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.
In the context of linear systems with known parameters and unbounded additive disturbances, the necessity of adaptive quantization for stabilization has been long recognized. Nair and Evans @cite_2 proved that time-invariant fixed-rate quantizers are unable to attain bounded cost if the noise is unbounded @cite_2 , regardless of their rate. The reason is that since the noise is unbounded, over time, a large magnitude noise realization will inevitably be encountered, and the dynamic range of the quantizer will be exceeded by a large margin, not permitting recovery. Adaptive quantizers of zooming type were originally proposed by Brockett and Liberzon @cite_5 , and further studied in @cite_2 @cite_16 @cite_14 . Such quantizers zoom out'' (i.e. expand their quantization intervals; this corresponds to our emergency mode'') when the system is far from the target and zoom in'' when the system is close to the target (this corresponds to our normal mode''). This paper extends our recent analysis @cite_14 of adaptive quantization for known linear systems to unpredictably varying ones.
@cite_5 @cite_14 @cite_16 @cite_2
2512605635 2963912378 2019116530 2161442655
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. Consider a target moving at a constant velocity on a unit-circumference circle, starting at an arbitrary location. To acquire the target, any region of the circle can be probed to obtain a noisy measurement of the target’s presence, where the noise level increases with the size of the probed region. We are interested in the expected time required to find the target to within some given resolution and error probability. For a known velocity and a given reliability, we provide an asymptotical characterization of the optimal tradeoff between time and resolution. Considering an asymptotically diminishing error probability, we derive the maximal targeting rate, and show that in contrast to the well-studied case of constant measurement noise, measurement dependent noise incurs a multiplicative gap in the maximal targeting rate between adaptive and non-adaptive search strategies. Moreover, for all rates below this maximal rate, our adaptive strategy attains the optimal rate-reliability tradeoff. We further show that accounting for a target moving at an unknown fixed velocity, the optimal non-adaptive search strategy incurs a factor of at least two in the maximal targeting rate. Abstract This paper investigates the stabilizability of a linear, discrete-time plant with a real-valued output when the controller, which may be nonlinear, receives observation data at a known rate. It is first shown that, under a finite horizon cost equal to the m th output moment, the problem reduces to quantizing the initial output. Asymptotic quantization theory is then applied to directly obtain the limiting coding and control scheme as the horizon approaches infinity. This is proven to minimize a particular infinite horizon cost, the value of which is derived. A necessary and sufficient condition then follows for there to exist a coding and control scheme with the specified data rate that takes the m th output moment to zero asymptotically with time. If the open-loop plant is finite-dimensional and time-invariant, this condition simplifies to an inequality involving the data rate and the unstable plant pole with greatest magnitude. Analagous results automatically hold for the related problem of state estimation with a finite data rate. This paper establishes the equivalence of three stabilizability-related properties for a class of linear impulsive systems. The first involves a gramian-based condition inspired by results for time-varying, discrete-time linear systems introduced decades ago. The second is the ability to achieve closed-loop exponential stability via state feedback. Finally, the third property is exponential stability of an ‘unreachable’ subsystem identified from a decomposition of the original system derived from an invariant subspace that characterizes the set of reachable states. A consequence of this analysis is that full state reachability of a linear impulsive system is not necessary for state feedback stabilization, a well-known fact for linear time-invariant systems. The main ideas of the paper are applied to the problem of synchronizing two Lorenz oscillators using underactuated impulsive control.