System Inversion Based Control of an Overhead Crane

Copyright @ IFAC Control Systems Design, Bratislava, Slovak Republic, 2000

SYSTEM INVERSION BASED CONTROL OF AN OVERHEAD CRANE Aurelio Piazzi * Antonio Visioli **

* Dipartimento di Ingegneria dell'Informazione, University of

Parma (Italy), e-mail: [email protected] ** Dipartimento di Elettronica per l'Automazione, University of

Brescia, (Italy), e-mail: [email protected]

Abstract: In this paper we describe a new methodology for the control of the transient sway and of residual oscillation of the payload carried by an overhead crane. The approach consists of dampening the system by a state-feedback closed-loop and applying a noncausal system inversion in order to assure a predetermined polynomial motion law for the payload. Polynomial functions are adopted in order to guarantee that the input function has a continuous derivative of an arbitrary order. Moreover, the motion time can be minimized, taking into account constraints on the actuators, by means of a simple bisection algorithm. Simulation results, based on a nonlinear crane model, show how the method is effective also when the payload is hoisted or lowered during the motion, and how it is inherently robust to parameter uncertainties. Copyright @2000 IFAC

Keywords: Overhead crane, system inversion, robust control, time optimization.

1. INTRODUCTION

increase the throughput of the crane. Different open-loop and closed-loop control solutions have been proposed in the literature, addressing the problem (Sakawa and Shindo, 1982), (Auering and Troger, 1987), (Moustafa and Ebeid, 1988), (Strip, 1989), (Butler et al., 1991), (Yoshida and Kawabe, 1992), (Cheng and Chen, 1996), (Corriga et al., 1998), (Levine, 1999). However, in many cases system uncertainties and actuators constraints are not taken into account and in other cases the controller design is done optimizing only one side of the problem, that is the travelling time of the payload is minimized or the swinging effect is reduced. In fact, these two aspect are conflicting, since the sway of the load depends on the acceleration of the trolley. In this paper we propose a new approach, based on system inversion, for the design of a robust feedforward/feedback control loop which allows to significantly reduce the transient sway and the residual oscillation of the payload, minimizing the travelling time taking into account trolley's actu-

The safety and the efficiency of the operations of an overhead crane is generally reduced by the transient sway and the residual oscillation of either the empty hook or the payload. In general, this problem is tackled by the experience and skill of the operators, which try to impose a decelerat~on that reduces the oscillation caused by the acceleration. Moreover, a man is often adopted in stopping the hook and fixing the payload. Hence, it is desirable to design a controller to reduce the swing effect. This might be considered a challenging problem, since the system, which can be regarded as a single-pendulum, is nonlinear (and hence, if a linearized model is considered for controller design, it has to be verified that the assumptions made in linearization hold during the task) and the value of certain parameters such as the rope length and the payload mass may significantly vary. Moreover, the motion time of the payload has to be minimized, subject to the trolley driving motor constraints, in order to

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ator constraints. The methodology basically consists of dampening the system by means of a statefeedback controller and then defining a desired smooth polynomial motion law of the payload. The reference input is subsequently determined by inverting the nominal closed-loop system. Finally, with a worst-case approach, the motion time can be opportunely minimized ensuring that limits on the trolley's actuator are not exceeded. Even though the entire design procedure is based on a linearized model of the crane, the achieved control performances are quite valuable, due to the inherent quasi-linearization of the high-gain feedback loop. This has been shown by using a full nonlinear crane model in simulations. The paper is organized as follows. In Section 2 the overhead crane control problem is formalized. In Section 3 the the control design methodology is explained. In Section 4 an illustrative simulation example is exposed and then conclusions are drawn in the last section.

X3. Moreover, assuming small deflection angles X3'S and small angular velocities X4 's, the dy-

namic model can be linearized by simply imposing O. Hence, the following linear state space model results:

COSX3 ~ 1, sinx3 ~ X3, sin2x3 ~ 0 and x~ ~

x = Ax+bu o1 o0 A=

0

0 0

mL -g

me 00 0 1 o 0 _ (mL + me)g 0

b=

me1

(3)

0 1 me 0 1

.(4)

me1

It has to be noted that the eigenvalues of the system are 81,2

= 0

and 2. PROBLEM FORMULATION An overhead crane can be sketched as in Figure 1, where me is the trolley mass, I is the rope length, mL is the hook/payload mass, Xl is the trolley position, X3 is the rope angle and u is the force applied to the trolley. Some assumptions can be made to simplify the model (Ackermann, 1993):

that is, a double pole in the origin of the 8plane and a couple of pure imaginary poles. It is of interest, for the following controller design approach, the evaluation of the transfer function from u to the hook/payload position YL = Xl + I sin X3 ~ Xl + IX3; it results:

a) dynamics and nonlinearities of the driving motor are neglected; b) the trolley moves along the track without friction or slip; c) the rope has no mass and elasticity; d) there is no damping of the pendulum; e) the load is regarded as a point mass.

The addressed problem is to design a feedforward/feedback strategy in order to obtain a minimum-time movement of the payload starting from a given initial position YLi = 0 to a final one YL, subject to (i) constraints on the force exerted by the actuator on the trolley, for example the values of luj and lul have to be limited, and (ii) the transient sway and the residual oscillation of the payload have to be reduced as much as possible. Moreover, it has to be taken into account that the values of the mass of the hook/payload and of the rope length are uncertain, although it is known that they belong to a given interval, that is mL E [mL:, m!] and I E [1-,1+]. Finally, the rope length can vary during the motion, in case the payload is hoisted or lowered.

It has to be noted that assumption a) is justified if the controller design assures that lul and lul are not excessively large and assumption b) is generally not very restrictive is the trolley position is somehow controlled. Hence, the plant can be modelled by two nonlinear second order differential equations (g is the gravity acceleration):

(m~. + me)XI + mL.~(x3 COSX3 -.x~ sinx3) mLxI cos X3 + m L lx3 = -mLg SIll X3

= u (1)

which, solving for Xl and X3, can be rewritten as: mL COSX3 sinx3g

+ mLlx~ sinx3 + u

-mf - me + mL cos 2 X3 cos x3mL1x3 sin X3 + u cos X3 +

-

(2)

3. SYSTEM INVERSION BASED CONTROL

1( -mL - me + mL cos 2 X3) mLg sin X3 + meg sin X3

1( -mL - me

Basically, the controller design consists of determining a suitable state-feedback control law u = k T x + v where the state-feedback vector k is necessary to dampen the system and the reference input vC) is determined by means of a noncausal

+ mL cos 2 X3)

The nonlinear model can be expressed into a state-space form by defining X2 = Xl and X4 = 188

u

g

m cls 4 + (k 4 - k21)s3+ (mLg + mcg + k3 - k l l)s2 - k 2gs - klg

(6)

where it has been emphasized the dependence of G VYL on the uncertain parameters mL and l.

3.2 Choosing the output The output function of the system (6), i.e. the payload motion law yL(t), has to be defined before applying the noncausal system inversion. The polynomial function class appears to be very suitable for this purpose since it can assure the smoothness of the input function and its derivatives of an arbitrarily prefixed order (Piazzi and Visioli, 2000). In particular, denote by G(s) the transfer function of a generic scalar system with input function u(t) and output function y(t); denote also by r the relative order of the system. Then, the following property holds.

Fig. 1. Sketch of an overhead crane. system inversion on the linearized system, once an appropriate polynomial output function has been defined. Then, the travelling time of the payload can be minimized taking into account the full nonlinearities of the crane dynamics and constraints on the control variable u and its time derivatives until a predefined order n. The different steps of the controller design are analyzed in details in the following.

