Adaptive backstepping control of Separately Excited DC Motor with Uncertainties

Jianguo Zhou, Youyi Wang and Rujing Zhou School of Electrical and Electronic Engineering

Nanyang Technological University Nanyang Avenue, Republic of Singapore

Singapore, 639798

Abstract: In this paper, a nonlinear adaptive backstepping based speed controller is designed for the field weakening area of a separately excited DC motor. The nonlinear dynamic model of the motor is firstly derived, in which the parameter uncertainties such as the inertia and load torque are considered. Then by using linear reference model to design the transient performance and through suitable nonlinear change of coordinates, a nonlinear adaptive backstepping control algorithm is derived step by step and the stability analysis is also carried out. The mechanical speed and the back EMF tracking objectives are satisfied. Simulation results clearly show that the proposed controller is of good performance and robust to the parameter uncertainties.

Keywords: nonlinear control, adaptive backstepping, robustness, separately excited DC motor.

I. INTRODUCTION

Over the past century many kinds of electic machines have been widely used in the field of electric dnve and servo control systems [I]. Of these machines, such as DC motor, induction motor, permanent magnetic brushless DC motor, although the AC motor became more and more popular in the past two decades, DC motor is still the most common choice if wide range of adjustable speed drive operation is specified. Of the three kinds of DC motors, such as series, shunt and separately excited DC motors, Separately excited DC motors has two winding circuits, one is field circuit, which is used to establish a magnetic field with the motor, another is armature circuit, containing a current, which interacts with the magnetic field produced by the field circuit to produce the electic torque resulting in the mechanical rotation. The magnitude of the torque and the motor speed depend on the armature current and the strength of the magnetic field. Different speed can be obtained by changing the armature current and the field current. The significant feature of separately excited DC motor configuration is its ability to produce high starting torque at low operation speed [2]. Although the conventional cascade PID technique is widely used in DC motor speed and position control, it isnt suitable for the high performance cases, because of the low robustness of PID controller. Many researches have been studying the various new control techniques in order to improve the system performance [3-lo].

In recently years, feedback linearization approach have been used to design the nonlinear controller by changing the original nonlinear dynamics into linear one, thus all the

standard linear control techniques can be used [Il-121. But the performance of feedback linearization approach may deteriorate because of the parameter variation and the load disturbance. Some other researchers have employed sliding mode variable structure control technique to electrical drive systems to obtain the robust control of motor speed and position [13-141. However, the main disadvantage of VSC is that it results in chattering, which is often unacceptable in high performance drive systems.

The separately excited DC motor has nonlinear dynamic model in the field weakening area. It is known that the normal SISO nonlinear output feedback linearization has zero dynamic unstable problem [12]. Even more, although the ordinary state feedback linearization method is feasible to linearize the nonlinear dynamic model of DC motor in field weakening area, the transformed output is the function of armature induction, inertia and motor speed [12]. Thus the speed may become sensitive to the uncertainty of the parameter J . Therefore in practical application, this method may not be very acceptable When the back EMF is added as another output, the resulted dynamic model can be of no zero dynamic unstable problem [IS].

Backstepping is a newly developed nonlinear control technique. The most appealing aspect of the backstepping design method is that this approach provides a systematic procedure to design stabilizing controllers, following a step- by-step algorithm [ 16-20].

In this paper, Backstepping control theory has been used to design a nonlinear controller to obtain high speed field weakening operation. It clearly shown that the general dynamic model of the motor is highly nonlinear. Even more, the parametic uncertainties will significantly affect the dynamic performance and the stability for practical application. Thus we have to design a nonlinear robust controller, which can compensate the nonlinearities and the uncertainties simultaneously. In this paper, the uncertainties we considered are the inertia and load torque, which are the major concerns in high performance electric drive systems. By using linear reference model to design the transient performance and through suitable nonlinear change of coordinates, a nonlinear adaptive backstepping control algorithm is derived step by step and the stability analysis is also carried out. The mechanical speed and the back EMF

0-7803-6338-8/00/$10.00(~)2000 IEEE 91

traclung objectives are satisfied. Simulation results clearly show that the proposed controller is of good performance and robust to the parameter uncertainties.

ILDYMAMIC MODEL

Separately excited DC motors has two winding circuits. The winding on the rotor is referred to as the armature circuit while the winding on the stator is referred to as field circuit. The field and armature circuits have independent voltage sources. That is to say both of the two voltages can be used as the control inputs.

