3690 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 64, NO. 9, SEPTEMBER 2019 Control Over Gaussian Channels With and Without Source–Channel Separation Anatoly Khina , Member, IEEE, Elias Riedel Garding° , Gustav M. Pettersson , Victoria Kostina , Member, IEEE, and Babak Hassibi, Member, IEEE Abstract—We consider the problem of controlling an un- first, the quantizer needs to be capable of zooming in and stable linear plant with Gaussian disturbances over an ad- out to be able to track unbounded system disturbances, and ditive white Gaussian noise channel with an average trans- second, the channel code must be capable of improving mit power constraint, where the signaling rate of commu- its estimates of the past transmissions exponentially with nication may be different from the sampling rate of the time, a characteristic known as anytime reliability. We im- underlying plant. Such a situation is quite common since plement a separated scheme by leveraging recently devel- sampling is done at a rate that captures the dynamics of oped techniques for control over quantized-feedback chan- the plant and that is often lower than the signaling rate of nels and for efficient decoding of anytime-reliable codes. the communication channel. This rate mismatch offers the We further propose an alternative, namely, to perform ana- opportunity of improving the system performance by us- log joint source–channel coding, by this avoiding the digital ing coding over multiple channel uses to convey a single domain altogether. For the case where the communication control action. In a traditional, separation-based approach signaling rate is twice the sampling rate, we employ ana- to source and channel coding, the analog message is first log linear repetition as well as Shannon–Kotel’nikov maps quantized down to a few bits and then mapped to a channel to show a significant improvement in stability margins and codeword whose length is commensurate with the num- linear-quadratic costs over separation-based schemes. We ber of channel uses per sampled message. Applying the conclude that such analog coding performs better than sep- separation-based approach to control meets its challenges: aration, and can stabilize all moments as well as guarantee almost-sure stability. Manuscript received August 11, 2018; accepted November 3, 2018. Index Terms—Channel coding, combined source– Date of publication April 19, 2019; date of current version August 28, channel coding, Gaussian channel, Lloyd–Max algorithm, 2019. This work was supported by the European Union’s Horizon 2020 networked control systems, quantization, tree codes. research and innovation program under the Marie Skłodowska-Curie Grant 708932. The work of E. Riedel Garding° was supported by the National Science Foundation (NSF) under Grant CCF-1566567 through the SURF program. The work of G. M. Pettersson was supported by The I. INTRODUCTION Boeing Company under the SURF program. The work of V. Kostina was HE current technological era of ubiquitous wireless con- supported in part by the NSF under Grant CCF-1566567. The work of nectivity and the Internet of Things exhibits an ever- B. Hassibi was supported in part by the NSF under Grant CNS-0932428, T Grant CCF-1018927, Grant CCF-1423663, and Grant CCF-1409204, growing demand for new and improved techniques for stabi- by a grant from Qualcomm Inc., by NASA’s Jet Propulsion Laboratory lizing cyber-physical and networked control systems, which as through the President and Director’s Fund, and by King Abdullah Univer- a result have been the subject of intense recent investigations sity of Science and Technology. This paper was presented in part at the [1]–[4]. Unlike traditional control with colocated plant, observer IEEE Conference on Decision and Control, Las Vegas, NV, USA, Dec. 2016. This work was done in part while A. Khina and V. Kostina were and controller, the components of such systems are separated by visiting the Simons Institute for the Theory of Computing. Recommended unreliable communication links. In many of these systems, the by Associate Editor K. Kashima. (Corresponding author: Anatoly Khina.) rate at which the output of the plant is sampled and observed, as A. Khina was with the Department of Electrical Engineering, California well as the rate at which control inputs are applied to the plant, Institute of Technology, Pasadena, CA 91125 USA. He is now with the Department of Electrical Engineering—Systems, Tel Aviv University, Tel is different from the signaling rate with which communication Aviv 6997801, Israel (e-mail:,[email protected]). occurs. The rate at which the plant is sampled and controlled is E. R. Garding° was with the Department of Applied Mathematics and often governed by how fast the dynamics of the plant is, whereas Theoretical Physics, Centre for Mathematical Sciences, University of the signaling rate of the communication depends on the band- Cambridge, Cambridge CB3 0WA, U.K. He is