Mohammad Javad Khojasteh, Massimo Franceschetti, Gireeja …mjkhojas/Papers/timing_journal.pdf · 2020-02-21 · Mohammad Javad Khojasteh, Massimo Franceschetti, Gireeja Ranade Abstract—We

1

Stabilizing a linear system using phone calls:when time is information

Mohammad Javad Khojasteh, Massimo Franceschetti, Gireeja Ranade

Abstract—We consider the problem of stabilizing an undis-turbed, scalar, linear system over a “timing” channel, namelya channel where information is communicated through thetimestamps of the transmitted symbols. Each symbol transmittedfrom a sensor to a controller in a closed-loop system is receivedsubject to some to random delay. The sensor can encode messagesin the waiting times between successive transmissions and thecontroller must decode them from the inter-reception times ofsuccessive symbols. This set-up is analogous to a telephonesystem where a transmitter signals a phone call to a receiverthrough a “ring” and, after the random delay required toestablish the connection; the receiver is aware of the “ring” beingreceived. Since there is no data payload exchange between thesensor and the controller, this set-up provides an abstraction forperforming event-triggering control with zero-payload rate. Weshow the following requirement for stabilization: for the stateof the system to converge to zero in probability, the timingcapacity of the channel should be at least as large as theentropy rate of the system. Conversely, in the case the symboldelays are exponentially distributed, we show a tight sufficientcondition using a coding strategy that refines the estimate ofthe decoded message every time a new symbol is received.Our results generalize previous zero-payload event-triggeringcontrol strategies, revealing a fundamental limit in using timinginformation for stabilization, independent of any transmissionstrategy.

Index Terms—Timing channel, control with communicationconstraints, event-triggered control.

I. INTRODUCTION

A networked control system with a feedback loop over acommunication channel provides a first-order approximationof a cyber-physical system (CPS), where the interplay betweenthe communication and control aspects of the system leads tonew and unexpected analysis and design challenges [3], [4].In this setting, data-rate theorems quantify the impact of thecommunication channel on the ability to stabilize the system.Roughly speaking, these theorems state that stabilization re-quires a communication rate in the feedback loop at least aslarge as the intrinsic entropy rate of the system, expressed bythe sum of the logarithms of its unstable eigenvalues [5]–[12].

We consider a specific communication channel in the loop— a timing channel. Here, information is communicated

The material in this paper was presented in part at the 18th EuropeanControl Conference (ECC), 2019 [1], and at 57th IEEE Conference on onDecision and Control, 2018 [2]. This research was partially supported byNSF awards CNS-1446891 and ECCS-1917177.

M. J. Khojasteh is with the Department of Electrical Engineering, Cali-fornia Institute of Technology. Some of this work was performed while atUniversity of California, San Diego, and also visiting Microsoft Research.([email protected])

M. Franceschetti is with the Department of Electrical and ComputerEngineering, University of California, San Diego. ([email protected])

G. Ranade is with EECS department at the University of California,Berkeley. Some of this work was performed while at Microsoft Re-search, and also at the Simons Institute for the Theory of Computing.([email protected])

through the timestamps of the symbols transmitted over thechannel; the time is carrying the message. This formulation ismotivated by recent works in event-triggering control, showingthat the timing of the triggering events carries information thatcan be used for stabilization [13]–[22]. By encoding informa-tion in timing, stabilization can be achieved by transmittingadditional data at a rate arbitrarily close to zero. However, inthese works, the timing information was not explicitly quanti-fied, and the analysis was limited to specific event-triggeringstrategies. In this paper, our goal is to determine the value ofa timestamp from an information-theoretic perspective, whenthis timestamp is used for control. We are further motivated bythe results on the impact of multiplicative noise in control [23],[24], since timing uncertainty can lead to multiplicative noisein systems and thus can serve as an information bottleneck.

To illustrate the proof of concept that timing carries in-formation useful for control, we consider the simple caseof stabilization of a scalar, undisturbed, continuous-time, un-stable, linear system over a timing channel and rely on theinformation-theoretic notion of timing capacity of the channel,namely the amount of information that can be encoded usingtime stamps [25]–[39]. In this setting, the sensor can commu-nicate with the controller by choosing the timestamps at whichsymbols from a unitary alphabet are transmitted. The controllerreceives each transmitted symbol after a random delay is addedto the timestamp. We show the following data-rate theorem.For the state to converge to zero in probability, the timingcapacity of the channel should be at least as large as theentropy rate of the system. Conversely, in the case the randomdelays are exponentially distributed, we show that when thestrict inequality is satisfied, we can drive the state to zero inprobability by using a decoder that refines its estimate of thetransmitted message every time a new symbol is received [40].We also derive analogous necessary and sufficient conditionsfor the problem of estimating the state of the system with anerror that tends to zero in probability.

The books [5], [6], [50] and the surveys [7], [8] providedetailed discussions of data-rate theorems and related resultsthat heavily inspire this work. A portion of the literaturestudied stabilization over “bit-pipe channels,” where a rate-limited, possibly time-varying and erasure-prone communica-tion channel is present in the feedback loop [41], [46]–[48],[51]. For more general noisy channels, Tatikonda and Mitter[42] and Matveev and Savkin [43] showed that the state ofundisturbed linear systems can be forced to converge to zeroalmost surely (a.s.) if and only if the Shannon capacity ofthe channel is larger than the entropy rate of the system.In the presence of disturbances, in order to keep the statebounded a.s., a more stringent condition is required, namelythe zero-error capacity of the channel must be larger thanthe entropy rate of the system [44]. Nair derived a similar

2

TABLE I: Capacity notions used to derive data-rate theorems in the literature under different notions of stability, channel types,and system disturbances.

Work Disturbance Channel Stability condition Capacity[41] NO Bit-pipe |X(t)| → 0 a.s. Shannon[42], [43] NO DMC |X(t)| → 0 a.s. Shannon[44] bounded DMC P(supt |X(t)| <∞) = 1 Zero-Error[6, Ch. 8] bounded DMC P(supt |X(t)| < Kε) > 1− ε Shannon[45] bounded DMC supt E(|X(t)|m) <∞ Anytime[46] unbounded Bit-Pipe supt E(|X(t)|2) <∞ Shannon[47]–[49] unbounded Var. Bit-pipe supt E(|X(t)|m) <∞ AnytimeThis paper NO Timing |X(t)| P→ 0 Timing

information-theoretic result in a non-stochastic setting [52].Sahai and Mitter [45] considered moment-stabilization overnoisy channels and in the presence of system disturbances ofbounded support, and provided a data-rate theorem in termsof the anytime capacity of the channel. They showed thatto keep the mth moment of the state bounded, the anytimecapacity of order m should be larger than the entropy rate ofthe system. The anytime capacity has been further investigatedin [49], [53]–[55]. Matveev and Savkin [6, Chapter 8] havealso introduced a weaker notion of stability in probability,requiring the state to be bounded with probability (1− ε) bya constant that diverges as ε → 0, and showed that in thiscase it is possible to stabilize linear systems with boundeddisturbances over noisy channels provided that the Shannoncapacity of the channel is larger than the entropy rate of thesystem. The various results, along with our contribution, aresummarized in Table I. The main point that can be drawnfrom all of these results is that the relevant capacity notion forstabilization over a communication channel critically dependson the notion of stability and on the system’s model.

From the system’s perspective, our set-up is closest to theone in [41]–[43], as there are no disturbances and the objectiveis to drive the state to zero. Our convergence in probabilityprovides a stronger necessary condition for stabilization, buta weaker sufficient condition than the one in these works.We also point out that our notion of stability is considerablystronger than the notion of probabilistic stability proposedin [6, Chapter 8]. Some additional works considered nonlinearplants without disturbances [56]–[58], and switched linearsystems [59], [60] where communication between the sensorand the controller occurs over a bit-pipe communication chan-nel. The recent work in [61] studies estimation of nonlinearsystems over noisy communication channels and the workin [62] investigates the trade-offs between the communicationchannel rate and the cost of the linear quadratic regulator forlinear plants.

Parallel work in control theory has investigated the possibil-ity of stabilizing linear systems using timing information. Oneprimary focus of the emerging paradigm of event-triggeredcontrol [63]–[75] has been on minimizing the number oftransmissions while simultaneously ensuring the control ob-jective [16], [76], [77]. Rather than performing periodic com-munication between the system and the controller, in event-

triggered control communication occurs only as needed, inan opportunistic manner. In this setting, the timing of thetriggering events can carry useful information about the stateof the system, that can be used for stabilization [13]–[22].In this context, it has been shown that the amount of timinginformation is sensitive to the delay in the communicationchannel. While for small delay stabilization can be achievedusing only timing information and transmitting data payload(i.e. physical data) at a rate arbitrarily close to zero, forlarge values of the delay this is not the case, and the datapayload rate must be increased [15], [21]. In this paper, weextend these results from an information-theoretic perspective,as we explicitly quantify the value of the timing information,independent of any transmission strategy. To quantify theamount of timing information alone, we restrict to transmittingsymbols from a unitary alphabet, i.e. at zero data payload rate.Research directions left open for future investigation includethe study of “mixed” strategies, using both timing informationand physical data transmitted over a larger alphabet, as wellas generalizations to vector systems and the study of systemswith disturbances. In the latter case, it is likely that the usageof stronger notions of capacity, or weaker notions of stability,will be necessary.

The rest of the paper is organized as follows. Section IIintroduces the system and channels models. The main re-sults are presented in Section III. Section IV considers theestimation problem, and Section V considers the stabilizationproblem. Section VI provides a comparison with related work,and Section VII presents a numerical example. Conclusions aredrawn in Section VIII.

A. NotationLet Xn = (X1, · · · , Xn) denote a vector of random

variables and let xn = (x1, · · · , xn) denote its realization. IfX1, · · · , Xn are independent and identically distributed (i.i.d)random variables, then we refer to a generic Xi ∈ Xn byX and skip the subscript i. We use log and ln to denote thelogarithms base 2 and base e respectively. We use H(X) todenote the Shannon entropy of a discrete random variable Xand h(X) to denote the differential entropy of a continuousrandom variable X . Further, we use I(X;Y ) to indicate themutual information between random variables X and Y . Wewrite Xn

P−→ X if Xn converges in probability to X . Similarly,

3

Fig. 1: Model of a networked control system where the feedback loop isclosed over a timing channel.

we write Xna.s.−−→ X if Xn converges almost surely to X . For

any set X and any n ∈ N we let

πn : X N →X n (1)

be the truncation operator, namely the projection of a sequencein X N into its first n symbols.

