Control with Minimum Communication Cost per Symbol

Transcription

1 Contro with Minimum Communication Cost per ymbo Justin Pearson, João P. Hespanha, Danie Liberzon Abstract We address the probem of stabiizing a continuous-time inear time-invariant process under communication constraints. We assume that the sensor that measures the state is connected to the actuator through a finite capacity communication channe over which an encoder at the sensor sends symbos from a finite aphabet to a decoder at the actuator. We consider a situation where one symbo from the aphabet consumes no communication resources, whereas each of the others consumes one unit of communication resources to transmit. This paper expores how the imposition of imits on an encoder s bit-rate and average resource consumption affect the encoder/decoder/controer s abiity to keep the process bounded. The main resut is a necessary and sufficient condition for a bounding encoder/decoder/controer which depends on the encoder s average bit rate, its average resource consumption, and the unstabe eigenvaues of the process. I. IRODUCTIO This paper addresses the probem of stabiizing a continuous-time inear time-invariant process under communication constraints. As in [1 7], we assume that the sensor that measures the state is connected to the actuator through a finite capacity communication channe. At each samping time, an encoder sends a symbo through the channe. The probem of determining whether or not it is possibe to bound the state of the process under this type of encoding scheme is not new; it was estabished in [2 4] that a necessary and sufficient condition for stabiity can be expressed as a simpe reationship between the unstabe eigenvaues of A and the average communication bit-rate. We expand upon this resut by considering the notion that encoders can effectivey save communication resources by not transmitting information, whie noting that the absence of an expicit transmission nevertheess conveys information. To capture this, we suppose that one symbo from the aphabet consumes no communication resources to transmit, whereas each of the others consumes one unit of communication resources. We then proceed to define the average cost per symbo of an encoder, which is essentiay the average fraction of non-free symbos emitted. This paper s main technica contribution is a necessary and sufficient condition for the existence of an encoder/decoder/controer that bounds the state of the process. This condition depends on the channe s average bit rate, the encoder s average cost per symbo, and the unstabe eigenvaues of A. Our resut extends [2] in the sense that as the constraint on the average cost per symbo is aowed to increase (becomes ooser), our necessary and sufficient condition becomes the condition from [2]. As with [2 4], our resut is constructive in the sense that we describe a famiy of encoder/decoder pairs that bound the process when our condition hods. A counterintuitive coroary to our main resut shows that if the process may be bounded with average bit-rate r, then there exists a bounding encoder/decoder/controer with average bit-rate r which uses no more than 50% non-free symbos in its symbo-stream. The remainder of this paper is organized as foows. ection III contains the main negative resut of the paper, namey that boundedness is not possibe when our condition does not hod. To prove this resut we actuay show that it is not possibe to bound the process with a arge cass of encoders which we ca M-of- encoders that incudes a the encoders with average cost per symbo not exceeding a given threshod. ection IV contains the positive resut of the paper, showing that when our condition does hod, there is an encoder/decoder pair that can bound the process; we provide the encoding scheme. II. PROBLEM TATEME Consider a stabiizabe inear time-invariant process 9x Ax Bu, x P R n, u P R m, (1) for which it is known that xp0q beongs to some bounded set X 0 R n. A sensor that measures the state xptq is connected to the actuator through a finite-data-rate, error-free, and deay-free communication channe. An encoder coocated with the sensor sampes the state once every T time units, and from this sequence of measurements txpkt q : k P 0 u causay constructs a sequence of symbos ts k P A : k P 0 u from a finite aphabet A. The encoder sends this symbo sequence through the channe at a rate of 1 symbo every T time units to a decoder/controer coocated with the actuator, which causay constructs the contro signa uptq, t 0 from the sequence of symbos ts k P A : k P 0 u that arrive at the decoder. We assume the channe faithfuy transmits each symbo without error. The positive time T between successive sampings is caed the samping period and has units of time units. The average bit-rate of an encoder/decoder pair is the amount of information that the encoder transmits in units of bits per time unit. For an encoder/decoder pair whose encoder transmits one symbo from an aphabet A every T time units, the pair s average bit-rate is given by r og 2 A. (2) T We consider encoder/decoder pairs whose aphabets each contain one specia symbo 0 P A that can be transmitted without consuming any communication resources, and

