Side-information in control and estimation

Abstract

Abstract-As in portfolio theory, we can think of the value of side-information in a control system as the change in the "growth rate" due to side-information. A scalar counterexample (motivated by carry-free deterministic models) shows the value of side-information for control does not exactly parallel the value of side-information for portfolios. Mutual-information does not seem to be a bound here.The concept is further explored through a spinning vector control system that is re-oriented at each time so that the control or observation direction is partially unknown. The value of sideinformation can be calculated in this setup and it behaves quite differently in a control vs. estimation context. A second example considers the problem of vector control over a (scalar) erasure channel, the dual problem to the estimation problem of intermittent Kalman Filtering. The value of information here is measured through the change in the critical packet-drop probability for the system. While non-causal side-information regarding the packet arrivals does not affect the critical probability for the estimation problem, we find that it can generically be very valuable for the control problem -it seems to change the scaling behavior for the control counterpart to what would be considered the "high SNR limit" in communication problems. I. INTRODUCTIONParameter uncertainty has a long history in control theory -the very idea of robust control is about dealing with it. Recently, the advent of networked control systems has made stochastic uncertainty models more relevant. There is now a real need to have a theory capable of dealing with sideinformation in control. As just one example, control theorists are interested in knowing how networked control systems behave with or without acknowledgements of dropped packets since this is relevant for choosing among practical protocols like TCP vs. UDP [1]. Acknowledgements are a kind of sideinformation about control channel state, but as of now, there is no theoretical guidance for how to think about it in a principled way.Fortunately, such things have long been studied in information theory in the context of unknown fading channels [2]. Medard in [3] examines the effect of imperfect channel knowledge on capacity, and Lapidoth and Shamai quantify the degradation in performance due to channel-state estimation errors by the receiver [4]. Pradhan et al. show that the duality between source and channel coding in fact extends to the case with side-information under certain conditions [5]: this is particularly interesting given the well-known parallel between source coding and portfolio theory, which we will connect to here. Further, Kotagiri and Laneman [6] study the impact of non-causal knowledge of the state in a multiple-access setting. There are many more interesting results as well, but space precludes any serious discussion here.

Stabilizability of Vector Systems with Uniform Actuation Unpredictability

“…Many previous works around parameter uncertainty in control systems highlight the importance of the controller to plan and have worked towards developing an understanding of the "value of information in control" [24]- [29]. More recent work has examined the importance of side-information in control problems [6], [8], [30]- [33].…”

“…The impact of parameter uncertainty in control and estimation is highlighted by the intermittent Kalman filtering model [1], [2] and related works on dropped control packets [3]- [7]. In particular, these works show how critical the ability to plan is for control systems -while the critical erasure probability for estimation in the intermittent Kalman filtering setting depends only on the maximal eigenvalue of the system, the critical erasure probability of the dual problem with dropped controls depends on the product of all eigenvalues -thus control can tolerate far less uncertainty than estimation [8]. This makes intuitive sense, since a controller must commit to an action at every timestep without knowledge of the future, whereas the observer in the estimation problem can revisit its decisions in the future.…”

Real-Time Cooperative Communication for Automation Over Wireless

“…Fundamental limits for control and estimation of systems over both noiseless rate-limited channels [45], [46] and noisy channels [47] have been established. A series of works [48]- [50] established the limits of control and estimation over packet dropping networks and it was recently shown that control and communication co-design could provide unbounded performance gains in such settings [51].…”

Control over networks with fading communication channels

“…Indeed, the right hand side of (2.2) denotes the information generating speed of the LTI system [8,23], which is generating information about the unknown initial system state. This resemblance also motivated researchers to study control systems from the perspective of information theory, e.g., see [24][25][26][27][28][29][30][31][32][33][34].…”