NONLINEAR OBSERVERS APPLIED TO THE CONTROL OF AN OVERHEAD CRANE

Copyright 0 2002 IFAC 15th Triennial World Congress, Barcelona, Spain

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NONLINEAR OBSERVERS APPLIED TO THE CONTROLOFANOVERHEADCRANE E. Aranda-Bricaire C.P. GonzBlez-Inda * F. Plestan ** M. Velasco-Villa*>2

* Departamento d e Ingenieria Ele'ctrica, Seccio'n d e Mecatro'nica CINVESTAV, Me'xico. { earanda, cpgonzal, velasco} @mail.cinvestav. m x ** IRCCYN, UMR 6597, ECN, France. Franck. [email protected] r

Abstract: This paper addresses the asymptotic output tracking problem for the model of an overhead crane. It is shown that the model is linearizable by dynamic state feedback when the full state is available for measurement. In the more realistic situation when the angle that the rope forms with respect to the vertical axis cannot be measured, two different nonlinear observers are used in order to estimate the actual value. These are: a numerical observer and a high gain observer. Numerical experiments serve to compare the performance of the proposed schemes. Copyright 0 2002 IFAC Keywords: Observers, Mechanical systems, Feedback linearization.

1. INTRODUCTION

proposed an observer which, under the knowledge of the car position and the rope angle, estimates their velocities. In (Altafini et al., 2000), the rope angle is estimated from the knowledge of the position and velocity of the car, but assuming that the rope length is constant and the overhead crane moves the load just along one axis.

An overhead crane is an element present in most industrial environments. Precise position control of the load becomes crucial when these environments are crowed by machines, production lines, stocks, and workers. Also, production rates may be affected by slow or inaccurate displacement of raw materials or manufactured goods. The control of an overhead crane is difficult because some state variable are not available for measurement. For instance, the angle that the rope forms with respect to the vertical axis is either unavailable for measurement, or so noised that it becomes unusable for control purposes.

The mathematical model of an overhead crane is known to be linearizable by dynamic state feedback when the full state is available for measurement (Cheng et al., 1996; d'Andr6a- Novel et al., 1989). The contribution of this paper is to propose two different observers which allow to estimate the unmeasurable components of the state. More precisely, it is assumed that the trolley position, its velocity, the rope length, and its rate of change are available for measurement. The unmeasurable states are the angle that the rope forms with respect to the vertical axis, and its angular velocity.

Some works related to this problem can be found in the literature. In (Cheng et al., 1996), a dynamic control law is applied assuming that the full state is measurable. In (Bornard et al., 1993), it is Partially supported by Fkench Ministry of Research and Technology. Partially supported by CNRS, Fkance, and CONACYT, Mexico.

The first observer is directly derived from the observability property. Basically, if a system is

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observable, the state can be written as a function of the input, the output, and a finite number of their time derivatives, (the so-called generic observability property (Plestan et al., 1997)). By using this basic property, an estimation of the unmeasurable state variables can be made. This approach was introduced in (Diop et al., 1994). Regardless of its simplicity, this approach is very general, flexible and straightforward to implement in practice. Among the shortcomings of this approach, one must note that the numerical derivatives are quite sensitive to noise measurement.

1

~ ~

The second solution is based on the high gain observer theory (Gauthier et al., 1992). This class of observers is interesting because it concerns a quite general class of nonlinear systems.

-

Q

0 0 0 J + mr2 -mr2 sin 2 3 - cos 2 3 (J mr2)

+

x2

2. MODELING

1

-

Q

An example of two-dimensional overhead crane system is depicted in Figure 1.

0 0 0 m r sin 2 3 -r ( M + m sin2 2 3 ) -mr sin 2 3 cos 2 3

Xl

x2

+ -g sin 2 3 [ J m -

+M

(J+ m r 2 ) ]

Jmx2x; sin 2 3 cos 23.

Q = M J + M r 2 m + Jmsin2x3 From practical considerations, a natural choice for the system output functions is y1 = hl(x) = x1

y2 = ha(.)

= x2

+

x2 sin23 cos 2 3 .

Clearly, the pair (yl, y2) represents the position of the load on the vertical plane.

Fig. 1. Overhead Crane System The overhead crane is provided with two actuators. The first one is aimed to control the position of the trolley in the horizontal direction. The second one is aimed to control the length of the rope. However, the model of the overhead crane is underactuated because there is a third non actuated degree of freedom, corresponding to the angle formed by the rope with respect t o the vertical axis.