now with the School of width available, the noise levels, etc. As a result, there is no Engineering Sciences, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden (e-mail:,[email protected]). inherent reason why these two rates should be related and, in G. M. Pettersson is with the School of Engineering Sciences, KTH fact, the communication signaling rate is almost always higher Royal Institute of Technology, SE-100 44 Stockholm, Sweden (e-mail:, than the control sampling rate. [email protected]). This latest fact clearly gives us the opportunity to improve the V. Kostina is with the Department of Electrical Engineering, Cal- ifornia Institute of Technology, Pasadena, CA 91125 USA (e-mail:, performance of the system by conveying the information about [email protected]). each sampled output of the plant, and/or each control signal, B. Hassibi is with the Department of Electrical Engineering, California through multiple uses of the communication channel. Institute of Technology, Pasadena, CA 91125 USA (e-mail:, hassibi@ The standard information-theoretic approach suggests quan- caltech.edu). tizing the analog messages (the sampled output or state signal) Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. and then protecting the quantized bits with a channel error- Digital Object Identifier 10.1109/TAC.2019.2912255 correcting code whose block length is commensurate with the 0018-9286 © 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information. Authorized licensed use limited to: CALIFORNIA INSTITUTE OF TECHNOLOGY. Downloaded on February 16,2020 at 05:48:56 UTC from IEEE Xplore. Restrictions apply. KHINA et al.: CONTROL OVER GAUSSIAN CHANNELS WITH AND WITHOUT SOURCE–CHANNEL SEPARATION 3691 number of channel uses available per sample. This approach the digital domain altogether, and can therefore be viewed as a relies on the source–channel separation principle, which prof- simple instance of joint source–channel coding (JSCC) [24]. fers that quantization of the messages and channel coding of the Surprisingly, in the Gaussian rate-matched case, in which one quantized bits can be done independently of one another. additive white Gaussian noise (AWGN) channel use is avail- Nonetheless, while source–channel separation-based able per one white Gaussian source sample, a simple amplifier schemes become optimal in communication systems where achieves the Shannon limit with zero delay [25], [26]. The op- large blocks of the message and the channel code are processed timality of linear schemes extends further to the case where together (necessitating noncausal knowledge of all the message KC > 1 uses of an AWGN channel with perfect instantaneous signals and entailing large delays)—a celebrated result [5], [6, feedback are available per one white Gaussian source sam- Ch. 3.9]—it is not true for control systems that require real-time ple [26]–[30], the reason being that a Gaussian source is prob- (low-delay) communication of causally available messages. abilistically matched to a Gaussian channel [31]—uncommon Furthermore, since any error made in the past is magnified coincidence. Tatikonda and Mitter [32] exploited a special prop- in each subsequent time step due to the unstable nature of erty of the erasure channel with feedback, in which a retransmis- the plant, the source–channel separation principle requires a sion scheme attains its capacity without delay. A related example stronger notion of error protection, termed anytime reliability is control over a packet-drop channel, considered by Sinopoli by Sahai and Mitter [7]. Anytime reliability guarantees that the et al. [33]. There, a simple retransmission scheme attains the error probability of causally encoded quantized (“information”) optimum, as long as the packet drop probability is not too high. bits decays faster than the inflation factor at each step. Sahai Coding of Gauss–Markov sources over a packet erasure channel and Mitter [7] further observed that anytime-reliable codes with feedback is studied in [34]. have a natural tree code structure reminiscent of the codes de- JSCC in the absence of probabilistic matching is challenging. veloped by Schulman [8] for the related problem of interactive In the Gaussian rate-mismatched case with no communication communication. feedback, in which KC > 1 AWGN channel uses are available Sukhavasi and Hassibi [9] showed that anytime reliability can per one source sample, repetitive transmission of the source sam- be guaranteed with high probability by concentrating on the fam- ple is suboptimal. Non-linear mappings are known to achieve ily of linear time-invariant (LTI) codes and choosing their co- better performance, as noted originally by Shannon [35] and Ko- efficients at random. Unfortunately, maximum-likelihood (ML) tel’nikov [36], and in the context of control—in [37]–[39] (and (optimum) decoding of tree codes is infeasible.1 To overcome is akin to Witsenhausen’s celebrated counterexample [40]).