Parabolic Boundary Control Problems with Delayed Actuator Dynamics
John Burns, Virginia Tech
Abstract: In this talk we discuss some control, optimization and design problems for a convection diffusion equation and investigate the impact of including actuator dynamics and delays. The problem is motivated by applications to control and design of energy efficient buildings where actuation is provided by a HVAC system. To provide some indication of the scope of this problem, it is helpful to note that buildings worldwide account for a approximately 40% of global energy consumption, and the resulting greenhouse gas emissions, significantly exceed those of all transportation combined. In the United States a 50% reduction in buildings energy usage is equivalent to taking every passenger vehicle and small truck in the United States off the road and a 70% reduction in buildings energy usage is equivalent to eliminating the entire energy consumption of the U.S. transportation sector.
As a mathematical object, a whole building system is the composition of diverse dynamic subsystems and is a complex, multi-scale, nonlinear, and uncertain dynamical system. Recent results have shown that by considering the whole building as an integrated system and applying modern estimation and control techniques to optimize this system, one can achieve greater efficiencies than obtained by optimizing individual building components such as lighting and HVAC. In order to control a whole building system for energy minimization one must address a variety of theoretical and computational science problems at various levels from room to complete building envelopes. We focus on a single room example where the basic model is governed by a parabolic partial differential equation which is augmented to include a model of an actuator with delays. We show that under suitable conditions, the coupled system is well posed in a standard Hilbert space and we use this corresponding abstract formulation to construct numerical methods for control design.