The Control System Design of a Traveling Crane using H, Control Theory

Kunihito Matsuki, Noriyuki Kikuti, Shigeto Ouchi Department of Electrical Engineering

Tokai University

1117 Kitakaname, Hiratsuka-shi, Kanagwa 259-1292, Japan

TekO463-58-1211, Fax:0463-59-4014, Email:oushis@keyaki.cc.u-tokai.ac.jp

Yuji Todaka Fuji Electric Co.,Ltd

1, Fuji-cho, Hino-shi, Tokyo 191-8502, Japan

Abstract : On the traveling crane the various methods such as fuzzy control and optimum control are studied. I n the control of actual traveling crane, i t is important to accurately stop the crane for the goal position b y positioning control and anti-sway control. I n this paper, we used the H, control theory which can keep the good performance, even i f there are modeling errors and parameter variations, and designed the control system by the loop shaping method. The simulation using the non-linear approximation model indicates the good performance.

Key Words: Crane, State feedback control, H, control, Anti - sway control

1 Introduction

For the anti-sway control of traveling cranes, there are several solutions, i.e., by fuzzy control['], optimal control theory[", etc. and each of them is reported to be effectivec3]. The development of the control system design method comparatively simple is an important theme for the field engineers. In the control system design, the controller is generally designed by using the mathematical model of the controlled object. On the other hand, it is impossible to make the model in which accurately exepresses the characteristics of the controlled object. Therefore, the controller obtained by the design is not always good for the accrual system, even if it is good for the mathematical model.

In this paper, we used the H, control theory which is a robust control technique for modeling error and parameter variation: and we contirmed the effect by simulations.

2 Controlled object

Fig. 1 is a model of controlled object, where x is the position of the trolley, 1 the rope length, m the mass of load, M the mass of trolley, 8 the sway angle of load, and U the external force. For the model shown in Fig. 1, the equation of motion for the trolley can be expressed by

Mjt=u-mjt ,

x, =x-ZsinO (1) Also the equation of motion relating to the load can be expressed by

(2) m ~ e = -mg sin e + F, F, = lnjtcose

From equations (1),(2), equations of motion of a traveling crane is expressed by

(3) (M + m ) i +mZb2 sin 8 = mlecos 8 +U + g sin e = i cos e

Upon approximating of these equations around e=O and using the stepping motor for driving the trolley, equation(3) is transformed into the following equation.

z e + g e = i (4)

From the above equation, the state-space equation of traveling crane can be expressed by

(5)

j j = U

x = A,X + B,U Y=C,X >

upon ;upposing that the state variable is Fig.1 Crane Model

0-7803-5976-3/00/$10.00 02000 IEEE. AMC2000-NAGOYA

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x=[x li- e e ] , where

-0 1 0 0- 0 0 0 0

Ap = 0 0 0 1 B P = o g 0 0 - - 0 1 -

Given plant P and weights W,, W,, W,, and P, are as

rO- -1 0 0 0- 1 0 1 0 0

-1 - 0 0 1 0 0 0 0 1

c P = 1

P, = w31 =

3 H, control system design

*Y

Fig.2 Block diagram Let us consider a system shown in Fig.2, in order to design a control system which follow a reference input and is robust against disturbances and parameter variation. In Fig.2, P is the plant, K the controller, W,, W,, W31 and P, the weighting functions, r the control input, d the disturbance, y the measured output, and zl, z, the controlled outputs. , _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Generalized Plant : G .

Fig.3 Block diagram of the &,control From Fig.2, H, control system can be expressed as shown in Fig.3. From Fig.3, the generalized plant G including the weights can be expressed by

l y w 3 i K p w -Fp1 G = 0 0 w2 (7)

p = [*] we have a state-space equation as

1 1 A,, 1 1 Bi2 c w 2 I Dw2 0 C p 0 Dw3, DW3, i 0

G = ' w l D w l c p Dw1Dw31 D w 1 D w 3 2 ,

..................................................... ........................................................ * .............

Given the equaton (8) is as

[ :] = G [ :] = [ G1l ' I 2 ] [ :] G21 G22

[zl z 2 ] ' = z , [r d I T = W let the control law be

Then transfer function from w to z can be expressed by

u = K e . (12)

z = QW = { G,, + G,,K(I - G,,K)-'G,, ).. (13) As well using H, control theory we can obtain such controller K as satisfy the condition that 1) @ is internally stable, 2) on the assumptions that a l ) (A B , ) is stabilizable, (C2 A ) is detectable, a2) D,, is column full rank, D,, is row full rank a3) G,, and G,, have no invariant zeros on the

IIQllr

W, is determined so as to satisfy the quick response. Particularly for Wll, a pseudo integration is used so that the trolley positon x can follow the reference value.

