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) .

Advantage: computation with unitary matrices

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:


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