Linear optimal control of continuous time chaotic systems

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Research Article

Linear optimal control of continuous time chaotic systems Kaveh Merat, Jafar Abbaszadeh Chekan, Hassan Salarieh n, Aria Alasty Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11155-9567, Tehran, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 11 June 2013 Received in revised form 4 December 2013 Accepted 17 January 2014 This paper was recommended for publication by Mohammad Haeri.

In this research study, chaos control of continuous time systems has been performed by using dynamic programming technique. In the first step by crossing the response orbits with a selected Poincare section and subsequently applying linear regression method, the continuous time system is converted to a discrete type. Then, by solving the Riccati equation a sub-optimal algorithm has been devised for the obtained discrete chaotic systems. In the next step, by implementing the acquired algorithm on the quantized continuous time system, the chaos has been suppressed in the Rossler and AFM systems as some case studies. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Chaos Optimal control Continuous time system Rossler system AFM system

1. Introduction The nonlinear dynamical systems which exhibit chaos phenomenon are appeared in many fields of science such as engineering, economy, ecology, and engineering. The concept of chaos has been introduced in 1975 by Li and Yorke for the first time [1]. The elimination of chaos has been studied in many researches and investigations, and different techniques including optimal approaches, and advanced nonlinear methods were utilized for chaos stabilization. The primary technique which is well known as the OGY method was proposed by Ott et al. for stabilizing the unstable periodic orbits embedded in a chaotic attractor [2]. In addition, some approaches based on OGY method such as SOGY have been presented to enhance the performance of control [3,4]. Pyragas [5] also proposed the delayed feedback technique to stabilize the unstable periodic orbits. Furthermore, the model free control of the Lorenz chaotic system is performed by utilizing an approach based on an approximate optimal control in [6]. In [7] an optimal control policy has been introduced to control a chaotic system via state feedback. In the mentioned study, at first the system has been converted to an uncertain piecewise linear model and then an optimal controller has been designed which minimizes the upper bound on cost function under constraints in the form of bilinear matrix inequality. In some other case studies, synchronization of chaos is concerned instead of controlling chaotic systems. For instance in [8], Jayaram and Tadi utilized State Dependent Riccati Equation (SDRE) method to synchronize chaotic systems.

n

Corresponding Author. Tel: þ 982166165538. E-mail address: [email protected] (H. Salarieh).

Furthermore, Park synchronized two chaotic systems using a nonlinear controller designed based on the Lyapunov stability theory [9]. In some other studies, Robust, adaptive and nonlinear control theory, also were applied. Cao introduced a nonlinear adaptive method for controlling a chaotic oscillator [10]. In [11] Layeghi et al. stabilized periodic orbits of chaotic systems by applying fuzzy adaptive sliding mode control. The Adaptive Lyapunov-based control which is another nonlinear control approach was hired by Salarieh and Shahrokhi in [12] to suppress the chaotic motion. Also, Fuh et al. introduced a Robust controller which combines feedback linearization and disturbance observer to suppress chaotic motion in a nonlinear system which is under external excitation [13]. Zhang and Tang [14] studied the dynamics of a new chaotic system containing two system parameters with nonlinear terms, using the Lyapunov exponents. They both stabilized and synchronized the mentioned chaotic system globally, using a linear state feedback controller, designed through the simple sufficient conditions resulted from the Lyapunov stability criteria. Dynamics of a new three dimensional chaotic system containing a nonlinear term in the form of arc-hyperbolic sine function was studied in [15], in which the system has been converted to an uncertain piecewise linear system. Using piecewise quadratic Lyapunov function method, the chaos phenomenon has been controlled globally in this system with alpha-stability constraint and via piecewise linear state feedback. The same authors in [16], by applying Lyapunov stability criteria, synchronized chaos in a chaotic system with a nonlinear term which does not satisfy Lipschitz continuity but satisfies 1/2- Hölder continuity. Since the target of our study is stabilizing chaotic systems via dynamic programming algorithm, reviewing some relevant works is useful. Dynamic programming concept was introduced in 1957 by

