Hankel-Norm Approximation

The Hankel-norm approximation method is a model reduction approach that solves the best-approximation problem in the Hankel semi-norm.

Description

Consider the standard linear-time invariant system

$G:\{\begin{array}{rl}\dot{x}(t)& =Ax(t)+Bu(t),\\ y(t)& =Cx(t)+Du(t),\end{array}$

with the matrices $A\in {\mathbb{ℝ}}^{n\times n}$, $B\in {\mathbb{ℝ}}^{n\times m}$, $C\in {\mathbb{ℝ}}^{p\times n}$ and $D\in {\mathbb{ℝ}}^{p\times m}$. For a system $G$, the Hankel operator ${\mathscr{H}}$ maps past inputs $u_{-}$ to future outputs $y_{+}$ of the system, i.e., $y_{+}={\mathscr{H}}{u}_{-}$. Then, the Hankel semi-norm of the system $G$ is defined as the ${{ℒ}}_2$-induced norm of the Hankel opertor

$\Vert G{\Vert }_H:=\sup {}_{u_{-}\in {{ℒ}}_2(-\mathrm{\infty },0]}\frac{\Vert y_{+}{\Vert }_{{{ℒ}}_2}}{\Vert u_{-}{\Vert }_{{{ℒ}}_2}}.$

If the system $G$ is stable, the controllability and observability Gramians ${{𝒢}}_c$ and ${{𝒢}}_o$ of the system above are given as the unique positive semidefinite solutions of the two Lyapunov equations

$\begin{array}{rl}A{{𝒢}}_c+{{𝒢}}_c{A}^T+B{B}^T& =0,\\ A^T{{𝒢}}_o+{{𝒢}}_{o}A+C^{T}C& =0.\end{array}$

The Hankel singular values of the system $G$ are then defined as the square-roots of the eigenvalues of the multiplied system Gramians, i.e., $\sqrt{\mathrm{\Lambda }({{𝒢}}_c{{𝒢}}_o)}=\{{\varsigma }_1,\dots ,{\varsigma }_n\}$. It can be shown, that the Hankel semi-norm of a system is given by the largest Hankel singular value $\Vert G{\Vert }_H={\varsigma }_{\text{max}}$.

The idea of the Hankel-norm approximation method is, to construct a reduced-order model $G_r$ of order $r$ such that the error system ${ℰ}=G-G_r$ has a scaled all-pass transfer function

${ℰ}(s){{ℰ}}^T(-s)={\varsigma }_{r+1}^2{I}_p,$

with ${\varsigma }_{r+1}$ the $(r+1)$-st Hankel singular value of the system $G$.

For such error systems, the Hankel semi-norm is known to be $\Vert {ℰ}{\Vert }_H={\varsigma }_{r+1}.$

Algorithm

Here, the algorithm of the Hankel-norm approximation method is shortly described :

1. Compute a minimal balanced realization $(\check{A},\check{B},\check{C},D)$ using the balanced truncation square-root method.
2. Choose the Hankel singular value ${\varsigma }_{r+1}$.
3. Permute the balanced realization such that the Gramians have the form
     $\begin{array}{rl}{\check{{𝒢}}}_c={\check{{𝒢}}}_o& =\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}({\varsigma }_1,\dots ,{\varsigma }_r,{\varsigma }_{r+k+1},\dots ,{\varsigma }_n,{\varsigma }_{r+1}{I}_k)\\ & =\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\mathrm{\Sigma },{\varsigma }_{r+1}{I}_k).\end{array}$
4. Partition the resulting permuted system according to the Gramians
     $\check{A}=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right],\check{B}=\left[\begin{array}{c}{B}_1\\ B_2\end{array}\right],\check{C}=\left[\begin{array}{cc}{C}_1& C_2\end{array}\right],$
   where $A_{22}\in {\mathbb{ℝ}}^{k\times k}$, $B_2\in {\mathbb{ℝ}}^{k\times m}$ and $C_2\in {\mathbb{ℝ}}^{p\times k}$.
5. Compute the transformation
     $\begin{array}{rl}\tilde{A}& ={\mathrm{\Gamma }}^{-1}({\varsigma }_{r+1}^2{A}_{11}^T+\mathrm{\Sigma }{A}_{11}\mathrm{\Sigma }+{\varsigma }_{r+1}{C}_1^{T}U{B}_1^T),\\ \tilde{B}& ={\mathrm{\Gamma }}^{-1}(\mathrm{\Sigma }{B}_1-{\varsigma }_{r+1}{C}_1^{T}U),\\ \tilde{C}& =C_1\mathrm{\Sigma }-{\varsigma }_{r+1}U{B}_1^T,\\ \tilde{D}& =D+{\varsigma }_{r+1}U,\end{array}$
   with $U={\left(C_2^T\right)}^{†}{B}_2$ and $\mathrm{\Gamma }={\mathrm{\Sigma }}^2-{\varsigma }_{r+1}^2{I}_{n-k}$.
6. Compute the additive decomposition
     $\tilde{G}(s)=\tilde{C}(s{I}_{n-k}-\tilde{A}{)}^{-1}\tilde{B}+\tilde{D}=G_r(s)+F(s),$
   where $F$ is anti-stable and $G_r$ is the $r$-th order stable Hankel-norm approximation.