3. DYNAMIC STATE FEEDBACK

In order t o achieve asymptotic output tracking of a prescribed trajectory, input-output linearization will be employed. It will be shown that inputoutput linearization can not be achieved by static state feedback because the system does not possess a well defined relative degree. Therefore, the well known dynamic extension algorithm will be used in order to achieve input-output linearization. We use the Dynamic Extension algorithm as presented in (Isidori, 1995). Instead of recalling the general theory, we will just present its application to the model of the overhead crane.

In the rest of this paper we adopt the following notations: X I denotes the trolley position, x2 the rope length, 2 3 the angle that the rope forms with respect to the vertical axis, 2 4 the trolley velocity, 2 5 the rate of change of the rope length, 2 6 the rope angular velocity, u1 the force applied to the trolley, u2 the torque applied to the winch, M the trolley mass, m the load mass, r the radius of the winch, and J the moment of inertia of the winch.

The control inputs appear at the second timederivatives of the outputs; namely, ij

With these notations, the following state space model can be obtained (d’Andr6a- Novel et al., 1989)

+ L,Lfh

= L2fh(x)

(x)u

where QL2h1(x) = - MJsinx3(x2xi +gcosx3) l = M J g s i n 2 x 3 -MJx2xicosx3 QLfh2(x)

+ M r 2 g m + J g m sin2 2 3

322

L&fh (xj =

7

Q

J sin2 2 3 - M r sin 2 3 J sin 2 3 cos 2 3 - M r cos 2 3

1

where a1 =

The rank of the decoupling matrix L , L f h ( x ) equals one, and therefore the relative degree is not well defined. The first step of the Dynamic Extension Algorithm (DEA) consists in linearizing the inputoutput behavior of one scalar output. This can be accomplished by means of the following preliminary feedback u1 = u2 =

w1 QL?h2(x) Mrcosx3

a2 =

a2

Mx2 (cos x3)3 g 2 M sin 23 cos 2 3 - 2 ~ x ~ cos x 2 3~ z ~ 2gMXsX.6 cos 2 3 mzf sin 2 3 cos 2 3 g 2 m sin 2 3 cos 2 3 - g M z 1 sin 2 3 cos 2 3 - 2 g ~ x 2 x sin ; 2 3 2 ~ x 2 x ; z sin i 23 2 M x 2 z 6 z 2 cos 2 3 - 2 g m z 1 sin 2 3 cos 23

+ +

+ +

+

The decoupling matrix for the extended system is now given by -

+-J sMi nrx 3 w 1 -

Q M r cos 2 3

(z1 - g )

sin23

w2

which is generically nonsingular

This step produces

From the above developments, the standard n o n interacting control law is given by

where w1 and w2 are new control variables. The second step of the DEA consists in adding two pure integrators before the input w1 in the following manner: w2 = z1,i 1 = % a , i 2 = 42, w1 = 41, (2) where 41 and 42 are new control variables, and z1 and 22 are extra state variables.

x1 = 2 4 x2 = 25

24 =

J m sin 2 3 ( g cos 2 3

n

+ ( J + mr2)41 + ( rQmsin

25 =

Jo

=

qi q2

= =

Mr2!m( g cos 2 3

v1 = y g - kl (y'l - y ' l d ) - kz (;jl - i l d ) -k3 (GI - G l d ) - k4 (Y1 - Y l d ) v2 = y g - kl ( Y ' 2 - Y 2 d ) - k2 ( ; j 2 - ; j M ) -k3 (Ga - G 2 d j - k4 (Y2 - Y 2 d j ,

+~ 2 x 2 )

Y

2 3 j a ( x ,41, z1

(4)

Y1

Finally, to achieve asymptotic tracking of a desired trajectory the new control inputs v ~ and , v2 are chosen as

The extended s y s t e m is then given by

i 3 = 26

The application of (3) decouples the input-output response of the extended nonlinear system, producing

j

(4)

where yld and y2d are the prescribed trajectories.

+ x2xZ) - ( r 2 msin 23) 42 Q

Under certain conditions the feedback law (3)(4) presents undesirable singularities. They occur when x2 = 0, ic3 = $, or z1 = g . Fortunately, these conditions do no occur in actual overhead cranes. The length of the rope can never be zero; the swing angle of the load is not expected to sway up to go", and the value of z1, the vertical acceleration, is smaller than gravity g .

-J + r 2 m Jmx2x; sin 2 3 cos 2 3 cos 2 3 (J+ r 2 m ) q 1 (rmsinx3cosx3)a(x,qi,zi)

Note that, for the extended system, the sum of the scalar relative degrees is equal to the dimension of the state. Therefore, the zero dynamics of the system are trivial.