2 additiona symbos that each require one unit of communication resources per transmission. One can think of the free symbo 0 as the absence of an expicit transmission. The communication resources at stake may be energy, time, or any other resource that may be consumed in the course of the communication process. In order to capture the average rate that an encoder consumes communication resources, we define the average cost per symbo of an encoder as foows: We say an encoder has average cost per symbo not exceeding γ max if for every symbo sequence ts k u the encoder may generate, we have 1 M 1 Ņ km I sk 0 γ max O 1 M 1 for a integers, M satisfying M 0, where I sk 0 1 if the kth symbo is not the free symbo, and 0 if it is. The summation in (3) captures the tota resources spent transmitting symbos s M, s M 1,..., s. Motivating this definition of average cost per symbo is the observation that the efthand side has the intuitive interpretation as the average cost per transmitted symbo between symbos s M and s. As M Ñ 8, the rightmost big-oh term vanishes, eaving γ max as an upper bound on the average ong-term cost per symbo of the symbo sequence. ote that the average cost per symbo of any encoder never exceeds 1 and does not depend on the samping period T. Whereas an encoder/decoder pair s average bit-rate r ony depends on its symbo aphabet A and samping period T, its average cost per symbo depends on every possibe symbo sequence it may generate, and therefore depends on the encoder/decoder pair, the controer, the process (1), and the initia condition xp0q. The specific question considered in this paper is: under what conditions on the bit-rate and average cost per symbo does there exist a controer and encoder/decoder pair that keep the state of process (1) bounded? III. ECEARY CODITIO FOR BOUDEDE WITH LIMITED-COMMUICATIO ECODER It is we known from [2 4] that it is possibe to construct a controer and encoder/decoder pair that bounds process (1) with average bit-rate r ony if r n 2 λ i ras, (4) where n denotes the base-e ogarithm, and the summation is over a eigenvaues of A with nonnegative rea part. The resut that foows shows that a arger bit-rate may be needed when one poses constraints on the encoder s average cost per symbo γ max. pecificay, when γ max {p 1q the (necessary) stabiity condition reduces to (4), but when γ max {p 1q a bit-rate arger than (4) is necessary to compensate for the diution of information content in the transmitted symbos due to the constraint on the average cost per symbo. Theorem 1: Consider an encoder/decoder pair with samping period T, an aphabet with nonfree symbos and one (3) free symbo, and an average cost per symbo not exceeding γ max. If this pair keeps the state of process (1) bounded for every initia condition x 0 P X 0, then we must have r fpγ max, q n 2 λ i ras, (5) where the bit-rate r is reated to and T via Equation (2), and the function f : r0, 1s r0, 8q Ñ r0, 8q is defined as # Hpγq γ og2 og fpγ, q 2 p 1q 0 γ γ 1, (6) and Hppq p og 2 ppq p1 pq og 2 p1 pq is the base-2 entropy of a Bernoui random variabe with parameter p. Remark 1: The function f is potted in Figure 1 for severa vaues of. It is worth making two observations regarding f. The first observation is that the function fpγ, q is monotone nonincreasing in for any fixed γ P r0, 1s, which impies that smaer aphabets are preferabe to arge ones when trying to satisfy (5) with a given fixed bit-rate. The second observation is that the average cost per time unit, which is γ{t, can be made arbitrariy sma in two different ways. 1) One coud pick T very arge, then everaging the fact that rf pγ, q is monotone increasing in, pick arge enough to satisfy (5). This approach has two downsides: First, with a arge samping period, the state, athough remaining bounded, can grow quite arge between transmissions. econd, arge means that the encoder/decoder pair must store and process a arge symbo ibrary, adding compexity to the pair s impementation. 2) Aternativey, one can make γ{t arbitrary sma by observing that the sequences γ k e k, T k e k? k, k P 0 have the property that γ k Ñ 0, T k Ñ 0, and γ k {T k Ñ 0, but Hpγ k q{t k Ñ 8, and hence r k fpγ k, q n 2 Ñ 8 (where r k og 2 p 1q{T k ). This means that one can find k P 0 to make the average cost per time unit γ k {T k arbitrariy sma, and aso satisfy the necessary condition (5). The drawback of this approach is that to achieve a sma samping period T in practice requires an encoder/decoder pair with a very precise cock. Remark 2: The addition of the free symbo effectivey increases the data rate without increasing the rate of resource consumption, as seen by the foowing two observations: Without the free symbos, the size of the aphabet woud be and the bit-rate woud be og 2 pq{t. It coud happen that this bit-rate is too sma to bound the pant, yet after the introduction of the free symbo, the condition (5) is satisfied. ince γ max is the fraction of non-free symbos, then the quantity r γ max is the number of bits per time unit spent transmitting non-free symbos. But since fpγ, q γ, again we see that the free symbos hep satisfy (5).

