  Structural symmetries of linear dynamical systems can be exploited for
decoupling the dynamics and reducing the computational complexity of the
controller implementation. However, in practical applications, inexact
structural symmetries undermine the ability to decouple the system, resulting
in the loss of any potential complexity reduction. To address this, we propose
substituting an approximation with exact structural symmetries for the original
system model, thereby introducing an approximation error. We focus on internal
model controllers for cross-directional systems encountered in large-scale and
high-speed control problems of synchrotrons or the process industry and
characterise the stability, performance, and robustness properties of the
resulting closed loop. While existing approaches replace the original system
model with one that minimises the Frobenius norm of the approximation error, we
show that this can lead to instability or poor performance. Instead, we propose
approximations that are obtained from semidefinite programming problems. We
show that our proposed approximations can yield stable systems even when the
Frobenius norm approximation does not. The paper concludes with numerical
examples and a case study of a synchrotron light source with inexact structural
symmetries. Exploiting structural symmetries in large-scale and high-speed
systems enables faster sampling times and the use of more advanced control
techniques, even when the symmetries are approximate.
