Pattern selection and control via localized feedback

Transcription

1 PHYSICAL REVIEW E 72, Pattern seection and contro via ocaized feedback Andreas Hande Deartment of Bioogy, Emory University, Atanta, Georgia 30322, USA Roman O. Grigoriev Schoo of Physics, Georgia Institute of Technoogy, Atanta, Georgia , USA Received 15 August 2005; ubished 14 December 2005 Many theoretica anayses of feedback contro of attern-forming systems assume that feedback is aied at every satia ocation, something that is often difficut to accomish in exeriments. This aer considers an exerimentay more feasibe scenario where feedback is aied at a sarse array of discrete satia ocations. We show how such feedback can be comuted anayticay for a cass of reaction-diffusion systems and use generaized inear stabiity anaysis to determine how dense the actuator array needs to be to seect or maintain contro of a given attern state in the resence of noise. The one-dimensiona Swift-Hohenberg equation is used to iustrate our theoretica resuts and exain earier exerimenta observations on the contro of the Rayeigh-Bénard convection. DOI: /PhysRevE PACS numbers: Gg, Yy I. INTRODUCTION Nonequiibrium satiay extended systems have a rominent ace in hysics, chemistry, bioogy, and engineering. Fuid fows, asmas, and wide-aerture semiconductor asers rovide a few technoogicay imortant exames. As these systems are driven farther from equiibrium, they tend to disay rogressivey more comicated dynamics, transitioning from satiay uniform to satiay atterned states and eventuay deveoing satiotemora chaos. To make the best use of these systems, we need to earn to contro them, forcing the system to choose one of the suitabe states, even though such states might be naturay unstabe. Fow contro for drag reduction or noise suression reresents arguaby the best known aication 1. Neither fuids nor asma can be racticay controed at every oint in sace. Athough recent advances in microeectro-mechanica systems technoogy in certain instances enabe rea-time indeendent contro at miions of ocations with satia resoution on the scae of tens of microns 2, more tyicay one is forced to contro the dynamics by aying feedback at reativey few ocations. A good exame woud be a tokamak with feedback aied by varying the current in a discrete and reativey sma set of magnetic cois 3. It is, therefore, imortant to understand the imitations of ocaized contro of satiay extended dynamics and the hysica mechanisms which determine the minima number or density of actuators that is needed for successfu stabiization of a given system. Serving as a motivation for this study is another exame of the ocaized contro of a attern forming system rovided by the exeriments of Tang and Bau 4, who attemted to maintain the ateray uniform no-motion state of the fuid in the Rayeigh-Bénard convection above the rimary instabiity threshod using an array of sensors therma diodes and actuators resistive heaters. The heat fux generated by each of the 24 heaters embedded in the bottom boundary was taken to be a inear function of the temerature deviation reative to the conductive rofie measured by the diode ocated at the midane of a cyindrica convection ce directy above the heater. The authors noted that the degree of stabiization obtained in the exeriments fe far short of their own theoretica rediction 5 that the critica Rayeigh number for the onset of convection in a ateray infinite fuid ayer can be ostoned by as much as an order of magnitude. The discrete nature of the sensor and actuator arrays was quoted as a ossibe reason: the theoretica work assumed that the actuators and sensors were continuousy distributed in sace, whie in the exeriment it was necessary to use a finite number of sensors and actuators. In the absence of environmenta noise, the minima number of actuators and sensors needed to contro the fuid inside a convection ce is, in fact, uniquey determined by the dimensionaity of the argest irreducibe reresentation of the symmetry grou of the inearized system 6. For a cyindrica convection ce the symmetry grou is O2, so just a air of sensors and a air of actuators is sufficient to stabiize a noiseess system athough a more eaborate feedback is needed than that used in the exeriments by Tang and Bau. In the resence of noise a much arger number of sensors and actuators might be necessary. The goa of the resent theoretica study is to determine how this number deends on the strength of the noise, the size of the system, and the secifics of the dynamics. Athough the equations describing the Rayeigh-Bénard convection are we known, they do not aow us to erform a comutations in urey anaytic form, so instead we wi use a one-dimensiona Swift-Hohenberg equation 7 which describes the emergence of a straight ro convective attern just above the onset of the rimary instabiity. Focusing on scaar one-dimensiona equations wi aso aow us to generaize the resuts for a much wider cass of reaction-diffusion systems. The aer is organized as foows. In Sec. II we describe our mode system and summarize revious research reevant to this study. Section III resents the comutation of the feedback gain and its scaing anaysis. In Sec. IV we characterize the transient growth of disturbances in the system, and use these resuts to determine the number of actuators re /2005/726/ /$ The American Physica Society

2 A. HANDEL AND R. O. GRIGORIEV PHYSICAL REVIEW E 72, a 1 = 2 3 1/2 cos + 3 3/2 cos cos2 +, /2 sin a 3 = 3 3/2 cos /2 sin cos2 +, FIG. 1. The eigenvaue sectrum of the Swift-Hohenberg equation with =0.5. The soid ine reresents the sectrum of a ateray infinite oen oo system. The circes reresent the sectrum of the system on a finite domain of ength =40. quired for successfu contro. We then roceed to discuss contro of atterned states in Sec. VI and resent our concusions in Sec. VII. The aendices contain the agebra omitted from the main text to streamine the exosition. II. THE MODEL Athough the foowing anaysis aies to a broad cass of attern-forming systems, for iustration uroses it wi be convenient for us to concentrate on a articuar exame of the Swift-Hohenberg equation: t = 1+ x 2 2 3, where x,t coud describe temerature and veocity disturbances reative to the ateray uniform no-motion state and is the reduced contro arameter describing how strongy the system is driven out of equiibrium by the imosed heat fux. The eigenvaues of the inearization about the uniform state 0 x,t0 are q = 1 q 2 2 with the argest one 