Stability of Discrete-Time Systems with Stochastically Delayed Feedback

Transcription

1 23 Europen Control Conference ECC July 7-9, 23, Zürich, Switzerlnd Stility of Discrete-Time Systems with Stochsticlly Delyed Feedck Mrcell M Gomez, Gáor Orosz 2 nd Richrd M Murry 3 Astrct This pper investigtes the stility of liner systems with stochstic dely in discrete time Stility of the men nd second moment of the non-deterministic system is determined y set of deterministic discrete-time equtions with distriuted dely A theorem is provided tht gurntees convergence of the stte with convergence of the second moment, ssuming tht delys re identiclly independently distriuted The theorem is pplied to sclr eqution where the stility of the equilirium is determined I INTRODUCTION Noise nd delys re often sources of concern for control engineers nd, most recently, concern in efforts to progress the field of synthetic iology In genetic regultory networks, delys rise in the trnscription nd trnsltion processes [] Often the production of protein induced y its ctivting trnscription fctor is modeled s n instntneous process In fct, there is dely in the process with some stochstic vrition Much investigtion hs een done on liner systems with stochstic stte mtrix [] ut little hs een done on nlysis of systems with stochsticlly vrying delys nd even less nlyzing the effects of stochstic dely vritions The ltter hs een ddressed in [6] using queuing theory Most results on stility use Lypunov pproches which result in theorems tht require the existence of positive definite mtrices stisfying liner mtrix inequlities; see [,9,2] However, these usully provide conservtive conditions for stility nd it is difficult to evlute how conservtive they re Similrly, tking the worst cse scenrio eg lrgest dely cn led to unnecessry conservtiveness or my simply give erroneous results Finlly, clculting stility for ech dely nd tking the intersection of the stle regimes in prmeter spce do not necessrily give the stility of the stochstic system [2] In this pper we present method for investigting the effects of stochstic delys in discrete-time dynmicl systems We derive set of deterministic systems contining distriuted delys which descrie the time evolution of the men nd second moment of the stochstic system Under certin conditions, we cn gurntee with proility one wp the stility of the equilirium We pply this method *This work ws supported y NSF grnt no # CNS-94 Division of Computer nd Network Systems M M Gomez is t the Deprtment of Mechnicl Engineering, Cliforni Institute of Technology, sden, CA 92, USA mgomez@cltechedu 2 G Orosz is t the Deprtment of Mechnicl Engineering, University of Michign, Ann Aror, MI 489, USA orosz@umichedu 3 R M Murry is t the Deprtment of Control nd Dynmicl Systems, Cliforni Institute of Technology, sden, CA 92, USA murry@cdscltechedu to simple sclr exmple nd demonstrte tht it is possile to otin noise-induced stility in some cses Tht is, system with single deterministic dely my e unstle ut upon introducing stochsticity in the dely, the system cn ecome symptoticlly stle II ROBLEM SET U In this pper, we consider the sclr stochstic system Xk + Xk + Uk, where Xk R is stochstic vrile t time k Z nd Uk represents the stochstic delyed feedck Uk Xk τk 2 The dely τk tkes finite positive integer vlues τk [,, N] nd N denotes the mximum dely The initil condition includes the stte vlues in the pst N time steps We cn generlize the prolem to include uncertinty in the initil condition, where X, X, X N re selected from known distriutions We consider the following proility density function for Uk p Uk u p Xk τk τk u p τk d, 3 where the density function p τk for the dely is with p τk w i δ τ i, 4 w i The function δ denotes the Dirc delt function The discrete stochstic vrile τk hs finite support, ecuse N is finite integer All the possile delys re given y positive integers τ i, nd w i represents their ssocited weights, or likelihood of occurring It is importnt to note tht the delys re identiclly independently distriuted t ech time step k For ese of nottion we tke τ i i nd w i The integrl in 3 is considered long the positive xis ecuse the delys re positive If we evlute 3 using 4 we otin p Uk u w i p Xk τiu 6 With this we cn proceed to nlyze the sttisticl properties of system, / 23 EUCA 269

2 III TIME EVOLUTION OF THE MEAN AND SECOND MOMENT First, we derive equtions governing the time evolution of the expected vlue, E[Xk] Tking the expecttion on oth sides of we otin E[Xk + ] E[Xk] + E[Uk], 7 where the expected vlue of the feedck term Uk is E[Uk] u p Uk u du [ N ] u w i p Xk τiu du w i u p Xk τiu du w i E[Xk τ i ] 8 Let us define the deterministic vrile yk E[Xk] 9 Sustituting this into 7 nd 8, the dynmics of the expecttion is descried y the deterministic system with distriuted dely yk + yk + w i yk τ i Since the system is discrete-time system with finite mximum dely, the stte spce is finite dimensionl By defining the stte vector Xk eqution cn e rewritten s Xk Xk Xk N, Xk + Ak Xk, 2 where Ak R N+ N+ is stochstic vrile whose proility distriution is independent of Xk So we hve p Xk,Ak X, A p Ak Xk A X p Xk X p Ak A p Xk X 3 Notice, tht the sequence { Xk} is Mrkov chin nd the sequence {Ak} is mutully independent Since the mtrix Ak cn only tke on finite set of vlues, its proility distriution ecomes p Ak A w i δa Λ i, 4 where I i I i 2 I i N Λ i with I i eing the indictor function { if j i, I i j if j i Indeed y tking the expected vlue of 2, one my lso derive : E[ Xk + ] 6 E[Ak Xk] AX p Xk,Ak X, A dxda R N+ N+ R N+ w i Λ ix p Xk X R dx N+ w i Λ i E[ Xk] 7 Using the vrile defined in 9, we define the deterministic stte vector yk yk yk, 8 yk N nd otin deterministic system with distriuted dely yk + w i Λ i yk, 9 where the stte trnsition mtrix is given y w w 2 w N w i Λ i 2 Indeed, nd 9,2 re equivlent We now determine the governing equtions for the second moment of Xk From 2 we hve Xk + X T k + Ak Xk X T ka T k 2 Tking the expected vlue on oth sides yields E[ Xk+ X T k+] E[Ak Xk X T ka T k], 22 where the expecttion opertor is tken element-wise, ut we use the short-hnd nottion ove The right hnd side 26

3 of 22 cn e evluted s [ E Ak Xk X ] T ka T k AX X T A T p Xk,Ak X, A dxda R N+ N+ R N+ w i Λ ix X R T Λ T i p Xk X dx N+ w i Λ i X X R T p Xk X dxλ T i N+ w i Λ i E[ Xk Xk T ]Λ T i 23 Defining the deterministic mtrix-vlued vrile k E[ Xk Xk T ], 24 nd sustituting this into 23 nd 22 we otin the deterministic system k + w i Λ i kλ T i 2 for the time evolution of k Note tht p i,j k E[Xk i + Xk j + ] for i, j,, N + The second moment E[Xk 2 ] is given y the mtrix element p, k E[Xk 2 ] C T k C, 26 where C [,,, ] T ut the time evolution of E[Xk 2 ] depends on other elements of the mtrix k Exploiting tht k is symmetric mtrix, ie p i,j k E[Xk i + Xk j + ] p j,i k, we crry out the mtrix multipliction in 2 nd otin set of discrete time systems tht descrie the time evolution of the elements of k This results in the distriuted dely system p, k + 2 p, k N w i p,i+ k, w i p, k i j 3 p,j k + p,j k + w i p,j i k i + ij 2 w i p,i j+3 k j + 2, 27 for j 2,, N + If for given j the suscript of w j is less thn one, then w j is considered nd if the upper vlue on the sum is less thn the lower vlue, then the sum is zero Now we show tht one cn otin Mrkovin structure for the system ove We define the stte vector p, k p,2 k k, 28 p i,n+ k nd with this we define super vector k k ˆ k 29 k N We cn now represent 27 in stte spce form: where  ˆ k +  ˆ k, 3 A B B 2 B N I I I 3 The sumtrices A, B,, B N R N+ N+ re given y 2 2w 2w 2 2w N w w 2 w N A, 32 B i 2 w i w i w i+ w N w N row w i i + 2 w i 33 nd I is the N + -dimensionl identity mtrix Notice tht  R N+2 N+ 2 IV STABILITY OF THE MEAN AND SECOND MOMENT To determine the stility of deterministic discrete time system one looks t the eigenvlues of the stte trnsition mtrix The mgnitude of ll eigenvlues must e less thn one for the system to e stle [,8] The stility of the men is derived from the eigenvlues of the stte trnsition 26

