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Control of the Lorenz system: Destroying the homoclinic orbits
Physics Letters A 338 (2005) 128–140 www.elsevier.com/locate/pla
Control of the Lorenz system: Destroying the homoclinic orbits Jose Alvarez-Ramirez 1 , Julio Solis-Daun 1 , Hector Puebla ∗ Corina 117-G3, Col. Del Carmen, 04100 Coyoacan, DF, Mexico Received 26 May 2003; received in revised form 17 May 2004; accepted 15 February 2005
Communicated by A.R. Bishop
Abstract The mechanisms leading to chaotic behavior in the Lorenz system are well understood. Basically, homoclinic connections induce a strange invariant set around the zero fluid motion stationary point. This set, associated with a Smale horseshoe, is in the heart of chaotic attractors. This Letter examines the application of a simple feedback controller to eliminate the chaotic behavior in a controlled Lorenz system. The main idea is to stabilize certain stationary points to destroy the homoclinic connections. In this way, stabilization of the Lorenz trajectories about non-chaotic motion is achieved. The effectivity of the feedback control strategy is illustrated by means of numerical simulations. 2005 Elsevier B.V. All rights reserved. Keywords: Lorenz system; Chaos control; Damping
1. Introduction Based on results due to Salzmann [1], in 1963 Edward Lorenz [2] derived and characterized a simple set of ordinary differential equations representing three modes (one in velocity and two in temperatures) of the * Corresponding author. Instrumentacion y Control, Instituto Mexicano del Petroleo, Eje Central Lazaro Cardenas No. 152, Col. San Bartolo Atepehuacan, 07730 México, DF, Mexico. E-mail address: [email protected] (H. Puebla). 1 J. Alvarez-Ramirez (Departamento de Ingenieria de Procesos) and J. Solis-Daun (Departamento de Matematicas) are in the Division de Ciencias Basicas e Ingenieria, Universidad Autonoma Metropolitana-Iztapalapa.
0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.02.024
Oberbeck–Boussinesq equations for fluid convection in a two-dimensional layer heated from below. Lorenz showed that the fluid motion in this system is chaotic for a sufficiently high rate of heating. Since then, the chaotic behavior of the Lorenz system has been studied and characterized (see, for instance, [3,4]). Physically, chaotic motion in the Lorenz system means that the fluid displays turbulent convection and nonperiodic temperature fluctuations. Its simplicity has made the Lorenz system a benchmark model for studying and understanding the stylized mechanisms that lead to turbulent motion in convection systems. Because of the possibility of operating a physical system at conditions where in a natural way the dynamics are turbulent, the use of feedback control has
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attracted the attention of the physics community. For instance, in the case of chaotic fluid phenomena, it is desirable to use a feedback control strategy to have less turbulent or even laminar motion at Reynold and Rayleigh numbers conditions well beyond the critical values. In practice, it would result in physical systems operating, e.g., at reduced drag conditions. Since the seminal paper by Ott et al. [5], in the last few years a large amount of results have been reported dealing with the application of feedback control techniques to regulate (i.e., reduce or even eliminate) the chaotic behavior (see the recent survey by Fradkov and Evans [6]). The underlying idea is to manipulate an accessible parameter of the system to induce a desired controlled dynamics. A drawback of most reported feedback controllers to control chaos is their lack of specificity. That is, most reported feedback control techniques labeled as “chaos controllers” are merely straightforward applications of well-known feedback control methodologies without a suitable exploitation and knowledge of the underlying chaos mechanisms. For instance, Bewley [7] proposed an observer-based feedback controller to stabilize a stationary convection state of the Lorenz system. The Bewley’s controller is based on an exact model of the system and requires the specification of the full stationary convection state. This is an important drawback for potential practical applications since specification of the stationary convective state is not an easy task due to the lack of an accurate knowledge of the system dynamics. In principle, a feedback control technique that exploit the chaos structure could lead to specific strategies focused on modifying or destroying the mechanisms responsible of chaotic/turbulent motion. Indeed, due to the lack of an exact knowledge of the system dynamics and due to the high dimensionality of convection systems, one should look for control strategies that rely on the particular structure of turbulence. This Letter uses the Lorenz system as a benchmark to study the feedback control of chaotic fluid phenomena. The main objective is to study a feedback control strategy derived from the idea of reducing the effects or even eliminating the mechanisms leading to chaotic motion. Specifically, certain homoclinic orbits are responsible of the presence of strange invariant sets which, eventually cause the erratic and unpredictable dynamics of the Lorenz system. The idea is to use
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feedback control to stabilize certain equilibrium points (corresponding to convective rollers) whose basin of attraction are able to erosion the basin of attraction of the chaotic attractor. The result is a simple feedback control strategy that makes use of measurements of the vertical temperature fluctuations in the fluid. With respect to existing results in the literature [6,7], our contribution can be summarized as follows: • Our control strategy leads to a specific chaos controller in the sense that it makes use of the structure of the chaotic attractor. • The resulting controller is reference-free. That is, an exogenous reference to the feedback compensator is not required. In this way, our feedback control strategy is “self-controlling”, as defined by Pyragas [8]. This feature is quite important for practical applications since a precise knowledge of the operating conditions (e.g., equilibrium intensity of the fluid motion and temperatures in convective systems) are hardly available. Numerical simulations are used along the Letter to illustrate the performance, including advantages and drawbacks, of the proposed feedback controller.
2. Chaotic dynamics in the Lorenz system The aim of this section is to provide a brief description of the mechanisms leading to chaotic motion in the Lorenz system. Indeed, the discussion will be focused on those results relevant for designing a chaos feedback controller. The equations derived by Lorenz are x˙1 = σ (x2 − x1 ), x˙2 = −x2 − x1 x3 , x˙3 = −bx3 + x1 x2 − br,
(1)
where x1 is proportional to the intensity of the fluid motion, x2 is proportional to the lateral temperature difference in the fluid, and x3 is proportional to the vertical temperature difference in the fluid. The loop Rayleigh number r is proportional to the heating rate at the bottom of the convective system, the loop Prandtl number σ is related to the fluid’s kinematic viscosity and thermal conductivity, the quantity b is
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related to the fluid’s thermal expansion coefficient, and all variables has been non-dimensionalized. When required, and for a more specific discussion, the values used by Lorenz and most other investigators are taken; namely, σ = 10.0 and b = 8/3, but similar behavior occurs for other values. Let us summarize the main results concerning the stability of equilibrium points of the Lorenz system [3]: (1) For r ∈ (0, 1), S = (0, 0, −r)T is the unique equilibrium point and is globally asymptotically stable. (2) For r > 1 the origin is non-stable. The flow linearized around the origin has two negative, and one positive, real eigenvalues. Furthermore, there are two additional equilibrium points C1√= (−d, −d, −1)T and C2 = (d, d, −1)T , where d = b(r − 1). (3) For 1 < r < rH ≈ 24.74, C1 and C2 are stable. All three eigenvalues of the flow, linearized about C1 and C2 , have negative real part. Providing r > 1.346 there is a complex conjugate pair of eigenvalues. (4) At r = rH , as the complex eigenvalues of C1 and C2 cross the imaginary axis, there is a subcritical Hopf bifurcation in which the equilibrium points C1 and C2 loss their stability. (5) For r > rH , C1 and C2 are non-stable. The flow, linearized around C1 and C2 , has one negative real eigenvalue and a complex conjugate pair of eigenvalues with positive real part. The equilibrium point S corresponds to zero fluid motion, and the equilibrium points C1 and C2 correspond respectively to uniform clockwise and anticlockwise fluid motion. In the sequel, we refer to C1 and C2 as convection stationary points because they corresponds to an effective (i.e., non-zero fluid motion) convection process. A central feature of the Lorenz system is its dissipativeness property. In fact, there exists a bounded set A ⊂ R3 , containing all the equilibrium points, such that any trajectory x(t, x(0)) with initial condition x(0) ∈ R3 will enter into A in finite time tf 0 and will remain there for all t > tf . Since for r > rH all three equilibrium points are non-stable, and since trajectories cannot escape from A, it is interesting to ask the following: where do most trajectories go? Instrumental in answering this question and on explain-
ing the advent of chaotic behavior is the existence of homoclinic orbits. In fact, for r = rhom 13.926, a global bifurcation appears where two homoclinic orbits, say H1 and H2 , of the equilibrium point S are found (see Fig. 2.3.9 in [4]). For r < rhom , the branches of the unstable manifold of S are trapped by either of the equilibrium points C1 and C2 . In this way, all trajectories tend to either of the equilibrium points C1 and C2 rather rapidly (see Fig. 1). For r slightly larger than rhom , the homoclinic orbits H1 and H2 induce a strange invariant set I in the vicinity of the equilibrium point S. This set consists of a countable infinity of periodic orbits, an uncountable infinity of aperiodic orbits, and an uncountable infinity of trajectories which terminates in S. All these orbits and trajectories are non-stable, as is the strange invariant set. In fact, almost all of them leave the vicinity of S and presumably spiral in towards either C1 or C2 (recall that these equilibrium points have a pair of complex eigenvalues). A pre-turbulence behavior is found in the range of values around r = rA ≈ 24.06 but smaller than rH . At this values of the parameter r, the stationary points C1 and C2 are still stable, but a small-size non-stable periodic orbit around each of them limits the rate at which trajectories are converged to either C1 or C2 . In pre-turbulence, though the trajectory eventually spirals in to either C1 or C2 , it wanders erratically near the strange invariant set I for a long time before being trapped by either C1 or C2 . For r > rH , the equilibrium points C1 and C2 are unstable and the strange invariant set I becomes stable. Trajectories are not longer trapped by the stationary points C1 and C2 , so that they wander into the invariant region A. Almost all trajectories starting in a vicinity of S are expelled along a tube of the unstable manifold of S towards the equilibrium points C1 and C2 . Around C1 and C2 , the trajectories spiral out. Since these equilibrium points are unstable, the trajectories are expelled along a tube around the homoclinic orbits H1 and H2 . Eventually, the trajectories enter in finite time a vicinity of the equilibrium point S where, due to the “initial conditions sensitivity” property of the strange invariant set I, are expelled out in an erratic way towards either C1 or C2 . This phenomenon is repeated again and again leading to what is called chaotic behavior (or non-periodic behavior).
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Fig. 1. Homoclinic orbits of the Lorenz system.
3. Chaos control for the Lorenz system The aim of this section is to exploit the structure of the chaotic attractor described in the above section to design a feedback controller whose objective is to eliminate the chaotic behavior. For control design, one has to identify a physically accessible parameter. As in Bewley [7], the heating rate at the bottom of the system, which is proportional to the parameter r, is used for control purposes. The idea is to modulate the heating rate about a steady-state value into the chaotic regimen r¯ > rH , so that r = r¯ + rf , where rf is the input used for feedback. In this way, the control problem can be formulated as designing a feedback control rf (x) such that the chaotic behavior displayed by the Lorenz system is eliminated and rf t → 0 as t → ∞. Here, rf t represents the mean of the signal rf (t). The condition rf t → 0 as t → ∞ is imposed to ensure that,
at steady state conditions, rt → r¯ as t → ∞. In this way, it is guaranteed that the controlled Lorenz convection system operates into a non-chaotic regimen at conditions that otherwise display turbulent behavior. That is, the turbulent behavior can be displaced by virtue of the feedback control action rf (x). As discussed in the above section, the homoclinic orbits H1 and H2 of the equilibrium point S are the responsible of the strange invariant set I. Eventually, as r is increased beyond rH , the set I becomes an strange attractor set where the chaotic behavior evolves. Since the aim of the feedback control rf (x) is to eliminate the chaotic behavior, an authentic chaos control strategy should rely on attacking the underlying mechanisms that generates the strange invariant attractor I. In this way, at the heart of the construction of I are the homoclinic orbits H1 and H2 . Consequently, if one is able to destroy the homoclinic orbits H1 and H2 , then
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one is in position to eliminate the chaotic convective turbulent behavior of the Lorenz system. In conclusion, the feedback controller rf (x) should focus on destroying the homoclinic connections H1 and H2 . Two ways to destroy, via feedback control, the orbits H1 and H2 can be described as follows: (a) To stabilize the equilibrium point S, so that no unstable manifolds, hence homoclinic orbits, exist. This option has the drawback that the equilibrium point S corresponds to a non-convection state, so that has a poor practical interest. (b) To stabilize the convection stationary points C1 and C2 . The idea is to induce, via feedback control, a locally exponentially stabilization such the homoclinic trajectories that spiral around C1 and C2 can be trapped and converged to these stationary points. If the homoclinic orbits H1 and H2 are destroyed, the former stranger invariant set I is also eliminated. In the sequel, we will approach the chaos control problem following the second option. Based on the above arguments, the control problem has been converted to the equivalent one of stabilizing the Lorenz system about the non-zero convection states C1 and C2 . 3.1. Feedback control design Let us proceed to design the feedback function rf (x). As discussed in the introduction, rf (x) must be self-controlling in the sense that stabilization of C1 and C2 must rely on none exogenous reference. That is, in practice the stationary states C1 and C2 are not known, so that the exact value of them cannot be provided to the controller. Indeed, this is a pitfall of most proposed chaos controllers (see [7] and references in [6]). In fact, the feedback controller rf (x) must stabilize the convective stationary states C1 and C2 without having access to their exact values. Actually, this corresponds to a robust stabilization problem without exact knowledge of the stabilization objective. The feedback control design is based on the following facts: Fact 1. Consider the modified Jacobian matrix J (α) of the Lorenz system about the stationary points C1
and C2 : J (α) =
−σ 1 αd
σ 0 −1 −d , αd −αb
(2)
where α is a positive constant and, as above, d = √ b(r − 1) (notice that the original Jacobian of the Lorenz system is recovered for α = 1). Then, there exists a positive constant αmin such that all the eigenvalues of J (α) have negative real part. To prove the this assertion, consider the characteristic polynomial P (s, α) of J (α): P (s, α) = s 3 + (αb + σ + 1)s 2 + α b + d 2 + bσ s + 2αd 2 σ.
