On the Real Stability Radius of Sparse Systems
For control theorists and network engineers, this work offers a method to compute the stability radius for sparse systems with arbitrary sparsity patterns, but the results are incremental as they extend existing optimization techniques to a specific problem formulation.
This paper studies robust stability of sparse LTI systems by formulating the stability radius problem as an equality-constrained minimization problem and developing a penalty-based gradient/Newton descent algorithm to find local minima. The approach provides structural insights into robust stability of sparse networks.
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.