Control of Chaos in Brushless DC Motor

2014 International Conference on Control, Instrumentation, Energy & Communication(CIEC)

41

Control of Chaos in Brushless DC Motor Design of Adaptive Controller Following Back-stepping Method

Partha Roy

Susanta Ray

Samar Bhattacharya

Dept. of Electrical Engineering Jadavpur University Kolkata-700032, India

Dept. of Electrical Engineering Jadavpur University Kolkata-700032, India

Dept. of Electrical Engineering Jadavpur University, Kolkata-700032, India

Abstract- This paper deals with the chaotic behavior of brushless dc(BLDC) motor and its adaptive control. The chaotic behavior of BLDC motor has a direct impact on the reliability andstability of the system with BLDC motor. The proposed adaptive controller is to suppress the chaotic behavior of BLDC motor and to terminate the motor in zero state. The adaptive back stepping controller is designed with modified care to obtain the guaranteed global stability. The simulation results of the controlled system show that the system can be escaped from the chaotic states and stay in zero states. Due to its inherent simplicity the controller can be implemented in a simple hardware. Key Words- BLDC Motor, Chaos Control, Back-stepping Method, Global Stability.

I.

INTRODUCTION

In 1963 Edward Lorenz found the first chaotic attractor in a three dimensional autonomous system while studying atmospheric convection [1]. Chaos is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions [2]. The chaos is an inherent phenomenon of the nonlinear systems. It has become a common practice to neglect it due to its unpredictable nature and unwanted oscillations. Yet the chaos is very much useful for the several physical applications [3]. The mathematical model of BLDC motor is an expression of nonlinear system and we face chaotic behavior of this system in different conditions [4]-[7]. To control the BLDC motor in chaotic condition is an attractive subject of the researchers, and there are several papers regarding this and from the different angle of views [8]-[12].

The back stepping technique is one of the most popular design methods for adaptive nonlinear control especially the control of chaos as it guaranties global stability. The back stepping control method can be implemented for the systems with strict feedback forms ; yet a class of non-strict feedback system can also be controlled following back stepping method [15], [16]. The objective of this paper is to terminate the chaotic behavior of BLDC motor and to fix it at simply idle state. To achieve this we develop our controller following modified back stepping method. This controller does not require any parameter of the considered model. It can efficiently estimate the parametersrequired for its controller. The designed controller works for very small duration, and this duration can also be adjusted by the gain parameter. As the idle state is also a fixed point to the mathematical model of the BLDC motor, after getting this idle point we do not further require the controller in action. The rest of the paper is organized as follows. In section II, we present the dynamic model of BLDC motor and we realize the chaotic behavior of the motor. In section III, we define the basics of back stepping control method. In section IV, we discuss the designed controller and its corresponding calculations. In section V we show the simulation results. This paper ends with section VI with a brief conclusion. II. DYNAMIC MODEL OF BLDC MOTOR The dynamic model of BLDC motor as in literatures [5] and [6] can be written on the d-q axis as:

Ouraim is not only to eliminate the chaotic behavior but to terminate the BLDC motor into zero state despite of its initial states, thus we get globally stable state after control.

ௗ௜೏

The control of chaotic state can be achieved by OttGrebogi-York (OGY)method [13], but this method is strict enough in real practice. For the case of parameter variations of the chaotic systems this method is hard to implement efficiently. In 1992 K. Pyragas introduced time delay feedback control method [14], this simple method does have excellent adaptability but there is a difficulty to calculate the proper time delay constant.

ௗ௧

978-1-4799-2044-0/14/$31.00©2014IEEE

ௗ௧ ௗ௜೜ ௗఠ ௗ௧

= =

ଵ ௅೏ ଵ ௅೜

[ ܸௗ െ ܴ݅ௗ ൅ ߱‫ܮ‬௤ ݅௤ ] [ܸ௤ െ ܴ݅௤ െ ߱‫ܮ‬ௗ ݅ௗ െȦߖ௥ ]

(1)

ଵ

= [݊௣ ߖ௥ ݅௤ ൅ ݊௣ ൫‫ܮ‬ௗ െ ‫ܮ‬௤ ൯݅ௗ ݅௤ െ ܶ௅ െ ߚ߱] ௃

Where, the variables with index q are referred to the quadratureaxis and the variables with index d are referred to the direct axis.


