passive control of overhead cranes - university at buffalocode.eng.buffalo.edu/jrnl/alli1999.pdf ·...

Passive Control of Overhead Cranes

HASAN ALLI

TARUNRAJ SINGH

Mechanical and Aerospace Engineering, SUNY at Buffalo, Buffalo, New York 14260, USA

(Received 18 February 1997; accepted 10 September 1997)

Abstract: The focus of this paper is on the design of optimal passive controllers for rest-to-rest maneuvers offlexible structures. The parameters of the passive controller are determined by formulating an optimizationproblem to minimize the integral of the time absolute error, subject to control constraints. Appropriate outputsare selected, resulting in passive input-output transfer functions for the plant. The design technique is firstillustrated on the benchmark floating oscillator problem. This technique is then used to design dissipativecontrollers for two models of overhead cranes. In the first model, a crane has a cable/mass combination thatis assumed to be a rigid link; in the second model, the wave equation is used to represent the dynamics of thecable. Numerical results illustrate the effectiveness of the proposed technique.

Key Words: Cranes, passive, optimized, control

1. INTRODUCTION

Most container cranes in use today are computer controlled using programmable logiccontrollers and encoders, to control and monitor the dynamics of the cranes. A variety ofenhancements, including fiber-optic communications and sway-damping controllers, are ofinterest to improve the performance and safety of the cranes. The fiber optics are beingproposed to avoid false signals generated by peripheral electrical equipment and sway-damping to permit operating the cranes in an automatic or semiautomatic fashion (Hubbell,1992). Control of cranes is a topic that has been addressed by numerous researchers overthe past decade. Noakes, Petterson, and Werner (1990) propose a technique to generateoscillation-damped transport and swing-free stop. Their technique consists of bang-off bangacceleration profiles in which the pulses are timed to minimize the cable sway during themaneuver and results in a swing-free stop. Experimental results corroborate the results ofthe open-loop control technique. Fliess, Levine, and Rouchon (1991) propose a feedbacklinearization technique to control the traversing and hoisting of an overhead crane. Theypropose tracking a C4 smooth reference profile to minimize the oscillations of the cableduring the maneuver. Their paper does not consider control constraints in the design of thecontrollers. d’Andrea-Novel, Boustany, and Rao (1991) design an asymptotically stablecollocated controller for an overhead crane in which the dynamics of the cable are modeledby the wave equation.

Journal of Vibration and Control, 5: 443-459, 1999C 1999 Sage Publications, Inc.

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Benhabib, Iwens, and Jackson (1981) propose a technique to design closed-loopcontrollers using the positivity concepts. The proposed technique does not require explicitanalysis of the closed-loop system if the plant and the controller’s transfer functions areensured to be strictly positive real (SPR) when the other is positive real (PR). This concept wasused to design controllers for large space structures. Wang and Vidyasagar (1992) demonstratethat the transfer function of a single flexible link is passive when the output considered isthe difference of the rigid body motion and the tip deflection. They further show that anystrictly passive controller with finite gain will result in a L2 stable system. They verify theirsimulation results by implementing the controller on an experimental setup. The limitationimposed on their work is that the link should be sufficiently rigid to result in a passivedynamical system. Juang et al. (1993) propose a controller that consists of passive second-order systems that ensure that the closed-loop system is stable even if the controlled systemis nonlinear. This controller also provides robust results with respect to uncertain systemparameters.

The focus of this paper is to design passivity-based controllers whose parameters areoptimized to minimize the integral of the time absolute error (ITAE). Section 2 briefly liststhe definitions and theorems that provide the motivation for this paper. Section 3 describes theformulation of the optimization problem. Three examples are used to illustrate the proposedtechnique in Section 4. The first example illustrates the design of a first- and second-orderpassive feedback compensator for the benchmark floating oscillator problem. The next twoexamples illustrate the design of passive controllers for overhead cranes. The paper concludeswith some remarks.

2. PASSIVITY BASED CONTROLLER

Passivity-based control design provides us an elegant technique to design energy dissipativecontrollers. Prior to developing the control technique, it would be apt to state somedefinitions.

Definition: A system with a transfer function

where n > m is PR if

It is SPR if h(s - E) is PR for some E > 0 (Slotine and Li, 1991).Remark: A linear system is passive if it is PR (Slotine and Li, 1991).Remark: A linear system is dissipative if it is SPR (Benhabib, Iwens, and Jackson, 1981).Definition: A reactance (lossless) function is a PR function that maps the imaginary axis intothe imaginary axis (Balabanian and Bickart, 1969).Theorem. A real rational function is a reactance function if and only if all the poles and zeros aresimple, lie on the j01 axis, and alternate with each other (Balabanian and Bickart, 1969).


