Chaos Synchronization of two Uncertain Chaotic System Using Genetic Based Fuzzy Adaptive PID Controller

Transcription

1 The Journal of Mathematics and Computer Science Available online at The Journal of Mathematics and Computer Science Vol. No.4 () Chaos Snchronization of two Uncertain Chaotic Sstem Using Genetic Based Fuzz Adaptive PID Controller Yaghoub Heidari,*, Soheil Salehi Alashti, Rouhollah Maghsoudi 3, Received: August, Revised: October Online Publication: December Department of Electrical, Nour Branch, Islamic Azad Universit, Nour, Iran Babol Noshirvan Universit of Technolog, Department of Computer, Nour Branch, Islamic Azad Universit, Nour, Iran Abstract A Genetic based Fuzz PID controller has been proposed to snchronization task of chaotic sstems in which one sstem has been considered as "master" whilst the other sstem has been treated as "slave" (a perturbed sstem with uncertaint and disturbance). Three PID control gains k, k, and k, will be updated online. An p adequate adaptation mechanism is used to minimize the sliding surface error with appropriate adaptive law. Using the gradient method, coefficients of the PID controller are updated. A supervisor controller has also been used to provide the stabilit. The proposed method has been found with a significant performance when it was implemented on the Van Der Pol oscillator chaotic equations. Kewords: Fuzz, Chaos snchronization; supervisor controller; Van der Pol; Adaptive Control. i d,* Corresponding Author. address: masudheidari@ahoo.com (Phone: ) - Babol Noshirvan Universit of Technolog, 3- Computer Department of Islamic Azad Universit of Nour (Phone: ) 73

2 . Introduction Feasibilit of control and snchronizing chaotic sstems in various tpes has attracted significant interest in the last few ears [-6]. Chaos is a peculiar random-like behavior in deterministic sstems. In order to achieve the snchronization, a closed loop nonlinear control sstem, which obtains signals from the master and controls the slave accordingl, will be designed. In the literature, different non linear methods such as periodic parametric perturbation [7, 8], driveresponse snchronization [9], adaptive control [ 4], variable structure (or sliding mode) control [5 7], backstepping control [8, 9], and H control [], have been used for snchronization. Chaos snchronization can also be considered as a design problem of a feedback law for an observer, using known information of the plant. This is to ensure that the controlled receiver is snchronized with the transmitter. In [] an adaptive robust PID controller of a chaotic sstem is presented. The aim is to improve and develop such a method to snchronize an unknown slave dnamic with a master sstem. A fuzz sliding-mode control (FSMC) strateg has been proposed for uncertain chaotic sstems in [], in which a fuzz controller is used to replace the discontinuous sign function of the reaching law in traditional sliding-mode control (SMC). An active sliding mode control for snchronizing two chaotic sstems with parametric uncertaint was proposed in [3]. An algorithm to determine parameters of active sliding mode controller, during snchronization of different chaotic sstems has been studied in [4]. In [5] an adaptive sliding mode controller is presented for class of master slave chaotic snchronization sstems with uncertainties. Fuzz sliding mode control has been used in man papers [6-9]. In this paper, parameters of PID controller are tuned adaptivel b the gradient method, according to a proper designed sliding surface. To guarantee the convergence, a supervisor control using a suitable Lapunov function has been designed. As a side effect, the signum function in the supervisor controller can cause chattering phenomenon in the control signal. To reduce this effect a Genetic Based Fuzz Controller (GBFC) will be proposed. The snchronization procedure and the stabilit requirements will be shortl introduced. This paper is organized as follows: In section, a quick introduction to the proposed method for snchronization of the chaotic sstem in presence of uncertaint is addressed. Section 3 is devoted on investigation of the application of the proposed method on chaotic groscope, described b Van Der Pol oscillator dnamic via simulation, when the sstem is perturbed with uncertainties. Finall, the result analsis and conclusion will be addressed in section 4.. Snchronization of uncertain chaotic sstem using adaptive PID controller Model reference adaptive control or model following task is one of usual tasks in control engineering which is called Master-Slave problem in electrical engineering. The ultimate goal of this task is to design a mechanism, which forces the slave, to follow the master regardless of some possible incompatibilit. If the slave is chaotic, uncertain or of a reduced order, the snchronization problem takes a complex form. Let us consider the following dnamic as a master: x x x f ( X, t ) Where, X is a two dimensional state vector as X [ x, x] T and f (.) is an unknown function. Suppose the following dnamic as slave in the snchronization task: () 74

