Time-Delayed Feedback Control Method

and

Unstable Controllers

Kestutis PyragasSemiconductor Physics Institute

11 A. Gostauto, 2600 Vilnius, Lithuaniapyragas@kes0.pfi.lt

Abstract

Time delayed-feedback control is an efficient methodfor stabilizing unstable periodic orbits of chaotic sys-tems. The method is based on applying feedbackproportional to the deviation of the current state ofthe system from its state one period in the past sothat the control signal vanishes when the stabiliza-tion of the desired orbit is attained. A brief review ofexperimental implementations, applications for theo-retical models, most important modifications as wellas recent advancements in the theory of the methodis presented. An idea of using unstable degrees offreedom in a feedback loop to avoid a well knowntopological limitation of the method is described indetails. This idea is extended for the problem ofadaptive stabilization of unknown steady states ofdynamical systems.

1 Introduction

An idea of controlling chaos has been first formu-lated by Ott, Grebogi, and Yorke [1] and attractedgreat interest among physicists over the past decade.Wy chaotic systems are interesting subjects for con-trol theory and applications? The major key ingre-dient for the control of chaos is the observation thata chaotic set, on which the trajectory of the chaoticprocess lives, has embedded within it a large numberof unstable periodic orbits (UPOs). In addition, be-cause of ergodicity, the trajectory visits or accessesthe neighborhood of each one of these periodic or-bits. Some of these periodic orbits may correspondto a desired system’s performance according to somecriterion. The second ingredient is the realizationthat chaos, while signifying sensitive dependence onsmall changes to the current state and henceforthrendering unpredictable the system state in the longtime, also implies that the system’s behavior can bealtered by using small perturbations. Then the ac-cessibility of the chaotic system to many different

period orbits combined with its sensitivity to smallperturbations allows for the control and manipula-tion of the chaotic process. These ideas stimulateda development of rich variety of new chaos controltechniques (see Refs. [2]–[4] for review), among whichthe delayed feedback control (DFC) method [5] hasgained widespread acceptance.

The DFC method is based on applying feedback pro-portional to the deviation of the current state ofthe system from its state one period in the past sothat the control signal vanishes when the stabiliza-tion of the desired orbit is attained. Alternativelythe DFC method is referred to as a method of time-delay autosynchronization, since the stabilization ofthe desired orbit manifests itself as a synchronizationof the current state of the system with its delayedstate. The DFC has the advantage of not requir-ing prior knowledge of anything except the periodof the desired orbit. It is particularly convenient forfast dynamical systems since does not require thereal-time computer processing. Experimental imple-mentations, applications for theoretical models, mostimportant modifications as as well as recent advance-ments in the theory of the DFC method are brieflylisted below.

Experimental implementations.— The time-delayedfeedback control has been successfully used inquite diverse experimental contexts including elec-tronic chaos oscillators [6]–[9], mechanical pendu-lums [10]–[11], lasers [12]–[14], a gas discharge sys-tem [15]–[17], a current-driven ion acoustic instabil-ity [18], a chaotic Taylor-Couette flow [19], chemi-cal systems [20]–[21], high-power ferromagnetic reso-nance [22], helicopter rotor blades [23], and a cardiacsystem [24].

Applications for theoretical models.— The DFCmethod has been verified for a large number of the-oretical models from different fields. Simmendingerand Hess [25] proposed an all-optical scheme based

on the DFC for controlling delay-induced chaoticbehavior of high-speed semiconductor lasers. Theproblem of stabilizing semiconductor laser arrays hasbeen considered as well [26]–[27]. Rappel, Fenton,and Karma [28] used the DFC for stabilization ofspiral waves in an excitable media as a model ofcardiac tissue in order to prevent the spiral wavebreakup. Konishi, Kokame, and Hirata [29]–[30] ap-plied the DFC in a model of a car-following traf-fic. Batlle, Fossas, and Olivar [31] implemented theDFC in a model of buck converter. Bleich and So-colar [32] showed that the DFC can stabilize regularbehavior in a paced, excitable oscillator describedby Fitzhugh-Nagumo equations. Holyst, Zebrowska,and Urbanowicz [33]–[34] used the DFC to controlchaos in economical model. Tsui and Jones investi-gated the problem of chaotic satellite attitude con-trol [35] and constructed a feedforward neural net-work with the DFC to demonstrate a retrieval behav-ior that is analogous to the act of recognition [36].The problem of controlling chaotic solitons by a time-delayed feedback mechanism has been considered byFronczak and Holyst [37]. Mensour and Longtin [38]proposed to use the DFC in order to store informa-tion in delay-differential equations. Galvanetto [39]demonstrated the delayed feedback control of chaoticsystems with dry friction. Lastly, Mitsubori and Ai-hara [40] proposed rather exotic application of theDFC, namely, the control of chaotic roll motion of aflooded ship in waves.

Modifications.— A reach variety of modifications ofthe DFC have been suggested in order to improve itsperformance. Adaptive versions of the DFC with au-tomatic adjustment of delay time [41]–[43] and con-trol gain [44]–[45] have been considered. Basso etal. [46]–[47] showed that for a Lur’e system (systemrepresented as feedback connection of a linear dy-namical part and a static nonlinearity) the DFC canbe optimized by introducing into a feedback loop alinear filter with an appropriate transfer function.For spatially extended systems, various modifica-tions based on spatially filtered signals have beenconsidered [48]–[50]. The wave character of dynam-ics in some systems allows a simplification of theDFC algorithm by replacing the delay line with thespatially distributed detectors. Mausbach et al. [17]reported such a simplification for a ionization waveexperiment in a conventional cold cathode glow dis-charge tube. Due to dispersion relations the delayin time is equivalent to the spatial displacement andthe control signal can be constructed without use ofthe delay line. Socolar, Sukow, and Gauthier [51] im-proved an original DFC scheme by using an informa-tion from many previous states of the system. Thisextended DFC (EDFC) scheme achieves stabilizationof UPOs with a greater degree of instability [52]–[53].

The EDFC presumably is the most important modi-fication of the DFC and it will be discussed at greaterlength in this paper.

Advancements in the theory.—The theory of theDFC is rather intricate since it involves nonlineardelay-differential equations. If the equations govern-ing the system dynamics are known the success of theDFC method can be predicted by a linear stabilityanalysis of the desired orbit. Several numerical meth-ods for the linear stability analysis of time-delayedfeedback systems have been developed. The maindifficulty of this analysis is related to the fact thatperiodic solutions of such systems have an infinitenumber of Floquet exponents (FEs), though onlyseveral FEs with the largest real parts are relevant forstability properties. Most straightforward methodfor evaluating several largest FEs is described inRef. [52]. It adapts the usual procedure of estimatingthe Lyapunov exponents of strange attractors. Thismethod requires a numerical integration of the vari-ational system of delay-differential equations. Bleichand Socolar [53] devised an elegant method to obtainthe stability domain of the system under EDFC inwhich the delay terms in variational equations areeliminated due to the Floquet theorem and the ex-plicit integration of time-delay equations is avoided.Unfortunately, this method does not define the val-ues of the FEs inside the stability domain and isunsuitable for optimization problems. An approx-imate analytical method for estimating the FEs oftime-delayed feedback systems has been developedin Refs. [54]–[56]. Here as well as in Ref. [53] thedelay terms in variational equations are eliminatedand the Floquet problem is reduced to the system ofordinary differential equations. However, the FEs ofthe reduced system depend on a parameter that isa function of the unknown FEs itself. In Refs. [54]–[56] the problem is solved on the assumption thatthe FE of the reduced system depends linearly onthe parameter. This method gives a better insightinto mechanism of the DFC and leads to reasonablequalitative results. Some recent theoretical resultson the DFC method are presented in Ref. [57]. Hereit is shown that the main stability properties of thesystem controlled by time-delayed feedback can besimply derived from a leading FE defining the sys-tem behavior under proportional feedback control.As a result the optimal parameters of the delayedfeedback controller can be evaluated without an ex-plicit integration of delay-differential equations.

Although linear stability analysis of the delayed feed-back systems is difficult, some general analytical re-sults have been obtained [54],[58]–[60]. It has beenshown that the DFC can stabilize only a certain classof periodic orbits characterized by a finite torsion.

More precisely, the limitation is that any UPOs withan odd number of real Floquet multipliers (FMs)greater than unity (or with an odd number of realpositive FEs) can never be stabilized by the DFC.This statement was first proved by Ushio [58] fordiscrete time systems. Just et al. [54] and Naka-jima [59] proved the same limitation for the continu-ous time DFC, and then this proof was extended fora wider class of delayed feedback schemes, includingthe EDFC [60]. Hence it seems hard to overcome thisinherent limitation. Two efforts based on an oscil-lating feedback [61]–[62] and a half-period delay [63]have been taken to obviate this drawback. In bothcases the mechanism of stabilization is rather un-clear. Besides, the method of Ref. [63] is valid onlyfor a special case of symmetric orbits. The limitationhas been recently eliminated in a new modificationof the DFC that does not utilize the symmetry ofUPOs [64]. The key idea is to introduce into a feed-back loop an additional unstable degree of freedomthat changes the total number of unstable torsion-free modes to an even number. Then the idea of usingunstable degrees of freedom in a feedback loop wasdrown on to construct a simple adaptive controllerfor stabilizing unknown steady states of dynamicalsystems [65].

Some recent theoretical results on the unstable con-troller are presented in more details in the rest ofthe paper. In Section 2 the problem of stabilizingtorsion-free periodic orbits is considered. We startwith a simple discrete time model and show that anunstable degree of freedom introduced into a feed-back loop can overcome the limitation of the DFCmethod. Then we propose a generalized modifica-tion of the DFC for torsion-free UPOs and demon-strate its efficiency for the Lorenz system. Section 3is devoted to the problem of adaptive stabilizationof unknown steady states of dynamical systems. Wepropose an adaptive controller described by ordinarydifferential equations and prove that the steady statecan never be stabilized if the system and controllerin sum have an odd number of real positive eigen-values. We show that the adaptive stabilization ofsaddle-type steady states requires the presence of anunstable degree of freedom in a feedback loop. Thepaper is finished with conclusions presented in Sec-tion 4.

2 Stabilizing torsion-free periodic orbits

Most investigations on the DFC method are re-stricted to the consideration of unstable periodic or-bits erasing from a flip bifurcation. The leading Flo-quet multiplier of such orbits is real and negative (orcorresponding FE lies on the boundary of the “Bril-

louin zone”, ImΛ = π/T ). Such a consideration ismotivated by the fact that the usual DFC and EDFCmethods work only for the orbits with a finite tor-sion, when the leading FE obeys ImΛ 6= 0. Unsuit-ability of the DFC technique to stabilize torsion-freeorbits (ImΛ = 0) has been over several years consid-ered as a main limitation of the method [54],[58]–[60].More precisely, the limitation is that any UPOs withan odd number of real Floquet multipliers greaterthan unity can never be stabilized by the DFC. Thislimitation can be explained by bifurcation theory asfollows. When an UPO with an odd number of realFMs greater than unity is stabilized, one of such mul-tipliers must cross the unite circle on the real axes inthe complex plane. Such a situation correspond to atangent bifurcation, which is accompanied with a co-alescence of T-periodic orbits. However, this contra-dicts the fact that DFC perturbation does not changethe location of T-periodic orbits when the feedbackgain varies, because the feedback term vanishes forT-periodic orbits.

Here we describe an unstable delayed feedback con-troller that can overcome the limitation. The idea isto artificially enlarge a set of real multipliers greaterthan unity to an even number by introducing into afeedback loop an unstable degree of freedom.

2.1 Simple example: EDFC for R > 1First we illustrate the idea for a simple unstable dis-crete time system yn+1 = µsyn, µs > 1 controlled bythe EDFC:

yn+1 = µsyn −KFn, (1)Fn = yn − yn−1 + RFn−1. (2)

The free system yn+1 = µsyn has an unstable fixedpoint y? = 0 with the only real eigenvalue µs > 1and, in accordance with the above limitation, cannot be stabilized by the EDFC for any values of thefeedback gain K. This is so indeed if the EDFCis stable, i.e., if the parameter R in Eq. (2) satis-fies the inequality |R| < 1. Only this case has beenconsidered in the literature. However, it is easy toshow that the unstable controller with the parame-ter R > 1 can stabilize this system. Using the ansatzyn, Fn ∝ µn one obtains the characteristic equation

(µ− µs)(µ−R) + K(µ− 1) = 0 (3)

defining the eigenvalues µ of the closed loop system(1,2). The system is stable if both roots µ = µ1,2

of Eq. (3) are inside the unit circle of the µ complexplain, |µ1,2| < 1. Figure 1 (a) shows the character-istic root-locus diagram for R > 1, as the parame-ter K varies from 0 to ∞. For K = 0, there aretwo real eigenvalues greater than unity, µ1 = µs and

Figure 1: Performance of (a,b) discrete and (c) contin-uous EDFC for R > 1. (a) Root loci of Eq.(3) at µs = 3, R = 1.6 as K varies from 0to ∞. (b) Stability domain of Eqs. (1,2) inthe (K, R) plane; Kmx = (µs +1)2/(µs−1),Rmx = (µs + 3)/(µs − 1). (c) Root loci ofEq. (6) at λs = 1, R = 1.6. The crosses andcircles denote the location of roots at K = 0and K →∞, respectively.

µ2 = R, which correspond to two independent sub-systems (1) and (2), respectively; this means thatboth the controlled system and controller are un-stable. With the increase of K, the eigenvalues ap-proach each other on the real axes, then collide andpass to the complex plain. At K = K1 ≡ µsR − 1they cross symmetrically the unite circle |µ| = 1.Then both eigenvalues move inside this circle, col-lide again on the real axes and one of them leavesthe circle at K = K2 ≡ (µs + 1)(R + 1)/2. In theinterval K1 < K < K2, the closed loop system (1,2)is stable. By a proper choice of the parameters Rand K one can stabilize the fixed point with an ar-bitrarily large eigenvalue µs. The corresponding sta-bility domain is shown in Fig. 1 (b). For a givenvalue µs, there is an optimal choice of the param-eters R = Rop ≡ µs/(µs − 1), K = Kop ≡ µsRop

leading to zero eigenvalues, µ1 = µ2 = 0, such thatthe system approaches the fixed point in finite time.

