Intelligent control of chaos using linear feedback controller and neural network identifier

Commun Nonlinear Sci Numer Simulat 17 (2012) 4731–4739

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Intelligent control of chaos using linear feedback controller and neural network identifier M. Sadeghpour, M. Khodabakhsh, H. Salarieh ⇑ Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11155-9657, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 9 July 2011 Received in revised form 23 November 2011 Accepted 22 March 2012 Available online 5 April 2012 Keywords: Intelligent control Chaos Linear feedback Neural network

a b s t r a c t A method for controlling chaos when the mathematical model of the system is unknown is presented in this paper. The controller is designed by the pole placement algorithm which provides a linear feedback control method. For calculating the feedback gain, a neural network is used for identification of the system from which the Jacobian of the system in its fixed point can be approximated. The weights of the neural network are adjusted online by the gradient descent algorithm in which the difference between the system output and the network output is considered as the error to be decreased. The method is applied on both discrete-time and continuous-time systems. For continuous-time systems, equivalent discrete-time systems are constructed by using the Poincare map concept. Two discrete-time systems and one continuous-time system are tested as examples for simulation and the results show good functionality of the proposed method. It can be concluded that the chaos in systems with unknown dynamics may be eliminated by the presented intelligent control system based on pole placement and neural network. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction A chaotic system is a deterministic system with great sensitivity to initial conditions. There are many nonlinear systems around us which behave chaotically. Economics, biology, chemistry, fluid dynamics, secure communication, and many other fields have already confronted with the chaos phenomenon. Thus, control and prediction of chaos have attracted great interests from researchers of different scientific fields. In chaos control, in general, the aim is to stabilize a chaotic system either in its unstable periodic orbit or in its unstable fixed point. Many methods and theories for chaos control and prediction have been developed by researchers [1–10]. Ott et al. [1] who were the first in introducing a method for chaos control, presented the OGY method. Feedback control methods [2–6] have been also utilized to control chaos by stabilizing a desired unstable periodic solution which is embedded in a chaotic attractor. In [11], Ushio and Yamamoto presented a static feedback control method where the difference between the predicted states and the current states of the system are used to produce a feedback control law. Control of chaos using a fuzzy estimating system based on batch training and recursive least squares method for a continuous-time dynamical system has been presented in [12]. In [13] a prediction-based chaos control scheme is proposed where to perform chaos prediction, a new neural network architecture is used. Controlling chaos in Bonhoeffer van der Pol system based on estimating the OGY and the delayed feedback control methods via neural networks is presented in [14]. In [15], an adaptive chaos control method by combining the sliding mode control and the fuzzy clustering identification method is introduced. ⇑ Corresponding author. Tel.: +98 21 6616 5538. E-mail addresses: [email protected], [email protected] (H. Salarieh). 1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2012.03.030

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In most chaos control techniques, it is assumed that the dynamical model of the system is known. But there are many systems that we do not have their mathematical model. Also many of the available dynamical models for systems are not exact. So, a chaos control technique which does not use the mathematical model of the system will be of great value. Some works attempt to control chaos while assuming the dynamical model of the system is unknown, such as in [16], where the authors use the idea of entropy minimization to suppressing chaos, and in [17], where they control chaos by a PSO-based neural network based on the technique of small perturbations. In [18], a fuzzy adaptive sliding mode control technique is presented for controlling chaos in un-modeled dynamical systems. Stabilizing unstable periodic orbits and unstable fixed points of chaotic systems with unknown models on their Poincare maps using fuzzy clustering technique is presented in [19]. In this paper, we propose an intelligent control system for chaos control. The controlling system consists of two main parts, i.e. the controller and the identifier. The structure of the identifier is based on neural network, and the controller is based on the linear feedback whose gain is obtained via pole placement technique. The proposed method provides a model-free controller in which the mathematical model of the system is not used directly to calculate the controller signal. The control and identification systems are utilized in an online manner which means that the neural network-based approximated model is updated simultaneously with the control signal applying to the system. The neural network model is used to estimate the Jacobian matrix of the system which is needed in the pole placement method. The effectiveness of the proposed method is examined by applying it to the Logistic and the Tent map as two discrete chaotic systems, and the Bonhoeffer van der Pol (BVP) system which is a continuous-time chaotic system. 2. Pole placement-based control of chaos using neural network Consider a chaotic system in the continuous form of:

x_ ¼ f ðx; uÞ

ð1Þ

or in the discrete form of:

xðk þ 1Þ ¼ f ðxðkÞ; uðkÞÞ

ð2Þ

where x 2 Rn is the state vector and u 2 Rm is the control vector. Suppose the dynamical model of the system – the function f – is not known. The objective is to control the chaotic behavior of the system-stabilize its unstable periodic orbit-by a linear feedback based on pole placement algorithm. Hence, it would be required to know the Jacobian of the system. To this end, a neural network (Fig. 1) is used as an identifier for the system. The system output, x(k + 1), and neural network output, ~ xðk þ 1Þ, are compared and the weights of the network are changed in a way that the difference between these outputs is decreased. In Fig. 1 a feed-forward neural network containing one hidden layer is depicted. The network gets x1(k), x2(k), and u(k) as inputs and gives ~ xi ðk þ 1Þ out as its output where i can be 1 or 2. The above network represents a two-dimensional function;

Fig. 1. The neural network used here.

M. Sadeghpour et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 4731–4739

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however, in general, a network can represent an n-dimensional function with x1(k), x2(k), . . . , xn(k), and u(k) as its inputs. For an n-dimensional function, n networks, each for one output of the system, can be used. The output of a feed-forward neural network may be expressed by considering a Multi-Layer (ML) network composed of 3 layers, each layer containing N(l) nodes where l = 1, 2, . . . , L. The input to the first layer is the input of the network, and the output of the last layer is the network output. The equations describing the ith node located at the lth layer of an ML network can be expressed using the following equations:

zl;i ðnÞ ¼

Nðl1Þ P

wðl;iÞðl1;jÞ  xl1;j

ð3Þ

j¼1

xl;i ðnÞ ¼ F l ðzl;i ðnÞÞ

ð4Þ

where zl,i(n) represents the internal state variable of the ith node at the lth layer, xl,i(n) is the output signal of the ith node at the lth layer, w(l,i)(l1,j) is the weight between the jth node of the (l  1)th layer and the ith node of the lth layer, and Fl(zl,i(n)) represents the activation function associated with the lth layer which is chosen a hyperbolic tangent function in this work. The gradient descent algorithm is used for updating the neural network weights as:

Dw ¼ g

@E @E ) wkþ1 ¼ wk  g @w @w

ð5Þ

where w is one of the neural network weights and g is the gradient descent coefficient or learning rate. E is the error defined by:

E¼

1 1 kxðk þ 1Þ  ~xðk þ 1Þk2 ¼ ½xðk þ 1Þ  ~xðk þ 1ÞT ½xðk þ 1Þ  ~xðk þ 1Þ 2 2

ð6Þ

Fig. 2 shows the neural network position as an online identifier during control procedure. As it is mentioned earlier, a linear feedback based on the pole placement algorithm is used for producing the controller value. According to this algorithm the control action is:

uðkÞ ¼ lk ½xðkÞ  xF 

ð7Þ

where xF is the fixed point of the system. lk is an m  n vector of coefficients which is computed in a way that the system is locally asymptotically stable around its fixed point. Because the function f is not known, the neural network is used for obtaining the approximate Jacobian matrix of the system. The estimation of f after kth step is denoted by ~f k , i.e. ~ xðk þ 1Þ ¼ ~f k ðxðkÞ; uðkÞÞ. Let