3 Fig. 1. A pot of f pγ, q for 1, 4, 20. A. Theorem 1 Proof etup We ead up to the proof of Theorem 1 by first estabishing three emmas centered around a restricted arge cass of encoders caed M-of- encoders. We first define M-of- encoders, which essentiay partition their symbo sequences into -ength codewords, each with M or fewer non-free symbos. Lemma 1 demonstrates that every encoder with a imited average cost per symbo is an M-of- encoder for appropriate and M. ext, in Lemma 2 we estabish a reationship between the number of codewords avaiabe to an M-of- encoder and the function f as defined in (6). Then, in Lemma 3 we everage previous work to estabish a necessary condition for an M-of- encoder to bound the state of the process. Finay, the proof of Theorem 1 everages these three resuts. To introduce the cass of M-of- encoders, we define an -symbo codeword to be a sequence ts 1, s 2,..., s u of consecutive symbos starting at an index k 1, with P 0. An M-of- encoder is an encoder for which every -symbo codeword has M or fewer non-free symbos, i.e., k 1 I sk 0 P 0. (7) The tota number of distinct -symbo codewords avaiabe to an M-of- encoder is thus given by Lp, M, q tm u i0 i i, (8) where the ith term in the summation counts the number of -symbo codewords with exacty i non-free symbos. ote that in keeping with the probem setup, the M-of- encoders considered here each draw their symbos from the symbo ibrary A t0, 1,..., u and transmit symbos with samping period T. An intuitive property of M-of- encoders is that an M-of- encoder has average cost per symbo not exceeding M{. To see this, suppose M and are fixed and et 1, 2 P 0 be arbitrary with The width of the interva r 1, 2 s may not be an integer mutipe of, but nevertheess it is between two integer mutipes of : we have pk 2q 2 1 ˆ 2 ˆ 1 k for some non-negative integer k P 0. From the first inequaity we obtain k , and from the second inequaity we have 2 k 1 I sk 0 km, because in each of the k -ength intervas containing r 1, 2 s there are at most M non-free symbos. Combining these yieds 2 k 1 I sk 0 M p 2 1 1q 0, (9) where 0 is some constant. Dividing both sides by p 2 1 1q and rearranging, (3) foows. The fact that an M-of- encoder refrains from sending expensive codewords effectivey reduces its abiity to transmit information. Indeed, since an M-of- encoder takes time units to transmit one of Lp, M, q codewords, the encoder transmits merey og 2 Lp,M,q bits per time unit. For M, this is stricty ess than the encoder s average bit-rate r. The first emma, proved in the appendix, shows that the set of M-of- encoders is compete in the sense that every encoder with average cost per symbo not exceeding a finite threshod γ max is actuay an M-of- encoder for sufficienty arge and M γ max. Lemma 1: For every encoder/decoder pair with average bit-rate r and average cost per symbo not exceeding γ max P r0, 1s, and every constant ɛ 0, there exist positive integers M and with M γ max p1 ɛq such that the encoder is an M-of- encoder. The next emma estabishes a reationship between the number of codewords Lp, M, q avaiabe to an M-of- encoder and the function f defined in (6). Lemma 2: For any, P 0 and γ P r0, 1s, the function L defined in (8) and the function f defined in (6) satisfy og 2 Lp, γ, q og 2 p 1qfpγ, q, (10) with equaity hoding ony when γ 0 or γ 1. Moreover, we have asymptotic equaity in the sense that og im 2 Lp, γ, q og Ñ8 2 p 1qfpγ, q. (11) Proof of Lemma 2. Let and be arbitrary positive integers. First we prove (10) for γ P p0, s. Appying the 1

