Doyle-Stein Criterion

SCALAR SISO SYSTEMS. Remember in the scalar SISO case we looked at the closedloop servo transfer function G B/ll + GuB). The peak in this curve, the maximum closedloop log modulus L (as shown in Fig. 16.9a), is a measure of the damping coefficient of the system. The higher the peak, the more underdamped die system and the less margin for changes in parameter values. Thus, in SISO systems the peak in the closedloop log modulus curve is a measure of robustness. 

If we plotted just the denominator of the right-hand side of Eq. 16.42, the curve would be the reeiproeal of the elosedloop servo transfer function and would look something like that shown in Fig. 16.96. The lower the dip in the curve, the lower the damping eoeffieient and the less robust the system. 

MATRIX MULTIVARIABLE SYSTEMS. For multivariable systems, the Doyle-Stein criterion for robustness is very similar to the reciprocal plot discussed above. The minimum singular value of the matrix given in Eq. (16.43) is plotted as a function of frequency (a. This gives a measure of the robustness of a closed- 

Doyle-Stein criterion minimum singular value of Q + (i i)] (16.43) 

The tuning and/or structure of the feedback controller matrix is changed until the minimum dip in the curve is something reasonable. Doyle and Stein gave no definite recommendations, but a value of about — 12 dB seems to give good results. 

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Doyle-Stein criterion for Wood and Berry column... 

The Doyle-Stein criterion discussed in the previous section gives us a simple way to evaluate quantitatively the robustness of a multivariable system. However, it assumes random variability in all the parameters of the system. 

In multivariable systems the question of robustness is very important. One method developed by Doyle and Stein (IEEE Trans., 1981, Vol. AC-26, p. 4) is quite usefiil and easy to use. It has the added advantage that it is quite similar to the maximuni closedloop log modulus criterion used in SISO systems. 

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