 @~f k  Ak ¼  @x 

x¼xF ;u¼0;w¼wðk1Þ

 @~f k  and Bk ¼  @u 

ð8Þ x¼xF ;u¼0;w¼wðk1Þ

then, according to the pole placement algorithm, the eigen-values of the matrix Ak  Bklk must be inside the unit circle in order for the system to be stable. Considering these eigen-values as ki ði ¼ 1; 2; . . . ; nÞ where n is the order of the system, lk will be a function of the neural network weights and ki ’s. In computation of the Jacobian matrices, we consider w = w(k  1). In other words lk and thereby u(k) are a function of weights in the time step k  1, i.e., w(k  1). With this assumption causality of the method will be satisfied. Noticing to this point that the system is locally asymptotically stable, the control action is applied in a small interval around the fixed point. In other words:



uðkÞ ¼ lk ½xðkÞ  xF  if juðkÞj 6 d uðkÞ ¼ 0

ð9Þ

if juðkÞj > d

Fig. 2. The neural network as an identifier.

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where d is a small positive constant. 2.1. Obtaining Dw Combining Eqs. (5) and (6) yields:

Dwk ¼ g

  @Ekþ1 @Ekþ1 @xðk þ 1Þ @Ekþ1 @ ~xðk þ 1Þ ¼ g þ @wk @wk @wk @xðk þ 1Þ @ ~xðk þ 1Þ

ð10Þ

The first term in the right hand side of Eq. (10) becomes zero, because considering Eq. (2), it can be written:

@xðk þ 1Þ @f @xðkÞ @f @uðkÞ @f @xðkÞ @f @uðkÞ @ lk ¼ þ ¼ þ @wk @xðkÞ @wk @uðkÞ @wk @xðkÞ @wk @uðkÞ @ lk @wk

ð11Þ

On the other hand, we have:

@xðkÞ ¼ 0 and @wk

@ lk ¼0 @wk

ð12Þ

since wk is only used in obtaining ~ xðk þ 1Þ. wk plays no role in obtaining x(k). x(k) is indirectly a function of network weights in previous time steps. x(k) is a function of u(k  1) when u(k  1) is a function of lk1. Also, as said earlier, lk1 is a function of wk2. Then x(k) will be a function of wk2. Similarly, olk/owk = 0; because, lk is a function of wk1. Considering Eq. (12), the first and the second terms in the right hand side of Eq. (11) will become zero, and consequently:

@xðk þ 1Þ ¼0 @wk

ð13Þ

Substituting Eq. (13) in Eq. (10) and using Eq. (6), the gradient descent algorithm reduces to:

Dwk ¼ g½xðk þ 1Þ  ~xðk þ 1ÞT 

@ ~xðk þ 1Þ @wk

ð14Þ

when @ ~ xðk þ 1Þ=@wk is known from the neural network structure. 3. Stability analysis of the proposed controller To investigate the stability of the closed loop system it is assumed that the identified map of the system, i.e. ~f k , converges to a function ~f as k converges to infinity. According to the Kolmogorov theorem [20], for each e > 0 there exists a neural structure for which the estimation error between the neural system and the real function is less than e. So, assume that the neural structure considered for estimating f(x, u), in sufficiently large values of k, e.g. k P K > 0, is represented by the function ~f and ~ and B. ~ the limits of Ak and Bk exist being denoted by A Introducing the error state e(k) = x(k)  xF and linearizing the nonlinear map of the system around the fixed point yields the following error dynamics:

eðk þ 1Þ ¼ AeðkÞ þ BuðkÞ þ OðeðkÞÞ

ð15Þ

~ and B ~ to and subtracting them from the right hand where O(e(k)) is the higher order terms of the error dynamics. Adding A side of Eq. (15), yields:

~ ~ ~ ~ eðk þ 1Þ ¼ AeðkÞ þ BuðkÞ þ ðA  AÞeðkÞ þ ðB  BÞuðkÞ þ OðeðkÞÞ

ð16Þ

~ and DB ¼ B  B ~ and substituting the control law of Eq. (7) into Eq. (16), the error dynamics is rewritten as Letting DA ¼ A  A

