UC Santa Barbara

We consider the optimal controller design problem for linear time-invariant, spatially-distributed systems. The controller to be designed is itself a distributed system; each subcontroller component is restricted to have access to only a local subset of system information, which is shared across the network according to an underlying communication graph. The design problem of interest is to synthesize optimal controllers (with respect to some performance measure) subject to this limited information sharing architecture. In this dissertation, we contribute to two directions of research in this setting: i) analysis of constraints that ensure such localization and can be imposed in a tractable manner, and ii) characterization of settings in which the unconstrained centralized optimal controller has an inherent degree of spatial localization.

For direction (i), we follow the `System Level Synthesis' (SLS) approach and consider directly designing the closed-loops as opposed to the controller or corresponding Youla parameter. Structural constraints on the closed-loop can be imposed in a convex manner, and we demonstrate that the optimal controller design problem subject to closed-loop transfer function sparsity constraints is a convex relaxation of the optimal controller design problem subject to structural constraints on a controller state-space realization (implementation). We provide an implicit parameterization of all achievable closed-loop mappings for

a broad class of systems, including continuous- or discrete-time spatially-invariant systems over an infinite spatial domain.

Under certain assumptions, in the state feedback setting we convert this implicit closed-loop parameterization to an explicit affine linear parameterization.

Our parameterizations allow for conversion of the closed-loop structured H2 optimal controller design problem to a standard model matching problem with finitely many transfer function parameters, allowing for analytic solutions in certain problem settings.

We further take a step toward quantifying the performance gap between structured closed-loop transfer function design and structured controller realization design by studying the setting of relative feedback controllers. To do so, we provide a compact and convex characterization of all relative feedback controllers, and demonstrate that the relative feedback requirement can be imposed as a convex constraint on the closed-loop in certain problem settings. We use this characterization to show that the optimal relative feedback controller design problem subject to closed-loop structural constraints may be infeasible.

In direction (ii), we consider the optimal control of PDEs over a Sobolev space. We demonstrate that the optimal state feedback is a spatial convolution operator given by an exponentially decaying convolution kernel, thus enabling implementation with a localized architecture, extending previous results in the L2 setting. The main tool we utilize is a transformation from a Sobolev to an L2 space, which is constructed from a spectral factorization of the spatial frequency weighting matrix of the Sobolev norm.