‫ܮ‬௤ and ‫ܮ‬ௗ are the stator inductances andܴis the stator resistance. The remaining variablesߖ௥ ǡ ߚand ‫ ܬ‬correspond to the permanent magnet flux, the viscous friction coefficient and the polar moment inertia respectively. The number of pole pairs is represented by݊௣ , angular velocity is represented by ߱ and finally external load torque is represented by ܶ௅ .

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considered parameters asμ= 1.00, ı = 5.58, Ȗ= 19.55 and ‫ ݒ‬ൌ Ͳ the motor experiences chaotic behavior. With the initial conditions ሾ‫ݔ‬ଵ ‫ݔ‬ଶ ‫ݔ‬ଷ ሿ் =ሾ͵ʹͳሿ் , Fig. 1 to Fig. 3 clearly illustrate the strange attractors of the BLDC motor.

Conventionally, the system (1) is expressed following simpler form after transformation. Let, ܾ݇ T: = Ͳ Ͳ

Ͳ ݇ Ͳ

ߖ

Ͳ Ͳ

; where, b: =

ோ

௅೜

௅೏

௅೜ ‫ݒ‬

ఉ௅

; k: =

ఉோ ௅೜ ௡೜ అ೜

‫ݒ‬

ߛǣ=‫ ݎܮܭ‬Ǣ ߪǣ= ோ௃೜ ; ‫ݑ‬ௗ ǣ=ܴ݇‫ݑ ; ݍ‬௤ ǣ=ܴ݇‫ݍ‬ ‫ݍ‬

݊‫ ʹ݇ ʹ ݍܮܾ ݌‬൫‫ ݀ܮ‬െ‫ ݍܮ‬൯

‫ܶ ʹݍ ܮ‬

‫ʹܴܬ‬

‫ʹܴܬ‬

‫ ݒ‬: =

variables as ( Ǥǁ) =ܶ

෥‫ ܮ‬ൌ ;ܶ

ିଵ

‫ܮ‬

ܴ‫ݐ‬

; ‫ݐ‬ǣƴ= and the state ‫ݍܮ‬

Fig. 1. Phase plane plot for the transformed angular velocity vs. quadratic axis current i.e. Ȧ vs. ݅௤ .

(.)

Performing these transformations we get system (1) in the dimensionless form as: ௗపǁ ೏ ௗ௧ሖ ௗపǁ ೜ ௗ௧ሖ ෥ ௗఠ ௗ௧ሖ

= ‫ݑ‬ௗ െ Ɋଓǁௗ ൅ ߱ ෥ଓ෥௤ = ‫ݑ‬௤ െ ଓǁ௤ െ ߱ ෥ଓ෥ௗ ൅ ߛ߱ ෥

(2)

= ߪሺଓ෥௤ െ ߱ ෥ሻ + ‫ݒ‬ଓ෥ௗ ଓ෥௤ െ ܶ෩௅

Where, μ, Ȗ, ı and ‫ ݒ‬are the structure parameters of the motor dynamic system after transformations. And‫ݑ‬ௗ is an input voltage on the direct axis, ‫ݑ‬௤ is input voltage on the quadratic axis and ܶ෩௅ is the load torque after transformation,ଓǁௗ is current on the direct axis and ଓǁ௤ is current on the quadratic axis and ߱ ෥ is the rotational speed after transformation. The state space ܶ ෥ ሿ as: model can be developed by defining ܺǣ ൌ ሾ݅ሚ݀ ݅ሚ‫߱ ݍ‬ ‫ݔ‬ଵሶ = െμ‫ݔ‬ଵ ൅ ‫ݔ‬ଷ ‫ݔ‬ଶ ൅ ‫ݑ‬ௗ ‫ݔ‬ଶሶ = െ ‫ݔ‬ଶ െ ‫ݔ‬ଷ ‫ݔ‬ଵ ൅ ߛ‫ݔ‬ଷ ൅ ‫ݑ‬௤ ‫ݔ‬ଷሶ = െ ߪሺ‫ݔ‬ଷ െ ‫ݔ‬ଶ ሻ െ ܶ෨௅ ൅ ‫ݔݒ‬ଵ ‫ݔ‬ଶ

