A Tiger by the Tail: When Multiplicative Noise Stymies Control

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1 A Tiger by he Tail: When Muliplicaive Noise Symies Conrol Jian Ding, Yuval Peres and Gireeja Ranade Microsof Research, Redmond, WA Universiy of Chicago, IL Absrac This paper considers he sabilizaion of an unsable discree-ime linear sysem ha is observed over a channel corruped by coninuous muliplicaive noise. The main resul is a converse bound ha shows ha if he sysem growh is large enough he sysem canno be sabilized in a mean-squared sense. This is done by showing ha he probabiliy of he sae magniude remains bounded mus go o zero wih ime. I was known ha a sysem wih muliplicaive observaion noise can be sabilized using a simple linear sraegy if he sysem growh is suiably bounded. However, i was no clear wheher non-linear conrollers could overcome arbirarily large growh facors. One difficuly wih using he sandard approach for a daa-rae heorem syle converse is ha he muual informaion per round beween he sysem sae and he observaion is poenially unbounded wih a muliplicaive noise observaion channel. Our proof echnique recursively bounds he condiional densiy of he sysem sae insead of focusing on he second momen o bound he progress he conroller can make. I. INTRODUCTION We consider he conrol of a sysem observed over a muliplicaive noise channel, and provide a converse bound o define a region in which he sysem canno be sabilized in he mean-square sense. Muliplicaive observaion noise can model he effecs of fas-fading communicaion channel or sampling and quanizaion errors [], [2]. A more deailed discussion of muliplicaive noise models is available in [3]. Specifically, we analyze he following sysem Fig. : X a,n+ = a X a,n U a,n, Y a,n = Z n X a,n. In he preceding formulaion, he sysem sae is represened by X a,n a ime n, and he conrol U a,n can be any funcion of he curren and previous observaions Y a,0 o Y a,n. The Z n s are i.i.d. random variables wih a known coninuous disribuion. The realizaion of he noise Z n is unknown o he conroller, much like he fading coefficien of a channel migh be unknown o he ransmier or receiver in non-coheren communicaion. The conroller s objecive is o sabilize he sysem in a mean-square sense, i.e. ensure ha sup n E[ X a,n 2 ] <. For a simple achievable sraegy, suppose he Z n s are i.i.d. wih mean and variance σ 2. I is easy o verify ha a simple memoryless linear sraegy can mean-square sabilize he sysem if and only if a 2 + σ. 2 X n Y n = X n Z n Channel Sysem Conroller Fig.. The sae X n is observed hrough a muliplicaive noise channel, modeled by Y n = X nz n. The gain a is suppressed in he figure noaion. A. Relaed work This fundamenal problem is inspired by many issues ha have been long sudied in conrol and we only menion some represenaive references here. I is conneced o he vas body of work on daa-rae heorems and conrol wih communicaion consrains such as [4], [5]. Nair e al. provide a survey of relaed problems in [6]. For a sysem ha is growing by a facor of a a each ime, he daa-rae heorems ell us ha a noiseless observaion daa rae R > log a is necessary and sufficien o sabilize he sysem [4]. A relaed problem is ha of esimaing a linear sysem over muliplicaive noise. While early work on his had been limied o eploring linear esimaion sraegies [7], some more recen work used a genie-based approach o show a general converse resul for he esimaion problem over muliplicaive noise for boh linear and non-linear sraegies [2]. The conrol problem in his paper can also be inerpreed as an acive esimaion problem for X 0, and our converse ha also uses a side-informaion genie applies o boh linear and non-linear conrol sraegies. However, echniques from he esimaion converse or he daa-rae heorems do no work for a converse here. Unlike he esimaion problem, we canno describe he disribuion of X n in our problem since he conrol U n is arbirary. For he same reason, we also canno bound he range of X n and we canno bound he rae across he channel o use a daa-rae heorem approach. Anoher inspiraion for our problem is he work on inermien Kalman filering [8], [9] and conrol over packe dropping neworks [0] i.e. esimaion and conrol over Bernoulli muliplicaive noise channels. The seup in his paper generalizes hose seups o consider a general coninuous muliplicaive noise on he observaion. U n