3 f ( Y, t) f ( Y, t) d( t) u( t) () [, ] T is the output vector, f (.) and dt () denote uncertaint and disturbance respectivel. The control signal ut () has a dut to snchronize the slave sstem () with the master dnamic in (). A schematic diagram of the snchronization, which is eventuall inspired from the adaptive control terminolog, is shown in Figure (). The difference between master and slave will be minimized via an adaptive controller. The snchronizer designation consists of the following sections:. PID controller. A supervisor controller to guarantee the stabilit. 3. A method to update PID coefficients. The gradient or a sliding surface will be such an update mechanism. Slave Sstem Master Sstem + - GBF Supervisor Controller + + Adaptive PID Controller Adaptation Law Figure : Schematic diagram of a snchronization mechanism The error vector of two states is defined for b e x, i,. Suppose that we can choose i i i a gain vector K = [k, k ] T such that roots of s k s k are in the left hand side of the s-plane. A feedback linearization technique is a possible choice, to combat both uncertaint and disturbance, which is as follows: e k e k e x k e k e (3) u ( t ) f ( Y, t ) f ( Y, t ) d ( t ) x k e k e FL.. It is of the goal to reduce the error asmptoticall. Therefore, the slave must follow the master dnamicall. It should be noted that nd sstem is unknown and the states and error are the onl available signals. These ma have been infected b some uncertainties and perturbations. The following assumptions are required to design an adaptive PID. Assumption. Let the constraint set for the state X is defined as: { Y : Y M }, (4) 75

4 Where, M is a pre-specified upper limit. It is desired that the state trajector of sstem X never reaches to the boundar, during the control action. For simplicit in analsis, we choose M. Let us assume the control input of the form: X u upid us where, u denotes the signal produced b PID controller which is defined as follows: PID t et () u PID K Pe ( t ) K I e( ) d K (6) D t Furthermore, u S supervisor controller signal [], is added to guarantee the stabilit. Then a sliding mode control is proposed. Accordingl, a proper adaptation law, based on the gradient method, is designed to minimize the error. Let us redefine the state in equation (3) using the control law in equation (5), b the following statement: f ( Y, t ) f ( Y, t ) d ( t ) u u f ( Y, t ) f ( Y, t ) d ( t ) u u u u PID x k e k e [ u u u ] PID s F. L. This immediatel follows that: e e s PID s F. L. F. L. e k e k e [ u PID us u F. L. ] The following gain vectors: A, B (9) k k Establish the state feedback. Therefore, the error dnamic will be of the form: E AE B[ u () F. L. upid us ] Where, E is[ e, e ]. The energ of the sstem will be candidate as a Lapunov function according to: T () V E PE Where p is a smmetric and positive definite matrix. This will be found via following Lapunov equation: T A P PA Q () Furthermore, Q is also a smmetric and positive definite matrix. This ma be designed b the user in advance, to meet such other criteria. Definition ofvm as: (3) min( ( ))( ) V M P M X leads us to the following equation: V min( ( P)) E min( ( P))( Y X ) (4) min( ( ))( ) P M X V M (5) (7) (8) 76