II. SYSTEM AND CHANNEL MODEL

We consider the networked control system depicted inFig. 1. The system dynamics are described by a scalar,continuous-time, noiseless, linear time-invariant (LTI) system

X(t) = aX(t) + bU(t), (2)

where X(t) ∈ R and U(t) ∈ R are the system state and thecontrol input respectively. The constants a, b ∈ R are suchthat a > 0 and b 6= 0. The initial state X(0) is random andis drawn from a distribution of bounded differential entropyand bounded support, namely h(X(0)) <∞ and |X(0)| <L, where L is known to both the sensor and the controller.Conditioned on the realization of X(0), the system evolutionis deterministic. Both controller and sensor have knowledgeof the system dynamics in (2). We assume the sensor canmeasure the state of the system with infinite precision, andthe controller can apply the control input to the system withinfinite precision and with zero delay.

The sensor is connected to the controller through a timingchannel (the telephone signaling channel defined in [25]). Theoperation of this channel is analogous to that of a telephonesystem where a transmitter signals a phone call to the receiverthrough a “ring” and, after a random time required to establishthe connection, is aware of the “ring” being received. Com-munication between transmitter and receiver can then occurwithout any vocal exchange, but by encoding messages in the“waiting times” between consecutive calls.

A. The channelWe model the channel as carrying symbols ♠ from a unitary

alphabet, and each transmission is received after a randomdelay. Every time a symbol is received, the sender is notifiedof the reception by an instantaneous acknowledgment. Thechannel is initialized with a ♠ received at time t = 0. Afterreceiving the acknowledgment for the ith ♠, the sender waitsfor Wi+1 seconds and then transmits the next ♠. Transmittedsymbols are subject to i.i.d. random delays {Si}. Letting Di

be the inter-reception time between two consecutive symbols,we have

Di = Wi + Si. (3)

It follows that the reception time of the nth symbol is

Tn =

n∑i=1

Di. (4)

Fig. 2 provides an example of the timing channel in action.

B. Source-channel encoderThe sensor in Fig. 1 can act as a source and channel

encoder. Based on the knowledge of the initial conditionX(0), system dynamics (2), and L, it can select the waitingtimes {Wi} between the reception and the transmission ofconsecutive symbols. As in [25], [28] we assume that thecausal acknowledgments received by the sensor every time a ♠is delivered to the controller are not used to choose the waitingtimes, but only to avoid queuing, ensuring that every symbolis sent after the previous one has been received. This is a usualpractice in TCP-based networks, where packet deliveries areacknowledged via a feedback link [78]–[82]. For networkedcontrol systems, this causal acknowledgment can be obtainedwithout assuming an additional communication channel in thefeedback loop. The controller can signal the acknowledgmentto the sensor by applying a control input to the system thatexcites a specific frequency of the state each time a symbolhas been received. This strategy is known in the literature as“acknowledgment through the control input” [6], [18], [42],[45].

C. Anytime decoderAt any time t ≥ 0, the controller in Fig. 1 can use the inter-

reception times of all the symbols received up to time t, alongwith the knowledge of L and of the system dynamics (2) tocompute the control input U(t) and apply it to the system. Thecontrol input can be refined over time, as the estimate of thesource can be decoded with increasing accuracy when moreand more symbols are received. The objective is to designan encoding and decoding strategy to stabilize the system bydriving the state to zero in probability, i.e. we want |X(t)| P−→0 as t→∞.

Although the computational complexity of differentencoding-decoding schemes is a key practical issue, in this pa-per we are concerned with the existence of schemes satisfyingour objective, rather than with their practical implementation.

D. Capacity of the channelIn the channel coding process, we assume the use of random

codebooks, namely the waiting times {Wi} used to encode anygiven message are generated at random in an i.i.d. fashion,and are also independent of the random delays {Si}. Thisassumption is made for analytical convenience, and it doesnot change the capacity of the timing channel. The followingdefinitions are derived from [25], incorporating our randomcoding assumption.

4

Fig. 2: The timing channel. Subscripts s and r are used to denote sent and received symbols, respectively.

Definition 1: A (n,M, T, δ)-i.i.d.-timing code for the tele-phone signaling channel consists of a codebook of M code-words {(wi,m, i = 1, . . . , n), m = 1 . . .M}, where thesymbols in each codeword are picked i.i.d. from a commondistribution as well as a decoder, which upon observation of(D1, . . . , Dn) selects the correct transmitted codeword withprobability at least 1− δ. Moreover, the codebook is such thatthe expected random arrival time of the nth symbol is at mostT , namely

E (Tn) ≤ T. (5)

Definition 2: The rate of an (n,M, T, δ)-i.i.d.-timing codeis

R = (logM)/T. (6)

Definition 3: The timing capacity C of the telephone sig-naling channel is the supremum of the achievable rates, namelythe largest R such that for every γ > 0 there exists a sequenceof (n,Mn, Tn, δTn)-i.i.d.-timing codes that satisfy

logMn

Tn> R− γ, (7)

and δTn → 0 as n→∞.The following result [25, Theorem 8] applies to our random

coding set-up, since the capacity in [25] is achieved by randomcodes.

Theorem 1 (Anantharam and Verdú): The timing capacityof the telephone signaling channel is given by

C = supχ>0

supW≥0

E(W )≤χ

I(W ;W + S)

E(S) + χ, (8)

and if S is exponentially distributed then

C =1

eE(S)[nats/sec]. (9)

III. MAIN RESULTS

A. Necessary conditionTo derive a necessary condition for the stabilization of the

feedback loop system depicted in Fig. 1, we first consider theproblem of estimating the state in open-loop over the timingchannel. We show that for the estimation error to tend to zeroin probability, the timing capacity must be greater than theentropy rate of the system. This result holds for any sourceand channel coding strategy adopted by the sensor, and for anystrategy adopted by the controller to generate the control input.Our proof employs a rate-distortion argument to compute alower bound on the minimum number of bits required torepresent the state up to any given accuracy, and this leads to

a corresponding lower bound on the required timing capacityof the channel. We then show that the same bound holds forstabilization, since in order to have |X(t)| P−→ 0 as t → ∞in closed-loop, the estimation error in open-loop must tend tozero in probability.

B. Sufficient conditionTo derive a sufficient condition for stabilization, we first

consider the problem of estimating the state in open-loop overthe timing channel. We provide an explicit source-channelcoding scheme which guarantees that if the timing capacity islarger than the entropy rate of the system, then the estimationerror tends to zero in probability. We then show that thiscondition is also sufficient to construct a control scheme suchthat |X(t)| P−→ 0 as t→∞. The main idea behind our strategyis based on the realization that in the absence of disturbancesall that is needed to drive the state to zero is communicatethe initial condition X(0) to the controller with accuracy thatincreases exponentially over time. Once this is achieved, thecontroller can estimate the state X(t) with increasing accuracyover time, and continuously apply an input that drives thestate to zero. This idea has been exploited before in theliterature [41], [42], and the problem is related to the anytimereliable transmission of a real-valued variable over a digitalchannel [40]. Here, we cast this problem in the framework ofthe timing channel. A main difficulty in our case is to ensurethat we can drive the system’s state to zero in probabilitydespite the unbounded random delays occurring in the timingchannel.

In the source coding process, we quantize the interval[−L,L] uniformly using a tree-structured quantizer [83]. Wethen map the obtained source code into a channel code suitablefor transmission over the timing channel, using the capacity-achieving random codebook of [25]. Given X(0), the encoderpicks a codeword from an arbitrarily large codebook and startstransmitting the real numbers of the codeword one by one,where each real number corresponds to a holding time, andproceeds in this way forever. Every time a sufficiently largenumber of symbols are received, we use a maximum likelihooddecoder to successively refine the controller’s estimate ofX(0). Namely, the controller re-estimates X(0) based onthe new inter-reception times and all previous inter-receptiontimes, and uses it to compute the new state estimate ofX(t) and control input U(t). We show that when the sensorquantizes X(0) at sufficiently high resolution, and when thetiming capacity is larger than the entropy rate of the system,the controller can construct a sufficiently accurate estimate ofX(t) and compute U(t) such that |X(t)| P−→ 0.

5

IV. THE ESTIMATION PROBLEM

We start considering the estimation problem depicted inFig. 3. By letting b = 0 in (2) we obtain the open-loop equation

Xe(t) = aXe(t). (10)

Our first objective is to obtain an estimate of the state Xe(tn),given the reception of n symbols over the telephone signalingchannel, such that |Xe(tn)− Xe(tn)| P→ 0 as n→∞, at anysequence of estimation times tn such that

1 < limn→∞

tnE(Tn)

≤ Γ. (11)

In practice, the condition (11) ensures that as n → ∞the estimation error is evaluated after n symbols have beenreceived, see Fig. 4. As before, we assume that the encoder hascausal knowledge of the reception times via acknowledgmentsthrough the system as depicted in Fig. 3.

A. Necessary conditionThe next theorem provides a necessary rate for the state

estimation error to tend to zero in probability.Theorem 2: Consider the estimation problem depicted in

Fig. 3 with system dynamics (10). Consider transmit-ting n symbols over the telephone signaling channel (3),and a sequence of estimation times satisfying (11). If|Xe(tn)− Xe(tn)| P→ 0, then

I(W ;W + S) ≥ a Γ E(W + S) [nats], (12)

and consequently

C ≥ Γa [nats/sec]. (13)

The proof of Theorem 2 is given in Appendix A.

Remark 1: The entropy-rate of our system is anats/time [56], [84]–[87]. This represents the amount ofuncertainty per unit time generated by the system in openloop. Letting Γ → 1, (12) recovers a typical scenario indata-rate theorems: to drive the error to zero the mutualinformation between an encoding symbol W and its receivednoisy version W + S should be larger than the average“information growth” of the state during the inter-receptioninterval D, which is given by

E(aD) = a E(W + S). (14)

On the other hand, for any fixed Γ > 1 our result showsthat we must pay a penalty of a factor of Γ in the case thereis a time lag between the reception time Tn of the last symboland the estimation time tn, see Fig. 4. Finally, the case Γ→∞ requires transmission of a codeword carrying an infiniteamount of information over a channel of infinite capacity, thusrevealing the initial state of the system with infinite precision.This case is equivalent to transmitting a single real numberover a channel without error, or a single symbol from a unitaryalphabet with zero delay. •

B. Sufficient conditionThe next theorem provides a sufficient condition for con-

vergence of the state estimation error to zero in probability

along any sequence of estimation times tn satisfying (11), inthe case of exponentially distributed delays.