2 Finally, he resul in his paper can also be hough of as eending classic uncerainy hreshold principle [] o also undersand muliplicaive observaion uncerainy. The uncerainy hreshold principle provides a hreshold for he sabilizabiliy of a sysem wih Gaussian uncerainy on he sysem gain and he conrol gain. We would evenually like o esablish he conrol capaciy of he sysem, as done for sysems wih muliplicaive noise on he acuaion channel conrol gain in [2], [3]. If he conroller for a sysem wih muliplicaive observaion noise is resriced o using linear conrol sraegies, hen is performance limi is he same as ha of sysem wih muliplicaive acuaion noise as in [2]. However, he approach from [2] does no seem o work for he sysem considered here. B. Proof approach This paper inroduces a non-sandard converse approach and we believe hese echniques are a primary conribuion of he work. Insead of focusing on he second-momen, our proof bounds he densiy of he sae and hus shows he insabiliy of any momen of he sae. A key elemen of he proof is ha a genie observes he sae of he sysem and provides a quanized version of he logarihm of he sae o he conroller a each ime as era side-informaion in addiion o he muliplicaive noise observaion. This side informaion bounds he sae in inervals of size 2 k around he iniial sae X 0 wih k increasing as ime increases. We know from resuls on non-coheren communicaion [3] and carry-free models [4] ha only he order of magniude of he message can be recovered from a ransmission wih muliplicaive noise. As a resul, his side-informaion does no effecively provide much era informaion, bu i allows us o quanify he rae a which he conroller may make progress. This paper focuses on a Gaussian disribuion for he muliplicaive noise, bu he ideas can be eended o general disribuions under mild assumpions. II. PROBLEM STATEMENT AND SETUP Consider he sysem S a in. For simpliciy, le he iniial sae X a,0 be disribued as X a,0 N 0,. Le Z n be i.i.d. Gaussian random variables such ha Z n N, σ 2, f Z z = 2π e z 2 2σ 2. Wihou loss of generaliy we can assume ha he Z n have mean. In his paper we show ha here eiss a > 0 such ha P X a,n < M 0 as n for all M. This implies ha for he same a, he second momen E[Xa,n] 2 is always unbounded, i.e. he sysem canno be sabilized in he secondmomen sense. To obain his hreshold i suffices o consider he sysem, S i.e. se a =. The disribuions on he Z n variables and he iniial sae X 0 are he same as sysem S a. X n+ = X n U n, Y n = Z n X n. 2 The sysem S a behaves as S scaled by a n when he conrol U a,n = a n U n is applied. A similar resul is given in [3, Ch. 5, Lemma 5.4.4]. Hence, we can wrie: P X a,n < M = P X n < a n M. A. Noaion and definiions Le S n = n i=0 U i for n. Hence, X n = X 0 S n. The goal of he conroller is o have S n be as close o X 0 as possible. We will rack he progress of he conroller hrough inervals I n ha conain X 0 and are decreasing in lengh. Le di n, S := inf In S denoe he disance of a poin S from he inerval I n. Definiion 2.: For all n 0 and for any k Z, here eiss a unique ineger hk such ha X 0 [ hk, hk+. 2 k 2 k Le Jk := [ hk, hk+. We now inducively define he 2 k 2 k inervals I n using he inervals Jk. Define K 0 := min{k 0 djk, 0 2 k }, and K n := min{k k > K n, djk, S n 2 k }. Wrie H n := hk n and I n := [ Hn 2 2 Le Y0 n indicae he observaions Y 0 o Y n, and le F n := {Y0 n, K0 n, H0 n }, which is he oal informaion available o he conroller a ime n. Le f Xn F n be he condiional densiy of X n given F n Kn Hn+ Kn, Kn. I n [ ] S n X 0 2 K n I n S n [ ] S n X n X 0 2 Kn Fig. 2. A caricaure illusraing he inervals I n and I n S n. B. Side-informaion lemma This lemma uses K n o bound how fas S n approaches X 0. The