The generalised singular perturbation approximation for bounded real and positive real control systems

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doi:10.3934/mcrf.2019016

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2019, Volume 9, Issue 2: 313-350. Doi: 10.3934/mcrf.2019016

This issue Previous Article Application of the boundary control method to partial data Borg-Levinson inverse spectral problem Next Article Nonlinear Schrödinger equations on a finite interval with point dissipation

The generalised singular perturbation approximation for bounded real and positive real control systems

Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

Received: June 2017

Published: June 2019

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Abstract

The generalised singular perturbation approximation (GSPA) is considered as a model reduction scheme for bounded real and positive real linear control systems. The GSPA is a state-space approach to truncation with the defining property that the transfer function of the approximation interpolates the original transfer function at a prescribed point in the closed right half complex plane. Both familiar balanced truncation and singular perturbation approximation are known to be special cases of the GSPA, interpolating at infinity and at zero, respectively. Suitably modified, we show that the GSPA preserves classical dissipativity properties of the truncations, and existing a priori error bounds for these balanced truncation schemes are satisfied as well.

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Mathematics Subject Classification: Primary: 34K07, 34K26, 93B11, 93C05, 93C70.

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Figure 5.1. Semi-log plot of combined errors on the real axis for the bounded real GSPA from Example 5.1, with $ r = 2 $. The lines numbered 1-4 correspond to $ \xi_1 = 0.1 $, $ \xi_2 = 1 $, $ \xi_3 = 10 $ and $ \xi_4 = 100 $, respectively. Note the interpolation properties (2.7) and (3.7) hold and are highlighted with vertical dotted lines. The dashed dotted line is the bound (3.3)

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Figure 5.2. Semi-log plot of combined errors on the real axis for the bounded real GSPA from Example 5.1, with $ r = 1 $. The lines numbered 1-4 correspond to $ \xi_1 = 0.1 $, $ \xi_2 = 1 $, $ \xi_3 = 10 $ and $ \xi_4 = 100 $, respectively. Note the interpolation properties (2.7) and (3.7) hold and are highlighted with vertical dotted lines. The dashed dotted line is the error bound (3.3)

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Figure 5.3. Plots of errors on the imaginary axis for the bounded real GSPA from Example 5.1, with $ r = 1 $ and $ r = 2 $ in panels (a) and (b), respectively. The lines numbered 1-4 correspond to $ \xi_1 = 0.1 $, $ \xi_2 = 1 $, $ \xi_3 = 10 $ and $ \xi_4 = 100 $, respectively, and are symmetric around $ \omega = 0 $. The dashed dotted lines are the bounds (3.3)

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Figure 5.4. Semi-log plot of combined errors on the real axis for the positive real GSPA from Example 5.2, with $ \xi = 10 $. The lines numbered 1-3 correspond to $ r \in \{1, 2, 3\} $ respectively. Note the interpolation property (2.7) holds

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Figure 5.5. Semi-log plot of gap metric error $ \hat \delta( \mathbf G , \mathbf G _r^\xi) $ (crosses) and error bounds (4.4) (circles) for extended circuit model from Example 5.2. Here $ \xi = 10 $

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