(3)

This dynamical model can be compared with the classical Lorenz chaotic system model [1] and Chen’s chaotic system model [17]. It is observed that there are similarities in between the BLDC motor model and the Lorenz and Chen’s models, but their topologies are not extremely equivalent. The chaotic behavior can be realized following the characteristic exponents of the nonlinear system. Calculations of Lyapunov exponents, Poincare map, power spectrum realization and the fractal dimension analysis are the conventional methodologies to realize the systems chaotic attitudes [2], [18]. We prefer to utilize the Lyapunov exponents to realize the chaotic mode of the system due to its straight forwardness indicating the range of variable parameters.According to [12] when the transformed BLDC motor model is assumed to be free from the external inputs such as the direct axis transformed input voltage (‫ݑ‬ௗ ), quadratic axis transformed input voltage (‫ݑ‬௤ ) and the external load torque (ܶ෨௅ ) with the remaining

Fig. 2. Phase plane plot for the transformed angular velocity vs. direct axis current i.e. Ȧ vs. ݅ௗ .

Fig. 3.Phase plane plot for the transformed direct axis current vs. quadratic axis current i.e.݅ௗ vs. ݅௤ .


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More specifically it can be written as: ‫ݕ‬ଵሶ = ߪሺ‫ݕ‬ଶ െ ‫ݕ‬ଵ ሻ ‫ݕ‬ଶሶ = െ ‫ݕ‬ଶ െ ‫ݕ‬ଵ ‫ݕ‬ଷ ൅ ߛ‫ݕ‬ଵ ൅ ‫ݑ‬Ǥ ‫ݕ‬ଵ ‫ݕ‬ଷሶ = ‫ݕ‬ଵ ‫ݕ‬ଶ െ ‫ݕ‬ଷ

(8)

Our objective is to define ‫ݑ‬without knowing the model parameters. In order to design the controller following back stepping method let us follow the following steps. In the Lyapunov based control technique we follow the Lyapunov function ܸ and we define derivative of this Lyapunov function ܸሶ as uniformly negative definite by choosing proper controller [15], [16], [18], [19]. Step 1: The Lyapunov function ܸଵ can be defined as:

Fig. 4. Time response of transformed angular velocity ɘ

ଵ

Fig. 1 to 3 illustrate the butterfly effects of the attractors of the BLDC motor and it obviously highlight the chaotic behavior of BLDC motor. In Fig. 4 the time response of transformed frequency of the BLDC motor is plotted. Our specific objective is to terminate this chaotic behavior and to set the states into the idle point. III.THE PROPOSED CONTROLLER In this section we discuss the chaos control of BLDC motor following improved back stepping algorithm. Back stepping design is a systematic Lyapunov based control technique, which can be applied to strict feedback systems, purefeedback systems and block strict-feedback systems [15],[16]. The transformed mathematical model of BLDC motor without any external inputs can be represented as: ‫ݔ‬ଵሶ = െμ‫ݔ‬ଵ ൅ ‫ݔ‬ଷ ‫ݔ‬ଶ ‫ݔ‬ଶሶ = െ ‫ݔ‬ଶ െ ‫ݔ‬ଷ ‫ݔ‬ଵ ൅ ߛ‫ݔ‬ଷ ‫ݔ‬ଷሶ = െ ߪሺ‫ݔ‬ଷ െ ‫ݔ‬ଶ ሻ ൅ ‫ݔݒ‬ଵ ‫ݔ‬ଶ

(4)