1=, so that aso measures the distance to the onset of rimary instabiity for the ateray infinite system. Once becomes ositive, the uniform state becomes unstabe and a eriodic attern emerges with a eriod corresonding to some wave numbers in the unstabe band q qq +, where q ± = 1± see Fig. 1. This attern can be easiy found in the form of an asymtotic Fourier series q x,t = a n cosnqx, n which quicky converges for moderate. A even coefficients vanish, a 2 =a 4 =a 6 = =0, whie the odd coefficients can be found using the method of dominant baance: a 5 = /2 cos 5 +,, where we have defined sin =1 q 2, such that the unstabe band of wave numbers q mas exacty onto the interva for any. Standard anaysis shows that the atterned states q near the edges of the stabe band i.e., /31 q 2 for an unbounded system, are unstabe and, without feedback, undergo a secondary Eckhaus instabiity toward another attern q with q coser to the center of the unstabe band. Both the uniform state 0 and the atterned states q coud be stabiized for 0 by aying feedback using a finite number of actuators whose action is assumed to be satiay ocaized, as is tyicay the case in ractice. In the foowing we wi concentrate on the imiting case where the region of the system affected by each actuator is so sma that the feedback can be described by a function. Aternativey, contro coud be envisioned as a way to seect a attern, if the system size and/or is arge enough for severa atterned states to coexist. Our goa here is to determine how the number of actuators deends on and, or more generay, on the choice of the evoution equation. Noninear contro of infinite-dimensiona dynamics such as those disayed by reaction-diffusion systems is sti in its infancy, desite some attemts to ay weaky noninear anaysis toos such as the center manifod reduction 8. For the most art feedback contro of this cass of systems is based on the same foundation as the theory of attern formation-inear stabiity anaysis. This more ouar aroach, however, has a significant imitation it describes ony the asymtotic stabiity of the system with resect to infinitesima disturbances. This resents a robem in describing rea e.g., finite size disturbances which can transienty grow, before the asymtotic exonentia decay sets in, to become arge enough to invaidate the inear aroximation on which the whoe anaysis is based, eading to a contro breakdown. Transient, or agebraic, growth of sontaneous disturbances has attracted a ot of attention recenty, mosty in the context of transition to turbuence in shear fows, eading to the deveoment of a generaized inear stabiity anaysis

3 PATTERN SELECTION AND CONTROL VIA LOCALIZED FIG. 2. Time evoution of an initia disturbance random noise with magnitude =10 4 for the Swift-Hohenberg equation 1 with eriodic boundary conditions and =0.5. Locaized feedback contro is aied at four oints marked with circes. For a system of size =40 the disturbance transienty grows by severa orders of magnitude before exonentia decay to the uniform target state x,t=0 takes over. We iustrate this henomenon in Figs. 2 and 3, which show the dynamics of the Swift-Hohenberg equation with ocaized feedback contro to be discussed in detai in the foowing sections. U to a certain system size, transient growth does not effect our abiity to stabiize the system and eventuay exonentia decay to the uniform target state is achieved, as Fig. 2 shows. However, for arger system sizes, transient growth becomes so strong that contro is no onger abe to suress the initia disturbance, as seen in Fig. 3. In fact, such transient growth shoud be found in most attern-forming systems evoving in the resence of ocaized feedback, as can be inferred from the foowing sime argument. Imagine a sontaneous disturbance ocaized between two actuators. Since the actuators cannot directy affect the region where the disturbance originated, the feedback has to roagate from the ocation of the actuators toward the disturbance. During the time it takes for the feedback to roagate towards and quench the disturbance, the atter grows in an essentiay uncontroed way. Once the feedback reaches the erturbed region of the system, the disturbance is being suressed unti it cometey vanishes or another disturbance is sontaneousy created somewhere in the system. FIG. 3. The evoution of a random initia disturbance same as in Fig. 2 in the Swift-Hohenberg equation on a arger domain, =45. Here the transient growth of the disturbance is strong enough for noninearities to become imortant, eading to the faiure of inear contro. The system eventuay diverges from the target state x,t=0. PHYSICAL REVIEW E 72, Extreme sensitivity of the controed cosed oo dynamics to sontaneous disturbances stemming from their transient growth has been discovered by one of us 12, whie Egof and Socoar, were the first to ink transient growth to the strong non-normaity of the evoution oerator describing the dynamics of disturbances in a coued ma attice CML 13. We have recenty extended their resuts to a system described by a artia differentia equation PDE 14. In articuar, we showed that non-normaity arises as a resut of imosing ocaized feedback, even when the uncontroed oen oo system is norma. Furthermore, we showed that transient growth increases exonentiay with the system size in both PDEs 14 and CMLs 12. In another study 15, we showed that contro faiure can be described by a combined action of transient growth and noninearity, which triggers a noninear instabiity. The work resented here significanty extends and generaizes the ideas resented in Ref. 14. III. FEEDBACK CONTROL OF THE UNIFORM STATE We wi start our anaysis by concentrating on the ocaized feedback contro of the uniform state 0 x,t=0. Feedback contro of other steady states is concetuay simiar, however, the high degree of symmetry of the uniform state makes it secia. On the one hand, contro of that state is, in some sense, the most chaenging, since it requires the argest number of indeendent actuators in the zero noise imit. On the other hand, this higher symmetry aows a much greater degree of anaytica rogress. We wi start by considering how the feedback contro setu needs to be modified as the system size is increased from very sma to very arge with 0 hed fixed. If the domain size is sufficienty sma e.g., 2/ 1+ for eriodic boundary conditions, the uniform state is stabe due to strong confinement effects and no contro is needed. A. Contro at one boundary Systems of a somewhat arger size can be successfuy controed by acing a singe actuator at one of the boundaries. Linear feedback contro can be imemented by moduating the fux of a quantity characterizing the state of the system, e.g., mass, temerature, or concentration, through a boundary here x=:,t kxx,tdx, 5 =0 where, as usua, the rime denotes the satia derivative. In the case of the Rayeigh-Bénard convection, for instance, one coud contro the dynamics of the fuid by changing the heat fux through a boundary of the convection ce 4,16. Then the feedback gain kx woud describe how temerature disturbances in different regions inside the ce affect the fux. There is a ot of fexibiity in choosing the three remaining boundary conditions since the Swift-Hohenberg equation is the fourth order, as ong as they are consistent with the target state. Here it is convenient for us to ick them as