If the field flux is of no saturation, the flux bf is directly proportional to i f , i.e. qjf = L i . [21]. Then the motor's dynamic equations can be written as:

di, 1 - = - (ua - E - Raia) dt La

E = K i p , Te = K i f i , , K =af.

where, w

ia

if

4f J B k 'e

TL K Ra Rf L a

L f U a

Motor angular velocity Current in the armature circuit Field current

Flux in the field windings

Inertia of motor Damping constant of motor Electromachnical transduction constant Electric torque generated by the motor Mechanical load torque The motor constant Resistance of the armature winding Resistance of the field winding

Armature inductance Field inductance

Voltage applied to armature winding (a control variable)

variable) , u.f Voltage applied to field winding (a control

Remarkl: In this section, fiom the electrical and mechanical characteristics of the separately excited DC motor, the general model of separately excited DC motor is derived. The resulted model includes both the armature and the field circuit voltages as the control inputs. It clearly shown that the general dynamic model of the motor is highly nonlinear. It can be seen from equation (1) to (3) that the

term kb w is the product of motor speed w and field flux If and the term kbfi, is the product of filed flux bf and armature current i, . Therefore, a nonlinear controller should be designed to compensate the nonlinearity.

111. NONLINEAR CONTROLLER DESIGN

Equations (1) to (3) can be written in the compact form as

where x = [x, x2 x3] = [io if 03 r . 1

The system parameters may deviate from the nominal values, especially the inertia and load torque.

Let a = - , b=--,then J J

a=a,+Aa, b=b,+Ab

1 TL

where a, and b, represent the nominal values of parameters a and b, while Aa and Ab represent the errors fiom their nominal values respectively.

Taking into account these uncertainties, the dnve system (4) can be changed as:

where j ( x ) and Af(x)are the nominal and uncertain matrices of f (x) respectively.

1 --if R f

L f a , (K i , i f -Bw)-b, a , (K i , i f -Bw)-b,

L J

0

Aa(iaif - Bw) - Ab

Since the control objective is to make the motor speed track the desired reference speed command, we should select

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the motor speed as one of the output variable. Further more, because in the field weakening area, the motor should works in the constant power mode, we should also choose the back EMF as another output variable. Therefore choosing the outputs variable of the motor drive system as

The following notation is used for the Lie derivative of a function h ( x ) : R + R along a vector field f ( x ) = ( f i (x), . . . > f n (x)) [111

Iteratively, we define the following notation

(7)

L>h(x) = L f (L?-h)

Then the dynamic model of the DC motor in the new coordmate is given by

where

L j h , = - f = a , (Ki,if - Bw) - b, ah1 - ax LAf hl = ax ah1 Af = Aa(i,if - Bw) - Ab L h --g,=O ah1

Lg,hl = x g f = o

- - -- E Rf +a,Kif(Ki, i f -Bw)-b,Kij

g , 1 - ax ah1

ah2 - L - h f 2 - F f -

L f ah ax

LAf h2 = 2 Af = AaKif (Ki,if - B o ) - AbKif

a , K i f a , K i f i , Lrl (-Kifu-R,i,)-- L f R f =-

-a, B[u, (Ki,if - Bw)- b, ]

In order to decouple the two control inputs, we construct the new control inputs as follows:

Thus the error-tracking model of the system can be written as

Because adaptive backstepping control theory requires that the uncertainties in the system must have linear forms, we have to change the uncertain items in (1 1) to the following forms:

Thus we obtain the compact form of the error-tracking model as follows

i = [ Z ( x ) + A A ( x ) ] + B(x)C (15)

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For the first two dynamic equations of (1 6), we define the virtual control Q for ez2 to stabilize the first equation as follows

where k , > 0, and i1 is the estimate of 0, .

time then become as The derivatives of the two new variables with respect to

If we look at the structural property of the equations (1 1) carefully, we can find out that the error-tracking model of DC motor contains two decoupled subsystems. The first subsystem consists of the first two equations and is controlled by iia ; the second subsystem is the third equation and U as its control input. This structure allows us to conveniently use adaptive backstepping design technique to obtain the desired controller.

In order to obtain good transient performance, a linear reference model is defined as [22]

where alm , a2m , and a3m are designed constants.

output of the system is defined as The tracking error ez between the reference model and the

Then the error dynamics becomes as follows:

e, = A ( x ) + A A ( x ) + B U (18)

where

Now, adaptive backstepping control method can be used to derive the nonlinear controllers for the above system systematically step by step.

Step 1 : Define new variables as

r -

Step 2: Define the first control Lyapunov function v d q , w $ ) as

The derivative of VI (e,, Z2 ) SI ) is

- 1 - 2 - PI (el, e2 ) e, ) = z1Z, + e2 g2 + - o1 el Y1

=-k,Zf +e1 E242 +Z1q+ + klZ24, +-el

+Z2[Z1 +L2-hl f +4p2 +Za +k1(Z2 - k , ? ) + p l ~ l ] (25)

-[ Y1 7 Step 3: For the h d dynamic equations of (16), the new

equation becomes as

$3 = ~ 7 ~ 2 ( x ) + Zf + e, p3 ( x ) (26)

We can easily define the control input for Uf as

i7f =-k3E3 + L f h z ( x ) - & l p 3 ( ~ ) (27)

where k , > 0, and is the estimate of el in the third equation.

