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Fuzzy control of an overhead crane performance comparison with classic control
ControlEng. Practice, Vol. 3, No. 12, pp. 1687-1696, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0967-0661/95 $9.50 + 0.00
Pergamon 0967-0661(95)00181-6
FUZZY CONTROL OF AN OVERHEAD CRANE PERFORMANCE COMPARISON WITH CLASSIC CONTROL A. Benhidjeb and G.L. Gissinger Universit~ de Haute Alsace, Laboratoire de Mod~lisation et d'ldentification en Automatique et M~canique, M.1.A.M/E.S.S.A.I.M/IAR, BP 2438-68067 Mulhouse Cedex, France
(Received December 1994; in final form September 1995)
Abstract: This paper describes the comparison of a fuzzy logic control system with the Linear Quadratic Gaussian control (LQG) of an overhead crane, considering the applicability of the control algorithms in real time and assuming that the model is representative of the real system. A number of possible perturbations are examined in a study of the robustness of the control algorithms. A complete reference trajectory model is also presented. Keywords: overhead crane, fuzzy control, LQG control, optimal control, reference trajectory.
1. INTRODUCTION The design of a controller is usually based on classic or modern control; adaptive control (Boustany and d'Andr6a-Novel, 1992), optimal control (Benhidjeb and Gissinger, 1993; Grassin, et al., 1991; Manson, 1982; Retz, 1992), and non-linear control (Siguerdidjane, 1991). Common to both approaches are the requirements that the plant to be controlled can be described in a rigid mathematical form. Real-time control requires some simplification of the experimental model. Human intervention is always necessary for this type of control. And the choice of the state to control is fundamental, since the choice of the angle of the pendulum and position of the load are essential for precise, robust control of the crane. This leads to a very complicated state representation and the given model is non-linear.
aforementioned imprecise information into a quantitative control system would prove very difficult. However, the theory of fuzzy sets can be used to describe such information directly and can be implemented in the development of a fuzzy controller. Yasunobu and Hasegawa (1986) and Yamada, et al., (1989) studied fuzzy control in order to minimise only the angle of the pendulum (which is not easily measured), but this is not sufficient to control the load position of the crane. This paper presents a comparison between the fuzzy control method and the classic control method with reference trajectories. Besides Linear Quadratic Gaussian control, there are other methods of classic control, such as adaptive control (Butler, et al., 1991), generalised predictive control (Retz, 1992), optimal control (Benhidjeb and Gissinger, 1993), etc. But to the authors' knowledge, only LQG control has been tried and tested on an industrial travelling crane. To be validated, the results obtained with the fuzzy controller must be compared with those obtained with LQG control.
In the case of classic control, robustness is not guaranteed for a variation of the load transport. Manson (1982) has concluded that "Optimal solutions are only attainable when dealing with a very idealised model of the overhead crane system". For all these reasons, a controller based on the experience of the human operator is desirable. Unfortunately, using the traditional control theory, the conversion of the
The present paper describes first the overhead crane and the mathematical model used by the classic 1687
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A. Benhidjeb and G.L. Gissinger
control law, second the complete reference trajectories developed considering the hoisting and lowering of the load, third an optimal linear quadratic control law, fourth the fuzzy controller and fifth the experimental crane and the experimental results obtained with both controllers.
2. SYSTEM ANALYSIS
X = (X1, X2, X3, X4, X5, X6) T ----(X, l, O, X', l', 0") r .
There are two control variables: the first one operates through the hoisting motor (winch), the second one through the hauling motor (trolley). Two types of control variables can be considered: -dynamic variables (force and torque) -kinematic variables (linear and angular accelerations).
2.1 System description The overhead crane considered is made up of a platform (Fig. 1) moving along the horizontal axis, equipped with a winch (Fig. 2) around which a cable holding the load is wound.
Ct or
cg
Fig. 1. Trolley
In the case of acceleration control, the control vector is: U = (Ul, U2) T ~-- (X'P, if2*) T
and the state-space equations are:
x
Y ~l-
The state-space vector is expressed as follows:
X{ = X4
(al)
X~ = X5
(a2)
X~ = X6
(a3)
X~ = u 1
(a4) (3)
X~ = - R u 2
(a5)
X~ =-~2 (Ul COS(X3)-gsin(X3)- XsX6)
(a6)
Fig. 2. Winch
2.2. Mathematical model for classic control law Differential equations describing the system mx" = F - Tsin(0) - Ff Mcx"c = Tsin(0)
(1)
tr
Mcy c = Mcg - Tcos(0) with: m the mass of the trolley, Mc the mass of the load, x the position of the trolley, x c the horizontal position of the load, Yc the vertical position of the load, F the external force generated by an electric motor, T the tension force, O the angle of the pendulum, and Ff the friction force. Differential equations describing the winch Jto' = C - RT l" = -Ro9 1"-
(2)
RC
R2T +.-J J
with: J the moment of inertia, m the angular velocity, R the radius of the winch, and C the applied torque. State-space equations of the process. In this case, the variables considered are: X the trolley position, 1 the rope length, and 19 the angle. This type of representation is based on the cylindrical co-ordinates.
It can be noted that the relations obtained from such a representation are only of the kinematic order. This state form is then linearised locally (constant cable length, low angular deviation) and leads to reference trajectories that are compatible with the input and output of the system. A robust control law can then be defined which will allow great variations of the load. It must also be noted that this system is entirely decoupled according to both inputs, since "u2" only affects (a2) and (a5) and the control "Ul" in (al), (a3), (a4) and (a6). So the cable length can be set separately and considered as a variable that is perfectly known at any time. Using kinematic variables as control variables requires motors which respond fast in speed with unit gains to input orders (the masses of the load and the trolley only play a part in the transfer functions of the hauling and hoisting motors). The input voltages of the motors are proportional to the rotation speeds of the motor-shafts at the output. The controls obtained with different algorithms are accelerations which must then be integrated. The input orders of the motors are speed targets, and servo-controlled motors are necessary to follow these orders properly. Two types of control can be used:
Fuzzy Control of an Overhead Crane Performance acceleration control, wlalch is current or power control and which allows the application of force control. speed control, which was chosen later on as it allows direct control of the motors' speed. -
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With these three equations and the differential equation (4), the behaviour of the processes can be described in three steps:
-
The motors are speed-looped by correctors with proportional and integral effect (PI).
first step: During the time T1, the trolley is accelerated, the cable length varies from L1 (initial position of the cable length) to L m (medium position of the cable length) with the hoisting speed Vl.
3. REFERENCE TRAJECTORIES
°91= ~
The reference trajectories represent the ideal behaviour of the crane (minimum transport time and minimum angular position of the load) and will be used in the controller in the case of optimal control. The non-linear kinematic equations (3) are used to determine a feedforward control which gives the reference trajectories for the state and control variables. The state-space representation obtained with acceleration control shows that the equation of the pendulum formed by the load and by the trolley is described by:
x"cos(O) = 2l'O" + lO" + gsin(O).
(4)
(5)
second step: The trolley moves with the maximum speed Vma x. The angle should be zero. This step should be avoided or minimised in order to optimise transport time (except for cases of high-risk load transport, such as moving uranium bars in a nuclear power station). third steo: During the time T 2, the trolley is decelerated, the cable length varies from Lm to L2 (final position of the cable length) with the lowering speed v2. It stops at the target position X T at the time Tf.
(.02 =11 °
vt
(6)
If the acceleration given to the trolley is constant during the period of oscillation of the pendulum (X" = F), the general solution of the differential equation (4) is:
21r , T2 --09 2
This leads to the reference trajectories represented in Fig. 12, Fig. 13, and Fig. 14.
where v is the speed variation of the cable length, 1o the initial cable length and t the time. When the cable length is constant, the equation of a simple pendulum is:
x" = O"lo + gO.