Bw31 =

O.Ool + 8 0 20 0

5 0 50 s +

0 0 0 0 0 1 0 0 D w 3 1 = o o l o ]

0 0 0 0 0 0 0 1

0 0 2 O

O 1 For W,, k,,, is determined so as to restrict the control input, and 1 and h f, determined so as not to be affected by noise. W, consequently is as follows.

5 0

0 0 0 8

Bw32 =

S + l f w2 = - km I, =30 h, =lo0 k , =50

s + h f

0 0 0 2 0 0

Dw32 =

W31 is chosen by considering the step reference as follows.

10 0 0 01 r2 0 0 01

For P,, Bw32 is determined by considering the disturbance and Dw32 is determined by considering the sensor noise.

ro 01 ro.5 01

4 Simulation results

Fig. 4 shows results of simulation. In Fig. 4, the reference position of 50 cm was given first, and after 20 seconds an impulsive disturbance was given. As a result, the position quickly followed the reference value, and even if a disturbance was added to the system, the error from the reference value was small. Fig. 5 shows the Bode diagram for the controller. I t is seen from the analysis of Fig.5 that the controller for the trolley position X has the function of PID controller, and the controllers for the trolley velocityi , the sway angle 8 and the sway angular velocity 6 have that of PD function respectively. Consequently, the quick response, the steady-state error and the stability are satisfied for the 8 , and it is seen for i , 8 , 6 that both the quick response and the stability are satisfied.

......... ........ ........

n ......... ......... ......... ......... ......... ..... .......... ........ ......... ......... ......... ......... .........

......... ......... ......... ......... ........ ......... .......... ..... ......... ......... ......... .........

, , , , , , , 0 5 IO I5 2 0 2 5 9 31 (0 45 91

time [sec] . . . . . . . . . . .

I 1 , , , , , , , , , , , , 1 , , , * , , , , , , , , , , , ~ , Sk ........ L ......... I .......... L ........ J ......... 1 ......... 1 ......... 1 ......... 1 ......... L ......... I

4k .................. , .......... , ................... > ......... ,. , ........ , , ......... , ......... , 1 ......... ......... : ......... : ......... : ......... : ......... ;

, , I I I I I I I I I I I . . . . . . . . . . .

I , , , , , , , , , , , , , , , I , , , , I , / , I I I I I ......... f ......... 1 ......... 1 ......... L ......... , I , , , , . . . . .

......... : ......... : ......... : ......... ; ......... ; , , , , , , , , , , , ~ , , , ~ $L ........ : ......... I .......... : ........ 1 ......... : ......... : ......... : ......... : ......... : ......... I , , , , , , , , , , , 0 5 10 I5 1 X 3 2 M 45 5D

time [sec] Fig.4 Response for H,control system

I I I I I I I I I I yo. lo4 .c .z .*I .J ."I ."I lo' frequency [radlsecl

frequency [radkec] Fig.4 Response for Hacontrol system

Fig.6 shows the poles of the controlled object described by 0 ,and the dominant poles of the closed loop system described by X . I t is seen that the poles of the controlled object on the imaginary axis are moved to the left-half plane by H, control.

133

4 I I 5 4 3 2 1 0

real axis (0;poles of plant, X idominant poles of closedk loop system)

Fig.6 Poles

5 Conclusion

The effect of H, control system was confirmed by simulation. And we will confinm the performance by experiments.

6 References

[11 Ito , Migita , Irie , Itou Nyuui Maihara Applivation of fuzzy control to automatic crane operation (in Japanese) Journal of Japan Society for Fuzzy Theory and Systems, vo1.6,N0.2,pp.402-411, 1994

[21 Yosida, kawaji, Mita, Hara, mechanical system control (in japanese) Ohmsha,Ltd, 1984

[31 Ouchi, IGkuchi, Todaka, Anti-sway Contlrol of a Traveling Crane using State Feedback Control (in Japanese) Proceedings of the 37 SICE Annual Conference, pp l l - 12, 1998

[41 K. Glover & J. C .Doyle, State - Sp ace Formulas for All Stabilizing Controllers that Satisfy an H a -norm Bound and Relation to Risk Sensitivity Systems & Control Letters, 11,167,1988

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