ଶ

Thetime derivative ofܸଵ is as: ܸଵሶ ൌ ‫ݕ‬ଵ ‫ݕ‬ଵሶ , Substituting the expression of ‫ݕ‬ଵሶ we get ܸଵሶ ൌ ߪ‫ݕ‬ଵ ሺ‫ݕ‬ଶ െ ‫ݕ‬ଵ ሻ, ܸଵሶ ൌ െߪ‫ ݕ‬ଶଵ ൅ ߪ‫ݕ‬ଶ ‫ݕ‬ଵ , Now ‫ݕ‬ଶ should be considered as‫ݕ‬ଶௗ௘௦ , the desired value of‫ݕ‬ଶ . In this case it is obvious that if ‫ݕ‬ଶௗ௘௦ ൌ Ͳthe expression of ܸଵሶ becomes uniformly negative definite (UND), it certainly makes this part of the system asymptotically stable. Step 2: We define a new variable as ‫ݖ‬ଵ where ‫ݖ‬ଵ ൌ ‫ݕ‬ଶ െ ‫ݕ‬ଶௗ௘௦ , In this case‫ݖ‬ଵ ൌ ‫ݕ‬ଶ െ ‫ݕ‬ଶௗ௘௦ =‫ݕ‬ଶ as ‫ݕ‬ଶௗ௘௦ ൌ Ͳ With this new variable the system equation (8) can be expressed as below: ‫ݕ‬ଵሶ = ߪሺ‫ݖ‬ଵ െ ‫ݕ‬ଵ ሻ ‫ݖ‬ଵሶ = െ ‫ݖ‬ଵ െ ‫ݕ‬ଵ ‫ݕ‬ଷ ൅ ߛ‫ݕ‬ଵ ൅ ‫ݑ‬Ǥ ‫ݕ‬ଵ ‫ݕ‬ଷሶ = ‫ݕ‬ଵ ‫ݖ‬ଵ െ ‫ݕ‬ଷ The Lyapunov function in this stage is defined as:

And we have already discussed that whenߤ=1.00, ߪ=5.58, ߛ=19.55 and ‫ ݒ‬ൌ Ͳthe motor experiences chaotic behavior. To express the system in formal strict feedback form we define[‫ݕ‬ଵ ‫ݕ‬ଶ ‫ݕ‬ଷ ] ‫ݔ[ؠ‬ଷ ‫ݔ‬ଶ ‫ݔ‬ଵ ] and then the model becomes: ‫ݕ‬ଵሶ = ߪሺ‫ݕ‬ଶ െ ‫ݕ‬ଵ ሻ ൅ ‫ݕݒ‬ଶ ‫ݕ‬ଷ (5) ‫ݕ‬ଶሶ = െ ‫ݕ‬ଶ െ ‫ݕ‬ଵ ‫ݕ‬ଷ ൅ ߛ‫ݕ‬ଵ ‫ݕ‬ଷሶ = ‫ݕ‬ଵ ‫ݕ‬ଶ െ Ɋ‫ݕ‬ଷ As Ɋ ൌ ͳǤͲͲand ‫ ݒ‬ൌ Ͳthe above model can further be simplified as: ‫ݕ‬ଵሶ = ߪሺ‫ݕ‬ଶ െ ‫ݕ‬ଵ ሻ ‫ݕ‬ଶሶ = െ ‫ݕ‬ଶ െ ‫ݕ‬ଵ ‫ݕ‬ଷ ൅ ߛ‫ݕ‬ଵ (6) ‫ݕ‬ଷሶ = ‫ݕ‬ଵ ‫ݕ‬ଶ െ ‫ݕ‬ଷ In this model ߪ and ߛare two unknowns and the controller is considered as‫ݑ‬௖ ൌ ‫ݑ‬Ǥ ‫ݕ‬ଵ . The model with proposed controller is as: ‫ݕ‬ଵሶ = ߪሺ‫ݕ‬ଶ െ ‫ݕ‬ଵ ሻ ‫ݕ‬ଶሶ = െ ‫ݕ‬ଶ െ ‫ݕ‬ଵ ‫ݕ‬ଷ ൅ ߛ‫ݕ‬ଵ ൅ ‫ݑ‬௖ ‫ݕ‬ଷሶ = ‫ݕ‬ଵ ‫ݕ‬ଶ െ ‫ݕ‬ଷ

ܸଵ ൌ ‫ ݕ‬ଶଵ ,

(7)

