CONTROLLING CHAOS IN A NONLINEAR PENDULUM USING AN ADAPTIVE FUZZY SLIDING MODE CONTROLLER Wallace Moreira Bessa∗, Aline Souza de Paula†, Marcelo Amorim Savi† ∗
CEFET/RJ, Federal Center for Technological Education Av. Maracan˜ a 229, CEP 20271-110, Rio de Janeiro, RJ, Brazil †
Universidade Federal do Rio de Janeiro COPPE – Department of Mechanical Engineering P.O. Box 68503, CEP 21945-970, Rio de Janeiro, RJ, Brazil Emails:
[email protected],
[email protected],
[email protected] Abstract— Chaos control may be understood as the use of tiny perturbations for the stabilization of unstable periodic orbits embedded in a chaotic attractor. Since chaos may occur in many natural processes, the idea that chaotic behavior may be controlled by small perturbations of physical parameters allows this kind of behavior to be desirable in different applications. In this work, a variable structure controller is employed to the chaos control problem in a nonlinear pendulum. The adopted approach is based on the sliding mode control strategy and enhanced by an adaptive fuzzy algorithm to cope with modeling inaccuracies. The convergence properties of the closed-loop system are analytically proven using Lyapunov’s direct method and Barbalat’s lemma. Numerical results are also presented in order to demonstrate the control system performance. Keywords—
Adaptive algorithms, Chaos control, Fuzzy logic, Nonlinear pendulum, Sliding modes.
Resumo— O controle do caos pode ser compreendido como a utiliza¸c˜ ao de pequenas perturba¸c˜ oes para a estabiliza¸c˜ ao de ´ orbitas peri´ odicas inst´ aveis imersas em um atrator ca´ otico. Tendo em vista que o caos pode ocorrer em v´ arios processos naturais, a id´ eia do comportamento ca´ otico poder ser controlado atrav´ es de pequenas perturba¸c˜ oes de certos parˆ ametros f´ısicos permite que este tipo de comportamento seja desej´ avel em diversas aplica¸c˜ oes. Neste trabalho, um controlador a estrutura vari´ avel ´ e aplicado ao controle do caos em um pˆ endulo n˜ aolinear. A abordagem adotada baseia-se na estrat´ egia de controle por modos deslizantes sendo por´ em aprimorada por um algoritmo difuso adaptativo para lidar com imperfei¸c˜ oes de modelagem. As propriedades de convergˆ encia do sistema em malha-fechada s˜ ao provadas analiticamente com o aux´ılio do m´ etodo direto de Liapunov e do lema de Barbalat. Resultados num´ ericos s˜ ao tamb´ em apresentados para demonstrar a performance do sistema de controle. Palavras-chave— Linear.
1
Algoritmos adaptativos, Controle do Caos, L´ ogica difusa, Modos deslizantes, Pˆ endulo N˜ ao-
Introduction
Chaotic response is related to a dense set of unstable periodic orbits (UPOs) and the system often visits the neighborhood of each one of them. Moreover, chaos has sensitive dependence to initial condition, which implies that the system evolution may be altered by small perturbations. Chaos control is based on the richness of chaotic behavior and may be understood as the use o tiny perturbations for the stabilization of an UPO embedded in a chaotic attractor. It makes this kind of behavior to be desirable in a variety of applications, since one of these UPO can provide better performance than others in a particular situation. Due to these characteristics, chaos and many regulatory mechanisms control the dynamics of living systems. Inspired by nature, it is possible to imagine situations where chaos control is employed to stabilize desirable behaviors of mechanical systems. Under this condition, these systems would present a great flexibility when controlled, being able to quickly change from one kind of response to another. Literature presents some contributions related