5 Where min( ( P)) denotes the minimum value of eigenvalue of matrix P. For the case V VM obtains X M. B taking the time derivative of () and using () leads us to: T T T V e E ( A P PA) E E PB[ uf. L. upid us ] T T E QE E PB[ uf. L. upid us ] (5) T T T E QE E PB ( uf. L. upid ) E PBuS An achievable approximation of inequalities f (.) f (.), f (.) f (.)and d( t) with aiding equation (3) provides a lower limit for the feedback linearization law. This immediatel follows that: u () t f f x k e k e (6) F. L. u u For the other case whenv VM, the following supervised controller guarantees the necessit condition of V. (7),, V VM us T sgn( E PB) f u fu x ke ke, V VM In order to have an adaptation mechanism to update PID coefficients, reference signal r can be defined as: (8) r x ke ke and the sliding surface b: S (9) r When the sliding mode is activated i.e. S, therefore we have: () r Replacing equation () in (8) approaches: e () ke ke A necessar and sufficient condition for the convergence needs the error to approach zero when the time tends to infinit. Furthermore, to meet the sliding condition, the following Lapunov function will be candied to realize a stable controller. V S () Therefore, differentiation of V would be as: V SS (3) It guarantees the convergence when St () as t. A proper adaptation mechanism needs to minimize the sliding condition SS. The gradient search algorithm is calculated in the opposite direction of the energ flow. Therefore, the convergence of the sstem can be obtained b a satisfactor adjustment of PID coefficients. Furthermore, it is quite intuitive to choose SS as an error function. From (9) and using (), obtains: S r f ( Y, t) f ( Y, t) d( t) u( t). (4) r Replacing (5) in (4) and multipling both sides of (4) in S, there will be: SS S[ f ( Y, t) f ( Y, t) d( t) u( t) r ]. (5) u u one 77

6 Appling the gradient method and the chain principle to equations (6) and (5), the adaptation law for determining PID parameters will be obtained as: SS SS upid K () P Se t (6) KP upid KP SS SS u t PID K I S e () d (7) K I upid K I SS SS upid d K D S e () t (8) K D upid K D dt where, is the learning rate. It should be noted that selection of and/or the initial setting of PID coefficients is ver crucial for the stabilit []. The discontinuit of the sign function in (7) causes the chattering phenomenon. Using saturation instead of a simple sign function in (3), to reduce the chattering, in (7) obtains: us Ksat( ) f u fu x ke ke (9) where: T E PB (3) and sgn( ) if (3) sat( ) if However to have a better compromise between small chattering and good tracking precision in the presence of parameter uncertainties, a fuzz controller is proposed. 3. Fuzz control The main advantage of the fuzz controller is its heuristic design procedure as a model-free approach. The combination of fuzz control strateg and (9) becomes a feasible approach to preserve advantages of these two approaches [3, 3] especiall to reduce the chattering. A fuzz controller is introduced to develop (9) control law. The IF-THEN rules of fuzz controller can be described as: R: If is NB then K is PB R: If is NM then K is PM R3: If is NS then K is PS (3) R4: If is ZE then K is ZE R5: If is PS then K is NS R6: If is PM then K is NM R7: If is PB then K is NB Where NB, NM, NS, ZE, PS, PM, PB are the linguistic terms of antecedent fuzz set. The mean Negative Big, Negative Medium, Negative Small, Zero, Positive Medium Positive Small and Positive Big, respectivel. The fuzz membership function for each fuzz term should be a proper design factor in the fuzz control problem. A general form is used to describe these fuzz rules as: Ri: If is Ai, then K is Bi (33) 78