2to O) 1
For a slight deviation of the load (O < 10°) and for linear variations of the cable length, equation (4) becomes:
x" = O"(lo + vt) + O'2v + gO
, T1
4. OPTIMAL CONTROL
4.1. Linearised state equations of the overhead crane In the case of a small angular deviation, system (3) becomes :
X" = A . X + B.U
(9)
with: X r = (6)', 6), X', X), U = P
O= A c o s ( ~ t ) + B s i n ( ~ t ) + C.
(7)
Assuming that the acceleration at the time t = 0 is 1", the angle and the linear speed are zero at this time. For the angle {9, the angular velocity O', and the angular acceleration O", are: 0
r
g
vt
0" =--/~lcos(~f~t)
g
,
o" =
F
.
g
(8)
A=
1
0 0
0 0
0
1
B=
The control principle with reference trajectories is to compare the outputs of the process with the reference trajectories as illustrated in Fig. 3. The aim is to minimise the error between the ideal trajectories and real trajectories of the crane.
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A. Benhidjeb and G.L. Gissinger XT
Lm
Vmax
measurements until the time t } which minimises the quadratic criterion. The optimal control is given by : UI
= -KaX
(13)
a
and the global acceleration of the control of the system (9): Fig. 3. Control with reference trajectories I
with: U T the trolley control, and U W the winch control.
EXc =Xc-Xcmf, eu=U-Ure f e'xc = A. exc + B. e u
(10)
(11)
4.2. Linear Quadratic Gaussian control Grassin, et al., (1991) studied Linear Quadratic Gaussian control in order to minimise the load sway and to stabilise the position of the trolley. Therefore, only the principle of this method and the global acceleration control of the system will be presented (for more details, see (Grassin, et al., 1991)). The principle of this method is to perform an unbiased state-space feedback with regard to constant disturbances of part of the state (in particular for O), and to develop an observer for the unmeasured variables of the system by assuming that the other variables are well known. Therefore a new extended state-space vector is introduced. It consists, first, of the derivative of eX(t) and second, of the tracking error eX(t) = X - Xrefi The state equation of the initial system with disturbances is : e'x = A e x + B e e = CE x
U+d
(12)
with: C = (0 0 0 1), d is a Wiener process (d' = ~), a white noise centred so that E{~(t+x).~(t) T} = M 8(0, M a positive definite matrix, 8 the Dirac distribution and e the tracking error (x-xref). The state variables are decomposed into two parts: a part Xr which has to be reconstructed (here Xr = e'0) and a part Xm (here Xm T = (e0 e'x ex)) which is supposed noise-free or only affected by a constant disturbance. 4.3. Optimal control The control uI is calculated in the admissible domain U defined as : U = { u I / uI(t) function of
U(t) = Uref (t) - Klr (Z(t)
+LrXra(t)) - KtmX m(t) -
K e
I e(v)dv
To calculate the control to implement, system (9) is linearised, assuming that the cable length is constant, which of course does not reflect the reality of the load transport. The LQG command is a quadratic one with an increased state and an optimal observer. Industrial implementation of such control without any simplification is complex and difficult, as some important variables for controlling the overhead crane are not easy to measure or not measured at all. In real cases, angular sensors are not used for different reasons, such as reliability, permanent offset on the measurement of the angle, noise, etc. Choosing the weighting matrices Q and R is an equally difficult task. Hence the necessary use of a controller which translates the orders of an experienced observer that appreciates the states as linguistic variables. Finally, the considered state representation for the development of the LQG controller is based on a state vector composed of the trolley position, load sway, cable-length and their respective first derivatives. Controlling the load position with an analytic controller implies the use of a dynamic mathematical model. Such control is obtained to the detriment of other state variables, among others the variation of the cable length and that of the load mass (Boustany and d'Andr6a-Novel, 1992; Retz, 1992).
5. FUZZY CONTROL METHOD
5.1. Fuzzy control Fuzzy control systems are rule-based systems in which a set of fuzzy rules represents a control decision mechanism to adjust the effects of any causes coming from the system. The aim of fuzzy control systems is normally to substitute a fuzzy rule-based system for a skilled human operator. Fig. 4 shows the basic structure of a fuzzy control system (described in (Jamshidi, et al., 1993)).
Fuzzy Control of an Overhead Crane Performance
+•eference Input
System Input
,I
--
I
System Output
Fig. 4 . Fuzzy control system In general, the presence of an experienced operator is often essential to make adjustments (to adapt the controller to a new running point, to correct perturbations which the mathematical model has not foreseen, etc.). Human judgement is required when a phenomenon or state cannot be quantified numerically without misrepresenting its significance according to its context. Such phenomena become even more acute when, after analysing or solving a problem, the information has to be transmitted to a human operator or a machine for a final decision. A human operator can qualify a state according to its environment and a dynamic phenomenon through tendencies. In general, a control engineer will use classic algorithms (of the classic PD type, for example) if the system can be modelled and fuzzy algorithms (of the fuzzy PD type, for example) if the system cannot be modelled (or can only be partly modelled). The first step in implementing the controller consists in fuzzifying either the states to control and their first derivatives, or the error and the error variation of the states considered. For the overhead crane, the angular error and the position error were considered as fuzzy state variables, since the measured error has to be minimised by comparing the measured values to the corresponding reference values. The four fuzzy state variables are: the angular error e0, the variation of the angular error AE0, the position error exc, the variation of the position error Aex c, and the linear acceleration F as the control variable. In practice, the angular speed 50 can be approximated by the expression: A t = 0 t - Or_ 1. Such an approximation is made possible since the variables are considered as fuzzy ones (imprecise data) and not as precise ones (even if measurement noises are amplified). C 0 = Omes -- Oref
~'Xc ----X c , m e s - X c , r e f
Z~E 0 = A O m e s - A O r e f
A e X c = Z ~ c , m e s -- L~Xc,re f
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Similarly, the control variables are quantified in five trapezoidal fuzzy sets : "medium acceleration" (MA), "slight acceleration" (SA), "zero acceleration" (ZA), "slight braking" (SB), "medium braking" (MB) (Fig. 5c). The terminals of the different fuzzy sets considered for each state variable are the following: NM=[-10
-10 -6.5]
NS=[-8.5
-0.5 -1.5]
ZR=[-3.5 0 +3.5] PS=[+I.5
+0.5 +8.5]
PM=[+6.5
+10 +10]
N M = [-0.1 - 0.1 - 0.065] NS = [-0.085
- 0.05 - 0.015]
ZR = [-0.035 0 + 0.035]
PS=[+0.015
+0.5 +0.085]
PM=[+0.065 MB=[-1
+1 +1]
-1 -0.9
SB = [-0.9
-0.55]
- 0.55 - 0.45
ZA=[-0.45
- 0.075]
-0.075 +0.075
SA=[+0.075 +0.45 +0.55 MA=[+0.55
-10
+0.45] +0.9]
+ 0 . 9 +1 +1]
-6.5
-3.5
0
3.5
6.5
+10 cm
Fig. 5a. Fuzzy performance index (Position)
0.5 0 -0.1
-0.065
-0.035
0
0.035
0.065
+0.1rd -
Fig. 5b. Fuzzy performance index (Angle) 1(1-3
F=Fme,-Frey
The state variables considered can be positive, negative or zero. If they are quantified as "small" and "medium", five triangular fuzzy sets are obtained (membership functions): "negative medium" (NM), "negative small" (NS), "zero" (ZR), "positive small" (PS), "positive medium" (PM) (Figs 5a and 5b).
