Distance to Internal Instability of Linear Time-Invariant Systems Under Structured Perturbations

In this article, uncertain continuous-time and discrete-time linear time-invariant systems are considered. The uncertainties are assumed to affect polynomially the dynamics of the system and they can be structured. The problem of computing the distance to internal instability of an internally exponentially stable nominal system is solved by using tools from algebraic geometry, thus extending previous results valid in case of unstructured uncertainties. The choice of the nominal system is formulated and solved as the choice of a point in the parameter space that is sufficiently far from the boundary of the domain of stability. A simple example of application to robust control is outlined.

Automated summary

This article discusses uncertain continuous-time and discrete-time linear time-invariant systems, addressing uncertainties that polynomially affect system dynamics and can be structured. By employing tools from algebraic geometry, the problem of computing distance to internal instability of an internally exponentially stable nominal system is solved, extending previous results applicable to unstructured uncertainties. The choice of the nominal system is framed and resolved as selecting a point in parameter space sufficiently distant from the stability domain boundary, with a simple example of robust control application outlined.