ଵ

ଵ

ଵ

ଶ

௔

௔

ܸଶ ൌ [‫ݕ‬ଵ ଶ ൅ ‫ݖ‬ଵ ଶ ൅ ሺߪ෤ െ ߪሻଶ ൅ ሺߛ෤ െ ߛሻଶ ] Where ߪ෤ and ߛ෤ are the estimated values of ߪ and ߛrespectively. And ܽ is the gain parameter of the controller. The time derivative of ܸଶ is defined as: ͳ ͳ ܸଶሶ ൌ ‫ݕ‬ଵ ‫ݕ‬ሶଵ ൅ ‫ݖ‬ଵ ‫ݖ‬ሶଵ ൅ ሺߪ෤ െ ߪሻǤ ߪ෤ሶ ൅ ሺߛ෤ െ ߛሻǤ ߛ෤ሶ ܽ ܽ With the expressions of ‫ݕ‬ሶଵ and ‫ݖ‬ሶଵ , ܸଶሶ can be defined as: ܸଶሶ ൌ ‫ݕ‬ଵ .ߪሺ‫ݖ‬ଵ െ ‫ݕ‬ଵ ሻ+‫ݖ‬ଵ ሾെ ‫ݖ‬ଵ െ ‫ݕ‬ଵ ‫ݕ‬ଷ ൅ ߛ‫ݕ‬ଵ ൅ ‫ݑ‬Ǥ ‫ݕ‬ଵ ሿ ͳ ͳ ൅ ሺߪ෤ െ ߪሻǤ ߪ෤ሶ ൅ ሺߛ෤ െ ߛሻǤ ߛ෤ሶ ܽ ܽ Which can be written as: ܸଶሶ ൌ െߪ‫ ݕ‬ଶଵ െ ‫ݖ‬ଵ ଶ ൅ ߪ‫ݕ‬ଵ ‫ݖ‬ଵ െ ‫ݕ‬ଵ ‫ݕ‬ଷ ‫ݖ‬ଵ ൅ ߛ‫ݕ‬ଵ ‫ݖ‬ଵ ൅ ‫ݑ‬Ǥ ‫ݕ‬ଵ ‫ݖ‬ଵ ͳ ͳ ൅ ሺߪ෤ െ ߪሻǤ ߪ෤ሶ ൅ ሺߛ෤ െ ߛሻǤ ߛ෤ሶ ܽ ܽ If we consider ߪ෤ሶ ൌ ߛ෤ሶ ൌ ܽǤ ‫ݕ‬ଵ ‫ݖ‬ଵ ,we get ܸଶሶ ൌ െߪ‫ ݕ‬ଶଵ െ ‫ݖ‬ଵ ଶ ൅ ߪ෤‫ݕ‬ଵ ‫ݖ‬ଵ െ ‫ݕ‬ଵ ‫ݕ‬ଷ ‫ݖ‬ଵ ൅ ߛ෤‫ݕ‬ଵ ‫ݖ‬ଵ ൅ ‫ݑ‬Ǥ ‫ݕ‬ଵ ‫ݖ‬ଵ , ܸଶሶ ൌ െߪ‫ ݕ‬ଶଵ െ ‫ݖ‬ଵ ଶ ൅ ሾߪ෤ ൅ ߛ෤ െ ‫ݕ‬ଷ ሻ ൅ ‫ݑ‬ሿ‫ݕ‬ଵ ‫ݖ‬ଵ .


