We consider the stabilization of an unstable discrete-time linear system that is observed over a channel corrupted by continuous multiplicative noise. Our main result shows that if the system growth is large enough, then the system cannot be stabilized in a second-moment sense. This is done by showing that the probability that the state magnitude remains bounded must go to zero with time. Our proof technique recursively bounds the conditional density of the system state (instead of focusing on the second moment) to bound the progress the controller can make. This sidesteps the difficulty encountered in using the standard data-rate theorem style approach; that approach does not work because the mutual information per round between the system state and the observation is potentially unbounded. It was known that a system with multiplicative observation noise can be stabilized using a simple memoryless linear strategy if the system growth is suitably bounded. In this paper, we show that while memory cannot improve the performance of a linear scheme, a simple non-linear scheme that uses one-step memory can do better than the best linear scheme.
The uncertainty threshold principle @cite_12 considers a systems with Gaussian uncertainty on the system growth factor and the control gain, and provides limits for when the system is stabilizable in a second-moment sense. Our work complements this result by considering uncertainty on the observation gain.
@cite_12
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This note shows that the optimal control of dynamic systems with uncertain parameters has certain limitations. In particular, by means of a simple scalar linear-quadratic optimal control example, it is shown that the infinite horizon solution does not exist if the parameter uncertainty exceeds a certain quantifiable threshold; we call this the uncertainty threshold principle. The philosophical and design implications of this result are discussed. Uncertainty principles such as Heisenberg's provide limits on the time-frequency concentration of a signal, and constitute an important theoretical tool for designing linear signal transforms. Generalizations of such principles to the graph setting can inform dictionary design, lead to algorithms for reconstructing missing information via sparse representations, and yield new graph analysis tools. While previous work has focused on generalizing notions of spreads of graph signals in the vertex and graph spectral domains, our approach generalizes the methods of Lieb in order to develop uncertainty principles that provide limits on the concentration of the analysis coefficients of any graph signal under a dictionary transform. One challenge we highlight is that the local structure in a small region of an inhomogeneous graph can drastically affect the uncertainty bounds, limiting the information provided by global uncertainty principles. Accordingly, we suggest new notions of locality, and develop local uncertainty principles that bound the concentration of the analysis coefficients of each atom of a localized graph spectral filter frame in terms of quantities that depend on the local structure of the graph around the atom's center vertex. Finally, we demonstrate how our proposed local uncertainty measures can improve the random sampling of graph signals. Control strategies for systems with information bottlenecks often follow an estimate-then-control paradigm. This paper presents a “non-coherent” system where this strategy cannot work and provides an alternative. The paper considers the estimation and control of a discrete-time linear system with continuous random observation gain, i.e. through a non-coherent channel. It is shown that such an unstable system is not mean-squared observable regardless of the density of the random observation gain: the mean-squared estimation error for any estimator must go to infinity. This is surprising in the context of threshold results for rate-limited estimation. In contrast to other results with rate-limited feedback, the paper shows that the system can be closed-loop mean-square stabilized in a certain parameter regime even though its open-loop counterpart is not mean-square observable. Finally, carry-free models (generalized deterministic models) provide an intuitive interpretation for the results. This paper addresses the optimal control problem known as the Linear Quadratic Regulator in the case when the dynamics are unknown. We propose a multi-stage procedure, called Coarse-ID control, that estimates a model from a few experimental trials, estimates the error in that model with respect to the truth, and then designs a controller using both the model and uncertainty estimate. Our technique uses contemporary tools from random matrix theory to bound the error in the estimation procedure. We also employ a recently developed approach to control synthesis called System Level Synthesis that enables robust control design by solving a convex optimization problem. We provide end-to-end bounds on the relative error in control cost that are nearly optimal in the number of parameters and that highlight salient properties of the system to be controlled such as closed-loop sensitivity and optimal control magnitude. We show experimentally that the Coarse-ID approach enables efficient computation of a stabilizing controller in regimes where simple control schemes that do not take the model uncertainty into account fail to stabilize the true system.