4 A. HANDEL AND R. O. GRIGORIEV PHYSICAL REVIEW E 72, ,t = 0,t =,t =0. 6 The sectrum of the oen oo kx=0 system inearized about the target state is discrete, with eigenfunctions 2n 1 f n x = sinq n x, q n =, n =1,2, and eigenvaues given by 2. The eigenfunctions f n x are orthogona which means the inearized evoution oerator for the oen oo system is norma and form a convenient basis for the stabiity anaysis of the cosed oo system. Projecting the inearized evoution equation onto the basis 7, we obtain an infinite system of coued ordinary differentia equations ODEs n = n n 1 n K m m M n, m=1 for the Fourier coefficients n t and K n of x,t and kx, resectivey, x,t = n tf n x, n=1 kx = K n f n x. n=1 As we show in Aendix A, the eigenvaues of the system 8 can be changed by modifying the feedback gain coefficients K m. For instance, reacing s consecutive eigenvaues a a,..., b b requires K m = m 1 =a b 1 m m =a b m =m+1 m for m=a,...,b and setting the rest to zero. Making a new eigenvaues negative and using 9 aows us to determine the feedback gain kx which renders the system ineary stabe with new eigenvaues m 0 that we are free to choose however we find convenient. In rincie, this resut aows one to find a stabiizing feedback for any system size. However, as we iustrated earier, contro eventuay fais if the system size is increased without adding additiona actuators. To understand the origin of contro faiure, we consider the behavior of the gain coefficients K m for arge. We first rewrite 10 in exonentia form b K m = ex =a m 1 n m n m =a b n m. 11 =m+1 In the arge imit, the eigenvaues are dense enough to aroximate the sums with integras. It is natura to choose the wave number q=q + a/ as the integration variabe, with integras going over the unstabe band of the oen oo system. To eading order in we obtain K m ex q q + n m q m q dq. 12 Note that this exression incudes a air of integrabe singuarities at q= m, which are excuded in 11. The singuarities contribute On terms in the argument of the exonentia, which are sma comared with the O dominant contribution and so can be ignored. The resuts are rather genera and ay to many different systems, not just the Swift-Hohenberg equation. This aows us to draw severa imortant concusions. First of a, since the terms inside the arentheses in 12 are indeendent of, one immediatey concudes that the Fourier coefficients K m, and hence the gain function kx, bow u exonentiay with the size of the system. Second, since a the m are negative, the strength of feedback, is minimized by choosing a new eigenvaues very cose to zero, m 0, m=a,...,b. The arger the absoute vaue of the new eigenvaues, the stronger the feedback is required, which is an intuitivey sensibe concusion. The scaing of kx is determined by the argest coefficient K m. Since dk m dm = K m dq q dm, 13 the maximum in K m with resect to m corresonds to the maximum in with resect to q. For the Swift-Hohenberg equation the maximum of q is achieved at q=1, for which m =. We can obtain a more exicit scaing reation if we use our freedom to choose the new eigenvaues and set a = = b = Substituting this into 12 yieds k max maxkx maxk m e / 0 15 x m with the characteristic ength scae 0 secific to a articuar evoution equation. For the Swift-Hohenberg equation, 0 is given by 0 = q + q 4+n + +2n 1+q q + 11 q 2 q 1+q + 1 q q q. 16 To check these anaytica resuts, we comare 15 with the maximum of the feedback gain comuted numericay using two different methods. In the first, the feedback is comuted directy from 10, where a ositive eigenvaues are set to 0 and the negatives ones are eft unchanged. This aroach is known as oe acement PP in contro theory. The second method uses the inear quadratic reguator LQR contro. In LQR the feedback is comuted as the soution minimizing a functiona quadratic in x,t and

5 PATTERN SELECTION AND CONTROL VIA LOCALIZED PHYSICAL REVIEW E 72, this woud generay ead to subotima contro 17. The remaining boundary conditions are chosen to be 0,t =,t=0. The eigenfunctions of the oen oo system are now n 1 f n x = cosq n x, q n =, n =1,2, and eigenvaues are given by 2 as usua. Note that the eigenfunctions 18 are again orthogona, so that the oen oo evoution oerator is norma. Proceeding as before see Aendix B for detais we find the feedback gain coefficients to be FIG. 4. Maxima feedback gain k max for =0.75. For LQR contro, the new eigenvaues k ie between 1 and 0.1. Straight ines show the theoretica resut 15. One actuator is aced on the right boundary.,t 17. In LQR, the new eigenvaues k cannot be exicity secified. Both PP and LQR are commony used in the contro theory The agreement between the anaytica and numerica scaing see Fig. 4 is quite imressive desite the fact that the asymtotic resuts are used for a reativey sma system size. The agreement woud imrove further for arger. In articuar, the fuctuations in the numerica resuts can be attributed to finite-size effects which woud become smaer for arger. Unfortunatey, the numerica soutions become unreiabe for 25, refecting the fact that the Jacobian of the cosed oo system becomes increasingy non-norma and hence near singuar. Nevertheess, it is cear that the soe is correcty redicted by our anaysis, even for system sizes that are not very arge. Our numerica resuts aso show that, according to 16, the soe increases with, as redicted by the genera scaing exression 12. B. Contro at two boundaries By increasing the size of the system further we find that the feedback gain grows exonentiay fast, eventuay becoming arge enough to vioate the inearity assumtion underying our stabiity anaysis. The most natura way to resove this robem is by adding an additiona actuator, acing it at the oosite boundary. With this setu the contro authority is sit in such a way that the eft actuator is rimariy resonsibe for suressing disturbances in the eft haf of the system, whie the right actuator is rimariy resonsibe for the right haf. Naivey this woud ead one to exect that this arrangement woud be abe to stabiize a system of about twice the size, a ese being equa. The foowing anaysis shows that this is indeed the case. Let us kee the the right boundary condition incororating the feedback term in the form 5. The action of the second actuator can be exressed in a symmetric form 0,t xx,tdx 17 =0 with x= k x. Athough, in rincie, one coud choose the gains x and kx differenty by ignoring the symmetry, K m = 1 m m m m 2 m, 19 with m=a,...,b. The rime on the roducts denotes that the index goes over integers =...,m 4,m 2,m+2,m +4,... in stes of two, with an additiona restriction a b. We further find for arge the feedback gain to scae as k max e /2 0, 20 where 0 is sti given by 16. This is exacty the same scaing reation as that found in the one-actuator case, but with a different characteristic ength. The additiona refactor of 1 2 in the exonent it arises due to the fact that the roduct over in 19 goes in stes of two, shows that doubing the number of actuators doubes the characteristic system size, as the naive argument made earier woud suggest. C. Contro with an array of actuators Increasing the system size yet further we find that the two actuator setu eventuay fais as we, requiring more actuators to be added. However, acing more than two actuators on the boundary is not going to imrove the erformance of our controer significanty, so we have to change our aroach. In order to contro a system of arbitrary size using a feedback gain of a fixed magnitude, we exect to need a number of actuators r on the order of / 0 as the maximum distance between actuators is restricted by the characteristic ength 0. In other words, contro authority is now sit in such a way that each actuator is resonsibe for an O 0 size segment of the system. With this more genera arrangement the system is described by t = 1+ 2 x d xu t, r =1 21 where the d x are the infuence functions describing the ocation and satia extent of the actuators. In the imit of erfecty satiay ocaized contro, d x reduce to functions. Again, to reserve the symmetry of the system, we wi assume eriodic boundary conditions and ace the actuators in a reguar array. However, we cannot arrange them eriodi

6 A. HANDEL AND R. O. GRIGORIEV PHYSICAL REVIEW E 72, cay, as that woud make the system uncontroabe 6. We can achieve controabiity or more recisey, stabiizabiity by using a eriodic array of airs of actuators, which corresonds to r, d x =x x 1+ r, -odd, 22 -even. x,t = n= k x = n= n tf n x, K n f n x, d x = 1 D n f n x. n 27 For instance, four actuators woud be aced as two airs, one air at x=1±/4 and the other at x=3±/4. The sacing 2/r between the two actuators in each air has to be chosen to satisfy the stabiizabiity constraints 6, but is otherwise arbitrary. As a rue of thumb, stabiizabiity can be achieved by making the sacing smaer than the waveength of the shortest unstabe mode of the oen oo system. The functions u t describe the strength of feedback and are chosen as u t k xx,tdx. 23 =0 Using the remaining symmetry of the robem a feedback gains k x can be exressed through k 1 x. Indeed, the system with the chosen arrangement of actuators remains invariant under transations by a mutie of the distance between airs of actuators, x x+n/r, n=2,4,... The feedback contro reserving this symmetry requires gain functions k +2 x to have the same shae as k x, ony shifted by 2/r. Further, the refection symmetry imies that for each actuator air the gain function for k +1 x shoud have the same shae as k x, refected at the midoint between the actuators. Combining these two resuts, we obtain k x = k 1 1 r 2r x + 2r 24 for =2,...,r. The resuting contro is transationay and refectionay symmetric, i.e., has a symmetry described by a discrete subgrou of the continuous symmetry grou O2 of the oen oo system. As shown in 17, such contro is otima with resect to genera measures that resect the symmetry of the system. With eriodic boundary conditions, the eigenfunctions of the oen oo system are Fourier modes f n x = e iq n x, q n = 2n, n =,..., 25 which are orthogona as we found reviousy, such that the Jacobian of the oen oo system is norma. Using this basis we can again convert the PDE 21 into a system of ODEs n = n n + D n r =1 m= K m m, 26 for the Fourier coefficients of x,t, d x and k x defined according to The cacuations simiar to those described reviousy see Aendix C for detais give K m 1 = K m m 2 m m 1 * = C m r m, 28 with m=a,...,b. The rimes on the roducts indicate that the index =...,m r,m r/2,m+r/2,m+r,... goes in stes of r/2 subject to an additiona restriction a b. The numerica refactor is given by ex 2i r m 1 C m = ex2ig m 1, 29 where g m is a ositive integer, which deends on both m and r. In the scaing anaysis this refactor is of no imortance, since it is indeendent of both the system size and the choice of new eigenvaues. Setting a unstabe eigenvaues to 0 we obtain the feedback gain scaing in this most genera case to be k max e /r 0 30 with 0 given by 16, indicating that the characteristic ength has grown by a factor of r. This resut again suorts our naive exectations: by emoying r actuators, the system size can be increased by a factor of r without increasing the magnitude of the feedback aied by each actuator. It is interesting to note how a articuar arrangement of actuators in the array i.e., the distance between actuators in each air affects the magnitude of the feedback signa. The atter deends on ony through the refactors C m, which diverge for =n/g m with any integer n, which corresonds to the oss of stabiizabiity. For instance, when 4m is an integer mutie of r, one obtains g m =4m/r, so that C m diverges when the ocations of a actuators coincide with the nodes of a Fourier mode with wave number q=±2m/. Choosing outside of sma neighborhoods of such vaues yieds C m =O1, making the coefficients K m very weaky deendent on the sacing between actuators in each air. IV. TRANSIENT GROWTH As the resuts of numerica simuations resented in Figs. 2 and 3 iustrate, arge feedback gain eads to strong tran

7 PATTERN SELECTION AND CONTROL VIA LOCALIZED sient growth of disturbances. Significant transient growth is exected in this robem due to the nonorthogonaity of the eigenfunctions of the cosed oo system which makes the inearization 8 non-norma. Consider, for instance the feedback at one boundary. As a resut of transationa invariance of our mode the eigenfunctions of the inearization of 1 subject to the boundary conditions 5 and 6 a have the form f q x=sin qx, regardess of whether contro is turned on or off. Cosing the feedback oo merey shifts a wave numbers q into the stabe band, at the same time destroying the mutua orthogonaity of the eigenfunction set. It is easy to see that certain choices of the cosed oo eigenvaues, such as 14, can force a number of cosed oo eigenfunctions to aign arbitrariy cosey. In this section we describe more exicity the connection between the magnitude of feedback gain and the strength of transient growth. The atter can be described quantitativey via the transient amification factor t 2 max = maxe Mt 2 = e Mt max 2, t,00 2 t 31 which measures the maximum amitude of an evoved disturbance t for a ossibe initia conditions 0. The initia condition roducing the maxima amification at time t max is often caed the otima disturbance ot and is given by the right singuar vector corresonding to the argest singuar vaue of e Mt max 10. For norma oerators =1, but for non-norma ones it can be arbitrariy arge. Severa authors have introduced quantities simiar to 31 to characterize transient growth 10,11,21,22. We shoud oint out that the transient amification factor is anaogous to transfer norms, which arise in the inut-outut descrition commony used in contro theoretic anayses, incuding those concerning transient growth Since both matrix exonentias and matrix norms are difficut to comute anayticay, as defined by 31 is usuay comuted numericay. Nevertheess, sometimes one can construct reasonaby tight bounds without resorting to numerics, an aroach that we wi foow here. To that end, we first note that for sma times e Mt 2 =1+t + Ot 2, 32 where = max W is the argest eigenvaue of W= 1 2 M +M 10. Athough this eigenvaue is aso difficut to comute exacty, a ower bound can be found by using the foowing roerty of Hermitian matrices: max W = maxu Mu. u=1 33 In the arge imit the eements of M vary by many orders of magnitude, so a tight ower bound is obtained by choosing a unit vector u which icks out the argest eement of M. This argest eement is M nn = n +Q nn rk max according to C7. Therefore, choosing u k = kn we obtain max Wrk max. Taking t=t max we obtain the foowing estimate for the maxima transient amification: k max t max. 34 Since 32 is ony accurate for sma times, 34 shoud rovide an accurate estimate for as ong as t max is reasonaby sma, i.e., when transient growth is a fast rocess. In order to make further anaytica rogress, et us again assume that the cosed oo system 21 has s identica eigenvaues k =, k=a,,b. The corresonding eigenmodes are identica and therefore do not form a comete basis. Hence the Jacobian matrix M is nondiagonaizabe. In this case, M can be converted into the Jordan norma form J = S 1 MS = J J s J 2, 35 where J 1 =diag..., a 2, a 1, J s is an ss Jordan bock with eigenvaues k =, J 2 =diag b+1, b+2,..., and S is the resective transformation matrix. The soution for the state t is t = a b c e t e + PHYSICAL REVIEW E 72, =a b m=0 t m c +m m!e t ê + c e t e, b 36 where e are eigenvectors corresonding to bocks J 1 and J 2, ê are the generaized eigenvectors such that Mê =ê +ê 1 for =a+1,...,b and ê a =e a is the ony eigenvector of J s. The c are integration constants that deend on the choice of the initia condition. The first and the ast sum in 36 monotonicay decay to zero the corresonding eigenvectors are norma and we can therefore negect them. The second sum exhibits both the initia agebraic growth and the asymtotic exonentia decay characteristic of transient dynamics. Each term in the second sum describes a mode of the cosed oo system with amitude ˆ mt tm m! et, m =1,...,s The maximum of ˆ m is achieved at t m = m. 38 Substituting 38 back into 37 we obtain ˆ mt m mm e m m! m. 39 This exression as a function of m has a singe extremum inside the domain, which is a minimum. The maximum occurs either on the eft or on the right boundary, that is either for m=1 or m=s 1, deending on the vaue of. Using the Stiring formua we can write 39 as 1 ˆ mt m. 40 2m m This exression shows that for 1, the goba maximum is achieved for m=1, whie for 1 the goba maximum wi be at m=s 1. In the intermediate region for

8 A. HANDEL AND R. O. GRIGORIEV PHYSICAL REVIEW E 72, FIG. 5. Transient growth of initia disturbances for a system with =0.5, two actuators, and =40 s=8. The soid and dashed ine show the numericay comuted exact resut 31 for = 0.1 and = 2, resectivey. The dash-dotted and dotted ine show the sma time aroximation 32 for = 0.1 and = 2, resectivey. neither maximum is dominant. Aroximating t max with the corresonding t m, we obtain t max =s 1 for sma and t max =1 for arge. Since for sma, the term with m=s 1 dominates, the magnitude of transienty amified disturbances cose to t max shoud be described by ˆ s 1t ts 1 s 1! et. 41 In this regime t max is arge and we shoud exect 34 to roduce a oor estimate of the maximum transient amification. Indeed, Fig. 5 shows that for = 0.1 the growth in e Mt 2 is inear ony for tt max and higher order oynomia terms dominate near t=t max. On the other hand, for arge the inear term with m = 1 dominates, so the magnitude of transienty amified disturbances is we aroximated by ˆ 1tte t. 42 In this regime t max t 1 =1/ is sma and we exect 34 to roduce an accurate estimate for. As Fig. 5 shows, one does indeed find a good agreement for = 2. For arge, we can therefore combine 34 with 38 and obtain to eading order in 1 e/r 0, 43 where 0 deends on both and. Just ike the feedback gain, the transient amification increases exonentiay fast with the ength of the system and decreases with the number of actuators. This scaing is comared with numerica resuts in Fig. 6. As exected, the agreement is better for = 2 than for = 0.1. However, even though the derivation eading to 43 breaks down for sma, we sti find that increases exonentiay fast with. FIG. 6. Transient amification as a function of system size. The resuts are shown for two actuators aced at the boundaries with =0.75 and a = 0.1 or b = 2. The cosed and oen circes show the numerica resuts comuted for an otima disturbance and for an ensembe of 100 random initia