It seems attractive to apply the EDFC with the pa-rameter R > 1 for continuous time systems. Unfor-tunately, this idea fails. As an illustration, let us

consider a continuous time version of Eqs. (1,2)

y(t) = λsy(t)−KF (t), (4)F (t) = y(t)− y(t− τ) + RF (t− τ), (5)

where λs > 0 is the characteristic exponent of thefree system y = λsy and τ is the delay time. By asuitable rescaling one can eliminate one of the pa-rameters in Eqs. (4,5). Thus, without a loss of gen-erality we can take τ = 1. Equations (4,5) can besolved by the Laplace transform or simply by thesubstitution y(t), F (t) ∝ eλt, that yields the charac-teristic equation:

1 + K1− exp(−λ)

1−R exp(−λ)1

λ− λs= 0. (6)

In terms of the control theory, Eq. (6) defines thepoles of the closed loop transfer function. Thefirst and second fractions in Eq. (6) correspond tothe EDFC and plant transfer functions, respectively.The closed loop system (4,5) is stable if all the rootsof Eq. (6) are in the left half-plane, Reλ < 0. Thecharacteristic root-locus diagram for R > 1 is shownin Fig. 1 (c). When K varies from 0 to ∞, theEDFC roots move in the right half-plane from loca-tions λ = ln R+2πin to λ = 2πin for n = ±1,±2 . . ..Thus, the continuous time EDFC with the parame-ter R > 1 has an infinite number of unstable degreesof freedom and many of them remain unstable in theclosed loop system for any K.

2.2 Usual EDFC supplemented by an unsta-ble degree of freedomHereafter, we use the usual EDFC at 0 ≤ R < 1,however introduce an additional unstable degree offreedom into a feedback loop. More specifically, fora dynamical system x = f(x, p) with a measurablescalar variable y(t) = g(x(t)) and an UPO of periodτ at p = 0, we propose to adjust an available systemparameter p by a feedback signal p(t) = KFu(t) ofthe following form:

Fu(t) = F (t) + w(t), (7)w(t) = λ0

cw(t) + (λ0c − λ∞c )F (t), (8)

F (t) = y(t)− (1−R)∞∑

k=1

Rk−1y(t− kτ), (9)

where F (t) is the usual EDFC described by Eq. (5)or equivalently by Eq. (9). Equation (8) defines anadditional unstable degree of freedom with parame-ters λ0

c > 0 and λ∞c < 0. We emphasize that when-ever the stabilization is successful the variables F (t)and w(t) vanish, and thus vanishes the feedback forceFu(t). We refer to the feedback law (7–9) as an un-stable EDFC (UEDFC).

Figure 2: Root loci of Eq. (11) at λs = 2, λ0c = 0.1,

λ∞c = −0.5, R = 0.5. The insets (a) and(b) show Reλ vs. K and the Nyquist plot,respectively. The boundaries of the stabilitydomain are K1 ≈ 1.95 and K2 ≈ 11.6.

To get an insight into how the UEDFC works let usconsider again the problem of stabilizing the fix point

y = λsy −KFu(t), (10)

where Fu(t) is defined by Eqs. (7–9) and λs > 0.Here as well as in a previous example we can take τ =1 without a loss of generality. Now the characteristicequation reads:

1 + KQ(λ) = 0, (11)

Q(λ) ≡ λ− λ∞cλ− λ0

c

1− exp(−λ)1−R exp(−λ)

1λ− λs

. (12)

The first fraction in Eq. (12) corresponds to thetransfer function of an additional unstable degree offreedom. Root loci of Eq. (11) is shown in Fig. 2. Thepoles and zeros of Q-function define the value of rootsat K = 0 and K →∞, respectively. Now at K = 0,the EDFC roots λ = lnR + 2πin, n = 0,±1, . . . arein the left half-plane. The only root λ0

c associatedwith an additional unstable degree of freedom is inthe right half-plane. That root and the root λs ofthe fix point collide on the real axes, pass to thecomplex plane and at K = K1 cross into the lefthalf-plane. For K1 < K < K2, all roots of Eq. (11)satisfy the inequality Reλ < 0, and the closed loopsystem (7–10) is stable. The stability is destroyed atK = K2 when the EDFC roots λ = ln R ± 2πi in

the second “Brillouin zone” cross into Reλ > 0. Thedependence of the five largest Reλ on K is shownin the inset (a) of Fig. 2. The inset (b) shows theNyquist plot, i.e., a parametric plot ReN(ω) versusImN(ω) for ω ∈ [0,∞], where N(ω) ≡ Q(iω). TheNyquist plot provides the simplest way of determin-ing the stability domain; it crosses the real axes atReN = −1/K1 and ReN = −1/K2.