to the analysis of chaos control in mechanical systems. Andrievskii and Fradkov (2004) present an overview of applications of chaos control in various scientific fields. Mechanical systems are included in this discussion presenting control of pendulums, beams, plates, friction, vibroformers, microcantilevers, cranes, and vessels. Savi et al. (2006) also present an overview of some mechanical system chaos control that includes system with dry friction (Moon et al., 2003), impact (Begley and Virgin, 2001) and system with non-smooth restoring forces (Hu, 1995). Spano et al. (1990) explores the ideas of chaos control applied to intelligent systems while Macau (2003) shows that chaos control techniques can be used in spacecraft orbits. Pendulum systems are analyzed in (Pereira-Pinto et al., 2004; Pereira-Pinto et al., 2005; Wang and Jing, 2004; Yagasaki and Uozumi, 1997; Yagasaki and Yamashita, 1999) using different approaches. There are different techniques employed to perform chaos control (Savi et al., 2006; Boccaletti et al., 2000; Ditto and Lindner, 1995; Pyragas, 1992), however, the inspirational idea of these methods is the well-known OGY method (Ott et al., 1990), which is a discrete technique that considers small perturbations promoted in the
neighborhood of the desired orbit when the trajectory crosses a specific surface, such as some Poincar´e section. This contribution proposes a robust controller to stabilize UPOs of a nonlinear pendulum that has a rich response, presenting chaos and transient chaos (De Paula et al., 2006). The adopted approach is based on the sliding mode control strategy and enhanced by a stable adaptive fuzzy inference system to cope with modeling inaccuracies. The convergence properties of the tracking error are analytically proven using Lyapunov’s direct method and Barbalat’s lemma. Numerical results are also presented in order to demonstrate the control system performance. Numerical simulations are carried out showing the stabilization of a period-1 UPO of the chaotic attractor showing an effective response. 2
Chaotic Pendulum
The nonlinear pendulum considered in this article is based on an experimental set up, previously analyzed by Franca and Savi (2001) and PereiraPinto et al. (2004). De Paula et al. (2006) presented a mathematical model to describe the dynamical behavior of the pendulum and the corresponding experimentally obtained parameters. The schematic picture of the considered nonlinear pendulum is shown in Fig. 1. Basically, the pendulum consists of an aluminum disc (1) with a lumped mass (2) that is connected to a rotary motion sensor (4). This assembly is driven by a string-spring device (6) that is attached to an electric motor (7) and also provides torsional stiffness to the system. A magnetic device (3) provides an adjustable dissipation of energy. An actuator (5) provides the necessary perturbations to stabilize this system by properly changing the string length. In order to obtain the equations of motion of the experimental nonlinear pendulum it is assumed that system dissipation may be expressed by a combination of a linear viscous dissipation together with dry friction. Therefore, denoting the angular position as φ, the following equation is obtained.
