# Algorithms

### Rational H2 approximation

Given a matrix-valued function $Latex formula$ whose entries are  assumed to be in the Hardy space 1 $Latex formula$ and a positive integer n, the problem is to find a stable2 rational matrix $Latex formula$ of McMillan degree3 at most n which minimizes the $Latex formula$ norm

$Latex formula$
It is known [1] that a global minimum does exist and has exact degree n unless $Latex formula$ is of lower degree. Moreover, if $Latex formula$ has degree n, then the only critical point of the criterion is $Latex formula$ itself [2]. The approach used in RARL2 was first developped for scalar functions [3] and then in the matrix case [4].

1. Hardy spaces: the space $Latex formula$ of square integrable functions on the unit disk splits into two orthogonal subspaces: the space $Latex formula$ of $Latex formula$ functions having an analytic continuation in the unit disk (negative Fourier coefficients are zero) and the space $Latex formula$ of $Latex formula$ functions having an analytic continuation outside the unit disk and vanishing at infinity (non-negative Fourier coefficients are zero)

2. stable: poles inside the unit disk

3. McMillan degree: size of the A-matrix in a minimal realization/ number of poles counting multiplicity

### Reduction of the optimization space

We use a special matrix fraction description known has Douglas-Shapiro-Shields factorization, and write a rational matrix of degree n as

$Latex formula$
where $Latex formula$ is a lossless matrix4, of same McMillan degree than $Latex formula$, and $Latex formula$ is an anti-stable matrix.

• $Latex formula$ and $Latex formula$: same observable pair $Latex formula$
• $Latex formula$ and $Latex formula$ bring the (left) pole structure
• unique up to a unitary matrix
• right generalization of an irreducible rational fraction

By the Hilbert projection theorem , at a local minimum $Latex formula$ of the criterion, $Latex formula$ has to be the projection $Latex formula$ of $Latex formula$ onto $Latex formula$. The approximation problem reduces to the minimization of a concentrated criterion [4,5]

$Latex formula$

over the (quotient) set of lossless matrices of McMillan degree n up to a right unitary matrix.

Advantage: minimization over a bounded set

4. lossless matrix: rationnal matrix analytic outside the unit disk which takes unitary values on the circle.

### State-space formulas

In practice, the concentrated criterion is computed using balanced realizations5 to represent lossless matrix functions. We have the nice following property

$Latex formula$

• Write the state-space realization of $Latex formula$ and $Latex formula$.
$Latex formula$

where the pair $Latex formula$ is assumed output normal : $Latex formula$
• The Hilbert projection property translates into the classical necessary conditions for optimality
$Latex formula$
• The concentrated criterion becomes
$Latex formula$

defined over the set of equivalence classes under similarity of balanced output pairs (BOP) .

5. A balanced realization is such that both grammians are equal and diagonal

### Parametrization of the optimization space

The optimization space has a manifold structure [6]. We use an atlas of charts, like for the earth, where each chart provides a local flat (Euclidien) representation which allows for the use of differential tools. In the actual version of RARL2, we use the atlas presented in [7], called lossless mutual encoding

• A chart is indexed by a lossless matrix/unitary realization $Latex formula$
• A BOP, given by a unitary realization $Latex formula$ can be parametrized in this chart iff the solution $Latex formula$ to
$Latex formula$

is positive definite.
• The BOP is represented in the chart by the parameters matrix
$Latex formula$

Advantage: use of differential tools safe

### Optimization over a manifold

In RARL2, the minimization process makes use of the Matlab solver fmincon.
The optimization is carried over the manifold structure as illustrated below: