Robust Control Over a Packet-based Network

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1 Robus Conrol Over a Packe-based Nework Ling Shi, Michael Epsein and Richard M. Murray Absrac In his paper, we consider a robus nework conrol problem. We consider linear unsable and uncerain discree ime plans wih a nework beween he sensors and conroller and he conroller and plan. We invesigae wo defining characerisics of nework conrolled sysems and he impac of uncerainy on hese. Namely, he minimum daa raes required for he wo neworks and he olerable daa drop ou in he form of packe losses. We are able o derive sufficien condiions in erms of he minimum daa rae and minimum packe arrival rae o ensure sabiliy of he closed loop sysem. I. INTRODUCTION Recenly, neworked conrol sysems (NCS) have gained grea aenion from boh he conrol communiy and he nework and communicaion communiy. When compared wih classical feedback conrol sysem, neworked conrol sysems have many advanages. For example, hey can reduce he sysem wiring, make he sysem easy o operae and mainain and laer diagnose in case of malfuncioning, and increase sysem agiliy [16]. In spie of he grea advanages ha he neworked conrol archiecure brings, insering a nework in beween he plan and he conroller inroduces many problems as well. For insance, zero-delayed sensing and acuaion, perfec informaion and synchronizaion are no longer guaraneed in he new sysem archiecure as only finie bandwidh is available and packe drops and delays may occur due o nework raffic condiions. These mus be revisied and analyzed before any pracical neworked conrol sysems are buil. In he pas decade, many researchers have spen effor on hose issues and a number of significan resuls were obained and many are in progress. Many of he aforemenioned issues are sudied separaely. Taikonda [15] and Sahai [11] have presened some ineresing resuls in he area of conrol under communicaion consrains. Specifically, Taikonda gave a necessary and sufficien condiion on he channel daa rae such ha a noiseless LTI sysem in he closed loop is asympoically sable. He also gave rae resuls for sabilizing a noisy LTI sysem over he digial channel. Sahai proposed he noion of anyime capaciy o deal wih real ime esimaion and conrol for a neworked conrol sysem. In our companion paper [13], he auhors have considered various rae issues under finie bandwidh, packe drops and finie conrols. The effec of pacek loss and delay on sae Conrol and Dynamical Sysems, California Insiue of Technology; Pasadena, CA {shiling, epsein, murray}@cds.calech.edu; Tel: (626) , Fax: (626) Work suppored in par by NSF ITR gran CCR The auhors are equally conribued o his work. esimaion was sudied by he work of Sinopoli, e. al. in [2]. I has furher been invesigaed by many researchers including he presen auhors in [12] and [5]. One of he hallmarks of a good conrol sysem design is ha he closed loop remain sable in he presence of uncerainy [3], [4]. While he researchers in [7] sudied he problem of LQG conrol across a packe dropping neworks, no many have considered he norm bounded uncerainy invesigaed in he presen paper. We examine he impac of a norm bounded uncerainy on he nework conrol sysem and provide sufficien condiions for sabiliy in erms of he minimum daa raes and packe arrival raes for he neworks. The paper is organized as follows. In Secion II, we presen he mahemaical model of he closed loop sysem and sae our assumpions. In Secion III, we sae he sufficien condiions for closed loop sabiliy for he case where a nework connecs he sensors o he conroller. In Secion IV, we sae he sufficien sabiliy condiions where in addiion here is a nework beween he conroller and he plan. For boh secions we obain resuls for scalar and general vecor cases. Conclusions and fuure work are given in he las secion. II. PROBLEM SET UP We consider linear discree ime sysems wih a norm bounded uncerainy in he A marix. We will invesigae wo NCS ha we will define by he ype of neworks embedded in he conrol loop. The firs NCS considered has a nework beween he measuremen sensors and he conroller, wih he conroller hen direcly conneced o he acuaors/plan. The second NCS will also include a nework beween he conroller and he acuaors/plan. These wo nework ypes and depiced in Figures 1 and 2. The neworks are defined in erms of heir daa raes and probabiliy of dropping packes. We would consider any packe delays as losses, i.e., we do no use delayed packes for esimaion or conrol. The following equaions represen he closed loop sysem for NCS I (Figure 1). x k+1 = (A + k )x k + Bu k (1) y k = λ k Cx k (2) where x k IR n is he sae of he sysem, u k IR m is he conrol inpu, y k IR p is he oupu of he sysem, and λ k are Bernoulli i.i.d random variable wih parameer λ, i.e., E[λ k ] = λ for all k. k saisfies T k k K 2 I for all k. We also assume he iniial condiion x IR n is bounded. The marix A is assumed o be unsable wihou loss of generaliy as for any marix A, we can always do some sae ransformaion o decompose he saes ino sable ones and

2 Plan Sensor Encoder Nework where he parameers are exacly he same as in (1) excep ha γ k are Bernoulli i.i.d random variable wih parameer γ, i.e., E[γ k ] = λ for all k. The nework one has daa rae R 1 + n and he nework wo has daa rae R 2 + n. The scalar version is x k+1 = (a + k )x k + γ k u k, (7) y k = λ k x k. (8) Fig. 1. Conroller Neworked Conrol Sysem I Decoder unsable ones. The sables ones converges o he origin for any given iniial condiion even wihou any conrol. The nework has daa rae R + n, i.e., he nework can deliver a packe of R + n bis of informaion per discree ime sep, which can be dropped depending on wha value λ is. The bis of he packe are allocaed such ha R bis are reserved for he magniude of he sae signals and n bis are o indicae he sign of each of he sae signals. The corresponding scalar sysem is represened by x k+1 = (a + k )x k + u k, (3) y k = λ k x k, (4) where a > 1 and k K for all k. Encoder 2 Nework 2 Decoder 2 Plan Conroller Sensor Encoder 1 Nework 1 Decoder 1 For all he res of he paper, if x IR n, means he Euclidian norm of i. If X IR n n, X means he induced marix norm of i. log is assumed o have base 2. As packe drops inroduce unavoidable randomness ino he sysem, he classical noion of sabiliy for deerminisic sysems in he sense of Lyapunov [14] is no adequae. The definiion of sabiliy in a probabilisic seing is no new. I is usually considered when here is inheren randomness in he sysem, for example, in he jump linear sysems [6] or in sochasic hybrid sysems [1]. In [6], he auhors have given he mos frequenly seen definiions of sochasic sabiliy. We use almos sure sabiliy in our problem formulaion which is defined below. Definiion 1: Sysem (1) is called almos sure sable if P { lim k x k(x, ω) = } = 1, where ω is he underlying randomness for he closed loop sysem. Sabiliy in he sense of Lyapunov requires ha for any ε >, here exiss a ime T, such ha for all k T, x k ε. For almos sure sabiliy, however, i is allowed ha x k > ε for any k > and for any ε > which may occur wih arbirary low probabiliy. III. GURANTEES FOR CLOSED LOOP STABILITY FOR NETWORK TYPE I A. Scalar Sysems All he lemmas in his secion are for he scalar sysem (3) in NCS I (Figure 1). Lemma 2: Assume λ = 1, i.e., here is no packe drop and R =. Then he closed loop sysem is exponenially sable if K < 1. Proof: A ime 1, he conroller knows x bu no. Le u = ax and so x 1 = x. Similarly, le we have u 1 = ax 1 = a x, x 2 = 1 x. Following his procedure, by leing u k = ax k, we have Fig. 2. Neworked Conrol Sysem II For NCS II (Figure 2), he closed loop sysem are represened by x k+1 = (A + k )x k + γ k Bu k, (5) y k = λ k Cx k, (6) x n = n n 1 x. Hence if K < 1, hen x n K n x, which converges o zero exponenially fas.

3 Lemma 3: Assume λ = 1 and K < 1. Then he closed loop sysem is exponenially sable if R saisfies R > R min = log a log(1 K). Proof: Encode x k o be he mos R significan bis of x k, hence if x k 2 M, hen ɛ k = x k x k 2 M R. Now le u k = a x k, we have ha x k+1 = aɛ k + k x k a2 M R + K2 M. Hence if x k+1 < 2 M, i.e., he upper bound on he sae norm is shrinking, or a R > R min = log = log a log(1 K), 1 K he closed loop sysem will be exponenially sable. Lemma 4: Assume λ < 1 and R =. Then he closed loop sysem is almos sure sable if K < 1 and λ > λ min = log K. (9) Proof: Suppose λ = 1, i.e., x is received. Le u = ax, hen x 1 = x. Now assume x 1 is no received. Applying no conrol we ge x 2 = (a + 1 ) x. Suppose x 2 is received, hen we can le Hence u 2 = ax 2 = a(a + 1 ) x. x 3 = 2 (a + 1 ) x. Following his procedure, whenever x k is received, u k = ax k and whenever x k is no received, u k =. Then we can wrie x n = i (a + j )x, i j where i indicaes ha packe a ime i is received and j indicaes ha packe a ime j is dropped. For n sufficienly large, from he weak law of large numbers [1], wih arbirary high probabiliy ha nλ packes are received and n(1 λ) packes are dropped. Hence x n K nλ (a + K) n(1 λ) x = [K λ (a + K) 1 λ ] n x, is rue almos surely for n sufficienly large. Therefore, if or λ > K λ (a + K) 1 λ < 1, log K, he closed loop sysem is almos surely sable. The lower bound on λ is shown in Figure 3 for differen values of (a, K). Noe ha λ min 1 when K 1 meaning ha zero drop rae is required and λ min when K meaning if here is no uncerainy we only require a nonzero arrival rae o be almos surely sable. In addiion λ min 1 as a, meaning all packes mus be received. This also means ha all he conour lines approach he y-axis in Figure 3. a Minimum λ Increasing λ min 1 K Fig. 3. Minimum packe arrival rae for NCS I. The conour lines are consan values of λ min given in Eqn. 1. The closed loop sysem is almos surely sable for all (a, K) pairs o he lef of he conour lines. Lemma 5: In proving Lemma 2 and Lemma 4, he proposed conrol law is opimal in he sense ha i minimizes he upper bound of he sae variable each ime sep. Proof: Le s firs revisi he proof in Lemma 2. A he firs ime sep, u = ax is indeed he bes conrol law as x 1 x K x. For any oher conrol law, i is no guaraneed ha x 1 K x as here always exiss allowable such ha For example, if we le hen leads o he fac ha x 1 > K x. u = ax +.5K, = sgn(x )K x 1 = K(.5 + x ) > K x. Therefore seing he conrol such ha i cancels only he known sae is he bes choice in he sense ha i minimizes he upper bound of he sae. Now consider he conrol sraegy in proving Lemma 4. Consider he same scenario ha x 1 is dropped. The conrol law u 1 = F (x ), as x is he only known facor o he conroller. Then x 2 = (a + 1 )x + F (x ).

4 If F (x ) =, hen we know for sure x 2 (ak + K 2 ) x. For any oher F (x ), here always exiss allowable and 1 such ha x 2 > (ak + K 2 ) x. Hence he proposed conrol law is opimal. Remark 6: The above lemma basically ells ha if we only know he upper bound bu no he disribuion of he uncerainies, he bes conrol law is ha i eiher compensae he known sae or does nohing. Lemma 7: Assume K < 1 and λ > log K. Then he closed loop sysem is almos sure sable if R saisfies R > R min = log a log(2 λ 1 λ log(a+k) K). Proof: Use he same encoding and decoding sraegy in proving Lemma 3. When here is a packe drop, applying no conrol so ha he norm of he sae expands a mos a+k imes. When here is no packe drop, he norm shrinks a leas a2 R + K imes. Hence by he weak law of large numbers, he criical value of R for he closed loop almos sure sabiliy saisfies (a + K) 1 λ (a2 R + K) λ < 1, which afer simplificaion gives R > R min = log a log(2 λ 1 λ Remark 8: Noice ha he condiion guaranees ha λ > And λ < 1 guaranees ha log K, 2 λ 1 λ log(a+k) K >. 2 λ 1 λ log(a+k) K < 1 K, log(a+k) K). hence he required bandwidh is bigger han ha in Lemma 3 which is as expeced. B. General Sysems The resuls for he scalar case are exended o he general vecor case and combined ino he heorem below. Theorem 9: Assume B, C are inverible and he sysem dimension is n. Then a sufficien condiion for he closed loop almos sure sabiliy (if here are no packe drops, i.e., λ = 1, change his noion o exponenial sabiliy) is ha he nework parameers and sysem parameers saisfy he inequaliy below ( A + K) 1 λ ( A 2 R n + K) λ < 1. Proof: The proof o his heorem is similar o he proofs for Lemma 2, 3, 4 and 7 and is omied here. Remark 1: The assumpions are in general very conservaive as we need boh he marices B and C o be inverible. We need his assumpion because if B is no inverible, we may no be able o compensae for he known sae in one ime sep and hence may make he uncerainies grow ou of conrol. If C is no inverible, we need o wai for long enough consecuive packes o ge complee sae informaion and hence make he probabilisic analysis difficul. The auhors are working owards relaxing hese condiions. I is ineresing o make a comparison beween our rae resuls wih hose in he lieraure, especially in Takionda s work [15], where various rae resuls were given for noiseless LTI sysems over digial communicaion channels. We briefly sae one of his main resuls below. Theorem 11: (Taikonda [15]) Consider he discree ime sysem (1) in Figure 1. Assume λ = 1, i.e., here is no packe drop and K =, i.e., here is no uncerainy abou he plan. Furher assume ha (A, B) is conrollable and (C, A) is observable. Then a sufficien and necessary condiion for he overall closed loop sysem o be asympoically sable is ha R saisfies R > R min = i log λ i (A), where λ i (A) are he unsable eigenvalues of A. One of he many reasons ha we canno direcly apply his resul o sysem (1) is ha (A, B) being conrollable does no imply (A + k, B) is conrollable for all k. If we apply his resul o he scalar case (3), we migh be able o say ha R > R min =, as R min log(a + k ) for all k. However as we see from Lemma 2, K < 1 is needed for closed loop sabiliy and direcly applying Taikonda s resul reveals his fac. The correc sufficien condiion is given by Lemma 3. IV. GURANTEES FOR CLOSED LOOP STABILITY FOR NCS II A. Scalar Sysems Lemma 12: Assume nework one has daa rae R 1 and nework wo has daa rae R 2. Furher assume ha λ = 1,

5 γ = 1 and K =. Then a sufficien condiion on R 1 and R 2 such ha he closed loop sysem (7) is exponenially sable is ha a(2 R1 + 2 R2 ) < 1 Proof: Use he same encoding and decoding sraegy in proving Lemma 3. Wihou loss of generaliy, le < x < 2 M and we wrie he binary expansion of x 1 as x 1 = M 1 i= α i 2 i, where α i s are eiher 1 or depending on he value of x. A ime 1, le x denoe he mos R 1 significan bis of x, i.e., Hence x = M 1 i=m R 1 1 ɛ = x x 2 M R1. Le u = a x, hen u a2 M. Le ū denoe he mos R 2 bis of i, hence Then u ū a2 M R2. x 1 = ax + ū = a x + aɛ + ū = u + ū + aɛ u ū + a ɛ In order ha he norm of x 1 condiion is hen a2 M R2 + a2 M R1. a2 M R2 + a2 M R1 < 2 M, which afer simplificaion gives a(2 R1 + 2 R2 ) < 1.. is shrinking, a sufficien Similarly if he above condiion holds, for all laer ime seps, he sae norm shrinks by a facor of a leas 1 a2 M R2 + a2 M R1 which guaranees he closed loop sysem is exponenially sable. Remark 13: In he above lemma, i is required ha R 1 > log a and R 2 > log a o make (1) hold. Furhermore, if we se eiher R 1 = or R 2 =, from (1), we ge R 1 > log a or R 2 > log a which is he same as in Taikonda s resul (Theorem 11). Figure 4 plos he conour lines of 2 R1 + 2 R2. Sabiliy regions are above he lines for fixed a and K. R2 Conour Lines of 2 R1 + 2 R2 Increasing a Increasing K Fig. 4. Conour plo of 2 R R 2. The lines depic fixed values of a and K. Regions above he lines are sable for hese fixed values. Lemma 14: Assume nework one has daa rae R 1 and nework wo has daa rae R 2. Furher assume ha λ = 1, γ = 1 and K < 1. Then a sufficien condiion on R 1 and R 2 such ha he closed loop sysem (7) is exponenially sable is ha a(2 R1 + 2 R2 ) < 1 K. Proof: The proof is similar o he proves for Lemma 4 and Lemma 12 and is omied here. R1 Lemma 15: Assume λ < 1, γ < 1, R 1 = and R 2 =. Then he closed loop sysem is almos sure sable if K < 1 and λγ > log K. (1) Proof: The proof is similar o he proves for Lemma 4 and is omied here. I is ineresing o noe he form of Eqn. 1 involves he produc of he wo nework arrival raes. This means he sabliiy condiions resul in a plo similar o ha in Figure 5. Theorem 16: Assume λ < 1, γ < 1, R 1 < and R 2 < and K < 1. Then he closed loop sysem is almos sure sable if he following inequaliy holds (a + K) 1 λγ (a2 R1 + a2 R2 + K) λγ < 1 Proof: The proof is similar o he proves for Lemma 4 and Lemma 7 and is omied here. Remark 17: The above heorem links all he differen pieces ogeher o produce a unified framework for closed loop almos sure sabiliy. Example 18: We wish o presen a simple numerical example o illusrae our resuls. Consider he closed loop scalar sysem (7) wih a = 2.75 and K =.4. We pick an iniial

6 1 Sabiliy Region λ Fig. 5. λ. 1 γ Sabiliy plo for he wo nework case packe arrival raes γ and Fig. 7. Sysem in Example 18 wih R 1 = R 2 = 5, λ = γ = 1. condiion of x = 1 and simulae he closed loop sysem for various values of he nework properies defined above. In Figure 6 we only consider packe drops, ha is le R 1 = R 2 =. We le λ =.85 and γ =.8, which saisfy he sufficien condiion from Lemma 4. The plo shows ha indeed he sysem is almos surely sable Fig. 8. Sysem in Example 18 wih R 1 = R 2 = 5, λ =.85, γ = he parameers chosen here were degraded far enough ha he sae did diverge Fig. 6. Sysem in Example 18 wih R 1 = R 2 =, λ =.85, γ = x 18 In Figure 7 we do no consider packe drops, ha is le λ = γ = 1 and le R 1 = 5 and R 2 = 5, which saisfy he sufficien condiion from Lemma 4 for exponenial sabiliy. Nex we assume boh packe drops and finie bandwidh, wih he nework parameers saisfying he sufficien condiions for almos sure sabiliy in Lemma 4. The resuls are ploed in Figure 8. Lasly, we again assume packe drops and finie bandwidh, bu se he parameers o R 1 = 5, R 2 = 3, λ =.5 and γ =.6. These values do no give sufficien condiions for almos sure sabiliy. Figure 9 shows he norm of he sae o diverge. Noe ha he condiions for sabiliy in his paper are only sufficien condiions, hence violaing hem does no necessarily mean he sae will diverge. I so happens ha Fig. 9. Sysem in Example 18 wih R 1 = 5, R 2 = 3, λ =.5, γ =.6.