conditions, resectivey. The straight ine reresents the theoretica estimate 43. Since the otima disturbances which roduce the argest transient amification might not be reresentative of tyica random noise in the system, we additionay comute the amount of transient amification max t t 2 /0 2 achieved for an ensembe of random initia conditions. Figure 6 shows the maximum over a moderate number of such initia conditions. Not surrisingy, since none of the random initia conditions are otima, they are amified ess than the otima ones. However, transient amification of the random initia conditions scaes in the same way as comuted from 31. This shows that 43 gives a good estimate for the transient growth of both otima and generic random initia conditions. This aso means that the above resuts aso generaize naturay to systems continuousy driven by stochastic noise, which is the case in the majority of exerimentay reevant situations. Assuming that the noise has standard deviation, transient growth wi amify it by a factor of order, such that the resuting disturbance about the target state wi have a standard deviation of order. As a resut, the noise with =O 1 wi be amified to O1 at which oint one can effectivey concude that the contro has faied. In the next section we wi see that contro can fai for even weaker

9 PATTERN SELECTION AND CONTROL VIA LOCALIZED noise once the effect of noninear terms is considered. The resuts obtained in this section sti ay if we reax some of the assumtions. In the anaysis resented above, we assumed that a new eigenvaues of the cosed oo system are identica. This aowed us to isoate the terms in the forma soution 36 resonsibe for transient growth. For a more tyica case when the new eigenvaues are different, the Jacobian matrix M is diagonaizabe and a eigenvectors of M are distinct. Therefore, the soution 36 is not vaid anymore. If, however, M is strongy non-norma with cosey aigned eigenvectors, the evoution wi sti be characterized by strong transient growth of disturbances. In this case 32 wi sti hod, rovided the transient growth haens on a fast enough time scae. Since t max is rimariy determined by the sectrum of the eigenvaues k, but is weaky deendent on the system size, we can exect 43 to hod as we. This concusion is suorted by simiar cacuations we erformed for other attern forming systems e.g., Ginzburg- Landau and Kuramoto-Sivashinsky equations, for a different number of actuators r as we as for different choices of the cosed oo eigenvaues k. V. ESTIMATION OF CONTROL FAILURE Now that we have understood the behavior of the cosed oo system in the inear aroximation, we can turn our attention to the effect of noninear terms. Loosey seaking, once the noninear terms become comarabe to the inear terms, the main assumtion on which the inear stabiity anaysis is based breaks down eading to faiure of contro. As a more carefu anaysis shows, at east for ower aw noninearities such as n, the destabiization in a cosed oo system can roceed aong two searate routes 15. Inthe first, transient dynamics i.e., urey inear effect amifies an initia disturbance of magnitude to the oint where the order of magnitude of the inear terms,, is comarabe to that of the noninear terms, n. Equating the two exressions we find a ower aw scaing of the critica noise eve eading to destabiization, 44 with exonent = 1. In another route, the noninear terms act as a secondary disturbance, which sustains transient growth after the rimary disturbance has been suressed by feedback. In this scenario, which is sometimes referred to as bootstraing 9,26,27, secondary disturbances can be further transienty amified by the inear art of the evoution oerator, cosing a ositive feedback oo, which can aso ead to destabiization of the cosed oo system for sufficienty arge magnitudes of disturbances. The ositive feedback oo roduces growth in the magnitude of secondary disturbances when the order of magnitude of noninear terms, n, is comarabe to that of the secondary or rimary disturbance itsef,. Again, comaring these exressions we find a ower aw scaing 44, but with a different exonent = n/n 1. A combination or rather a cometition of the two mechanisms is aso ossibe, eading to a ower aw deendence of the critica noise eve with a crossover between the PHYSICAL REVIEW E 72, FIG. 7. Length at which contro of the uniform state fais for a given number r of actuators. We used an initia disturbance of magnitude =10 3 and =0.75. Contro is cacuated with new eigenvaues k saced uniformy on the interva 2, 1. The ines reresents the anaytica resut 45 with a crossover from = 1 to = 3/2 and = 1.5. two vaues of the exonent. This haens, for instance, for the Swift-Hohenberg equation with ocaized feedback 21, where the noninearity is cubic, n=3, so that one finds a crossover between = 1 for smaer where the direct route is dominant and = 3 2 for arger where the bootstraing route dominates. Denoting the coefficient of roortionaity in 44 as C and substituting 43 we find the minima number of actuators necessary to avoid destabiization of the cosed oo system in the resence of noise to be given by r 0 n C +n 1 n. 45 As the numeric data shown in Figs. 7 and 8 iustrates, 45 accuratey describes the deendence of the minima number of actuators on both the system size and the distance from the onset of rimary instabiity, which is easiy accessibe exerimentay, unike the deendence on the FIG. 8. Critica arameter at which contro of the uniform state fais for a given number r of actuators. We used an initia disturbance of magnitude =10 3 and =300. Contro is cacuated with new eigenvaues k saced uniformy on the interva 2, 1. The curve reresents the anaytica resut 45 with = 1 and =