Figure 1: (a) Nonlinear pendulum – (1) metallic disc; (2) lumped mass; (3) magnetic damping device; (4) rotary motion sensor (PASCO CI-6538); (5) anchor mass; (6) string-spring device; (7) electric motor (PASCO ME-8750). (b) Parameters and forces on metallic disc. (c) Parameters from driving device. mass, ζ represents the linear viscous damping coefficient, while µ is the dry friction coefficient; g is the gravity acceleration, I is the inertia of the disk-lumped mass, k is the string stiffness, ∆l is the length variation in the spring provided by the linear actuator (5) and sgn(x) is defined as −1 if x < 0 0 if x = 0 sgn(x) = (2) +1 if x > 0 De Paula et al. (2006) show that this mathematical model is in close agreement with experimental data and, therefore, it will be used for the control purposes. 3
Adaptive fuzzy sliding mode control
In order to write Eq. (1) in a more convenient form, it is rewritten as follows: ˙ ζ kd2 µ sgn(φ) mgD sin(φ) φ¨ + φ˙ + φ+ + = I 2I I 2I kd p 2 a + b2 − 2ab cos(ωt) − (a − b) − ∆l (1) 2I where ω is the forcing frequency related to the motor rotation, a defines the position of the guide of the string with respect to the motor, b is the length of the excitation crank of the motor, D is the diameter of the metallic disc and d is the diameter of the driving pulley, m is the lumped
˙ t) + hu + p(φ, φ) ˙ φ¨ = f (φ, φ,
(3)
where h = kd/2I, u = −∆l, f can be obtained from Eq. (1) and Eq. (3), and the term p is added to represent unmodeled dynamics. Now, let S(t) be a sliding surface defined in the state space by the equation s(e, ˙ e) = 0, with the function s : R2 → R satisfying s(e, ˙ e) = e˙ + λe
(4)
where e = φ−φd is the tracking error, e˙ is the first time derivative of e, φd is the desired trajectory and λ is a strictly positive constant. The control of the system dynamics (3) is done by assuming a sliding mode based approach, defining a control law composed by an equivalent ˆ −1 (−fˆ − pˆ + φ¨d − λe) control u ˆ=h ˙ and a discontinuous term −K sgn(s): ˆ −1 (−fˆ − pˆ + φ¨d − λe) u=h ˙ − K sgn(s)
(5)
ˆ fˆ, and pˆ are estimates of h, f and p, where h, respectively, and K is a positive control gain. Regarding the development of the control law, the following assumptions should be made: Assumption 1 The function f is unknown but bounded, i.e. |fˆ − f | ≤ F. Assumption 2 The input gain h is unknown but positive and bounded, i.e. 0 < hmin ≤ h ≤ hmax . Assumption 3 The perturbation p is unknown but bounded, i.e. |p| ≤ P. Based on Assumption 2 and considering that ˆ could be chosen according to the the estimate h √ ˆ = hmax hmin , the bounds of geometric mean h −1 ˆ h may ≤ h/h ≤ H, where p be expressed as H H = hmax /hmin . Under this condition, the gain K should be chosen according to ˆ −1 (η + |ˆ K ≥ Hh p| + P + F) + (H − 1)|ˆ u|
(6)
here η is a strictly positive constant related to the reaching time. At this point, it should be highlighted that the control law (5), together with (6), is sufficient to impose the sliding condition 1 d 2 s ≤ −η|s| (7) 2 dt and, consequently, the finite time convergence to the sliding surface S. In order to obtain a good approximation to p, the estimate pˆ will be computed directly by an adaptive fuzzy algorithm. The adopted fuzzy inference system was the zero order TSK (Takagi– Sugeno–Kang), whose rules can be stated in a linguistic manner as follows: If φ is Φr and φ˙ is Φ˙ r then pˆ = Pˆr , r = 1, 2, . . . , N where Φr and Φ˙ r are fuzzy sets, whose membership functions could be properly chosen, and Pˆr is the output value of each one of the N fuzzy rules. Considering that each rule defines a numerical value as output Pˆr , the final output pˆ can be computed by a weighted average:
PN
(8)
˙ =P ˙ ˆ T Ψ(φ, φ) pˆ(φ, φ)
(9)
˙ = pˆ(φ, φ) or, similarly,
ˆ r=1 wr · Pr PN r=1 wr
