Minimum-Information LQG Control - Part I: Memoryless Controllers

With the increased demand for power efficiency in feedback-control systems, communication is becoming a limiting factor, raising the need to trade off the external cost that they incur with the capacity of the controller's communication channels. With a proper design of the channels, this translates into a sequential rate-distortion problem, where we minimize the rate of information required for the controller's operation under a constraint on its external cost. Memoryless controllers are of particular interest both for the simplicity and frugality of their implementation and as a basis for studying more complex controllers. In this paper we present the optimality principle for memoryless linear controllers that utilize minimal information rates to achieve a guaranteed external-cost level. We also study the interesting and useful phenomenology of the optimal controller, such as the principled reduction of its order

Design of State-based Schedulers for a Network of Control Loops

For a closed-loop system, which has a contention-based multiple access network on its sensor link, the Medium Access Controller (MAC) may discard some packets when the traffic on the link is high. We use a local state-based scheduler to select a few critical data packets to send to the MAC. In this paper, we analyze the impact of such a scheduler on the closed-loop system in the presence of traffic, and show that there is a dual effect with state-based scheduling. In general, this makes the optimal scheduler and controller hard to find. However, by removing past controls from the scheduling criterion, we find that certainty equivalence holds. This condition is related to the classical result of Bar-Shalom and Tse, and it leads to the design of a scheduler with a certainty equivalent controller. This design, however, does not result in an equivalent system to the original problem, in the sense of Witsenhausen. Computing the estimate is difficult, but can be simplified by introducing a symmetry constraint on the scheduler. Based on these findings, we propose a dual predictor architecture for the closed-loop system, which ensures separation between scheduler, observer and controller. We present an example of this architecture, which illustrates a network-aware event-triggering mechanism.Comment: 17 pages, technical repor

Algorithms for Optimal Control with Fixed-Rate Feedback

We consider a discrete-time linear quadratic Gaussian networked control setting where the (full information) observer and controller are separated by a fixed-rate noiseless channel. The minimal rate required to stabilize such a system has been well studied. However, for a given fixed rate, how to quantize the states so as to optimize performance is an open question of great theoretical and practical significance. We concentrate on minimizing the control cost for first-order scalar systems. To that end, we use the Lloyd-Max algorithm and leverage properties of logarithmically-concave functions and sequential Bayesian filtering to construct the optimal quantizer that greedily minimizes the cost at every time instant. By connecting the globally optimal scheme to the problem of scalar successive refinement, we argue that its gain over the proposed greedy algorithm is negligible. This is significant since the globally optimal scheme is often computationally intractable. All the results are proven for the more general case of disturbances with logarithmically-concave distributions and rate-limited time-varying noiseless channels. We further extend the framework to event-triggered control by allowing to convey information via an additional "silent symbol", i.e., by avoiding transmitting bits; by constraining the minimal probability of silence we attain a tradeoff between the transmission rate and the control cost for rates below one bit per sample

An integrative perspective to LQ and ℓ∞ control for delayed and quantized systems

Deterministic and stochastic approaches to handle uncertainties may incur very different complexities in computation time and memory usage, in addition to different uncertainty models. For linear systems with delay and rate constrained communications between the observer and the controller, previous work shows that a deterministic approach, the ℓ ∞ control has low complexity but can only handle bounded disturbances. In this article, we take a stochastic approach and propose a linear-quadratic (LQ) controller that can handle arbitrarily large disturbance but has large complexity in time and space. The differences in robustness and complexity of the ℓ ∞ and LQ controllers motivate the design of a hybrid controller that interpolates between the two: The ℓ ∞ controller is applied when the disturbance is not too large (normal mode) and the LQ controller is resorted to otherwise (acute mode). We characterize the switching behavior between the normal and acute modes. Using our theoretical bounds which are supplemented by numerical experiments, we show that the hybrid controller can achieve a sweet spot in the robustness-complexity tradeoff, i.e., reject occasional large disturbance while operating with low complexity most of the time