891 research outputs found
Type II balanced truncation for deterministic bilinear control systems
When solving partial differential equations numerically, usually a high order
spatial discretisation is needed. Model order reduction (MOR) techniques are
often used to reduce the order of spatially-discretised systems and hence
reduce computational complexity. A particular MOR technique to obtain a reduced
order model (ROM) is balanced truncation (BT), a method which has been
extensively studied for deterministic linear systems. As so-called type I BT it
has already been extended to bilinear equations, an important subclass of
nonlinear systems. We provide an alternative generalisation of the linear
setting to bilinear systems which is called type II BT. The Gramians that we
propose in this context contain information about the control. It turns out
that the new approach delivers energy bounds which are not just valid in a
small neighbourhood of zero. Furthermore, we provide an -error bound
which so far is not known when applying type I BT to bilinear systems
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Type II balanced truncation for deterministic bilinear control systems
When solving partial differential equations numerically, usually a high
order spatial discretisation is needed. Model order reduction (MOR)
techniques are often used to reduce the order of spatially-discretised
systems and hence reduce computational complexity. A particular MOR technique
to obtain a reduced order model (ROM) is balanced truncation (BT), a method
which has been extensively studied for deterministic linear systems. As
so-called type I BT it has already been extended to bilinear equations, an
important subclass of nonlinear systems. We provide an alternative
generalisation of the linear setting to bilinear systems which is called type
II BT. The Gramians that we propose in this context contain information about
the control. It turns out that the new approach delivers energy bounds which
are not just valid in a small neighbourhood of zero. Furthermore, we provide
an H1-error bound which so far is not known when applying type I BT to
bilinear systems
Model reduction of controlled Fokker--Planck and Liouville-von Neumann equations
Model reduction methods for bilinear control systems are compared by means of
practical examples of Liouville-von Neumann and Fokker--Planck type. Methods
based on balancing generalized system Gramians and on minimizing an H2-type
cost functional are considered. The focus is on the numerical implementation
and a thorough comparison of the methods. Structure and stability preservation
are investigated, and the competitiveness of the approaches is shown for
practically relevant, large-scale examples
Energy estimates and model order reduction for stochastic bilinear systems
In this paper, we investigate a large-scale stochastic system with bilinear drift and linear diffusion term. Such high dimensional systems appear for example when discretizing a stochastic partial differential equations in space. We study a particular model order reduction technique called balanced truncation (BT) to reduce the order of spatially-discretized systems and hence reduce computational complexity. We introduce suitable Gramians to the system and prove energy estimates that can be used to identify states which contribute only very little to the system dynamics. When BT is applied the reduced system is obtained by removing these states from the original system. The main contribution of this paper is an L2-error bound for BT for stochastic bilinear systems. This result is new even for deterministic bilinear equations. In order to achieve it, we develop a new technique which is not available in the literature so far
Energy estimates and model order reduction for stochastic bilinear systems
In this paper, we investigate a large-scale stochastic system with bilinear
drift and linear diffusion term. Such high dimensional systems appear for
example when discretizing a stochastic partial differential equations in space.
We study a particular model order reduction technique called balanced
truncation (BT) to reduce the order of spatially-discretized systems and hence
reduce computational complexity. We introduce suitable Gramians to the system
and prove energy estimates that can be used to identify states which contribute
only very little to the system dynamics. When BT is applied the reduced system
is obtained by removing these states from the original system. The main
contribution of this paper is an -error bound for BT for stochastic
bilinear systems. This result is new even for deterministic bilinear equations.
In order to achieve it, we develop a new technique which is not available in
the literature so far
Identification and data-driven model reduction of state-space representations of lossless and dissipative systems from noise-free data
We illustrate procedures to identify a state-space representation of a lossless- or dissipative system from a given noise-free trajectory; important special cases are passive- and bounded-real systems. Computing a rank-revealing factorization of a Gramian-like matrix constructed from the data, a state sequence can be obtained; state-space equations are then computed solving a system of linear equations. This idea is also applied to perform model reduction by obtaining a balanced realization directly from data and truncating it to obtain a reduced-order mode
Dual Pairs of Generalized Lyapunov Inequalities and Balanced Truncation of Stochastic Linear Systems
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