Then equation (26) can be written as

- Z3 = 4 3 ~ 3 - $ 1 ~ 3 ( X I (28)

Define the Lyapunov function V2(Z3,&) for the third equation of ( 16) as

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Now, lets integrate (33) and we can have (29)

1 - ~2 ( 3 , $1 = Z: + - 5;

2 2 Y 2

Rated Power Rated speed Field voltage Armature resistance

The derivative of V2 ( 4 , q ) is as follows: (39)

3.73kW 183.26 Rads 240V 1.20

Since V ( E l , Z 2 , Z 3 , gl, & ) is positive definite, we can obtain the following result:

1 -

lim M ( T ) ~ T I v(z(o), g1 (01, g1 (0)) < m (41) (30) 6 = - k 3 E : + g l ( Z 3 p 3 + - $ I )

Y 2 t+m

Step 4: To design the final adaptive backstepping nonlinear control for the system, we define the augmented Lyapunov function v ( z ~ , ~ 2 , ~ 3 , 5 1 , ti& as:

Via Barbalats Lemma [ 18][23], it can be obtained that

M ( t ) -+ 0 as t + m .

The derivative of V(E1, 3 , Z 3 , 5 1 , & ) is computed as

IV. STABILITY ANALYSIS

We consider the fact that V ( E l , Z 2 , E 3 ,F], ) is positive definite, and v ( E l , E 2 , E 3 , &, 51 ) is semi-defite. That is to say all the error variables are bounded.

Define the following new function:

M ( t ) = k l Z f + k 2 E ; +k3E: 2 0

Therefore we can conclude that all the error variables are bounded and converge to zero asymptotically. Further, since the motor speed and the back EMF tracking objectives are equivalent to that the error variable el and e 3 converge to zero, We can draw the conclusion that the design objectives have been satisfied.

V.SIMULATION RESULTS

In this section, some simulations have been carried out in MatlabJSimulink to evaluate the proposed nonlinear adaptive backstepping feedback controller for the field weakening operation of the separately excited DC motor.

The nominal values of the motor parameters are shown in the table below.

Field resistance

I Rated field current I 4A The controller gains k , , k , , k3 and the adaptation gains

y l , y 2 are chosen as follows: k, = 100, k , =200, k3 = 200

(38)

95

In order to test the robustness of the controller to the change of the system parameters, the inertia and the load torque is changed to J = 2JnOm and TL = respectively. The step speed command changed from the base speed 183.26 rads to 220 rads and then to 240 rads and finally to 260 rads at time instant 2 second, 4 second and 6 second respectively. The results are showed $Fig. 1 and Fig. 2. We can also see that the system speed can quickly follow the step command, and no steady error can be observed. Further, the field current satisfies the field weakening theory.

4

93.8

3.6

0 3.4

3.2

A

*

a Q) U .-

3 -

2.8

Fig. 1. Dynamic response of motor speed

t

-

-

- 1

-

4.2 I I

Fig. 2. Field current response

VI. CONCLUSION

A nonlinear adaptive backstepping based speed controller is designed for the field weakening area of a separately excited DC motor. Firstly the nonlinear model of the system is derived, at the same time the parameter uncertainties such as the inertia and load torque are considered. Then by using linear reference model to design the transient performance and through suitable nonlinear change of coordinates, a nonlinear adaptive backstepping control algorithm is designed systematically and the stability analysis is also

carried out. Simulation results clearly show that the proposed controller is of good performance and robust to the parameter uncertainties.

VI. REFERENCES

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J. Y. Hung, W. Gao and J. C. Hung, Variable structure control: a survey, IEEE Trans. on Industrial Electronics, vol. 40, no. 1, pp. 2-22, 1993. V. I. Utkin, Sliding mode control design principles and applications to electric drives, IEEE Trans. on Industrial Electronics, vol. 40, no.

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P.V. Kokotovic, The joy of feedback nonlinear and adaptive, IEEE Control System Magazine. 12, pp. 7-1 7, 1992 M. Krstic, 1. Kanellakopoulos and P. Kokotovic, Nonlinear And Adaptive Control Design, John Willey & Sons, Inc. 1995. H. Tan, Field orientation and adaptive backstepping for induction motor control, IEEE Industry Applications Conf., vol. 4, pp. 2357- 2363,1999. H. Tan and J. Chang, Adaptive backstepping control of induction motor with uncertainties, Proceedings of American Control Conf., vol. 1,pp. 1-5, 1999 B. Friedland, Advanced Control System Design, Prentice-Hall, Int, Inc. 1996. H.J. Shieh and K.K. S h y , Nonlinear sliding-mode torque control with adaptive backstepping approach for induction motor drive, IEEE Trans. on Industrial Electronics, vol. 46, no. 2, April 1999, pp. 380- 389. J.J. Slotine and W. Li, Applied Nonlinear Control, Prentice-Hall, New Jersey, 1991.

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