0.5 0
-1
-0.55
-0.075 0.075
0.55
Fig. 5c. Fuzzy performance index (Control)
+1 m/s2
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A. Benhidjeb and G.L. Gissinger
Experimentation as well as the precision and maximum values of the different position and speed sensors determined the choice of the terminals. For the crane, Mamdani's controller (1974) was used. Fuzzy control rules can be described by the triplet (ZR, PS; AS) for example. This triplet can be interpreted with the following implication:
If 0
IS ZR and AO IS PS then F IS AS
These control rules are represented in a matrix form called "Fuzzy Associative Memories" (FAM). Such representation is easy to implement and fast to perform in real time (Kosko, 1986). The two variables to control are the load sway and the load position. Two sets of rules in the form of FAM were chosen to control the overhead crane. These FAMs are a matrix representation of five lines and five columns (5,5). In each column a fuzzy set quantifies the angular error or the position error, and in each line a fuzzy set quantifies the variation of the angular error (angular speed) or the variation of the position error (linear trolley speed). The angle matrix representing the set of rules for the control of the load sway is shown in Fig. 6 and the position matrix representing the set of rules for the control of the load position is shown in Fig. 7. There is an output acceleration value for each pair formed by the angular error and its variation (position error and its variation). The 25 inputs of the matrix represent a subset of the 125 (53 ) possible rules of the two antecedents. In practice, most inputs are empty (white boxes in Figs 6 and 7). At the start, all the cells generate rules, which gives 25 control rules for each variable. But after several tests it appears that some control rules do not affect the behaviour of the overhead crane (low frequency of appearance) and bring no improvement. So, they may be left out. The simplest strategy to complete the cells is to select those which appear most often (high frequency of appearance). But if such a case occurs during transport, fuzzy logic does not allow the controller to react; this case is then called "pruned set" (Jamshidi, et al., 1993). In fuzzy control, control rules can be weighted, which gives them priority of performance. For the overhead crane, priorities were defined for each variable (load sway and position). From the hoisting to the target point more importance was given to the rules controlling the load sway. A minimum sway during this phase is essential for the safety of the load being transported. However, at the target point, the rules controlling the load position are given more weight. The principle is to place the load at the reference point even if the deviation angle is not zero.
Therefore, nine rules have been defined for the control of the deviation angle (grey boxes in Fig. 6), while sixteen have been defined for the control of the load position (grey boxes in Fig. 7). The fuzzy control rules can be described by the triplet (ZR,PS;SA) which can be expressed by the following implication:
If the angular error e o is ZR and the variation of the angular error AeO is PS, then the trolley acceleration F is SA I
I
NS ]ZIP.] PS PM NM
MB~
SA MA
NS SB
SA
ZR MB SB ~
SA MA
PS MB SB ~ M A
Im',M
PM MB
Fig. 6. "Angle" matrix
Fig. 7. "Position" matrix
5.2. Defuzzification There are several methods of defuzzification: the method of the centre of gravity, the method of the maximum, and the method of the mean value of the maxima. The choice of one of these methods mainly depends on the nature of the system and on the necessary computation time. Indeed, even if the centre of gravity method is most often used in fuzzy control, it could not be used for the overhead crane during tests because of the calculation time. That is why the maximum method was chosen; this, as its name indicates, gives the maximum values of fuzzy sets obtained by the implication. The example in Fig. 8 illustrates the maximum method for the rules (NS,ZR;SB) and (ZR,ZR;ZA) where the value of the angular error is found in both fuzzy sets SA and ZA (the variation of the angular error is ZR in both rules). As the output acceleration is different (SB for the first rule and ZA for the second one), the maximum value will be chosen. Rule I : (NS, ZR;$B) Rule 2 : (ZR, ZR;ZA) .for 5B , pO = 0.2 for ZA , ~0 = 0.8
then : pF = max[SB, ZA]=0.8
FP
AZ
Fig. 8. First maximum method
Fuzzy Control of an Overhead Crane Performance
1693
5. 3. Control of the load position at the target point
1. If ex IS PS Then F IS SB
Controlling the load position with an analytic controller implies the use of a dynamic mathematical model. Such control is performed to the detriment of other state variables, among others the variation of the cable length. The differential equations binding the different variable are strongly coupled.
The linguistic translation of this rule is as follows: If the position error is positive small (the load is ahead of the trolley position), the trolley must then be moved in the opposite direction (slight braking). 2. If e x IS NS Then F IS SA
The second problem is the variation of the load mass. Simulation tests showed that if the controller is computed for a 1,500-kilogramme load, a 100kilogramme variation of the load mass degrades all the results (horizontal and vertical position of the load, load sway, cable length and trolley position). Finally, the different tests that were carried out show that a single controller cannot control the trolley position and the load position at the same time. Indeed, the order necessary for the control of the load position is the opposite of the one necessary for the control of the trolley position, which is the case at the target point of the load. The basic idea is to link the fuzzy controller presented in Section 5.1 (which allows the control of the load position from the hoisting point to the target point) with a second one which is completely independent of the first one and only works at the target point of the load. As the orders of these two controllers are opposite, certain conditions can be defined to allow either controller to work: - if the load sway is -3 ° < O < +3 ° , the second controller does not work; the only fuzzy rules to play a part are the ones defined in the matrix position presented in Fig. 7. - if the load sway is -10 ° < O < -3 ° and +3 ° < O < +10 °, if the load position is (Xtarget - 5 cm) < Xc < (Xtarget + 5 cm) and if a constant disturbance maintains the load in this position, then the second controller starts working. In this case, two triangular fuzzy sets are defined for the load position : "positive small" (PS) and "negative small" (NS), and two trapezoidal fuzzy controller sets : "slight acceleration" (SA) and "slight braking" (SB). Fig. 9a and Fig. 9b show the shapes of these membership functions.
0.5 o
-10
-5
0
+5
+10 crn
Fig. 9a. Fuzzy performance index (Load Position)
The linguistic translation of this rule is as follows: If the position error is negative small (the load is behind the trolley position), the trolley must then move in the opposite direction (slight acceleration). Fig. 10 illustrates the principle of the second rule.
,-x
: 7....J
~11
~
-
-W -i n d
Fig. 10. Disturbance on the load The methods of implication and defuzzification that were used are the same as the ones described in Section 5.2, that is, Mamdani's implication and the maximum method. As may be noted, two fuzzy rules are sufficient to solve one of the crucial problems in controlling an overhead crane, which shows the advantage of fuzzy control over analytic control based on complex mathematical models. 6. EXPERIMENTAL RESULTS A field test was performed using an experimental crane, as shown in Fig. 11. The specifications are summarised in Table 1.
Table 1 Specifications of the experimental crane Crane span Crane height Maximum trolley velocity Maximum rope velocity Maximum load mass Angul~r sensor Transmission strap . . . . . . . Mo'cr 1 ~
900 [mm] 600 [mm] 0.6 [m/s] 2.0 [m/s] 480 [g] Lowering motor
"\
"\
I ',, [ ~
J
jl
Trolley /~ Djyp~acement bar
1 . . . . . .
A O' -1
-O.5
O
+O.5
+1 m / s 2
/
/o
Fig. 9b. Fuzzy performance index (Control) Two fuzzy control rules are sufficient to carry the load to its reference point.
trolley
/
~
Fig. 11. Experimental Model
,c
1694
A. Benhidjeb and G.L. Gissinger
Fig. 12 shows the behaviour of the crane in the extreme case of load transportation (480 g). These results show that the fuzzy controller and the optimal controller are not sensitive to significant variations of the load transportation; the results obtained coincide with the reference trajectories, which justifies the choice of a kinematic mathematical model (in the case of optimal command) where the mass does not explicitly appear in the equations.
the conditions of use and the normal wear of the railwheel contact points make such an operation impossible. In Fig. 14, at the time tl, when the outputs arrive at their target point, the load is displaced about a constant angle ®0, and maintained in this position. The results with the fuzzy controller show that, with the two rules described in Section 5.3, this displacement is compensated for, which is not the case for optimal control. When the controller is being designed, the mathematical model used is based on a state representation where only the deviation angle of the load and the trolley position are considered. Therefore, the load cannot reach its reference point.
Fig. 13 shows that for a small travelling speed the action of Coulomb's friction is more important; so, in the case of optimal control, the trolley does not arrive at the target point, contrary to the results obtained with the fuzzy controller, where the trolley and the load arrive at the target position.