(3)
Since σ , b and d are positive constants, all the coefficients of P (s, α) are positive. This is a necessary condition for stability of P (s, α). From the well-known Routh–Hurwitz criteria, stability of P (s, α) is guaranteed if (consider that α > 0) b + d 2 + bσ (αb + σ + 1) − 2d 2 σ > 0. Equivalently, if α > αmin , where 2d 2 σ 1 − σ − 1 . αmin = b b + d 2 + bσ
(4)
For r > rH , the effect of the parameter α is to move into the left-hand side of the complex plane the two unstable complex eigenvalues of J (1). In this way, continuity arguments imply that at α = αmin the complex eigenvalues of J (α) cross the imaginary axis. For the typical values σ = 10.0, b = 8/3 and r = 28.0, one has that αmin = 2.1617. If J (α) was the Jacobian of the controlled Lorenz system, then one has that the convective stationary points C1 and C2 are stable for α > αmin . This stabilization effect is obtained by means of the feedback function: rf (x) = −K x˙3 ,
(5)
b−1 =
where K is a positive constant, which is called as the controller gain. Notice that rf (x) feedbacks the time-derivative of vertical temperature fluctuations rather than the temperature-self. Since r = r¯ + rf (x) = r¯ − K x˙3 , one has that x˙3 = −bx3 + x1 x2 − br¯ − bK x˙3 , or x˙3 = α(−bx3 + x1 x2 − br¯ ),
(6)
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where 1 . α= 1 − bK One has the following result.
(7)
Fact 2. The following hold for the Lorenz system under the feedback function (5): (a) The former stationary points S, C1 and C2 remain as equilibrium points. The proof of this result is straightforward since 1 ), b1 ), the equilibrium α = 0. (b) For K ∈ ( b1 (1 − αmin points C1 and C2 are stable. The claim follows from the Fact 1; namely, α > αmin > 0. It should be stressed 1 ) < b1 , so that there are really posithat b1 (1 − αmin tive values of K such that the stationary points C1 and C2 are stable. In fact, since αmin > 0, one has that 1 < 1. 1 − αmin The above result ensures the existence of stabilizing feedback gains K. Given that the feedback function (5) is able to stabilize the stationary states C1 and C2 , what happens with the stability of the zero fluid motion stationary point S? Fact 3. There are not exists values of the control gain K such that the stationary point S is stable. The proof follows from analyzing the Jacobian matrix of the controlled Lorenz system about S. In such case, one has that −σ σ 0 J (α) = 1 −1 0 0 0 −αb and the corresponding characteristic polynomial is P (s, α) = s 3 + (αb + σ + 1)s 2 + αb(1 + σ )s, which has a zero eigenvalue regardless of the value of α. Summarizing, the feedback function (5) is able to stabilize the stationary states C1 and C2 for a suitable selection of the controller gain K. Besides, the zero fluid motion stationary state S is not stabilized by the controller rf (x) = −K x˙3 . It is also noted that the feedback controller (5) do not requires the specification of the coordinates of C1 or C2 (i.e., rf (x) is a self-stabilizing feedback function [8]). Finally, as one trajectory x(t, x(0)) converges to either the equilibrium points C1 or C2 , rf (x) → 0 as required (i.e., the feedback controller rf (x) vanishes at the stabilized stationary points).
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We know that during the chaotic regimen, trajectories passing close to the saddle point S are expelled out along a tube of the unstable manifold M u (S), then spiral quite rapidly towards a vicinity of the stationary points C1 and C2 . Subsequently, since such convective stationary points are non-stable, the trajectories are expelled out from the vicinity of C1 and C2 towards a vicinity of S. In principle, if the convective stationary points are stabilized by the action of the feedback control (5), the trajectories can be eventually trapped and converged towards either C1 or C2 . Fig. 2 presents the behavior of the controlled Lorenz system for two values of α (1.4 and 2.0) and two initial conditions close to the saddle point S. The value α = 1.4 is close to the minimum stability value αmin = 1.20394, so that C1 and C2 are only weakly stable. In this case, the trajectories display a behavior that is similar to a pre-turbulence regimen. In fact, the trajectory spirals slowly around the stationary point before being stabilized. However, for α = 2.0 the trajectory is rapidly trapped and stabilized because the convective equilibrium points have better stability properties. 3.2. Implementation of the feedback control rf (x) = K x˙3 The main result of the above analysis is the existence of a feedback controller rf (x) = K x˙3 that is able to stabilize the convective stationary points C1 and C2 without a knowledge of their coordinates. The controller (5) is quite simple since its action is based solely on feed-backing the fluctuations (represented by x˙3 ) of the vertical temperature difference x3 . In this way, since rf (x) vanishes at C1 and C2 , one is able to operate the Lorenz system at conditions r = r¯ that otherwise (i.e., without the action of the feedback control) induce turbulent behavior. These features make the feedback controller structure (5) quite attractive for potential applications. However, in practice the fluctuations x˙3 are not easily measured. Instead, what is actually measured is the vertical temperature difference x3 . Given this lack of measurement, the following question is posed: How the feedback controller (5) can be implemented based solely on measurements of vertical temperature difference measurements x3 ? The key point to address this question is to exploit the fact that x˙3 is the time-derivative of x3 . Then, the idea is to
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Fig. 2. Behavior of the controlled Lorenz system for two values of α and two initial conditions close to the saddle point S.
use an estimate x˙3e of x˙3 obtained from measures of x3 , so that the practical feedback function becomes rf (x) = K x˙3e .
(8)
Two procedures to obtain the estimate x˙3e can be depicted: • Finite difference approximation. Let T > 0 be a sampling period. Then, the simple approximation
x˙3e (t) = T1 (x3 (t) − x3 (t − T )). This leads to the feedback function rf (x) = KT x3 (t) − x3 (t − T ) , (9) where KT = K/T . Interestingly, the feedback function (9) corresponds to the delayed feedback function proposed by Pyragas [8]. In this way, one can see the proposed feedback function (5) as the limit T → 0 of the delayed feedback controller (9).
J. Alvarez-Ramirez et al. / Physics Letters A 338 (2005) 128–140 d • Continuous-time approximation. Let s = dt be the time-derivative operator. Then x˙3 = sx3 . It is noted that s is not a causal operator, so that an exact computation of the time derivative x˙3 cannot be obtained using measures of the state x3 only [9]. One can obtain a continuous-time approximation x˙3e by taking a causal approximation of the time-derivative operator s in the following form: x˙3e (t) = F (s)sx3 (t), where F (s) is a filtering operator satisfying F (0) = 1. The simplest one is F (s) = ωc /(s + ωc ) where ωc is called as the cutting frequency [9]. In this way, one obtains
x˙3e (t) =
ωc s x3 (t). s + ωc
(10)
It is noted that one recovers the exact time-derivative operator when the cutting frequency goes to larger ωc s → s as ωc → ∞. Actually, the values; that is, s+ω c operation (10) is equivalent to a high-pass filter with cutting frequency ωc . A realization of (9) in terms of a differential system can be made as follows. Expresd x˙ e 3 sion (10) implies that dt3 + ωc x˙3e = ωc dx dt (here we dx have used the notation dt instead of x˙ to avoid confusion with the notation used for the estimated timederivative x˙3e ). Introduce the variable z = x˙3e − ωc x3 , so that dz = −ωc (z + ωc x3 ), dt and x˙3e = z + ωc x3 .
