Theorem~ 4.1 in the 2011 paper "A Framework for Control System Design Subject
to Average Data-Rate Constraints" allows one to lower bound average operational
data rates in feedback loops (including the situation in which encoder and
decoder have side information). Unfortunately, its proof is invalid.
  In this note we first state the theorem and explain why its proof is flawed,
and then provide a correct proof under weaker assumptions.Comment: Submitted to IEEE Transactions on Automatic Contro

Derpich, Milan S.

Østergaard, Jan

English

arXiv

Theorem 4.1 in the 2011 paper "A Framework for Control System Design Subject to Average Data-Rate Constraints" allows one to lower bound average operational data rates in feedback loops (including the situation in which encoder and decoder have side information). Unfortunately, its proof is invalid. In this note we first state the theorem and explain why its proof is flawed, and then provide a correct proof under weaker assumptions

    Aalborg UniversitetComments on "A Framework for Control System Design Subject to Average Data-RateConstraints"Derpich, Milan S.; Østergaard, JanPublication date:2021Link to publication from Aalborg UniversityCitation for published version (APA):Derpich, M. S., & Østergaard, J. (2021). Comments on "A Framework for Control System Design Subject toAverage Data-Rate Constraints".General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            - Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            - You may not further distribute the material or use it for any profit-making activity or commercial gain            - You may freely distribute the URL identifying the publication in the public portal -Take down policyIf you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access tothe work immediately and investigate your claim.Downloaded from vbn.aau.dk on: September 08, 20231Comments on“A Framework for Control System DesignSubject to Average Data-Rate Constraints”Milan S. Derpich and Jan Østergaard†AbstractTheorem 4.1 in the 2011 paper “A Framework for Control System Design Subject to Average Data-Rate Constraints” allows one to lower bound average operational data rates in feedback loops (includingthe situation in which encoder and decoder have side information). Unfortunately, its proof is invalid.In this note we first state the theorem and explain why its proof is flawed, and then provide a correctproof under weaker assumptions.I. INTRODUCTIONThe paper [1] provides a set of theorethical tools to design networked stochastic control problems byusing linear time invariant systems and entropy-coded dithered quantizers. In this paradigm, the averagedata rate is precisely linked to the signal-to-noise ratio in an identical system where the encoder anddecoder are replaced by an additive white Gaussian noise channel. In this way, the design task can becarried out in a stochastic control arena.In this note we reveal that the proof of [1, Theorem 4.1], a key result of that paper, is flawed. Then weuse a recent result by the authors [2, Theorem 2] not only to derive a simple (and correct) proof of [1,Theorem 4.1], but also to generalize it.II. NOTATION AND PRELIMINARIESIf x denotes a random sequence, then x = x(0), x(1), . . . and xk , x(0), x(1), . . . , x(k).Department of Electronic Engineering, Universidad Técnica Federico Santa Marı́a, Valparaı́so, Chile. milan.derpich@usm.cl.†Department of Electronic Systems, Aalborg University, Fredrik Bajers Vej 7, DK-9220, Aalborg, Denmark; janoe@ieee.org.March 25, 2021 DRAFTarXiv:2103.12897v1  [math.OC]  23 Mar 20212Denote the marginal probability distributions of two random variables x and y by µ, ν, respectively.Define the product measure π , µ× ν. The mutual information between x and y is defined asI(x; y) ,∫log(dmdπ)dm, (1)where dmdπ is the Radon-Nikodym derivative of m with respect to π [3].Rather than giving a definition of the conditional mutual information I(x; y|z) in terms of conditionaldistributions, it suffices for our purposes to define it through the chain rule of mutual information [4,Corollaries Corollary 2.5.1 and 7.14]:I(x; y|z) , I(x; y, z)− I(x; z) = I(x, z; y)− I(z; y). (2)If y ∈ A is a discrete-valued random variable with alphabet A, thenI(x; y|z) = H(y|z)−H(y|x, z), (3)whereH(y|z) , −E[∑u∈APr{y = u|z} log2 (Pr{y = u|z})](4)is the conditional entropy of y given z, Pr{y = u|z} is the probability that y = u given z and theexpectation E[] is with respect to z.Definition 1 (Directed Information with Forward Delay). In this paper, the directed information from xkto yk through a forward channel with a non-negative time varying delay of dxy(i) samples is defined asI(xk → yk) ,k∑i=0I(y(i);xi−dxy(i)|yi−1). (5)For a zero-delay forward channel, the latter definition coincides with Massey’s [5].Likewise, we adapt the definition of causally-conditioned directed information to the definitionI(xk → yk ‖ ek) ,k∑i=0I(y(i);xi−dxy(i)|yi−1, ei). (6)where, as before, dxy(i) is the delay from x to y(i).III. THEOREM STATEMENT AND THE FLAW IN ITS PROOFThe networked control system considered in [1], depicted in Fig. 1, considers lossy encoders E ,reproduction decoders D and EC-ED pairs that are causal, operating without delay. It is also consideredthe possible availability of side information sequences SE(k) at the encoder and SD(k) at the decoder.Such side information is contained in suitably defined sets SD(k) and SE(k), where SE(k) ∈ SE(k) andMarch 25, 2021 DRAFT3SECSECEDslosslessdecoderEslossyencoderscFduySEsource encoder source decoderEClosslessencoderDreproductiondecoderSDFigure 1. General source-coding scheme for the networked control of an LTI plant F considered in [1].SD(k) ∈ SD(k). Next, [1] defines the set SEC(k) , SE(k) ∩ SD(k), which contains the common sideinformation that becomes available at both the encoder and decoder sides at instant k. More explicitly,what the authors of [1] meant wasSE(k) , {SE(k), SEC(k)} , k = 0, 1, . . . (7)SD(k) , {SD(k), SEC(k)} , k = 0, 1, . . . (8)The output y of the plant F satisfiesy(k) = Fk(uk−1, dk, xo), (9a)where Fk is a (possibly time varying) deterministic mapping, xo ∈ Rno is the initial state of F , d, withd(k) ∈ Rnd , is an exogenous random process and u is the plant input. In [1], u and y were real valuedscalars, but for its Theorem 4.1 and for this note, their samples may take their outcomes from arbitrarysets. The other sequences are generated ass(k) = Ek(yk, SkE ) ∈ As, (9b)sc(k) = Hk(sk, SkEC) ∈ A(k), (9c)s(k) = H−1k (skc , SkEC), (9d)u(k) = Dk(sk, SkD), (9e)where As is a fixed countable set and A(k) is a countable set of prefix-free binary words.The expected length R(k) of any binary description sc(k) of the lossy encoder output symbol s(k)satisfies (see [6, Chapter 5], [7] and also [8])H(s(k)|sk−1, SkEC) ≤ R(k). (10)From these definitions and results, [1, Theorem 4.1] can be stated as followsMarch 25, 2021 DRAFT4Theorem 1 ( [1, Theorem 4.1]). Consider a source-coding scheme inside a feedback loop, as depictedin Fig. 1, where the sequences are generated via (9). If SkD ⊥ (xo, dk), and D is conditionally invertible,then, for all k ∈ N0,∑ki=0R(i) ≥ I(yk → uk). Thus,limk→∞1k + 1∑ki=0R(i) ≥ limk→∞1k + 1I(yk → uk) (11)provided both limits exist.This is a key result, because, combined with [1, eq. 9], it yields1k + 1I(yk → uk) ≤ 1k + 1k∑i=0R(i) ≤ 1k + 1I(yk → uk) + 1 [bits/sample], k = 0, 1, 2, . . . . (12)This result highlights the operational meaning of the directed information as a lower bound (tight towithin one bit) to the data rate of any given source code in a closed-loop system. This fact has beena crucial ingredient in characterizing the best rate-performance achievable in Gaussian linear quadraticnetworked control [9], [10].The proof of Theorem 4.1 in [1] is invalid since it relies upon [1, Lemma 4.2], whose first claim doesnot hold. The latter claim was that the Markov chain SiD ←→ ui−1 ←→ yi held. The flawed reasoningin the proof of this result is the following:“Given ui−i, it follows from (9a) that there exists a deterministic mapping Ti such thatyi = Ti(di, xo). Since SiD ⊥ (di, xo), it immediately follows that yi and SiD are independentupon knowledge of ui−i”.The problem with this argument is that it neglects the relationship between SiD and yi through si. Specif-ically, given ui−1, one may gain information about si−1 from SiD. This, in turn, may give informationabout yi. More generally, two independent random variables may cease to be independent given a thirdone.IV. A MORE GENERAL THEOREMIn this section we dorive a proof for Theorem 1 which does not require a conditionally invertibledecoder. This result builds upon a