A dynamic mathematical model is necessary to control the load position, but, as specified before, the use of such a representation leads to a mathematical model that is strongly coupled and parametrized according to the load and, in that case, the performance of the controller decreases with variations of the load mass and of the cable length.
Indeed, in the case of the fuzzy controller, dry friction is considered as a measurement noise and so is included in the different membership functions. To overcome this problem in the case of analytic control this dry friction has to be compensated for. This can be done either by adding an integrator on the position that will increase the command so as to move the trolley (which only applies for one working point) or by using an identification and compensation method. But a mathematical model or identifications cannot always reflect the physical reality of the systems. Indeed, the nature of the physical system,
Indeed, most authors (Grassin, et al., (1991) among others) consider that the position of the load is not directly measurable (and can only be reconstructed in the case of dynamic mathematical models parametrized according to the load) and that this would necessarily lead to calculation or measurement errors.
LQG results Trolley position (m)
Fuzzy controller results qfirolley positio n (m)
t°IF 046
0.6 0.4 0.2 00
1 2 Ti3em(s) Load position (m)
4
6
O0
1
0.6 0.4 0.2 1 Angle (rd)
2
Ti3el~(s)
4
6
1
2
Load position (m)
0.20.40.60.8
00
........ 3 Time (s)
~
4
5
6
.......
00
1 Angle (rd)
2
3 Time (s) ,
4
6
,
0.050
oF
-0.05 t/
-0.05 0 i 2 Ti 3~,1,~(s) Trolley speed (m/s) 0.4 0.3 0.2 0.1 00
1
2
4
..... T i 3 e (s) 4
Fig. 12. Load Mass = 480 g
6 -0.1 0
-%. I
1
2
"l'i"~3~(s)
4
3 Time (s)
4
Trolley speed (m/s) 0.4 0.3 0.2 0.1 6 O0
1
2
6
Fuzzy Control of an Overhead Crane Performance
Fuzzy controller results
LQG results
T r o l l e y position (m)
o
0.4 0.3 0.2 0.1 00
0.4
Trolley position (m)
0.3 0.2" 0.1 2
4 Time (s)
L o a d position (m)
6
,"
00
2
2
4 Time (s)
Angle (rd) 0.04 0.02 0 -0.02 -0.04 0
4
6
8
4 Time (s)
6
8
Time (s)
L o a d position (m) 0.4 0.3 0.2 0.1
0.4 0.3 0.2 0.1 . 00
1695
~° "
°o
2
Angle (rd) 0.03. 0.02 / ",, 0.0~)
2
Trolley speed (m/s)
4 Time (s)
-0.01 0.02 8 -0.03 0 2 Tim 4,~(s) Trolley speed (m/s)
6
0.15 0.1 0.05
00
2
4 Time (s)
.
.
.
.
.
.
.
6
0
1
0.2 0.15 0.1 0.05
_ _ _ Y
6
8
, ;.
00
Ti4em (s)
2
6
Fig. 13. Travelling speed = 0.15 m/s, Target position = 0.4 m
Fuzzy controller results
LQG results
T r o l l e y p o s i t i o n (m) 0.8 0.6 0.4 0.2 00
2
T r o l l e y ~ 0.8 0.6 0.4
t
4
L o a d p o s i t i o n (m)
.
6 Time (s)
.
.
.
.
0.2
.
8
10
00
,
I tl
2
4
0.8 ["
'f"
...... 6
8
'
'---
6
8
10
6
8
10
10
Time (s)
L o a d position (m)
06 oo4 00
j
---'--~-'~
0.6 0.4 0.2 2
4
6
8
I0
Time (s)
Angle (rd) 0.2 0.1 o -o.1
00 2 A n g l e (rd) 0.2 0.1
4 Time (s)
_0. O
-0.2 0
2
4 Time (s~6
8
10 -0.2 0
o. --
6 Time (s)
H
0.4 " 0.2
~, 4'
Time (s)
o6L/
0.4 "
, 2
4
Trolley speed (m/s)
T r o l l e y s p e e d (m/s)
-0.2 / 0
2
8
10
Fig. 14. Disturbance on the load position at the time t l
_o.° . 0
, 2
~
4
....... ;
Time (s)
8
lo
1696
A. Benhidjeb and G.L. Gissinger
The results of the fuzzy controller are better in each case of load transportation, particularly for low-speed transportation. In the case of classic control, the different forms of friction have to be modelled appropriately when the controller is being designed. Modelling such perturbations perfectly is practically impossible. The primary motivation of a fuzzy controller is that it can exploit the tolerance for some inexactness and imprecision, such as Coulomb's friction. 7. CONCLUSION Two different ways of controlling an experimental crane were developed and applied : fuzzy control and optimal control. The state representation considered for the development of the LQG controller is based on a state vector composed of the trolley position, load sway, cable length and their respective first derivatives. Controlling the load position with an analytic controller implies the use of a dynamic mathematical model. Such control is obtained to the detriment of other state variables, among others the variation of the cable length and that of the load mass. Experimentation as well as the precision and maximum values of the different position and speed sensors determined the bounds between the fuzzy variable domains, in fuzzification as well as in defuzzification, for the design of the fuzzy controller. It is easier to obtain correct results rapidly, but it takes more time to optimise them because of the numerous parameters. The experimental test indicates three principal results. First, the fuzzy controller is fully capable of controlling the crane when the angle is difficult to measure or marred by noise, and when the second variable to control is the load position. Second, the fuzzy controller can be implemented without any mathematical model. The third and last result, perhaps the most important one, is that only the fuzzy controller can take account of different types of disturbances and needs no modelling to do so.
8. REFERENCES Benhidjeb, A. and G. Gissinger (1993). Commande automatique d'un pont roulant; validations temps rrel des algorithmes. J.T.E.A. HAMMAMET, TUNISIE. Vol. 1, pp. 195-199. Boustany, F. and B. d'Andrra-Novel (1992). Adaptive Control of an Overhead Crane using Dynamic Feedback Linearization and Estimation Design. Proc. of the 1992 IEEE; international Conference on Robotics and Automation. Nice, France. pp. 1963-1968.
Butler, H., G. Honderd and J. Van Amerongen (1991). Model Reference Adaptive Control of a Gantry Crane Scale Model. IEEE Control Systems Magazine. Vol. 11 No. 1, pp. 5762. Grassin, N., T. Retz, B. Caron, H. Boud~se and E. Irving (1991). Robust Control of a travelling crane. European Control Conference, Grenoble, France, pp. 2196-2201. Jamshidi, M., N. Vadiee and T. J. Ross (1993). Fuzzy Logic and Control, Prentice Hall series on Environmental and Intelligent Manufacturing Systems, New Jersey. Kosko, B. (1986). Fuzzy Associative Memory Systems, Fuzzy Expert Systems, A. Kandel (Ed.), Addison-Wesley, Reading, MA. Mamdani, E. H. (1974). Applications of Fuzzy algorithms for control of a simple dynamic plant. Proc. of IEEE, Control and Science, Vol. 121, No 12, pp. 1585-1588. Manson, G. A. (1982). Time-Optimal Control of an Overhead Crane Model. Optimal Control Applications & Methods, Vol. 3, No. 2, pp. 115-120. Retz, T. (1992). Commande automatique optimale d'un pont roulant. Thdse de Doctorat. Centre de Recherche en Automatique de Nancy. Ecole sup~rieure des sciences et technologie de l'ing~nieur de Nancy. Siguerdidjane, H. B. (1991). Non-linear Control Tracking of an overhead crane. European Control Conference, Grenoble, France, pp. 1137-1138. Yamada, S., H. Fujikawa, O. Takeuchi and Y. Wakasugi (1989). Fuzzy Control of the Roof Crane. IECON Proceedings. 15th Annual Conference oflEEE, Philadelphia, pp. 709-714. Yasunobu, S. and T. Hasegawa (1986). Evaluation of an Automatic Crane Operation System based on Predictive Fuzzy Control. Control Theory and advanced Technology, Vol. 2, No 3 , pp. 419-432.