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Bellman [17] and subsequently many modifications on this approach have been proposed by other researchers [18,19]. Merat et al. applied the dynamic programming approach to stabilize n-th order unstable periodic orbits of discrete chaotic systems in [20] where a linear feedback control with its applicable domain is obtained. In this research study, the control of continuous time chaotic systems has been done through using a stabilizing algorithm specialized for discrete systems. In order to apply this optimal control algorithm on a continuous time system, it should be quantized in time domain. To this aim, a Poincare section is selected which crosses all of system response trajectories where each cross point is mapped on the Poincare surface. Consequently, a nonlinear discrete map from a continuous time system that gives a comprehensive knowledge about the dynamic behavior of the main system is achieved. So omitting the chaos phenomenon from the acquired discrete Poincare map leads to regular behavior of the corresponding continuous system. The only complexity that arises is that obtaining the nonlinear equations of Poincare map for a chaotic system is a difficult task and in some cases is not possible. So, as a solution, local linearization of the Poincare map is suggested. Then through solving Riccati equation, a linear controller is presented which can stabilize the acquired quantized continuous time chaotic system around its fixed points. Hence, utilizing the presented optimal linear controller for the obtained discrete system, results in the stabilizing the system unstable fixed point (UFP) and subsequently stabilizing the corresponding unstable periodic orbit (UPO) in the continuous time system. In this article, first a method based on regression approach is suggested which gives a linear approximation for the Poincare map around its fix point and then by solving the Ricatti equation, a linear optimal control technique is introduced to stabilize the obtained discrete system on its UFP which leads to suppression of chaos. It is notable that, the introduced method can be used for autonomous and non-autonomous chaotic systems. At the end, the presented approach is implemented for controlling two continuous time chaotic systems namely the Rossler System and an AFM (Atomic Force Microscope) system in its tapping mode. Being invented by Bining et al. [21], AFM made an impressive revolution in the topography of different surfaces in the small scale. Due to high demands in having pictures with further quality control AFM systems with chaotic behavior have attracted great attention. Elimination of chaos for this system can be accomplished through two ways. The first method is changing some parameters of the system to the values in which no chaotic behavior occurs [22]. The second approach consists of implementing an active controller through piezo-actuators to stabilize systems UPO [23]. There are some studies which are done in controlling the chaotic AFM systems. In [24], the chaos control of continuous AFM system in its tapping mode has been analyzed. Assuming model parameters uncertainties, the sliding nonlinear delayed feedback control approach is applied to stabilize the first order periodic orbit of the system. Salarieh and Alasty [25] presented an active control for the AFM system in its tapping mode. They applied delayed feedback method in which the feedback gain is obtained and adopted according to the Minimum Entropy algorithm. In their investigation the target was stabilizing the system on its unstable fixed point. In [26] Wang and Yau studied the nonlinear dynamic behavior of the probe tip of an AFM system with a PD feedback controller using differential transformation method.

2. Continuous time chaotic system quantization using regression method By selecting an appropriate Poincare section and specifying the cross points of the system trajectories with this section, the

resultant Poincare map can be defined and introduced as; zk þ 1 ¼ Pðzk ; uk Þ

ð1Þ m1

is the k-th point of the system response that in which zk A ℜ crossed the Poincare map (m indicates the number of state variables of continues system including the explicit term of time in the equations). uk A ℜp is the control signal, exerted to the continues system between two subsequent touch of the system response with the Poincare map. By linearization of Eq.(1) around its fixed point ðjÞ Z nF and using the variable change zk ¼ zk  ðjÞ Z nF we may write zk þ 1 ¼ ðjÞ Azk þ ðjÞ Buk ; j ¼ 1; …; n

ð2Þ

ðjÞ

In the above equation A is a matrix with the dimension of ðm  1Þ  ðm 1Þ and ðjÞ B is a ðm  1Þ  p matrix, both correspond to the j-th fixed point of the system with the order of “n”, ðjÞ Z nF . The target is to find the so-called matrices. For this aim, different points on the Poincare surface and in the vicinity of the corre1 sponding fixed point, ðjÞ Z nF , are chosen which are shown by ξi . In the next step, each of these points is set as the initial condition for uncontrolled system trajectory. Then the next cross point of the system response with the Poincare section is acquired which is 1 2 2 marked by ξi . The couples of subsequent points, ðζi ; ζi Þ are on the Poincare section, hence, all couples fulfill the equation:

ζ2i ¼ Pðζ1i ; 0Þ

ð3Þ

With the assumption that, ξ are close enough to the fixed point. It 1 2 is assumed that the couple ðξi ; ξi Þ satisfies the linear uncontrolled Poincare map: 1 i

ζ 2i ¼ ðjÞ A ζ 1i

ð4Þ

For appropriate number of ðξi ; ξi Þ, using the linear regression method the matrix ðjÞ A can be obtained. Generally the regression approach is applicable for the multi input (n)- multi output (m) systems in the shape of 1

Y mq ¼ β mn X nq þ εmq

2

ð5Þ

in which, Y is the output matrix for the input matrix X, and q is the number of couples ðx^ i ; y^ i Þ, where x^ i and y^ j are input and output vectors respectively. The matrices X, Y can be built as

ð6Þ

The goal is finding the coefficient matrix β which minimizes the value of S defined in q

m

S ¼ ∑ ∑ εij 2

ð7Þ

j¼1i¼1

This problem is categorized into the class of the least square problems which has a known solution. 1 2 Considering ζ i and obtained ζ i for the uncontrolled system the input and output matrices are ð8Þ Using the linear regression technique for this multi-input multioutput system, the coefficient matrix β ¼ ðjÞ Aðm  1Þðm  1Þ can be acquired. For extraction of matrix, ðjÞ B in Eq.(2), similar to the calculation n 1 ðjÞ of A a few points, ζ i close to the fixed point ðjÞ Z F on the Poincare surface are selected, then by exerting control signals, ηi , to the system, the continuous time equations are solved for the initial 1 2 conditions, ζ i . The next crossed points, ζ i are obtained. Now,

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3

where

regarding Eq.(2), the input and output matrices X; Y can be written as

G ¼  ðBT KB þ Λ Þ  1 ðBT KAÞ T

ð18Þ

is asymptotically stable and all of the eigen-values of the closed-loop system matrix ACL is inside the unique circle. ð9Þ Applying the linear regression method, the coefficient matrix β ¼ ðjÞ Bðm  1Þp can be attained. 3. Stabilizing the first order unstable fixed point of a discrete chaotic system via linear optimal control algorithm in infinite time domain In this section, the goal is stabilizing the first order UFP, x1F , for a chaotic system using dynamic programming in infinite time domain. Similar to the research in [20], the system, xk þ 1 ¼ f ðxk ; uk Þ is linearized around its UFP, x1F , while the control signal is set to zero. Considering only the first order terms in the Taylor Extension, it can be written as xk þ 1  x1F

¼ Aðxk  x1F Þ þ Buk m

ð10Þ

p

where xk A ℜ and uk A ℜ are the state vector and control signal in the k-th step respectively, and the matrices A and B are constant. Considering the change of variable, ek ¼ ðxk  x1F Þ, Eq.(10) can be rewritten as ek þ 1 ¼ Aek þ Buk

ð11Þ

and the objective scalar function for this system for infinite time domain is introduced by   1 1 T 1 ek W ek þ uk T Λuk J¼ ∑ ð12Þ 2 k¼0 2 In which W and Λ represent constant weighting matrices that report the significance of state variables and control signal with respect to each other. The weighting matrices are positive definite and symmetric. The following theorem presented in [27] gives the linear controller for stabilizing the first order fixed point of the chaotic system. Theorem. [27] Considering the m  m matrix A, and m  p matrix B, if the m  m matrices W and Λ are symmetric and positive definite, the Riccati equation can be written as; K k þ 1 ¼ W þ A K k A  ðA K k BÞ ðB K k B þ Λ Þ T

T

T 1

T

T

ðB K k AÞ

ð13Þ

where its initial value, K 0 , can be every arbitrary symmetric, positive definite matrix.

