Robust output tracking control of a linear brushless DC motor with time-varying disturbances

N.C. Shieh, P.C. Tung and C.L. Lin

Abstract: Robust output tracking control of a linear brushless DC motor for transportation within a manufacturing system is presented. System identification for a linear motor with a driver is performed by a dynamic signal analyser, thus providing a linearised model for the motor. External disturbances corrupting the driver and modelling uncertainties due to payload variations and identification errors are considered. A tracking control scheme with a guaranteed H , tracking performance is proposed. It is shown that a state feedback control law acheving the desired objective can be obtained by solving a Riccati matrix equation. Computer simulation and experimental implementation are completed and compared. These results demonstrate the effectiveness of the proposed scheme for output tracking control of the linear brushless DC motor, and also confirm the feasibility of its implementation in industrial applications.

1 Introduction

Many manufacturing systems require fast and accurate linear motion. Consequently, the direct drive design of mechanical applications, based on the linear brushless DC motor (LBDCM), plays an important role in meeting these demands [l, 21. However, because it is not equipped with auxiliary mechanisms such as gears or ball screws, the LBDCM is greatly affected by the uncertainties of the plant, which comprise plant parameter variations, external load disturbances, etc. Therefore, it is very important to compensate these disturbances which directly affect the mover of the LBDCM. In related works, many modern control theories, such as nonlinear control [3], optimal control [4], variable structure control [5], adaptive control [6], and robust H , control [7], have been proposed for the related motors to deal with these uncertainties. However, they are either based on complex theories or very difficult to implement.

On the other hand, there has been research developing disturbance suppression control systems with force or torque feedforward action to suppress the effect of disturbances [S, 91.

In this study to obtain the mathematical model for the linear motor, system identification is first executed with the necessary equipment, including a digital signal analyser and a PC-based controller. The identified objects are a driver together with the LBDCM. From this, the required numerical model parameters are obtained. However, during identification, several uncertainties arise, which are due to friction force, the precision of the experimental equipment, and motor parameters, such as resistance, etc. Furthermore, more severe variations arise from the possibility of a

0 IEE, 2002 IEE Proceedings online no. 20020027 DOL IO. 1049/ip-epa:20020027 Paper first received 9th April 2001 and in revised form 20th September 2001 N.C. Shieh and P.C. Tung are with the Department of Mechanical Engineering, National Central University, Chung Li, Taiwan, ROC C.L. Lin is with the Department of Automatic Control Engineering, Feng Chid University. Taichung, Taiwan, ROC

variable payload parameter. In addition, time-varying disturbances may corrupt the driver.

In the work of [lo], an optimisation solution for the output tracking problem was proposed, but it could not deal with time-varying uncertainties. Hence, an effective method is proposed to suppress the time-varying distur- bances. It is applied to the design and implementation of robust output tracking control of LBDCM with variable payloads and time-varying disturbances.

2

The complete system including a LBDCM, a carriage, a box, a granite bed, a driver and a PC-based controller is shown in Fig. 1 [lo]. The system configuration of the LBDCM drive is shown in Fig. 2, consisting of a position control loop, a speed control loop, a sinusoidal current command generator, a current control loop, a PWM inverter and the LBDCM. The position and speed control loops are realised in the industrial PC at a sampling rate of lkHz and programmed in the C language. The PWM inverter regulates the primary winding currents to closely follow sinusoidal commands. The sinusoidal current command generator generates three-phase current com- mands, with their amplitude determined by the speed control loop and their argument synchronised to the moving-member’s position. A linear encoder and three Hall effect sensors are employed to obtain the position and electrical position information of the moving-member, respectively. With the Hall effect signal synchronised to the back-EMFs, a basic concept of the field orientation and a zero current value id of the field component, the correct current commands can be generated to let the primary moving-magnet field be kept perpendicular to the secondary magnetic flux. Hence, the thrust force is directly propor- tional to the current [ l l , 121. According to the physical dynamic modelling derived in [l 11, the electromagnetic force equation of the LBDCM drive can be expressed as

LBDCM drive and system identification

d2x & dt2 dt

F x = K ~ i q = M - + p - + F

IEE Proc.-Electr. Power Appl., Vol. 149, No. I , January 2002 39

where Fx is the thrust force, KT is the thrust force constant, Mis the total mass of the moving part of the motor, p is the viscous friction constant, F is the external disturbance force, and i, denotes the force current component, which can be considered as i, Y it for the close current-tracking char- acteristics.

