20.8: Appendix

Necessity of the small gain condition for robust stability can be proved by showing that if \(\sigma_{\max }\left[M\left(j \omega_{0}\right)\right]>1\) for some \(\omega_{0}\), we can construct a \(\Delta\) of norm less than one, such that the resulting closed-loop map \(G_{zv}\) is unstable. This is done as follows. Take the singular value decomposition of \(M\left(j \omega_{0}\right)\),

\[M\left(j \omega_{0}\right)=U \Sigma V^{\prime}=U\left[\begin{array}{ccc}
\sigma_{1} & & \\
& \ddots & \\
& & \sigma_{n}
\end{array}\right] V^{\prime} \label{20.17}\]

Since \(\sigma_{\max }\left[M\left(j \omega_{0}\right)\right]>1, \sigma_{1}>1\). Then \(\Delta\left(j \omega_{0}\right)\) can be constructed as:

\[\Delta\left(j \omega_{0}\right)=V\left[\begin{array}{cccc}
1 / \sigma_{1} & & \\
& 0 & \\
& & \ddots \\
& & & 0
\end{array}\right] U^{\prime} \label{20.18}\]

Clearly, \(\sigma_{\max } \Delta\left(j \omega_{0}\right)<1\). We then have

\[(I-M \Delta)^{-1}\left(j \omega_{0}\right)= I-U\left[\begin{array}{cccc}
\sigma_{1} & & \\
& \sigma_{2} & \\
& & \ddots \\
& & & \sigma_{n}
\end{array}\right] \quad V^{\prime} V\left[\begin{array}{cccc}
1 / \sigma_{1} & & & \\
& 0 & & \\
& & \ddots & \\
& & & 0
\end{array}\right] U^{\prime} \

=U\left[I-\left[\begin{array}{cccc}
1 & & \\
& 0 & \\
& & \ddots & \\
& & & 0
\end{array}\right]\right] U^{\prime}

=U\left[\begin{array}{cccc}
0 & & & \\
& 1 & & \\
& & \ddots & \\
& & & 1
\end{array}\right] U^{\prime} \label{20.19}\]

which is singular. Only one problem remains, which is that \(\Delta\left(s\right)\) must be legitimate as the transfer function of a stable system, evaluating to the proper value at \(s=j \omega_{0}\), and having its maximum singular value over all \(\omega\) bounded below 1. The value of the destabilizing perturbation at \(\omega_{0}\) is given by

\[\Delta_{0}\left(j \omega_{0}\right)=\frac{1}{\sigma_{\max }\left(M\left(j \omega_{0}\right)\right)} v_{1} u_{1}^{\prime} \nonumber\]

Write the vectors \(v_{1}\) and \(u_{1}^{\prime}\) as

\[v_{1}=\left[\begin{array}{c}
\pm\left|a_{1}\right| e^{j \theta_{1}} \\
\pm\left|a_{2}\right| e^{j \theta_{2}} \\
\vdots \\
\pm\left|a_{n}\right| e^{j \theta_{n}}
\end{array}\right], \quad u_{1}^{\prime}=\left[\begin{array}{ccc}
\pm\left|b_{1}\right| e^{j \phi_{1}} & \pm\left|b_{2}\right| e^{j \phi_{2}} & \cdots & \pm\left|b_{n}\right| e^{j \phi_{n}}
\end{array}\right] \label{20.20}\]

where \(\theta_{i}\) and \(\phi_{i}\) belong to the interval [0, \(pi\)). Note that we used \(\pm\) in the representation of the vectors \(v_{1}\) and \(u_{1}^{\prime}\) so that we can restrict the angles \(\theta_{i}\) and \(\phi_{i}\) to the interval [0, \(pi\)). Now we can choose the nonnegative constants \(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{n} \text { and } \beta_{1}, \beta_{2}, \cdots, \beta_{n}\) such that the phase of the function \(\frac{s-\alpha_{i}}{s+\alpha_{i}}\) at \(s=j \omega_{0}\) is \(\theta_{i}\), and the phase of the function \(\frac{s-\beta_{i}}{s+\beta_{i}}\) at \(s=j \omega_{0}\) is \(\phi_{i}\). Now the destabilizing \(\Delta\left(s\right)\) is given by

\[\Delta(s)=\frac{1}{\sigma_{\max }\left(M\left(j \omega_{0}\right)\right)} g(s) h^{T}(s)\]

where

\[g(s)=\left[\begin{array}{c}
\pm\left|a_{1}\right| \frac{s-\alpha_{1}}{s+\alpha_{n}} \\
\pm\left|a_{2}\right| \frac{s-\alpha_{2}}{s+\alpha_{2}} \\
\vdots \\
\pm\left|a_{n}\right| \frac{s-\alpha_{n}}{s+\alpha_{n}}
\end{array}\right], \quad h(s)=\left[\begin{array}{c}
\pm\left|b_{1}\right| \frac{s-\beta_{1}}{s+\beta_{1}} \\
\pm\left|b_{2}\right| \frac{s-\beta_{2}}{s+\beta_{2}} \\
\vdots \\
\pm\left|b_{n}\right| \frac{s-\beta_{n}}{s+\beta_{n}}
\end{array}\right]\]