3.1. Chaos control of the Rossler system In this section the stated procedure is applied for stabilizing the first order UPO of the Rossler chaotic system where its state space equations are x_ ¼ y  x y_ ¼ x þ ay z_ ¼ xz  cz þ b

ð19Þ

In [28], for the following parameters, the chaotic behavior has been reported for this system. a ¼ 0:2;

b ¼ 0:2;

c ¼ 5:7

ð20Þ

The chaotic attractor of this system for the parameters mentioned in Eq.(20) has been plotted in phase space and shown in Fig. 1. It can be seen that if the plane, y ¼ 0 is chosen as the Poincare section, it will cross all the trajectories of system solution. In this analysis, to suppress chaos in the system, the first order UPO has been considered which its corresponding unstable fixed point is x1F ¼ ½  7:21138

0:01556 

0

ð21Þ

To obtain the linearized Poincare map around the fixed point represented in Eq. (21), a 5  5 square mesh with the length of 1 0:005 is defined. By choosing the center of each square as ζ i , there will be 25 points. For each of these points, considering Eqs. (3) and 2 (19), the next crossed point, ζ i is found. In this way, regarding Eq.(8) the input and output matrices can be built and subsequently the matrix ð1Þ A for the linearized Poincare map represented in Eq.(2) can be achieved, which is    2:4065 0:18629 ð1Þ A¼ ð22Þ 3  0:29134 0:22554  10 Similar to the matrix ð1Þ A, the procedure is done for extraction of 1 2 matrix ð1Þ B. For the selected ζ i , the next touch points, ζ i are computed for the control signals, ηj A f  0:25 ;  0:1 ; 0:1 ; 0:25g.

Assuming the couples ðA; BÞ and ðA; W 1=2 Þ, if the following conditions hold: ðA; BÞ is Controllable;

ðA; W 1=2 Þ is observable

ð14Þ

25

then: 20

(i) There is a symmetric, positive definite matrix K such that: ð15Þ

that for an arbitrary initial value K 0 , is the unique solution of the following Riccati equation: K ¼ W þ AT KA  ðAT KBÞ ðBT KB þ Λ Þ  1 ðBT KAÞ T

ð16Þ

(ii) Furthermore, the closed loop system with the linear optimal control, uk ¼ G xk , ek þ 1 ¼ ðA þ BGÞek ¼ ACL ek

ð17Þ

15

z

lim K k ¼ K

k-1

10 5

15 10

0 10

5 5

0 0

y

-5

-5 -10

-15

x

-10

Fig. 1. Chaotic attractor of the Rossler system.

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Assembling the input and output matrices as stated in Eq.(9) and using the regression method give   1:35457 ð1Þ ð23Þ B¼ 2  0:25846  10

0.3

0.25

0.2

To design the linear controller, the weighting parameters, W ¼ Identity; Λ ¼ 0:1 are chosen, then the Riccati equations, presented in Eq.(16), is solved for K. Finally the linear optimal controller coefficient, G, is acquired by Eq.(18). Note that the solution of the Riccati equations is unique because the couples ðA; BÞ and ðA; W 1=2 Þ are controllable and observable respectively. Fig. 2 shows the result of simulation in the phase space, in which the system's stabilized periodic orbit has been shown with bold line and the transient response of the system has been marked with dotted lines. The time series of the system states, x, y, z have been plotted and represented in Fig. 3 under the implementation of the acquired linear controller. The results show after about 25 s, the system has been stabilized on its periodic orbit. Fig. 4 illustrates the control signal. It is notable that control signal is constant within the time interval between two successive response touches to the Poincare section. 3.2. Chaos control in atomic force microscope (AFM) In this section, the active chaos control of AFM in its Tapping mode has been inspected. This has been performed through stabilizing its fixed point located on the Poincare section via the presented method in this investigation. AFM is a time periodic system which is excited by an external actuator with constant

0.15

uk 0.1

0.05

0

-0.05 0

5

10

15

20

25

30

35

Time (s) Fig. 4. The exerted control signal in stabilizing the Rossler system first order UPO.