Since the accurate dynamic model of the driver together with the LBDCM is difficult to obtain analytically,

Fig. 1 LBDCM. carriage, box, yrunite bed, driver, PC-bused controller)

Trunsportution facilities ofmunufucturing system (including

rectifier

36 AC source

I I filter

I

identification is performed to find the dynamic model [lo]. When the payload is 1.5kg, the following transfer function is obtained:

X ( s ) 34.24 U ( s ) - s(s + 13.82) ( m / V

where X(s) is the position of the carriage and U(s) is the voltage input of the driver. Notably, the voltage input represents the force current component into the driver. Clearly, the gain of the driver is a constant value and the inductance of the motor can be ignored. For the cases of Okg and 3kg payloads, the transfer functions are identified, respectively, as xfi = *(m/V) and

driver’s input voltage U(s) to the motor carriage’s position X(s) can be expressed as [lo, 131:

xo- - ,(,yyi96,,(m/V). The transfer function from the

b (” U ( s ) - s(s + a , )

-

where al and b have distinct values for various loading conditions.

3 Construction for H, tracking control law

Consider a generalised representation of the motor dynamics with modelling uncertainties and external dis- turbance:

( 3 ) x = - (U1 + d a , ( t ) ) i - (U0 + duo(t))x

+ ( b + db(t))U + d

where x is the carriage’s position, U is the control input, the coefficients a l , a. and b are the nominal plant parameters, and the linear or nonlinear time-varying parametric

inverter m LBDCM

triangular PWM switching and driving circuit

software

Hall encoder signals signals

7 7 position detector

I’ Fig. 2

40

System conJgurution of LBDCM drive

IEE Proc.-Electr. Power Appl., Vol. 149, No. I , January 2002

uncertainties Aa,(t), dao(t) and db(t), denoting the uncertain damping effect, stiffness variations and perturbations of the driver, respectively, satisfy

sup l’4cI0 ( t ) 1 < 00, sup I A a1 ( t ) I < 00, sup Id b( t ) I < 00 t t t

The external disturbance d satisfies Jr d2(t)dt< ca. In addition, ~ the mass variation effect has been included in dal(t), &(t) and db(t).

Define the tracking error and error rate as

e=x-x , . , e = X - x , (4) where x,. is the desired position. For this problem, a state feedback control input for the uncertain system in eqn. 3 is proposed as follows:

(5) 1 b

U = - ( a , i + aox + U + U,)

where U , is an auxiliary control variable yet to be determined, and the variable U is chosen to be

v = x,. - ale - 02e

in which the positive constants a I and a2 are chosen so that the following error dynamics possess the desired response:

e + ale + a2e = 0 Substituting U into eqn. 3 yields the closed-loop system of

the following form:

x = --Aa1i - daox + [i,. + Q I (i,. - X) + Q2(X,. - x)]

+ U, + Abu + d (6)

(7)

This can be further expressed in a compact form as follows: e = - a l k - a2e + ua + w

where w = -dalX - Aaox + Abu + d, which is the sum of the modelling uncertainties and external disturbances. It is crucial and is to be attenuated by U,.

We can rewrite eqn. 7 in an augmented state-space representation as

z ( t ) = A,z(t) + B,u, + Bzw (8) where the augmented error state vector z= [e &IT, and

The linear motor positioning control problem is now turned to consider the condition ensuring tracking perfor- mance and stability of the perturbed system (eqn. 8). To deal with the attenuation of the total perturbation w, the H , control design method can be efficiently applied. The performance index regarding this issue is defined as [14, 151:

jtfIl 0 44 I @ < I I 4 0 ) II:, + P 2 f l l w ( t ) 112dt (9)

where Q= QT>O and P = PT>O are the weighting matrices, O < p < 1 is a prescribed attenuation level, and the weighted Euclidean norm 1 1 z [/e= a. It should be noted that the smaller values of p will result in larger magnitude of the control gain K when the control design is based on th s performance index. In addition, the values of E, and a2 will also affect the total control input U .

The problem under investigation becomes that of finding a control law to achieve the H , tracking performance (eqn. 9). The following theorem establishes the result.

Theorem I: Consider the uncertain linear motor dynamics described by eqn. 3, with the control input U defined as eqn.

5. If the auxiliary control input U , is chosen as

where K = -(l/p2)BFP, with P= PT>O being a solution to the following Riccati matrix inequality:

A:P+PA, --PB,BTP+Q<O

then the closed-loop system achieves the tracking perfor- mance described by eqn. 9.