z(0) = z0
(11)
(12)
Observe that the estimate x˙3e (t) is obtained on the basis of measures of the state x3 (t) only. That is, the differential system (11)–(12) offers a continuous-time approximation of the time-derivative x˙3 based only on measures of the state x3 (t). In the above subsection, we have proven that there are values of the control gain K such that the controlled Lorenz system is stable about the convective stationary points C1 and C2 (see Facts 1 and 2). The following result states that the continuous-time approximation (10) can retain the stability of the controlled Lorenz system. Fact 4. Assume that the control gain K is chosen such that the convective equilibrium points C1 and C2 are stable under the exact feedback controller (5). Then,
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there exists a positive constant ωc,min such the equilibrium points C1 and C2 are stable by virtue of the approximate feedback controller (8), for all ωc > ωc,min . To prove this result, we proceed as follows. Under ωc s )x3 , the effect of the controller rf (x) = −K( s+ω c the Jacobian matrix of the controlled Lorenz system about C1 and C2 can be written as −σ σ 0 , J (K, τc ) = (13) 1 −1 −d d d −bQ(s, K, τc ) where τc = ωc−1 and Q(s, K, τc ) = N (s, K, τc )/ (τc s + 1) and N (s, K, τc ) = (τc + K)s + 1. The characteristic polynomial of (13) can be expressed as P1 (s, K, τc ) = (τc s + 1)R(s, α, τc ), where R(s, α, τc ) = s 3 + bN (s, K, τc ) + σ + 1 s 2 + [α + σ ]bN (s, K, τc ) + d 2 s + 2αd 2 σ. One has that, since N (s, K, 0) = Ks, R(s, K, 0) is equivalent (via a scaling of the coefficient of s 3 ) to the characteristic polynomial P (s, α). By virtue of Fact 2, P (s, α) is stable for all α > αmin . From the continuity property of the roots of polynomials, one conclude the existence of a positive constant τc,min such that R(s, α, τc ) is stable for α > αmin + δ (where δ is a positive constant) and all positive τc < τc,min . The claim of the Fact 4 is finally proven by observing that the product P1 (s, α, τc ) of two stable polynomials τc s + 1 and R(s, α, τc ) is also stable. 3.3. Discussion The results described above can be summarized as follows. We have proven that the feedback controller given by rf (x) = −K x˙3e ,
(14)
where x˙3e is computed from (11)–(12) is able to stabilize the convective stationary points C1 and C2 for suitable choices of the controller parameters K and ωe . Key features of such proposed controller are that its implementation rely on measurements of vertical temperature difference and does not require specification of the coordinates of the stationary points. Besides, the control signal rf (x(t)) vanishes as x(t)
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converges to either C1 or C2 . We think these features make attractive the proposed feedback controller for potential applications in real buoyancy-driven systems. To account for the fact that the chaotic instability is constrained to a bounded region [3], the following modification to the feedback control function (14) is proposed: rf (x) = rf,max tanh −K x˙3e , (15) where rf,max > 0 is the maximum amplitude of the control input. The rationale behind the modified control function (15) is that the aim of the feedback action is to stabilize locally the stationary points C1 and C2 , so that such objective can be achieved with a bounded stabilizing signal. It should be stressed that the conditions required to ensure the result in Fact 4 are only sufficient. In the numerical simulations to be described in the following section, we will show that the selection of the control gain K is not so tight. In fact, by fixing a value K, one can tune the cutting frequency ωc to sufficiently large values in order to obtain stability of the convection stationary points C1 and C2 . Finally, the structure of the feedback function ωc s )x3 allows a nice interpretation rf (x) = −K( s+ω c in the frequency-domain terminology. The operator ωc s H (s) = s+ω is actually a high-pass filter with cutc ting frequency ωc . H (s) removes modes evolving at frequencies much smaller than ωc , passing only modes at higher frequencies than ωc . In this way, the effectivity of the feedback controller (14) can be explained from this property: the action of the conωc s )x3 can be seen as a feedtroller rf (x) = −K( s+ω c back damping of the high-frequency modes of the vertical temperature difference fluctuations. As a consequence, high-frequency oscillation modes of the chaotic Lorenz system are effectively damped, resulting in the stability of the stationary points C1 and C2 .
4. Numerical simulations This section is devoted to illustrate the performance of the proposed chaos controller. All simulations below are based on the typical parameter set used in above sections; that is, σ = 10.0, b = 8/3 and r¯ = 28.0.
For the exact feedback controller (5), one has that K < b1 = 3/8. However, the use of the approximate ωc s )x3 allows more feedback function rf (x) = −K( s+ω c flexibility and less tightness in the selection of the control gain K. This can be explained by the fact that conditions for stability under this controller are only sufficient (Fact 4). Fig. 3 presents the behavior of the controlled Lorenz system for K = 2.0, ωc = 10.0 and rf,max = 4.0. Simulations are shown for two initial conditions close to the zero fluid motion stationary point S; the trajectory of one of them converges to C1 and the other to C2 . Similar behavior has been found for other values of K; for instance, for K = 1.0 and 5.0. The value of rf,max plays a central role on the rate of stabilization of the stationary points. In fact, we have found that the minimum value of rf,max at which stabilization is obtained is about 0.142. In this way, the larger the value of rf,max is, the faster the trajectory convergence. This is illustrated in Fig. 4 for three values of rf,max . It is noted that, because C1 and C2 have two stable complex eigenvalues, the trajectories converge in an spiral manner. These spiraling dynamics are reduced as the maximum value of the control input rf,max is increased. For the Lorenz system values σ = 10.0, b = 8/3 and r¯ = 28.0, we have not found stable periodic orbits under the feedback control action. In fact, we have found a direct transition from an apparently chaotic motion to a stable dynamics regimen about C1 and C2 as the controller parameters (K, ωc , rf,max ) are varied in the correct direction. An interesting bifurcation phenomenon is found for large values of r¯ . For values of r¯ larger than about 120.0, the behavior of the Lorenz system is dominated by large stable periodic orbits [3]. For r¯ = 150.0, Fig. 5 shows the deformation of the attractor for the uncontrolled Lorenz system (rf,max = 0.0) and two different values of rf,max . For rf,max = 0.0, the trajectories are attracted to a large period-2 periodic orbits. The feedback controller stabilizes the convective stationary points C1 and C2 . However, if the maximum control amplitude rf,max is not sufficiently large, the basin of attraction of these points are not sufficiently large to trap the periodic orbit. In this case, there is a conflict between the attracting power of the former periodic orbit and the stabilized stationary points, and an apparent chaotic attractor is displayed. The form
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Fig. 3. Behavior of the controlled Lorenz system for K = 2.0, ωc = 10.0 and rf,max = 4.0.
of such chaotic attractor is induced by the feedback action and is different from those found for the uncontrolled Lorenz system. As rf,max is increased, the basin of attraction of the stationary points becomes larger and is able to capture the trajectories of the system.
5. Conclusions It has been shown that a close knowledge of the mechanisms leading to chaotic motions allows to design simple, specific and effective feedback controllers
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Fig. 4. Stabilization of the Lorenz system for three values of rf,max .
to reduce and even eliminate the chaos dynamics from a physical system. In the particular case of the Lorenz system, it has been shown that a simple feedback controller with the structure of a high-gain filter on the vertical temperature difference is able to stabilize
the convection stationary points. Although the numerical simulations revealed the effectivity of the proposed control strategy, it should be stressed that the Lorenz system is only a very simplistic model of actual buoyancy-driven flows. In this form, the potential
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Fig. 5. Deformation of the attractor for the uncontrolled Lorenz system (rf,max = 0.0) and two different values of rf,max .
application of the results reported in this Letter should be evaluated carefully because of the high complexity of actual flow systems. However, as mentioned by Be-
wley [7], it is important to understand the control problems and the remedies available in a low-dimensional framework before moving to more complex systems.
