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A priori error bounds have been derived for different balancing-related model reduction methods. The classical result is a bound for balanced truncation and singular perturbation approximation that is applicable for asymptotically stable linear time-invariant systems with zero initial conditions. Recently, there have been a few attempts to generalize the balancing-related reduction methods to the case with inhomogeneous initial conditions. Those strongly rely on the assumption that the space of initial conditions of interest is known a priori In this paper, we show how the exact error representation in terms of the Gramians can be used as a sharp and efficient error bound. In particular, by exploiting an appropriate offline–online decomposition of the computation, this approach is feasible for arbitrary initial conditions. This is in contrast to previous error bounds, which are valid only for an a priori restricted set of initial conditions. Furthermore, our approach can be realized in a large-scale setting, in which case the resulting error estimator is as accurate as the underlying low-rank approximation of the Gramian allows. The effectivity, accuracy and applicability of our bound/estimator for a posteriori estimation and certified model selection are also demonstrated numerically. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present a finite-time model order reduction method for linear systems via shifted Legendre polynomials. The main idea of the approach is to use finite-time empirical Gramians, which are constructed from impulse responses by solving block tridiagonal linear systems, to generate approximate balanced system for the large-scale system. The balancing transformation is directly computed from the expansion coefficients of impulse responses in the space spanned by shifted Legendre polynomials, without individual reduction of the Gramians and a separate eigenvector solve. Then, the reduced-order model is constructed by truncating the states corresponding to the small approximate Hankel singular values (HSVs). The stability preservation of the reduced model is briefly discussed. And in combination with the dominant subspace projection method, we modify the reduction procedure to alleviate the shortcomings of the above method, which may unexpectedly lead to unstable systems even though the original one is stable. Furthermore, the properties of the resulting reduced models are considered. Finally, numerical experiments are given to demonstrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of finding surrogate models for large-scale second-order linear time-invariant systems with inhomogeneous initial conditions. For this class of systems, the superposition principle allows us to decompose the system behavior into three independent components. The first behavior corresponds to the transfer between the input and output having zero initial conditions. In contrast, the other two correspond to the transfer between the initial position or the initial velocity and the output when no input is applied. Based on this superposition of systems, our goal is to propose model reduction schemes that allow to preserve the second-order structure in the surrogate models. To this aim, we introduce tailored second-order Gramians for each system component and compute them numerically, solving Lyapunov equations. As a consequence, two methodologies are proposed. The first one consists in reducing each of the components independently using a suitable balanced truncation procedure. The sum of these reduced systems provides an approximation of the original system. This methodology allows flexibility on the order of the reduced-order model. The second proposed methodology consists in extracting the dominant subspaces from the sum of Gramians to build the projection matrices leading to a surrogate model. Additionally, we discuss error bounds for the overall output approximation. Finally, the proposed methods are illustrated by means of benchmark problems. [ABSTRACT FROM AUTHOR]

Abstract When solving partial differential equations numerically, usually a high order spatial discretization is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretized systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model (ROM) is balanced truncation (BT). However, if one aims at finding a good ROM on a certain finite time interval only, time-limited BT (TLBT) can be a more accurate alternative. So far, no error bound on TLBT has been proved. In this paper, we close this gap in the theory by providing an output error bound for TLBT with two different representations. The performance of the error bound is then shown in several numerical experiments. [ABSTRACT FROM AUTHOR]

To solve a stochastic linear evolution equation numerically, finite dimensional approximations are commonly used. For a good approximation, one might end up with a sequence of ordinary stochastic linear differential equations of high order. To reduce the high dimension for practical computations, model order reduction is frequently used. Balanced truncation (BT) is a well-known technique from deterministic control theory and it was already extended for controlled linear systems with Lévy noise. Recently, a new ansatz was investigated which provides an alternative way to generalize BT for stochastic systems. There, the question of the existence of an H 2 -error bound was asked which we answer in this paper. We discuss how this bound can be computed practically and how to use it to find a suitable reduced order dimension. [ABSTRACT FROM AUTHOR]

We consider the model reduction problem for linear time-invariant dynamical systems having nonzero (but otherwise indeterminate) initial conditions. Building upon the observation that the full system response is decomposable as a superposition of the response map for an unforced system having nontrivial initial conditions and the response map for a forced system having null initial conditions, we develop a new approach that involves reducing these component responses independently and then combining the reduced responses into an aggregate reduced system response. This approach allows greater flexibility and offers better approximation properties than other comparable methods. [ABSTRACT FROM AUTHOR]

This paper is concerned with the eigenvalue decay of the solution to operator Lyapunov equations with right-hand sides of finite rank. We show that the k th (generalized) eigenvalue decays exponentially in k , provided that the involved operator A generates an exponentially stable analytic semigroup, and A is either self-adjoint or diagonalizable with its eigenvalues contained in a strip around the real axis. Numerical experiments with discretizations of 1D and 2D PDE control problems confirm this decay. [ABSTRACT FROM AUTHOR]