Proposition 1. Assume that h ~ T. If y (t) E C( h) then it is functional reproducible by a unique u(t) E c(h-r). Considering the overhead crane system and the transfer function expressions (5)-(6), from the above property it results that yL(t) E C(h) in order to obtain u(t) E C(h-4) and v(t) E C(h-4j (note that with respect to Property 1 it is T = 4). Specifically, consider over the domain [0, T] the polynomial expression of yL(t) of order 2h + 1:

3.1 Choosing the state-feedback control vector The state-feedback control vector k = [k l k 2 k 3 k 4 ]T is necessary to dampen the system and has to be calculated in order to guarantee the robust stability of the uncertain system. Different algorithms can be employed with this purpose (see for example (Ackermann, 1993)), assuring also that the real part of the characteristic polynomial roots is less than an arbitrarily chosen real number p < O. It is important to stress that the resulting robust pole placement is not a critical issue for the performance of the overall control system, due to the adopted smooth system inversion concept, as it will be illustrated in the following. Moreover, it has also to be highlighted that, the trolley position and the rope angle can be measured by appropriate sensors and consequently also the other state vector elements can be reconstructed (Corriga et al., 1998; Virkkunen et al., 1990). Alternatively, all the state vector components can be obtained measuring only the trolley position, by means of a simple Luenberger observer, based on the nominal values of the system parameters, again without impairing the control performances, as it will be shown in the simulation examples of Section 4. Once the state-feedback controller u = k T x + v has been determined, the transfer function of the closed-loop linearized system between the reference input v and the hook/payload position YL results to be

y(t)

= C2h+lt2h+l + C2ht2h + ... + clt + Co

Determine the 2h + 2 coefficients by solving the following parameterized system with 2h + 2 linear equations:

yL(O)

= 0; = 0;

~il)(O) {

yi

h

) (0)

= 0;

yL(T)

= YL, =0

Yil)(T)

yi

h

\ T)

=0

The above algebraic system always admits a unique solution which is given by the following closed-form expression:

f t

yL(t; T)

= YL,{3(h, T)

{h(T - {)hd{,

(7)

o where { is the real integrand variable. The positive coefficient {3(h, T) is given by:

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Derivation of (7) is easy by considering that

full nonlinearities of the crane dynamics and the trolley's actuator constraints. Specifically, choosing a worst-case approach, the absolute values of the manipulative input u and its derivatives (till a prefixed order) have to be limited for any possible values of the uncertain parameters mL and 1. Formally, the following semi-infinite optimization problem has to be solved:

Moreover, yi1l(t; T) is always strictly positive in the open interval (0, T) so that yL(t; T) does not exhibit swing modes. Outside the interval [0, TJ the function yL(t; T) is equal to 0 for t < 0 and equal to y L, for t > T, so that no residual vibration occurs as well.

minT subject to (i = 0, ... , n):

3.3 Inverting the system Once the non oscillatory output function ydt; T) has been selected, the corresponding reference input function v(t; T) can be determined by means of a noncausal system inversion on the nominal linearized closed-loop system. Consider the transfer function (6) corresponding to the nominal parameters = (m"L + mt)/2, 1° = (1- + 1+)/2 and denote by V(s; T) and YL(s; T) the Laplace transforms of v(t; T) and ydt; T) respectively. Considering that both v(t; T) = 0 and Ydt; T) = 0 for all t < 0 (at t = 0- all initial conditions are zeros), it follows that

lu(i)(tjT,mL,I)I::; u~ax, Vt E [O,T], VmL E [mi, mtJ and VI E [1-,1+], where U~ax, i = 0, ... , n (0 ::; n ::; h-3) are fixed in such a way that they correspond to physical limits of the trolley's actuator. The derivatives of the control function u(t; T, mL, I) are to be evaluated considering the exact nonlinear crane model as described in (1) and (2). The optimal solution T* of the above problem can be found by means of the following typical bisection algorithm:

m1

The required system inversion on the nominal closed-loop linearized system can be simply performed by computing the inverse of G VYL (s; m1, 1°):

V(SjT) = G VyL (s;m1,10)-lYds;T) Hence, applying the inverse Laplace transform on V(Sj T) we obtain:

melo (4) V(t; T) = --YL (t; T) g

+

+ ( m1 + me +

(9)

k4

k 2 lO

-

9

k

3

~ k1 )

(3)

YL (t; T)+

YP)(t; T)+ (8)