period. This tool has a vast application in different realms such as inspecting surface properties, changing the material in Nano scale, identification, and assembling nano-particles. AFM in its tapping mode is composed of a micro-cantilever beam which is under the periodic base excitation. There is nonlinear forces resultant from internal molecular forces exerted on the tip of this beam. To achieve high quality of picture using AFM, the base excitation of system should be intensified, because further amplitude of excitation, gives picture with high horizontal and vertical resolution. An increase in the amplitude of excitation can affect the stability of system where in some cases results in chaotic behavior [29,30] and consequently low quality in achieved pictures. This is the essence why the act of inspecting AFM's performance and eliminating the chaotic behavior is vital. 3.2.1. Governing equations of AFM A forced dynamical system which resembles the performance of AFM in its tapping mode has been introduced and studied in [22]. The cantilever-tip-sample system is modeled by a sphere which is suspended by a spring and damper. Accordingly, the dynamics of this system is formulated by the equation: meq x€ ðt Þ þ beq x_ ðt Þ þ keq xðt Þ ¼ F o cos ðωt Þ þ f IL

Fig. 2. Transient response of the Rossler system during stabilizing its first order UPO. The transient response (dotted line) and the 1st order UPO (bold line).

20

x

10 0 -10 0

20

40

60

80

100

120

140

160

180

y

in which, considering spring-sphere mass-damper model for cantilever, The cantilever tip position is given by x measured from the equilibrium position, ω is the excitation frequency and meq is the equivalent mass at the tip of beam. beq , keq and F o CosðωtÞ represent the equivalent damping coefficient, spring coefficient and exerted force on the base respectively. Finally based on the Lennard-Jones potential [31], f IL that comprises the Van der Waals forces can be described by

200

f IL ¼

10 0 -10 0

20

40

60

80

0

20

40

60

80

100

120

140

160

180

200

100

120

140

160

180

200

20

z

10 0 -10

time (s) Fig. 3. Time series of state variables for the Rossler system obtained through stabilizing its first order UPO.

ð24Þ

A1 R 180ðZ þ xÞ8



A2 R 6ðZ þ xÞ2

ð25Þ

where Z is the distance between the equilibrium position of the cantilever tip point and the sample when only the gravity is acting on it. x is the distance of the case from the fixed frame (Base) and A1 ; A2 are Hamacker constants and, R is the radiuses of the sphere mass which represents the tip point. Eq. (24) in the nondimensional form can be formulated in a similar way as represented in [31]:

ξ€ 1 ðτÞ ¼  δξ_ 1 ðτÞ  ξ1 ðτÞ þ f~ cos ðΩτÞ þ

s~ 6 d d  30ðα þ ξ1 ðτÞÞ8 ðα þ ξ1 ðτÞÞ2 ð26Þ

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A2 R ; Z s ¼ 32 ð2DÞ1=3 ; ω D ¼ 6k eq

keq 2 0 ¼ meq ;

μ¼

beq meq ;

s¼

 1=2 A1 A2

4 Ω ¼ ωω0 ; f~ ¼ ωF20Zs ; d ¼ 27 ; α ¼ ZZs ; s~ ¼ Zss ; δ ¼ ωμ0 Z s ; ξ1 ¼ Zxs ; ξ2 ¼ ω0x_Z s 0

ð27Þ where the derivatives also have been taken with respect to nondimensional time ðω0 t ¼ τÞ where ω0 is the system natural frequency. Substituting τ with t gives:

ξ€ 1 ðtÞ ¼  δξ_ 1 ðtÞ  ξ1 ðtÞ þ f~ cos ðΩtÞ þ f^ IL ðtÞ

ð28Þ

where it can be written in the shape of a 3rd order differential equation as

ξ_ 1 ðtÞ ¼ ξ2 ðtÞ 6 ξ_ 2 ðtÞ ¼  δξ2 ðtÞ  ξ1 ðtÞ þ f~ cos ðΘÞ þ 30ðαsþ~ ξd ðtÞÞ8  ðα þ ξd ðtÞÞ2 1

_ ¼Ω Θ

ð29Þ

1

ξ11F ¼ 0:62199830098610; ξ12F ¼ 0:09047978513394

3.2.2. Stabilizing AFM system's first order fixed point For the parameters:

δ ¼ 0:04;

s~ ¼ 0:3;

α ¼ 0:8;

f~ ¼ 2:0;

Ω ¼ 1;

