Externally positive systems : Analysis and control based on combinatorial polynomials

Abstract: Monotonic tracking is required in many control systems, including those that cannot tolerate any overshoots and undershoots in their closed-loop responses. Classical examples are found in vehicle cruise control and liquid tank level control. In the former, an overshoot happens when the speed of the vehicle goes beyond the set value violating the safety measures, and in the latter an overshoot is considered as filling up the tank with an excessive amount of liquid which leads to a waste of resources. Controllers that eliminate overshoots are undeniably more desirable in these examples. In fact, the same requirement is in place for many more engineering applications, including biological systems, robotics and process control, indicating widespread benefits of controllers which can guarantee monotonicity in the system response. Formally, linear systems that exhibit monotonically increasing step responses are called externally positive. Designing controllers that render the closed-loop system externally positive requires a thorough understanding of this property in linear systems. In this thesis, we leverage combinatorial polynomials and their properties to study external positivity in both discrete-time and continuous-time linear systems modelled by transfer functions or impulse responses. Several conditions are provided that are either necessary, sucientor both necessary and sufficient for a linear system to be externally positive. These conditions are then used to synthesize controllers that ensure external positivity in closed-loop systems and hence, eliminate both overshoots and undershoots in the system response. In particular, we provide synthesis techniques based on convex optimization that ensure stability, robustness and offset-free monotonic tracking in the closed-loop system and improve its decay rate and sensitivity. We compare the results with the state-of-the-art in the literature and demonstrate the efficacy of the proposed controller synthesis methods through several numerical examples.