  In this paper, we study robust stability of sparse LTI systems using the
stability radius (SR) as a robustness measure. We consider real perturbations
with an arbitrary and pre-specified sparsity pattern of the system matrix and
measure their size using the Frobenius norm. We formulate the SR problem as an
equality-constrained minimization problem. Using the Lagrangian method for
optimization, we characterize the optimality conditions of the SR problem,
thereby revealing the relation between an optimal perturbation and the
eigenvectors of an optimally perturbed system. Further, we use the Sylvester
equation based parametrization to develop a penalty based gradient/Newton
descent algorithm which converges to the local minima of the optimization
problem. Finally, we illustrate how our framework provides structural insights
into the robust stability of sparse networks.