As a more involved example let us consider theLorenz system under the UEDFC:

xyz

=

−σx + σyrx− y − xzxy − bz

−KFu(t)

010

.

(13)We assume that the output variable is y and the feed-back force Fu(t) [Eqs. (7–9)] perturbs only the secondequation of the Lorenz system. Denote the variablesof the Lorenz system by ρ = (x, y, z) and those ex-tended with the controller variable w by ξ = (ρ, w)T .For the parameters σ = 10, r = 28, and b = 8/3,the free (K = 0) Lorenz system has a period-oneUPO, ρ0(t) ≡ (x0, y0, z0) = ρ0(t+τ), with the periodτ ≈ 1.5586 and all real FMs: µ1 ≈ 4.714, µ2 = 1 andµ3 ≈ 1.19 × 10−10. This orbit can not be stabilizedby usual DFC or EDFC, since only one FM is greaterthan unity. The ability of the UEDFC to stabilizethis orbit can be verified by a linear analysis of Eqs.(13) and (7–9). Small deviations δξ = ξ − ξ0 fromthe periodic solution ξ0(t) ≡ (ρ0, 0)T = ξ0(t + τ)may be decomposed into eigenfunctions according tothe Floquet theory, δξ = eλtu, u(t) = u(t + τ),where λ is the Floquet exponent. The Floquet de-composition yields linear periodically time depen-dent equations δξ = Aδξ with the boundary con-dition δξ(τ) = eλτδξ(0), where

A =

−σ σ 0 0r − z0(t) −(1 + KH) −x0(t) −K

y0(t) x0(t) −b 00 (λ0

c − λ∞c )H 0 λ0c

.

(14)Due to equality δy(t − kτ) = e−kλτδy(t), the delayterms in Eq. (9) are eliminated, and Eq. (9) is trans-formed to δF (t) = Hδy(t), where

H = H(λ) = (1−exp(−λτ))/(1−R exp(−λτ)) (15)

is the transfer function of the EDFC. The price forthis simplification is that the Jacobian A, definingthe exponents λ, depends on λ itself. The eigenvalueproblem may be solved with an evolution matrix Φt

that satisfies

Φt = AΦt, Φ0 = I. (16)

The eigenvalues of Φτ define the desired exponents:

det[Φτ (H)− eλτI] = 0. (17)

Figure 3: Stabilizing an UPO of the Lorenz system.(a) Six largest Reλ vs. K. The boundaries ofthe stability domain are K1 ≈ 2.54 and K2 ≈12.3. The inset shows the (x, y) projection ofthe UPO. (b) and (c) shows the dynamics ofy(t) and Fu(t) obtained from Eqs. (13,7–9).The parameters are: λ0

c = 0.1, λ∞c = −2,R = 0.7, K = 3.5, ε = 3, λr = 10.

We emphasize the dependence Φτ on H conditionedby the dependence of A on H. Thus by solvingEqs. (15–17), one can define the Floquet exponentsλ (or multipliers µ = eλτ ) of the Lorenz system un-der the UEDFC. Figure 3 (a) shows the dependenceof the six largest Reλ on K. There is an intervalK1 < K < K2, where the real parts of all exponentsare negative. Basically, Fig. 3 (a) shows the resultssimilar to those presented in Fig. 2 (a). The unstableexponent λ1 of an UPO and the unstable eigenvalueλ0

c of the controller collide on the real axes and passinto the complex plane providing an UPO with a fi-nite torsion. Then this pair of complex conjugateexponents cross into domain Reλ < 0, just as theydo in the simple model of Eq. (10).

Direct integration of the nonlinear Eqs. (13, 7–9)confirms the results of linear analysis. Figures 3 (b,c)show a successful stabilization of the desired UPOwith an asymptotically vanishing perturbation. Inthis analysis, we used a restricted perturbation sim-ilar as we did in Ref. [5]. For |F (t)| < ε, the controlforce Fu(t) is calculated from Eqs. (7–9), howeverfor |F (t)| > ε, the control is switched off, Fu(t) = 0,and the unstable variable w is dropped off by replac-

ing Eq. (8) with the relaxation equation w = −λrw,λr > 0.

To verify the influence of fluctuations a small whitenoise with the spectral density S(ω) = a has beenadded to the r.h.s. of Eqs. (8,13). At every stepof integration the variables x, y, z, and w wereshifted by an amount

√12haξi, where ξi are the ran-

dom numbers uniformly distributed in the interval[−0.5, 0.5] and h is the stepsize of integration. Thecontrol method works when the noise is increased upto a ≈ 0.02. The variance of perturbation increasesproportionally to the noise amplitude, 〈F 2

u(t)〉 = ka,k ≈ 17. For a large noise a > 0.02, the system inter-mittently loses the desired orbit.

