2,325 research outputs found
Empirical balanced truncation of nonlinear systems
Novel constructions of empirical controllability and observability gramians
for nonlinear systems for subsequent use in a balanced truncation style of
model reduction are proposed. The new gramians are based on a generalisation of
the fundamental solution for a Linear Time-Varying system. Relationships
between the given gramians for nonlinear systems and the standard gramians for
both Linear Time-Invariant and Linear Time-Varying systems are established as
well as relationships to prior constructions proposed for empirical gramians.
Application of the new gramians is illustrated through a sample test-system.Comment: LaTeX, 11 pages, 2 figure
Balanced truncation for linear switched systems
In this paper, we present a theoretical analysis of the model reduction
algorithm for linear switched systems. This algorithm is a reminiscence of the
balanced truncation method for linear parameter varying systems. Specifically
in this paper, we provide a bound on the approximation error in L2 norm for
continuous-time and l2 norm for discrete-time linear switched systems. We
provide a system theoretic interpretation of grammians and their singular
values. Furthermore, we show that the performance of bal- anced truncation
depends only on the input-output map and not on the choice of the state-space
representation. For a class of stable discrete-time linear switched systems (so
called strongly stable systems), we define nice controllability and nice
observability grammians, which are genuinely related to reachability and
controllability of switched systems. In addition, we show that quadratic
stability and LMI estimates of the L2 and l2 gains depend only on the
input-output map.Comment: We have corrected a number of typos and inconsistencies. In addition,
we added new results in Theorem
Improved results on frequency-weighted balanced truncation and error bounds
In this paper, we present some new results on frequency-weighted balanced truncation which is a significant improvement on Lin and Chiu's frequency-weighted balanced truncation technique. The reduced-order models, which are guaranteed to be stable in the case of double-sided weighting, are obtained by direct truncation. Two sets of simple, elegant and easily calculatable a priori error bounds are also derived. Numerical examples and comparison with other well-known techniques show the effectiveness of the proposed technique
Balanced Truncation of Networked Linear Passive Systems
This paper studies model order reduction of multi-agent systems consisting of
identical linear passive subsystems, where the interconnection topology is
characterized by an undirected weighted graph. Balanced truncation based on a
pair of specifically selected generalized Gramians is implemented on the
asymptotically stable part of the full-order network model, which leads to a
reduced-order system preserving the passivity of each subsystem. Moreover, it
is proven that there exists a coordinate transformation to convert the
resulting reduced-order model to a state-space model of Laplacian dynamics.
Thus, the proposed method simultaneously reduces the complexity of the network
structure and individual agent dynamics, and it preserves the passivity of the
subsystems and the synchronization of the network. Moreover, it allows for the
a priori computation of a bound on the approximation error. Finally, the
feasibility of the method is demonstrated by an example
Empirical Model Reduction of Controlled Nonlinear Systems
In this paper we introduce a new method of model reduction for nonlinear systems
with inputs and outputs. The method requires only standard matrix computations, and
when applied to linear systems results in the usual balanced truncation. For nonlinear
systems, the method makes used of the Karhunen-Lo`eve decomposition of the state-space,
and is an extension of the method of empirical eigenfunctions used in fluid dynamics. We
show that the new method is equivalent to balanced-truncation in the linear case, and
perform an example reduction for a nonlinear mechanical system
Balanced truncation model reduction of periodic systems
The balanced truncation approach to model reduction is considered for linear discrete-time periodic systems with time-varying dimensions. Stability of the reduced model is proved and a guaranteed additive bound is derived for the approximation error. These results represent generalizations of the corresponding ones for standard discrete-time systems. Two numerically reliable methods to compute reduced order models using the balanced truncation approach are considered. The square-root method and the potentially more accurate balancing-free square-root method belong to the family of methods with guaranteed enhanced computational accuracy. The key numerical computation in both methods is the determination of the Cholesky factors of the periodic Gramian matrices by solving nonnegative periodic Lyapunov equations with time-varying dimensions directly for the Cholesky factors of the solutions
Balanced truncation of perturbative representations of nonlinear systems
The paper presents a novel approach for a balanced truncation style of model reduction of a perturbative representation of a nonlinear system. Empirical controllability and observability gramians for nonlinear systems are employed to define a projection matrix. However, the projection matrix is applied to the perturbative representation of the system rather than directly to the exact nonlinear system. This is to achieve the required increase in efficiency desired of a reduced-order model. Application of the new method is illustrated through a sample test-system. The technique will be compared to the standard approach for reducing a perturbative representation of a nonlinear system
Time-limited Balanced Truncation for Data Assimilation Problems
Balanced truncation is a well-established model order reduction method which
has been applied to a variety of problems. Recently, a connection between
linear Gaussian Bayesian inference problems and the system-theoretic concept of
balanced truncation has been drawn. Although this connection is new, the
application of balanced truncation to data assimilation is not a novel idea: it
has already been used in four-dimensional variational data assimilation
(4D-Var). This paper discusses the application of balanced truncation to linear
Gaussian Bayesian inference, and, in particular, the 4D-Var method, thereby
strengthening the link between systems theory and data assimilation further.
Similarities between both types of data assimilation problems enable a
generalisation of the state-of-the-art approach to the use of arbitrary prior
covariances as reachability Gramians. Furthermore, we propose an enhanced
approach using time-limited balanced truncation that allows to balance Bayesian
inference for unstable systems and in addition improves the numerical results
for short observation periods.Comment: 24 pages, 5 figure
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