A Popov criterion for networked systems

Abstract

We consider robustness analysis of heterogeneous and homogeneous networked systems based on integral quadratic constraints (IQCs). First, we show how the analysis decomposes into lower dimensional problems if the interconnection structure is exploited. This generally leads to a significant reduction of the computational complexity. Secondly, by considering a set of IQCs that characterizes the eigenvalues of the interconnection matrices of symmetrically networked systems, we derive a Popov-like criterion for such systems. In particular, when the nodes of the networked system are single-input-single-output linear time-invariant operators, the criterion can be illustrated using a generalized Popov plot. In such cases, the Popov criterion is also a necessary condition in the sense that if the criterion is violated then a destabilizing network with the specified eigenvalue distribution can be constructed.

A scalable robust stability criterion for systems with heterogeneous LTI components

“…The result is proven in the same way as the corresponding result in [8] and the result is also related to earlier contributions in [4], [5].…”

A Scalable Robust Stability Criterion for Systems With Heterogeneous LTI Components

“…In the case where , Nyquist curve of must have a zero net encirclement of the points . The normality assumption on the interconnection matrix can in this case be removed by using a reasoning along the lines of Proposition 2 of [7]. Example 6: Let us illustrate Theorem 2 by applying it to the system considered in Example 1.…”

Characterization of Robust Stability of a Class of Interconnected Systems

“…Notice that the dimension of the left-hand-side of (4) is nm × nm, while it is m × m in the case of (5) (and there are n such inequalities). Hence the computational complexity of verifying condition (5) grows linearly in n (after finding all eigenvalues ofΓ, which has complexity O(n 3 )), while the complexity of verifying condition (4) in general grows as O(n β ), where β is between 4.5 to 6.5, see [7].…”