ICCAS2004 August 25-27, The Shangri-La Hotel, Bangkok, THAILAND
1. INTRODUCTION
There are several methods of controlling the sway of traveling cranes, such as fuzzy control[1] and optimal control[2], each of which is reported to be effective[3]. The development of a simple method for designing control systems is important for field engineers. In a control system, the controller is generally designed by using a mathematical model of the controlled object. However, it is impossible to make a model which accurately expresses the characteristics of the controlled object. Therefore, the controller obtained by this method is not always appropriate for actual systems, even if it is valid theoretically. That is: the models for actual systems have always uncertainties. H infinity control or H2 control is well known as the method which overcome such uncertainties.
However, the conventional H infinity control or H2 control are made on the basis of the mathematical models and the full order observers. As a result, the obtained control system has a redundancy. It is considered that we use the minimal order observer for the control system design instead of the full order one.
In this paper, the control system is designed by using an H2 controller reduced by the minimal order observer, which is a new approach that we propose. The results of simulations and experiments show that this system has good performance.
2. REDUCED ORDER H2 CONTROL 2.1 Minimal order observer
Let us consider a system such as
+
= +
=
+ +
=
u D x C z
w D x C y
u B w B Ax x
12 1
21 2
2
1
(1)
For the controlled object shown in the above equation, weapply a transfer matrix as equation (1) based on the design method of Gopinath
C T
C
S=[ 2 #]
(2)
where C# is chosen so that S is full rank.Putting x:=Sx, we obtain
+
= +
=
+ +
=
u D x C z
w D x C y
u B w B x A x
12 1
21 2
2 1
.
(3)
where[
0]
, : .:
: , :
:
1 1 1 1
2 2
22 21 2 2 12 11 1 1
22 21
12 1 11
−
−
−
=
=
=
=
=
=
=
=
=
S C C I S C C
B SB B B B
SB B B
A A
A SAS A
A
p
(4)
In equation (1), we consider estimating the unknown state variables except y, which is observable, by using
[
L In P]
U x
U = − −
= :
ω
. (5)
Expressed the minimal order observer which estimates ω asu J y B Aˆ + ˆ + ˆ
= ω
ω
, (6)
we putˆ . ˆ
2 2
B U J
A U C B U A
=
=
+
(7)
Assuming disturbance w as step one, we can obtain x
=U
ω as t→∞.
Next, we put the estimate of xas ω
C y D
xˆ= ˆ + ˆ
. (8)
Then, we also obtain xˆ=x by putting. ˆ 0 ˆ ˆ
21 2
=
= + D D
I U C C
D n
(9)
From equations (6) and (8), the minimal order observer isexpressed as
ˆ , ˆ ˆ
ˆ ˆ ˆ
ω ω ω
C y D x
u J y B A
+
=
+ +
=
(10)
where. ˆ 0 ˆ ,
, ˆ ˆ 0
ˆ ˆ ˆ
ˆ ˆ
21 2
21 11
22 12
=
=
=
=
= + +
−
=
= +
−
=
−
D D B U J
L D I C I
D A U L A A LA B
C A U A LA A
P
P n
(11)
2.2 Reduced order H2 controller
When the transfer functionΦ(s)fromw(s)toz(s)is defined,
Design of Reduced Order H2 Controller
-Application to Anti-Sway-Control of a Traveling Crane-
Nariyuki Kodani*, Shigeto Ouchi*, Yuji Todaka*** Department of Applied Computer Engineering, Tokai University, Kanagawa, Japan (Tel : +81-463-58-1211; E-mail: ouchis@keyaki.cc.u-tokai.ac.jp)
**Fuji Electric Co., Ltd., Tokyo, Japan
Abstract: For the anti-sway control of traveling cranes, there are several solutions, i.e., by fuzzy control, by optimal control theory, etc. Each of them is reported to be effective. And, H infinity control and H2 control can be also used. However, the full order observer which estimates all states in the controlled object is used in these methods. Therefore, the orders of these controllers are apt to be higher than that of the optimal controller, etc. Because the conventional H2 controller which minimizes H2 norm consists of two parts, that is: feedback gains which make the controlled object stable and the full order observer which estimate those states. If the minimal order observer is used instead of the full order one, the order of the controller can be reduced.
In this paper, we propose a new method based on the minimalization of H2 norm using the minimal order observer. And, we confirm the effect of a new H2 controller in the experiments of the anti-sway control of a traveling crane.
Keywords: reduced controller, H2 control, minimal order observer, anti-sway, crane
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our purpose is to obtain an H2controller which satisfies the following conditions:
1) Φ(s) is internally stable, and
2) ( ) min
2→
Φ s ,
on the assumptions that
a1)
(
A B2)
can be stabilized and(
A12 A22)
is detectable, a2) D is column full rank and 12 B11 is row full rank, anda3) ω ∀ω
−
12 1
2
D C
B I j
A and ω ω
∀
−
11 12
12 22
B A
B I j A
are column full rank and row full rank respectively.
Here, assume the control law u=−Fxˆ for equation (14).
