12.4: Design for Robustness

It is ubiquitous that models of plants degrade with increasing frequency. For example, the DC gain and slow, lightly-damped modes or zeros are easy to observe, but higher-frequency components in the response may be hard to capture or even to excite in a repeatable manner. Higher-frequency behavior may have more nonlinear properties as well.

The effects of modeling uncertainty can be considered to enter the nominal feedback system as a disturbance at the plant output, \(d_y\). One of the most useful descriptions of model uncertainty is the multiplicative uncertainty:

\[ \tilde{P}(s) \, = \, (1 + \Delta(s) W_2 (s)) P(s). \]

Here, \(P(s)\) represents the nominal plant model used in the design of the control loop, and \(\tilde{P}(s)\) is the actual, perturbed plant. The perturbation is of the multiplicative type, \(\Delta(s) W_2 (s) P(s)\), where \(\Delta(s)\) is an unknown but stable function of frequency for which \( |\Delta(s) \leq 1|. \) The weighting function \(W_2(s)\) scales \(\Delta(s)\) with frequency; \(W_2(s)\) should be growing with increasing frequency, since the uncertainty grows. However, \(W_2(s)\) should not grow any faster than necessary, since it will turn out to be at the cost of nominal performance.

In the scalar case, the weight can be estimated as follows: since \(\tilde{P} / P-1 = \Delta W_2\), it will suffice to let \(|\tilde{P} / P-1| < |W_2|\).

Example \(\PageIndex{1}\):

Let \(\tilde{P} = k/(s-1)\), where \(k\) is in the range 2-5. We need to create a nominal model \(P = k_0 / (s-1)\), augmented with the smallest possible value of \(W_2\), which will not vary with frequency in this case. Two equations can be written using the above estimate, for the two extreme values of \(k\), yielding \(k_0 = 7/2\), and \(W_2 = 3/7\). In particular, \(k_0 \pm W_2 = [2, \, 5]\).

Solution

For constructing the Nyquist plot, we observe that \(\tilde{P}(s) C(s) = (1 + \Delta(s) W_2(s)) P(s)C(s).\) The path of the perturbed plant could be anywhere on a disk of radius \(|W_2(s) P(s)C(s)|\), centered on the nominal loci \(P(s)C(s)\). The robustness condition is that this disk should not intersect the critical point. This can be written as

\begin{align*} 1 + PC \,\, &> \,\, |W_2 PC| \longleftrightarrow \\[4pt][4 pt] 1 \,\, &> \,\, \frac{|W_2 PC|}{1 + PC} \longleftrightarrow \\[4pt][4 pt] 1 \,\, &> \,\, |W_2 T|, \end{align*}

where \(T\) is the complementary sensitivity function. The last inequality is thus a condition for robust stability in the presence of multiplicative uncertainty parameterized with \(W_2\).