Theorem 3: Consider the estimation problem depicted inFig. 3 with system dynamics (10). Consider transmitting nsymbols over the telephone signaling channel (3). Assume{Si} are drawn i.i.d. from exponential distribution with meanE(S). If the capacity of the timing channel is at least

C > aΓ [nats/sec], (15)

then for any sequence of times {tn} that satisfies (11), we cancompute an estimate Xe(tn) such that as n→∞, we have

|Xe(tn)− Xe(tn)| P→ 0. (16)

The proof of Theorem 3 is given in Appendix A. The resultis strengthened in the next section (see Corollary 1), showingthat C > a is also sufficient to drive the state estimation errorto zero in probability for all t→∞.

V. THE STABILIZATION PROBLEM

A. Necessary conditionWe now turn to consider the stabilization problem. Our first

lemma states that if in closed-loop we are able to drive thestate to zero in probability, then in open-loop we are also ableto estimate the state with vanishing error in probability.

Lemma 1: Consider stabilization of the closed-loop sys-tem (2) and estimation of the open-loop system (10) overthe timing channel (3). If there exists a controller such that|X(t)| P→ 0 as t → ∞, in closed-loop, then there exists anestimator such that |Xe(t)− Xe(t)|

P→ 0 as t→∞, in open-loop.

Proof: From (2), we have in closed loop

X(t) = eatX(0) + ζ(t), (17)

ζ(t) = eat∫ t

0

e−a%bU(%)d%. (18)

It follows that if

limt→∞

P (|X(t)| ≤ ε) = 1, (19)

then we also have

limt→∞

P(∣∣eatX(0) + ζ(t)

∣∣ ≤ ε) = 1. (20)

On the other hand, from (10) we have in open loop

Xe(t) = eatX(0), (21)

and we can choose Xe(t) = −ζ(t) so that

|Xe(t)− Xe(t)| = |eatX(0) + ζ(t)| P→ 0, (22)

where the last step follows from (20).The next theorem provides a necessary rate for the stabi-

lization problem.Theorem 4: Consider the stabilization of the closed-loop

system (2). If |X(t)| P→ 0 as t→∞, then

I(W ;W + S) ≥ a E(W + S) [nats], (23)

6

Fig. 3: The estimation problem.

Fig. 4: Codeword transmission and state estimation for different estimationtime sequences {tn}.

and consequently

C ≥ a [nats/sec]. (24)

Proof: By Lemma 1 we have that if |X(t)| P→ 0, then|Xe(t)− Xe(t)|

P→ 0 for all t → ∞, and in particular alonga sequence {tn} satisfying (11). The result now follows fromTheorem 2 letting Γ→ 1.

B. Sufficient conditionOur next lemma strengthens our estimation results, stating

that it is enough for the state estimation error to convergeto zero in probability as n → ∞ along any sequence ofestimation times {tn} satisfying (11), to ensure it convergesto zero for all t→∞.

Lemma 2: Consider estimation of the system (10) over thetiming channel (3). If there exists Γ0 > 1 such that alongthe sequence of estimation times tn = Γ0E(Tn) we have|Xe(tn)− Xe(tn)| P→ 0 as n → ∞, then for all t → ∞ wealso have |Xe(t)− Xe(t)|

P→ 0.Proof: We have that for tn = Γ0E(Tn) and for all ε′ > 0,

and φ > 0, there exist nφ such that for all n ≥ nφ

P(|Xe(tn)− Xe(tn)| > ε′

)≤ φ. (25)

Let tnφ = Γ0E(Tnφ) be the time at which we estimate the statefor the nφth time. We want to show that for all t ∈ [tnφ , tnφ+1]and ε > 0, we also have

P(|Xe(t)− Xe(t)| > ε

)≤ φ. (26)

Consider the random time Tnφ at which ♠ is received for the

nφth time. We have

tnφ+1 − tnφ = Γ0 E(Tnφ+1)− Γ0 E(Tnφ)

= (nφ + 1)Γ0 E(D)− nφΓ0 E(D)

= Γ0 E(D). (27)

For all t ∈ [tnφ , tnφ+1], from the open-loop equation (10) wehave

Xe(t) = ea(t−tnφ )Xe(tnφ). (28)

We then let

Xe(t) = ea(t−tnφ )Xe(tnφ). (29)

Combining (28) and (29) and using (27), we obtain that forall t ∈ [tnφ , tnφ+1]

|Xe(t)− Xe(t)| ≤ eaΓ0 E(D)|Xe(tnφ)− Xe(tnφ)|. (30)

From which it follows that

P(|Xe(t)− Xe(t)| > ε′eaΓ0 E(D)

)≤ P

(|Xe(tnφ)− Xe(tnφ)| > ε′

). (31)

Since (25) holds for all n ≥ nφ, we also have

P(|Xe(tnφ)− Xe(tnφ)| ≥ ε′

)≤ φ. (32)

We can now let ε′ < εe−aΓ0 E(D) and the result follows.

Lemma 2 yields the following corollary, which is an imme-diate extension of Theorem 3.

Corollary 1: Consider the estimation problem depicted inFig. 3 with system dynamics (10). Consider transmitting nsymbols over the telephone signaling channel (3). Assume{Si} are drawn i.i.d. from exponential distribution with meanE(S). If the capacity of the timing channel is at least C > aΓ,then we have |Xe(t)− Xe(t)|

P→ 0 as t→∞.Proof: The result follows by noticing that if C > a

then there exists a Γ0 > 1 such that C > aΓ0, andhence by Theorem 3 along the sequence of estimation timestn = Γ0E(Tn) we have |Xe(tn)− Xe(tn)| P→ 0 as n → ∞.Then, by Lemma 2 we also have |Xe(t)− Xe(t)|

P→ 0 ast→∞.

The next key lemma states that if we are able to estimatethe state with vanishing error in probability, then we are alsoable to drive the state to zero in probability.

Lemma 3: Consider stabilization of the closed-loop sys-tem (2) and estimation of the open-loop system (10) overthe timing channel (3). If there exists an estimator such that|Xe(t)− Xe(t)|

P→ 0 as t → ∞, in open-loop, then there

7

exists a controller such that |X(t)| P→ 0 as t→∞, in closed-loop.

Proof: We start by showing that if there exists an open-loop estimator such that |Xe(t)− Xe(t)|

P→ 0 as t → ∞,then there also exists a closed-loop estimator such that|X(t)− X(t)| P→ 0 as t → ∞. We construct the closed-loopestimator based on the open-loop estimator as follows. Thesensor in closed-loop runs a copy of the open-loop system byconstructing the virtual open-loop dynamic

Xe(t) = X(0)eat. (33)

Using the open-loop estimator, for all t > 0 the con-troller acquires the open-loop estimate Xe(t) such that|Xe(t)− Xe(t)|

P→ 0. It then uses this estimate to constructthe closed-loop estimate

X(t) = Xe(t) + eat∫ t

0

e−a%bU(%)d%. (34)

Since from (2) the true state in closed loop is

X(t) = X(0)eat + eat∫ t

0

e−a%bU(%)d%, (35)

it follows by combining (33), (34) and (35) that

|X(t)− X(t)| = |Xe(t)− Xe(t)|P→ 0. (36)

What remains to be proven is that if |X(t)− X(t)| P→ 0,then there exists a controller such that |X(t)| P→ 0.

Let b > 0 and choose k so large that a − bk < 0. LetU(t) = −kX(t). From (2), we have

X(t) = (a− bk)X(t) + bk[X(t)− X(t)]. (37)

By solving (37) and using the triangle inequality, we get

|X(t)| ≤|e(a−bk)tX(0)|+∣∣∣∣∫ t

0

e(t−%)(a−bk)bk(X(%)− X(%))d%

∣∣∣∣ . (38)

Since |X(0)| < L and a− bk < 0, the first term in (38) tendsto zero as t→∞. Namely, for any ε > 0 there exists a numberNε such that for all t ≥ Nε, we have

|e(a−bk)tX(0)| ≤ ε. (39)

Since by (36) we have that |X(t)− X(t)| P→ 0, we also havethat for any ε, δ > 0 there exist a number N ′ε such that for allt ≥ N ′ε, we have

P(|X(t)− X(t)| ≤ ε

)≥ 1− δ. (40)

It now follows from (38) that for all t ≥ max{Nε, N ′ε} thefollowing inequality holds with probability at least (1− δ)

|X(t)| ≤ ε+ bket(a−bk)

∫ N ′ε

0

e−%(a−bk)|X(%)− X(%)|d%

+ εbket(a−bk)

∫ t

N ′ε

e−%(a−bk)d%. (41)

Since both sensor and controller are aware that |X(0)| < L, by(33) we have that for all t ≥ 0 the open-loop estimate acquired

by the controller satisfies Xe(t) ∈ [−Leat, Leat]. By (36)the closed-loop estimation error is the same as the open-loopestimation error, and we then have that for all % ∈ [0, N ′ε]

|X(%)− X(%)| = |Xe(%)− Xe(%)| ≤ 2LeaN′ε . (42)

Substituting (42) into (41), we obtain that with probability atleast (1− δ)

|X(t)| ≤ε+ 2Lbke[t(a−bk)+aN ′ε]e−N

′ε(a−bk) − 1

−(a− bk)

+ εbket(a−bk) e−t(a−bk) − e−N ′ε(a−bk)

−(a− bk). (43)

By first letting ε be sufficiently close to zero, and then lettingt be sufficiently large, we can make the right-hand side of (43)arbitrarily small, and the result follows.

The next theorem combines the results above, providing asufficient condition for convergence of the state to zero inprobability in the case of exponentially distributed delays.

Theorem 5: Consider the stabilization of the system (2).Assume {Si} are drawn i.i.d. from an exponential distributionwith mean E(S). If the capacity of the timing channel is atleast

C > a [nats/sec], (44)

then |X(t)| P→ 0 as t→∞.Proof: The result follows by combining Corollary 1 and

Lemma 3.

