7.2 Model Reduction Using Schur and SquareRoot Methods
Given an order continuoustime statespace system
the model reduction problem is one of finding a statespace system of order , such that the error with respect to the norm
is minimized over all statespace systems of order ; where is the transfer function of the original model and is the transfer function of the reducedorder model. The exact minimization of is a complicated task and, in practice, one has to be satisfied with a less strict requirement, such as the guaranteed upper bound on
where are the Hankel singular values. The dominant Hankel singular values are used to determine .
Advanced Numerical Methods provides two new numerical methods, namely the Schur method and the squareroot method. Both are accessible using the function DominantSubsystem, introduced in Section 8.5 of Control System Professional. The Schur method, invoked with ReductionMethod SchurDecomposition, is based on the reduction of the product of to the ordered real Schur form. The reducedorder model is then formed by finding the orthonormal bases for the right and left invariant subspaces corresponding to the dominant eigenvalues of the matrix using orthogonal transformations.
This procedure does not give an internally balanced realization; however, the order transfer function , obtained by the method, has the same properties as the transfer function of the internal balancing procedure. Specifically, the matrix of the reducedorder model is stable and the transfer function satisfies the error bound Eq. (7.9).
The squareroot method is invoked with ReductionMethod SquareRoot. The method results from incorporating the Hammarling algorithm to compute the Cholesky factor of a Gramian into the Schur method. The squareroot method gives a partial internal balancing in the sense that the only retained strong states are balanced.
Although both the Schur method and the squareroot method provide reasonable fits in the frequency domain (at least for lower frequencies), the time responses of the reducedorder models are not always as good fits to those of the original higherorder systems. However, both these methods provide stable reducedorder models in the case when the original model is stable. The details of these methods can be found in Safonov and Chiang (1989) and Datta (2003).
Model reduction methods.
Make sure the application is loaded.
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Load the collection of test examples.
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This experiment uses the statespace model of an absorption column.
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Here are the numbers of states, inputs, and outputs of the system.
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Here are the ratios of each Hankel singular value to the largest one. The Hankel singular values are computed using the Cholesky factors of the controllability and observability Gramians.
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Selecting the threshold in the ratios retains only the first six strongest modes. This is the corresponding reducedorder model obtained by the Schur method.
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The reducedorder model obtained by the Schur method is not internally balanced. The controllability Gramian is not diagonal.
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Here is the reducedorder model of the absorption column system obtained by the squareroot method.
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The reducedorder model obtained by the squareroot method is internally balanced.
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The frequency response from the first input of the original system (solid lines) is compared with that of each of the reducedorder models obtained by the Schur method (dotted lines) and the squareroot method (thick dotted lines). The graphs show that the transfer functions of the reducedorder models in both cases match closely at the low frequencies, while they differ considerably at the higher frequencies.
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