4 Binomia Theorem to the identity 1 pγ p1 γqq, we obtain Ņ 1 γ i p1 γq i i i0 ince each term in the summation is positive, keeping ony the first tγu terms yieds the inequaity 1 tγu i0 i γ i p1 γq i (12) ext, a cacuation presented as Lemma 5 in the appendix reveas that γ i p1 γq i Hpγq i 2 γ (13) for a, P 0, γ P p0, {p 1qs, and i P r0, γs. Using this in (12) and simpifying yieds og 2 Lp, γ, q Hpγq γ og 2 (14) for a, P 0 and γ P p0, {p 1qs. ince og 2 p 1qfpγ, q Hpγq γ og 2 when γ P, (14) 0, 1 proves (10) for γ P p0, 1s. ext, suppose γ P p 1, 1q and observe from (8) that Lp, M, q is a sum of positive terms whose index reaches tmu, hence Lp, γ, q is stricty ess than Lp,, q for any γ 1. We concude that og 2 Lp, γ, q og 2 Lp,, q (15) og 2 p 1q (16) ince og 2 p 1qfpγ, q og 2 p 1q for γ P p 1, 1q, this concudes the proof of (10) for γ P p0, 1q. The proof of (10) for γ 0, foows merey from inspection of (10), and the γ 1 case foows from the equaity in (15). The proof of the asymptotic resut (11) appears in [8]. This concudes the proof of Lemma 2. The foowing emma provides a necessary condition for an M-of- encoder to bound process (1). Lemma 3: Consider an M-of- encoder/decoder pair using symbos t0,..., u with samping period T. If the pair keeps the state of (1) bounded for every initia condition, then we must have n Lp, M, q λ i ras. (17) Proof of Lemma 3. Consider an encoder/decoder/controer tripe that bounds the state of the process (1) and whose encoder is an M-of- encoder using symbos t0,..., u and samping period T. ince the encoder sends one of LpM,, q codewords every time units, the average bitrate of the encoder is og 2 LpM,, q{. Theorem 1 of [2] proves that if the state of the process (1) is bounded by an encoder/decoder/controer tripe under the communication constraints described in our probem setup, then the pair s average bit-rate must not be ess than 1 n 2 i:rλ λ iras 0 iras. Hence, we have og 2 LpM,, q 1 λ i ras, n 2 from which (17) foows. This proves the emma. ow we are ready to prove Theorem 1. Proof of Theorem 1. By Lemma 1, for any ɛ 0 there exist M, P 0 with M γ max p1 ɛq for which the encoder/decoder is an M-of- encoder. ince the state of the process is kept bounded, by Lemma 3 we have og λ i ras 2 Lp, M, q n 2. (18) ince L is monotonicay nondecreasing in its second argument and M γ max p1 ɛq, we have og 2 Lp, M, q og 2 Lp, γ max p1 ɛq, q. (19) Lemma 2 impies that og 2 Lp, γ max p1 ɛq, q rfpγ max p1 ɛq, q. (20) Combining these and etting ɛ Ñ 0, we obtain (5). This competes the proof of Theorem 1. IV. UFFICIE CODITIO FOR BOUDEDE WITH LIMITED-COMMUICATIO ECODER The previous section estabished a necessary condition (5) on the average bit-rate and average cost per symbo of an encoder/decoder pair in order to bound the state of (1). In this section, we show that that condition is aso sufficient for a bounding encoder/decoder to exist. The proof is constructive in that we provide the encoder/decoder. Theorem 2: Assume that A is diagonaizabe. For every P 0, T 0, and γ max P r0, 1s satisfying rfpγ max, q n 2 λ i ras, (21) where r is defined in (2) and the function f is defined in (6), there exists a controer and an encoder/decoder pair using nonfree symbos, samping period T, and average cost per symbo not exceeding γ max, that keeps the state of the process (1) bounded for every initia condition x 0 P X 0. The proof of Theorem 2 reies on the foowing emma, which provides a sufficient condition for the existence of an M-of- encoder to bound the state of process (1). Lemma 4: Assume that A is diagonaizabe. For every T 0 and, M, P 0 with M 0 satisfying n Lp, M, q λ i ras (22) there exists an M-of- encoder on aphabet t0,..., u with samping period T that keeps the state of the process (1) bounded for every initia condition.