7 Where Ai has a triangle membership function (depicted in figure ) and Bi is a fuzz singleton. The modified controller (9), invites an idea to restrict the width of boundar laer Φ, and uses a continuous function (3) to smoothen the control action. Therefore, the problem of the discontinuousness of the sign function can be solved, and the chattering phenomena will be decreased. From the control point of view, the parameters of structures should be automaticall modified b evaluating the results of fuzz control in (3). Hitting time and chattering phenomenon are two important factors that influence the performance of proposed Controller. The width of boundar laer, Φ, will influence the chattering magnitude of control signal, and the gain K, will influence how soon the states of Slave reach to that of in Master. NB NM NS ZE PS PM PB -Φ/4 Φ/4 z (a) NB NM NS ZE PS PM PB k k k /4 k /4 k (b) Figure. (a): The input membership function of the Fuzz supervisor Controller (b): The output membership Fuzz supervisor Controller The reaching time can also be reduced via a suitable selection of parameter K. GA is used to search for a best fit for k,, k, k parameters in (9). The tracking error and existence of chattering in the controlled response are chosen as a performance index to select the parameters. The proposed fitness function is defined in such a wa that the selected parameters can force the state to reach together fast and then keep the control signal with less chattering and snchronization error. In this manner, GA is used to search the parameter space to find appropriate values of the parameters, k,, k, k in (9). A fitness function is defined as follows: W e e W dt (34) wheree, e is defined in (3), and W and W are the weight factors. The design parameters of the GBFCassociated with the above control rules are specified as follows: Population size = 6 Crossover probabilit =.8 (35) Generations = 7 Mutation probabilit =. k belongs to [, ] belongs to [, ] k K 79

8 (t) These are popular setting and can be easil found b a minor modification. The Fuzz designed controller has applied to some case studies in the next sections. 4. Simulation 4. Snchronization of Uncertain Hard Spring Φ6 -Van der Pol Oscillator 4.. Sstem description Consider the following sstem namel Φ 6 Van Der Pol oscillator [34] as another case stud: 3 5 x ( x ) x (36) x x x fcost The above sstem for f, is a complete oscillator sstem. Increasing f appears chaotic 6 characteristic of the sstem [34]. In Figure (3), phase portrait of the chaotic sstem namel φ Van Der Pol oscillator with f 4.5 has been shown. Simultaneousl, parameters of sstem have been selected as M.4,,., wo.46, w.86. In Figure (3), chaotic behavior of the sstem is completel evident. Correspondingl, parameter f is treated and designed as an uncertaint of the sstem. To show the effectiveness of the proposed controller, the procedure is implemented here. 6 phase plane (t) Figure 3: Phase portrait of Φ6-Van Der Pol oscillator 4.. Implementation The master sstem is defined b the following dnamic: x x 3 5 x ( x ) x x x x Similarl, the slave is defined as follows: 3 5 ( ) f cost (4) (4) 8

9 x, Sets of initial conditions of master and slave sstems are respectivel defined as x()., x(). and ().5, (). Considering the disturbance d( t) f cost and 3 5 f ( x, x ) M ( ) w, an upper limit of related functions are obtained as follows: The value of f (.) ( ) M 3 5 ( ) 3 5 d ( t ) f cost f is similarl selected to be large enough such that the sstem state x never reach the boundar of. Initial setting of PID coefficients are equal to k (), k (), () K and the learning rate has been selected as. Also, k and k have been assigned as.5 and respectivel. In comparison with equation (9), the state feedback matrix will be found as: A (43).5 Regarding to equations (9) and (4), choosing Q, ields the smmetric and definite positive.375 matrix p as: p.75. Parameters of the sstem, similar to section 3.., have been selected as f 4.5, M.4,,.,.46, w.86. Appling the sign function (7) to the master and slave sstem of Φ6 Van Der Pol results the oscillation, which have been shown in Figures (4) to (6). In Figure (4), snchronization of x, and x,, in Figure (5), the sliding surface and ultimatel in Figure (6). Simulation results show the chattering phenomenon in the control signal, this can wear and tear the actuators. To reduce this undesired effect in control signal the GBFC has applied to snchronize the master and slave sstem of Φ6 Van Der Pol. The relevant responses are shown in Figure (7) to (). Figure (7) shows the snchronization of x, and x,, whereas, Figure (8) shows the sliding surface and ultimatel in Figure (9), the control signal can be seen. The snchronization error has also been seen in Figure (). From Figure (9) can be seen that the control signal is smoother than that of in the sign function application. It should be noted that in this case, the control i.e. ut (), has been activated at t=5s. The performance of the GBFC is seen much improved. p D 3 Slave sstem Master sstem