recent theorem by the authors that applies to the general feedbacksystem shown in Fig. 2. In this diagram, the blocks S1, . . . ,S4 represent possibly non-linear and time-varying causal discrete-time systems such that the total delay of the loop is at least one sample. In thesame figure, r, p, s, q are exogenous random signals (scalars, vectors or sequences), which could represent,for example, any combination of disturbances, noises, random initial states or side informations. Notethat any of these exogenous signals, in combination with its corresponding deterministic mapping Si, canMarch 25, 2021 DRAFT5S1 S2S3S4ey xur psqFigure 2. The general system considered by Theorem 2.yield any desired stochastic causal mapping (for example, a noisy communication channel, a zero-delaysource coder or decoder, or a causal dynamic system with disturbances and a random initial state).Theorem 2 (Full Conditional Closed-Loop Directed Data-Processing Inequality from [2, Thm. 2]).Consider the system shown in Fig. 2.If (q, s) ⊥ (r, p) and1 qki+1 ↔ qi ↔ si for i = 0, 1, . . . , k − 1, thenI(xk → yk‖qk) ≥ I(ek → uk). (13)NNow we can prove the following more general version of Theorem 1.Theorem 3. Consider a source-coding scheme inside a feedback loop, as depicted in Fig. 1, where thesequences are generated via (9). If (SkE ,SkD) ⊥ (xo, dk), and(SD(i+ 1),SD(i+ 2), . . . ,SD(k))↔ SiD ↔ SiE , i = 0, 1, . . . , k − 1, (14)then, for all k ∈ N0,∑ki=0R(i) ≥ I(yk → uk). Thus,lim supk→∞1k + 1∑ki=0R(i) ≥ lim supk→∞1k + 1I(yk → uk) (15)lim infk→∞1k + 1∑ki=0R(i) ≥ lim infk→∞1k + 1I(yk → uk) (16)1The Markov chain notation a ↔ b ↔ c means “a and c are independent when b is given”.March 25, 2021 DRAFT6Proof. We have thatk∑i=0R(i)(a)≥k∑i=0H(s(i)|si−1,SkD) (17)(b)≥k∑i=0(H(s(i)|si−1,SiD)−H(s(i)|si−1,SiD, yi))(18)(3)=k∑i=0I(s(i); yi|si−1,SiD) (19)(6)= I(yk → sk‖SkD) (20)≥ I(yk → uk) (21)where (a) follows from (10) and (8), (b) is from the non-negativity of entropy and the last inequalityfollows directly from Theorem 2.Remark 1. The additional independency assumptions made in Theorem 3 with respect to Theorem 1 arenot a problem for the other results in [1] that rely upon Theorem 4.1. This is so because they are met bythe only side information signals considered in [1]: identical dither sequences available to the encoderand decoder. NV. CONCLUSIONSWe have revealed a flaw in the proof of Theorem 4.1 in [1] and derived a proof for it without requiringan invertible decoder. Thus, we have extended the validity of [1, Theorem 4.1] in such a way that all theother results in [1] which depend on that theorem continue to hold.REFERENCES[1] E. I. Silva, M. S. Derpich, and J. Østergaard, “A framework for control system design subject to average data-rateconstraints,” IEEE Transactions on Automatic Control, vol. 56, no. 8, pp. 1886–1899, June 2011.[2] M. S. Derpich and J. Østergaard, “Directed data-processing inequalities for systems with feedback,” Submitted to Entropy,2021.[3] J. Yeh, Real analysis, 3rd ed. World Scientific, 2014.[4] R. M. Gray, Entropy and Information Theory, 2nd ed., ser. Science+Business Media. New York: Springer, 2011.[5] J. L. Massey, “Causality, feedback and directed information,” in Proc. Intl. Symp. Inf. Theory and its Appl., Hawaii, USA,Nov. 1990, pp. 303–305.[6] T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed. Hoboken, N.J: Wiley-Interscience, 2006.[7] C. Shannon, “A mathematical theory of cummunication,” in Bell Syst. Tech. Journal, vol. 27, no. 3, 1948, pp. 379–423.[Online]. Available: http://cm.bell-labs.com/cm/ms/what/shannonday/shannon1948.pdf[8] T. Linder, V. Tarokh, and K. Zeger, “Existence of optimal prefix codes for infinite source alphabets,” IEEE Transactionson Information Theory, vol. 43, no. 6, pp. 2026–2028, 1997.March 25, 2021 DRAFT7[9] E. I. Silva, M. S. Derpich, J. Østergaard, and M. A. Encina, “A characterization of the minimal average data rate thatguarantees a given closed-lop performance level,” IEEE Transactions on Automatic Control, vol. 61, no. 8, pp. 2171–2186,Aug. 2016.[10] T. Tanaka, P. M. Esfahani, and S. K. Mitter, “LQG control with minimum directed information: Semidefinite programmingapproach,” IEEE Transactions on Automatic Control, vol. 63, no. 1, pp. 37–52, 1 2018.March 25, 2021 DRAFT

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