4 mtrix N w iλ i in 9,2 The chrcteristic eqution ecomes det si w i Λ i s w i s i 34 To determine the stility oundries in the prmeter spce,, we evlute the chrcteristic eqution so tht s In prticulr, s, s nd s e ±iθ for θ, π re considered Ech of these provide different set of stility curves [,8] Notice tht for s, one otins delyindependent condition Stility for the vrince is determined in the sme wy using the stte trnsition mtrix in 3 At first glnce, it ppers tht nlyzing the stility of the second moment involves mtrix of dimension N + 2, ut it cn e reduced to nlyzing n N + -dimensionl mtrix We denote the sumtrices delimitted y the lines in 3 s [ ] A B Â C D, 3 which yields detsî Â detsĩ D det si A BsĨ D C s NN+ det si A BsĨ D C, 36 where Ĩ denotes the NN + dimensionl identity mtrix We re left with determining the eigenvlues given y det si A BsĨ D C detm + M 2, 37 where V NOTIONS OF STABILITY FOR STOCHASTIC SYSTEMS We hve provided deterministic discrete time equtions whose stility determine the stility of the men nd second moment for the non-deterministic system,2 However, the first nd second moments converging to zero does not gurntee tht the stte converges to zero with proility one wp in ll circumstnces We restte theorem tht cn e found in [4]: The following implictions hold Xk wp X Xk X Xk D X Xk r X for ny r Also, if r > s then Xk r X Xk s X No other implictions hold in generl Here, Xk r X denotes tht the sequence Xk converges to constnt X in r th order, for r, which holds if E[ Xk r ] < for ll k nd E[ Xk X r ] s k, while Xk X nd Xk D X denote convergence in proility nd distriution [4] Notice tht convergence in r th order only gurntees convergence in proility nd M pointwise towrds X s 2 2w 2w 2 2w N Convergence of the second moment Xk 2 is then equiv- s w w 2 w N lent to convergence in 2 nd order since Xk 2 is positive s definite This is why convergence of the men is insufficient w s s ut the convergence of the second moment my e enough w 2 w s 2 s s Given the generl vector cse Xk + Ak Xk, where {Ak} re mutully independent rndom mtrices, [7] provides the following theorem, using Lypunov function of the form X T QX, where Q is positive definite w N 2 w N 3 w s N 2 s N 3 s s 38 denoted s Q > nd 2 Let Q >, C nd N w i s i E[Ak T QAk] Q C 4 w w 2 w 3 w s s s N s Then E[ Xk T CXk] nd Xk T CXk wp M 2 w 2 w 3 w s 2 s N 2 s 2 Let the Ak e identiclly distriuted If { Xk} is men squre stle tht is, E[ Xk T Xk], then for ny w N C >, there is Q > stisfying 4 w N s N s N 39 Given this theorem, if {Ak} re identiclly distriuted which give the chrcteristic eqution for the second moment for 4 if we choose C I According to the theorem, nd mutully independent in 2, there exists solution Q the 262 wp distriution Finlly, Xk X denotes convergence with proility one wp, tht is, for every ɛ >, Xk X ɛ occurs only finitely often Consequently, for ech pth ω, there is numer kω so tht Xk X ɛ, for ll k > kω, see [7] We my sy tht, with the exception of finite set of sequences, ll sequences {Xk} converge

5 existence of the solution implies Xk T Xk wp This is sufficient condition for wp stility when the delys τk re chosen independently of ech other nd from the sme distriution t ech k in,2 VI EXAMLES Here we pply the stility conditions derived for men nd second moment to exmples with different dely distriutions Figure shows uniform dely distriutions left nd distriutions with two eqully prole delys right, which we refer to s toggle distriutions E nd V refer to the expected vlue nd the vrince of the dely distriutions V The lck dsh-dot nd red dotted curves indicte n eigenvlue crossings of the unit circle on the complex plne t nd The green solid curves indicte pir complex conjugte eigenvlues crossing the unit circle One cn see tht s the vrince is incresed, the region of stility shded region increses It is importnt to point out the regions of stility for single dely is not contined in the regions of stility for the distriuted delys V V E 3 V E 4 V E 3 V 2 3 E 4 V 2 3 V 2 3 V E 3 V 2 E 4 V V V 4 Fig Left: Discrete uniform dely distriution with expected vlue E 3 Right: Discrete toggle distriution with E 3 The vrince V is listed in ech pnel Although stility of the second moment implies stility of the men, it is interesting to tke look t the region of stility for the men since it provides necessry conditions for stility In [3] we showed tht introducing dditionl delys to n lredy delyed continuous-time system my stilize n unstle system It is interesting to see tht similr result cn e otined for discrete-time system Figure 2 shows the stility region for system with uniform dely distriution of expected vlue E nd vrince Fig 2 Stility chrts for the men for uniform dely distriutions Shding indictes stility When crossing lck dsh-dot curve from stle to unstle n eigenvlue crosses the unit circle t outwrd, while crossing red dotted curve indictes tht n eigenvlue crosses the unit circle t Crossing green solid curve indictes tht pir of complex conjugte eigenvlues crosses the unit circle Next, we look t wp stility region Recll tht system with identiclly independently distriuted delys is stle wp if the second moment is stle We first consider such systems with uniform dely distriution, then look t systems where the dely toggles etween two vlues, ech with equl proility The left pnels in Fig 3 show the stility oundries of the non-deterministic system with uniform dely distriution The curves indicte the stility losses of the men s in Fig 2 ut here the shded region indictes the region of wp stility ie stility of the second moment The shded region ws found y sweeping cross the prmeter spce, nd checking the eigenvlues of the system 3,3 263