7 B. General Sysems The resuls for he scalar case are exended o he general vecor case and combined ino he heorem below. Theorem 19: Assume B, C are inverible and he sysem dimension is n. Then a sufficien condiion for he closed loop almos sure sabiliy (if here are no packe drops, i.e., λ = 1 and γ = 1, change his noion o exponenial sabiliy) is ha he nework parameers and sysem parameers saisfy he inequaliy below ( A + K) 1 λγ ( A 2 R 1 n + B B 1 A 2 R 2 n + K) λγ < 1. Proof: The proof o his heorem is similar o he proofs for Lemma 2, 3, 4 and 7 and is omied here. Remark 2: We can in fac recover any of he above lemmas or heorems from Theorem 19 only. For example, ake R 2 = and γ = 1, i.e., he NCS has changed from II o I. Then we ge Theorem 9 which summarizes all he lemmas in secion III. V. CONCLUSION AND FUTURE WORK In his paper we analyzed conrolling linear discree ime sysems wih norm bounded uncerainy in he plan marix over packe dropping neworks. We considered he effec of finie bandwidh and packe losses on closed loop sabiliy. Sufficien condiions for sabiliy are given in erms of he minimum daa raes and packe arrival probabiliy as well as he norm of he uncerainy. The mos obvious exension o his work is o relax he resricion ha B and C are inverible for he general case. Likewise o exend he NCS II resuls o he vecor case. As mos neworks experience no only finie bandwidh and packe drops bu delays as well [8], we would like o include his effec in he sabiliy analysis. Alhough he norm bounded uncerainy is frequenly used, here are oher ypes of uncerain models ha migh be more applicable in cerain cases. For example, when he uncerainy is described by a convex se [9]. We would like o obain similar resuls for oher ypes of uncerainy as well. This paper has given sufficien condiions for sabiliy and cerainly some of hem are no neccesary. I is ineresing o find necessary condiions for sabiliy as well. Lasly, his paper has deal wih guaranees for sabiliy bu has no addressed he issue of performance. Invesigaing how he embedded neworks degrade performance is a ineresing area for fuure work. [5] Michael Epsein, Ling Shi, Abhishek Tiwari, and Richard M. Murray. Esimaion of linear sochasic sysems over a queueing nework. Inernaional Conference on Sensor Neworks, 25. [6] Yuguang Fang and Kenneh A. Loparo. Sochasic sabiliy of jump linear sysems. IEEE Transacions on Auomaic Conrol, 47(7): , 22. [7] Vijay Gupa, Demeri Spanos, Babak Hassibi, and Richard M. Murray. On lqg conrol across a sochasic packe-dropping link. American Conrol Conference, 25. [8] J.Nilsson. Real Time Conrol Sysems wih Delays. PhD hesis, Lund Insiue of Technology, Lund, Sweden, [9] Valer Leie and Pedro Peres. Robus conrol hrough piecewise lyapunov funcions for discree ime-varying uncerain sysems. Inernaional Journal of Conrol, 77(3):23 238, 24. [1] Alber Leon-Garcia. Probabiliy and Random Processes for Elecrical Engineering. Addison-Wesley Pub Co, 2 ediion, [11] A. Sahai. Anyime Informaion Theory. PhD hesis, Massachuses Insiue of Technology, 21. [12] Ling Shi, Michael Epsein, Ahbishek Tiwari, and Richard.M.Murray. Esimaion wih informaion loss: Asympoic analysis and error bounds. CDC-ECC5, 25. [13] Ling Shi and Richard.M.Murray. Towards a packe-based conrol heory - par wo: Rae issues. American Conrol Conference, Minneapolis, Minnesoa, Submied, 26. [14] S.S.Sasry. Nonlinear Sysems; Analysis, Sabiliy, and Conrol. Springer Verlag, [15] Sekhar Taikonda and Sanjoy Mier. Conrol under communicaion consrains. IEEE Trans. Auoma. Conr., 49: , July 24. [16] Wei Zhang. Sabiliy Analysis of Neworked Conrol Sysems. PhD hesis, Case Wesern Reserve Universiy, 21. REFERENCES [1] A.Abae, Ling Shi, S.Simic, and S.Sasry. A sabiliy crierion on sochasic hybrid sysems. Inernaional Symposium of Mahemaical Theory of Neworks and Sysems, Leuven, 24. [2] B.Sinopoli, L.Schenao, M.Franceschei, K.Poolla, M.Jordan, and S.Sasry. Kalman filering wih inermien observaions. IEEE Transacions on Auomaic Conrol, 49(9): , 24. [3] John Doyle, Bruce Francis, and Allen Tannenbaum. Feedback Conrol Theory. Macmillan Publishing, 199. [4] Geir Dullerud and Fernando Paganini. A Course in Robus Conrol Theory. Springer, 1999.