U , = Kz (10)

(11) 1

P2

Pvooj We take a quadratic Lyapunov function candidate as follows:

V ( t ) = z T ( t ) B ( t )

V ( t ) = 2 ( t ) P . ( t ) + z T ( t ) P i ( t )

where the matrix P = PT> 0. The time derivative of V(t) is

(12) By substituting eqn. 8 into eqn. 12, we obtain successively

V =zT(ATP + PAZ)z + uTBTB + zTPB,u, + 2zTPBzw

< z ~ ( A , T P + PA,)Z + Z ~ K ~ B , T B + Z~PB,KZ 1 + - z T ~ ~ z ~ , T ~ + p2w2

P2

=zT ApP +PA, f pK + -B:P pK + -B:P [ ( ; r( : . ) - p 2 ~ T ~ + p2w2 (13) I

where the following quadratic inequality has been used in the second derivation:

1 6 ~ X ~ D E ~ U 5 S ~ ~ D D ~ X + - U ~ E E ~ U

where x and U are vectors, D and E are matrices with compatible dimensions, and 6 is any positive real constant.

From eqn. 13, we observe that if the control gain matrix K is chosen in the form of eqn. 10, then

Let the matrix P be the positive d e h t e solution to the Riccati matrix inequality described by eqn. 11; then eqn. 14 can be written as

5 -zTQz + p2w2 (15)

Next, integrating both sides of eqn. 15 from t = 0 to t ,>O gives

/ t f ~ T Q ~ d t + v(t) 5 V ( 0 ) + p2/ tf w2dt

0 0

Since V(t) 2 0 for all t 2 0 then tf / zTQzdt 5 zT(0)B(O) + p 2 1 t f w 2 d t (16)

This shows that the H , performance index defined by eqn. 9 is guaranteed with a prescribed level of p2.

It is impractical to analytically find the solution P= PT>O from eqn. 11. In addition, the solution may. not be unique. Fortunately, it can be formulated in a linear matrix inequality and efficiently solved numerically [ 161. A

0 0

IEE Proc.-Elect,.. Power Appl., Vol. 149, No. 1. Junuary 2002 41

substitutive way is to introduce an arbitrarily chosen constant < > 0, convert eqn. 11 into the following matrix equality:

1 ATP+PA,--PB,B,TP+Q+[I, = o (17) P2

and solve this via the available software. One may view the constant t as a design parameter and use it to tune the control gain matrix.

4

4.1 Step command Based on the identification results, the second-order linear uncertain system for the LBDCM can be described as

Robust controller design and numerical results

i = - (a1 + dal )x + (b + db(t))Zl = - ( U , + A u ~ ) X + [b + Ab1 + Abz(t)]u (18)

where Aal and Abl are due to payload variations, and Ab2(t) is assumed as 10 * sin(27r * 60 * t ) to represent the noise with 60Hz frequency corrupting the driver. The constant b is concerned with the driver’s gain. Hence, the imperfection of the driver’s current loop, noise in the electric circuit, etc.. may induce a perturbation affecting the driver’s gain. The external disturbance d is not present because there is no external force. The purpose of output tracking control is to drive the carriage from the home position (i.e. x = Om) to the desired position. The constant reference output is set as 0.2m. Consider the case with a 1.5kg load as the nominal design; thus al = 13.82 and b= 34.24. The parameters p, &, c(1 and a2 are set as 0.1, 0.1, 1 and 50, respectively. For the control design, the matrix Q is chosen to be 312. Using the result of theorem 1, the feedback control gain K can be found as -[3.0094 16.8051. The control input is thus generated by U = &( 1 3 . 8 2 - .4 - 50e-

(8.9383 0.0301 1 0.0301 0.1681 ‘

[3.0094 16.8051~). The matrix P is First, consider the case Wit; a 1.5kg load, i.e.

Aal = Ab, = 0; Ab2(t) is assumed as 10 * sin(27r * 60 * t ) . The linear motor dynamics is given by

X =z - 13.82i + 34.24[1 + 10 * s i n ( 2 ~ * 60 * t)/34.24]u (19)

For the case with Okg load, eqn. 18 becomes X = - 20.04X

+ 70.62[1 + I O * sin(27r * 60 * t) /70.62]~ (20)

For the case with a 3kg load, eqn. 18 becomes X = -9.67.i + 19.9[I + 10 * s i n ( 2 ~ * 60 * t)/l9.9]u (21 J