(x,4 1 , 4

+

=

Jsinx3ql M r cos 2 3

-

mQ[zi

+ L;h2(x)]

4. DESIGN OF NONLINEAR OBSERVERS

M r cos 2 3

After this dynamic extension, the new control variables ( 4 1 , ~ )appear at the fourth timederivatives of the outputs: -

In this Section, two solutions based on nonlinear observers are proposed for the estimation of 2 3 and 2 6 . In order to design these observers, some assumptions will be made. As a matter of fact, due to mechanical properties and operation conditions of actual overhead cranes (see Appendix

(zl- g ) sinx3 1

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Figure 2 shows the desired load trajectories along the horizontal and vertical axis. Figure 3 displays the tracking errors when the numerical observer is used in the closed loop. Finally, Figure 4 shows the corresponding control inputs.

A), these assumptions remain reasonable. These assumptions allows to obtain a simplified model for the overhead crane.

4.1 Simplified model

Desired X-Position of the load 1

In practice, due to the mechanical properties of the system, the angle 2 3 and the angular velocity 2 6 stay small even when the trolley is moving. Therefore, the following approximations will be adopted: sin23 z 23, cosx3 z 1, sin2 2 3 z 0, and 22 z 0.

08

-E 0 6 -0 4 02 0 0

1

Using these approximations, the model of the overhead crane becomes

iI

2

3

4

5

6

7

8

9

10

[SI

Desired Y-Position of the load I

I

I

I

1

2

3

4

I

I

I

,

,

5

6

7

8

9

10

7

8

9

10

08-

= 24

x 2 = 25

x3

= 26

x4 = x5 =

.

26 =

+

-06E

+ +

[ m ~Jg( ru2) (J m r 2 ) ~ 1 ] / Q [ m M r2 g - mr”3ul - rMu2]/Q (5) Ci - 2M(J m r 2 ) x 5 x 6

+

22

-

Q

(J+ mr2)u1+ rmx3u2 22

where Q = M J M(J+mr2)).

Q

-

02

0

[SI

,

Fig. 2. Prescribed trajectory

+ M r 2 m and Ci = -g23( J m +

T r x k i w emor along X-coordinates

From practical considerations, it is possible to assume that the state variables X I , 2 2 , 2 4 , and 2 5 are available for measurement. Therefore, for the

purpose of observer design, the following output function is defined: =

[GI, 5 2 , Y3, &IT

0

1

2

3

4

T

= [XI, 2 4 , 2 2 , 2 5 1

5

PI

6

T r x k i w emor along Y-coodinates

4.2 N u m e r i c a l observer -0 06

The system (5) is generically observable. More precisely, the state vector is a function of the output, the input and a finite number of their time derivatives. Explicitely, we have Fig. 3. Tracking errors with a numerical estimation of 2 3 and 2 6 . The design of a numerical observer is carried out by using directly equation (6) in order to estimate the value of 2 3 which depends on the variables &, u ~and , u2. Then, by numerical differentiation of 23, one gets 2 6 . This estimation has been used in closed loop, together with the linearizing controller, based on the certainty equivalence principle. In odrer to test the robustness of the control scheme, the measurments y3 and y 4 were contam-

4.3 H i g h g a i n observer Some basic facts about high gain observers are recalled below. For further details, the reader is referred to (Gauthier e t al., 1992; Bornard e t al., 1993). Consider the nonlinear system

inated with noise. The Simulations presented below have been made using the parameters stated in Appendix A.

(7)

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variables 2 3 and 2 6 can be observed from the output Y 2 only. This property will be exploited to obtain a high gain observer for the following subsystem

Control input u,

1

x3 = 26

.

24 = -151 0

:

1

2

4

3

6

5

PI

8

7

9

mx3(Jg

+ ru2) + ( J + mr2)u1 n ~

10

(9)

Control input u1 265

Recall that the variables Y 3 = x2 and Y 4 = 2 5 are available for measurement. Therefore, they can be viewed as known inputs signals. The first step is to find a change of coordinates which transforms system (9) into (7). For doing that, consider the following change of coordinates:

-

A=

0 1 0 0 0 1

0 0

" ' " '

<

-

. :. . . , ' . . . 0 0 0 1 0 0 0 0-

In coordinates, system (9) takes the following form

, C=[l,O,...,O]

+ ( J + mr2)u1 n ~

el = <2

" ' " '

Suppose that the following two assumptions are satisfied: Assumption A1 The function 'p is globally Lipschitzian with respect to <, uniformly with respect to u.

2; = 0, for i

Assumption A2 and j = i 1, . . . , n.

+

=

1 , . . . ,n

-

1,

Let K denote a matrix of appropriate dimensions, such that A - K C is Hurwitz, and let A be a diagonal matrix defined by

rX

o

...

