Control and tracking with communication constraints

Most of the work in my Ph.D. dissertation is on data compression for feedback measurements in LQG control.  

The goal is to control the state of a Gauss/Markov plant (an autoregressive linear system with Gaussian process noise). In particular, we want to obtain a prespecified "control performance" in terms of the standard LQG (quadratic) cost. We assume that the plant's controller cannot measure the the plant state directly. Instead, we assume that a remote sensor platform can make arbitrary measurements of the plant, but must encode these measurements into packets of bits to convey them to the controller. Our goal is to design a sensor/encoder block (that maps measurements into bitstrings) and a decoder/controller block (that maps bitstrings into controls ) that jointly minimize the required bitrate of the feedback channel while achieving the desired LQG control performance. 

In our work on lower bounds on the bitrate/LQG cost tradeoff, we rigorously defined prefix constraints that can be imposed on the feedback packets. These constraints allow the decoder, and possibly other users on the communication network, to identify when the encoder is done transmitting. We showed that under two extremal notions of these constraints, the bitrate is lower bounded in terms of the directed information from the sequence of plant states to the control inputs. This lower bound holds even when the encoder and decoder share randomness. Shared randomness is a resource used in the design and analysis of quantizers that use dithering. This result proves that the availability of a dither signal does not change the fundamental lower bound. A prerecording of my CDC presentation on lower bounds is here. It's a little rushed; hopefully it was better live... 

In our work on upper bounds, we developed and analyzed a quantizer design and lossless encoding that provably nearly achieves a time-average expected codeword length close to the directed information lower bound. Our source-coding approach uses a dithered innovations quantizer together wtih zero-delay variable-length lossless source coding (e.g. Shannon-Fano-Elias coding). In particular, using some tools from ergodic theory, we proved that, without loss of generality, the lossless coding can be time-invariant. While quantizer's output is a nonstationary Markov chain on a countable state space, we proved that it has a limiting distribution,  and furthermore that using a lossless encoding optimized for the limiting distribtion for all time does not result in an appreciable increase in the long-term time-average bitrate.  I presented some of this work at MeditCom 2022; you can see the prerecording here.

While in our prior work we showed that a particular quantizer/lossless source codec design could nearly achieve minimum bitrate LQG control, our approach was not strictly practical. In our recent work, we developed a practical source coding algorithm based on the aformentioned nearly-theoretically-optimal approach using ideas from adaptive universal source coding. Our algorithm demonstrated good performance numerically, and we proved some technical lemmas that we hope can be used to gain more theoretical insight into its performance. 

I'll post the code sometime soon; in the meantime, I'd be happy to send it to you via email.

With some collaborators, I helped develop a heuristic scheme for sensor selection in distributed tracking over wireless sensor networks. We assume we are trying to track a Gauss-Markov plant with several sensors that have an "uplink" connection to a fusion center. Each sensor in the network is assigned a communication cost per bit, and, given the sensors' observation models, our algorithm assigns bitrates to each uplink connection (via tuning quantizers) to try and minimize the total network communication cost for a fixed tracking performance. The formulation is versatile; for a wide variety of communication architectures, bitrate can be shown to be a useful surrogate for "real" physical communication costs (e.g. time, bandwidth, transmit power). There is an example of how to assign these costs for a narrowband TDMA system in Section 2.5 of the paper.