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Therefore,‫ݕ =ݑ‬ଷ െ ሺߪ෤ ൅ ߛ෤ሻ makes ܸଶሶ negative definite and it tends ܸଶ asymptotically stable.By making ܸଵ ܸܽ݊݀ଶ asymptotically stable we can make ‫ݕ‬ଵ and ‫ݕ‬ଶ asymptotically stable, i.e. ௧՜ஶ ‫ݕ‬ଵ ൌ Ͳand ௧՜ஶ ‫ݖ‬ଵ ൌ Ͳor ௧՜ஶ ‫ݕ‬ଶ ൌ Ͳ. As‫ݕ‬ሶ ଷ ൌ ሺ‫ݕ‬ଵ ‫ݕ‬ଶ െ ‫ݕ‬ଷ ሻ, when ‫ݕ‬ଵ and ‫ݕ‬ଶ becomes zero, we have,‫ݕ‬ሶ ଷ ൌ െ‫ݕ‬ଷ , it certainly makes ௧՜ஶ ‫ݕ‬ଷ ൌ Ͳ Therefore, the designed controller becomes ‫ݑ‬௖ =‫ݑ‬Ǥ ‫ݕ‬ଵǡ where,‫ݕ =ݑ‬ଷ െ ሺߪ෤ ൅ ߛ෤ሻ and ߪ෤ሶ ൌ ߛ෤ሶ ൌ ܽǤ ‫ݕ‬ଵ ‫ݖ‬ଵ it implies ‫ݑ‬௖ ൌ ‫ݕ‬ଵ Ǥ[‫ݕ‬ଷ െ2.ܽ.‫ݕ ׬‬ଵ Ǥ ‫ݕ‬ଶ ݀‫]ݐ‬, as ‫ݖ‬ଵ ൌ ‫ݕ‬ଶ , where, ܽ is considered as a gain parameter of the controller. IV.SIMULATION RESULTS

Fig. 7. Plot for transformed quadratic axis current‫ݕ‬ଶ ൫ൌ ݅௤ ൯. The controller is switched at 40 second.

In this part we investigate the effect of controller which has been calculated. MATLAB-SIMULINK̻ is used to check our controller in numeric simulation. In the figures from Fig.5 to Fig.9 we have taken the parameters as ı = 5.58, Ȗ= 19.55and the corresponding estimated parameters are revealed as ߪ෤ = 20, ߛ෤= 20. The gain parameter of the controller ܽ=10. The initial conditions are considered asሾ‫ݕ‬ଵ ሺͲሻ‫ݕ‬ଶ ሺͲሻ‫ݕ‬ଷ ሺͲሻሿ் =ሾͲǤͳͲǤʹͲǤ͵ሿ்

Fig. 8. Response of the estimator as to estimate required ߪand ߛ for the controller. The controller is switched at 40 second.

Fig. 5. Plot for transformed rotational speed ‫ݕ‬ଵ ሺൌ ߱ሻ. The controller is switched at 40 second.

Fig. 9. Response of the controller. The controller is switched at 40 second.

Fig. 6. Plot for transformed direct axis current‫ݕ‬ଶ ሺൌ ݅ௗ ሻ. The controller is switched at 40 second.

For the corresponding MATLAB simulations the controller is switched at 40 second. It is clear from the simulation results that The controllertakes about two seconds to neutralize the chaotic states correspond to the considered transformed rotational speed߱, the transformed direct and quadratic axis currents ݅ௗ and ݅௤ respectively. The response of the controller in Fig. 9 shows that the specific control action is required for about 2 seconds. It can be adjusted by varying gain parameter of the controller.

2014 International Conference on Control, Instrumentation, Energy & Communication(CIEC) [6]

The unknown parameters ߪand ߛ for the required controller are estimated as about 20. It is obvious from (6) that as the value of ߛis considered as 19.55 for this case, estimated ߛ must be more than the considered value and it yields‫ݕ‬ሶ ଶ ൌ െ ݇Ǥ ‫ݕ‬ଶ , where,݇is a positive number. Therefore ultimately it reveals ௧՜ஶ ‫ݕ‬ଶ =0 and consequently we have ௧՜ஶ ‫ݕ‬ଵ = 0 as ‫ݕ‬ଵሶ = ߪሺ‫ݖ‬ଵ െ ‫ݕ‬ଵ ሻ and following ‫ݕ‬ଷሶ = ‫ݕ‬ଵ ‫ݕ‬ଶ െ ‫ݕ‬ଷ we get,‫ݕ‬ሶ ଷ ൌ െ‫ݕ‬ଷ , it certainly makes ௧՜ஶ ‫ݕ‬ଷ ൌ Ͳ V. CONCLUSION In this paper we have discussed the design and simulation of a controller to terminate the chaotic behavior of BLDC motor. The controller is designed following adaptive backstepping method which does not require the parameters of the BLDC motor under control. The simulation results show that the controller takes few seconds to terminate the chaotic state to the stable state. The simple mathematical model of the controller concludes that it can be implemented in a simple hardware. REFERENCES [1] [2]

[3]

[4] [5]

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