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References [1] B. Salzmann, J. Atmos. Sci. 19 (1963) 239. [2] E.N. Lorenz, J. Atmos. Sci. 20 (1963) 130. [3] C.S. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Springer-Verlag, New York, 1983. [4] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
[5] E. Ott, C. Grebogi, Y.A. Yorke, Phys. Rev. Lett. 64 (1990) 1196. [6] A.L. Fradkov, R.J. Evans, in: Preprints of 15th IFAC World Congress on Automatic Control, Barcelona, 2002, p. 143. [7] T.R. Bewley, Phys. Fluids 11 (1999) 1169. [8] K. Pyragas, Phys. Lett. A 170 (1992) 421. [9] K. Ogata, Modern Control Engineering, second ed., Prentice Hall, Englewood Cliffs, NJ, 1990.
Control of the Lorenz system: Destroying the homoclinic orbits Jose Alvarez-Ramirez 1 , Julio Solis-Daun 1 , Hector Puebla ∗ Corina 117-G3, Col. Del Carmen, 04100 Coyoacan, DF, Mexico Received 26 May 2003; received in revised form 17 May 2004; accepted 15 February 2005
Communicated by A.R. Bishop
Abstract The mechanisms leading to chaotic behavior in the Lorenz system are well understood. Basically, homoclinic connections induce a strange invariant set around the zero fluid motion stationary point. This set, associated with a Smale horseshoe, is in the heart of chaotic attractors. This Letter examines the application of a simple feedback controller to eliminate the chaotic behavior in a controlled Lorenz system. The main idea is to stabilize certain stationary points to destroy the homoclinic connections. In this way, stabilization of the Lorenz trajectories about non-chaotic motion is achieved. The effectivity of the feedback control strategy is illustrated by means of numerical simulations. 2005 Elsevier B.V. All rights reserved. Keywords: Lorenz system; Chaos control; Damping
1. Introduction Based on results due to Salzmann [1], in 1963 Edward Lorenz [2] derived and characterized a simple set of ordinary differential equations representing three modes (one in velocity and two in temperatures) of the * Corresponding author. Instrumentacion y Control, Instituto Mexicano del Petroleo, Eje Central Lazaro Cardenas No. 152, Col. San Bartolo Atepehuacan, 07730 México, DF, Mexico. E-mail address: [email protected] (H. Puebla). 1 J. Alvarez-Ramirez (Departamento de Ingenieria de Procesos) and J. Solis-Daun (Departamento de Matematicas) are in the Division de Ciencias Basicas e Ingenieria, Universidad Autonoma Metropolitana-Iztapalapa.
0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.02.024
Oberbeck–Boussinesq equations for fluid convection in a two-dimensional layer heated from below. Lorenz showed that the fluid motion in this system is chaotic for a sufficiently high rate of heating. Since then, the chaotic behavior of the Lorenz system has been studied and characterized (see, for instance, [3,4]). Physically, chaotic motion in the Lorenz system means that the fluid displays turbulent convection and nonperiodic temperature fluctuations. Its simplicity has made the Lorenz system a benchmark model for studying and understanding the stylized mechanisms that lead to turbulent motion in convection systems. Because of the possibility of operating a physical system at conditions where in a natural way the dynamics are turbulent, the use of feedback control has
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attracted the attention of the physics community. For instance, in the case of chaotic fluid phenomena, it is desirable to use a feedback control strategy to have less turbulent or even laminar motion at Reynold and Rayleigh numbers conditions well beyond the critical values. In practice, it would result in physical systems operating, e.g., at reduced drag conditions. Since the seminal paper by Ott et al. [5], in the last few years a large amount of results have been reported dealing with the application of feedback control techniques to regulate (i.e., reduce or even eliminate) the chaotic behavior (see the recent survey by Fradkov and Evans [6]). The underlying idea is to manipulate an accessible parameter of the system to induce a desired controlled dynamics. A drawback of most reported feedback controllers to control chaos is their lack of specificity. That is, most reported feedback control techniques labeled as “chaos controllers” are merely straightforward applications of well-known feedback control methodologies without a suitable exploitation and knowledge of the underlying chaos mechanisms. For instance, Bewley [7] proposed an observer-based feedback controller to stabilize a stationary convection state of the Lorenz system. The Bewley’s controller is based on an exact model of the system and requires the specification of the full stationary convection state. This is an important drawback for potential practical applications since specification of the stationary convective state is not an easy task due to the lack of an accurate knowledge of the system dynamics. In principle, a feedback control technique that exploit the chaos structure could lead to specific strategies focused on modifying or destroying the mechanisms responsible of chaotic/turbulent motion. Indeed, due to the lack of an exact knowledge of the system dynamics and due to the high dimensionality of convection systems, one should look for control strategies that rely on the particular structure of turbulence. This Letter uses the Lorenz system as a benchmark to study the feedback control of chaotic fluid phenomena. The main objective is to study a feedback control strategy derived from the idea of reducing the effects or even eliminating the mechanisms leading to chaotic motion. Specifically, certain homoclinic orbits are responsible of the presence of strange invariant sets which, eventually cause the erratic and unpredictable dynamics of the Lorenz system. The idea is to use
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feedback control to stabilize certain equilibrium points (corresponding to convective rollers) whose basin of attraction are able to erosion the basin of attraction of the chaotic attractor. The result is a simple feedback control strategy that makes use of measurements of the vertical temperature fluctuations in the fluid. With respect to existing results in the literature [6,7], our contribution can be summarized as follows: • Our control strategy leads to a specific chaos controller in the sense that it makes use of the structure of the chaotic attractor. • The resulting controller is reference-free. That is, an exogenous reference to the feedback compensator is not required. In this way, our feedback control strategy is “self-controlling”, as defined by Pyragas [8]. This feature is quite important for practical applications since a precise knowledge of the operating conditions (e.g., equilibrium intensity of the fluid motion and temperatures in convective systems) are hardly available. Numerical simulations are used along the Letter to illustrate the performance, including advantages and drawbacks, of the proposed feedback controller.
