Representation of Uncertainty

As we have seen, H ∞ optimization is an adequate tool for controller design if the specification of the feedback loop is formulated in the frequency domain. On the other hand, the H ∞ optimal controller is not necessarily robust. A robust controller design can be achieved by the so-called D-K iteration, which will be introduced in the next chapter. The application of this method requires a precise mathematical framework of system robustness, which will be given now. One source of uncertainty is neglected high-frequency dynamics. This will be described in Sect. 12.2, after the basic definitions concerning robust stability and robust performance are introduced (Sect. 12.1). Then uncertainty is modeled by one unknown transfer function together with some weights and is called unstructured uncertainty. Another type of uncertainty arises if one or more parameters of the plant are not known precisely. This leads to structured uncertainty, whose mathematical description is given in Sect. 12.3. Of course it may occur that parameter uncertainty arises in combination with uncertain dynamics. A mathematical framework covering all kinds of uncertainty will be described in Sect. 12.4. It is the basis for the analysis and design methods for robust feedback loops, which we will develop in Chap. 13. The chapter ends with discussing uncertainty for a simple SISO second-order plant, which prepares the way for the complex case studies in Chap. 15.