easy proof is deferred o he Appendi. Lemma 2.: 2 Kn X 0 S n, and if K n > K n +, hen X 0 S n 2 2 Kn. III. CONVERSE RESULT Theorem 3.: There eiss a R, 0 < a < such ha P X a,n < M 0 for all M <. Sraegy: The proof provides an eponenial upper bound on he densiy f Xn F n, which implies ha he conroller canno localize he sae wihin an eponenially shrinking bo. The bound on he densiy is obained by esimaing he change in he densiy from ime n o n + due o he observaion Y n. The inerval I n i.e. H n and K n is provided o he conroller as side informaion a ime n, which helps generae he bound. The firs sep of he proof uses Lemma 3.2 o recursively bound he raio f Xn Fn f Xn w F n. This leads o an eponenial bound

3 on he densiy f Xn F n in erms of he side-informaion K n in 8. This bound helps us generae bounds on he probabiliy he even of ineres { X n < a n M}, which we show hen mus go o 0 as n. Proof: Consider f Xn F n = f Xn Y n, K n, H n, F n = f Y n,k n,h n Y n, K n, H n X n =, F n f Xn F n f Yn,K n,h n Y n, K n, H n F n Formally, f Yn,K n,h n is a densiy wih respec o a produc of he Lebesgue measure and wo couning measures. Since X 0 I n, he conroller knows ha X n I n S n, where I n S n represens he inerval I n shifed by S n. We can calculae he raio of he densiies a, w I n S n as: f Xn F n f Xn w F n = f X n F n f Yn Y n X n =, F n f Xn w F n f Yn Y n X n = w, F n. 3 Since K n and H n are defined by I n, he condiional disribuions of K n and H n given X n = and X n = w are equal for, w I n S n. So hese erms cancel when we consider a raio, giving 3. Taking logarihms and using he riangle inequaliy gives he following recursive lemma, which we prove in he Appendi. Lemma 3.2: F n f Xn w F n 4 σ 2 σ 2 + 2Z n 2Z n 2 Kn w + F n f Xn w F n. Now, based on he conrol law we know ha f Xn F n = f Xn + U n F n, since U n is F n measurable. Subsiuing his ino 4 and unfolding recursively gives: F n f Xn w F n 5 n σ 2 σ 2 + 2Z i 2Z i 2 Ki w + log f X 0 + S n f X0 w + S n. i= The inequaliy 5 separaes he effec of he uncerainy due o X 0 and he subsequen uncerainy due o he observaions and conrol. To bound he effec of he iniial sae le η n = log f ma X0 +S n,w In S n f X0 w+s n. Since I n is an inerval of size a mos 2 n ha conains X 0 we ge ha: η n 2 X n 2 X 0 2 n 2 2 n X 0. 6 Now, we define Ψ n = 2 n Kn σ 2 σ 2 + 2Z i 2Z i 2 Ki, 7 i= and rewrie 5 as: F n f Xn w F n Ψ n 2 Kn w + η n. Finally, we are in a posiion o bound f Xn F n : f Xn F n ep{ψ n 2 Kn w + η n }f Xn w F n. 8 We inegrae 8 over an inerval of lengh γ = 2 Kn T wih a one end poin. So w 2 Kn T. Here, T R is a consan ha we will choose laer. Such an inerval can be fi ino I n o he lef or righ of any depending on where is in he inerval. Assuming wihou loss of generaliy ha is he lef endpoin of he inegraion inerval we compue ha f Xn F n dw e Ψn2Kn w +η n f Xn w F n dw Bound w on he RHS by 2 Kn T. Inegraing gives γ f Xn F n e Ψn2Kn 2 Kn T +η n f Xn w F n dw e Ψn2 T +η n, since he densiy inegraes ou o. Hence, f Xn F n e Ψn2 T +η n 2 Kn+T. 9 This gives us a bound on he densiy of X n in erms of K n. Since he K n s are racking he magniude of X n wha remains o be done is o bound he growh of he K n s. To do his we firs sae he following lemma abou he crucial quaniy Ψ n. Lemma 3.3: For T > C σ, where C σ is a consan ha depends only on σ, he epecaion E[e Ψn2 T ] is uniformly bounded for all n. The ne lemma uses Lemma 3.3 and 9 o show ha he K n grow a mos linearly. Lemma 3.4: There eiss a consan C = C σ,t depending only on σ, T such ha PK n K 0 > C σ,t n 0. The proofs of Lemmas 3.3, 3.4 are deferred o he Appendi. Le G n denoe he even ha K n K 0 > Cn, and G c n is complemen. Then he even { X n < a n M} is covered by: P X n < a n M 0 PG n + PK 0 > n + P X n a n M, G c n, K 0 n. We evaluae he hree erms one by one. For he firs erm in 0, we have PG n = PK n K 0 > Cn 0 as n from Lemma 3.4. The second erm, PK 0 > n, capures he case where he iniial sae X 0 migh be very close o zero. However, evenually his advanage dies ou for large enough n, since PX 0 < 2 n 0 as n.

