I am having trouble following the derivation of the $H_{\infty}$ norm subdifferential described in this paper Sections 3 and 4.
The paper claims all subgradients of the $\|.\|_\infty$ norm at a stable transfer function $G$ have the form of Eq. (9) when evaluated at a particular set of frequencies: $$\phi(H) = \|G\|_{\infty}^{-1} \text{ Re } \sum_{\nu=1}^{p} \text{ Tr } G(j\omega_\nu)^HQ_{\nu}Y_{\nu}Q_{\nu}^H H(j\omega_\nu)$$ where the columns of $Q_{\nu}$ form an orthonormal basis of the eigenspace $G(j\omega_\nu)G(j\omega_\nu)^H$ associated with the leading eigenvalue, and where $Y_\nu\succeq 0, \sum_\nu^p \text{ Tr}(Y_\nu) =1.$
I have two questions about this: (1) How does one form the orthonormal eigenspace $Q_{\nu}Y_{\nu}Q_{\nu}^H$ associated with the leading eigenvalue of $G$ ? And (2) why can Eq. (9) be reduced to a sum evaluated at certain frequencies?
Also, the paper claims that the set of subgradients (the subdifferential) for $f(x)=\|\mathcal{G}(x)\|_\infty$ is given by Eq. (10) : $$ \partial{f(x)} = \mathcal{G}'(x)^\star[\partial{\|.\|_\infty(\mathcal{G}(x)]} $$ where $\partial{\|.\|_\infty}$ is the subdifferential of the $H_\infty$ norm from Eq.(9), $\mathcal{G}$ is a smooth operator mapping $\mathbb{R}^n$ to the space of stable transfer functions, and $\mathcal{G}'(x)^\star$ is the adjoint of dual of $\mathcal{G}$.
In section 4, they compute the subdifferential for the $H_\infty$ norm of the closed loop system $T_{zw}$ evaluated at the controller $K$. They use a 'chain rule' to compute Eq (10) and eventually derive Eq (13), which gives the set of subgradients for the $H_\infty$ norm of $T_{zw}$ at $K$. That is: $$f:=\|.\|_\infty \circ T_{zw}(K)\\ \partial{f(K)}= \Phi_Y:=T_{zw}'(K)^\star\phi_Y=\langle T_{zw}'(K)\delta K, \phi_Y \rangle\\ \Phi_Y = \|T_{zw}(K)\|_{\infty}^{-1} \sum_{\nu=1}^p \text{ Re}((I-P_{22}(j\omega_\nu)K)^{-1}P_{21}(j\omega_{\nu})T_{zw}(K,j\omega_nu)^H\\\ Q Y_{\nu} Q^H P_{12}(j\omega_{\nu})(I-KP_{22}(j\omega_\nu))^{-1})^T$$ where $\phi_Y$ is a subgradient of $\|.\|_\infty$ at $T_{zw}(K)$ as in Eq.(9), $Y = (Y_1,...Y_p), Y_\nu\succeq 0, \sum_\nu^p \text{ Tr}(Y_\nu) =1$.
I have two questions about this: (1) how does the chain rule yield the set of all subgradients of $T_{zw}$ at $K$? And (2) again, how can Eq (13) reduce to a sum evaluated at certain frequencies?
Any help would be greatly appreciated.