ˆ = [Pˆ1 , Pˆ2 , . . . , PˆN ] is the vector conwhere, P taining the attributed values Pˆr to each rule r, ˙ = [ψ1 , ψ2 , . . . , ψN ] is a vector with comΨ(φ, φ) ˙ = wr / PN wr and wr is the ponents ψr (φ, φ) r=1 firing strength of each rule, which can be computed from the membership values with any fuzzy intersection operator (t-norm). To ensure the best possible estimate pˆ, the vector of adjustable parameters can be automatically updated by the following adaptation law: ˙ ˆ˙ = ϕsΨ(φ, φ) P
(10)
where ϕ is a strictly positive constant related to the adaptation rate. It is important to emphasize that the chosen adaptation law, Eq. (10), must not only provide a good approximation to p but also assure the convergence of the state variables to the sliding surface S(t), for the purpose of trajectory tracking. In this way, in order to evaluate the stability of the closed-loop system, let a positive-definite function V be defined as V (t) =
1 2 1 T s + δ δ 2 2ϕ
(11)
ˆ −P ˆ ∗ and P ˆ ∗ is the optimal paramewhere δ = P ter vector, associated to the optimal estimate pˆ∗ . Thus, the time derivative of V is V˙ (t) = ss˙ + ϕ−1 δ T δ˙ = (φ¨ − φ¨d + λe)s ˙ + ϕ−1 δ T δ˙ = (f + hu + p − φ¨d + λe)s ˙ + ϕ−1 δ T δ˙ ˆ −1 (−fˆ − pˆ + φ¨d − λe) = f + hh ˙ ¨ − hK sgn(s) + p − φd + λe˙ s + ϕ−1 δ T δ˙ Defining a minimum approximation error as ˆ −1 (−fˆ− pˆ+ φ¨d −λe) ε = pˆ∗ −p, recalling that u ˆ=h ˙ ˙ ˆ f = fˆ − (fˆ − f ) and and noting that δ˙ = P, p = pˆ − (ˆ p − p), V˙ becomes: ˆ u − hˆ V˙ (t) = − (fˆ − f ) + ε + (ˆ p − pˆ∗ ) + hˆ u ˙ ˆ + hK sgn(s) s + ϕ−1 δ T P ˙ + hˆ ˆ u − hˆ = − (fˆ − f ) + ε + δ T Ψ(φ, φ) u ˙ ˆ + hK sgn(s) s + ϕ−1 δ T P ˆ u − hˆ = − (fˆ − f ) + ε + hˆ u + hK sgn(s) s ˙ ˙ ˆ − ϕsΨ(φ, φ) + ϕ−1 δ T P
By applying the adaptation law, Eq (10), to ˙ ˙ ˆ P, V (t) becomes: ˆ u −hˆ V˙ (t) = − (fˆ−f )+ε+ hˆ u +hK sgn(s) s (12) Furthermore, considering assumptions 1–3, defining K according to (6) and verifying that |ε| = |ˆ p∗ − p| ≤ |ˆ p − p| ≤ |ˆ p| + P, it follows V˙ (t) ≤ −η|s|
(13)
which implies V (t) ≤ V (0) and that s and δ are bounded. Integrating both sides of (13) shows that
lim
t→∞
t
Z
η|s| dτ ≤ lim [V (0) − V (t)] ≤ V (0) < ∞ t→∞
0
Now Barbalat’s lemma is evoked establishing that s → 0 as t → ∞, which ensures the convergence of the states to the sliding surface S and to the desired trajectory. In spite of the demonstrated properties of the controller, the presence of a discontinuous term in the control law leads to the well known chattering phenomenon. To overcome the undesirable chattering effects, a thin boundary layer, S , in the neighborhood of the switching surface can be adopted (Slotine, 1984): S = (e, e) ˙ ∈ R2 |s(e, e)| ˙ ≤
where is a strictly positive constant that represents the boundary layer thickness. The boundary layer is achieved by replacing the sign function by a continuous interpolation inside S . It should be noted that this smooth approximation must behave exactly like the sign function outside the boundary layer. There are several options to smooth out the ideal relay but the most common choice is the saturation function: sat(s/) =
sgn(s) if |s/| ≥ 1 s/ if |s/| < 1
In this way, to avoid chattering, a smooth version of Eq. (5) is defined: ˆ −1 (−fˆ − pˆ + φ¨d − λe) u=h ˙ − K sat(s/)
(14)
Nevertheless, it should be emphasized that the substitution of the discontinuous term by a smooth approximation inside the boundary layer turns the perfect tracking into a tracking with guaranteed precision problem, which actually means that a steady-state error will always remain. For a second order system with a smooth sliding mode controller, the tracking error vector will exponentially converge to a closed region Γ = {(e, e) ˙ ∈ R2 | |s(e, e)| ˙ ≤ and |e| ≤ −1 λ and |e| ˙ ≤ 2 } (Bessa, 2008).