VI. COMPARISON WITH PREVIOUS WORK

A. Comparison with stabilization over an erasure channelIn [42] the problem of stabilization of the discrete-time

version of the system in (2) over an erasure channel has beenconsidered. In this discrete model, at each time step of thesystem’s evolution the sensor transmits I bits to the controllerand these bits are successfully delivered with probability 1−µ,or they are dropped with probability µ, in an independentfashion. It is shown that a necessary condition for X(k)

a.s−−→ 0is that the capacity of this I-bit erasure channel is

(1− µ)I ≥ log a [bits/sec]. (45)

Since almost sure convergence implies convergence in prob-ability, by Theorem 4 we have that the following necessarycondition holds in our setting for X(t)

a.s.−−→ 0:

I(W ;W + S)

E(W + S)≥ a [nats/sec]. (46)

We now compare (45) and (46). The rate of expansion ofthe state space of the continuous system in open loop is anats per unit time, while for the discrete system is log a bitsper unit time. Accordingly, (45) and (46) are parallel to eachother: in the case of (46) the controller must receive at leastaE(W + S) nats representing the initial state during a timeinterval of average length E(W + S). In the case of (45) thecontroller must receive at least log a/(1−µ) bits representingthe initial state over a time interval whose average lengthcorresponds to the average number of trials before the first

8

successful reception

(1− µ)

∞∑k=0

(k + 1)µk =1

1− µ. (47)

B. Comparison with event triggering strategiesThe works [13]–[22] use event-triggering strategies that

exploit timing information for stabilization over a digitalcommunication channel. These strategies encode informationover time in a specific state-dependent fashion and use acombination of timing information and data payload to conveyinformation used for stabilization. Our framework, by consid-ering the transmission of symbols from a unitary alphabet, usesonly timing information for stabilization. In Theorem 4 weprovide a fundamental limit on the rate at which informationcan be encoded in time, independent of any transmissionstrategy. Theorem 5 then shows that this limit can be achieved,in the case of exponentially distributed delays.

The work [14] shows that using event triggering it ispossible to achieve stabilization with any positive transmissionrate over a zero-delay digital communication channel. Indeed,for channels without delay achieving stabilization at zero rateis easy. One could for example transmit a single symbol at atime equal to any bijective mapping of x(0) into a point of thenon-negative reals. For example, we could transmit ♠ at timet = tan−1(x(0)) for t ∈ [0, π]. The reception of the symbolwould reveal the initial state exactly, and the system could bestabilized.

The work in [15] shows that when the delay is positive, butsufficiently small, a triggering policy can still achieve stabi-lization with any positive transmission rate. However, as thedelay increases past a critical threshold, the timing informationbecomes so much out-of-date that the transmission rate mustbegin to increase. In our case, since the capacity of our timingchannel depends on the distribution of the delay, we mayalso expect that a large value of the capacity, correspondingto a small average delay, would allow for stabilization tooccur using only timing information. Indeed, when delays aredistributed exponentially, from (9) and Theorems 4 and 5 itfollows that as longs as the expected value of delay is

E(S) <1

ea, (48)

it is possible to stabilize the system by using only timinginformation. On the other hand, the system is not stabilizableusing only timing information if the expected value of thedelay becomes larger than (ea)−1.

VII. NUMERICAL EXAMPLE

We now present a numerical simulation of stabilizationover the telephone signaling channel. While our analysis isfor continuous-time systems, the simulation is performed indiscrete time, considering the system

X[m] = aX[m] + U [m], for m ∈ N, (49)

where a > 1 so that the system is unstable.In this case, assuming i.i.d. geometrically distributed delays

{Si}, the sufficient condition for stabilization becomes

C > log a [nats/sec], (50)

where C is the timing capacity of the discrete telephonesignaling channel [26]. The timing capacity is achieved inthis case using i.i.d. waiting times {Wi} that are distributedaccording to a mixture of a geometric and a delta distribution.This results in {Di} also being i.i.d. geometric [26], [28].

Assuming that a decoding operation occurs at time musing all km symbols received up to this time, and followingthe source-channel coding scheme described in the proof ofTheorem 3, the controller decodes an estimate Xm[0] of theinitial state and estimates the current state as

X[m] = amXm[0] +

m−1∑j=0

am−1−jU [j]. (51)

The estimate Xm[0] corresponds to the binary representationof X(0) using dkmE(D)Ce bits, provided that there is nodecoding error in the tranmsission. Accordingly, in our sim-ulation, we let η > 0 and Pe = e−ηkm , and we assumethat at every decoding time, with probability (1 − Pe) weconstruct a correct quantized estimate of the initial state Xm[0]using dkmE(D)Ce bits. Alternatively, with probability Pe weconstruct an incorrect quantized estimate. In the case of acorrect estimate, we apply the asymptotically optimal controlinput U [m] = −KX[m], where K > 0 is the control gainand X[m] is obtained from (51). In the case of an incorrectestimate, the state estimate used to construct the control inputcan be arbitrary. We consider three cases: (i) we do not applyany control input and let the system evolve in open loop, (ii)we apply the control input using the previous estimate, (iii)we apply the opposite of the asymptotically optimal controlinput: U [m] = KX[m]. In all cases, the control input remainsfixed to its most recent value during the time required for anew estimate to be performed.

Fig. 5 pictorially illustrates the evolution of our simulationin an error-free case in which the binary representation of X[0]is refined by E(D)C = 3 bits at each symbol reception.

Numerical results are depicted in Fig. 6, showing conver-gence of the state to zero in all cases, provided that the timingcapacity is above the entropy rate of the system. In contrast,when the timing capacity is below the entropy rate, the statediverges. The plots also show the absolute value of the controlinput used for stabilization in the various cases.

Fig. 7 illustrates the percentage of times at which thecontroller successfully stabilized the plant versus the capacityof the channel in a run of 500 Monte Carlo simulations. Thephase transition behavior at the critical value C = log a isclearly evident.

VIII. CONCLUSIONS

In the framework of control of dynamical systems overcommunication channels, it has recently been observed thatevent-triggering policies encoding information over time ina state-dependent fashion can exploit timing information forstabilization in addition to the information traditionally carriedby data packets [13]–[22]. In a more general framework, thispaper studied from an information-theoretic perspective thefundamental limitation of using only timing information forstabilization, independent of any transmission strategy. Weshowed that for stabilization of an undisturbed scalar linear

9

Fig. 5: Evolution of the channel used in the simulation in an error-free case. Each time ♠ is received, a new codeword is decoded using all the symbolsreceived up to that time. The decoded codeword represents the initial state X[0] with a precision that increases by E(D)C bits at each symbol reception. Inthe figure, for illustration purposes we have assumed E(D)C = 3 bits.

system over a channel with a unitary alphabet, the timingcapacity [25] should be at least as large as the entropy rate ofthe system. In addition, in the case of exponentially distributeddelays, we provided a tight sufficient condition using a codingstrategy that refines the estimate of the decoded message asmore and more symbols are received. Important open problemsfor future research include the effect of system disturbances,understanding the combination of timing information andpackets with data payload, and extensions to vector systems.

Our derivation ensures that when the timing capacity islarger than the entropy rate, the estimation error does not growunbounded, in probability, even in the presence of the randomdelays occurring in the timing channel. This is made possibleby communicating a real-valued variable (the initial state) at anincreasingly higher resolution and with vanishing probabilityof error. This strategy has been previously studied in [40] in thecontext of estimation over the binary erasure channel, ratherthan over the timing channel. It is also related to communica-tion at increasing resolution over channels with feedback viaposterior matching [88], [89]. The classic Horstein [90] andSchalkwijk-Kailath [91] schemes are special cases of posteriormatching for the binary symmetric channel and the additiveGaussian channel respectively. The main idea in our settingis to employ a tree-structured quantizer in conjunction to acapacity-achieving timing channel codebook that grows expo-nentially with the tree depth, and re-compute the estimate ofthe real-valued variable as more and more channel symbols arereceived. The estimate is re-computed for a number of receivedsymbols that depends on the channel rate and on the averagedelay. In contrast to posterior matching, we are not concernedwith the complexity of the encoding-decoding strategy, butonly with its existence. We also do not assume a specificdistribution for the real value we need to communicate, andwe do not use the feedback signal to perform encoding, butonly to avoid queuing [25], [28]. We point out that our controlstrategy does not work in the presence of disturbances: in thiscase, one needs to track a state that depends not only on theinitial condition, but also on the evolution of the disturbance.This requires to update the entire history of the system’s statesat each symbol reception [45], leading to a different, i.e. non-classical, coding model. Alternatively, remaining in a classicalsetting one could aim for less, and attempt to obtain resultsusing weaker probabilistic notions of stability, such as the onein [6, Chapter 8].

Finally, by showing that in the case of no disturbances and

exponentially distributed delay it is possible to achieve stabi-lization at zero data-rate only for sufficiently small averagedelay E(S) < (ea)−1, we confirmed from an information-theoretic perspective the observation made in [15] regardingthe existence of a critical delay value for stabilization at zerodata-rate.

APPENDIX APROOFS OF THE ESTIMATION RESULTS

A. Proof of Theorem 2We start by introducing a few definitions and proving some

useful lemmas.Definition 4: For any ε > 0 and φ > 0, we define the rate-

distortion function of the source Xe = aXe(t) at times {tn}as

Rεtn(φ) = infP(Xe(tn)|Xe(tn))

{I(Xe(tn); Xe(tn)

): (52)

P(|Xe(tn)− Xe(tn)| > ε

)≤ φ

}.

The proof of the following lemma adapts an argumentof [42] to our continuous-time setting.

Lemma 4: We have

Rεtn(φ) ≥ (1− φ) [atn + h(X(0))] (53)

− ln 2ε− ln 2

2[nats].

Proof: Let

ξ =

{0 if |Xe(tn)− Xe(tn)| ≤ ε1 if |Xe(tn)− Xe(tn)| > ε.

(54)

Using the chain rule, we have

I(Xe(tn); Xe(tn))

= I(Xe(tn); ξ, Xe(tn))− I(Xe(tn); ξ|Xe(tn)) (55)

= I(Xe(tn); ξ, Xe(tn))−H(ξ|Xe(tn))

+H(ξ|Xe(tn), Xe(tn)).

