Optimal distributed control and estimation for systems with spatiotemporal dynamics
Author(s)
Arbelaiz Mugica, Juncal
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Advisor
Hosoi, Anette E.
Jadbabaie, Ali
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Traditional optimal control and estimation synthesis often relies on the implicit assumption that sufficient computational power and bandwidth are available to implement centralized communication architectures. However, this is often not the case in large-scale spatially distributed systems. In these, relying only on spatially localized information transfer might be preferred or even imperative for control and estimation. Motivated by this challenge, this dissertation analyzes optimal distributed control and estimation synthesis for systems with spatiotemporal dynamics. Emphasis is placed on the state estimation problem for dynamical systems with shift-invariances over unbounded spatial domains and with distributed noisy measurements. Dimensional analysis and scaling arguments are used to define physically interpretable dimensionless groups, which are found to be related to the measurement signal-to-noise ratio and to decentralization measures. The branch point locus is introduced as a useful tool to systematically explore the sensitivity of the spatial localization of the feedback operator to the relevant dimensionless groups.
After some background is provided and mathematical preliminaries are introduced, the information structures of the optimal (in the sense of error variance minimization) state estimator for infinite-dimensional spatially invariant systems over 𝐿² (R) spaces are analyzed. The optimal state estimate is described by a spatially invariant distributed-parameter Kalman-Bucy filter. Its Kalman gain operator is a spatial convolution. Hence, the spatial decay of its kernel determines the information structures of the filter. In the problem set-up considered, such kernel exhibits asymptotic exponential spatial decay, which implies that estimation of the state at each spatial site heavily relies on local measurements. The role that the statistical properties of the noise processes perturbing the plant play in such spatial localization is analyzed. Under certain assumptions, it is shown that noise can make the optimal estimator rely more heavily on local information. A matching condition for which the filter gain is completely decentralized is found. Two case studies illustrate the theoretical results: i) optimal estimation of a diffusion process over the real line, and ii) optimal estimation of a linearized Swift-Hohenberg equation over the real line. Second, a similar analysis and spatial localization results for systems over Sobolev spaces are presented. Typically, these are necessary to synthesize filters for plants with higher order temporal dynamics, such as elastic waves. Third, the optimal estimation problem with hard information constraints for spatially invariant systems over 𝐿² (R) spaces is studied. Based on insights from analyzing the information structures of the distributed-parameter Kalman-Bucy filter, a convex functional optimization is proposed to design an optimal information-constrained filter gain for a subclass of spatially invariant plants. Such a gain operator minimizes estimation error variance, is stabilizing, and is compactly supported in space (i.e., the filter is enforced to share information only locally). The size of such support is defined a priori and determines the communication burden of the filter. The method is applied to design an optimal information-constrained Kalman-Bucy filter for a diffusion process over the real line, for which performance-locality trade-offs are discussed. Finally, conclusions are drawn and related future research directions discussed.
Date issued
2022-09Department
Massachusetts Institute of Technology. Department of MathematicsPublisher
Massachusetts Institute of Technology