The iterative rational Krylov algorithm (IRKA) is an iterative algorithm, useful for model order reduction (MOR) of single-input single-output (SISO) linear time-invariant dynamical systems. At each iteration, IRKA does an Hermite type interpolation of the original system transfer function. Each interpolation requires solving r {\displaystyle r} shifted pairs of linear systems, each of size n × n {\displaystyle n\times n} ; where n {\displaystyle n} is the original system order, and r {\displaystyle r} is the desired reduced model order (usually r ≪ n {\displaystyle r\ll n} ). The algorithm was first introduced by Gugercin, Antoulas and Beattie in 2008. It is based on a first order necessary optimality condition, initially investigated by Meier and Luenberger in 1967. The first convergence proof of IRKA was given by Flagg, Beattie and Gugercin in 2012, for a particular kind of systems.
MOR as an optimization problem Consider a SISO linear time-invariant dynamical system, with input v ( t ) {\displaystyle v(t)} , and output y ( t ) {\displaystyle y(t)} :
{ x ˙ ( t ) = A x ( t ) + b v ( t ) y ( t ) = c T x ( t ) A ∈ R n × n , b , c ∈ R n , v ( t ) , y ( t ) ∈ R , x ( t ) ∈ R n . {\displaystyle {\begin{cases}{\dot {x}}(t)=Ax(t)+bv(t)\\y(t)=c^{T}x(t)\end{cases}}\qquad A\in \mathbb {R} ^{n\times n},\,b,c\in \mathbb {R} ^{n},\,v(t),y(t)\in \mathbb {R} ,\,x(t)\in \mathbb {R} ^{n}.}
Applying the Laplace transform, with zero initial conditions, we obtain the transfer function G {\displaystyle G} , which is a fraction of polynomials:
G ( s ) = c T ( s I − A ) − 1 b , A ∈ R n × n , b , c ∈ R n . {\displaystyle G(s)=c^{T}(sI-A)^{-1}b,\quad A\in \mathbb {R} ^{n\times n},\,b,c\in \mathbb {R} ^{n}.}
Assume G {\displaystyle G} is stable. Given r < n {\displaystyle r<n} , MOR tries to approximate the transfer function G {\displaystyle G} , by a stable rational transfer function G r {\displaystyle G_{r}} , of order r {\displaystyle r} :
G r ( s ) = c r T ( s I r − A r ) − 1 b r , A r ∈ R r × r , b r , c r ∈ R r . {\displaystyle G_{r}(s)=c_{r}^{T}(sI_{r}-A_{r})^{-1}b_{r},\quad A_{r}\in \mathbb {R} ^{r\times r},\,b_{r},c_{r}\in \mathbb {R} ^{r}.}
A possible approximation criterion is to minimize the absolute error in H 2 {\displaystyle H_{2}} norm:
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