(1) Set Tmin = O. (2) Determine an initial value for Tmax such as maxtE[O,rjlu(i)(t;Tmax,mL,I)1 ::; U~ax, i = 1, ... ,n, VmL E [m"L,mtJ and VI E [l-,I+J. (3) Set T = (Tmin + Tmax )/2. (4) If maxtE[O,r]lu(i)(tjT,mL,I)! ::; U~ax, i = 1, ... ,n, VmL E [m"L,mtJ and VI E [1-,I+J then set Tmax = T else set Tmin = T. (5) If (Tmax - Tmin) > c then goto 3. (6) Set T* = Tmax . (7) End. Note that the precision parameter c, which determines the terminal condition of the algorithm at step 5, is arbitrarily fixed. Remark 2. The initial value of Tmax can be easily found starting from a reasonable value and if it doesn't satisfy the constraints of step 2, multiplying it repeatedly by a constant >. > 1 until the condition is true. Remark 3. In order to verify the conditions of step 4 (or 2), the determination of the maximum

1 -k2 yi \t; T) - k1ydt; T) Remark 1. It is worth stressing that the expression (8) is simply formed by an algebraic linear combination of yL(tj T) and its derivatives (till to the 4-th order) without any inclusion of zero dynamics modes, due to the absence of finite zeros in G VYL (s; m1, 1°). Moreover, the final position parameter YL, appears just as a scaling factor in the expression of v(tj T) so that the reference function is easy to calculate once the travelling time T is known.

max{!u(i)(tjT,mL,I)I: t E [O,T], mL E [m"L,mt]1 E [l-,l+]} is required. This is a difficult problem because, in general, uti) (t; T, mL, I) does not admit a closedform expression in the arguments t, mL and 1. However, the problem can be reasonably relaxed by replacing the intervals [m"L, mtJ and [1-, l+j with finite discretizations.

3.4 Minimizing the travelling time

Once the reference input v(tj T) has been determined by means of expression (8), the travelling time T can be minimized taking into account the

190

4. SIMULATION RESULTS

System parameters values mL = m" 1 = 1-

As an illustrative example we considered a crane with trolley mass me = 1000kg. The nominal payload mass value is = 1500kg and the rope nominal length is lO = 8m. The uncertainty on these two values is ±50% for the payload mass, i.e. m"L = 750kg, = 2250kg, and ±25% for the rope length, i.e. l- = 6m and l+ = 10m. We considered the following payload motion function YL(t; 7) E C(4), so that v(t; 7) E e(O) and u(t; 7, mL, l) E e(O) (h = 4):

mL=m"I=I+ mL=m+,I=ImL = m+, 1 = 1+

m1

Closed-loop poles of linearized system -1.03 ± 0.96i, -2.01, -0.43 -0.45 ± 0.80i, -3.03, -0.40 -1.85 ± 1.92i, -0.41 ± 0.28i -1.72 ± 0.52i, -0.45 ± 0.34i

Table 1. Poles locatIOn for different values of system parameters.

mt

'0

yL(t; 7) = YL

70

9

315

- t - -8t

( 9 '7

7

8

540 7 +t 7 7

420

- -6t 7

6

126 5) +t 5

.

7

The absolute value of the maximum force lul that can be applied to the trolley is 1000N, whilst the maximum value of its derivative is 600N/s. The payload has to be moved from 0 to 10m and during the motion it is hoisted with a constant velocity equal to O.lm/s, starting at time t = 0 from an unknown initial rope length l E [l-, l+]. The statefeedback vector has been calculated in such a way that the poles location in the s-plane for the nominal closed-loop system is [-1.5-1.2-0.8-0.9]. It results k = [-1057.9, - 4084.9, 23916.3, 2520.8]. Closed-loop poles location for limit values of system parameters are reported in Table 1. In the simulation results, the state vector has been reconstructed by means of a classic Luenberger observer, based on the nominal Iinearized system, using only the measure of the trolley position. The poles location of the observer system is [-10, 15, - 20, - 25]. The optimization procedure presented in Section 3.4 has been applied with E = 0.1 taking into account limits on the force to be exerted on the trolley and its first derivative (n = 1). The minimum motion time results to be 7· = 16.3s. All the results showed in the following have been obtained by means of simulations performed taking into account the full nonlinearities of the crane dynamics. In Figure 2 the nominal payload motion law is reported, whilst in Figure 3 it is reported u( t; m l+), that is the case in which the force limit is attained. The optimal reference input v(t; 7·), obtained by inverting the nominal system is shown in Figure 4. In order to verify the performances of the devised overall controller, several simulations have been done, with different values of the payload mass and of the initial rope length, over their range of uncertainty. As an example, the actual payload motion when mL = and l = 1+ is shown in Figure 5. With a tight gridding over the uncertain parameter domain, the maximum overshoot of the payload has been plotted in Figure 6. Moreover, in Figure 7 it has been plotted also the following index:

7"

Fig. 2. Nominal output function.