5 Proof of Lemma 4. This proof buids on Theorem 2 from [2], which provides sufficient conditions on an encoder s average bit-rate to ensure the existence of a stabiizing controer and encoder/decoder pair. The resut states that if an encoder s average bit-rate r satisfies r 1 n 2 λ i ras, (23) where A is the continuous-time process matrix, then there exists a controer and encoder/decoder pair that bound the state of the process. An outine of the proof of this resut from [2] foows. The encoder works by pacing a bounding rectange around the voume where the state is known to ie, and partitions it into two pieces aong its argest axis. Then the encoder transmits a 0 or 1, depending on which subrectange the state ies in. ince the decoder can cacuate the bounding rectanges with its knowedge of X 0 and process dynamics in (1), it estimates the state to be the centroid of whichever sub-rectange the received symbo corresponds to. The decoder transmits this state estimate to the controer, which can be any stabiizing inear state-feedback controer. We now provide a summary of the proof of Lemma 4, and refer the reader to [8] for the detaied proof. uppose T,, M, satisfy the constraints in the statement of the emma as we as (22). Theorem 2 from [2] guarantees the existence of a stabiizing encoder/decoder/controer tripe with average bit-rate og 2 Lp, M, q{. A that remains is to adapt this encoder/decoder pair into our framework, i.e., an M-of- encoder that uses 1 symbos with samping period T. We do this by buiding an encoder which runs a copy of the encoder from [2] internay. The outer encoder feeds sampes of the state to the inner encoder, from which the inner encoder generates a string of 1 s and 0 s. Due to the bit-rate being og 2 Lp, M, q{, after T time units the inner encoder has generated a string of ength og 2 Lp, M, q. There are Lp, M, q possibe such strings, and so the outer encoder can map this string uniquey to one of the Lp, M, q possibe -ength codewords with M or fewer non-free symbos from the aphabet t0,..., u. The outer encoder transmits this codeword across the channe to the decoder, which performs the inverse mapping to recover the string of og 2 Lp, M, q 1 s and 0 s, which it deivers to its inner decoder. From this bit-string and knowedge of the process dynamics, the inner decoder computes a state estimate, which it deivers to the stabiizing inear state-feedback controer. This process repeats each T time units. ince the inner encoder/decboder pair bound the process, the outer M-of- encoder described here does as we. This proves the emma. ow we are ready to prove Theorem 2. Proof of Theorem 2. Assume that, T, and γ max satisfy (21), so that ɛ rfpγ max, q n 2 λ i ras 0. (24) n Lp,γmax,q Equation (11) estabishes that gets arbitrariy cose to rfpγ max, q as we increase, so we pick sufficienty arge to satisfy rfpγ max, q n 2 n Lp, γ max, q ɛ{2. (25) By (8), we have Lp, tγ max u, q Lp, γ max, q for every, γ max, and. etting M tγ max u, then by (24) and (25) we have found and M satisfying n Lp,M,q i:rλ λ iras 0 iras. Hence by Lemma 4, there exists an M-of- encoder/decoder which bounds the state of (1). ince a M-of- encoder/decoders have average cost per symbo not exceeding M{, and this encoder/decoder satisfies M{ γ max, we concude that this encoder has an average cost per symbo not exceeding γ max. This concudes the proof, since we have found the desired encoder. An unexpected consequence of Theorems 1 and 2 is that when it is possibe to keep the state of a process bounded with a given bit-rate r og 2 p 1q{T, one can aways find M-of- encoders that bound it for (essentiay) the same bit-rate and average cost per symbo not exceeding {p 1q, i.e., approximatey a fraction 1{p 1q of the symbos wi not consume communication resources. In the most advantageous case, the encoder/decoder use the aphabet t0, 1u and the encoder s symbo stream consumes no more than 50% of the communication resources. The price paid for using an encoder/decoder with average cost per symbo near {p 1q is that it may require prohibitivey ong codewords (arge ) as compared to an encoder with higher average cost per symbo. This is due to the fact that n Lp, γ max, q{ q is monotonicay nondecreasing in γ max, so that a smaer γ max generay requires a arger to satisfy (25). Coroary 1: If the process (1) can be bounded with an encoder/decoder pair with symbo aphabet t0, 1,..., u and samping period T, then for any ɛ 0 with T ɛ, there exists an M-of- encoder with aphabet t0, 1,..., u, samping period T ɛ, and average cost per symbo not exceeding {p 1q that bounds its state. Proof of Coroary 1. Let ɛ be arbitrary in p0, T q. ince the controer and encoder/decoder pair bound the system and the average cost per symbo never exceeds 1, by Theorem 1 we have rfp1, q n 2 λ i ras, (26) where r og 2 p 1q{T. ince fp{p 1q, q fp1, q for a P 0, we have og 2 p 1q f T ɛ 1, r f 1, rfp1, q n 2 λ i ras. (27) By Theorem 2, there exists an encoder/decoder pair with symbo aphabet t0,..., u, samping period T ɛ, and average cost per symbo not exceeding {p 1q which