Given X(tn) and X(tn), there is no uncertainty in ξ, hencewe deduce

I(Xe(tn); Xe(tn))

= I(Xe(tn); ξ, Xe(tn))−H(ξ|Xe(tn))

= h(Xe(tn))− h(Xe(tn)|ξ, Xe(tn))−H(ξ|Xe(tn))

10

Case I: decoding error → open loop

C = 1.2 log a C = 1.2 log a C = 0.9 log a

Case II: decoding error → previous estimate

C = 1.2 log a C = 1.2 log a C = 0.9 log a

Case III: decoding error → opposite of the optimal control

C = 1.2 log a C = 1.2 log a C = 0.9 log a

Fig. 6: Here we show the evolution of a single run of a system with different capacities for the timing channel. The first and second columns representthe absolute value of the state and control input, respectively, when the timing capacity is larger than the entropy rate of the system (C > log a). The thirdcolumn represents the absolute value of the state when the timing capacity is smaller than the entropy rate of the system (C < log a). In the first row, in thepresence of a decoding error, we do not apply any control input and let the system evolve in open-loop; in the second row, we apply the control using theprevious estimate; the third row, we apply the opposite of the optimal control. The simulation parameters were chosen as follows: a = 1.2, E(D) = 2, andPe = e−ηkm , where η = 0.09. For the optimal control gain we have chosen K = 0.4, which is optimal with respect to the (time-averaged) linear quadraticregulator (LQR) control cost (1/200)E[

∑199m=0(0.01X

2k + 0.5U2

k ) + 0.01X2200].

= h(Xe(tn))− h(Xe(tn)|ξ = 0, Xe(tn))P(ξ = 0) (56)

− h(Xe(tn)|ξ = 1, Xe(tn))P(ξ = 1)−H(ξ|Xe(tn)).

Since H(ξ|Xe(tn)) ≤ H(ξ) ≤ ln 2/2 [nats], P(ξ = 0) ≤ 1,and P(ξ = 1) ≤ φ, it then follows that

I(Xe(tn); Xe(tn)) ≥

h (Xe(tn))− h(Xe(tn)− Xe(tn)|ξ = 0, Xe(tn)

)− h

(Xe(tn)|ξ = 1, Xe(tn)

)φ− ln 2

2. (57)

Since conditioning reduces the entropy, we have

I(Xe(tn); Xe(tn)) ≥ h(Xe(tn)) (58)

− h(Xe(tn)− Xe(tn)|ξ = 0)− h(Xe(tn))φ− ln 2

2

= (1− φ)h(Xe(tn))− h(Xe(tn)− Xe(tn)|ξ = 0)− ln 2

2.

By (54) and since the uniform distribution maximizes thedifferential entropy among all distributions with boundedsupport, we have

I(Xe(tn); Xe(tn)) ≥ (1− φ)h(Xe(tn))− ln 2ε− ln 2

2.

(59)

Since Xe(tn) = X(0) eatn , we have

h(Xe(tn)) = ln eatn + h(X(0)) = atn + h(X(0)). (60)

Combining (59), and (60) we obtain

I(Xe(tn); Xe(tn)) ≥ (1− φ) (atn + h(X(0)))− ln 2ε− ln 2

2.

(61)

Finally, noting that this inequality is independent ofP(Xe(tn)|Xe(tn)) the result follows.

11

Fig. 7: Here we show the fraction of times stabilization was achieved versusthe capacity of the channel across a run of 500 simulations for each valueof the capacity. Successful stabilization is defined in these simulations as|X[250]| ≤ 0.05. In the case of a decoding error, no control input is appliedand we let the system evolve in open loop. The simulation parameters werechosen as follows: a = 1.2, E(D) = 2, and Pe = e−ηkm , where η =0.09. For the control gain, we have chosen K = 0.4, which is optimal withrespect to the (time-averaged) linear quadratic regulator (LQR) control cost(1/200)E[

∑199m=0(0.01X

2k + 0.5U2

k ) + 0.01X2200].

Remark 2: By letting φ = ε in (53), we have

Rεtn(ε) ≥ (1− ε)atn + ε′, (62)

where

ε′ = (1− ε)h (X(0))− ln 2ε− ln 2

2. (63)

For sufficiently small ε we have that ε′ ≥ 0, and hence

Rεtn(ε)

tn≥ (1− ε)a. (64)

It follows that for sufficiently small ε the rate-distortion perunit time of the source must be at least as large as the entropyrate of the system. Since the rate-distortion represents thenumber of bits required to represent the state of the process upto a given fidelity, this provides an operational characterizationof the entropy rate of the system. •

The proof of the following lemma follows a converseargument of [25] with some modifications due to our differentsetting.

Lemma 5: Under the same assumptions as in Theorem 2, ifby time tn, κn symbol is received by the controller, we have

I(Xe(tn); Xe(tn)

)≤ κnI(W ;W + S). (65)

Proof: We denote the transmitted message by V ∈{1, . . . ,M} and the decoded message by U ∈ {1, . . . ,M}.Then

Xe(tn)→ V → (D1, . . . , Dκn)→ U → Xe(tn), (66)

is a Markov chain. Therefore, using the data-processing in-equality [92], we have

I(Xe(tn); Xe(tn)

)≤ I(V ;U) ≤ I(V ;D1, . . . , Dκn). (67)

By the chain rule for the mutual information, we have

I(V ;D1, . . . , Dκn) =

κn∑i=1

I(V ;Di|Di−1). (68)

Since Wi is uniquely determined by the encoder from V , usingthe chain rule we deduce

κn∑i=1

I(V ;Di|Di−1) =

κn∑i=1

I(V,Wi;Di|Di−1). (69)

In addition, again using the chain rule, we haveκn∑i=1

I(V,Wi;Di|Di−1) =

κn∑i=1

I(Wi;Di|Di−1) (70)

+

κn∑i=1

I(V ;Di|Di−1,Wi).

Di is conditionally independent of V when given Wi. Thus:κn∑i=1

I(V ;Di|Di−1,Wi) = 0. (71)

Combining (69), (70), and (71) it follows thatκn∑i=1

I(V ;Di|Di−1) =

κn∑i=1

I(Wi;Di|Di−1). (72)

Since the sequences {Si} and {Wi} are i.i.d. and independentof each other, it follows that the sequence {Di} is also i.i.d.,and we have

κn∑i=1

I(Wi;Di|Di−1) = κnI(W ;D). (73)

By combining (67), (68), (72) and (73) the result follows.We are now ready to finish the proof of Theorem 2.

Proof: If E(W + S) = 0, (12) is straightforward. Thus,for the rest of the proof, we assume E(W + S) > 0.

Recall that Γ > 1. Thus, we can choose ` > 1 suchthat for κn = n + `, we have ΓE(Tn) ≤ E(Tκn). Sincelimn→∞

tnE(Tn) ≤ Γ, it follows that for sufficiently large n

we have

P(Tκn+1 ≤ tn) ≤ P[Tκn+1 ≤ ΓE(Tn)]

≤ P[Tκn+1 ≤ E(Tκn)]

≤ P[Tκn+1 ≤ E(Tκn+1)]. (74)

Since the waiting times {Wi} and the random delays {Si}are i.i.d. sequences and independent of each other, it followsby the strong law of large numbers that (74) tends to zero asn → ∞. Thus, as n → ∞ with high probability at most κnsymbols are received by the controller, and using Lemma (5),it follows that

(n+ `)I(W ;W + S) ≥ I(Xe(tn); Xe(tn)

). (75)

By the assumption of the theorem, for any ε > 0 we have

limn→∞

P(|Xe(tn)− Xe(tn)| ≤ ε

)= 1. (76)

Hence, for any ε > 0 and any φ > 0 there exist nφ such that

12

for n ≥ nφ

P(|Xe(tn)− Xe(tn)| > ε

)≤ φ. (77)

Using (77), (52), and Lemma 4 it follows that for n ≥ nφ

Rεtn(φ) ≥ (1− φ) [atn + h(X(0))]− ln 2ε− ln 2

2. (78)

By (52), we have

I(Xe(tn); Xe(tn)) ≥ Rεtn(φ), (79)

and combining (78), and (79) we obtain that for n ≥ nφ

I(Xe(tn); Xe(tn)

)n

≥ (80)

(1− φ)atnn

+(1− φ)h(X(0))− ln 2ε− ln 2

2

n.

We now let φ → 0, so that n → ∞. Sincelimn→∞(n+ `)/n = 1, using (75) we have

I(W ;W + S) ≥ a limn→∞

tnn. (81)

Since, E(Tn) = nE(Dn) from (11) it follows that

E(D) ≤ limn→∞

tnn≤ ΓE(D). (82)

Since |Xe(tn)− Xe(tn)| P→ 0 for all the measurement timestn satisfying (82), we let limn→∞ tn/n = ΓE(D) in (81)and (12) follows. Finally, using (8) and noticing

supW≥0

E(W )≤χ

I(W ;W + S)

E(S) + χ≥ sup

W≥0E(W )=χ

I(W ;W + S)

E(S) + χ, (83)

we deduce that if (12) holds then (13) holds as well.

B. Proof of Theorem 3Proof: If E(S) = 0 the timing capacity is infinite, and the

result is trivial. Hence, for the rest of the proof, we assumethat

E(S +W ) ≥ E(S) > 0, (84)

which by (4) implies that E(Tn) → ∞ as n → ∞. As aconsequence, by (11) we also have that tn →∞ as n→∞.

The objective is to design an encoding and decoding strat-egy, such that for all ε, δ > 0 and sufficiently large n, wehave

P(|Xe(tn)− Xe(tn)| > ε) < δ. (85)

We start by bounding the probability of the event that the nthsymbol does not arrive by the estimation deadline tn. Sincelimn→∞ tn/E(Tn) > 1, it follows that there exists ν > 0 suchthat for large enough n we have

tn > (1 + ν)E(Tn). (86)

Hence, for large enough n, we have that the probability ofmissing the deadline is

P(Tn > tn) ≤ P[Tn > (1 + ν)E(Tn)]. (87)

Since the waiting times {Wi} and the random delays {Si}

are i.i.d. sequences and independent of each other, it followsby the strong law of large numbers that (87) tends to zero asn→∞. We now have

P(|Xe(tn)− Xe(tn)| > ε) =

P(|Xe(tn)− Xe(tn)| > ε | tn ≥ Tn)P(tn ≥ Tn)

+ P(|Xe(tn)− Xe(tn)| > ε | tn < Tn)P(tn < Tn)

≤ P(|Xe(tn)− Xe(tn)| > ε | tn ≥ Tn) + P(tn < Tn), (88)

where the second term in the sum (88), tends to zero asn→∞. It follows that to ensure (85) it suffices to design anencoding and decoding scheme, such that for all ε, δ > 0 andsufficiently large n, we have that the conditional probability

P(|Xe(tn)− Xe(tn)| > ε | tn ≥ Tn) < δ. (89)

From the open-loop equation (10), we have

Xe(tn) = eatnX(0), (90)

from which it follows that the decoder can construct theestimate

Xe(tn) = eatnXtn(0), (91)

where Xtn(0) is an estimate of X(0) constructed at time tnusing all the symbols received by this time.