4 The las erm in 0 remains. By he law of ieraed epecaion: P X n < a n M, G c n, K 0 n = E[P X n < a n M, G c n, K 0 n F n ]. We focus on he erm condiioned on F n : P X n < a n M, G c n, K 0 n F n = E[ { Xn <a n M} {G c n } {K0n} F n ] = P X n < a n M F n {G c n } {K0n}. Now, we can apply 9 o ge P X n < a n M F n = Then we can bound as a n M f Xn F n d a n M a n M e η+ψn2 T a n M 2 Kn+T d = 2Ma n e ηn+ψn2 T 2 Kn+T. P X n < a n M F n {G c n } {K0n} 2Ma n e ηn+ψn2 T 2 C+n+T, 2 since K n Cn + K 0 and K 0 n implies K n C + n. Taking epecaions on boh sides we ge: P X n < a n M, G c n, K 0 n 2Ma n 2 C+n+T E[e ηn e Ψn2 T ]. 3 By Lemma 3.3 and 6, he above epression 3 ends o 0 for a > 2 C+. Thus, all hree probabiliies in 0 converge o 0 as n. Hence, if a > 2 C+ hen P X n < a n M 0 for all M. IV. CONCLUSION AND FUTURE WORK This paper provides a firs proof-of-concep converse for a conrol sysem observed over coninuous muliplicaive noise. However, here is an eponenial gap beween he scaling behavior of he achievable sraegy and he converse. I sill remains o be seen if he achievable region can grow wih nonlinear conrollers o significanly bea he a 2 < + σ bound, 2 and compuing he conrol capaciy of he sysem remains open. ACKNOWLEDGEMENTS The auhors would like o hank Anan Sahai for iniial discussions regarding his problem. REFERENCES [] H. Meyr, M. Moeneclaey, and S. A. Fechel, Digial communicaion receivers: synchronizaion, channel esimaion and signal processing. John Wiley and Sons, New York, USA, 998. [2] G. Ranade and A. Sahai, Non-coherence in esimaion and conrol, in Communicaion, Conrol, and Compuing, 5s Annual Alleron Conference on, 203. [3] G. Ranade, Acive sysems wih uncerain parameers: an informaionheoreic perspecive, Ph.D. disseraion, Universiy of California, Berkeley, 204. [4] S. Taikonda and S. Mier, Conrol under communicaion consrains, Auomaic Conrol, IEEE Transacions on., vol. 49, no. 7, pp , [5] P. Minero, M. Franceschei, S. Dey, and G. N. Nair, Daa-rae heorem for sabilizaion over ime-varying feedback channels, Auomaic Conrol, IEEE Transacions on., vol. 54, no. 2, pp , [6] G. N. Nair, F. Fagnani, S. Zampieri, and R. J. Evans, Feedback conrol under daa rae consrains: An overview, Proceedings of he IEEE, vol. 95, no., pp , [7] P. Rajasekaran, N. Sayanarayana, and M. Srinah, Opimum linear esimaion of sochasic signals in he presence of muliplicaive noise, Aerospace and Elecronic Sysems, IEEE Transacions on., no. 3, pp , 97. [8] B. Sinopoli, L. Schenao, M. Franceschei, K. Poolla, M. I. Jordan, and S. S. Sasry, Kalman filering wih inermien observaions, Auomaic Conrol, IEEE Transacions on., vol. 49, no. 9, pp , [9] S. Park and A. Sahai, Inermien Kalman filering: Eigenvalue cycles and nonuniform sampling, in American Conrol Conference ACC., 20, pp [0] L. Schenao, B. Sinopoli, M. Franceschei, K. Poolla, and S. S. Sasry, Foundaions of conrol and esimaion over lossy neworks, Proceedings of he IEEE, vol. 95, no., pp , [] M. Ahans, R. Ku, and S. Gershwin, The uncerainy hreshold principle: Some fundamenal limiaions of opimal decision making under dynamic uncerainy, Auomaic Conrol, IEEE Transacions on., vol. 22, no. 3, pp , 977. [2] G. Ranade and A. Sahai, Conrol capaciy. Inernaional Symposium on Informaion Theory ISIT, 205. [3] A. Lapidoh and S. Moser, Capaciy bounds via dualiy wih applicaions o muliple-anenna sysems on fla-fading channels, Informaion Theory, IEEE Transacions on., vol. 49, no. 0, pp , [4] S. Park, G. Ranade, and A. Sahai, Carry-free models and beyond, in Informaion Theory, IEEE Inernaional Symposium on., 202, pp A. Proof of Lemma 2. APPENDIX From he definiion of I n, we know ha di n, S n 2 Kn. Hence, I n canno conain S n. This gives 2 Kn X 0 S n, since X 0 I n. To show he second half of he inequaliy, suppose ha X 0 S n > 2 2 Kn. Then, 2 Kn < X 0 S n 2 Kn. Hence, here eiss a smaller inerval JK n ha conains X 0 such ha 2 Kn < djk n, S n, where JK n is an inerval of lengh 2 Kn > 2 Kn. Since we also assumed ha K n > K n +, his conradics he assumpion ha K n was he minimal k > K n such ha djk, S n 2 k. B. Proof of Lemma 3.2 We ake logarihms on boh sides of 3 and apply he riangle inequaliy o ge F n f Xn w F n 4 log f Y n Y n X n =, F n f Yn Y n X n = w, F n + log f X n F n f Xn w F n.