4
Numerical simulations
The numerical simulations are carried out considering the fourth order Runge-Kutta method. The model parameters are chosen according to De Paula et al. (2006): I = 1.738 × 10−4 kg m2 ; m = 1.47 × 10−2 kg; k = 2.47 N/m; ζ = 2.368 × 10−5 kg m2 /s; µ = 1.272 × 10−4 N m; a = 1.6e×10−1 m; b = 6.0×10−2 m; d = 4.8×10−2 m; D = 9.5 × 10−2 m and ω = 5.61 rad/s. To ratify the robustness of the proposed control scheme against modeling imprecisions, the ˙ term µ sgn(φ)/I in Eq. (1) will be treated as unmodeled dynamics and not considered in controller design. In this way, regarding controller parameters, the following values were chosen: µ ˆ = 0, λ = 0.8; η = 0.5; ϕ = 2.8; = 1.0 and F = 0.73. Concerning the fuzzy system, triangular and trapezoidal membership functions are adopted for both Φr and Φ˙ r , with the central values defined respectively as C0 = {0.0} and C1 = {−10.0; −1.0; −0.1; 0.0; 0.1; 1.0; 10.0}×10−1 . The chosen fuzzy intersection operator was the product t-norm. It is also important to emphasize that the vector of adjustable parameters is initialized ˆ = 0, and updated at each itwith zero values, P eration step according to Eq. (10). In order to evaluate the control system performance, a period-1 UPO was identified using the close return method (De Paula et al., 2006) and chosen to be stabilized. The obtained results are presented in Fig. 2. As observed in Fig. 2, even in the presence of modeling inaccuracies, the adaptive fuzzy sliding mode controller (AFSMC) is capable to provide the trajectory tracking with a small associated error. It should be emphasized that the control action u represents the length variation in the string and only tiny variations are required to provide such different dynamic behaviors, which actually allows a great flexibility for the controlled nonlinear system. It can be also verified that the proposed control law provides a smaller tracking error when compared with the conventional sliding mode controller (SMC), Fig. 2(c). By considering simulation purposes, the AFSMC can be easily converted to the classical SMC by setting the adaptation rate to zero, ϕ = 0. The improved performance of AFSMC over SMC is due to its ability to recognize and compensate the modeling imprecisions, Fig. 2(d). The time evolution of the input-output surface of the fuzzy inference system is shown in Fig. 3 after four different iteration steps. To demonstrate the effectiveness of the proposed control scheme in controlling orbits with initial conditions that are not embedded in the strange attractor, a numerical simulation was performed with φ = 7 and φ˙ = 0. Figure 4 shows the related Poincar´e section.
0 Iterations
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pˆ [rad/s2]
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φ [rad] pˆ [rad/s2]
(a) Phase space.
0.8 0.4 0.0 -0.4 -0.8
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u [m]
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(b) Control action.
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pˆ [rad/s2]
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Figure 3: Convergence of the input-output surface of the fuzzy inference system after four different iteration steps.
(c) Tracking error.
1
Actual Estimated
0.8
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µsgn(φ)/I [rad/s2]
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˙ (d) Convergence of pˆ to µ sgn(φ)/I.
Figure 2: Tracking of period-1 UPO.