Fig. 3. Force applied to the trolley when mL = and 1 = l+. J:= max JyL(t) - YL,I .100 t~r'

YL,

'

mt (10)

which might be considered a better measurement of the residual amplitude, because it however somehow penalizes a slow motion even if it does not present an overshoot. From these results it appears how the control strategy is effective since it assures a small swing effect also in presence of large uncertainties, without impairing the efficiency of the operations. Besides, the transient sway and the residual oscillation are eliminated in practice when the mass of the payload is known, which is a highly desirable feature because sometimes this information is available (for example when no payload is carried and the only mass is the one of the hook).

t'

mt

5. CONCLUSIONS In this paper a new feedforward/feedback control design strategy for the reduction of the transient sway and of the residual oscillation of an overhead

191

crane is proposed. Despite system nonlinearities are considered, the devised methodology is simple and permits to consider at the same time different desirable features of the system. Specifically, the motion time is minimized taking into account actuator limits, only the trolley's position needs to be measured and the hoisting of the load does not results in a significant worsening of the performances. Moreover, large uncertainties in the knowledge of the payload mass and of the rope length are well-compensated. The achieved performances of the methodology have been validated taking into account the full nolinear crane model in simulations.

H'OOO

.... .... .... 2000

°0l.L..~~--:-------:----710::---:':'2:---:-:---:'';--' --7"~20 *"-r'J

Fig. 4. The optimal reference input function.

6. REFERENCES

I.

Ackermann, J. (1993). Robust control - Systems with uncertain physical parameters. SpringerVerlag. London. Auering, J.W. and H. Troger (1987). Timeoptimal control of overhead cranes with hoisting of the load. Automatica pp. 437-446. Butler, H., G. Honderd and J. Van Amerongen (1991). Model reference adaptive control of a gantry crane scale model. IEEE Control Systems Magazine pp. 57-62. Cheng, C.-C. and C.-Y. Chen (1996). Controller design of an overhead crane system with uncertainty. Cont. Eng. Practice 4,645-653. Corriga, G., A. Giua and G. Usai (1998). An implicit gain-scheduling controller for cranes. IEEE Trans. on Cont. Syst. Tech. 6, 15-20. Levine, J. (1999). Are there new industrial perspectives in the control of mechanical systems? In: Advances in Control (P.M. Frank, Ed.). pp. 197-226. Springer. London. Moustafa, K.A.F. and A.M. Ebeid (1988). Nonlinear modeling and control of overhead load sway. Trans. of ASME, J. of Dynamic Systems, Measurement, and Control 110, 266271. Piazzi, A. and A. Visioli (2000). Minimum-time system-inversion-based motion planning for residual vibration reduction. IEEE/ASME Trans. on Mechatronics. Sakawa, Y. and Y. Shindo (1982). Optimal control of container cranes. Automatica pp. 257-266. Strip, D.R. (1989). Swing-free transport of suspended objects: a general treatment. IEEE Trans. on Rob. and Aut. 5, 234-236. Virkkunen, J., A. Marttinen, K. Rintanen, R. Salminen and J. Seitsonen (1990). Computer control of over-head and gantry cranes. Proc. of the IFAC 11th World Congress, Tallin (Estonia) pp. 401-405. Yoshida, K. and H. Kawabe (1992). A design of saturating control with a guaranteed cost and its application to the crane control system. IEEE Trans. on Automatic Control 37, 121127.

°0~---"'=----;---::'0-----'1~' ----:20::--------1'·. trne 'a)

Fig. 5. Payload motion for mL

= mt

and 1 = l+.

..,

-20 -50

Fig. 6. Overshoot (%) ofthe payload with different values of the system parameters.

Fig. 7. J index with different values of the system parameters.

192