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Figure 1. Feedback configuration.

Theorem. The closed-loop system illustrated in Figure 1 is asymptotically stable in the input-output sense if at least one of the transfer functions is SPR and the other is PR (Balabanian andBickart, 1969; Krstic, Kanellakopoulos, and Kokotovic, 1995).

These definitions and theorems form the foundation of the design of asymptoticallystable controllers in this paper. For a system that can be shown to be reactive, the designof stabilizing controllers can be reduced to the requirement that the transfer function of thecompensator is SPR. The gains of the SPR function can then be optimized to minimize a costfunction subject to control and state constraints.

3. OPTIMIZATION OF CONTROLLER PARAMETERS

Having established the fact that feedback systems, in which the plant and controller transfermatrices are square and at least one of them is SPR and the other is PR, are asymptoticallystable, we can formulate a parameter optimization problem to determine the controller gains.In this work, we impose saturation constraints on the actuator to arrive at a realistic controllerwith the objective of minimizing the integral of the ITAE. The optimization problem can bestated as,

Minimize.)

where y(t) represents the output of interest, subject to the constraints

andJ

and

where x represents the system states, u the control input, and umax the maximum control. Thecontrol is the output of a PR or SPR dynamical system


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Figure 2. Two mass-spring system.

where r is the reference input to the system and p is the vector that parametrizes the controller.Constraints are imposed onp, to guarantee that the controller is PR or SPR.

An automated approach to optimize for structural and controller parameters was

developed by Ducourau, Singh, and Mayne (1996). This approach automatically evaluatesthe analytical gradients using the symbolic manipulator MAPLE and generates a Fortranprogram that uses the recursive quadratic programming function NCONG, which is a

gradient-based optimizer developed by IMSL. This program, which provides a user-friendlyinterface, is used to expedite the process of optimizing for the controller parameters.

4. NUMERICAL EXAMPLES

The passivity-based control design approach is first illustrated on the floating mass bench-mark problem. Here, we consider a first-order compensator and a second-order compensatorfor rest-to-rest maneuvers. Following the benchmark problem, which captures the dynamicsof a flexible structure, the proposed approach is illustrated on two models of overhead cranes.

4.1. Floating Oscillator

A two mass-spring system shown in Figure 2 is characterized by one rigid body mode andone flexible mode. The equations of motion of the floating oscillator are

where u is the control input and xl and X2 represent the position of the two masses from aninertial frame of reference. The transfer function relating the displacements of the two massesare ,


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Figure 3. Passive feedback control structure.

and

Defining the output of the system to be

the input-output transfer function is

The poles of the system are located at

and zeros at

For a system to be PR, the poles on tJ1e jw axis are simple, with real positive residues(Balabanian and Bickart, 1969). Therefore, we concatenate the input-output transfer functionwith a derivative operator, which adds a zero at the origin (see Figure 3). The resulting transferfunction maps the imaginary axis into the imaginary axis and has alternating poles and zeros,


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indicating that the input-output transfer function is passive. If we now design a feedbackcontroller that is strictly PR, we are guaranteed that the system is stable in the ,C2 sense. Forthe numerical simulation, we assume that ml = m2 = k = 1, and the objective is to move thesystem states from

and:

to the origin of the state space.The first controller considered is parametrized as

where the parameters a and b are constrained to be positive by the SPR requirement of h2.The resulting closed-loop transfer function of the system is

Optimizing for the parameters a and b, which minimize the ITAE, with the constraint that

results in

and

with an ITAE cost of 8.8362. We now parametrize a second-order SPR function as

where the variables a, b, c, and d are constrained to be positive. The result of the optimizationis

and

with an ITAE cost of 7.224 representing a cost reduction of 18.3%.Figures 4, 5, and 6 illustrate the results of the numerical simulation of these two

controllers. The dashed and solid lines represent the results of the first- and second-order

compensators, respectively It is evident from Figures 4 and 5 that the positions of the twomasses approach the final position without any overshoot when the second-order compensatoris used compared to the case in which a first-order compensator is used. Figure 6 illustratesthat the control reaches its limit rapidly, unlike the first-order compensator, which neverreaches the saturation value of the control input.


