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

$G:\left\{ \begin{align} \dot{x}(t) & = Ax(t) + Bu(t),\\ y(t) & = Cx(t) + Du(t), \end{align} \right.$

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 $\mathcal{H}$ maps past inputs $u_{-}$ to future outputs $y_{+}$ of the system, i.e., $y_{+} = \mathcal{H}u_{-}$. Then, the Hankel semi-norm of the system $G$ is defined as the $\mathcal{L}_{2}$-induced norm of the Hankel opertor

$\lVert G \rVert_{H} := \sup\limits_{u_{-} \in \mathcal{L}_{2}\left(-\infty, 0\right]}\frac{\lVert y_{+} \rVert_{\mathcal{L}_{2}}}{\lVert u_{-} \rVert_{\mathcal{L}_{2}}}.$

If the system $G$ is stable, the controllability and observability Gramians $\mathcal{G}_{c}$ and $\mathcal{G}_{o}$ of the system above are given as the unique positive semidefinite solutions of the two Lyapunov equations

$\begin{align} A\mathcal{G}_{c} + \mathcal{G}_{c}A^{T} + BB^{T} & = 0,\\ A^{T}\mathcal{G}_{o} + \mathcal{G}_{o}A + C^{T}C & = 0. \end{align}$

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., $\sqrt{\Lambda(\mathcal{G}_{c}\mathcal{G}_{o})} = \{ \varsigma_{1}, \ldots, \varsigma_{n} \}$. It can be shown, that the Hankel semi-norm of a system is given by the largest Hankel singular value $\lVert G \rVert_{H} = \varsigma_{\text{max}}$.

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 $\mathcal{E} = G - G_{r}$ has a scaled all-pass transfer function

$\mathcal{E}(s)\mathcal{E}^{T}(-s) = \varsigma_{r + 1}^{2} I_{p},$

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 $\lVert \mathcal{E} \rVert_{H} = \varsigma_{r + 1}.$

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
     $\begin{align}\check{\mathcal{G}}_{c} = \check{\mathcal{G}}_{o} & = \mathrm{diag}(\varsigma_{1}, \ldots, \varsigma_{r}, \varsigma_{r + k + 1}, \ldots, \varsigma_{n}, \varsigma_{r + 1}I_{k})\\ & = \mathrm{diag}(\Sigma, \varsigma_{r + 1}I_{k}).\end{align}$
4. Partition the resulting permuted system according to the Gramians
     $\check{A} = \begin{bmatrix} A_{11} & A_{12}\\ A_{21} & A_{22}\end{bmatrix}, ~~~ \check{B} = \begin{bmatrix} B_{1}\\ B_{2}\end{bmatrix}, \check{C} = \begin{bmatrix} C_{1} & C_{2}\end{bmatrix},$
   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
     $\begin{align}\tilde{A} & = \Gamma^{-1}(\varsigma_{r+1}^{2}A_{11}^{T} + \Sigma A_{11} \Sigma + \varsigma_{r+1}C_{1}^{T}UB_{1}^{T}),\\ \tilde{B} & = \Gamma^{-1}(\Sigma B_{1} - \varsigma_{r+1}C_{1}^{T}U),\\ \tilde{C} & = C_{1}\Sigma - \varsigma_{r+1}UB_{1}^{T},\\ \tilde{D} & = D + \varsigma_{r+1}U,\end{align}$
   with $U = \left(C_{2}^{T}\right)^{\dagger}B_{2}$ and $\Gamma = \Sigma^{2} - \varsigma_{r+1}^{2}I_{n-k}$.
6. Compute the additive decomposition
     $\tilde{G}(s) = \tilde{C}(sI_{n-k} - \tilde{A})^{-1}\tilde{B} + \tilde{D} = G_{r}(s) + F(s),$
   where $F$ is anti-stable and $G_{r}$ is the $r$-th order stable Hankel-norm approximation.