frequency domain balanced truncation within a desired frequency bound and inner-outer factorization techniques. The proposed method in this paper is a carry over of the frequency-domain balanced stochastic truncation and is of interest for practical model order reduction because in this context it shows to keep the accuracy of the approximation as high as possible without sacrificing the computational efficiency and important system properties. It is shown that some important properties of the frequency domain stochastic balanced reduction technique are extended to the proposed reduction method by using the concept and properties of the reciprocal systems. Numerical results show the accuracy, simplicity and flexibility enhancement of the method.

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.

A standard approach for model reduction of linear input-output systems is balanced truncation, which is based on the controllability and observability properties of the underlying system. The related dominant subspaces projection model reduction method similarly utilizes these system properties, yet instead of balancing, the associated subspaces are directly conjoined. In this work, we extend the dominant subspace approach by computation via the cross Gramian for linear systems, and describe an a-priori error indicator for this method. Furthermore, efficient computation is discussed alongside numerical examples illustrating these findings.

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.

Considering the use of dynamical systems in practical applications, often only limited regions in the time or frequency domain are of interest. Therefore, 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 to second-order systems by using the different second-order balanced truncation formulas from the literature. Also, we present numerical methods for solving the arising large-scale sparse matrix equations and give numerical modifications to deal with the problematic case of second-order systems. The results are then illustrated on three numerical examples.

Numerical algorithms based on linear matrix in-equalities is presented for model order reduction of stable discrete-time linear systems using $\mathbf{H}_{2}$ norm principle. The set of policies is formulated to obtain matrix parameters of the state-space representation of the reduced-order discrete-time system. It is reiterated that the minimization of $\mathbf{H}_{2}$ norm of the approximation error leads to satisfactory approximation. The results and proposed regularization are listed to illustrate the validity of the method.

In this paper we study a numerical algorithm for model reduction problem of polytopic uncertain linear systems. The polytopic system has its state-space data contained in a convex polytope, a situation that often arises. The problem we are trying to solve is to find a lower order polytopic uncertain linear system with a guaranteed induced /spl Lscr//sub 2/ norm error. A sufficient solvability condition was provided in terms of LMIs with one extra coupling rank constraint, which generally leads to a non-convex feasibility problem. To deal with this problem, a cone complementarity algorithm is developed, although only local convergence is guaranteed.

In this paper we propose a simplified method utilizing a variation of balanced truncation method, for model reduction of unstable system. This method can be applied to all unstable system. A numerical example is given to illustrate the results and compare the error between the actual system and the reduced order model.

<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.