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Figure 4. Evolution of the position of the first mass.

Figure 5. Evolution of the position of the second mass.


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Figure 6. Control history of the passive controller

, . ,~ .

4.2. Overhead Cranes: Rigid Approximation

Having illustrated the ease with which the controllers can be designed, we now study theproblem of controlling an overhead crane performing rest-to-rest maneuvers. We considertwo cases: the first models the cable and the suspended mass as a rigid link that correspondsto the case when the suspended mass is much larger than the cable mass, and the second casemodels the cable as a flexible structure.

Hamilton’s approach is used to derive the equations of motion of the crane (see Figure 7).The crane consists of a horizontally moving platform of mass ml to which is connected thecable, which carries a suspended mass m2. The kinetic and potential energy of the systemwhen the cable-suspended mass pair is considered to be rigid is

and

where 0 is the angle the cable makes with respect to the vertical axis,g is the acceleration dueto gravity, and I is the length of the cable. The Euler-Lagrange equations of motion, assumingsmall angular displacement of the cable, are _

.


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Figure 7. Rigid overhead crane.

The transfer function relating the displacement states to the input are

and

To make the plant passive, we define the output

and the resulting input-output transfer function is

whose poles are located at


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Figure 8. Time response of x, tip sway, and the control.

and zeros are located at

Similar to the first problem, we concatenate the system transfer function with a differentiatorto form a reactive system since the complex poles will always have a magnitude greater thanthe zeros, leading to an alternating pattern of poles and zeros. Since the plant is passive, anySPR compensator will stabilize the system in a IC2 sense.

We select a second-order feedback compensator and optimize for the dynamics of thecompensator to minimize the ITAE with the constraint that the transfer function of thecompensator is SPR, and the control does not violate the saturation constraints of 2,000 N.Assuming that ml = 12,000 kg, m2 = 3,000 kg, and I = 5 m, the resulting optimum transferfunction is


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Figure 9. Overhead crane with a flexible rope.

Figures 8a and 8b illustrate the evolution of the position of the platform, which reachesits final position with no overshoot, and the tip sway of the pendulum, which indicates small-amplitude oscillations. Figure 8c indicates that the control force does not violate the controlconstraints and saturates early in the maneuver.

4.3. Overhead Cranes: Distributed Model

When the cable length is large and the suspended mass is comparable to the mass of the cableor smaller than the cable mass, the dynamics of the cable cannot be ignored. We now needto model the dynamics of the cable by the wave equation. The equations of motion of theoverhead crane with a flexible cable (see Figure 9) are derived using Hamilton’s principle.The kinetic energy of the system is

and the potential energy is


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where p is unit mass per length for rope, T is tension of the cable, and 0 and w are the rigidand flexible displacements, respectively. (:) indicates the derivative of (.) with respect to thetime, and (.)’ indicates the partial derivative of (.) with respect to the spatial coordinate x.

The Lagrangian of the system, ignoring the axial deformation of the cable, is

The Euler-Lagrange equations can be shown to be

where u(t) is the control force, and

Assuming C2 = If, where .1 c is the wave speed, and substituting this definition into equation(35), we arrive at

with the associated boundary conditions

and -

Laplace transformation of equations (34), (36), and (37) results in

and

and -

The homogeneous solution of equation (39) is


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Table 1. Alternating open-loop poles and zeros.

Substituting the boundary conditions (40) and solving for the particular solution, we have

Rearranging equations (38) and (42), we have

If we define the output as the rigid-body displacement minus a weighted flexible rope tipdeflection

the input-output transfer function can be shown to be

To study a case in which the suspended mass is small compared to the actuator mass, weassume that the masses are

; and i

the length of the cable 1 = 5 m, and p = 1 kg/m. For a = 100, and with a maximum controlinput of 2,000 N, the resulting dominant open-loop poles and zeros are tabulated in Table 1,which reveals that the poles and zeros alternate if a differentiator is concatenated to the input-output transfer function. Since the plant is passive, any SPR compensator will asymptotically


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Figure 10. Time response of the rigid body

stabilize the system. We optimize for the gains of a second-order SPR compensator, resultingin the compensator transfer function

A finite element model with linear shape functions is used to derive a 52nd-order model.Numerical simulation of the finite element model of the overhead crane with a flexible cable

subject to a passive controller is illustrated in Figures 10, 11, and 12. Numerical simulationsrevealed that any model greater than a 12th-order model did not significantly change thesystem response. For a rest-to-rest maneuver of 10 m, in which the maximum force is

±2,000 N, Figure 10 illustrates that the actuator mass moves to the desired final positionwith no overshoot.