The control input U obtained above is applied to the motor driver. The initial conditions of the carriage are set a:s x(0) = Om and x(0) = Om/s. The constant reference output x, = 0.2m. Figs. 3 and 4 show the position and velocity responses of the carriage by simulation for three load cases. They are very similar and achieve the desired reference position is less than 2.0s. In Fig. 4, the case with the Okg load has the largest maximum value, 0.626m/s; the 1.5kg load has the median maximum value, 0.471m/s; and the 3kg load has the smallest maximum value, 0.393m/s. To check the H , performance index defined by eqn. 9, I( z ( t ) 11; and p2 )I w( t ) \ I 2 are displayed in Fig. 5 for the Okg load case. After integration by Simpson’s method, we get sL’11 z(t) lli5 dt = 0.2498 and p2JfII w(t) 1I2dt = 0.0414. In addition, we find that 1 1 z(0) (I;= 0.3575. Hence, the inequality in eqn. ‘3 is satisfied. Moreover. the calculation is also done for the

1.5kg and 3kg load cases. All the results are listed in Table I. Obviously, they all satisfy the inequality in eqn. 9.

. .

load = 0 kg load = 1 5 kg - load=3 kg

” 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

time, s

Fig.3 Simulated position responses of step command case ( p = 0. I )

0.7

0.6

0.5

q 0.4 E .< 0.3

0.2

0 a, -

0.1

n

. - load=Okg -.- -. - load=3 kg

” 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

time, s

Fig. 4 Simulated velocity responses of step command case ( p = 0 . 1 )

- - -.. . . .

- . . . . __ - . - - . -. . .

- I I z(t) 11; - --- p2w2

- -_ - .. . . . - .

0 0 2 0 4 0 6 0 8 1 0 1 2 1 4 1 6 1 8 2 0 time, s

Fig.5 load= 0 kg)

IIz(t)II; and p2vd of step command case (p=O. I ,

To verify the influence of p on tracking performance, we keep all the parameters invariant, except that p increases to 0.224. The feedback control gain K can be found as -[0.6162 7.01451. Figs. 6 and 7 show the position and velocity responses of the carriage for three load cases. It is obvious that the motor overshoots. Next, we also keep all

IEE Proc.-Electr. Power Appl., Vol. 149, No. I , January 2002 42

Table 1: Results of H , performance index

Load, kg J;' I/ z 11; dt 11 z(0) 11; f p 2 11 w / I 2 dt

0 0.2498 0.3575 0.0414

1.5 0.2021 0.3575 0.0062

3.0 0.1825 0.3575 0.0194 2 0.15 .- 0 0 -

0.10

load = 0 kg load = 1 5 kg

load = 3 kg

time, s

Fig. 6 Simulated position responses of step command case ( p = 0.224)

1.5

1 .o

e E .- 2 0.5 - 8 al

0

-0.5

load = 0 kg - - load=3 kg

0 0.2 0 4 0.6 0 8 1 0 1 2 1 4 1.6 1 8 2 0 time, s

Fig. 7 Simulated velocity responses of step command case (p = 0.224)

0.20

0.18

0.16

0.14

E 0.12

.E 0.10 c .- (0

8 0.08

0.06

0.04

0.02

n

load = 0 kg -.

- - - . .

" 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

time, s

Fig. 8 Simulated position responses of step command case (p = 0.032)

"0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 time, s

Fig.9 Simulated velocity responses of step command case ( p = 0.032)

the parameters invariant, except that p reduces to 0.032. The feedback control gain K can be found as -[24.8331 55.13081. Figs. 8 and 9 show the position and velocity responses of the carriage for three load cases. The motor acheves the desired reference position in more than 4s. Although smaller values of p will result in a larger magnitude of the control gain K, the values of a1 and a2 also influence the total control input U and the traclung performance. Hence, the values of all parameters must be designed consistently to get a good traclung performance.

4.2 Periodic sinusoidal command A periodic sinusoidal command with amplitude of 0.02m and period of 0.5s is used as the motion profile to examine the effectiveness of the proposed controller. Consider the case with a 1.5kg load as the nominal design; thus al = 13.82 and b = 34.24. The parameters p , (, al and a2 are set as 0.01, 0.1, 1 and 1, respectively. For the control design, the matrix Q is chosen to be 512. Using the result of theorem 1, the feedback control gain K can be found as -[224.834 225.8271. The control input U = A{ 13.82i- 0.02 * 16 * 7r2 c sin(4rt) - 2 - e - [224.834 225.82712).

The control input U obtained above is applied to the motor driver. The initial conditions of the carriage are set as x(O)=Om and i(O)=Om/s. Figs. 10 and 11 show the position and velocity responses of the carriage by simulation for three load cases. They are very similar, and follow the desired position profile after 1 Ss.

time, s

load = 0 kg load = 1.5 kg - load = 3 kg - Fig. 10 Simulated position responses of sinusoidal command case

IEE Proc.-Electr. Power Appl.. Vol. 149, No. I , January 2002 43

2.5 2.0

1.5

1 .o -? 0.5 E .- 2 0

-0.5

-1 .o -1.5

-2.0

-2.5

?

