Collaborative Research: Robust Control of Large-Scale Networks with Variable Delays using Partial Integral Equations

Concerns over energy security and global warming have driven a recent dramatic increase in installed wind capacity. Moreover, the need for optimal wind conditions and limited availability of suitable land has resulted in the aggregation of turbines into high-density wind farms. In such farms, aerodynamic coupling between turbines strongly influences the efficiency of the turbines. Because the coupling between turbines is mediated by wind, however, there is significant and variable delay in the interaction between turbines–the upstream turbines affect downstream turbines with a delay determined by velocity of the wind exiting the upstream turbines. Thus, our ability to efficiently control wind farms is fundamentally limited by our ability to control large-scale networks with uncertain, time-varying and even state-dependent delay. The project will develop methods to design and analyze controllers for such systems with networked dynamics and delays. This will enable a host of benefits for wind energy including improved power capture, reduced loading, and active power control for grid services. The work will also enable the design of safe and efficient networked controllers in other domains including fleets of autonomous vehicles and swarms of uninhabited aerial vehicles. The goal of the project is to develop new theory and algorithms which allow for robust analysis and control of nonlinear systems with uncertain and variable delay. To do this, we combine the Integral Quadratic Constraint (IQC) framework for robust analysis and control Integral with the Partial Integral Equation (PIE) framework for optimal control of fixed-delay linear systems. Unlike previous work which considered the entire delay to be a source of uncertainty, we partition the delay into nominal and uncertain/variable parts. We then use PIE’s for the known/fixed-delay linear part of the system and IQC’s for the uncertain/nonlinear part. This approach reduces the uncertainty in the system and thereby increases the accuracy/performance of the resulting analysis/controllers. The technical approach of the project is divided into 4 Technical Challenges. (T1) First, we consider uncertain/nonlinear dynamics and known, fixed delay and formulate a convex stability test. (T2) Next, we model static uncertainty in delay using a nominal PIE subsystem and an unstructured parametric uncertainty acting on infinite-dimensional channels. We generalize the PIE and IQC frameworks to handle infinite-dimensional channels including a parameterization of dynamic PIE multipliers. (T3) Third, we consider time-varying and state-dependent delay. These cases are modeled by a nominal PIE coupled with either linear parameter-varying uncertainty or a nonlinearity. (T4) Finally, for robust control design, we use an alternation between a synthesis step and an IQC analysis step. The results are applied to models of wind farm control. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.