Pergamon 0967-0661(95)00181-6
FUZZY CONTROL OF AN OVERHEAD CRANE PERFORMANCE COMPARISON WITH CLASSIC CONTROL A. Benhidjeb and G.L. Gissinger Universit~ de Haute Alsace, Laboratoire de Mod~lisation et d'ldentification en Automatique et M~canique, M.1.A.M/E.S.S.A.I.M/IAR, BP 2438-68067 Mulhouse Cedex, France
(Received December 1994; in final form September 1995)
Abstract: This paper describes the comparison of a fuzzy logic control system with the Linear Quadratic Gaussian control (LQG) of an overhead crane, considering the applicability of the control algorithms in real time and assuming that the model is representative of the real system. A number of possible perturbations are examined in a study of the robustness of the control algorithms. A complete reference trajectory model is also presented. Keywords: overhead crane, fuzzy control, LQG control, optimal control, reference trajectory.
1. INTRODUCTION The design of a controller is usually based on classic or modern control; adaptive control (Boustany and d'Andr6a-Novel, 1992), optimal control (Benhidjeb and Gissinger, 1993; Grassin, et al., 1991; Manson, 1982; Retz, 1992), and non-linear control (Siguerdidjane, 1991). Common to both approaches are the requirements that the plant to be controlled can be described in a rigid mathematical form. Real-time control requires some simplification of the experimental model. Human intervention is always necessary for this type of control. And the choice of the state to control is fundamental, since the choice of the angle of the pendulum and position of the load are essential for precise, robust control of the crane. This leads to a very complicated state representation and the given model is non-linear.
aforementioned imprecise information into a quantitative control system would prove very difficult. However, the theory of fuzzy sets can be used to describe such information directly and can be implemented in the development of a fuzzy controller. Yasunobu and Hasegawa (1986) and Yamada, et al., (1989) studied fuzzy control in order to minimise only the angle of the pendulum (which is not easily measured), but this is not sufficient to control the load position of the crane. This paper presents a comparison between the fuzzy control method and the classic control method with reference trajectories. Besides Linear Quadratic Gaussian control, there are other methods of classic control, such as adaptive control (Butler, et al., 1991), generalised predictive control (Retz, 1992), optimal control (Benhidjeb and Gissinger, 1993), etc. But to the authors' knowledge, only LQG control has been tried and tested on an industrial travelling crane. To be validated, the results obtained with the fuzzy controller must be compared with those obtained with LQG control.
In the case of classic control, robustness is not guaranteed for a variation of the load transport. Manson (1982) has concluded that "Optimal solutions are only attainable when dealing with a very idealised model of the overhead crane system". For all these reasons, a controller based on the experience of the human operator is desirable. Unfortunately, using the traditional control theory, the conversion of the
The present paper describes first the overhead crane and the mathematical model used by the classic 1687
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A. Benhidjeb and G.L. Gissinger
control law, second the complete reference trajectories developed considering the hoisting and lowering of the load, third an optimal linear quadratic control law, fourth the fuzzy controller and fifth the experimental crane and the experimental results obtained with both controllers.
2. SYSTEM ANALYSIS
X = (X1, X2, X3, X4, X5, X6) T ----(X, l, O, X', l', 0") r .
There are two control variables: the first one operates through the hoisting motor (winch), the second one through the hauling motor (trolley). Two types of control variables can be considered: -dynamic variables (force and torque) -kinematic variables (linear and angular accelerations).
2.1 System description The overhead crane considered is made up of a platform (Fig. 1) moving along the horizontal axis, equipped with a winch (Fig. 2) around which a cable holding the load is wound.
Ct or
cg
Fig. 1. Trolley
In the case of acceleration control, the control vector is: U = (Ul, U2) T ~-- (X'P, if2*) T
and the state-space equations are:
x
Y ~l-
The state-space vector is expressed as follows:
X{ = X4
(al)
X~ = X5
(a2)
X~ = X6
(a3)
X~ = u 1
(a4) (3)
X~ = - R u 2
(a5)
X~ =-~2 (Ul COS(X3)-gsin(X3)- XsX6)
(a6)
Fig. 2. Winch
2.2. Mathematical model for classic control law Differential equations describing the system mx" = F - Tsin(0) - Ff Mcx"c = Tsin(0)
(1)
tr
Mcy c = Mcg - Tcos(0) with: m the mass of the trolley, Mc the mass of the load, x the position of the trolley, x c the horizontal position of the load, Yc the vertical position of the load, F the external force generated by an electric motor, T the tension force, O the angle of the pendulum, and Ff the friction force. Differential equations describing the winch Jto' = C - RT l" = -Ro9 1"-
(2)
RC
R2T +.-J J
with: J the moment of inertia, m the angular velocity, R the radius of the winch, and C the applied torque. State-space equations of the process. In this case, the variables considered are: X the trolley position, 1 the rope length, and 19 the angle. This type of representation is based on the cylindrical co-ordinates.
It can be noted that the relations obtained from such a representation are only of the kinematic order. This state form is then linearised locally (constant cable length, low angular deviation) and leads to reference trajectories that are compatible with the input and output of the system. A robust control law can then be defined which will allow great variations of the load. It must also be noted that this system is entirely decoupled according to both inputs, since "u2" only affects (a2) and (a5) and the control "Ul" in (al), (a3), (a4) and (a6). So the cable length can be set separately and considered as a variable that is perfectly known at any time. Using kinematic variables as control variables requires motors which respond fast in speed with unit gains to input orders (the masses of the load and the trolley only play a part in the transfer functions of the hauling and hoisting motors). The input voltages of the motors are proportional to the rotation speeds of the motor-shafts at the output. The controls obtained with different algorithms are accelerations which must then be integrated. The input orders of the motors are speed targets, and servo-controlled motors are necessary to follow these orders properly. Two types of control can be used:
Fuzzy Control of an Overhead Crane Performance acceleration control, wlalch is current or power control and which allows the application of force control. speed control, which was chosen later on as it allows direct control of the motors' speed. -
1689
With these three equations and the differential equation (4), the behaviour of the processes can be described in three steps:
-
The motors are speed-looped by correctors with proportional and integral effect (PI).
first step: During the time T1, the trolley is accelerated, the cable length varies from L1 (initial position of the cable length) to L m (medium position of the cable length) with the hoisting speed Vl.
3. REFERENCE TRAJECTORIES
°91= ~
The reference trajectories represent the ideal behaviour of the crane (minimum transport time and minimum angular position of the load) and will be used in the controller in the case of optimal control. The non-linear kinematic equations (3) are used to determine a feedforward control which gives the reference trajectories for the state and control variables. The state-space representation obtained with acceleration control shows that the equation of the pendulum formed by the load and by the trolley is described by:
x"cos(O) = 2l'O" + lO" + gsin(O).
(4)
(5)
second step: The trolley moves with the maximum speed Vma x. The angle should be zero. This step should be avoided or minimised in order to optimise transport time (except for cases of high-risk load transport, such as moving uranium bars in a nuclear power station). third steo: During the time T 2, the trolley is decelerated, the cable length varies from Lm to L2 (final position of the cable length) with the lowering speed v2. It stops at the target position X T at the time Tf.
(.02 =11 °
vt
(6)
If the acceleration given to the trolley is constant during the period of oscillation of the pendulum (X" = F), the general solution of the differential equation (4) is:
21r , T2 --09 2
This leads to the reference trajectories represented in Fig. 12, Fig. 13, and Fig. 14.
where v is the speed variation of the cable length, 1o the initial cable length and t the time. When the cable length is constant, the equation of a simple pendulum is:
x" = O"lo + gO.
