We study the stabilization of an unpredictable linear control system where the controller must act based on a rate-limited observation of the state. More precisely, we consider the system @math , where the @math 's are drawn independently at random at each time @math from a known distribution with unbounded support, and where the controller receives at most @math bits about the system state at each time from an encoder. We provide a time-varying achievable strategy to stabilize the system in a second-moment sense with fixed, finite @math . While our previous result provided a strategy to stabilize this system using a variable-rate code, this work provides an achievable strategy using a fixed-rate code. The strategy we employ to achieve this is time-varying and takes different actions depending on the value of the state. It proceeds in two modes: a normal mode (or zoom-in), where the realization of @math is typical, and an emergency mode (or zoom-out), where the realization of @math is exceptionally large.
We also build on a series of investigations around model unpredictability and multiplicative noise in control systems @cite_1 @cite_19 @cite_10 . In this work, we take a stochastic control approach, which complements the investigations around parameter uncertainty in the robust control literature @cite_12 @cite_18 .
@cite_18 @cite_1 @cite_19 @cite_10 @cite_12
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1. Introduction. 2. Linear Algebra. 3. Linear Systems. 4. H2 and Ha Spaces. 5. Internal Stability. 6. Performance Specifications and Limitations. 7. Balanced Model Reduction. 8. Uncertainty and Robustness. 9. Linear Fractional Transformation. 10. m and m- Synthesis. 11. Controller Parameterization. 12. Algebraic Riccati Equations. 13. H2 Optimal Control. 14. Ha Control. 15. Controller Reduction. 16. Ha Loop Shaping. 17. Gap Metric and ...u- Gap Metric. 18. Miscellaneous Topics. Bibliography. Index. 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 considers the stabilization of an unstable discrete-time linear system that is observed over a channel corrupted by continuous multiplicative noise. The main result is a converse bound that shows that if the system growth is large enough the system cannot be stabilized in a mean-squared sense. This is done by showing that the probability of the state magnitude remains bounded must go to zero with time. It was known that a system with multiplicative observation noise can be stabilized using a simple linear strategy if the system growth is suitably bounded. However, it was not clear whether non-linear controllers could overcome arbitrarily large growth factors. One difficulty with using the standard approach for a data-rate theorem style converse is that the mutual information per round between the system state and the observation is potentially unbounded with a multiplicative noise observation channel. 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 paper studies robust control problems under the setting of quantized feedback. We consider both the static and dynamic logarithmic quantizers. In the static quantization case, the quantizer has an infinite number of levels, and the design problem is to find the minimal quantization density required to achieve a given control objective. In the dynamic quantization case, the problem is to minimize the number of quantization levels to achieve a given control objective. We present a number of results for different controller-quantizer configurations. These results are developed using the so-called sector bound approach for quantized feedback control, which was initiated by the authors previously for systems without uncertainties. Copyright © 2009 John Wiley & Sons, Ltd.