10 u(t) S(t) x, 6 4 Slave sstem Master sstem Figure 4: Time response of controlled chaotic Φ6 van der pol snchronization sstem Surface Figure 5: The sliding surface behavior Control Signal Figure 6: The control signal with sign function 8

11 S(t) x, x, 3 Slave sstem Master sstem Slave sstem Master sstem Figure 7: Time response of GBFC snchronization Φ6 van der pol Surface Figure 8: The sliding surface behavior 83

12 e e u(t) 5 Control Signal t Figure 9: The control signal with GBFC Figure: The snchronization error of Master and Slave Φ6 van der pol 5. Conclusion A methodolog for snchronization of chaotic sstem in presence of uncertaint has been proposed. To deal the uncertaint, an Adaptive PID controller has been used on the slave sstem. A supervisor control has been used to guarantee the stabilit of the global sstem through energ like Lapunov function. Furthermore, a sliding mode controller has been designed using another Lapunov function to update the coefficients of the PID controller in an adaptive schema. The gradient method has been chosen to establish the adaptation law. The sign function in the supervisor controller cause the chattering phenomenon; to reduce this undesired effect a GBFC was proposed. The proposed snchronization method has been simulated on Φ 6 Van Der Pol oscillator. The results confirm the significance of the proposed control technique. 84

13 References [] Pecora, L.M. and T.L. Carroll, 99. Snchronization in chaotic sstems. Phs. Rev. Lett., 64: [] Yassen, M.T., 5.The Chaos snchronization between two different chaotic sstems using active control. Chaos Solitons Fractals, 3() :3. [3] Abdous, F., A. Ranjbar N., S. H. Hosein Nia, A. Sheikhol Eslami and B. Abdous, 8. The Hopf Bifurcation control in Internet Congestion Control Sstem. Second International Conference on Electrical Engineering (ICEE), Lahore (Pakistan(. [4] Abdous, F., A. Ranjbar N., S. H. Hosein Nia and A. Sheikhol Eslami, 8. Chaos Control of Voltage Fluctuations in DC Arc Furnaces Using Time Dela Feedback Control. Second International Conference on Electrical Engineering (ICEE), Lahore (Pakistan). [5] Xie, Q. and G. Chen, 996.Snchronization stabilit analsis of the chaotic Rossler sstem. Int. J. Bifur. Chaos 6():53-6. [6] Kapitaniak, T., 995.Continuous control and snchronization in chaotic sstems. Chaos Solitons Fractals, 6(3): 37. [7] Astakhov, VV., VS. Anishchenko, T. Kapitaniak and AV. Shabunin, 997.Snchronization of chaotic oscillators b periodic parametric perturbations. Phs D 9: 6. [8] Blazejczk-Okolewska, B., J. Brindle, K. Czolcznski and T. Kapitaniak,.Antiphase snchronization of chaos b noncontinuous coupling: two impacting oscillators. Chaos, Solitons & Fractals :83 6. [9] Yang, XS., CK. Duan and XX. Liao,999. A note on mathematical aspects of drive-response tpe snchronization. Chaos Solitons & Fractals : [] Wang, Y., ZH. Guan and X. Wen, 4. Adaptive snchronization for Chen chaotic sstem with full unknown parameters. Chaos, Solitons & Fractals 9 : [] Chua, LO., T. Yang, GQ. Zhong and CW. Wu, 996. Adaptive snchronization of Chua s oscillators. Int J. Bifurc. Chaos 6(): 89. [] Liao, TL., 998. Adaptive snchronization of two Lorenz sstems. Chaos, Solitons & Fractals 9 : [3] Lian, KY., P. Liu, TS. Chiang and CS. Chiu,. Adaptive snchronization design for chaotic sstems via a scalar driving signal. IEEE Trans Circuits Sst I 49():7 7. [4] Wu, CW., T. Yang and LO. Chua, 996. On adaptive snchronization and control of nonlinear dnamical sstems. Int J Bifurc Chaos 6: [5] Fang JQ., Y. Hong and G. Chen,999.Switching manifold approach to chaos snchronization. Phs Rev E 59: [6] Yin, X., Y. Ren and X. Shan,.Snchronization of discrete spatiotemporal chaos b using variable structure control. Chaos, Solitons &Fractals 4 :77 8. [7] Yu, X. and Y. Song,. Chaos snchronization via controlling partial state of chaotic sstems. Int J Bifurc Chaos (6): [8] Wang, C. and SS. Ge,. Adaptive snchronization of uncertain chaotic sstems via backstepping design. Chaos, Solitons & Fractals : [9] Lu, J. and S. Zhang,. Controlling Chen s chaotic attractor using backstepping design based on parameters identification. Phs Lett A 86:

14 [] Sukens, JAK., PF. Curran, J. Vandewalle, 997. Robust nonlinear snchronization of chaotic Lurenze sstems. IEEE Trans Circuits Sst I 44() : [] Wei-Der, Ch. and Y. Jun-Juh, 5.Adaptive robust PID controller design based on a sliding mode for uncertain chaotic sstems. Chaos, Solitons & Fractals, 6(): [] Yau, H-T and Ch. Chieh-Li, 6.Chattering-free fuzz sliding-mode control strateg for uncertain chaotic sstems. Chaos, Solitons and Fractals 3: [3] Zhang, H., X-K. Ma and W-Z. Liu, 4.Snchronization of chaotic sstems with parametric uncertaint using active sliding mode control. Chaos, Solitons and Fractals : [4] Tavazoei, M. S. and M. Haeri, 7.Determination of active sliding mode controller parameters in snchronizing different chaotic sstems. Chaos, Solitons and Fractals 3: [5] Yau, H-T., 4. Design of adaptive sliding mode controller for chaos snchronization with uncertainties. Chaos, Solitons and Fractals, : [6] Liang, C-Y. and J-P. Su, 3. A new approach to the design of a fuzz sliding mode controller. Fuzz Sets and Sstems 39: 4. [7] Hung, JY.,995. Magnetic bearing control using fuzz logic. IEEE Trans Indust Appl 3: [8] Li, THS. and MY. Shieh,. Switching-tpe fuzz sliding mode control of a cart-pole sstem. Mechatronics : 9 9. [9] Li, JH., THS. Li and TH. Ou, 3. Design and implementation of fuzz sliding-mode controller for a wedge balancing sstem. J. Intell. Robotic Sst., 37(3): [3] Delavari, H. and A. Ranjbar, 7. Robust Intelligent Control of Coupled Tanks. WSEAS International Conferences, Istanbul, Turke, Ma -6. [3] Delavari, H. and A. Ranjbar, 7. Genetic-based Fuzz Sliding Mode Control of an Interconnected Twin-Tanks. IEEE Region 8 EUROCON 7 conference, Poland, September, [3] Chen, H.K.,. Chaos and chaos snchronization of a smmetric gro with linearplus-cubic damping. Journal of Sound and Vibration, 55: [33] Yau, H-T., 8. Chaos snchronization of two uncertain chaotic nonlinear gros using fuzz sliding mode control. Mechanical Sstems and Signal Processing, (): [34] F. M. Moukam Kakmeni and S. Bowong, C. Tchawoua and E. Kaptouom, 4.Chaos control and snchronization of a Φ6-Van der Pol oscillator. Phsics Letters A, 3(5-6):