CPS: Small: A Convex Framework for Control of Interconnected Systems over Delayed Networks

Recent years have seen an explosion in the use of ad-hoc, IP-based and Wifi networks for control of spatially-distributed physical systems, with applications including automotive fleets; swarms of UAVs; remote surgery; and optimization of sensor networks. These new forms of communication have dramatically decreased the cost, energy, and maintenance associated with remote regulation, but have added fundamental challenges in the form of delay, packet drops, and intermittent feedback. However, despite the increasing importance of these Networked Control Systems (NCS), there is no accurate and reliable method of designing controllers in the presence of delayed and lossy feedback for systems with multiple states and multiple feedback channels (MIMO). In this project, we combine a new theory of duality with a computational framework for solving Linear Operator Inequalities (LOIs) to obtain convex, non-conservative conditions for optimal output-feedback control of MIMO systems with multiple, time-varying, uncertain and input delays and extend this framework to sampled-data systems with asynchronous sampling. Key Words: Networked Control; Control over Networks; Delay; Sampled Data; LMIs Target Area: Technology of Cyber-Physical Systems

Intellectual Merit In this project, we use a new principle of duality for linear systems with delay which allows the optimal output-feedback control problem to be expressed as an LOI. LOIs are a form of convex optimization where the decision variables parameterize classes of operators and inequalities are defined using the inner product of an infinite-dimensional space. In this project, we define an LOI and establish several classes of LOI which are solvable using LMIs. By solvable we mean the existence of a sequence of algorithms, each an LMI, each sufficient for feasibility of the LOI and whose limit approaches necessity. For example, the existence of a Lyapunov-Krasovskii functional for a delayed system can be posed as an LOI and the use of this LOI for stability analysis is known to be non-conservative. Unlike stability, the controller synthesis problem is Bilinear Operator Inequality since we search for both a feedback operator and a Lyapunov operator. Using as inspiration the LMI approach to controller synthesis, we extend a newly developed duality theorem to several classes of system and obtain an LOI for controller synthesis in each case. In each case, our approach is to: 1. Use polynomials to parameterize classes of operators which satisfy the conditions of the duality theorem.; 2. Use a Sum-of-Squares (SOS) - based approach to solve the LOI.; Invert the Lyapunov operator and obtain the controller. The classes of systems are: uncertain and time- varying delay; multiple delays; and sampled-data systems. We next extend the LOI framework to output feedback control which includes the problem of delay in the input. We also use a differential-difference formulation to manage complexity in NCSs with large numbers of delayed channels. Preliminary testing suggests that the algorithms will synthesize controllers for 30+ states and 10+ delays. Next, we extend  the results to optimal control using an H∞ gain metric. We test the controllers on the Pheeno wifi-based wheeled robotic testbed at ASU. The project includes a strong international collaboration, with researchers from Oxford, Grenoble and LAAS.

Broader Impacts The project will provide tractable and non-conservative algorithms for the optimal dy- namic output-feedback control of 30+ state linear systems with multiple delays, uncertain or time-varying delay, input delay, and sampled-data systems. The result will be a fundamental shift in our ability to safely and efficiently control physical systems over unreliable communication networks - thereby benefiting all forms of control which utilize ad-hoc, IP-based, or WiFi-based networks. Furthermore, the LOI framework developed in this project can be used for more general classes of infinite-dimensional systems beyond Net- worked Control Systems. To enhance distribution, we will develop a Matlab toolbox and post it on the project website for free distribution via public license. To enhance outreach, the educational part of the project will develop a graduate class and minicourse/tutorial on LOIs for cyber-physical systems.