10 A. HANDEL AND R. O. GRIGORIEV PHYSICAL REVIEW E 72, noise magnitude or the transient amification factor. In articuar, we find that r increases with ineary with a crossover in the soe, as redicted, as we as with. This resut aso eads to another interesting consequence. According to for and 0 const for 0, so that r diverges for both and 0. The otima choice that minimizes r for a given eve of noise corresonds to a finite vaue of that can be found by trivia differentiation of 45. In the concusion of this section we shoud mention that 45 is exected to describe the minima number of actuators for a broad cass of scaar reaction-diffusion equations, not just the Swift-Hohenberg equation 1. The deendence on the choice of the evoution equation and secific arameters is contained entirey in the characteristic ength 0 and the coefficient C. The characteristic ength can be comuted using 12 and describes the inear art of the evoution equation, whie the noninear art manifests through the coefficient C defined by 44. Comuting C quantitativey for a genera noninearity remains an oen robem. VI. FEEDBACK CONTROL OF PATTERNS So far we have focused on controing the uniform state. However, one might aso be interested in using feedback to seect one of the atterned nonuniform states, such as 3, that exist above the onset of rimary instabiity. Without feedback the Swift-Hohenberg equation 1 tyicay evoves toward a eriodic state q with the base wave number q near the center of the unstabe band, seected as a resut of noninear cometition invoving secondary instabiities. In this section we restrict our attention to systems with eriodic boundary conditions, for which the eigenmodes of the oen oo system are given by 25. Therefore the infinite domain resuts summarized in Sec. II ay with the restriction that the base wave number has to be quantized. A. Pattern seection Locaized feedback contro can be used to bias the attern seection in favor of a articuar state with minor modifications. Indeed, a ateray confined system ossesses a finite discrete set of unstabe modes f m, amb for any 0 see Fig. 9. The feedback designed to stabiize the uniform state suresses the growth of a these modes by shifting their wave numbers into the stabe band. If the feedback for one of the originay unstabe modes is turned off, e.g., we set K 1 ±m =0, then the mode m wi start to grow, unti noninear saturation sets in, resuting in a stationary attern 3 with the base wave number q m. Since, for moderate, a higher harmonics 3m,5m,... of the mode m ie in the stabe band, the feedback wi change neither their temora dynamics nor their saturated amitudes. Furthermore, we exect the feedback to be abe to suress one attern e.g., m ros in favor of a different attern e.g., n ros by making m 0 and setting K 1 ±n =0, so that ±n = ±n. This attern switching is exected to work ony for reasonaby sma, since the amitude of the saturated state, and with it the deviation from the uniform state, FIG. 9. The eigenvaue sectrum of the Swift-Hohenberg equation with =0.25. The cosed oo eigenvaues are chosen such that the five ro attern is seected on the domain of size =40. reative to which the feedback is comuted, scaes ike 1/2. For arger vaues of the noninear terms in 1 become significant enough to invaidate the inear stabiity anaysis and to ead contro faiure. The attern seection and switching based on the aroach described in this section are iustrated in Fig. 10. The initia condition is taken as the uniform state 0 with a sma amount of broad sectrum noise added to it. Initiay, the feedback is chosen to suress a unstabe eigenmodes excet f ±5, eading to the emergence of a five ro attern. After that attern has saturated, the feedback is switched to suress a modes excet f ±7, resuting in the five ro attern being reaced with a seven ro attern. As exected, the attern seection becomes unreiabe for arge vaues of, roducing stationary but midy disordered atterns not shown. Since the asymtotic state in this case differs from the target nonineary saturated attern 3, the feedback signa does not vanish even for very arge times. This is in contrast to a other instances considered in this aer, where the feedback disaears when the target state is reached. B. Suression of secondary instabiities As is we known, the eriodic atterns q near the edges of the unstabe wave number band are unstabe towards the FIG. 10. Pattern switching from a five ro to a seven ro state for a system with four actuators, =40 and =0.25. The initia condition was chosen as a random noise with magnitude =

11 PATTERN SELECTION AND CONTROL VIA LOCALIZED PHYSICAL REVIEW E 72, contro breakdown is aicabe in a more genera context and hence the scaing reation 45 ikey describes the actuator sacing irresective of the target state. FIG. 11. Length at which the contro of a steady atterned state fais for a given number r of actuators. The wave number q ies in the unstabe band, near q +. The initia disturbance is of magnitude =10 3 and =0.75. New cosed oo eigenvaues are chosen to ie in the interva k 2, 1. For comarison we aso show the scaing resut 45 for the uniform target state with a crossover from = 1 to = 2 and = 1.5, chosen as the mean of the k. Eckhaus instabiity see Sec. II. This secondary instabiity can aso be suressed using ocaized feedback contro we wi restrict our attention to the mutie actuator case here. The comutation of the feedback gain again starts with rewriting the cosed oo evoution equation 21 for disturbances ˆ = q about the target attern: t ˆ = ˆ 1+ 2 x 2 ˆ 3 2 q ˆ 3 q ˆ 2 ˆ 3 + d xu t. r =1 46 Just ike in the case of the uniform target state, we can roject the inearized version of this equation onto the basis of Fourier modes 25. Since these are not the eigenmodes of the evoution oerator of the oen oo system anymore, the corresonding Jacobian matrix A wi not be diagona see Aendix D. Another way to ook at this is by noting that the inearization of 46 contains an extra term 3 q 2 ˆ which reresents noninear couing between the disturbance wave number k and the target attern base wave number q, roducing secondary disturbances with wave number ±2q ± k manifesting as the off-diagona terms in the Jacobian. These secondary disturbances wi affect the inear stabiity of the attern since, for k cose to q the mode with wave number 2q k ies inside the unstabe band. Our anaytica aroach cannot be easiy extended to this case, however the feedback gain coefficients that can be easiy comuted numericay using standard oe acement techniques. Comuting the ength of the system for which the contro with r actuators breaks down we again find that it is consistent with a inear deendence with a crossover in the soe see Fig. 11. What is erhas more unexected, we find the soes to be aroximated we by the exression 45 derived for the uniform state we take the crossover to be between the vaues of = 1 and = 2 here, because 46 contains a quadratic, instead of a cubic, noninearity. This suggests that our descrition of the mechanism for the VII. CONCLUSIONS AND OUTLOOK To summarize, we have shown that a broad cass of satiay homogeneous extended systems scaar onedimensiona reaction-diffusion PDEs-can, in rincie, be controed or more recisey, their uniform and atterned states can be made ineary stabe by aying