2to O) 1
For a slight deviation of the load (O < 10°) and for linear variations of the cable length, equation (4) becomes:
x" = O"(lo + vt) + O'2v + gO
, T1
4. OPTIMAL CONTROL
4.1. Linearised state equations of the overhead crane In the case of a small angular deviation, system (3) becomes :
X" = A . X + B.U
(9)
with: X r = (6)', 6), X', X), U = P
O= A c o s ( ~ t ) + B s i n ( ~ t ) + C.
(7)
Assuming that the acceleration at the time t = 0 is 1", the angle and the linear speed are zero at this time. For the angle {9, the angular velocity O', and the angular acceleration O", are: 0
r
g
vt
0" =--/~lcos(~f~t)
g
,
o" =
F
.
g
(8)
A=
1
0 0
0 0
0
1
B=
The control principle with reference trajectories is to compare the outputs of the process with the reference trajectories as illustrated in Fig. 3. The aim is to minimise the error between the ideal trajectories and real trajectories of the crane.
1690
A. Benhidjeb and G.L. Gissinger XT
Lm
Vmax
measurements until the time t } which minimises the quadratic criterion. The optimal control is given by : UI
= -KaX
(13)
a
and the global acceleration of the control of the system (9): Fig. 3. Control with reference trajectories I
with: U T the trolley control, and U W the winch control.
EXc =Xc-Xcmf, eu=U-Ure f e'xc = A. exc + B. e u
(10)
(11)
4.2. Linear Quadratic Gaussian control Grassin, et al., (1991) studied Linear Quadratic Gaussian control in order to minimise the load sway and to stabilise the position of the trolley. Therefore, only the principle of this method and the global acceleration control of the system will be presented (for more details, see (Grassin, et al., 1991)). The principle of this method is to perform an unbiased state-space feedback with regard to constant disturbances of part of the state (in particular for O), and to develop an observer for the unmeasured variables of the system by assuming that the other variables are well known. Therefore a new extended state-space vector is introduced. It consists, first, of the derivative of eX(t) and second, of the tracking error eX(t) = X - Xrefi The state equation of the initial system with disturbances is : e'x = A e x + B e e = CE x
U+d
(12)
with: C = (0 0 0 1), d is a Wiener process (d' = ~), a white noise centred so that E{~(t+x).~(t) T} = M 8(0, M a positive definite matrix, 8 the Dirac distribution and e the tracking error (x-xref). The state variables are decomposed into two parts: a part Xr which has to be reconstructed (here Xr = e'0) and a part Xm (here Xm T = (e0 e'x ex)) which is supposed noise-free or only affected by a constant disturbance. 4.3. Optimal control The control uI is calculated in the admissible domain U defined as : U = { u I / uI(t) function of
U(t) = Uref (t) - Klr (Z(t)
+LrXra(t)) - KtmX m(t) -
K e
I e(v)dv
To calculate the control to implement, system (9) is linearised, assuming that the cable length is constant, which of course does not reflect the reality of the load transport. The LQG command is a quadratic one with an increased state and an optimal observer. Industrial implementation of such control without any simplification is complex and difficult, as some important variables for controlling the overhead crane are not easy to measure or not measured at all. In real cases, angular sensors are not used for different reasons, such as reliability, permanent offset on the measurement of the angle, noise, etc. Choosing the weighting matrices Q and R is an equally difficult task. Hence the necessary use of a controller which translates the orders of an experienced observer that appreciates the states as linguistic variables. Finally, the considered state representation for the development of the LQG controller is based on a state vector composed of the trolley position, load sway, cable-length and their respective first derivatives. Controlling the load position with an analytic controller implies the use of a dynamic mathematical model. Such control is obtained to the detriment of other state variables, among others the variation of the cable length and that of the load mass (Boustany and d'Andr6a-Novel, 1992; Retz, 1992).
5. FUZZY CONTROL METHOD
5.1. Fuzzy control Fuzzy control systems are rule-based systems in which a set of fuzzy rules represents a control decision mechanism to adjust the effects of any causes coming from the system. The aim of fuzzy control systems is normally to substitute a fuzzy rule-based system for a skilled human operator. Fig. 4 shows the basic structure of a fuzzy control system (described in (Jamshidi, et al., 1993)).
Fuzzy Control of an Overhead Crane Performance
+•eference Input
System Input
,I
--
I
System Output
Fig. 4 . Fuzzy control system In general, the presence of an experienced operator is often essential to make adjustments (to adapt the controller to a new running point, to correct perturbations which the mathematical model has not foreseen, etc.). Human judgement is required when a phenomenon or state cannot be quantified numerically without misrepresenting its significance according to its context. Such phenomena become even more acute when, after analysing or solving a problem, the information has to be transmitted to a human operator or a machine for a final decision. A human operator can qualify a state according to its environment and a dynamic phenomenon through tendencies. In general, a control engineer will use classic algorithms (of the classic PD type, for example) if the system can be modelled and fuzzy algorithms (of the fuzzy PD type, for example) if the system cannot be modelled (or can only be partly modelled). The first step in implementing the controller consists in fuzzifying either the states to control and their first derivatives, or the error and the error variation of the states considered. For the overhead crane, the angular error and the position error were considered as fuzzy state variables, since the measured error has to be minimised by comparing the measured values to the corresponding reference values. The four fuzzy state variables are: the angular error e0, the variation of the angular error AE0, the position error exc, the variation of the position error Aex c, and the linear acceleration F as the control variable. In practice, the angular speed 50 can be approximated by the expression: A t = 0 t - Or_ 1. Such an approximation is made possible since the variables are considered as fuzzy ones (imprecise data) and not as precise ones (even if measurement noises are amplified). C 0 = Omes -- Oref
~'Xc ----X c , m e s - X c , r e f
Z~E 0 = A O m e s - A O r e f
A e X c = Z ~ c , m e s -- L~Xc,re f
1691
Similarly, the control variables are quantified in five trapezoidal fuzzy sets : "medium acceleration" (MA), "slight acceleration" (SA), "zero acceleration" (ZA), "slight braking" (SB), "medium braking" (MB) (Fig. 5c). The terminals of the different fuzzy sets considered for each state variable are the following: NM=[-10
-10 -6.5]
NS=[-8.5
-0.5 -1.5]
ZR=[-3.5 0 +3.5] PS=[+I.5
+0.5 +8.5]
PM=[+6.5
+10 +10]
N M = [-0.1 - 0.1 - 0.065] NS = [-0.085
- 0.05 - 0.015]
ZR = [-0.035 0 + 0.035]
PS=[+0.015
+0.5 +0.085]
PM=[+0.065 MB=[-1
+1 +1]
-1 -0.9
SB = [-0.9
-0.55]
- 0.55 - 0.45
ZA=[-0.45
- 0.075]
-0.075 +0.075
SA=[+0.075 +0.45 +0.55 MA=[+0.55
-10
+0.45] +0.9]
+ 0 . 9 +1 +1]
-6.5
-3.5
0
3.5
6.5
+10 cm
Fig. 5a. Fuzzy performance index (Position)
0.5 0 -0.1
-0.065
-0.035
0
0.035
0.065
+0.1rd -
Fig. 5b. Fuzzy performance index (Angle) 1(1-3
F=Fme,-Frey
The state variables considered can be positive, negative or zero. If they are quantified as "small" and "medium", five triangular fuzzy sets are obtained (membership functions): "negative medium" (NM), "negative small" (NS), "zero" (ZR), "positive small" (PS), "positive medium" (PM) (Figs 5a and 5b).
