Real-time chaotic circuit stabilization via inverse optimal control

INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONSInt. J. Circ. Theor. Appl. 2009; 37:887–898Published online 9 July 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cta.500

Real-time chaotic circuit stabilization via inverse optimal control

Alexander Jimenez1,2,∗,†, Edgar N. Sanchez3, Guanrong Chen4 and Jose P. Perez5

1Facultad Tecnologica, Universidad Distrital Francisco Jose de Caldas, Cll 74 Sur No. 68A-20, Bogota, Colombia2Facultad de Ingenierıa, Universidad de los Andes, Calle 19A #1-37 este, Bogota, Colombia3CINVESTAV, Unidad Guadalajara, Apartado Postal 31-438, Plaza La Luna, Guadalajara,

Jalisco C.P. 45081, Mexico4Department of Electronic Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon,

Hong Kong SAR, People’s Republic of China5School of Physics and Mathematics, Universidad Autonoma de Nuevo Leon, Pedro de Alba s/n, San Nicolas de

los Garza, Nuevo Leon, Mexico

SUMMARY

In this paper, an efficient approach is developed for real-time global asymptotic stabilization of the chaoticChen’s circuit, as a typical example for chaotic circuit control. Based on a recently introduced methodologyof inverse optimal control for nonlinear systems, a very simple stabilization control law, a linear statefeedback, is electronically implemented for the desired global asymptotic stabilization. Both Chen’schaotic system and the designed controller are synthesized and realized by analog electronic components,with the aim of evaluating the physical performance of the real-time control law and demonstrating thepracticality of the control method, which is robust to some input uncertainties. Copyright q 2008 JohnWiley & Sons, Ltd.

Received 8 September 2007; Revised 10 April 2008; Accepted 13 April 2008

KEY WORDS: real-time control; chaos control; Lyapunov function; inverse optimal control; circuitimplementation

1. INTRODUCTION

Chaotic systems have been studied for quite a long time, particularly in the mathematical andphysical communities, and controlling this kind of complex dynamical systems has attracted a

∗Correspondence to: Alexander Jimenez, Facultad Tecnologica, Universidad Distrital Francisco Jose de Caldas,Cll 74 Sur No. 68A-20, Bogota, Colombia.

†E-mail: [email protected]

Contract/grant sponsor: Universidad Distrital Francisco Jose de CaldasContract/grant sponsor: Universidad de los AndesContract/grant sponsor: CONACYT; contract/grant number: 39866Y

Copyright q 2008 John Wiley & Sons, Ltd.

888 A. JIMENEZ ET AL.

great deal of attention within the engineering society. In many areas such as telecommunications,electronics, Internet technology, chemical processes, biomedical systems, and so on, real-timechaos control is required, for which different techniques have been proposed to achieve chaoscontrol; for instance, linear state feedback [1], Lyapunov function methods [2], adaptive control[3], and bang-bang control [4], among many others [5, 6].

On the other hand, control methods for general nonlinear systems have been extensively devel-oped since the early 1980s, for example, based on differential geometry theory [7], and the recentpassivity approach for synthesizing control laws for nonlinear systems [8–10], to name just acouple. An important problem in this field is how to achieve robust nonlinear control in the pres-ence of unmodelled dynamics and external disturbances; along the same line there is the so-calledH∞ nonlinear control technique [11]. It was noticed that one major difficulty with this approach,alongside its possible system structural instability, seems to be caused by the requirement of solvingthe associated partial differential equations. In order to alleviate this computational problem, theso-called inverse optimal control technique was developed based on the input-to-state stabilityconcept [12], which extends the previous inverse optimal control method [13, 14] to deal withsome more general nonlinear systems.

This paper presents a hardware implementation of real-time chaos stabilization by inverse optimalcontrol. The main engineering significance of the inverse optimal approach is in the stabilitymargins that it guarantees for the closed-loop system [15]. For that reason the system can toleratesome uncertainties at the input without causing the loss of stability.

This paper is organized as follows. Firstly, the chaotic Chen’s system and its attractor [16]are introduced briefly. Then, a theorem from [17], which establishes a stabilizing control lawusing inverse optimal control technique, is discussed. Next, an analog electronic circuit design isdescribed for implementing Chen’s circuit realizing its chaotic attractor, followed by the hardwareimplementation of the intended real-time stabilizing optimal controller. Throughout, computersimulation is also given for illustration and verification.

