  Bilinear dynamical systems are commonly used in science and engineering
because they form a bridge between linear and non-linear systems. However,
simulating them is still a challenge because of their large size. Hence, a lot
of research is currently being done for reducing such bilinear dynamical
systems (termed as bilinear model order reduction or bilinear MOR). Bilinear
iterative rational Krylov algorithm (BIRKA) is a very popular, standard and
mathematically sound algorithm for bilinear MOR, which is based upon
interpolatory projection technique. An efficient variant of BIRKA, Truncated
BIRKA (or TBIRKA) has also been recently proposed.
  Like for any MOR algorithm, these two algorithms also require solving
multiple linear systems as part of the model reduction process. For reducing
very large dynamical systems, which is now-a-days becoming a norm, scaling of
such linear systems with respect to input dynamical system size is a
bottleneck. For efficiency, these linear systems are often solved by an
iterative solver, which introduces approximation errors. Hence, stability
analysis of MOR algorithms with respect to inexact linear solves is important.
In our past work, we have shown that under mild conditions, BIRKA is stable (in
the sense as discussed above). Here, we look at stability of TBIRKA in the same
context.
  Besides deriving the conditions for a stable TBIRKA, our other novel
contribution is the more intuitive methodology for achieving this. This
approach exploits the fact that in TBIRKA a bilinear dynamical system can be
represented by a finite set of functions, which was not possible in BIRKA
(because infinite such functions were needed there). The stability analysis
techniques that we propose here can be extended to many other methods for doing
MOR of bilinear dynamical systems, e.g., using balanced truncation or the ADI
methods.