It can also be seen from Figure 11 that the flexible deflection of the tip of the cable isonly 4 cms, providing us with an excellent sway-minimizing controller. The evolution of thecontrol (see Figure 12) shows that the control input reaches its maximum values quickly tominimize the maneuver time.

To gauge the performance of the proposed passivity-based noncollocated controller, asimple Lyapunov-based collocated control strategy is studied. This controller only requiressensors to measure the states of the actuator mass. Optimizing for the collocated controllerresulted in a passive controller whose parameters are nearly identical to those of thenoncollocated case. Plotting the time response of the flexible tip deflection in Figure 13, it canbe seen that although the maximum cable deflection is slightly smaller than the noncollocatedcase, the settling time has been significantly increased.


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Figure 11. Cable tip sway.

Figure 12. Control input.


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Figure 13. Comparison of flexible tip displacement for the collocated and noncollocated sensor-actuatorcases.

° °

Figure 14. Time responses with uncertain system parameters.


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Finally, the robustness of the passivity-based controller is studied by perturbing thesuspended mass by 50%. Figure 14 illustrates that the system response does not changesignificantly from the case for which the passive controller was optimized.

5. CONCLUSIONS

A technique to design optimal passive controllers is proposed in this paper. The fact that asystem with alternating poles and zeros is passive is exploited to determine an output variablethat forces the input-output transfer function to be passive. Following the establishment ofan output, passive controllers of different orders are optimized to minimize the integral of thetime absolute error.

Numerical simulations illustrate the improvement of second-order controllers over afirst-order passive controller for the benchmark problem. The same approach is followedto design passive controllers for an overhead crane. The first simulation assumes that thesuspension cable is rigid, followed by a model in which the wave equations are used to modelthe dynamics of the cable. Second-order passive controllers are designed for both models,and the simulation results illustrate small sway rest-to-rest motion of the payload. Finally, acollocated controller and a noncollocated controller are compared based on the sway of thepayload, and it is shown that the passive noncollocated controller outperforms the collocatedcontroller. Numerical simulations are also used to illustrate the robustness of the passivecontroller.

REFERENCES

Balabanian, N. and Bickart, T., 1969, Electrical Network Theory, John Wiley, New York.

Benhabib, R. J., Iwens, R. P., and Jackson, R. L., 1981, "Stability of large space structure control systems using positivityconcepts," Journal of Guidance, Control and Dynamics 4(5), 487-494.

d’Andrea-Novel, B., Boustany, F., and Rao, B. P, 1991, "Control of an overhead crane: Feedback stabilization of anhybrid PDE-ODE system," in Proceedings of the 1991 European Control Conference, Grenoble, France, July2-5, pp. 2244-2248.

Ducourau, L., Singh, T., and Mayne, R. W, 1996, "Automated parameter optimization for structural and controllerdesign," in Proceedings of the 1996 CSME Mechanics in Design Forum, Toronto, Canada, May 6-9.

Fliess, M., Levine, J., and Rouchon, P., 1991, "A simplified approach of crane control via a generalized state-spacemodel," m Proceedings of IEEE International Conference on Decision and Control, Brighton, England, De-cember, pp. 736-741.

Hubbell, J. T, 1992, "Modern crane control enhancements," in Proceedings of Ports 92, Seattle, WA, July 20-22, pp.757-767.

Juang, J., Wu, S.-C., Phan, M., and Longman, R. W, 1993, "Passive dynamic controllers for nonlinear mechanicalsystems," Journal of Guidance, Control and Dynamics 16(5), 845-851.

Krstic, M., Kanellakopoulos, I., and Kokotovic, P., 1995, Nonlinear and Adaptive Control Design, Wiley-Interscience,New York.

Noakes, M. W, Petterson, B. J., and Werner, J. C., 1990, "An application of oscillation damped motion for suspendedpayloads to advanced integrated maintenance systems," in Proceedings of the 38th Conference on RemoteSystems Technology 1, pp. 63-68.

Slotine, J. E. and Li, W, 1991, Applied Nonlinear Control, Prentice Hall, Englewood Cliffs, NJ.

Wang, D. and Vidyasagar, M., 1992, "Passive control of a stiff flexible link," International Journal ofRobotics Research11(6), 572-578.


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