3 Stabilizing and tracking unknown steadystates

Although the field of controlling chaos deals mainlywith the stabilization of unstable periodic orbits, theproblem of stabilizing unstable steady states of dy-namical systems is of great importance for varioustechnical applications. Stabilization of a fixed pointby usual methods of classical control theory requiresa knowledge of its location in the phase space. How-ever, for many complex systems (e.g., chemical orbiological) the location of the fixed points, as wellas exact model equations, are unknown. In this caseadaptive control techniques capable of automaticallylocating the unknown steady state are preferable. Anadaptive stabilization of a fixed point can be attainedwith the time-delayed feedback method [5],[52],[66].However, the use of time-delayed signals in this prob-lem is not necessary and thus the difficulties relatedto an infinite dimensional phase space due to delaycan be avoided. A simpler adaptive controller forstabilizing unknown steady states can be designedon a basis of ordinary differential equations (ODEs).The simplest example of such a controller utilizes aconventional low pass filter described by one ODE.The filtered dc output signal of the system estimatesthe location of the fixed point, so that the differencebetween the actual and filtered output signals canbe used as a control signal. An efficiency of such asimple controller has been demonstrated for differentexperimental systems [66]–[68]. Further examples in-volve methods which do not require knowledge of theposition of the steady state but result in a nonzerocontrol signal [69].

In this section we describe a generalized adaptivecontroller characterized by a system of ODEs andprove that it has a topological limitation concern-ing an odd number of real positive eigenvalues of thesteady state [65]. We show that the limitation can

Figure 4: Stabilizing an unstable fixed point withan unstable controller in a simple model ofEqs. (18) for λs = 1 and λc = 0.1. (a)Root loci of the characteristic equation ask varies from 0 to ∞. The crosses and soliddot denote the location of roots at k = 0and k → ∞, respectively. (b) Reλ vs. k.k0 = λs + λc, k1,2 = λs + λc ∓ 2

√λsλc.

be overcome by implementing an unstable degree offreedom into a feedback loop. The feedback producesa robust method of stabilizing a priori unknown un-stable steady states, saddles, foci, and nodes.

3.1 Simple exampleAn adaptive controller based on the conventionallow-pass filter, successfully used in several experi-ments [66], is not universal. This can be illustratedwith a simple model:

x = λs(x− x?) + k(w − x), w = λc(w − x). (18)

Here x is a scalar variable of an unstable one-dimensional dynamical system x = λs(x−x?), λs > 0that we intend to stabilize. We imagine that the lo-cation of the fixed point x? is unknown and use afeedback signal k(w−x) for stabilization. The equa-tion w = λc(w − x) for λc < 0 represents a conven-tional low-pass filter (rc circuit) with a time constantτ = −1/λc. The fixed point of the closed loop sys-tem in the whole phase space of variables (x,w) is(x?, x?) so that its projection on the x axes corre-sponds to the fixed point of the free system for anycontrol gain k. If for some values of k the closed loopsystem is stable, the controller variable w convergesto the steady state value w? = x? and the feedbackperturbation vanishes.

The closed loop system is stable if both eigenvaluesof the characteristic equation λ2 − (λs + λc − k)λ +λsλc = 0 are in the left half-plane Reλ < 0. Thestability conditions are: k > λs + λc, λsλc > 0. Wesee immediately that the stabilization is not possiblewith a conventional low-pass filter since for any λs >0, λc < 0, we have λsλc < 0 and the second stability

criterion is not met. However, the stabilization canbe attained via an unstable controller with a positiveparameter λc. Electronically, such a controller can bedevised as the RC circuit with a negative resistance.Figure 4 shows a mechanism of stabilization. For k =0, the eigenvalues are λs and λc, which correspondto the free system and free controller, respectively.With the increase of k, they approach each other onthe real axes, then collide at k = k1 and pass to thecomplex plane. At k = k0 they cross symmetricallyinto the left half-plane (Hopf bifurcation). At k =k2 we have again a collision on the real axes andthen one of the roots moves towards −∞ and anotherapproaches the origin. For k > k0, the closed loopsystem is stable. An optimal value of the controlgain is k2 since it provides the fastest convergence tothe fixed point.

