Rational H2 approximation
Given a matrix-valued function whose entries are assumed to be in the Hardy space 1 and a positive integer n, the problem is to find a stable2 rational matrix of McMillan degree3 at most n which minimizes the norm
1. Hardy spaces: the space of square integrable functions on the unit disk splits into two orthogonal subspaces: the space of functions having an analytic continuation in the unit disk (negative Fourier coefficients are zero) and the space of 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
- and : same observable pair
- and 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 of the criterion, has to be the projection of onto . The approximation problem reduces to the minimization of a concentrated criterion [4,5]
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.
In practice, the concentrated criterion is computed using balanced realizations5 to represent lossless matrix functions. We have the nice following property
- Write the state-space realization of and .
where the pair is assumed output normal :
- The Hilbert projection property translates into the classical necessary conditions for optimality
- The concentrated criterion becomes
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 . 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 , called lossless mutual encoding
- A chart is indexed by a lossless matrix/unitary realization
- A BOP, given by a unitary realization can be parametrized in this chart iff the solution to
is positive definite.
- The BOP is represented in the chart by the parameters matrix
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:
- Initialization: computed by balanced truncation [Kun, Lin, 1981]
- Optimization with respect to the local coordinates using fmincon
- Change of chart handled by a non-linear constraint:
- Changing chart is easy: adapted chart at
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