feedback at a few satia ocations with the ony restriction that these ocations have to satisfy the stabiizabiity condition. The tradeoff, comared with contro schemes emoying satiay continuous feedback so-caed fuy actuated systems, is that the basin of attraction of the uniform state shrinks exonentiay fast with increasing distance between actuators due to a strong transient growth of both initia and sontaneous disturbances. As a resut, desite forma stabiity, systems controed by a sarse array of actuators become extremey sensitive to noise, if the distance between actuators exceeds a certain characteristic distance secific to a articuar system. This exonentia sensitivity is indeendent of the tye of the evoution equation or the choice of feedback and effectivey sets a imit on the minima number of actuators er unit ength of the system in the resence of noise. Noise sensitivity, and hence the minima number of actuators is aso indeendent for systems of ength 0 on the choice of boundary conditions. Since there is no mean fux and the diffusion is the ony effective transort mechanism, the infuence of boundary conditions any boundary conditions can extend no further than the characteristic ength 0. Noise sensitivity can be reduced, and hence the robustness of contro imroved, by either using a denser array of actuators or, to a esser degree, by an aroriate choice of the eigenvaues of the cosed oo system. The otima choice of the new eigenvaues which eads to the smaest transient amification deends on a deicate baance between the strength of the feedback gain and the time required to suress a disturbance. The exicit anaytica soutions for the feedback gain and the scaing reations for the transient amification factor resented in this aer shoud he obtain a deeer insight into contro theoretic methods, such as H contro 28, whose objective is to increase robustness with resect to worst-case disturbances. Furthermore, in comuting a stabiizing feedback we used the knowedge of the system state on the whoe domain 0 x. This requirement can be easiy reaxed by introducing a finite number of sensors that can measure the oca state of the system at an array of oints inside the domain. The feedback can then be comuted using the state estimate constructed based on these satiay ocaized measurements. Mathematicay, the contro robems and the state estimation robems are dua, so by soving the former we aso sove the atter 18. As a resut, one finds that the transient amification shoud scae ike e /r 0 +/ 0, 47 where r and are the number of actuators and sensors, resectivey, so that by using as many sensors as we have

12 A. HANDEL AND R. O. GRIGORIEV PHYSICAL REVIEW E 72, actuators haves the characteristic ength of the system. Finay, whie we restricted ourseves to evoution equations for scaar fieds in one satia dimension, we fuy exect the main quaitative resuts resented here, such as the fast growth of the minima number of actuators with the distance from the onset of rimary instabiity, to be aicabe aso to vector or tensor fieds and in higher dimensions e.g., to the Boussinesque equations for Rayeigh-Bénard convection in a three-dimensiona ce, thus roviding an exanation to the emirica observations of Tang and Bau 4 that the contro fais much sooner than the theoretica estimates 5 based on what the satiay continuous feedback woud imy. APPENDIX A: COMPUTING FEEDBACK FOR A SINGLE ACTUATOR The matrix M in 8 is the Jacobian of the cosed oo system and can be reresented as the sum of the Jacobian A=diag 1, 2, 3,... of the oen oo system and the matrix = K1 Q K2 K3 K 1 K 2 K 3 A1 K 1 K 2 K 3 ] ] ] ] reresenting the action of the feedback. Our goa here is to find a set of coefficients K m which wi render the matrix M stabe. For a system of a given ength, we ony have a finite number of ositive eigenvaues corresonding to unstabe modes of the oen oo system. We wi assume that n 0 for n=a,...,b and n 0 otherwise see Fig. 1. The eigenvaues n can aways be reindexed to achieve such an ordering. To render the matrix M stabe, we ony need to make these s=b a+1 eigenvaues negative. We assume for simicity that the eigenvaues of both the cosed oo and the oen oo system are rea; generaization to the case of comex eigenvaues is straightforward. This can be achieved by setting a K m =0 with the excetion of K a,k a+1,...,k b.to see this, et us write the matrix M in bock form as M = A 1 Q A s + Q s 0 0 Q 2 A 2. A2 A 1 and A 2 denote the diagona matrices containing the stabe eigenvaues of A, A s is the art of A containing the s unstabe eigenvaues, and Q 1, Q 2, and Q s are the resective nonzero bocks of A1. The bock structure of A2 shows that the change in the eigenvaues of A s through an aroriate choice of Q s does not affect the rest of the eigenvaues of M. We, therefore, need to focus on the s-dimensiona bock matrix M s =A s +Q s and choose the coefficients K a,...,k b such that M s becomes stabe. This automaticay renders the infinite matrix M stabe. The coefficients K m can be comuted anayticay. To that end, consider choosing the s new negative eigenvaues of M s as a sequence a,..., b. We then need to find K m that satisfy the set of equations detm s m I =0, m = a,...,b. A3 Due to the secia structure of A1, one can sove A3 for K m and obtains the resut 10. APPENDIX B: COMPUTING FEEDBACK FOR A PAIR OF ACTUATORS Projecting the inearization of the evoution equation 1 onto the basis 18 we obtain n = n n 1 m + 1 n K m m M n m=1 where n t and K n are again the Fourier coefficients of x,t, and kx, resectivey, defined as in 9. The matrix M = A + Q is again comosed of the Jacobian A =diag 1, 2, 3,... and 0 2K3 0 2K 2 0 Q =2K1 B1 2K 1 0 2K 3 ] ] ]. Reeating the rocedure described in Aendix A yieds the resut 19. To obtain the scaing of the feedback gain with, we again rewrite 19 in exonentia form K m = ex n m m + n m B2 2 m and aroximate the sums with integras in the arge imit. Integrating over the unstabe wave number band, we obtain to eading order in K m ex q + m q n B3 2q m q dq, from which 20 immediatey foows uon substitution of m =. APPENDIX C: COMPUTING FEEDBACK FOR MULTIPLE ACTUATORS Our choice 24 of the individua feedback gains means that the resective Fourier coefficients K m can be exressed in terms of the Fourier coefficients K m 1 K 1 m = K 1 m ex i m r C1 A simiar reation exists for the Fourier coefficients of the infuence functions 22: D n = ex i n r C2 Using these two reations, the evoution equation 26 can be written in the form