Figure 4: Poincar´e section of the tracking of a period-1 UPO with an initial condition that is not embedded in the strange attractor
5
Conclusions
The present contribution considers the orbit stabilization employing an adaptive fuzzy sliding mode controller. The stability and convergence properties of the closed-loop system is analytically proven using Lyapunov stability theory and Barbalat’s lemma. As an application of the general formulation, numerical simulations of a nonlinear pendulum with chaotic response is of concern. The control system performance is investigated showing the stabilization of a period-1 UPO. The improved performance over the conventional sliding mode controller is demonstrated. Acknowledgements The authors acknowledge the support of the Brazilian Research Council (CNPq) and the State of Rio de Janeiro Research Foundation (FAPERJ). References Andrievskii, B. R. and Fradkov, A. L. (2004). Control of chaos: Methods and applications, II - applications, Automation And Remote Control 65(4): 505–533. Begley, C. J. and Virgin, L. N. (2001). On the ogy control of an impact-friction oscillator, Journal of Vibration and Control 7(6): 923– 931. Bessa, W. M. (2008). A remark on the boundedness and convergence properties of smooth sliding mode controllers. Submitted for publication. ArXiv:0802.2978v3[math.OC]. Boccaletti, S., Grebogi, C., Lai, Y.-C., Mancini, H. and Maza, D. (2000). The control of chaos: Theory and applications, Physics Reports 329: 103–197. De Paula, A. S., Savi, M. A. and Pereira-Pinto, F. H. I. (2006). Chaos and transient chaos in an experimental nonlinear pendulum, Journal of Sound and Vibration 294: 585–595. Ditto, W. L. and Lindner, M. L. S. J. F. (1995). Techniques for the control of chaos, Physica D 86: 198–211. Franca, L. F. P. and Savi, M. A. (2001). Distinguishing periodic and chaotic time series obtained from an experimental pendulum, Nonlinear Dynamics 26: 253–271. Hu, H. Y. (1995). Controlling chaos of a periodically forced nonsmooth mechanical system, Acta Mechanica Sinica 11(3): 251–258.
Macau, E. E. N. (2003). Exploiting unstable periodic orbits of a chaotic invariant set for spacecraft control, Celestial Mechanics & Dynamical Astronomy 87(3): 291–305. Moon, F. C., Reddy, A. J. and Holmes, W. T. (2003). Experiments in control and anticontrol of chaos in a dry friction oscillator, Journal of Vibration and Control 9(3/4): 387–397. Ott, E., Grebogi, C. and Yorke, J. A. (1990). Controlling chaos, Physical Review Letters 64(11): 1196–1199. Pereira-Pinto, F. H. I., Ferreira, A. M. and Savi, M. A. (2004). Chaos control in a nonlinear pendulum using a semi-continuous method, Chaos, Solitons and Fractals 22(3): 653–668. Pereira-Pinto, F. H. I., Ferreira, A. M. and Savi, M. A. (2005). State space reconstruction using extended state observers to control chaos in a nonlinear pendulum, International Journal of Bifurcation and Chaos 15(12): 4051– 4063. Pyragas, K. (1992). Continuous control of chaos by self-controlling feedback, Physics Letters A 170: 421–428. Savi, M. A., Pereira-Pinto, F. H. I. and Ferreira, A. M. (2006). Chaos control in mechanical systems, Shock and Vibration 13(4/5): 301– 314. Slotine, J.-J. E. (1984). Sliding controller design for nonlinear systems, International Journal of Control 40(2): 421–434. Spano, M. L., Ditto, W. L. and Rauseo, S. N. (1990). Exploitation of chaos for active control: An experiment, Journal of Intelligent Material Systems and Structures 2(4): 482– 493. Wang, R. and Jing, Z. (2004). Chaos control of chaotic pendulum system, Chaos, Solitons and Fractals 21(1): 201–207. Yagasaki, K. and Uozumi, T. (1997). Controlling chaos in a pendulum subjected to feedforward and feedback control, International Journal of Bifurcation and Chaos 7(12): 2827–2835. Yagasaki, K. and Yamashita, S. (1999). Controlling chaos using nonlinear approximations for a pendulum with feedforward and feedback control, International Journal of Bifurcation and Chaos 9(1): 233–241.