Abstract: The paper deals with the balanced truncation and coprime factors reduction of Markovian jump linear (MJL) systems, which can have mode-varying state, input, and output dimensions. We develop machinery for balancing mean square stable MJL system realizations using generalized Gramians and strict Lyapunov inequalities, and provide an improved a priori upper bound on the error induced in the balanced truncation process. We also generalize the coprime factors reduction method and, in doing so, extend the applicability of the balanced truncation technique to the class of mean square stabilizable and detectable MJL systems. We provide tools to establish mean square stabilizability and detectability of the considered MJL systems. In addition, a notion of right-coprime factorization of MJL systems and methods to construct such factorizations are given. The error measure in the coprime factors reduction approach, while still norm-based, does not directly capture the mismatch between the nominal system and the reduced-order model, as is the case in the balanced truncation approach where mean square stable models are considered. Instead, the error measure is given in terms of the distance between the coprime factors realizations, and thus has an interpretation in terms of robust feedback stability. The paper concludes with an illustrative example which demonstrates how to apply the coprime factors model reduction approach. [Copyright &y& Elsevier]

Abstract: We introduce an approach to model reduction that is founded on ideas originating in realization theory, interpolatory -optimal model reduction, and complex Chebyshev approximation. Within this new framework, we are able to formulate a method that remains effective in large-scale settings with the main cost dominated by sparse linear solves. By employing Loewner “data-driven” partial realizations within each optimization cycle, computationally demanding norm calculations can be completely avoided. Several numerical examples illustrate that our approach will produce high fidelity reduced models consistently exhibiting better performance than those produced by balanced truncation; these models often are as good as (and occasionally better than) those models produced by optimal Hankel norm approximation. In all cases, reduced models are produced at far lower cost than is possible either with balanced truncation or with optimal Hankel norm approximation. [Copyright &y& Elsevier]

Abstract: In this paper, stability and synchronization preserving model reduction schemes are developed for linear multi-agent systems. The multi-agent systems that are considered here are composed of general, yet identical linear subsystems, and the communication topology is assumed to be time-independent. First, under the assumption that the agents have stable internal dynamics and the network is stable, the dynamic order of the agents is reduced such that the corresponding reduced order network is again stable. Then, starting from a synchronized network where agents are allowed to have unstable dynamics, a reduced order model for the network which preserves synchronization is obtained. In addition, model reduction error bounds are established to compare the behavior of the original network to that of the reduced order model. The proposed results are illustrated through a numerical example. [Copyright &y& Elsevier]

Abstract: In this article, we will extend the method of balanced truncation using normalised right coprime factors of the system transfer matrix by Meyer (1990) to balanced truncation with preservation of half line dissipativity. Special cases are preservation of positive realness and bounded realness. We consider a half line dissipative input–output system, with quadratic supply rate given by a nonsingular symmetric matrix with the property that its positive signature is equal to the number of input components of the system. The transfer matrix of such system allows a -normalised right coprime factorisation. We associate with such factorisation two Lyapunov equations, one of which is nonstandard, involving the indefinite matrix . Balancing will be based on making the unique solutions of these two Lyapunov equations equal and diagonal. The diagonal elements will be called the Hankel -singular values, because their squares are the nonzero eigenvalues of the composition of the ‘graph’ Hankel operator, multiplication by , and the adjoint graph Hankel operator. This method of balanced truncation will be shown to preserve stability, minimality, and half line dissipativeness. We will characterize the ‘classical’ positive real and bounded real characteristic values in terms of the new Hankel -singular values. Finally, we will derive one-step error bounds for the special case of balanced truncation of bounded real systems. [Copyright &y& Elsevier]

Abstract: We present a generalization of the coprime factors model reduction method of Meyer and propose a balanced truncation reduction algorithm for a class of systems containing linear parameter varying and uncertain system models. A complete derivation of coprime factorizations for this class of systems is also given. The reduction method proposed is thus applicable to linear parameter varying and uncertain system realizations that do not satisfy the structured -induced stability constraint required in the standard nonfactored case. Reduction error bounds in the -induced norm of the factorized mapping are given. [Copyright &y& Elsevier]

This paper investigates the decay rate of the Hankel singular values of linear dynamical systems. This issue is of considerable interest in model reduction by means of balanced truncation, for instance, since the sum of the neglected singular values provides an upper bound for an appropriate norm of the approximation error. The decay rate involves a new set of invariants associated with a linear system, which are obtained by evaluating a modified transfer function at the poles of the system. These considerations are equivalent to studying the decay rate of the eigenvalues of the product of the solutions of two Lyapunov equations. The related problem of determining the decay rate of the eigenvalues of the solution to one Lyapunov equation will also be addressed. Very often these eigenvalues, like the Hankel singular values, are rapidly decaying. This fact has motivated the development of several algorithms for computing low-rank approximate solutions to Lyapunov equations. However, until now, conditions assuring rapid decay have not been well understood. Such conditions are derived here by relating the solution to a numerically low-rank Cauchy matrix determined by the poles of the system. Bounds explaining rapid decay rates are obtained under some mild conditions. [Copyright &y& Elsevier]