Algebraic Tools to Study Stability and Control of Linear Time-Invariant Systems with Multiple Time Delays

Guest Speaker: Rifat Sipahi, Assistant Professor, Mechanical and Industrial Engineering, Northeastern University

Abstract: Time delays are present in many dynamical systems. For instance, inter-planetary communication delays make maneuvering the Rover on Mars from Earth a challenge; in tele-surgery, visual and force feedback are delayed at different amounts, confusing the surgeon; and in biological systems, maturation of species requires time, adding delays into the way a population grows/vanishes. To obtain a baseline understanding of the effects of delays to dynamical behavior, and to eventually sprout from this to more complicated systems, we start studying “seemingly” manageable linear time-invariant (LTI) systems with multiple delays. In this research, we wish to answer several WH-questions pertaining to the mechanisms of stability and control. For instance, what parameters in the system should we change so that we can improve stability; in what sense should these parameters be changed; how could we make the system stable regardless of the amount of delay; with which delays could the system remain stable. Answering these questions calls for non-conservative analyses. However, in the presence of multiple delays, performing non-conservative analyses is extremely difficult as it requires solving nonlinear infinite-dimensional eigenvalue problems in the domain of multiple delays. Implementation of algebraic tools can actually be a promising direction as we discuss in this talk. We first present, using algebra, how one can extract the stability maps of LTI systems with multiple delays. These maps are quite useful in system design as they display the decomposition of stability/instability features of the system with respect to delays. We next study the stability-descriptor of the system. We know that when the range of the stability-descriptor is empty set, then the system cannot transition from stability to instability no matter what the delays are. We create an algebraic technique to identify whether or not the stability-descriptor has a feasible range. With this technique, we develop controllers that can make the system to remain stable no matter how large/small the delays are. That is, we can design controllers to render a system delay- independent stable. We then move on to our results on the interplay between network topology, delays, and stability. On a benchmark consensus dynamics with inter-agent communication delays, we first present the Responsible Eigenvalue (RE) concept. RE turns out to be the one and only one eigenvalue of the graph Laplacian determining the amount of delay that the entire consensus system can tolerate without becoming unstable. With RE at hand, we are able to analyze the delay tolerance of large-scale consensus networks; we are able to tailor various graphs in order to design the delay tolerance of larger scale consensus systems; and we are able to construct controllers that tune the responsible eigenvalue in real-time such that the consensus system attains autonomy.