4 d¼ 27

ð1Þ

 A¼

0

 0:9388

0:1433

0:6135

 0:9206

 ð33Þ

In similar way, the process is accomplished for matrix ð1Þ B. For the 1 2 chosen ζ i , the next cross points, ζ i , are computed for the control signal values, ηj equal to f  0:1;  0:05; 0:05; 0:1g, then building input and output matrices, presented in Eq.(9) the matrix ð1Þ B is obtained. 6

Targeted UPO

4

2

0

-2

ð30Þ

The chaotic behavior has been reported for this dynamical system in [22]. The chaotic attractor region of this system has been plotted in the phase space in Fig. 5, besides its fractal structure on the stroboscopic Poincare map with the specification of ∑ ¼ fðx; θÞ A ℝ2  S1 jθ ¼ 0 A ½0; 2π Þg

ð32Þ

Is this step, the linearized Poincare map around the system fixed point is obtained. By having a 10  10 square mesh with the 1 length of 0:01 around the fixed point and using 100 points as ζ i , 2 the next crossed points ζ i are calculated. Now, regarding Eq.(8) the input and output matrices are built and subsequently through the regression method, the matrix ð1Þ A for the linearized Poincare map represented in Eq.(2) is computed.

ξ

Generally due to the presence of the harmonic base excitation, the governing differential equations of the AFM system have harmonic term which causes the system even in the absence of nonlinear terms, oscillate with a combination of natural and excitation frequencies. Taking into account this fact that the AFM system works in tapping mode, the excitation frequency is close to its natural frequency, consequently the system in the absence of nonlinear force oscillates with a frequency, close to the natural Frequency. At the presence of nonlinear force, when the amplitude of excitation is low, the system still will have oscillation with natural frequency [32], but increase in the excitation amplitude leads to appearance of chaos phenomenon in system [33,34]. In this domain, different frequencies appear dynamically in the system where the resultant perturbed motion is a combination of periodic and semi-periodic responses that is representative of the presence of a rich structure from different frequencies in chaotic attractor region.

Note that the Poincare section in (31) crosses all the system trajectories. The first order unstable periodic orbit (UPO) which has been chosen to be stabilized has the same period as the excitation period, 2Ωπ . Fig. 6 shows the target UPO on the chaotic attractor region plane. The first order fixed point of the system that corresponds to the point that the target UPO crosses the Poincare section is

2

in which

5

ð31Þ

-4

-6 -1

-0.5

0

0.5

1

1.5

ξ

2

2.5

3

3.5

4

1

Fig. 6. The target first order unstable periodic orbit of AFM [23].

Fig. 5. The chaotic attractor of AFM, (a) plotted in phase plane. (b) plotted in Poincare section (with 10,000 points) [23].

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As it was stated, an AFM micro-beam is clamped and in its tapping mode is excited via a piezo-actuator periodically. Suppressing the chaos phenomenon in this system can be done applying the presented linear control method through adding a feedback control signal to the exciting signal, which is created by piezoactuator. It is assumed that both state variables ξ1 ðtÞ and ξ2 ðtÞ, which represent the position and velocity of the tip respectively, are measurable. In this way, the non-dimensional governing equation of motion in the presence of control force u can be written as

ξ€ 1 ðtÞ ¼  δξ_ 1 ðtÞ  ξ1 ðtÞ þ f~ cos ðΩtÞ þ f^ IL ðtÞ þ uðtÞ

ð34Þ

As it was pointed out before, via the stroboscopic Poincare map, the system is quantized in the time t ¼ k2Ωπ ; k ¼ 1; 2; …. Supposing ξ1 ðkÞ and ξ2 ðkÞ are the state variables in the kth cross, the control

1.5

ξ

1

1

uPoincare ðkÞ ¼ Gðx  xf Þ ¼  G1 ½ξ1 ðkÞ  ξ1F   G2 ½ξ2 ðkÞ  ξ2F  k2Ωπ r t o ðk þ 1Þ2Ωπ

uðtÞ ¼ uPoincare ðkÞ

ð35Þ

Using the introduced linear optimal control algorithm, with consideration of weighting parameters W ¼ Identity; Λ ¼ 0:5, and substituting the obtained result into Eq.(18), the optimal linear coefficient, G is acquired: G ¼ ½  0:2236; 0:2141