By (90) and (91), we now have that (89) is equivalent to

P(|X(0)− Xtn(0)| > εe−atn | tn ≥ Tn) < δ, (92)

namely it suffices to design an encoding and decoding schemeto communicate the initial condition with exponentially in-creasing reliability in probability. Our coding procedure thatachieves this objective is described next.

Source coding: We let the source coding map

Q : [−L,L]→ {0, 1}N (93)

be an infinite tree-structured quantizer [83]. This map con-structs the infinite binary sequence Q (X(0)) = {Q1, Q2, . . .}as follows. Q1 = 0 if X(0) falls into the left-half of theinterval [−L,L], otherwise Q1 = 1. The sub-interval whereX(0) falls is then divided into half and we let Q2 = 0 ifX(0) falls into the left-half of this sub-interval, otherwiseQ2 = 1. The process then continues in the natural way, andQi is determined accordingly for all i ≥ 3.

Using the definition of truncation operator (1), for any n′ ≥1 we can define

Qn′ = πn′ ◦ Q. (94)

It follows that Qn′ (X(0)) is a binary sequence of lengthn′ that identifies an interval of length L/2n

′−1 that containsX(0). We also let

Q−1n′ : {0, 1}n

′→ [−L,L] (95)

be the right-inverse map of Qn′ , which assigns the middlepoint of the last interval identified by the sequence thatcontains X(0). It follows that for any n′ ≥ 1, this procedureachieves a quantization error

|X(0)−Q−1n′ ◦ Qn′(X(0))| ≤ L

2n′. (96)

13

•Channel coding: In order to communicate the quantized

initial condition over the timing channel, the truncated binarysequence Qn′(X(0)) needs to be mapped into a channelcodeword of length n.

We consider a channel codebook of n columns and Mn

rows. The codeword symbols {wi,m, i = 1, · · · , n; m =1 · · ·Mn} are drawn i.i.d. from a distribution which is mix-ture of a delta function and an exponential and such thatP(Wi = 0) = e−1, and P(Wi > w | Wi > 0) = exp{ −weE(S)}.By Theorem 3 of [25], if the delays {Si} are exponentiallydistributed, using a maximum likelihood decoder this construc-tion achieves the timing capacity. Namely, letting

Tn = E(Tn) = nE(D), (97)

using this codebook we can achieve any rate

R = limn→∞

logMn

Tn< C (98)

over the timing channel.Next, we describe the mapping between the source coding

and the channel coding constructions. •Source-channel mapping: We first consider the direct map-

ping. For all i ≥ 1, we let n′ = diRE(D)e and consider the2n′

possible outcomes of the source coding map Qn′(X(0)).We associate them, in a one-to-one fashion, to the rows of acodebook Ψn′ of size 2n

′ × dn′/RE(D)e. This mapping isdefined as

En′ :{0, 1}n′→ Ψn′ . (99)

By letting i → ∞, the codebook becomes a double-infinitematrix Ψ∞, and the map becomes

E : {0, 1}N → Ψ∞. (100)

Thus, as i→∞, X(0) is encoded as

X(0)Q−→ {0, 1}N E−→ Ψ∞. (101)

We now consider the inverse mapping. Since the elements ofΨn′ are drawn independently from a continuous distribution,with probability one, no two rows of the codebook are equal toeach other, so for any i ≥ 1 and number of received symbolsn = di/RE(D)e we define

E−1n′ : Ψn′ → {0, 1}n

′, (102)

where n′ = dnRE(D)e. This map associates to every row inthe codebook a corresponding node in the quantization tree atlevel n′.

Figures 8 and 9 show the constructions described abovefor the cases RE(D) = 2 and RE(D) = 0.5, respec-tively. In Fig. 8, the nodes in the quantization tree at leveln′ = diRE(D)e = 2, 4, 6, . . . , are mapped into the rowsof a table of Mn = 22, 24, 26, . . . rows and n = 1, 2, 3 . . .columns. Conversely, the rows in each table are mapped intothe corresponding nodes in the tree. In Fig. 9, the nodes in thequantization tree at level n′ = diRE(D)e = 1, 2, 3, . . . , aremapped into the rows of a table of Mn = 2, 22, 23, . . . rowsand n = 2, 4, 6, . . . columns. Conversely, the rows in eachtable are mapped into the corresponding nodes in the tree.

Next, we describe how the encoding and decoding opera-tions are performed using these maps and how transmissionoccurs over the channel. •

One-time encoding: The encoding of the initial state X(0)occurs at the sensor in one-shot and then the correspondingsymbols are transmitted over the channel, one by one. GivenX(0), the source encoder computes Q(X(0)) according tothe source coding map (93) and the channel encoder picksthe corresponding codeword E(Q(X(0))) from the doubly-infinite codebook according to the map (100). This codewordis an infinite sequence of real numbers, which also corre-sponds to a leaf at infinite depth in the quantization tree.Then, the encoder starts transmitting the real numbers of thecodeword one by one, where each real number corresponds toa holding time, and proceeds in this way forever. Accordingto the source-channel mapping described above, transmittingn = dn′/RE(D)e symbols using this scheme correspondsto transmitting, for all i ≥ 1, n′ = diRE(D)e source bits,encoded into a codeword En′(Qn′(X(0))), picked from atruncated codebook of 2n

′rows and n columns. •

Anytime Decoding: The decoding of the initial state X(0)occurs at the controller in an anytime fashion, refining theestimate of X(0) as more and more symbols are received.

For all i ≥ 1 the decoder updates its guess for the value ofX(0) any time the number of symbols received equals n =di/RE(D)e. Assuming a decoding operation occurs after nsymbols have been received, the decoder picks the maximumlikelihood codeword from a truncated codebook of size Mn×nand by inverse mapping, it finds the corresponding node inthe tree. It follows that at the nth random reception time Tn,the decoder utilizes the inter-reception times of all n symbolsreceived up to this time to construct the estimate XTn(0).First, a maximum likelihood decoder Dn is employed to mapthe inter-reception times (D1, . . . , Dn) to an element of Ψn′ .This element is then mapped to a binary sequence of lengthn′ using E−1

n′ . Finally, Q−1n′ is used to construct XTn(0). It

follows that at the nth reception time where decoding occurs,we have

(D1, . . . , Dn)Dn−−→ Ψn′

E−1

n′−−→ {0, 1}n′ Q−1

n′−−−→ [−L,L], (103)

and we let

XTn(0) = Q−1n′

(E−1n′ (Dn(D1, . . . , Dn))

). (104)

Thus, as n→∞ the final decoding process becomes

(D1, Dn, . . . )D−→ Ψ∞

E−1

−−→ {0, 1}N Q−1

−−−→ [−L,L]. (105)

•To conclude the proof, we now show that if C > Γa, then it

is possible to perform the above encoding and decoding oper-ations with an arbitrarily small probability of error while usinga codebook so large that it can accommodate a quantizationerror at most L/2n

′< εe−atn .

Since the channel coding scheme achieves the timing ca-pacity, we have that for any R < C, as n→∞ the maximumlikelihood decoder selects the correct transmitted codewordwith arbitrarily high probability. It follows that for any δ > 0and n sufficiently large, we have with probability at least

14

Fig. 8: Tree-structured quantizer and the corresponding codebook for RE(D) = 2. In this case, every received channel symbolrefines the source coding representation by two bits. Here the black nodes in the quantization tree at level n′ = diRE(D)e =2, 4, 6, . . . , are mapped into the rows of the codebook

Fig. 9: Tree-structured quantizer and the corresponding codebook for RE(D) = 1/2. In this case, every two received channelsymbols refine the source coding representation by one bit.

(1− δ) that

Qn′(X(0)) = E−1n′ (Dn(D1, . . . , Dn)) , (106)

and then by (96) we have

|X(0)− XTn(0)| ≤ L

2n′. (107)

We now consider a sequence of estimation times {tn} sat-isfying (11) and let the estimate at time tn ≥ Tn in (92)be Xtn(0) = XTn(0). By (107) we have that the sufficient

condition for estimation reduces toL

2n′≤ εe−atn , (108)

which means having the size of the codebook Mn be such that

L

Mn≤ εe−atn , (109)

or equivalently

logMn − logL+ log ε

tn≥ a. (110)

15

Using (97), we have

logMn − logL+ log ε

tn=

logMn − logL+ log ε

Tn· Tntn

=logMn − logL+ log ε

Tn

· E(Tn)

tn. (111)

Taking the limit for n→∞, we have

limn→∞

logMn − logL+ log ε

Tn· E(Tn)

tn≥ R · 1

Γ. (112)

It follows that as n → ∞ the sufficient condition (110) canbe expressed in terms of the rate as

R ≥ Γa. (113)

It follows that the rate must satisfy

C > R ≥ Γa (114)

and since C > Γa, the proof is complete.

REFERENCES

[1] M. J. Khojasteh, M. Franceschetti, and G. Ranade, “Stabilizing a linearsystem using phone calls,” in 18th Eur. Cont. Conf. (ECC). IEEE,2019, pp. 2856–2861.

[2] ——, “Estimating a linear process using phone calls,” in IEEE Conf.Decis. and Cont. (CDC), 2018, pp. 127–131.

[3] K.-D. Kim and P. R. Kumar, “Cyber–physical systems: A perspectiveat the centennial,” Proc. IEEE, vol. 100 (Special Centennial Issue), pp.1287–1308, 2012.

[4] J. P. Hespanha, P. Naghshtabrizi, and Y. Xu, “A survey of recent resultsin networked control systems,” Proceedings of the IEEE, vol. 95, no. 1,pp. 138–162, 2007.

[5] S. Yüksel and T. Basar, Stochastic Networked Control Systems: Sta-bilization and Optimization under Information Constraints. SpringerScience & Business Media, 2013.

[6] A. S. Matveev and A. V. Savkin, Estimation and control over commu-nication networks. Springer Science & Business Media, 2009.

[7] M. Franceschetti and P. Minero, “Elements of information theory fornetworked control systems,” in Information and Control in Networks.Springer, 2014, pp. 3–37.

[8] B. G. N. Nair, F. Fagnani, S. Zampieri, and R. J. Evans, “Feedbackcontrol under data rate constraints: An overview,” Proceedings of theIEEE, vol. 95, no. 1, pp. 108–137, 2007.