~B ~l ~ ÞeðkÞ þ ðDA  DBl ~ ÞeðkÞ þ OðeðkÞÞ eðk þ 1Þ ¼ ðA

ð17Þ D

~B ~l ~ lie inside the unit circle of the complex plane. Since all As stated before, l is chosen such that all eigen-values of G ¼ A eigen-values of the matrix G are inside the unit circle, there exists a positive definite symmetric matrix P which satisfies the following Lyapunov equation, for any positive definite symmetric matrix Q [21]:

P  GT PG ¼ Q

ð18Þ D

~ , can violate the asympThe error between the linearized part of the real system and the estimated system, i.e. Ee ¼ DA  DBl totic stability of the closed-loop system. For analyzing the effect of the estimation error, Ee, on the stability of the closed loop system, consider the following Lyapunov function:

VðkÞ ¼ eT ðkÞPeðkÞ

ð19Þ

To have an asymptotically stable error dynamics, DV(k) = V(k + 1)  V(k) must be always negative [21]. Then we can write:

DVðkÞ ¼ Vðk þ 1Þ  VðkÞ ¼ eT ðk þ 1ÞPeðk þ 1Þ  eT ðkÞPeðkÞ

ð20Þ

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Substituting e(k + 1) from Eq. (17) into Eq. (20), it is concluded that:

DVðkÞ ¼ eT ðkÞGT PGeðkÞ þ eT ðkÞETe PEe eðkÞ þ eT ðkÞGT PEe eðkÞ þ eT ðkÞETe PGeðkÞ  eT ðkÞPeðkÞ þ OðeT ðkÞÞPOðeðkÞÞ ¼ eT ðkÞðGT PG  PÞeðkÞ þ eT ðkÞETe PEe eðkÞ þ eT ðkÞGT PEe eðkÞ þ eT ðkÞETe PGeðkÞ þ OðeT ðkÞÞPOðeðkÞÞ ¼ eT ðkÞQeðkÞ þ eT ðkÞETe PEe eðkÞ þ eT ðkÞGT PEe eðkÞ þ eT ðkÞETe PGeðkÞ þ OðeT ðkÞÞPOðeðkÞÞ 6 kmin ðQ ÞkeðkÞk2 þ kmax ðPÞkEe k2 eðkÞ2 þ 2kEe kkGkeig kPkeig keðkÞk2 þ MkeðkÞk4

ð21Þ

where kmin and kmax are the minimum and maximum eigen-value functions, and kkeig is the eigen norm of matrix, and |||| is the Euclidian vector norm. Now letting a ¼ 2kGkeig kPkeig and noticing that we should have DV(k) < 0, the following equation is obtained:

kmin ðQ Þ þ kmax ðPÞkEe k2 þ akEe k < 0

ð22Þ

The fact that kEe k, kmin ðQ Þ, and kmax ðPÞ are all positive (P and Q are positive definite matrices) implies that for Eq. (22) to hold it is required that:

kEe k <

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ 4kmax ðPÞkmin ðQ Þ  a 2kmax ðPÞ

ð23Þ

Eq. (23) is a sufficient condition for the asymptotic stability of the controlled system. It is not, however, a necessary condition. In other words, the system may be stabilized without this condition holding. This bound for the estimation error, Ee, is investigated in one of the simulations below. 4. Simulations In all of the simulations, neural networks with one hidden layer are used. Also, for the weights between the input layer and the hidden layer and the weights between the hidden layer and the output layer, different learning rates g1 and g2 are considered for using in Eq. (14). The number of neurons in the hidden layer and the learning rates for each simulation can be found in Table 1. The range of applying control action, ±d, is also mentioned in Table 1. 4.1. Stabilizing chaotic logistic map on its first order fixed point The chaotic logistic map is a first order discrete-time system which has the following dynamic equation:

xðk þ 1Þ ¼ rxðkÞ½1  xðkÞ

ð24Þ

x(k) is between zero and one, and for r = 3.7 this map shows chaotic behavior [22]. The 1-cycle fixed point of the system occurs at xF = 0.7297. As said before, it is supposed that we have not got the dynamical model of the system. Hence, to obtain the feedback gain, l, of the pole placement-based controller, we shall use the structure of neural network to compute the Jacobian of the system in its fixed point (see Eq. (8)). The dimension of the logistic map is one, so one neural network is utilized for producing the approximate output of the system, ~ xðk þ 1Þ. The results of applying the proposed method to the logistic map are illustrated in Fig. 3. As shown in Fig. 3 the map is stabilized after about 20 iterations and the controller value becomes zero. Also ~ x (denoted by xnet in Fig. 3) becomes equal to x which implies that the neural network has accurately identified the dynamical system. To investigate whether the condition of Eq. (23) holds or not, we calculate the value of the right hand side of it which equals to 0.4 for any Q where, in this case, P, Q, and G are 1  1 matrices. Fig. 4 shows the graph of the estimation error, kEe k, with respect to k. As shown in Fig. 4, the estimation error has become smaller than 0.4 after a transient period when the weights of the neural network are properly trained and the feedback gain of the controller, l, has taken an appropriate stabilizing value.

Table 1 Parameters used in simulations. Number of neurons in the hidden layer

Learning rate Range of control signal, d

g1 g2

Logistic map J = 15

Tent map J = 15

BVP oscillator J1 = 15 J2 = 15

2.65 2.65 0.05

0.65 0.65 0.08

0.65 0.65 0.5

M. Sadeghpour et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 4731–4739

1

0.8

0.8

0.6

x (k)

0.6

net

x(k)

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0.4 0.2

0

20

40

0.2 0

60

0

20

40

60

20

40

60

0.8

0.02

0.6

E(k)

0.01

u(k)

0.4

0

-0.01

0.4 0.2

-0.02 0

20

40

k (number of iterations)

0

60

0

k (number of iterations)

Fig. 3. Stabilizing the 1-cycle fixed point of the logistic map.

1.6

Estimation error

1.4 1.2 1 0.8 0.6 0.4 0.2

10

20

30

40

50

60

k (number of iterations) Fig. 4. Estimation error for the logistic map.

4.2. Stabilizing chaotic Tent map on its first order fixed point The Tent map is also a discrete-time one-dimensional dynamical system. Its dynamic equations are as follows:

 xðk þ 1Þ ¼

qxðkÞ; xðkÞ < 1=2 qð1  xðkÞ; xðkÞ P 1=2

ð25Þ

x(k) is between zero and one. If q = 2 the behavior of this map becomes chaotic [23].The 1-cycle fixed point of the map is xF = 0.6667. Fig. 5 shows the results of the stabilizing the Tent map. In this case, the map is stabilized after about 35 iterations. 4.3. Stabilizing 1-cycle fixed point of the chaotic BVP oscillator The Bonhoeffer-van der Pol (BVP) oscillator is a biologically and physically important dynamical system exhibiting chaos. The governing equations of BVP oscillator are [12,24,25]:

x31  x2 þ fcosðtÞ þ uðtÞ 3 x_ 2 ¼ cðx1 þ a  bx2 Þ

x_ 1 ¼ x1 

ð26Þ

where a, b, c, and f are constants. For a = 0.7, b = 0.8, c = 0.1, f = 0.74, and u(t) = 0, the behavior of this system is chaotic. To apply the proposed method to this system, we shall first construct an equivalent discrete-time system for the continuous-time system using the Poincare map of the system. To this aim, a stroboscopic transversal of t = 4kp, k = 0, 1, 2, . . . is used in a way that the intersections of the trajectory of the continuous-time system with the transversals t = 4kp, k = 0, 1, 2, . . ., constitute the states of the discrete system. As a result, the following nonlinear maps are obtained:

M. Sadeghpour et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 4731–4739

0.8

xnet(k)

x(k)

1

4737

0.5

0.6 0.4 0.2

0 0

20

40

60

0 0

80

E(k)

u(k)

0.05 0

-0.05 0

20

40

60

k (number of iterations)

40

60

80

20

40

60

80

0.2 0.1 0 0

80

20

k (number of iterations)

2

2

1

1

xnet1(k)

x1(k)

Fig. 5. Stabilizing the 1-cycle fixed point of the Tent map.