0.5 0
-1
-0.55
-0.075 0.075
0.55
Fig. 5c. Fuzzy performance index (Control)
+1 m/s2
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A. Benhidjeb and G.L. Gissinger
Experimentation as well as the precision and maximum values of the different position and speed sensors determined the choice of the terminals. For the crane, Mamdani's controller (1974) was used. Fuzzy control rules can be described by the triplet (ZR, PS; AS) for example. This triplet can be interpreted with the following implication:
If 0
IS ZR and AO IS PS then F IS AS
These control rules are represented in a matrix form called "Fuzzy Associative Memories" (FAM). Such representation is easy to implement and fast to perform in real time (Kosko, 1986). The two variables to control are the load sway and the load position. Two sets of rules in the form of FAM were chosen to control the overhead crane. These FAMs are a matrix representation of five lines and five columns (5,5). In each column a fuzzy set quantifies the angular error or the position error, and in each line a fuzzy set quantifies the variation of the angular error (angular speed) or the variation of the position error (linear trolley speed). The angle matrix representing the set of rules for the control of the load sway is shown in Fig. 6 and the position matrix representing the set of rules for the control of the load position is shown in Fig. 7. There is an output acceleration value for each pair formed by the angular error and its variation (position error and its variation). The 25 inputs of the matrix represent a subset of the 125 (53 ) possible rules of the two antecedents. In practice, most inputs are empty (white boxes in Figs 6 and 7). At the start, all the cells generate rules, which gives 25 control rules for each variable. But after several tests it appears that some control rules do not affect the behaviour of the overhead crane (low frequency of appearance) and bring no improvement. So, they may be left out. The simplest strategy to complete the cells is to select those which appear most often (high frequency of appearance). But if such a case occurs during transport, fuzzy logic does not allow the controller to react; this case is then called "pruned set" (Jamshidi, et al., 1993). In fuzzy control, control rules can be weighted, which gives them priority of performance. For the overhead crane, priorities were defined for each variable (load sway and position). From the hoisting to the target point more importance was given to the rules controlling the load sway. A minimum sway during this phase is essential for the safety of the load being transported. However, at the target point, the rules controlling the load position are given more weight. The principle is to place the load at the reference point even if the deviation angle is not zero.
Therefore, nine rules have been defined for the control of the deviation angle (grey boxes in Fig. 6), while sixteen have been defined for the control of the load position (grey boxes in Fig. 7). The fuzzy control rules can be described by the triplet (ZR,PS;SA) which can be expressed by the following implication:
If the angular error e o is ZR and the variation of the angular error AeO is PS, then the trolley acceleration F is SA I
I
NS ]ZIP.] PS PM NM
MB~
SA MA
NS SB
SA
ZR MB SB ~
SA MA
PS MB SB ~ M A
Im',M
PM MB
Fig. 6. "Angle" matrix
Fig. 7. "Position" matrix
5.2. Defuzzification There are several methods of defuzzification: the method of the centre of gravity, the method of the maximum, and the method of the mean value of the maxima. The choice of one of these methods mainly depends on the nature of the system and on the necessary computation time. Indeed, even if the centre of gravity method is most often used in fuzzy control, it could not be used for the overhead crane during tests because of the calculation time. That is why the maximum method was chosen; this, as its name indicates, gives the maximum values of fuzzy sets obtained by the implication. The example in Fig. 8 illustrates the maximum method for the rules (NS,ZR;SB) and (ZR,ZR;ZA) where the value of the angular error is found in both fuzzy sets SA and ZA (the variation of the angular error is ZR in both rules). As the output acceleration is different (SB for the first rule and ZA for the second one), the maximum value will be chosen. Rule I : (NS, ZR;$B) Rule 2 : (ZR, ZR;ZA) .for 5B , pO = 0.2 for ZA , ~0 = 0.8
then : pF = max[SB, ZA]=0.8
FP
AZ
Fig. 8. First maximum method
Fuzzy Control of an Overhead Crane Performance
1693
5. 3. Control of the load position at the target point
1. If ex IS PS Then F IS SB
Controlling the load position with an analytic controller implies the use of a dynamic mathematical model. Such control is performed to the detriment of other state variables, among others the variation of the cable length. The differential equations binding the different variable are strongly coupled.
The linguistic translation of this rule is as follows: If the position error is positive small (the load is ahead of the trolley position), the trolley must then be moved in the opposite direction (slight braking). 2. If e x IS NS Then F IS SA
The second problem is the variation of the load mass. Simulation tests showed that if the controller is computed for a 1,500-kilogramme load, a 100kilogramme variation of the load mass degrades all the results (horizontal and vertical position of the load, load sway, cable length and trolley position). Finally, the different tests that were carried out show that a single controller cannot control the trolley position and the load position at the same time. Indeed, the order necessary for the control of the load position is the opposite of the one necessary for the control of the trolley position, which is the case at the target point of the load. The basic idea is to link the fuzzy controller presented in Section 5.1 (which allows the control of the load position from the hoisting point to the target point) with a second one which is completely independent of the first one and only works at the target point of the load. As the orders of these two controllers are opposite, certain conditions can be defined to allow either controller to work: - if the load sway is -3 ° < O < +3 ° , the second controller does not work; the only fuzzy rules to play a part are the ones defined in the matrix position presented in Fig. 7. - if the load sway is -10 ° < O < -3 ° and +3 ° < O < +10 °, if the load position is (Xtarget - 5 cm) < Xc < (Xtarget + 5 cm) and if a constant disturbance maintains the load in this position, then the second controller starts working. In this case, two triangular fuzzy sets are defined for the load position : "positive small" (PS) and "negative small" (NS), and two trapezoidal fuzzy controller sets : "slight acceleration" (SA) and "slight braking" (SB). Fig. 9a and Fig. 9b show the shapes of these membership functions.
0.5 o
-10
-5
0
+5
+10 crn
Fig. 9a. Fuzzy performance index (Load Position)
The linguistic translation of this rule is as follows: If the position error is negative small (the load is behind the trolley position), the trolley must then move in the opposite direction (slight acceleration). Fig. 10 illustrates the principle of the second rule.
,-x
: 7....J
~11
~
-
-W -i n d
Fig. 10. Disturbance on the load The methods of implication and defuzzification that were used are the same as the ones described in Section 5.2, that is, Mamdani's implication and the maximum method. As may be noted, two fuzzy rules are sufficient to solve one of the crucial problems in controlling an overhead crane, which shows the advantage of fuzzy control over analytic control based on complex mathematical models. 6. EXPERIMENTAL RESULTS A field test was performed using an experimental crane, as shown in Fig. 11. The specifications are summarised in Table 1.
Table 1 Specifications of the experimental crane Crane span Crane height Maximum trolley velocity Maximum rope velocity Maximum load mass Angul~r sensor Transmission strap . . . . . . . Mo'cr 1 ~
900 [mm] 600 [mm] 0.6 [m/s] 2.0 [m/s] 480 [g] Lowering motor
"\
"\
I ',, [ ~
J
jl
Trolley /~ Djyp~acement bar
1 . . . . . .
A O' -1
-O.5
O
+O.5
+1 m / s 2
/
/o
Fig. 9b. Fuzzy performance index (Control) Two fuzzy control rules are sufficient to carry the load to its reference point.
trolley
/
~
Fig. 11. Experimental Model
,c
1694
A. Benhidjeb and G.L. Gissinger
Fig. 12 shows the behaviour of the crane in the extreme case of load transportation (480 g). These results show that the fuzzy controller and the optimal controller are not sensitive to significant variations of the load transportation; the results obtained coincide with the reference trajectories, which justifies the choice of a kinematic mathematical model (in the case of optimal command) where the mass does not explicitly appear in the equations.
the conditions of use and the normal wear of the railwheel contact points make such an operation impossible. In Fig. 14, at the time tl, when the outputs arrive at their target point, the load is displaced about a constant angle ®0, and maintained in this position. The results with the fuzzy controller show that, with the two rules described in Section 5.3, this displacement is compensated for, which is not the case for optimal control. When the controller is being designed, the mathematical model used is based on a state representation where only the deviation angle of the load and the trolley position are considered. Therefore, the load cannot reach its reference point.
Fig. 13 shows that for a small travelling speed the action of Coulomb's friction is more important; so, in the case of optimal control, the trolley does not arrive at the target point, contrary to the results obtained with the fuzzy controller, where the trolley and the load arrive at the target position.