2. Chaotic dynamics in the Lorenz system The aim of this section is to provide a brief description of the mechanisms leading to chaotic motion in the Lorenz system. Indeed, the discussion will be focused on those results relevant for designing a chaos feedback controller. The equations derived by Lorenz are x˙1 = σ (x2 − x1 ), x˙2 = −x2 − x1 x3 , x˙3 = −bx3 + x1 x2 − br,
(1)
where x1 is proportional to the intensity of the fluid motion, x2 is proportional to the lateral temperature difference in the fluid, and x3 is proportional to the vertical temperature difference in the fluid. The loop Rayleigh number r is proportional to the heating rate at the bottom of the convective system, the loop Prandtl number σ is related to the fluid’s kinematic viscosity and thermal conductivity, the quantity b is
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related to the fluid’s thermal expansion coefficient, and all variables has been non-dimensionalized. When required, and for a more specific discussion, the values used by Lorenz and most other investigators are taken; namely, σ = 10.0 and b = 8/3, but similar behavior occurs for other values. Let us summarize the main results concerning the stability of equilibrium points of the Lorenz system [3]: (1) For r ∈ (0, 1), S = (0, 0, −r)T is the unique equilibrium point and is globally asymptotically stable. (2) For r > 1 the origin is non-stable. The flow linearized around the origin has two negative, and one positive, real eigenvalues. Furthermore, there are two additional equilibrium points C1√= (−d, −d, −1)T and C2 = (d, d, −1)T , where d = b(r − 1). (3) For 1 < r < rH ≈ 24.74, C1 and C2 are stable. All three eigenvalues of the flow, linearized about C1 and C2 , have negative real part. Providing r > 1.346 there is a complex conjugate pair of eigenvalues. (4) At r = rH , as the complex eigenvalues of C1 and C2 cross the imaginary axis, there is a subcritical Hopf bifurcation in which the equilibrium points C1 and C2 loss their stability. (5) For r > rH , C1 and C2 are non-stable. The flow, linearized around C1 and C2 , has one negative real eigenvalue and a complex conjugate pair of eigenvalues with positive real part. The equilibrium point S corresponds to zero fluid motion, and the equilibrium points C1 and C2 correspond respectively to uniform clockwise and anticlockwise fluid motion. In the sequel, we refer to C1 and C2 as convection stationary points because they corresponds to an effective (i.e., non-zero fluid motion) convection process. A central feature of the Lorenz system is its dissipativeness property. In fact, there exists a bounded set A ⊂ R3 , containing all the equilibrium points, such that any trajectory x(t, x(0)) with initial condition x(0) ∈ R3 will enter into A in finite time tf 0 and will remain there for all t > tf . Since for r > rH all three equilibrium points are non-stable, and since trajectories cannot escape from A, it is interesting to ask the following: where do most trajectories go? Instrumental in answering this question and on explain-
ing the advent of chaotic behavior is the existence of homoclinic orbits. In fact, for r = rhom 13.926, a global bifurcation appears where two homoclinic orbits, say H1 and H2 , of the equilibrium point S are found (see Fig. 2.3.9 in [4]). For r < rhom , the branches of the unstable manifold of S are trapped by either of the equilibrium points C1 and C2 . In this way, all trajectories tend to either of the equilibrium points C1 and C2 rather rapidly (see Fig. 1). For r slightly larger than rhom , the homoclinic orbits H1 and H2 induce a strange invariant set I in the vicinity of the equilibrium point S. This set consists of a countable infinity of periodic orbits, an uncountable infinity of aperiodic orbits, and an uncountable infinity of trajectories which terminates in S. All these orbits and trajectories are non-stable, as is the strange invariant set. In fact, almost all of them leave the vicinity of S and presumably spiral in towards either C1 or C2 (recall that these equilibrium points have a pair of complex eigenvalues). A pre-turbulence behavior is found in the range of values around r = rA ≈ 24.06 but smaller than rH . At this values of the parameter r, the stationary points C1 and C2 are still stable, but a small-size non-stable periodic orbit around each of them limits the rate at which trajectories are converged to either C1 or C2 . In pre-turbulence, though the trajectory eventually spirals in to either C1 or C2 , it wanders erratically near the strange invariant set I for a long time before being trapped by either C1 or C2 . For r > rH , the equilibrium points C1 and C2 are unstable and the strange invariant set I becomes stable. Trajectories are not longer trapped by the stationary points C1 and C2 , so that they wander into the invariant region A. Almost all trajectories starting in a vicinity of S are expelled along a tube of the unstable manifold of S towards the equilibrium points C1 and C2 . Around C1 and C2 , the trajectories spiral out. Since these equilibrium points are unstable, the trajectories are expelled along a tube around the homoclinic orbits H1 and H2 . Eventually, the trajectories enter in finite time a vicinity of the equilibrium point S where, due to the “initial conditions sensitivity” property of the strange invariant set I, are expelled out in an erratic way towards either C1 or C2 . This phenomenon is repeated again and again leading to what is called chaotic behavior (or non-periodic behavior).
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Fig. 1. Homoclinic orbits of the Lorenz system.
3. Chaos control for the Lorenz system The aim of this section is to exploit the structure of the chaotic attractor described in the above section to design a feedback controller whose objective is to eliminate the chaotic behavior. For control design, one has to identify a physically accessible parameter. As in Bewley [7], the heating rate at the bottom of the system, which is proportional to the parameter r, is used for control purposes. The idea is to modulate the heating rate about a steady-state value into the chaotic regimen r¯ > rH , so that r = r¯ + rf , where rf is the input used for feedback. In this way, the control problem can be formulated as designing a feedback control rf (x) such that the chaotic behavior displayed by the Lorenz system is eliminated and rf t → 0 as t → ∞. Here, rf t represents the mean of the signal rf (t). The condition rf t → 0 as t → ∞ is imposed to ensure that,
at steady state conditions, rt → r¯ as t → ∞. In this way, it is guaranteed that the controlled Lorenz convection system operates into a non-chaotic regimen at conditions that otherwise display turbulent behavior. That is, the turbulent behavior can be displaced by virtue of the feedback control action rf (x). As discussed in the above section, the homoclinic orbits H1 and H2 of the equilibrium point S are the responsible of the strange invariant set I. Eventually, as r is increased beyond rH , the set I becomes an strange attractor set where the chaotic behavior evolves. Since the aim of the feedback control rf (x) is to eliminate the chaotic behavior, an authentic chaos control strategy should rely on attacking the underlying mechanisms that generates the strange invariant attractor I. In this way, at the heart of the construction of I are the homoclinic orbits H1 and H2 . Consequently, if one is able to destroy the homoclinic orbits H1 and H2 , then
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one is in position to eliminate the chaotic convective turbulent behavior of the Lorenz system. In conclusion, the feedback controller rf (x) should focus on destroying the homoclinic connections H1 and H2 . Two ways to destroy, via feedback control, the orbits H1 and H2 can be described as follows: (a) To stabilize the equilibrium point S, so that no unstable manifolds, hence homoclinic orbits, exist. This option has the drawback that the equilibrium point S corresponds to a non-convection state, so that has a poor practical interest. (b) To stabilize the convection stationary points C1 and C2 . The idea is to induce, via feedback control, a locally exponentially stabilization such the homoclinic trajectories that spiral around C1 and C2 can be trapped and converged to these stationary points. If the homoclinic orbits H1 and H2 are destroyed, the former stranger invariant set I is also eliminated. In the sequel, we will approach the chaos control problem following the second option. Based on the above arguments, the control problem has been converted to the equivalent one of stabilizing the Lorenz system about the non-zero convection states C1 and C2 . 3.1. Feedback control design Let us proceed to design the feedback function rf (x). As discussed in the introduction, rf (x) must be self-controlling in the sense that stabilization of C1 and C2 must rely on none exogenous reference. That is, in practice the stationary states C1 and C2 are not known, so that the exact value of them cannot be provided to the controller. Indeed, this is a pitfall of most proposed chaos controllers (see [7] and references in [6]). In fact, the feedback controller rf (x) must stabilize the convective stationary states C1 and C2 without having access to their exact values. Actually, this corresponds to a robust stabilization problem without exact knowledge of the stabilization objective. The feedback control design is based on the following facts: Fact 1. Consider the modified Jacobian matrix J (α) of the Lorenz system about the stationary points C1
and C2 : J (α) =
−σ 1 αd
σ 0 −1 −d , αd −αb
(2)
where α is a positive constant and, as above, d = √ b(r − 1) (notice that the original Jacobian of the Lorenz system is recovered for α = 1). Then, there exists a positive constant αmin such that all the eigenvalues of J (α) have negative real part. To prove the this assertion, consider the characteristic polynomial P (s, α) of J (α): P (s, α) = s 3 + (αb + σ + 1)s 2 + α b + d 2 + bσ s + 2αd 2 σ.
