## 5 Robust Control  
### Uncertainty Models  
When the exact model of a physical system is not known due to for example unmodeled dynamics or nonlinearities, the unstructured uncertainty can be useful for modeling the errors. There are several types of unstructured uncertainty models in literature, e.g., [32; 33; 34]:  
* Additive Uncertainty \[G(s)=\widetilde{G}(s)+W(s)\Delta(s).\] (8)
* Multiplicative Uncertainty \[G(s)=\widetilde{G}(s)\left(1+W(s)\Delta(s)\right).\] (9)  
Figure 4: Simplified structure of the closed control loop. Figure 5: Deflection of the controlled system for \(22\,\mathrm{mm}\) reference with classically designed proportional integral (PI) controller. * Feedback Uncertainty \[G(s)=\frac{\widetilde{G}(s)}{1+W(s)\Delta(s)\widetilde{G}(s)}.\] (10)
* Multiplicative Feedback Uncertainty \[G(s)=\frac{\widetilde{G}(s)}{1+W(s)\Delta(s)}.\] (11)  
In these models, \(\widetilde{G}\) denotes the transfer function of the nominal plant, \(W(s)\) is a proper and stable weight function characterizing the uncertainty dynamics, and \(\Delta(s)\) contains the uncertainty, which can be an arbitrary stable transfer function fulfilling the inequality:  
\[\left\|\Delta\right\|_{\infty}\leq 1, \tag{12}\]  
where  
\[\left\|\Delta\right\|_{\infty}=\sup_{\omega\in\mathbb{R}}\left|\Delta(j\omega)\right| \tag{13}\]  
is the norm of the Hardy space \(\mathcal{H}_{\infty}\). Additive uncertainty can be used when the numerator of the transfer function \(G\) contains uncertain terms while the feedback uncertainty is useful when the denominator has uncertainties. Also, multiplicative uncertainty is suitable for gain uncertainties. The block diagrams of these uncertainty models are shown in Figure 6.  
Now, we want to describe the plant Equation (1) with these uncertainty models using the identified interval uncertainty Equation (2). With this extension we also cover nonlinearities such as hysteresis and other unmodeled dynamics. To do so, an appropriate weighting function \(W\) for each case should be found such that the condition in Equation (12) holds. First, we compute \(W\) for additive uncertainty case. Solving Equation (8) for the product term \(W(s)\Delta(s)\) we get  
\[W(s)\cdot\Delta(s)=G(s)-\widetilde{G}(s). \tag{14}\]  
Together with the plant's transfer functions Equation (1) and Equation (4) this results in  
\[G(s)=\widetilde{G}(s)\left(1+W(s)\Delta(s)\right).\]0  
\[G(s)=\widetilde{G}(s)\left(1+W(s)\Delta(s)\right).\]1 Uncertainty models of the plant’s transfer function \(G(s)\). (**a**) Additive uncertainty; (**b**) Multiplicative uncertainty; (**c**) Feedback uncertainty; (**d**) Multiplicative feedback uncertainty. In order to fulfill the bound in Equation (12) we replace the unknown parameters \(K,T\) by their lower or upper bound as in Equation (2), respectively. A possible solution fulfilling the condition in Equation (12) is then given by  
\[G(s)=\widetilde{G}(s)\left(1+W(s)\Delta(s)\right).\]2 \[G(s)=\widetilde{G}(s)\left(1+W(s)\Delta(s)\right).\]3  
Applying the same procedure to other uncertainty models, the weighting function \(W\) and the uncertainty function \(\Delta\) can be described as follows:  
* Multiplicative Uncertainty \[G(s)=\widetilde{G}(s)\left(1+W(s)\Delta(s)\right).\]4 (17)
* Feedback Uncertainty \[G(s)=\widetilde{G}(s)\left(1+W(s)\Delta(s)\right).\]5 (18)
* Multiplicative Feedback Uncertainty \[G(s)=\widetilde{G}(s)\left(1+W(s)\Delta(s)\right).\]6 (19)  
As mentioned before, the transfer functions \(W\) and \(\Delta\) must be stable. In order to be stable, the transfer functions must also be proper, i.e., the degree of the numerator polynomial should not exceed the degree of the denominator polynomial. This structural condition is violated for the weighting function \(W\) of the feedback uncertainty model in Equation (18). Therefore, this model is not taken into account for the robust controller design.