0 -1 -2

0

50

100

150

-2

200

1

50

100

150

200

50

100

150

200

50

100

150

200

0

xnet2(k)

0 -0.5 0

50

100

150

200

-0.2 -0.4 -0.6 -0.8 0 4

0.3 0.2 0.1

3

E(k)

x2(k) u(k)

0

0.2

0.5

-1

0 -1

0 -0.1 0

50

100

150

200

k (number of iterations)

2 1 0

0

k (number of iterations)

Fig. 6. Stabilizing the 1-cycle fixed point of the BVP oscillator.

xp1 ðk þ 1Þ ¼ F 1 ðxp1 ðkÞ; xp2 ðkÞ; up ðkÞÞ xp2 ðk þ 1Þ ¼ F 2 ðxp1 ðkÞ; xp2 ðkÞ; up ðkÞÞ

ð27Þ

where ðxp1 ðkÞ; xp2 ðkÞÞ is the kth point of intersection to the transversal t = 4kp, that is, xp1 ðkÞ ¼ x1 ð4kpÞ, xp2 ðkÞ ¼ x2 ð4kpÞ, and up(k) = u(4kp) .The functions F1 and F2 are not known. However, the proposed control method can also be used here. The controller value on the Poincare section, i.e., the transversal t = 4kp, k = 0, 1, 2, . . . is determined as follows:

" up ðkÞ ¼ lk

xp1 ðkÞ  xp1F

# ð28Þ

xp2 ðkÞ  xp2F

where lk is a 1  2 vector and xp1F and xp2F are the coordinates of the 1-cycle fixed point of the Poincare map of the system. The controller value remains constant between two successive intersects, that is:

" uðtÞ ¼ lk where T = 4p.

xp1 ðkÞ  xp1F xp2 ðkÞ  xp2F

# kT < t 6 ðk þ 1ÞT

ð29Þ

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-0.2 -0.25

x2(t)

-0.3 -0.35 -0.4 -0.45 -0.5 -0.55 -2

-1.5

-1

-0.5

0

0.5

x1(t) Fig. 7. The stabilized periodic orbit of the BVP oscillator.

As stated before, lk is determined from the pole placement method. Because the BVP system is a two-dimensional system, two neural networks, each for one output of the system, are used. The number of neurons in the hidden-layers of these networks, J1 and J2, are shown in Table 1.By applying the proposed approach, the period 4p unstable orbit of the BVP system is stabilized. The results are depicted in Figs. 6 and 7. As seen in Fig. 6, the system is stabilized after about 40 iterations in its fixed point ðxF1 ; xF2 Þ ¼ ð0:9041788; 0:5094619Þ: The neural network outputs ~ x1 and ~ x2 (denoted, respectively, by xnet1 and xnet2 in Fig. 6) become finally equal to the system outputs x1 and x2 and thereby the identification error, E, becomes zero. Moreover, the control action u finally becomes zero as x equals xF.

5. Conclusion In this paper, a neural network-based linear feedback control via pole placement technique for stabilizing unstable periodic orbits of chaotic systems with unknown mathematical models is proposed. The outputs of the system and the neural network are compared and the weights of the network are adjusted to minimize this error in an online manner. Having no information of the system dynamics, the neural network structure is used as a representative of the system model to extract the Jacobian of the system in its fixed point and obtain an appropriate feedback gain to stabilize the system behavior. The method is applied on both discrete and continuous-time systems. In the latter, an equivalent discrete system to the continuous-time system is created defining a proper Poincare section and producing the Poincare map of the system. The discrete Logistic map and Tent map and the continuous Bonhoeffer-van der Pol oscillator system are tested as simulation experiments. The results in each case corroborate the efficiency of this method in eliminating chaos and regulating system’s behavior.

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