Robust and Decentralized Control of Positive Systems: A Convex Approach

This thesis focuses on robust and decentralized control problems for positive systems. We begin by revisiting a classical modeling framework that allows us to model uncertain systems in a systematic way and, using tools from convex analysis, we derive tractable necessary and sufficient conditions for robust stability of uncertain positively dominated systems. These conditions involve only the system’s static gain and can be verified using convex optimization. We illustrate our results by deriving conditions for the robust stability of the Foschini-Miljanic algorithm, with applications to power control for communication systems with uncertain interference matrix. We then derive equivalent necessary and sufficient state-space conditions for the robust stability of positive systems. We use these conditions to show that designing structured controllers that enforce robust stability and close-loop positivity is a tractable problem that can be solved using convex optimization. We further extend our results to a class of systems with sector-bound nonlinearities, proving that the S-Procedure is lossless for positive systems. In the second part of the thesis we study a class of decentralized control problems for positive systems with application to biology and network theory. We show that, despite the structural constraints, these problems are convex both in the nominal and in the robust case and can be solved efficiently using custom descent algorithms. We provide several examples from leader selection in directed networks to robust optimal drug therapy for HIV. In the final part of the thesis we tackle the problem of analyzing the robustness properties of a class of systems that do not fall into the standard modeling framework for uncertain systems. These problems arise naturally in the study of directed networks with uncertain edge weights. Exploiting powerful results from duality theory of linear programming, we show that the problem of assessing robust performance for transportation and consensus networks with respect to the induced L1 and L∞ norms is tractable and can be solved using linear programming. We further extend this result to optimal robust network design in case the weights of a number of edges are left as decision variables.