[9] N. C. Martins, M. A. Dahleh, and N. Elia, “Feedback stabilization ofuncertain systems in the presence of a direct link,” IEEE Trans. Autom.Control, vol. 51, no. 3, pp. 438–447, 2006.

[10] D. F. Delchamps, “Stabilizing a linear system with quantized statefeedback,” IEEE Trans. Autom. Control, vol. 35, no. 8, pp. 916–924,1990.

[11] W. S. Wong and R. W. Brockett, “Systems with finite communicationbandwidth constraints. II. stabilization with limited information feed-back,” IEEE Trans. Autom. Control, vol. 44, no. 5, pp. 1049–1053, 1999.

[12] J. Baillieul, “Feedback designs for controlling device arrays with com-munication channel bandwidth constraints,” in ARO Workshop on SmartStructures, Pennsylvania State Univ, 1999, pp. 16–18.

[13] M. J. Khojasteh, M. Hedayatpour, J. Cortés, and M. Franceschetti,“Event-triggering stabilization of complex linear systems with distur-bances over digital channels,” in IEEE Conf. Decision and Cont. (CDC),2018, pp. 152–157.

[14] E. Kofman and J. H. Braslavsky, “Level crossing sampling in feedbackstabilization under data-rate constraints,” in 45st IEEE Conf. Decisionand Cont. (CDC), 2006, pp. 4423–4428.

[15] M. J. Khojasteh, P. Tallapragada, J. Cortés, and M. Franceschetti, “Thevalue of timing information in event-triggered control,” IEEE Trans.Autom. Control, to appear, 2020.

[16] M. J. Khojasteh, P. Tallapragada, J. Cortés, and M. Franceschetti, “Time-triggering versus event-triggering control over communication channels,”in 56th IEEE Conf. Decision and Cont. (CDC), Dec 2017, pp. 5432–5437.

[17] S. Linsenmayer, R. Blind, and F. Allgöwer, “Delay-dependent datarate bounds for containability of scalar systems,” IFAC-PapersOnLine,vol. 50, no. 1, pp. 7875–7880, 2017.

[18] M. J. Khojasteh, M. Hedayatpour, J. Cortés, and M. Franceschetti,“Event-triggered stabilization of disturbed linear systems over digitalchannels,” in Information Sciences and Systems (CISS), 2018 52ndAnnual Conference on. IEEE, 2018, pp. 1–6.

[19] Q. Ling, “Bit-rate conditions to stabilize a continuous-time linear systemwith feedback dropouts,” IEEE Trans. Autom. Control, vol. 63, no. 7,pp. 2176–2183, July 2018.

[20] H. Yildiz, Y. Su, A. Khina, and B. Hassibi, “Event-triggered stochasticcontrol via constrained quantization,” in Data Comp. Conf. IEEE, 2019,pp. 612–612.

[21] M. J. Khojasteh, M. Hedayatpour, and M. Franceschetti, “Theory andimplementation of event-triggered stabilization over digital channels,” inIEEE Conf. Decis. and Cont. (CDC), 2019, pp. 4183–4188.

[22] N. Guo and V. Kostina, “Optimal causal rate-constrained sampling ofthe wiener process,” in 57th Ann. Allerton Conf. on Comm., Cont., andComp. (Allerton). IEEE, 2019, pp. 1090–1097.

[23] G. Ranade and A. Sahai, “Non-coherence in estimation and control,” in51st Communication, Annu. Allerton Conf. Commun., Control, Comput.IEEE, 2013, pp. 189–196.

[24] ——, “Control capacity,” IEEE Transactions on Information Theory,vol. 65, no. 1, pp. 235–254, 2018.

[25] V. Anantharam and S. Verdú, “Bits through queues,” IEEE Trans. Inf.Theory, vol. 42, no. 1, pp. 4–18, 1996.

[26] A. S. Bedekar and M. Azizoglu, “The information-theoretic capacityof discrete-time queues,” IEEE Trans. Inf. Theory, vol. 44, no. 2, pp.446–461, 1998.

[27] E. Arikan, “On the reliability exponent of the exponential timingchannel,” IEEE Trans. Inf. Theory, vol. 48, no. 6, pp. 1681–1689, 2002.

[28] R. Sundaresan and S. Verdú, “Robust decoding for timing channels,”IEEE Trans. Inf. Theory, vol. 46, no. 2, pp. 405–419, 2000.

[29] C. Rose and I. S. Mian, “Inscribed matter communication: Part I,” IEEETransactions on Molecular, Biological and Multi-Scale Communications,vol. 2, no. 2, pp. 209–227, 2016.

[30] A. Gohari, M. Mirmohseni, and M. Nasiri-Kenari, “Information theoryof molecular communication: Directions and challenges,” IEEE Transac-tions on Molecular, Biological and Multi-Scale Communications, vol. 2,no. 2, pp. 120–142, 2016.

[31] T. J. Riedl, T. P. Coleman, and A. C. Singer, “Finite block-lengthachievable rates for queuing timing channels,” in Information TheoryWorkshop (ITW), 2011 IEEE. IEEE, 2011, pp. 200–204.

[32] A. B. Wagner and V. Anantharam, “Zero-rate reliability of theexponential-server timing channel,” IEEE Trans. Inf. Theory, vol. 51,no. 2, pp. 447–465, 2005.

[33] M. Tavan, R. D. Yates, and W. U. Bajwa, “Bits through bufferlessqueues,” in 51st Annu. Allerton Conf. Commun., Control, Comput., Oct2013, pp. 755–762.

[34] B. Prabhakar and R. Gallager, “Entropy and the timing capacity ofdiscrete queues,” IEEE Trans. Inf. Theory, vol. 49, no. 2, pp. 357–370,2003.

[35] L. Aptel and A. Tchamkerten, “Feedback increases the capacity ofqueues,” in 2018 IEEE International Symposium on Information Theory(ISIT). IEEE, 2018, pp. 1116–1120.

[36] J. Giles and B. Hajek, “An information-theoretic and game-theoreticstudy of timing channels,” IEEE Trans. Inf. Theory, vol. 48, no. 9, pp.2455–2477, 2002.

[37] T. P. Coleman and M. Raginsky, “Mutual information saddle points inchannels of exponential family type,” in Information Theory Proceedings(ISIT), 2010 IEEE International Symposium on. IEEE, 2010, pp. 1355–1359.

[38] X. Liu and R. Srikant, “The timing capacity of single-server queueswith multiple flows,” DIMACS Series in Discrete Mathematics andTheoretical Computer Science, 2004.

[39] S. H. Sellke, C.-C. Wang, N. Shroff, and S. Bagchi, “Capacity boundson timing channels with bounded service times,” in 2007 IEEE Interna-tional Symposium on Information Theory. IEEE, 2007, pp. 981–985.

[40] G. Como, F. Fagnani, and S. Zampieri, “Anytime reliable transmission ofreal-valued information through digital noisy channels,” SIAM J. ControlOptimiz., vol. 48, no. 6, pp. 3903–3924, 2010.

[41] S. Tatikonda and S. Mitter, “Control under communication constraints,”IEEE Trans. Autom. Control, vol. 49, no. 7, pp. 1056–1068, 2004.

[42] ——, “Control over noisy channels,” IEEE Trans. Autom. Control,vol. 49, no. 7, pp. 1196–1201, 2004.

[43] A. S. Matveev and A. V. Savkin, “An analogue of shannon informationtheory for detection and stabilization via noisy discrete communication

16

channels,” SIAM J. Control Optimiz., vol. 46, no. 4, pp. 1323–1367,2007.

[44] ——, “Shannon zero error capacity in the problems of state estima-tion and stabilization via noisy communication channels,” InternationalJournal of Control, vol. 80, no. 2, pp. 241–255, 2007.

[45] A. Sahai and S. Mitter, “The necessity and sufficiency of anytimecapacity for stabilization of a linear system over a noisy communicationlink. Part I: Scalar systems,” IEEE Trans. Inf. Theory, vol. 52, no. 8,pp. 3369–3395, 2006.

[46] G. N. Nair and R. J. Evans, “Stabilizability of stochastic linear systemswith finite feedback data rates,” SIAM J. Control Optimiz., vol. 43, no. 2,pp. 413–436, 2004.

[47] P. Minero, M. Franceschetti, S. Dey, and G. N. Nair, “Data rate theoremfor stabilization over time-varying feedback channels,” IEEE Trans.Autom. Control, vol. 54, no. 2, p. 243, 2009.

[48] P. Minero, L. Coviello, and M. Franceschetti, “Stabilization over Markovfeedback channels: the general case,” IEEE Trans. Autom. Control,vol. 58, no. 2, pp. 349–362, 2013.

[49] P. Minero and M. Franceschetti, “Anytime capacity of a class of markovchannels,” IEEE Trans. Autom. Control, vol. 62, no. 3, pp. 1356–1367,2017.

[50] S. Fang, J. Chen, and I. Hideaki, Towards integrating control andinformation theories. Springer, 2017.

[51] J. Hespanha, A. Ortega, and L. Vasudevan, “Towards the control oflinear systems with minimum bit-rate,” in Proc. 15th Int. Symp. onMathematical Theory of Networks and Systems (MTNS), 2002.

[52] G. Nair, “A non-stochastic information theory for communication andstate estimation,” IEEE Trans. Autom. Control, vol. 58, pp. 1497–1510,2013.

[53] R. Ostrovsky, Y. Rabani, and L. J. Schulman, “Error-correcting codes forautomatic control,” IEEE Trans. Inf. Theory, vol. 55, no. 7, pp. 2931–2941, 2009.

[54] R. T. Sukhavasi and B. Hassibi, “Linear time-invariant anytime codesfor control over noisy channels,” IEEE Trans. Autom. Control, vol. 61,no. 12, pp. 3826–3841, 2016.

[55] A. Khina, W. Halbawi, and B. Hassibi, “(Almost) practical tree codes,”in Information Theory (ISIT), 2016 IEEE International Symposium on.IEEE, 2016, pp. 2404–2408.

[56] G. N. Nair, R. J. Evans, I. M. Mareels, and W. Moran, “Topologicalfeedback entropy and nonlinear stabilization,” IEEE Trans. Autom.Control, vol. 49, no. 9, pp. 1585–1597, 2004.

[57] D. Liberzon and J. P. Hespanha, “Stabilization of nonlinear systems withlimited information feedback,” IEEE Trans. Autom. Control, vol. 50,no. 6, pp. 910–915, 2005.