2. CHAOTIC CHEN’S SYSTEM

A chaotic system, referred to as Chen’s system by many researchers [16], is described by

⎡⎢⎣x

y

z

⎤⎥⎦=

⎡⎢⎣

a(y−x)

(c−a)x−xz+cy

xy−bz

⎤⎥⎦ (1)

or, in the state-space form,

X = f (X) (2)

where X =(x, y, z)T and f (X) is given by Equation (1).With a=35,b=3, and c=28, Chen’s attractor is obtained as presented in Figure 1. It has been

widely experienced that this chaotic system is relatively difficult to control as compared with theLorenz system and Chua’s circuit due to its prominent three-dimensional and rapidly changingcomplex topological features.

Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2009; 37:887–898DOI: 10.1002/cta

REAL-TIME CHAOTIC CIRCUIT STABILIZATION 889

Figure 1. Chen’s chaotic attractor.

The interest here is to globally asymptotically stabilize Chen’s system to one of its unstableequilibrium points, (0,0,0). Henceforth, a controller is added to the second state, so the controlledsystem becomes

⎡⎢⎣x

y

z

⎤⎥⎦=

⎡⎢⎣

a(y−x)

(c−a)x−xz+cy

xy−bz

⎤⎥⎦+

⎡⎢⎣0

1

0

⎤⎥⎦u (3)

or

X = f (X)+g(X)u (4)

where u is the control input.

3. INVERSE OPTIMAL CONTROL

The main approach is based on the following theorem, with a complete proof given in [17].Theorem 1Chen’s system can be globally asymptotically stabilized by the following linear state-feedbackcontrol law:

u=−(2c+ c2

2a+1

)y=−Ky (5)

Outline of proofOne first finds a Lyapunov function candidate that satisfies all the requirements to be an input-to-state control Lyapunov function [12]:

V = 12 (x

2+ y2+z2) (6)

Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2009; 37:887–898DOI: 10.1002/cta

890 A. JIMENEZ ET AL.

its time derivative is

V = �V�X

X = �V�X

( f (X)+g(X)u)

= L f V +(LgV )u (7)

where (�V /�X) f (X)= L f V and (�V /�X)g(X)= LgV . After some simple calculations [17], oneobtains

V =−a(x− cy

2a

)2−bz2+(c+ c2

4a

)y2+ yu (8)

next, define the following simple linear state-feedback control law:

u=−�R(X)−1(LgV )=−(c+ c2

4a+k0

)y (9)

where k0 and � are positive constants and R(X)−1 is a positive-definite function of X in general.In order to retain the notation used in [14], R(X)−1 is used here although it is actually a constant:

R(X)−1= 1

�

(c+ c2

4a+k0

)(10)

now, substituting Equation (9) into Equation (8) gives

V =−a(x− cy

2a

)2−k0y2−bz2 (11)

which implies V <0 for all X �=0. This means that the proposed control law (9) can globallyasymptotically stabilize system (3). For a reason of assigning the control gain, see [12].

To this end, consider the control law (9) and define a cost functional as follows:

J (u)= limt→∞

{2�V (X)+

∫ t

0(l(X)+uTR(X)u)d�

}(12)

with

l(x)=−2�L f V +�2R(X)−1(LgV )2 (13)

According to the basic idea of the inverse optimal control theory, it is required that l(X) be radiallyunbounded, i.e. l(X)>0 for all X �=0 and l(X)→∞ as X →∞. Hence, select

k0=(c+ c2

4a

)+1 (14)

so that

l(X)=2a�(x− cy

2a

)2+2b�z2+�y2 (15)

which satisfies the required condition.

Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2009; 37:887–898DOI: 10.1002/cta

REAL-TIME CHAOTIC CIRCUIT STABILIZATION 891

Figure 2. Simulation results of applying the control law to the chaotic Chen’s system,with different initial conditions.

Figure 3. Blocks diagram for the controlled Chen’s system. When the switch is on, thefeedback control signal is applied.

Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2009; 37:887–898DOI: 10.1002/cta

892 A. JIMENEZ ET AL.

Figure 4. Analog electronic implementation of the chaotic Chen’s system and the controller: R1= R11=R35=R39=R42=30K , R2=R4=R13=R22=R24=R25=R26=R27= R28= R31= R34= R46= R47=R48=1K , R3=R6=R7=R9=R23=R16=R18=R19=R20=R21=R36=R37=R41=R43=R44=R45=R49=R50=10K , R5=R14=7K , R8=R15=2K , R10=40K , R12=4K , R17=70K , R29=25K , R30=R33=5K , R38=80K , R40=8K , C1=C2=C3=100u f, R52/R51=K (Equation (4)). The circuit can be

scaled in both time and state variables.

Finally, after some calculations, one obtains

J (u)= limt→∞

(2�V (X (t))+

∫ t

0−2�V d�

)(16)

thus, the minimum of the cost functional is given by J (u)=2�V (X (0)) for the optimal controllaw (9).

Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2009; 37:887–898DOI: 10.1002/cta

REAL-TIME CHAOTIC CIRCUIT STABILIZATION 893

Figure 5. Chen’s circuit implementation: the printed circuit board is a double-sided circuit,low cost, and very robust. Some resistors can be easily changed in order to scale the system

in its time and state variables.

In summary, taking into account Equation (15), the optimal and stabilizing control law is finallyobtained as

u=−(2c+ c2

2a+1

)y (17)

This is a very simple linear state-feedback controller.The stabilizing control law that minimizes the cost (12) achieves a sector margin ( 12 ,∞). This

means that the controlled chaotic system remains asymptotically stable with respect to staticnonlinearities at the input in the sector ( 12 ,∞) [18].

4. SIMULATION RESULTS

In order to verify the applicability of the proposed control law (17), consider system (1), currentlyin its chaotic state, i.e. with a=35, b=3, and c=28.

Under the control law (5), i.e. u=−68.2y, the chaotic orbit of the system is quickly drivento the originally unstable zero equilibrium point of the system, as expected. Figure 2 shows thesimulation results for different initial conditions.

Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2009; 37:887–898DOI: 10.1002/cta

894 A. JIMENEZ ET AL.

Figure 6. Controller implementation: the circuit has three configurable inverters to feedbackthe three states; however, according to Equation (3), only one is used here.

Figure 7. Chaotic time series from Chen’s circuit.

5. EXPERIMENTAL RESULTS

In order to implement the above-derived controller, the first step is to recognize the main functionalblocks of the circuit that implement equation (1), as presented in Figure 3.

Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2009; 37:887–898DOI: 10.1002/cta

REAL-TIME CHAOTIC CIRCUIT STABILIZATION 895

Figure 8. Real-time phase portrait on the x–y and the x–z planes.

Each block of this system can be realized by analog electronic components, as depicted inFigure 4. However, it is required to scale system (1) in its state variables (to slow down), inorder to avoid saturations of the operational amplifiers (OpAmps). Hence, all the state variablesare reduced by K =30 times. In the system equation (1), it is amplified by K times the productterms xz and xy and is made in the circuit of Figure 4 by using resistors R26–R29, R33, R34 andthe OpAmps connected in a manner like inverting the amplifiers. To get the best performance ofthe analog multipliers, the signal has been amplified by 10 times using the resistors R22–R25,R46–R49 and the OpAmps connected in a manner like inverting amplifiers. Additional amplifiershave been connected to scale system (1) in time variable to visualize the responses of the controllerin different scales of time, R1–R2, R5–R6, R38–R39 and the associated OpAmps are used, alsoconnected like some amplifiers. The other components in the circuit, Figure 4, are connected inorder to obtain the electronic representation of Equation (1), as usual [19].

Based on the nature of the control law, which uses feedback signals, the controller has beenimplemented using an amplifier and resistors R51–R52, as in Figure 4.

Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2009; 37:887–898DOI: 10.1002/cta

896 A. JIMENEZ ET AL.

Figure 9. Stabilizing time evolution in time, X and Y states.

Figure 10. Stabilizing phase portrait on the x–y plane.

Figure 5 is the chaotic circuit as implemented in the printed circuit board, and Figure 6 is thecontroller. Only one OpAmp has been used because only one state variable is fed back. Figure 7shows the real-time time series of x , y states, obtained by the implemented circuit. Figure 8 showsthe real-time phase portraits of states x–y and x–z, respectively.

6. REAL-TIME STABILIZATION

Once Chen’s system has been implemented by analog electronic components, one can proceedto its stabilization by using control law (5). Different initial conditions, as well as different timeinstants, have been tested. The attractor is stabilized to (0,0,0) in all the experiments.

Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2009; 37:887–898DOI: 10.1002/cta

REAL-TIME CHAOTIC CIRCUIT STABILIZATION 897

Figure 11. Control signal u for the controlled system.

For illustration, a particular experiment is shown in Figure 9 (time evolution) and Figure 10(phase portrait). Here, it is easy to visualize and verify the satisfactory performance of the proposedcontrol law.

Figure 11 shows the control signal applied for stabilizing the chaotic circuit. In all tested cases,stabilization was done satisfactorily. Besides, the circuit and the control law are both robust in thepresence of parameter uncertainties.

7. CONCLUSIONS

This paper has presented a simple and effective control law for real-time stabilizing of the chaoticChen’s circuit, as a typical chaotic circuit control, based on the inverse optimal control technique.This control law is remarkably simple as compared with other existing chaos control methods.Thanks to its generic nature, this control approach can be applied to many other complex dynamicalsystems as well.

Owing to the characteristics of the circuits, which are constructed by using low-cost componentswith limited precisions, it has actually proved the robustness of the system to some extent. Usingthe domination redesign as introduced by Sepulchre et al. [14], it is possible to guarantee robustnessagainst a class of dynamic input uncertainties at the input.

REFERENCES

1. Chen G, Dong X. On feedback control of chaotic continuous-time systems. IEEE Transactions on Circuits andSystems I 1993; 40(9):591–601.

2. Nijmeijer H, Berghuis H. On Lyapunov control of the Duffing equation. IEEE Transactions on Circuits andSystems I 1995; 42(8):473–477.

3. Zeng Y, Singh SN. Adaptive control of chaos in Lorenz system. Dynamics and Control 1997; 7(2):143–154.4. Vincent L, Yu J. Control of a chaotic system. Dynamics and Control 1991; 1(1):35–52.5. Chen G, Dong X. From Chaos to Order: Methodologies, Perspectives, and Applications. World Scientific:

Singapore, 1998.

Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2009; 37:887–898DOI: 10.1002/cta

898 A. JIMENEZ ET AL.

6. Wu XF, Cai JP, Zhao Y. Some new algebraic criteria for chaos synchronization of Chua’s circuits by linear stateerror feedback. International Journal of Circuit Theory and Applications 2006; 34(3):265–280.

7. Isidori A. Nonlinear Control Systems (3rd edn). Springer: New York, 1995.8. Byrnes C, Isidori A, Willems J. Passivity, feedback equivalence, and the global stabilization of minimum phase

nonlinear systems. IEEE Transactions on Automatic Control 1991; 36(11):1228–1240.9. Lin W. Feedback stabilization of general nonlinear control systems: a passive system approach. Systems and

Control Letters 1995; 25:41–52.10. Jiang ZP, Hill DJ. Passivity and disturbance attenuation via output feedback for uncertain nonlinear systems.

IEEE Transactions on Automatic Control 1998; 43(7):992–997.11. Knobloch W, Isidori A, Flockerzi D. Topics in Control Theory. Birkhauser: Boston, MA, 1993.12. Kristic M, Deng H. Stabilization of Nonlinear Uncertain Systems. Springer: New York, 1998.13. Freeman RA, Kokotovic PV. Robust Nonlinear Control Design. Birkhauser: Boston, MA, 1996.14. Sepulchre R, Jankovic M, Kokotovic PV. Constructive Nonlinear Control. Springer: New York, 1997.15. Magni L, Sepulchre R. Stability margins of nonlinear receding-horizon control via inverse optimality. Systems

and Control Letters 1997; 32(4):241–245.16. Chen G, Ueta T. Yet another chaotic attractor. International Journal of Bifurcation and Chaos 1999; 9(7):

1465–1466.17. Sanchez EN, Perez JP, Martinez M, Chen G. Chaos stabilization: an inverse optimal control approach. Latin

American Applied Research 2002; 32:111–114.18. Glad ST. Robustness of nonlinear state feedback—a survey. Automatica 1987; 23:425–435.19. Zhong GQ, Tang WKS. Circuitry implementation and synchronization of Chen’s attractor. International Journal

of Bifurcation and Chaos 2002; 12(6):1423–1427.

Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2009; 37:887–898DOI: 10.1002/cta