The name is absent

32

The controllability gramian T⁵ and observability gramian Q are defined via

P = [ e^A4BB*e^A’X Q = [ е^А‘‘С*Се^А‘сЙ,
Jo                          Jo

but typically are computed as solutions to the Lyapunov equations

AP + PA* + BB* = O, A*Q+QA + C*C = O.

We gather the Cholesky factors of the gramians,

P = UU*, and Q = LL*,

and compute the singular value decomposition of the mixed product

U*L = ZYY*.

Here Σ is a diagonal matrix whose entries are the eigenvalues of U* QU, Z is an or-
thogonal matrix whose columns are the eigenvectors of U* QU, and Y is an orthogonal
matrix whose columns are the eigenvectors of L*PL (for details on the SVD₁ see (Tre-
fethen and Bau, 1997)). The diagonal elements of Σ, nonnegative and in descending
order, are known as the Hankel Singular Values (HSVs) of the dynamical system

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