Communication Delay Co-Design in $\mathcal{ H}_{2}$-Distributed Control Using Atomic Norm Minimization

Abstract

When designing distributed controllers for large-scale systems, the actuation, sensing and communication architectures of the controller can no longer be taken as given. In particular, controllers implemented using dense architectures typically outperform controllers implemented using simpler ones -however, it is also desirable to minimize the cost of building the architecture used to implement a controller. The recently introduced Regularization for Design (RFD) framework poses the controller architecture/control law co-design problem as one of jointly optimizing the competing metrics of controller architecture cost and closed loop performance, and shows that this task can be accomplished by augmenting the variational solution to an optimal control problem with a suitable atomic norm penalty. Although explicit constructions for atomic norms useful for the design of actuation, sensing and joint actuation/sensing architectures are introduced, no such construction is given for atomic norms used to design communication architectures. This paper describes an atomic norm that can be used to design communication architectures for which the resulting distributed optimal controller is specified by the solution to a convex program. Using this atomic norm we then show that in the context of H 2 distributed optimal control, the communication architecture/control law co-design task can be performed through the use of finite dimensional second order cone programming. * N. Matni is with the architectures -this family of simple controllers is then used to construct various atomic norms [3][4][5] that penalize the use of specific architectural resources, such as actuators, sensors or additional communication links. These atomic norms are then added as a penalty function to the variational solution to an optimal control problem (formulated in the model matching framework), allowing the controller designer to explore the tradeoff between architectural complexity and closed loop performance by varying the weight on the atomic norm penalty in the resulting convex optimization problem.In [2] we give explicit constructions of atomic norms useful for the design of actuation, sensing and joint actuation/sensing architectures, but do not address how to construct an atomic norm for communication architecture design. Indeed constructing a suitable atomic norm for communication architecture design has substantial technical challenges that do not arise in actuation and sensing architecture design: we address these challenges in this paper. We model a distributed controller as a collection of sub-controllers, each equipped with a set of actuators and sensors, that exchange their respective measurements with each other subject to communication delays imposed by an underlying communication graph. Keeping with the philosophy adopted in RFD [2], we view dense communication architectures, i.e., ones with a large number of communication links between subcontrollers, as being composed of multiple simple atomic communication architectures, i.e., ones...

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