ð36Þ

Similar to the discussed Rossler system, the couples ðA; BÞ and ðA; W 1=2 Þ for AFM system are also controllable and observable respectively, hence uniqueness of solution for the Riccati equation is assured. The obtained results from the simulation with the parameter value in Eq.(30) have been plotted and shown in Figs. 7 and 8. In order to evaluate the presented algorithm, here we have compared it with another method. In [35] the Pyragas controller has been applied to the same AFM system as our case study. The time delayed control law in the mentioned study was uðtÞ ¼ 0:2½ξ2 ðtÞ  ξ2 ðt  TÞ

0.5 0

low is considered to have the following feedback form:

ð37Þ

the results corresponding to the Pyragas method have been shown in Fig. 9. 0

5

10

15

0

5

10

15

1.5 1

ξ

2

0.5 0 0.3 0.2

u

0.1 0 -0.1

0

2

4

6

8

10

12

14

16

Time (s) Fig. 7. Time series of the control signal and quantized state variables on the Poincare section.

Fig. 9. Stabilizing the UPO of the AFM system using the Pyragas controller.

2

4

1.5

3

ξ1

1 0.5

2

0 -0.5

1 70

75

80

85

ξ2

-1 65

0

4

-1 2

ξ2

-2 0

-3

-2 -4

65

70

75

time

80

85

-4 -1

-0.5

0

0.5

ξ

1

1.5

2

1

Fig. 8. Stabilized UPO of the AFM system. Phase plane behavior (right) time series of state variables (left).

Please cite this article as: Merat K, et al. Linear optimal control of continuous time chaotic systems. ISA Transactions (2014), http://dx. doi.org/10.1016/j.isatra.2014.01.003i

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As it can be seen, both methods, including the presented approach in this study and the Pyragas method have stabilized the target UPO, but the comparison indicates, even though the Pyragas technique has stabilized the system faster, but the magnitude of control parameter in our introduced technique is lower. 4. Conclusion In this paper, the first order periodic orbit stabilization of a continuous time system was accomplished using a suboptimal controller. At first, using the concept of Poincare map the system was converted to a discrete time system. Because of the difficulty in extraction of the nonlinear Poincare map, the regression method was used for extracting a linear approximation of the Poincare map around a fixed point. By solving the Riccati equation, a linear optimal control algorithm in infinite time horizon was obtained for stabilizing the discrete chaotic system on its first order fixed point. At the end, the presented procedure was applied on the continuous Rossler system which is an autonomous system and also on the chaotic AFM system which is non-autonomous. The simulation results showed that this algorithm can stabilize continuous time chaotic systems on their UPOs. Finally the performance of the proposed technique was compared with the Pyragas method. References [1] Li TY, York JA. Period three imply chaos. Am Math Mon 1975;82:985–92. [2] Ott E, Grebogi C, Yorke JA. Controlling chaos. Phys Rev Lett 1990;64:1196–9. [3] Shinbort T, Ott E, Grebogi C, Yorke JA. Using chaos to direct trajectories to targets. Phys Rev Lett 1990;65:3215–8. [4] Shinbort T, Grebogi C, Ott E, Yorke JA. Using small perturbation to control chaos. Nature 1993;386:411–7. [5] Pyragas K. Continuous control of chaos by self-controlling feedback. Phys Lett A 1992;170:421–8. [6] Li S, Li Y, Liu B, Murray T. Model-free control of Lorenz chaos using an approximate optimal control strategy. Commun Nonlinear Sci Numer Simul 2012;17(12):4891–900. [7] Zhang J, Tang W. Optimal control for a class of chaotic systems. J Appl Math. 2012;2012. [8] Jayaram A, Tadi M. Synchronization of chaotic systems based on SDRE method. Chaos Solitons Fractals 2006;28:707–15. [9] Hilborn RC. Chaos and nonlinear dynamics. New York: Oxford University Press; 1994. [10] Cao YJ A. Nonlinear adaptive approach to controlling chaotic oscillators. Phys Lett A 2000;270:171–6.

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Please cite this article as: Merat K, et al. Linear optimal control of continuous time chaotic systems. ISA Transactions (2014), http://dx. doi.org/10.1016/j.isatra.2014.01.003i