5 Using f Yn Y n X n =, F n = 2πσ e Yn 2 2σ 2, and he applying he riangle inequaliy again we ge, log f Y n Y n X n =, F n f Yn Y n X n = w, F n log log w + 2σ 2 Yn 2 Yn 2 w. 5 We can use he derivaives of he funcions o bound he wo funcion differences above. Using d d log = and d Yn d 2 = 2 Yn Y n, we can bound 5 as: 2 log f Y n Y n X n =, F n f Yn Y n X n = w, F n ma w + I n S n ma I n S n 2σ 2 2 Xn Z n Xn Z n 2 w. 6 Since X n I n S n, he maimizaions are over I n S n in 6. Firs, noe ha ma In S n 2 Kn. Second, for all I n S n, we have 2 Xn 2. Hence, he second erm on he RHS of 6 can upper bounded by: ma I n S n 2σ 2 22Z 2Zn n w. Combining hese wo ideas we upper bound 6 as: 2 Kn w + 2σ 2 4Z n2z n 2 Kn w = + 2Z n2z n σ 2 2 Kn w. This implies he desired bound for 4. C. Proof of Lemma 3.3 E[e Ψn2 T ] = E[ep{2 T n Kn σ 2 σ 2 + 2Z i 2Z i 2 Ki }]. i= Since he K i s increase by a leas each ime, we can replace 2 Kn 2 Ki by 2 i n. We ake absolue values and use he riangle inequaliy o ge he upper bound E[e Ψn2 T ] E[ep{2 T n i= σ 2 σ 2 + 2Z i 2Z i 2 i n }]. Subsiuing Z i as Z i = σ Z i +, noing ha he Z i are independen and some simple algebra gives he bound below. α, β are consans ha depend on σ. E[e Ψn2 T ] n E j= [ ep { 2 T α Z 2 j + β For each erm in he produc of 7: [ { E ep 2 T α Z j 2 + β 2 j}] 2 j}]. 7 = ep { β2 T j} 2π e α2 T j 2 z2 dz. Choose T > log 2 2α, so ha ξ := α2 T < 2. Noe 0 < ξ <. The inegral is equal o. Then, ξ2 j n n ξ2 j j= j= e ξ2 j = e 2ξ n j= 2 j e 4ξ. A similar bound can be obained for he erms wih β. The deails are omied and will be included in he full version. D. Proof of Lemma 3.4 By consrucion, K n+ K n +. In he case where K n+ > K n +, we can apply Lemma 2. and ge ha for l 2 PK n+ K n l F n P X n+ 2 2 Kn l F n = P X n U n 2 2 Kn l F n. This is because he conrol U n mus have been very close o X n for K n+ o be much larger han K n. Then we calculae his probabiliy by inegraing ou he densiy. Un+2 2 Kn l P X n U n 2 2 Kn l F n = f Xn F n d U n 2 2 Kn l Kn l ma f Xn F n. Combined wih 9, his gives us ha PK n+ K n l F n Kn l e ηn+ψn2 T Le D n = K n+ K n and 2 Kn+T = 2 3 l+t e ηn+ψn2 T. 8 n K n = D i E[D i F i ]. i=0 I is clear ha K n is a maringale wih respec o F n. In addiion, 8 yields ha he condiional disribuion of D n given F n is sochasically dominaed by he disribuion of G T + η n + Ψ n 2 T, 9 ln 2 where G is an independen geomeric variable wih mean 2. Combined wih 6 and 7, i yields ha ED i E[D i F i ] 2 ED 2 i C, where C is a consan depending on T, σ. This implies P K n n 0 as n. 20 Now, using 9 again, we ge ha E[D i F i ] 25 + T + η i + Ψ i 2 T. 2 A careful analysis of he quaniy n = η n + Ψ n 2 T can be used o show ha n P E[D i F i ] T + σ 2 n 0, i= which combined wih 20 proves he lemma. Deails will be included in he full version.