[58] C. De Persis, “n-bit stabilization of n-dimensional nonlinear systemsin feedforward form,” IEEE Trans. Autom. Control, vol. 50, no. 3, pp.299–311, 2005.

[59] D. Liberzon, “Finite data-rate feedback stabilization of switched andhybrid linear systems,” Automatica, vol. 50, no. 2, pp. 409–420, 2014.

[60] G. Yang and D. Liberzon, “Feedback stabilization of switched linearsystems with unknown disturbances under data-rate constraints,” IEEETrans. Autom. Control, vol. 63, no. 7, pp. 2107–2122, 2018.

[61] V. Sanjaroon, A. Farhadi, A. S. Motahari, and B. H. Khalaj, “Estimationof nonlinear dynamic systems over communication channels,” IEEETrans. Autom. Control, vol. 63, no. 9, pp. 3024–3031, 2018.

[62] V. Kostina and B. Hassibi, “Rate-cost tradeoffs in control,” IEEE Trans.Autom. Control, vol. 64, no. 11, pp. 4525–4540, 2019.

[63] P. Tabuada, “Event-triggered real-time scheduling of stabilizing controltasks,” IEEE Trans. Autom. Control, vol. 52, no. 9, pp. 1680–1685, 2007.

[64] W. P. M. H. Heemels, K. H. Johansson, and P. Tabuada, “An introduc-tion to event-triggered and self-triggered control,” in 51st IEEE Conf.Decision and Cont. (CDC), 2012, pp. 3270–3285.

[65] K. J. Astrom and B. M. Bernhardsson, “Comparison of riemann andlebesgue sampling for first order stochastic systems,” in 41st IEEE Conf.Decision and Cont. (CDC), vol. 2, 2002, pp. 2011–2016.

[66] X. Wang and M. D. Lemmon, “Event-triggering in distributed networkedcontrol systems,” IEEE Trans. Autom. Control, vol. 56, no. 3, p. 586,2011.

[67] D. V. Dimarogonas, E. Frazzoli, and K. H. Johansson, “Distributedevent-triggered control for multi-agent systems,” IEEE Trans. Autom.Control, vol. 57, no. 5, pp. 1291–1297, 2012.

[68] B. A. Khashooei, D. J. Antunes, and W. Heemels, “A consistentthreshold-based policy for event-triggered control,” IEEE Control Sys-tems Letters, vol. 2, no. 3, pp. 447–452, 2018.

[69] W. H. Heemels, M. Donkers, and A. R. Teel, “Periodic event-triggeredcontrol for linear systems,” IEEE Trans. Autom. Control, vol. 58, no. 4,pp. 847–861, 2013.

[70] L. Li, X. Wang, and M. Lemmon, “Stabilizing bit-rates in quantizedevent triggered control systems,” in Proceedings of the 15th ACMinternational conference on Hybrid Systems: Computation and Control.ACM, 2012, pp. 245–254.

[71] B. Demirel, V. Gupta, D. E. Quevedo, and M. Johansson, “On the trade-off between communication and control cost in event-triggered dead-beatcontrol,” IEEE Trans. Autom. Control, vol. 62, no. 6, pp. 2973–2980,2017.

[72] D. E. Quevedo, V. Gupta, W.-J. Ma, and S. Yüksel, “Stochastic sta-bility of event-triggered anytime control,” IEEE Trans. Autom. Control,vol. 59, no. 12, pp. 3373–3379, 2014.

[73] L. Lindemann, D. Maity, J. S. Baras, and D. V. Dimarogonas, “Event-triggered feedback control for signal temporal logic tasks,” in IEEEConf. Decision and Cont. (CDC), 2018, pp. 146–151.

[74] A. Girard, “Dynamic triggering mechanisms for event-triggered control,”IEEE Trans. Autom. Control, vol. 60, no. 7, pp. 1992–1997, 2015.

[75] A. Seuret, C. Prieur, S. Tarbouriech, and L. Zaccarian, “LQ-based event-triggered controller co-design for saturated linear systems,” Automatica,vol. 74, pp. 47–54, 2016.

[76] P. Tallapragada and J. Cortés, “Event-triggered stabilization of linearsystems under bounded bit rates,” IEEE Trans. Autom. Control, vol. 61,no. 6, pp. 1575–1589, 2016.

[77] J. Pearson, J. P. Hespanha, and D. Liberzon, “Control with minimal cost-per-symbol encoding and quasi-optimality of event-based encoders,”IEEE Trans. Autom. Control, vol. 62, no. 5, pp. 2286–2301, 2017.

[78] L. Schenato, B. Sinopoli, M. Franceschetti, K. Poolla, and S. S.Sastry, “Foundations of control and estimation over lossy networks,”Proceedings of the IEEE, vol. 95, no. 1, pp. 163–187, Jan 2007.

[79] K. You and L. Xie, “Minimum data rate for mean square stabilizationof discrete LTI systems over lossy channels,” IEEE Tran. Auto. Cont.,vol. 55, no. 10, pp. 2373–2378, 2010.

[80] S. Yuksel and S. P. Meyn, “Random-time, state-dependent stochasticdrift for Markov chains and application to stochastic stabilization overerasure channels,” IEEE Tran. on Auto. Cont., vol. 58, no. 1, pp. 47–59,2012.

[81] V. Gupta, A. F. Dana, J. P. Hespanha, R. M. Murray, and B. Hassibi,“Data transmission over networks for estimation and control,” IEEETran. on Auto. Cont., vol. 54, no. 8, pp. 1807–1819, 2009.

[82] A. Khina, V. Kostina, A. Khisti, and B. Hassibi, “Tracking and controlof Gauss–Markov processes over packet-drop channels with acknowl-edgments,” IEEE Tran. on Cont. of Net. Sys., vol. 6, no. 2, pp. 549–560,2018.

[83] A. Gersho and R. M. Gray, Vector quantization and signal compression.Springer Science & Business Media, 2012, vol. 159.

[84] D. Liberzon and S. Mitra, “Entropy and minimal bit rates for stateestimation and model detection,” IEEE Trans. Autom. Control, 2017.

[85] F. Colonius and C. Kawan, “Invariance entropy for control systems,”SIAM J. Control Optimiz., vol. 48, no. 3, pp. 1701–1721, 2009.

[86] F. Colonius, “Minimal bit rates and entropy for exponential stabiliza-tion,” SIAM J. Control Optimiz., vol. 50, no. 5, pp. 2988–3010, 2012.

[87] M. Rungger and M. Zamani, “On the invariance feedback entropy oflinear perturbed control systems,” in 56th IEEE Conf. Decision and Cont.(CDC), 2017, pp. 3998–4003.

[88] O. Shayevitz and M. Feder, “Optimal feedback communication viaposterior matching,” IEEE Trans. Inf. Theory, vol. 57, no. 3, pp. 1186–1222, 2011.

[89] “extrinsic jensen–shannon divergence: Applications to variable-lengthcoding.”

[90] M. Horstein, “Sequential transmission using noiseless feedback,” IEEETrans. Inf. Theory, vol. 9, no. 3, pp. 136–143, 1963.

[91] J. Schalkwijk and T. Kailath, “A coding scheme for additive noisechannels with feedback–i: No bandwidth constraint,” IEEE Trans. Inf.Theory, vol. 12, no. 2, pp. 172–182, 1966.

[92] T. M. Cover and J. A. Thomas, Elements of information theory. JohnWiley & Sons, 2012.

17

Mohammad Javad Khojasteh (S’14) did his under-graduate studies at Sharif University of Technologyfrom which he received double-major B.Sc. degreesin Electrical Engineering and in Pure Mathematics,in 2015. He received the M.Sc. and Ph.D. degreesin Electrical and Computer Engineering from Uni-versity of California San Diego (UCSD), La Jolla,CA, in 2017, and 2019, respectively. Currently, heis a Postdoctoral Scholar in the Department of Elec-trical Engineering and Department of Control andDynamical Systems (CDS) at California Institute

of Technology, Pasadena, CA. His research interests include cyber-physicalsystems, machine learning, and robotics.

Massimo Franceschetti (M’98-SM’11-F’18) re-ceived the Laurea degree (with highest honors) incomputer engineering from the University “Fed-erico II” of Naples, Italy, in 1997, the M.S. andPh.D. degrees in electrical engineering from theCalifornia Institute of Technology, Pasadena, CA,in 1999, and 2003, respectively. He is Professor ofElectrical and Computer Engineering at the Uni-versity of California at San Diego (UCSD). Beforejoining UCSD, he was a postdoctoral scholar at theUniversity of California at Berkeley for two years.

He has held visiting positions at the Vrije Universiteit Amsterdam, theÉcole Polytechnique Fédérale de Lausanne, and the University of Trento.His research interests are in physical and information-based foundations ofcommunication and control systems. He is co-author of the book “RandomNetworks for Communication” (2007) and author of “Wave Theory ofInformation” (2018), both published by Cambridge University Press. Dr.Franceschetti served as Associate Editor for Communication Networks ofthe IEEE Transactions on Information Theory (2009-2012), as associateeditor of the IEEE Transactions on Control of Network Systems (2013-2016), as associate editor for the IEEE Transactions on Network Science andEngineering (2014-2017), and as Guest Associate Editor of the IEEE Journalon Selected Areas in Communications (2008, 2009). He also served as generalchair of the North American School of Information Theory in 2015. He wasawarded the C. H. Wilts Prize in 2003 for best doctoral thesis in electricalengineering at Caltech; the S.A. Schelkunoff Award in 2005 for best paperin the IEEE Transactions on Antennas and Propagation, a National ScienceFoundation (NSF) CAREER award in 2006, an Office of Naval Research(ONR) Young Investigator Award in 2007, the IEEE Communications SocietyBest Tutorial Paper Award in 2010, and the IEEE Control theory societyRuberti young researcher award in 2012. He has been elected fellow of theIEEE in 2018, and became a Guggenheim fellow in the category of naturalsciences, engineering in 2019.

Gireeja Ranade is an Assistant Teaching Profes-sor in the EECS department at the University ofCalifonia at Berkeley. Before this, she was a Re-search at Microsoft Research AI, Redmond. Shereceived her BS in EECS from the MassachusettsInstitute of Technology and MS and Ph.D. in EECSfrom the University of California at Berkeley. Shedesigned and taught the first offerings for a newcourse introductory course sequence that introducesfirst-year undergraduates to machine learning anddesign at UC Berkeley (EECS 16A and 16B). She

is the recipient of the 2017 UC Berkeley Electrical Engineering Award forOutstanding Teaching.