Straightforward computations show that Assumptions A1 and A2 are satisfied for system (10) on the whole domain of operation. Therefore, a high gain observer of the form (8) can be employed for system ( 1 0 ) .

o1

The gain matrices are arbitrarily chosen in such a way that the observer converges. The estimated values of 2 3 and 2 6 , denoted 23 and 2 6 , are derived from the observer by using the inverse state coordinates transformation

Under Assumptions A1 and A2, system (7) is uniformly locally observable. Moreover, there exists XO such that, for all X satisfying 0 < X 5 XO, the system

<

t

= At

+ p(t, + AplK(Y ct) U)

-

(8)

with E En, is an asymptotic observer for (7). Furthermore, the observation error norm is bounded by an exponential function whose decrease speed can be chosen.

Simulations have been made introducing the observer in the closed-loop system, and under the same conditions than previously (stated in Appendix A). As for the previous scheme, the measurments Y 3 and Y 4 were contaminated with noise.

The observer developed in the sequel can be viewed as a "reduced-order" one. In fact, the state

325

,

sensitivity to noisy measurements. On the other hand, the high gain observer is relatively more complicated, but much more robust versus initial condition errors and noised measurements.

Figure 5 displays the horizontal and vertical tracking errors. Figure 6 displays the control inputs. ,

05,

,

,

,

Trackmg erroral07 X-cca?

,

,

6. REFERENCES

-015' 0

1

2

3

4

5

7

6

8

9

Altafini C., R. Frezzam (2000), Observing the load dynamic of an overhead crane with minimal sensor equipment, Proc. IEEE Int. Conf. Robotics and Automation, Vol. 2, pp. 18761881. Bornard, G., F. Celle-Couenne, G. Gilles (1993), Observabiliti: et observateurs, in SystZmes non line'aires, Chapter 5, Vol. 1, A.J. Fossard and D. Normand-Cyrot, Eds., Masson, Paris. Cheng C. and C-Y. Chen (1996), Controller design for an overhead crane system with uncertainty, Control Eng. Practice, Vol. 4, No. 5, pp. 645-653. d'Andri:a- Novel B., J . Li:vine (1989), Modeling and nonlinear control of an overhead crane, Proc. Int. Symp. MTNS, Vol. 11, pp. 523-529. Diop, S., J.W. Grizzle, A. Stefanopoulou (1994), Interpolation and numerical differentiation for observer design, Proc. American Control Conference, Evanston, Illinois, pp. 1329-1333. Gauthier J.P., H. Hammouri, S. Othman (1992), A simple observer for nonlinear systems. Applications to bioreactors, IEEE Trans. Autom. Control, Vol. 37, No. 6, pp. 875-880. Isidori A. (1995), Nonlinear Control Systems, 2nd. Edition, Springer-Verlag, New York. Plestan, F., and A. Glumineau (1997), Linearization by generalized input-output injection, Syst. Contr. Lett., Vol. 31, pp. 115-128.

I

10

[SI

Trackmg erroralow Y-coordinates

I

-0 12 0

1

2

3

4

5

7

6

8

9

10

[SI

Fig. 5. Plot of tracking errors using a high gain observer Control input u,

-10 -

-15.

'

Appendix A. NUMERICAL PARAMETERS 2 35' 0

1

2

3

4

5

6

7

8

9

I

10

The physical parameters of the model (1) are M = 3kg, m = 5kg, r = 0.05m, and J = 0.001kg. These parameters correspond to the prototype of an overhead crane that is being built at the Electrical Engineering Department of CINVESTAV. The gains of the linear feedback (4) are given by kl = 25, k2 = 221, k3 = 815, k4 = 1050. The prescribed trajectory correspond to a displacement from the point y(0) = (0,0.25) to the point y(6) = (1,0.75) (in six units of time), along a smooth polynomial trajectory of 9th. order. The initial conditions for all the simulations were taken as X I = 0.1, 2 2 = 0.35, 2 3 = 0.0, 2 4 = 0.0, 2 5 = 0.0, 2 6 = 0.5. In order to test the robustness of the proposed schemes versus noised measurements, all the simulations were carried out supposing that the variables y3 and y4 were contaminated by white noise of power p = 1 x lop6. The high gain observer ( 8 ) was tuned with the following numerical parameters: X = 0.1 and K = [3,3,1IT.

[SI

Fig. 6. Plot of the control inputs versus time, with a high gain observer for 2 3 and 2 6 . 5. CONCLUSIONS A dynamic state feedback that linearizes the model of an overhead crane has been derived. This dynamic control law depends of the angle that the rope forms with respect t o the vertical axis and its rate of change. Since these variables are not available for measurement, two different nonlinear observers have been proposed to estimate their actual values. The estimated values have been fed into the control loop. Numerical simulations show that both schemes present good performance. The main advantage of the numerical observer is its simplicity, while its main drawback is its

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