6 bounds the state of (1). By Lemma 1, this encoder is an M-of- encoder for appropriatey chosen M and. V. COCLUIO AD FUTURE WORK In this paper we considered the probem of bounding the state of a continuous-time inear process under communication constraints. We considered constraints on both the average bit-rate and the average cost per symbo of an encoding scheme. Our main contribution was a necessary and sufficient condition on the process and constraints for which a bounding encoder/decoder/controer exists. In the absence of a imit on the average cost per symbo, the conditions recovered previous work. A surprising coroary to our main resut was the observation that one may impose a constraint on the average cost per symbo without necessariy needing to oosen the bit-rate constraint. pecificay, we proved that if a process may bounded with a particuar bit-rate, then there exists a (possiby very compex) encoder/decoder that can bound it using no more than 50% non-free symbos on average, yet obeying the same bit-rate. This was surprising because one woud expect the prohibition of some codewords woud require that the encoder aways compensate by transmitting at a higher bit-rate. We observed in Remark 1 that smaer aphabets incur a smaer penaty on the conditions for boundedness in Theorems 1 and 2. This suggests that encoding schemes with sma aphabets may be abe to bound the state of the process with bit-rates and average costs not far above the minimum theoretica bounds as estabished in Theorems 1 and 2. Eventbased contro strategies comprise one such cass of encoders; they use a sma number of non-free symbos to notify the decoder/controer about certain state-dependent events. We have preiminary resuts showing that event-based encoders can indeed be used to produce encoder/decoder pairs that amost achieve the minimum achievabe average cost per symbo that appears in Theorems 1 and 2. Finay, our probem setup considered merey wherether there exists a bounding encoder/decoder/controer tripe. It seems natura to extend this setup to finding stabiizing tripes. APPEDIX Proof of Lemma 1. By the definition of average cost per symbo not exceeding γ max in (3) there exists an integer 0 P 0 such that for any symbo sequence ts k u that the encoder generates, we have Ņ i1 I si0 0 γ P 0. (28) Pick P 0 arge enough to satisfy 0 2 ɛγ max and pick M t 0 2 γ max u. Combining these with (28), we obtain Ņ i1 I si0 0 γ max M 0 2 γ max γ max p1 ɛq, which estabishes that M γ max p1 ɛq. This competes the proof. Lemma 5: The foowing inequaity hods for a, P 0, q P p0, {p 1qs, and i P r0, qs: q i p1 qq i Hpqq i 2 q (29) where Hpqq q og 2 q p1 qq og 2 p1 qq is the the base- 2 entropy of a Bernoui random variabe with parameter q. Proof of Lemma 5. Let,, q, and i take arbitrary vaues from the sets described in the emma s statement. ince og 2 is a monotone increasing function, og 2 pq{p1 qqq for q 0 is maximized at the right endpoint vaue, q {p 1q, where it equas og 2. This eads to og 2 q og 2 p1 qq og 2 (30) for a P 0 and q P p0, {p 1qs. ext, since i P r0, qs by assumption, then i q 0. Mutipying (30) by i q, standard agebraic manipuations yied i og 2 q p iq og 2 p1 qq from which (29) foows. Hpqq pi qq og 2, REFERECE [1] R. Brockett and D. Liberzon, Quantized feedback stabiization of inear systems, Automatic Contro, IEEE Transactions on, vo. 45, no. 7, pp , Ju [2] J. P. Hespanha, A. Ortega, and L. Vasudevan, Towards the contro of inear systems with minimum bit-rate, in Proc. of the Int. ymp. on the Mathematica Theory of etworks and yst., Aug [3] G. air and R. Evans, Communication-imited stabiization of inear systems, in Decision and Contro, Proceedings of the 39th IEEE Conference on, vo. 1, 2000, pp vo.1. [4]. Tatikonda and. Mitter, Contro under communication constraints, Automatic Contro, IEEE Transactions on, vo. 49, no. 7, pp , juy [5] G.. air and R. J. Evans, Exponentia stabiisabiity of finite-dimensiona inear systems with imited data rates, Automatica, vo. 39, no. 4, pp , [6] A. ahai and. Mitter, The necessity and sufficiency of anytime capacity for stabiization of a inear system over a noisy communication ink part i: caar systems, Information Theory, IEEE Transactions on, vo. 52, no. 8, pp , Aug [7] A. Matveev and A. avkin, Mutirate stabiization of inear mutipe sensor systems via imited capacity communication channes, IAM Journa on Contro and Optimization, vo. 44, no. 2, pp , [8] J. Pearson, J. P. Hespanha, and D. Liberzon, Contro with minimum communication cost per symbo, University of Caifornia, anta Barbara, Tech. Rep., Mar [Onine]. Avaiabe: edu/ hespanha/techrep.htm