3.2 Generalized adaptive controllerNow we consider the problem of adaptive stabiliza-tion of a steady state in general. Let

x = f(x, p) (19)

be the dynamical system with N -dimensional vectorvariable x and and L-dimensional vector parameterp available for an external adjustment. Assume thatan n-dimensional vector variable y(t) = g

(x(t)

)(a

function of dynamical variables x(t)) represents thesystem output. Suppose that at p = p0 the sys-tem has an unstable fixed point x? that satisfiesf(x?,p0) = 0. The location of the fixed point x?

is unknown. To stabilize the fixed point we perturbthe parameters by an adaptive feedback

p(t) = p0 + kB[Aw(t) + Cy(t)] (20)

where w is an M -dimensional dynamical variable ofthe controller that satisfies

w(t) = Aw + Cy. (21)

Here A, B, and C are the matrices of dimensionsM ×M , M × L, and n×M , respectively and k is ascalar parameter that defines the feedback gain. Thefeedback is constructed in such a way that it does notchange the steady state solutions of the free system.For any k, the fixed point of the closed loop sys-tem in the whole phase space of variables {x,w} is{x?,w?}, where x? is the fixed point of the free sys-tem and w? is the corresponding steady state valueof the controller variable. The latter satisfies a sys-tem of linear equations Aw? = −Cg(x?) that hasunique solution for any nonsingular matrix A. Thefeedback perturbation kBw vanishes whenever thefixed point of the closed loop system is stabilized.

Small deviations δx = x−x? and δw = w−w? fromthe fixed point are described by variational equations

δx = Jδx + kPBδw, δw = CGδx + Aδw, (22)

where J = Dxf(x?,p0), P = Dpf(x?, p0), andG = Dxg(x?). Here Dx and Dp denote the vec-tor derivatives (Jacobian matrices) with respect tothe variables x and parameters p, respectively. Thecharacteristic equation for the closed loop systemreads:

∆k(λ) ≡∣∣∣∣

Iλ− J −kλPB−CG Iλ−A

∣∣∣∣ = 0. (23)

For k = 0 we have ∆0(λ) = |Iλ − J ||Iλ − A|and Eq. (23) splits into two independent equations|Iλ − J | = 0 and |Iλ − A| = 0 that define N eigen-values of the free system λ = λs

j , j = 1, . . . , Nand M eigenvalues of the free controller λ = λc

j ,j = 1, . . . , M , respectively. By assumption, at leastone eigenvalue of the free system is in the right half-plane. The closed loop system is stabilized in an in-terval of the control gain k for which all eigenvaluesof Eq. (23) are in the left half-plane Reλ < 0.

The following theorem defines an important topolog-ical limitation of the above adaptive controller. It issimilar to the Nakajima theorem [59] concerning thelimitation of the time-delayed feedback controller.

Theorem.— Consider a fixed point x? of a dynamicalsystem (19) characterized by Jacobian matrix J andan adaptive controller (21) with a nonsingular matrixA. If the total number of real positive eigenvalues ofthe matrices J and A is odd, then the closed loopsystem described by Eqs. (19)-(21) cannot be stabi-lized by any choice of matrices A, B, C and controlgain k.

Proof.— The stability of the closed loop system isdetermined by the roots of ∆k(λ). Writing Eq. (23)for k = 0 in the basis where martcies J and A arediagonal, we have

∆0(λ) =∏N

j=1(λ− λsj)

∏Mm=1(λ− λc

m). (24)

Here λsj and λc

m are the eigenvalues of the matricesJ and A, respectively. Now from Eq. (23), we alsohave ∆k(0) = ∆0(0), so Eq. (24) implies

∆k(0) =∏N

j=1(−λsj)

∏Mm=1(−λc

m) (25)

for all k. Since the total number of eigenvalues λsj

and λcm that are real and positive is odd and other

eigenvalues are real and negative or come in complexconjugate pairs, ∆k(0) must be real and negative.On the other hand, from the definition of ∆k(λ) wesee immediately that when λ → ∞ then ∆k(λ) →λN+M > 0 for all k. ∆k(λ) is an N + M orderpolynomial with real coefficients and is continuousfor all λ. Since ∆k(λ) is negative for λ = 0 and ispositive for large λ, it follows that ∆k(λ) = 0 forsome real positive λ. Thus the closed loop system

always has at least one real positive eigenvalue andcannot be stabilized, Q.E.D.

This limitation can be explained by bifurcation the-ory, similar to Ref. [59]. If a fixed point with an oddtotal number of real positive eigenvalues is stabilized,one of such eigenvalues must cross into the left half-plane on the real axes accompanied with a coales-cence of fixed points. However, this contradicts thefact that the feedback perturbation does not changelocations of fixed points.

From this theorem it follows that any fixed point x?

with an odd number of real positive eigenvalues can-not be stabilized with a stable controller. In otherwords, if the Jacobian J of a fixed point has an oddnumber of real positive eigenvalues then it can be sta-bilized only with an unstable controller whose matrixA has an odd number (at least one) of real positiveeigenvalues.