Then, we obtain the following equation substituting equation (18) for the above control law:
) ˆ 0 ˆ (
ˆ 2 − 21=
−
= FDC x FC DD
u ω
. (11)
The following equation is obtained by substituting equation(11) into equation (3):
[ ]
[ ]
=
−
=
+
−
=
w D x C y
e C x F D C z
B w B e x A
C F B A e x
F
K K F
21 2
12 2 1
ˆ 0
ˆ
From the above equation, the transfer function Φ(s) is expressed by
ˆ , :
) (
) ( ) ( ) (
1 c f
cB U FCG G
s
s w s s z
−
=
= Φ
Φ
(12)
whereˆ . :
ˆ , :
, :
, :
, , 0 0
21 1
12 1 2
12 2
D B B U B A A
F D C C F B A A
D C
B U A
I B G A
C I G A
K K
F F
F F c
K K f F
F c
+
−
=
=
−
=
−
=
=
=
=
(13)
When A ,F AK are stable, the 2 norm of the transfer function
Φ
(s)above is calculated as(
BTXB)
RF trace(
FCYCTFT)
trace
s 1 ˆ ˆ
1 1 2
) 2
( = + −
Φ
, (14)
where, 0
0
12 12
2 12
I D D
X B C D
C C X A XA
T T F T
F T F T F F
δ
=
= +
= + +
(15)
and
.
=0 + + K K TK T
K A Y B B
YA
(16)
From equation (15), we obtain( ) ( )
( )
. 0 :
0
12 12
2 1 12 1
1 1 2 1 12 1 2
1 12
>
=
+
=
= + + +
− +
−
−
D D R
X B C D R F
C C
X B C D R X B C D X A A X
T F
T T F
T T T F T T T T
(17)
and from equation (16), we also obtain
( ) ( )
( )
. 0 :
,
0
11 11
1 11 12 12
12 12
12 11 12 1 11 12 12 22 22
>
= +
=
= +
+ +
− +
−
−
T K
K T T
T T K
T T
T
B B R
R B B YA L
B B
B B Y A R B B YA Y A YA
(18)
From equation (11), the H2controller is given by
)
ˆ ( ˆ
ˆ ) ˆ
(
y s D F C FD UA C s UA
u F F
−
= , (19)
where[ ]
ˆ . 0 , ˆ
: ,
: 2
=
=
−
=
−
=
−
−
L D I C I
F B A A I L U
P
p n
F p n
(20)
Here, feedback gainF and observer gainLare given by equation (17) and equation (18) respectively.
Further, equation (19) is transformed as the following equation:
( )
. ] [
:
) ˆ (
) ˆ
( 1
p n p
F F p
n p
F F F
s y L D UA C UA sI F s y F u
−
−
−
=
− +
−
−
=
From the result above, we can see that this control law consists of the measured outputs and the estimated outputs.
3. SIMULATION AND EXPERIMENT 3.1 Controlled object
We confirm the effectiveness of a reduced order H2 controller using a traveling crane in this chapter. Figure 1 shows a model of the controlled object, where x is the position of the trolley, l the lope length, m the mass of a load, M the mass of the trolley, θ the sway angle of a load and u the external force. For fig. 1, the motion of a traveling crane is expressed by
−
−
=
+
−
= +
θ θ θ
θ θ θ θ
sin cos
cos sin
)
( 2
g x l
u ml
ml x m M
. (21)
Upon approximating these equations around θ=0 andassuming u=Kv(v−x), equation (21) is transformed into the following equation:
− +
−
=
+ +
−
=
v K g m M x K Ml
v K mg x K x M
v v
v v
θ θ
θ )
(
(22)
x
u
θ l M
m
F
Mx m
θ
Fig. 1 Crane model
1755
Considering controlled outputs and disturbances for above equations, the state equation is expressed as
+
=
=
+ +
=
r D x C y
x C y
v B d B x A x
rp P
z z
P d
P
(23)
where,, ) 0
0 (
0 0
1 0 0
0
0 1 0
0 I : 0
, :
, : , : , : , :
3 2
2 2 1 1
2 1 2 1
+ −
−
−
= −
=
=
=
=
=
=
Ml K Ml
g m M
M K M
mg A
A A x x x
x x x x
v v P
P P
θ θ
θ θ θ
θ
. ] / [ 8 . 9 , ] [ 1 , ] [ 18 , ] [ 415 . 0
0 , 0 0
0 0 : 0
, : 0 , 0 1 0
0 0 0 : 1
, 0
0 0
0 0
0
0 0 0
0 0 0 :
, 0 0 :
2 2 1 4
4
4 3 2 1
2 1
s m g m l kg M kg m
d D d
I C C
b b b b
B B B
Ml K M B K
rp rp rp z
P
d d d d
d d d
v P v
=
=
=
=
=
=
=
−
−
−
−
=
=
−
=
×
From equation (23), a transfer function from input w(s) to output z(s) is expressed by
=
=
=
) (
) ( ) ( )
(
) ( ) ( ) ( ) (
) ( ) (
) ( ) ( ) (
) ) ( (
1 1
2 1
2 1
s u
s w s P s
P
s P s W s P s W
s u s W
s y s W s z
s s z z
w
z wz
z
, (24)
where
=
=
=
=
=
=
=
2 2
2 2 2
1 1
1 1
1( ) , ( )
0 , 0 : 0 ) ( 0 ,
: 0 ) (
0 , :
) ( 0 , :
) ( ) , (
) : ( ) (
w w
w w
w w
w w
z d P wz rp P
d P w
z P P z P
P P
D C
B s A
D W C
B s A
W
C B s A
D P C
B s A
P
C B s A
C P B s A
s P r
s s d w
Here, we apply a transfer matrix based on the design method of Gopinath for the equation (24), and we can obtain the following equation.
