Recent results on linear systems with multiple delays: high-dimensional stability maps; delay-independent stability; delay-tolerant topology design

This talk covers our most recent results on coupled linear systems with multiple delays. In particular, we discuss the state-of-the-art in extracting stability maps in delay parameter space, and how the limitations in the existing studies can be circumvented with our new method called Advanced Clustering with Frequency Sweeping ACFS (IEEE TAC Feb 2011). This method is a non-trivial cross-fertilization of root clustering paradigms and frequency sweeping techniques, and allows us taking cross-sectional views of high dimensional stability maps. The proposed method is based on algebra, is computationally efficient, and satisfies the necessary and sufficient conditions of stability.

In ACFS, we reveal that the upper and lower bounds of the frequency parameter we sweep can actually be calculated non-conservatively. This new result therefore allows frequency sweeping techniques to sweep the frequency only within these bounds, bringing simplification to existing practice. We then convert the bound calculations to testing delay-independent stability. By using algebraic geometry, we build a mathematical approach that can test if a linear system with multiple delays is delay-independent stable. This becomes possible primarily by identifying whether or not the frequency upper/lower bounds ever exist. Although the existence of such bounds needs to be confirmed in infinite-dimensional analysis, our developed delay-independent stability test requires us checking the roots of finite number of single-variable polynomials. With such a dramatic simplification, the test becomes tractable, computationally efficient, while still remaining necessary and sufficient in terms of stability (IEEE TAC, accepted).

We next discuss our most recent results on the interplay between network topology, delays, and stability. On a benchmark consensus dynamics, we demonstrate the concepts of Responsible Eigenvalue, which becomes the one and only one eigenvalue of the graph Laplacian determining the delay margin of the entire consensus system, which has otherwise infinite-dimensional stability problem (IET Control Theory & Applications, accepted). With this simplification at hand, we are able to analyze the delay margins of large scale consensus networks, we are able to design the network structure such that the delay margin becomes larger (delay-tolerant topology design), and we are able to construct controllers that tune the responsible eigenvalue in real-time such that the consensus system attains autonomy.