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Original Articles

A note on the cross Gramian for non-symmetric systems

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Pages 199-208 | Received 04 Nov 2015, Accepted 18 Jul 2016, Published online: 10 Aug 2016

Figures & data

Figure 1. Relative output error of reduced-order models for reduced orders up to one hundred by balanced truncation, cross Gramian and non-symmetric cross Gramian for the state-space symmetric MIMO system.

Figure 1. Relative output error of reduced-order models for reduced orders up to one hundred by balanced truncation, cross Gramian and non-symmetric cross Gramian for the state-space symmetric MIMO system.

Figure 2. Relative output error of reduced-order models for reduced orders up to one hundred by balanced truncation, embedding cross Gramian and non-symmetric cross Gramian for a non-square SIMO system.

Figure 2. Relative output error of reduced-order models for reduced orders up to one hundred by balanced truncation, embedding cross Gramian and non-symmetric cross Gramian for a non-square SIMO system.

Figure 3. Relative output error of reduced-order models for reduced orders up to one hundred by balanced truncation, cross Gramian and non-symmetric cross Gramian for a non-symmetric, but square, MIMO system.

Figure 3. Relative output error of reduced-order models for reduced orders up to one hundred by balanced truncation, cross Gramian and non-symmetric cross Gramian for a non-symmetric, but square, MIMO system.

Figure 4. Relative output error of reduced-order models for reduced orders up to one hundred by balanced truncation, cross Gramian and non-symmetric cross Gramian for the ISS-1r benchmark.

Figure 4. Relative output error of reduced-order models for reduced orders up to one hundred by balanced truncation, cross Gramian and non-symmetric cross Gramian for the ISS-1r benchmark.

Figure 5. Relative output error of reduced-order models for reduced orders up to one hundred by balanced truncation and non-symmetric cross Gramian for the non-square variant of the FOM benchmark.

Figure 5. Relative output error of reduced-order models for reduced orders up to one hundred by balanced truncation and non-symmetric cross Gramian for the non-square variant of the FOM benchmark.