0 0.5 1.0 1.5 2.0 2.5 3.0 time, s

o load = O kg __ load = 1.5 kg * load = 3 kg

Fig. 11 Simulated velocity responses of sinusoidal command case

5 Experimental results

5. I Step command The experimental setup for output tracking and system identification setup are similar, as stated in [lo], and the experiment is finished digitally. The experiment is imple- mented on a Pentium PC using the Turbo C language. The state variables x and k are the position and velocity of the carriage, respectively. The actual position is measured with an optical linear scale. The carriage's velocity is obtained by. differentiating the measured position state x. Because the disturbance &(t) is assumed to directly corrupt the driver, we must enforce such a perturbation to be realised in the experiment. Hence, the modified control inputs in the experiment are [ l + 10 * s i n ( 2 ~ * 60 * t)/34.24]~, [l + 10 * sin(2~60t)/70.62]~ and [l + 10 * sin(2~60t) /19.9]~ , for the 1.5kg, Okg, and 3kg load case, respectitively. The control input U = h(13.82X - e - 50e - [3.0094 16.805]~). The position and velocity responses of the carriage with 0, 1.5 and 3kg payloads are shown in Figs. 12-17. These Figures show that the experimental results are similar to those obtained from the' numerical simulations. Furthermore, they all attain the desired reference position in less than 2.0s. For the velocity responses, the case with the Okg load has the largest maximum value, 0.68m/s; the 1.5kg load has the median maximum value, 0.525m/s; and the 3kg load has the smallest maximum value, 0.44m/s. They are very similar to the maximum value of the simulated results.

~ . - - .

...._ .__. ~ ., C-

_. .

.- - simulation - experiment 006 -

. _. . - - . . .- . . 1 0.2 0.'4 0'6 0:8 1'0 1 2 1'4 1:6 1:8 2 0

time, s

Fig. 12 command case (load= 0 kg)

Simulated and experimental positlon responses of step

time, s

Fig. 13 command case (load= 0 kg)

Simulated and experimental velocity responses of step

. . - I

. . . .

simulation - experiment

0 0 2 0 4 0 6 0.8 1 0 1.2 1.4 1.6 1.8 2.0 time, s

Fig. 14 command case (load = 1 5 kg)

Simulated and experimental position responses of step

0.6

0.5

0.4 VI . E 5 0.3 .- - 8 9 0.2

0.1

0

... .... . . ..... *. .... . -..- ..

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 time, s

Fig. 15 command case (load = I . 5 kg)

Simulated and experimental velocity responses of step

5.2 Periodic sinusoidal command The periodic sinusoidal command and the control input in the experiment are the same as those in Section 4.2. The simulated and experimental position and velocity responses

IEE Proc -Electr Power Appl, Vol 149, No I , Junwry 2002 44

0.20

0.18

0.16

0.14

E 0.12

.E 0.10 - .- g 0.08

0.06

0.04

0.02

0 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

time, s

Fig. 16 command case (load= 3 kg)

Simulated and experimental position responses of step

0.50

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 time, s

Simulated and experimental velocity responses of step Fig. 17 command case (load=3 kg)

of the carriage with 1.5kg payload are shown in Figs. 18 and 19. They are very similar, and the motor can follow the motion profile correctly.

0.20

0.15

0.10

E 0.05

.- - 0

-0.05

0 ._

z

-0.10

-0.15

-0.20

0 0 5 1.0 1.5 2.0 2.5 3.0 time, s

- - simulation - experiment

Simulated and experimental position responses of sinusoi- Fig. 18 dal command case (load= 1.5 kg)

6 Conclusions

A dynamic signal analyser was first used to identify the LBDCM for transportation within a manufacturing system,

I 0 0 5 1.0 1.5 2.0 2.5 3.0

time, s

simulation - experiment

Fig. 19 command case (load= I 5 kg)

Simulated and experimental velocity responses ofsinusoidal

and to obtain the open loop transfer function from the driver input to the carriage position. Next, a tracking control scheme with a guaranteed H , tracking perfor- mance was proposed. Based on the defined performance requirement, a robust control law was designed to efficiently suppress the uncertainties due to payload variations and time-varying disturbances. A computer simulation and experimental implementation were also conducted and compared. These results demonstrate the effectiveness of the proposed approach for output tracking control of LBDCM, and the ease of implementation for application in manufacturing systems.

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