Algorithms

Rational H2 approximation

Given a matrix-valued function \(F(z)\) whose entries are  assumed to be in the Hardy space 1 \(H^2\) and a positive integer n, the problem is to find a stable2 rational matrix \(H(z)\) of McMillan degree3 at most n which minimizes the \(L^2\) norm

\( \|F-H\|_2^2=\frac{1}{2\pi} {\rm Tr} \int_0^{2\pi}(F-H)(e^{it}) {(F-H)(e^{it})}^*dt.\)
It is known [1] that a global minimum does exist and has exact degree n unless \(F(z)\) is of lower degree. Moreover, if \(F(z)\) has degree n, then the only critical point of the criterion is \(F(z)\) 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 \( L^2 \) of square integrable functions on the unit disk splits into two orthogonal subspaces: the space \(H^2\) of \(L^2\) functions having an analytic continuation in the unit disk (negative Fourier coefficients are zero) and the space \(H_-^2\) of \(L^2\) 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

\(H= G C\)
where \(G(z)\) is a lossless matrix4, of same McMillan degree than \(H(z)\), and \(C(z)\) is an anti-stable matrix.

  • \(H(z)\) and \(G(z)\): same observable pair \((C,A)\)
  • \(G(z)\) and \((C,A)\) 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 \(H(z)\) of the criterion, \(C(z)\) has to be the projection \(P^+\) of \(G^{-1}F\) onto \(H^2\). The approximation problem reduces to the minimization of a concentrated criterion [4,5]

\( G \mapsto \|F-G\,P^+(G^{-1}F)\|^2\)

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

\(\left\{\begin{array}{cl}
D+C(zI-A)^{-1}B &\hbox{ lossless }\\
(A,B,C,D) &\hbox{ balanced }\end{array}\right. \Leftrightarrow \begin{bmatrix} D & C \\
B & A \end{bmatrix} \hbox{ unitary }\)

  • Write the state-space realization of \(F(z)\) and \(H(z)\).
    \(F(z)= {\cal C} (zI- {\cal A})^{-1} {\cal B}~~~ H(z)={C}(zI-{ A})^{-1}{\hat B}\)

    where the pair \((C,A)\) is assumed output normal : \(A^*A+C^*C=I\)
  • The Hilbert projection property translates into the classical necessary conditions for optimality
    \({\hat B}=-{\cal Q}^*{\cal B}~~~ \hbox{ where }~~~ {\cal A}^* {\cal Q} A +{\cal C}^* C= {\cal Q}. \)
  • The concentrated criterion becomes
    \(J(A,C)=\|F\|_2^2-\hbox{Tr}({\cal B}^* {\cal Q}{\cal Q}^* {\cal B})\)

    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 \( (W,X,Y,Z) \)
  • A BOP, given by a unitary realization \((A,B,C,D)\) can be parametrized in this chart iff the solution \( \Sigma \) to
    \(\Sigma – A^* \Sigma W = C^* Y \)

    is positive definite.
  • The BOP is represented in the chart by the parameters matrix
    \( V= D^*Y+B^*\Sigma W \)

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:

manifold