(3)
Since σ , b and d are positive constants, all the coefficients of P (s, α) are positive. This is a necessary condition for stability of P (s, α). From the well-known Routh–Hurwitz criteria, stability of P (s, α) is guaranteed if (consider that α > 0) b + d 2 + bσ (αb + σ + 1) − 2d 2 σ > 0. Equivalently, if α > αmin , where 2d 2 σ 1 − σ − 1 . αmin = b b + d 2 + bσ
(4)
For r > rH , the effect of the parameter α is to move into the left-hand side of the complex plane the two unstable complex eigenvalues of J (1). In this way, continuity arguments imply that at α = αmin the complex eigenvalues of J (α) cross the imaginary axis. For the typical values σ = 10.0, b = 8/3 and r = 28.0, one has that αmin = 2.1617. If J (α) was the Jacobian of the controlled Lorenz system, then one has that the convective stationary points C1 and C2 are stable for α > αmin . This stabilization effect is obtained by means of the feedback function: rf (x) = −K x˙3 ,
(5)
b−1 =
where K is a positive constant, which is called as the controller gain. Notice that rf (x) feedbacks the time-derivative of vertical temperature fluctuations rather than the temperature-self. Since r = r¯ + rf (x) = r¯ − K x˙3 , one has that x˙3 = −bx3 + x1 x2 − br¯ − bK x˙3 , or x˙3 = α(−bx3 + x1 x2 − br¯ ),
(6)
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where 1 . α= 1 − bK One has the following result.
(7)
Fact 2. The following hold for the Lorenz system under the feedback function (5): (a) The former stationary points S, C1 and C2 remain as equilibrium points. The proof of this result is straightforward since 1 ), b1 ), the equilibrium α = 0. (b) For K ∈ ( b1 (1 − αmin points C1 and C2 are stable. The claim follows from the Fact 1; namely, α > αmin > 0. It should be stressed 1 ) < b1 , so that there are really posithat b1 (1 − αmin tive values of K such that the stationary points C1 and C2 are stable. In fact, since αmin > 0, one has that 1 < 1. 1 − αmin The above result ensures the existence of stabilizing feedback gains K. Given that the feedback function (5) is able to stabilize the stationary states C1 and C2 , what happens with the stability of the zero fluid motion stationary point S? Fact 3. There are not exists values of the control gain K such that the stationary point S is stable. The proof follows from analyzing the Jacobian matrix of the controlled Lorenz system about S. In such case, one has that −σ σ 0 J (α) = 1 −1 0 0 0 −αb and the corresponding characteristic polynomial is P (s, α) = s 3 + (αb + σ + 1)s 2 + αb(1 + σ )s, which has a zero eigenvalue regardless of the value of α. Summarizing, the feedback function (5) is able to stabilize the stationary states C1 and C2 for a suitable selection of the controller gain K. Besides, the zero fluid motion stationary state S is not stabilized by the controller rf (x) = −K x˙3 . It is also noted that the feedback controller (5) do not requires the specification of the coordinates of C1 or C2 (i.e., rf (x) is a self-stabilizing feedback function [8]). Finally, as one trajectory x(t, x(0)) converges to either the equilibrium points C1 or C2 , rf (x) → 0 as required (i.e., the feedback controller rf (x) vanishes at the stabilized stationary points).
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We know that during the chaotic regimen, trajectories passing close to the saddle point S are expelled out along a tube of the unstable manifold M u (S), then spiral quite rapidly towards a vicinity of the stationary points C1 and C2 . Subsequently, since such convective stationary points are non-stable, the trajectories are expelled out from the vicinity of C1 and C2 towards a vicinity of S. In principle, if the convective stationary points are stabilized by the action of the feedback control (5), the trajectories can be eventually trapped and converged towards either C1 or C2 . Fig. 2 presents the behavior of the controlled Lorenz system for two values of α (1.4 and 2.0) and two initial conditions close to the saddle point S. The value α = 1.4 is close to the minimum stability value αmin = 1.20394, so that C1 and C2 are only weakly stable. In this case, the trajectories display a behavior that is similar to a pre-turbulence regimen. In fact, the trajectory spirals slowly around the stationary point before being stabilized. However, for α = 2.0 the trajectory is rapidly trapped and stabilized because the convective equilibrium points have better stability properties. 3.2. Implementation of the feedback control rf (x) = K x˙3 The main result of the above analysis is the existence of a feedback controller rf (x) = K x˙3 that is able to stabilize the convective stationary points C1 and C2 without a knowledge of their coordinates. The controller (5) is quite simple since its action is based solely on feed-backing the fluctuations (represented by x˙3 ) of the vertical temperature difference x3 . In this way, since rf (x) vanishes at C1 and C2 , one is able to operate the Lorenz system at conditions r = r¯ that otherwise (i.e., without the action of the feedback control) induce turbulent behavior. These features make the feedback controller structure (5) quite attractive for potential applications. However, in practice the fluctuations x˙3 are not easily measured. Instead, what is actually measured is the vertical temperature difference x3 . Given this lack of measurement, the following question is posed: How the feedback controller (5) can be implemented based solely on measurements of vertical temperature difference measurements x3 ? The key point to address this question is to exploit the fact that x˙3 is the time-derivative of x3 . Then, the idea is to
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Fig. 2. Behavior of the controlled Lorenz system for two values of α and two initial conditions close to the saddle point S.
use an estimate x˙3e of x˙3 obtained from measures of x3 , so that the practical feedback function becomes rf (x) = K x˙3e .
(8)
Two procedures to obtain the estimate x˙3e can be depicted: • Finite difference approximation. Let T > 0 be a sampling period. Then, the simple approximation
x˙3e (t) = T1 (x3 (t) − x3 (t − T )). This leads to the feedback function rf (x) = KT x3 (t) − x3 (t − T ) , (9) where KT = K/T . Interestingly, the feedback function (9) corresponds to the delayed feedback function proposed by Pyragas [8]. In this way, one can see the proposed feedback function (5) as the limit T → 0 of the delayed feedback controller (9).
J. Alvarez-Ramirez et al. / Physics Letters A 338 (2005) 128–140 d • Continuous-time approximation. Let s = dt be the time-derivative operator. Then x˙3 = sx3 . It is noted that s is not a causal operator, so that an exact computation of the time derivative x˙3 cannot be obtained using measures of the state x3 only [9]. One can obtain a continuous-time approximation x˙3e by taking a causal approximation of the time-derivative operator s in the following form: x˙3e (t) = F (s)sx3 (t), where F (s) is a filtering operator satisfying F (0) = 1. The simplest one is F (s) = ωc /(s + ωc ) where ωc is called as the cutting frequency [9]. In this way, one obtains
x˙3e (t) =
ωc s x3 (t). s + ωc
(10)
It is noted that one recovers the exact time-derivative operator when the cutting frequency goes to larger ωc s → s as ωc → ∞. Actually, the values; that is, s+ω c operation (10) is equivalent to a high-pass filter with cutting frequency ωc . A realization of (9) in terms of a differential system can be made as follows. Expresd x˙ e 3 sion (10) implies that dt3 + ωc x˙3e = ωc dx dt (here we dx have used the notation dt instead of x˙ to avoid confusion with the notation used for the estimated timederivative x˙3e ). Introduce the variable z = x˙3e − ωc x3 , so that dz = −ωc (z + ωc x3 ), dt and x˙3e = z + ωc x3 .