A dynamic mathematical model is necessary to control the load position, but, as specified before, the use of such a representation leads to a mathematical model that is strongly coupled and parametrized according to the load and, in that case, the performance of the controller decreases with variations of the load mass and of the cable length.
Indeed, in the case of the fuzzy controller, dry friction is considered as a measurement noise and so is included in the different membership functions. To overcome this problem in the case of analytic control this dry friction has to be compensated for. This can be done either by adding an integrator on the position that will increase the command so as to move the trolley (which only applies for one working point) or by using an identification and compensation method. But a mathematical model or identifications cannot always reflect the physical reality of the systems. Indeed, the nature of the physical system,
Indeed, most authors (Grassin, et al., (1991) among others) consider that the position of the load is not directly measurable (and can only be reconstructed in the case of dynamic mathematical models parametrized according to the load) and that this would necessarily lead to calculation or measurement errors.
LQG results Trolley position (m)
Fuzzy controller results qfirolley positio n (m)
t°IF 046
0.6 0.4 0.2 00
1 2 Ti3em(s) Load position (m)
4
6
O0
1
0.6 0.4 0.2 1 Angle (rd)
2
Ti3el~(s)
4
6
1
2
Load position (m)
0.20.40.60.8
00
........ 3 Time (s)
~
4
5
6
.......
00
1 Angle (rd)
2
3 Time (s) ,
4
6
,
0.050
oF
-0.05 t/
-0.05 0 i 2 Ti 3~,1,~(s) Trolley speed (m/s) 0.4 0.3 0.2 0.1 00
1
2
4
..... T i 3 e (s) 4
Fig. 12. Load Mass = 480 g
6 -0.1 0
-%. I
1
2
"l'i"~3~(s)
4
3 Time (s)
4
Trolley speed (m/s) 0.4 0.3 0.2 0.1 6 O0
1
2
6
Fuzzy Control of an Overhead Crane Performance
Fuzzy controller results
LQG results
T r o l l e y position (m)
o
0.4 0.3 0.2 0.1 00
0.4
Trolley position (m)
0.3 0.2" 0.1 2
4 Time (s)
L o a d position (m)
6
,"
00
2
2
4 Time (s)
Angle (rd) 0.04 0.02 0 -0.02 -0.04 0
4
6
8
4 Time (s)
6
8
Time (s)
L o a d position (m) 0.4 0.3 0.2 0.1
0.4 0.3 0.2 0.1 . 00
1695
~° "
°o
2
Angle (rd) 0.03. 0.02 / ",, 0.0~)
2
Trolley speed (m/s)
4 Time (s)
-0.01 0.02 8 -0.03 0 2 Tim 4,~(s) Trolley speed (m/s)
6
0.15 0.1 0.05
00
2
4 Time (s)
.
.
.
.
.
.
.
6
0
1
0.2 0.15 0.1 0.05
_ _ _ Y
6
8
, ;.
00
Ti4em (s)
2
6
Fig. 13. Travelling speed = 0.15 m/s, Target position = 0.4 m
Fuzzy controller results
LQG results
T r o l l e y p o s i t i o n (m) 0.8 0.6 0.4 0.2 00
2
T r o l l e y ~ 0.8 0.6 0.4
t
4
L o a d p o s i t i o n (m)
.
6 Time (s)
.
.
.
.
0.2
.
8
10
00
,
I tl
2
4
0.8 ["
'f"
...... 6
8
'
'---
6
8
10
6
8
10
10
Time (s)
L o a d position (m)
06 oo4 00
j
---'--~-'~
0.6 0.4 0.2 2
4
6
8
I0
Time (s)
Angle (rd) 0.2 0.1 o -o.1
00 2 A n g l e (rd) 0.2 0.1
4 Time (s)
_0. O
-0.2 0
2
4 Time (s~6
8
10 -0.2 0
o. --
6 Time (s)
H
0.4 " 0.2
~, 4'
Time (s)
o6L/
0.4 "
, 2
4
Trolley speed (m/s)
T r o l l e y s p e e d (m/s)
-0.2 / 0
2
8
10
Fig. 14. Disturbance on the load position at the time t l
_o.° . 0
, 2
~
4
....... ;
Time (s)
8
lo
1696
A. Benhidjeb and G.L. Gissinger
The results of the fuzzy controller are better in each case of load transportation, particularly for low-speed transportation. In the case of classic control, the different forms of friction have to be modelled appropriately when the controller is being designed. Modelling such perturbations perfectly is practically impossible. The primary motivation of a fuzzy controller is that it can exploit the tolerance for some inexactness and imprecision, such as Coulomb's friction. 7. CONCLUSION Two different ways of controlling an experimental crane were developed and applied : fuzzy control and optimal control. The state representation considered for the development of the LQG controller is based on a state vector composed of the trolley position, load sway, cable length and their respective first derivatives. Controlling the load position with an analytic controller implies the use of a dynamic mathematical model. Such control is obtained to the detriment of other state variables, among others the variation of the cable length and that of the load mass. Experimentation as well as the precision and maximum values of the different position and speed sensors determined the bounds between the fuzzy variable domains, in fuzzification as well as in defuzzification, for the design of the fuzzy controller. It is easier to obtain correct results rapidly, but it takes more time to optimise them because of the numerous parameters. The experimental test indicates three principal results. First, the fuzzy controller is fully capable of controlling the crane when the angle is difficult to measure or marred by noise, and when the second variable to control is the load position. Second, the fuzzy controller can be implemented without any mathematical model. The third and last result, perhaps the most important one, is that only the fuzzy controller can take account of different types of disturbances and needs no modelling to do so.
8. REFERENCES Benhidjeb, A. and G. Gissinger (1993). Commande automatique d'un pont roulant; validations temps rrel des algorithmes. J.T.E.A. HAMMAMET, TUNISIE. Vol. 1, pp. 195-199. Boustany, F. and B. d'Andrra-Novel (1992). Adaptive Control of an Overhead Crane using Dynamic Feedback Linearization and Estimation Design. Proc. of the 1992 IEEE; international Conference on Robotics and Automation. Nice, France. pp. 1963-1968.
Butler, H., G. Honderd and J. Van Amerongen (1991). Model Reference Adaptive Control of a Gantry Crane Scale Model. IEEE Control Systems Magazine. Vol. 11 No. 1, pp. 5762. Grassin, N., T. Retz, B. Caron, H. Boud~se and E. Irving (1991). Robust Control of a travelling crane. European Control Conference, Grenoble, France, pp. 2196-2201. Jamshidi, M., N. Vadiee and T. J. Ross (1993). Fuzzy Logic and Control, Prentice Hall series on Environmental and Intelligent Manufacturing Systems, New Jersey. Kosko, B. (1986). Fuzzy Associative Memory Systems, Fuzzy Expert Systems, A. Kandel (Ed.), Addison-Wesley, Reading, MA. Mamdani, E. H. (1974). Applications of Fuzzy algorithms for control of a simple dynamic plant. Proc. of IEEE, Control and Science, Vol. 121, No 12, pp. 1585-1588. Manson, G. A. (1982). Time-Optimal Control of an Overhead Crane Model. Optimal Control Applications & Methods, Vol. 3, No. 2, pp. 115-120. Retz, T. (1992). Commande automatique optimale d'un pont roulant. Thdse de Doctorat. Centre de Recherche en Automatique de Nancy. Ecole sup~rieure des sciences et technologie de l'ing~nieur de Nancy. Siguerdidjane, H. B. (1991). Non-linear Control Tracking of an overhead crane. European Control Conference, Grenoble, France, pp. 1137-1138. Yamada, S., H. Fujikawa, O. Takeuchi and Y. Wakasugi (1989). Fuzzy Control of the Roof Crane. IECON Proceedings. 15th Annual Conference oflEEE, Philadelphia, pp. 709-714. Yasunobu, S. and T. Hasegawa (1986). Evaluation of an Automatic Crane Operation System based on Predictive Fuzzy Control. Control Theory and advanced Technology, Vol. 2, No 3 , pp. 419-432.