6 The right pnels in Fig 3 show stility chrts for the toggle distriution Agin, we plot the men stility curves nd indicte the second moment stility regions y shding tht imply wp stility Here, the wp stility region is dominted y the region of stility for the men of the system x E 3 curves: V shding: V 2 3 V V 2 3 time E 3 V 2 3 E 3 V Fig 4 Left: Compring stility for the cse of deterministic single dely of τ 3 curves with wp stility region shded of uniformly stochsticlly vrying dely with men E 3 Right: Simultion compring the cses deterministic nd stochstic dely for 79 nd 3 s indicted y dimensionl systems nd finding generl reltionships etween distriution types nd size or shpe of stility regions ACKNOWLEDGMENT The uthors would like to thnk Mrk J Bls, Enoch Yeung nd Mehdi Sdeghpour for helpful discussion nd feedck E 3 V 2 E 3 V 4 Fig 3 Left: Stility oundries for uniform dely distriution Right: Stility oundries for toggle distriution In ll pnels, the curves represent the stility losses of the men s in Fig2, while the shded region indictes wp stility given y the second moment The introduction of stochsticity in the dely distorts the stility region when compred to the cse of single deterministic dely s cn e seen in Fig 4 Since some of the wp stility regions extend outside the stility ounds for the deterministic system we cn stilize the system y introducing uncertinty in the dely We demonstrte this y numericl simultion in Fig 4 where the prmeters correspond to the mrk in the left pnel VII CONCLUSION In this pper, we defined notion of stility nd performed stility nlysis for clss of liner system with stochstic dely We investigted stility regions for sclr stochsticlly delyed feedck systems nd mde some interesting oservtions We looked t two different types of dely distriutions nd found they hd very different effects on the region of stility In the cse of the uniformly distriuted delys, worst cse scenrio would certinly e conservtive However, for the cses with two eqully prole delys, the stility region of the men seemed to provide good pproximtion of the wp stility region We lso demonstrted tht introducing stochsticity in the dely my stilize the system tht my look counter intuitive Future work includes generlizing results for higher REFERENCES [] H Go nd T Chen, New results on stility of discrete-time systems with time-vrying stte dely, IEEE Trnsctions on Automtic Control, vol 2, no 2, pp , 27 [2] M Ghsemi, S Zho, T Insperger, nd T Klmár-Ngy, Act-nd-wit control of discrete systems with rndom delys, Americn Control Conference, pp , 22 [3] M M Gomez nd R M Murry, Stiliztion of feedck systems vi distriution of delys, in th IFAC Workshop on Time Dely Systems, IFAC pp 23-28, 22 [4] G R Grimmet nd D R Stirzker, roility nd Rndom rocesses Oxford University ress, 2 [] J Guckenheimer nd Holmes, Nonliner Oscilltions, Dynmicl Systems, nd Bifurctions of Vector Fields, ser Applied Mthemticl Sciences Springer, 983, vol 42 [6] K Josić, J M López, W Ott, L Shiu, nd M R Bennett, Stochstic dely ccelerted signling in gene networks, LoS Computtionl Biology, vol 7, no, Novemer 2 [7] H Kushner, Introduction to Stochstic Control Holt, Rinehrt nd Winston, Inc, 97 [8] Y A Kuznetsov, Elements of Applied Bifurction Theory, 3rd ed, ser Applied Mthemticl Sciences Springer, 24, vol 2 [9] G rk, A dely-dependent stility criterion for systems with uncertin time-invrint delys, IEEE Trnsctions on Automtic Control, vol 44, no 4, pp , 999 [] T-J Su nd C-G Hung, Roust stility of dely dependence for liner uncertin systems, IEEE Trnsctions on Automtic Control, vol 37, no, pp 66 69, 992 [] U Vogel nd K F Jensen, The RNA chin elongtion rte in Escherichi coli depends on the growth rte, Journl of Bcteriology, vol 76, no, pp , 994 [2] D Yue, Y Zhng, E Tin, nd C eng, Dely-distriution-dependent exponentil stility criteri for discrete-time recurrent neurl networks with stochstic dely, IEEE Trnsctions on Neurl Networks, vol 9, no 7, pp ,