z(0) = z0
(11)
(12)
Observe that the estimate x˙3e (t) is obtained on the basis of measures of the state x3 (t) only. That is, the differential system (11)–(12) offers a continuous-time approximation of the time-derivative x˙3 based only on measures of the state x3 (t). In the above subsection, we have proven that there are values of the control gain K such that the controlled Lorenz system is stable about the convective stationary points C1 and C2 (see Facts 1 and 2). The following result states that the continuous-time approximation (10) can retain the stability of the controlled Lorenz system. Fact 4. Assume that the control gain K is chosen such that the convective equilibrium points C1 and C2 are stable under the exact feedback controller (5). Then,
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there exists a positive constant ωc,min such the equilibrium points C1 and C2 are stable by virtue of the approximate feedback controller (8), for all ωc > ωc,min . To prove this result, we proceed as follows. Under ωc s )x3 , the effect of the controller rf (x) = −K( s+ω c the Jacobian matrix of the controlled Lorenz system about C1 and C2 can be written as −σ σ 0 , J (K, τc ) = (13) 1 −1 −d d d −bQ(s, K, τc ) where τc = ωc−1 and Q(s, K, τc ) = N (s, K, τc )/ (τc s + 1) and N (s, K, τc ) = (τc + K)s + 1. The characteristic polynomial of (13) can be expressed as P1 (s, K, τc ) = (τc s + 1)R(s, α, τc ), where R(s, α, τc ) = s 3 + bN (s, K, τc ) + σ + 1 s 2 + [α + σ ]bN (s, K, τc ) + d 2 s + 2αd 2 σ. One has that, since N (s, K, 0) = Ks, R(s, K, 0) is equivalent (via a scaling of the coefficient of s 3 ) to the characteristic polynomial P (s, α). By virtue of Fact 2, P (s, α) is stable for all α > αmin . From the continuity property of the roots of polynomials, one conclude the existence of a positive constant τc,min such that R(s, α, τc ) is stable for α > αmin + δ (where δ is a positive constant) and all positive τc < τc,min . The claim of the Fact 4 is finally proven by observing that the product P1 (s, α, τc ) of two stable polynomials τc s + 1 and R(s, α, τc ) is also stable. 3.3. Discussion The results described above can be summarized as follows. We have proven that the feedback controller given by rf (x) = −K x˙3e ,
(14)
where x˙3e is computed from (11)–(12) is able to stabilize the convective stationary points C1 and C2 for suitable choices of the controller parameters K and ωe . Key features of such proposed controller are that its implementation rely on measurements of vertical temperature difference and does not require specification of the coordinates of the stationary points. Besides, the control signal rf (x(t)) vanishes as x(t)
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converges to either C1 or C2 . We think these features make attractive the proposed feedback controller for potential applications in real buoyancy-driven systems. To account for the fact that the chaotic instability is constrained to a bounded region [3], the following modification to the feedback control function (14) is proposed: rf (x) = rf,max tanh −K x˙3e , (15) where rf,max > 0 is the maximum amplitude of the control input. The rationale behind the modified control function (15) is that the aim of the feedback action is to stabilize locally the stationary points C1 and C2 , so that such objective can be achieved with a bounded stabilizing signal. It should be stressed that the conditions required to ensure the result in Fact 4 are only sufficient. In the numerical simulations to be described in the following section, we will show that the selection of the control gain K is not so tight. In fact, by fixing a value K, one can tune the cutting frequency ωc to sufficiently large values in order to obtain stability of the convection stationary points C1 and C2 . Finally, the structure of the feedback function ωc s )x3 allows a nice interpretation rf (x) = −K( s+ω c in the frequency-domain terminology. The operator ωc s H (s) = s+ω is actually a high-pass filter with cutc ting frequency ωc . H (s) removes modes evolving at frequencies much smaller than ωc , passing only modes at higher frequencies than ωc . In this way, the effectivity of the feedback controller (14) can be explained from this property: the action of the conωc s )x3 can be seen as a feedtroller rf (x) = −K( s+ω c back damping of the high-frequency modes of the vertical temperature difference fluctuations. As a consequence, high-frequency oscillation modes of the chaotic Lorenz system are effectively damped, resulting in the stability of the stationary points C1 and C2 .
4. Numerical simulations This section is devoted to illustrate the performance of the proposed chaos controller. All simulations below are based on the typical parameter set used in above sections; that is, σ = 10.0, b = 8/3 and r¯ = 28.0.
For the exact feedback controller (5), one has that K < b1 = 3/8. However, the use of the approximate ωc s )x3 allows more feedback function rf (x) = −K( s+ω c flexibility and less tightness in the selection of the control gain K. This can be explained by the fact that conditions for stability under this controller are only sufficient (Fact 4). Fig. 3 presents the behavior of the controlled Lorenz system for K = 2.0, ωc = 10.0 and rf,max = 4.0. Simulations are shown for two initial conditions close to the zero fluid motion stationary point S; the trajectory of one of them converges to C1 and the other to C2 . Similar behavior has been found for other values of K; for instance, for K = 1.0 and 5.0. The value of rf,max plays a central role on the rate of stabilization of the stationary points. In fact, we have found that the minimum value of rf,max at which stabilization is obtained is about 0.142. In this way, the larger the value of rf,max is, the faster the trajectory convergence. This is illustrated in Fig. 4 for three values of rf,max . It is noted that, because C1 and C2 have two stable complex eigenvalues, the trajectories converge in an spiral manner. These spiraling dynamics are reduced as the maximum value of the control input rf,max is increased. For the Lorenz system values σ = 10.0, b = 8/3 and r¯ = 28.0, we have not found stable periodic orbits under the feedback control action. In fact, we have found a direct transition from an apparently chaotic motion to a stable dynamics regimen about C1 and C2 as the controller parameters (K, ωc , rf,max ) are varied in the correct direction. An interesting bifurcation phenomenon is found for large values of r¯ . For values of r¯ larger than about 120.0, the behavior of the Lorenz system is dominated by large stable periodic orbits [3]. For r¯ = 150.0, Fig. 5 shows the deformation of the attractor for the uncontrolled Lorenz system (rf,max = 0.0) and two different values of rf,max . For rf,max = 0.0, the trajectories are attracted to a large period-2 periodic orbits. The feedback controller stabilizes the convective stationary points C1 and C2 . However, if the maximum control amplitude rf,max is not sufficiently large, the basin of attraction of these points are not sufficiently large to trap the periodic orbit. In this case, there is a conflict between the attracting power of the former periodic orbit and the stabilized stationary points, and an apparent chaotic attractor is displayed. The form
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Fig. 3. Behavior of the controlled Lorenz system for K = 2.0, ωc = 10.0 and rf,max = 4.0.
of such chaotic attractor is induced by the feedback action and is different from those found for the uncontrolled Lorenz system. As rf,max is increased, the basin of attraction of the stationary points becomes larger and is able to capture the trajectories of the system.
5. Conclusions It has been shown that a close knowledge of the mechanisms leading to chaotic motions allows to design simple, specific and effective feedback controllers
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Fig. 4. Stabilization of the Lorenz system for three values of rf,max .
to reduce and even eliminate the chaos dynamics from a physical system. In the particular case of the Lorenz system, it has been shown that a simple feedback controller with the structure of a high-gain filter on the vertical temperature difference is able to stabilize
the convection stationary points. Although the numerical simulations revealed the effectivity of the proposed control strategy, it should be stressed that the Lorenz system is only a very simplistic model of actual buoyancy-driven flows. In this form, the potential
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Fig. 5. Deformation of the attractor for the uncontrolled Lorenz system (rf,max = 0.0) and two different values of rf,max .
application of the results reported in this Letter should be evaluated carefully because of the high complexity of actual flow systems. However, as mentioned by Be-
wley [7], it is important to understand the control problems and the remedies available in a low-dimensional framework before moving to more complex systems.
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