20.4: More General Representation of Uncertainty

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Consider a nominal interconnected system obtained by interconnecting various (reachable and observable) nominal subsystems. In general, our representation of the uncertainty regarding any nominal subsystem model such as \(P_{0}\) involves taking the signal \(\psi\) at the input or output of the nominal subsystem, feeding it through an "uncertainty block" with transfer function \(W \Delta\) or \(W_{1} \Delta W_{2}\), where each factor is stable and \(\|\Delta\|_{\infty} \leq 1\), and then adding the output \(\theta\) of this uncertainty block to either the input or output of the nominal subsystem. The one additive and two multiplicative representations described earlier are special cases of this construction, but the construction actually yields a total of three additional possibilities with a given uncertainty block. Specifically, if the uncertainty block is \(W \Delta\), we get the following additional feedback representations of uncertainty:

\(P=P_{0}\left(I-W \Delta P_{0}\right)^{-1}\)
\(P=P_{0}(I-W \Delta)^{-1}\)
\(P=(I-W \Delta)^{-1} P_{0}\)

A useful feature of the three uncertainty representations itemized above is that the unstable poles of the actual plant \(P\) are not constrained to be (a subset of) those of the nominal plant \(P_{0}\).

Note that in all six representations of the perturbed or actual system, the signals \(\psi\) and \(\theta\) become internal to the actual subsystem model. This is because it is the combination of \(P_{0}\) with the uncertainty model that constitutes the representation of the actual model \(P\), and the actual model is only accessed at its (overall) input and output.

In summary, then, perturbations of the above form can be used to represent many types of uncertainty, for example: high-frequency unmodeled dynamics, unmodeled delays, unmodeled sensor and/or actuator dynamics, small nonlinearities, parametric variations.