+
= +
=
+ +
=
w D x C y
u D x
C z
u B w B x A x
21 2
12 1
2
1
(25)
where,[ ]
[
0]
.0 , :
0 0 : 0 ,
0 : 0
0 , 0 0
] 0 [ : , 0 0
0 0 0 :
21 2 12
2 2 1 1 1
2 2 1
2 1 1
rp w
p w
w z w
w p d
w w z w
p
D D D
D
C C C
C C C D
B B B B B A A C B
A A
=
=
=
=
=
=
=
The block diagram in Fig.2 is H2 control system.
yp
W1 Gneralized Plant G
z1
z2
䋫䋫 䋫䋫 䋫䋫䋫䋫
Pw
P
yz
u
Pz
W2
㪄 K Pwz
y
w z
䋫 䋫 䋫 䋫 䋫䋫䋫䋫
P :Plant for Control,:Plant for Output-evaluation Pw:Plant for Disturbance,
Pwz:Plant for Disturbance-evaluation, K:Controller,u:Control Input, w:Disturbance, yp:Measured Output, z:Controlled Outputs, W1, W2:Weighting Functions
Fig.2 H2 control system 3.2 Selection of weights
For the weighting function, we chose the following weights accordingly:
• Selection of weight W1
W11, W22, W33 and W44 are respectively weights for the position of a trolley, the angle of a load, the velocity of a trolley and the angular velocity of a load. Particularly for W11, pseudo integrations are used so that the trolley position x can follow the reference value. And the gain of W44 has a large value so that it can reduce the quickly motion and sway of a load.
+ + +
= +
1 ,100 1 , 30 1 . 50 9 . 0 20
1 diag s s s s
W
(26)
• Selection of weight W2
For W2, km is determined so as to restrict the control input, and lf and hf are determined so as not to be affected by noise.
30 , 100 , 50
2 ⋅ , = = =
+
= + m f f m
f
f k l h k
h s
l
W s
(27)
• Selection of weights Bd and Drp
Bd11, Bd22, Bd33 and Bd44 are respectively weights for a trolley velocity, torque disturbances and a load velocity and torque ones. And we chose Drp so as to become zero, because Drp corresponding D21 in the conventional H2 control is not necessary to satisfy row full rank.
0 30
30 1
1 − − − =
−
= rp
d diag D
B ( , , , ) ,
(28)
3.3 Experimental & simulation results
Figure 3 and Fig. 4 show the simulation and experimental results by the reduced order H2 control system based on the minimal order observer that we propose. In these results, the desired position of 0.5 [m] was given first, and after about 25 seconds an impulse disturbance 4 ~ 5 [deg] was given. As a result, we can obtain similar results for simulations and experiments. And, you can see 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, and the sway of the load was quickly reduced.
Next, fig. 5 shows experimental results by the H2 control
1756
system with the full order observer (conventional system).
From the comparison Fig.3 and Fig. 5, you can see that both positioning controls are almost same performances, but the anti-sway for disturbance is better performance in the reduced order H2 control system rather than in the conventional system.
Finally, fig. 6 shows that the crane is in control after 15 [sec].
You can see that the sway of a load is not quickly reduced in the non-control case, however done in the control case.
Fig. 3 Simulation results of minimal order observer
Fig. 4 Experimental results for full-order observer
Fig. 5 Experimental results for minimal-order observer
Fig. 6 The responses of non-control and control cases 4. CONCLUSIONS
We proposed a reduced order H2controller by using the minimal order observer for a system with disturbances. We confirmed that the responses by the H2 controller based on the minimal order observer are the same as those by the conventional H2 controller in the experiments. Future tasks are to apply other systems and design H infinity control system using the minimal order observer.
REFERENCES
[1] M. Ito, I. Itou, and N. Maihara “Application of fuzzy control to automatic crane operation” (in Japanese), Journal of Japan Society for Fuzzy Theory and Systems, Vol. 6, No. 2, pp. 402-411, 1994
[2] K. Yoshida and H. Mita, “Mechanical system control”
(in Japanese), Ohmsha, Ltd., 1984
[3] K. Glover and J.C. Doyle, “State-Space Formulas for All Stabilizing Controllers that Satisfy an H-norm Bound and Relation to Risk Sensitivity,” Systems &
Control Letters, 11,167, 1988
[4] N. Kodani, S. Ouchi, and Y. Todaka, “Control System Design of a Traveling Crane using HControl Theory”, Proceedings of SICE Annual Conference 2002 (2002)
