5.3 Ways of Achieving Diagonal Dominance

5.3.1 The Perron-Frobenius Eigenvalue and Constant Scaling Compensators

For any p × p transfer-function matrix G(s), evaluated at some frequency s = j, a corresponding comparison matrix M can be defined as

where the elements m_ij are the moduli of the corresponding elements of the matrix G. By definition, the comparison matrix M is element-wise nonnegative.

A normalized form of M can be constructed as

where = diag{m_ii}. If is square, positive, and irreducible, that is, there is no permutation matrix P such that

where X and Z are square matrices, then there exists an eigenvalue of called the Perron-Frobenius eigenvalue, _pf, that is always real and positive. Furthermore, if the eigenvalues are ordered so that |₁| |₂| ... |_p|, then _pf > _i for i ≠ 1.

If _pf < 2, then

is both row and column diagonal dominant at s = j, and in a nearby neighborhood, and the required scaling matrix S^-1 is formed from the elements of the corresponding Perron-Frobenius right eigenvector w_pf of , as

When working with the inverse system, Eq. (5.NumberedEquationNormalized M) is replaced by

and the matrix S is formed from the elements of the corresponding Perron-Frobenius left eigenvector w_pf of .

If a constant input scaling compensator, as defined by Eq. (5.NumberedEquationPerron-Frobenius scaled G), is implemented, then the same dominance ratio, introduced in Section 5.2.3, is obtained in all diagonal elements of the resulting Nyquist array at the design frequency s = j. The best row (or column) dominance ratio that can be achieved with a constant diagonal scaling compensator is given by _pf(j)-1.

You should note that although the compensator designed using the Perron-Frobenius approach is implemented as an input-scaling compensator, manipulating the columns of the Nyquist array, it attempts to achieve the best row diagonal dominance.

A plot of the Perron-Frobenius eigenvalue with frequency is therefore a useful scalar indicator of the level of interaction, or diagonal dominance, in a system that can be achieved by implementing a diagonal input-scaling compensator. The function PerronFrobeniusEigenvaluePlot generates this plot.

Creating the Perron-Frobenius eigenvalue plot.

Make sure the application is loaded.

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This is the two-input, two-output transfer-function object, considered earlier.

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The direct-system Nyquist array of this system, determined in Section 5.2.1, was found not to be diagonal dominant, and a first step in the design of a compensator for this system might be to determine whether, or not, it can be made diagonal dominant using a simple diagonal constant input-scaling compensator.

This creates the Perron-Frobenius eigenvalue plot for the transfer-function object.

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Since the Perron-Frobenius eigenvalue is less than 2 over the bandwidth of interest, the system can be made diagonal dominant at any frequency in this bandwidth, and in the nearby neighborhood, by a simple constant diagonal scaling matrix. The function PerronFrobeniusCompensator can be used to determine such a compensator.

The Perron-Frobenius constant compensator functions.

This determines a Perron-Frobenius constant scaling compensator at zero frequency.

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This applies the compensator to the system.

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Here is the resulting Nyquist array.

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Comparing this with the Nyquist array plot in Section 5.2.1, you can see that this compensator has significantly improved the low-frequency row diagonal dominance of element {1, 1} of the Nyquist array, but has made the high-frequency row diagonal dominance of element {2, 2} much worse. With this input scaling compensator, the row dominance ratio that will be achieved in both diagonal elements of the Nyquist array at zero frequency is _pf(0)-1. Since _pf(0) is 1.079, the best row dominance ratio at zero frequency is 0.079.

This can be confirmed by applying the function RowDominanceRatioPlot, introduced in Section 5.2.3, to the scaled system.

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While the dominance ratio in each row is 0.08 at = 0 rad/s, as predicted, the dominance ratio of row 2 at the intermediate and high frequencies is unacceptable, and a scaling compensator at another frequency might yield better results.