3.3 Controlling an electrochemical oscillatorThe use of an unstable degree of freedom in a feed-back loop is now demonstrated with control in anelectrodissolution process, the dissolution of nickel insulfuric acid. The main features of this process canbe qualitatively described with a model proposed byHaim et al. [70]. The dimensionless model togetherwith the controller reads:

e = i− (1−Θ)[

Ch exp(0.5e)1 + Ch exp(e)

+ a exp(e)]

(26)

Θ =1Γ

[exp(0.5e)(1−Θ)

1 + Ch exp(e)− bCh exp(2e)Θ

Chc + exp(e)

](27)

w = λc(w − i) (28)

Here e is the dimensionless potential of the electrodeand Θ is the surface coverage of NiO+NiOH. An ob-servable is the current

i = (V0 + δV − e)/R, δV = k(i− w), (29)

where V0 is the circuit potential and R is the seriesresistance of the cell. δV is the feedback perturbationapplied to the circuit potential, k is the feedbackgain. From Eqs. (29) it follows that i = (V0 − e −kw)/(R − k) and δV = k(V0 − e − wR)/(R − k).We see that the feedback perturbation is singular atk = R.

In a certain interval of the circuit potential V0, afree (δV = 0) system has three coexisting fixedpoints: a stable node, a saddle, and an unstable focus[Fig. 5 (a)]. Depending on the initial conditions, thetrajectories are attracted either to the stable node orto the stable limit cycle that surrounds an unstablefocus. As is seen from Figs. 5 (b) and 5 (c) the coex-isting saddle and the unstable focus can be stabilized

Figure 5: Results of analysis of the electrochemicalmodel for R = 50, Ch = 1600, a = 0.3, b =6× 10−5, c = 10−3, Γ = 0.01. (a) Steady so-lutions e? vs. V0 of the free (δV = 0) system.Solid, broken, and dotted curves correspondto a stable node, a saddle, and an unsta-ble focus, respectively. (b) and (c) Eigenval-ues of the closed loop system as functions ofcontrol gain k at V0 = 63.888 for the saddle(e?, Θ?) = (0, 0.0166) controlled by an unsta-ble controller (λc = 0.01) and for the unsta-ble focus (e?, Θ?) = (−1.7074, 0.4521) con-trolled by a stable controller (λc = −0.01),respectively. (d) Stability domain in (k, V0)plane for the saddle (crossed lines) at λc =0.01 and for the focus (inclined lines) atλc = −0.01.

with the unstable (λc > 0) and stable (λc < 0) con-troller, respectively if the control gain is in the inter-val k0 < k < R = 50. Figure 5 (d) shows the stabilitydomains of these points in the (k, V0) plane. If thevalue of the control gain is chosen close to k = R, thefixed points remain stable for all values of the poten-tial V0. This enables a tracking of the fixed pointsby fixing the control gain k and varying the potentialV0. In general a tracking algorithm requires a con-tinuous updating of the target state and the controlgain. Here described method finds the position ofthe steady states automatically. The method is ro-bust enough in the examples investigated to operatewithout change in control gain. We also note thatthe stability of the saddle and focus points can beswitched by a simple reversal of sign of the parame-ter λc.

Laboratory experiments for this system have been

successfully carried out by I. Z. Kiss and J. L. Hud-son [65]. They managed to stabilize and track boththe unstable focus and the unstable saddle steadystates. For the focus the usual rc circuit has beenused, while the saddle point has been stabilized withthe unstable controller. The robustness of the con-trol algorithm allowed the stabilization of unstablesteady states in a large parameter region. By map-ping the stable and unstable phase objects the au-thors have visualized saddle-node and homoclinic bi-furcations directly from experimental data.

4 Conclusions

The aim of this paper was to review experimentalimplementations, applications for theoretical mod-els, and modifications of the time-delayed feedbackcontrol method and to present some recent theoreti-cal ideas in this field.

In Section 2 we discussed the main limitation of thedelayed feedback control method, which states thatthe method cannot stabilize torsion-free periodic or-bits, ore more precisely, orbits with an odd numberof real positive Flocke exponents. We have shownthat this topological limitation can be eliminated byintroduction into a feedback loop an unstable degreeof freedom that changes the total number of unstabletorsion-free modes to an even number. An efficiencyof the modified scheme has been demonstrated forthe Lorenz system. Note that the stability analy-sis of the torsion-free orbits controlled by unstablecontroller can be performed in a similar manner asdescribed in Ref. [57]. This problem is currently un-der investigation and the results will be publishedelsewhere.

In Section 3 the idea of unstable controller has beenused for the problem of stabilizing unknown steadystates of dynamical systems. We have consideredan adaptive controller described by a finite set ofordinary differential equations and proved that thesteady state can never be stabilized if the system andcontroller in sum have an odd number of real posi-tive eigenvalues. For two dimensional systems, thistopological limitation states that only an unstable fo-cus or node can be stabilized with a stable controllerand stabilization of a saddle requires the presence ofan unstable degree of freedom in a feedback loop.The use of the controller to stabilize and track sad-dle points (as well as unstable foci) has been demon-strated numerically with an electrochemical Ni dis-solution system.

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