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 of flexible structures. The parameters of the passive controller are determined by formulating an optimization problem to minimize the integral of the time absolute error, subject to control constraints. Appropriate outputs are selected, resulting in passive input-output transfer functions for the plant. The design technique is first illustrated on the benchmark floating oscillator problem. This technique is then used to design dissipative controllers for two models of overhead cranes. In the first model, a crane has a cable/mass combination that is assumed to be a rigid link; in the second model, the wave equation is used to represent the dynamics of the cable. 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 logic controllers and encoders, to control and monitor the dynamics of the cranes. A variety of enhancements, including fiber-optic communications and sway-damping controllers, are of interest to improve the performance and safety of the cranes. The fiber optics are being proposed to avoid false signals generated by peripheral electrical equipment and swaydamping 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 over the past decade. Noakes, Petterson, and Werner (1990) propose a technique to generate oscillation-damped transport and swing-free stop. Their technique consists of bang-off bang acceleration profiles in which the pulses are timed to minimize the cable sway during the maneuver and results in a swing-free stop. Experimental results corroborate the results of the open-loop control technique. Fliess, Levine, and Rouchon (1991) propose a feedback linearization technique to control the traversing and hoisting of an overhead crane. They propose tracking a C4 smooth reference profile to minimize the oscillations of the cable during the maneuver. Their paper does not consider control constraints in the design of the controllers. d Andrea-Novel, Boustany, and Rao (1991) design an asymptotically stable collocated controller for an overhead crane in which the dynamics of the cable are modeled by the wave equation. Journal of Vibration and Control, 5: 443-459, 1999 C 1999 Sage Publications, Inc.
444 Benhabib, Iwens, and Jackson (1981) propose a technique to design closed-loop controllers using the positivity concepts. The proposed technique does not require explicit analysis of the closed-loop system if the plant and the controller s transfer functions are ensured to be strictly positive real (SPR) when the other is positive real (PR). This concept was used to design controllers for large space structures. Wang and Vidyasagar (1992) demonstrate that the transfer function of a single flexible link is passive when the output considered is the difference of the rigid body motion and the tip deflection. They further show that any strictly passive controller with finite gain will result in a L2 stable system. They verify their simulation results by implementing the controller on an experimental setup. The limitation imposed on their work is that the link should be sufficiently rigid to result in a passive dynamical system. Juang et al. (1993) propose a controller that consists of passive secondorder systems that ensure that the closed-loop system is stable even if the controlled system is nonlinear. This controller also provides robust results with respect to uncertain system parameters. The focus of this paper is to design passivity-based controllers whose parameters are optimized to minimize the integral of the time absolute error (ITAE). Section 2 briefly lists the definitions and theorems that provide the motivation for this paper. Section 3 describes the formulation of the optimization problem. Three examples are used to illustrate the proposed technique in Section 4. The first example illustrates the design of a first- and second-order passive feedback compensator for the benchmark floating oscillator problem. The next two examples illustrate the design of passive controllers for overhead cranes. The paper concludes with some remarks. 2. PASSIVITY BASED CONTROLLER Passivity-based control design provides us an elegant technique to design energy dissipative controllers. Prior to developing the control technique, it would be apt to state some definitions. 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 into the imaginary axis (Balabanian and Bickart, 1969). Theorem. A real rational function is a reactance function if and only if all the poles and zeros are simple, lie on the j01 axis, and alternate with each other (Balabanian and Bickart, 1969).
445 Figure 1. Feedback configuration. Theorem. The closed-loop system illustrated in Figure 1 is asymptotically stable in the inputoutput sense if at least one of the transfer functions is SPR and the other is PR (Balabanian and Bickart, 1969; Krstic, Kanellakopoulos, and Kokotovic, 1995). These definitions and theorems form the foundation of the design of asymptotically stable controllers in this paper. For a system that can be shown to be reactive, the design of stabilizing controllers can be reduced to the requirement that the transfer function of the compensator is SPR. The gains of the SPR function can then be optimized to minimize a cost function subject to control and state constraints. 3. OPTIMIZATION OF CONTROLLER PARAMETERS Having established the fact that feedback systems, in which the plant and controller transfer matrices are square and at least one of them is SPR and the other is PR, are asymptotically stable, 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 controller with the objective of minimizing the integral of the ITAE. The optimization problem can be stated as, Minimize.) where y(t) represents the output of interest, subject andj and to the constraints where x represents the system states, u the control input, and umax the maximum control. The control is the output of a PR or SPR dynamical system
446 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 evaluates the analytical gradients using the symbolic manipulator MAPLE and generates a Fortran program that uses the recursive quadratic programming function NCONG, which is a gradient-based optimizer developed by IMSL. This program, which provides a user-friendly interface, 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 benchmark problem. Here, we consider a first-order compensator and a second-order compensator for rest-to-rest maneuvers. Following the benchmark problem, which captures the dynamics of 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 and one 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 an inertial frame of reference. The transfer function relating the displacements of the two masses are,
447 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 function with a derivative operator, which adds a zero at the origin (see Figure 3). The resulting transfer function maps the imaginary axis into the imaginary axis and has alternating poles and zeros,
448 indicating that the input-output transfer function is passive. If we now design a feedback controller that is strictly PR, we are guaranteed that the system is stable in the,c2 sense. For the numerical simulation, we assume that ml = m2 = k = 1, and the objective is to move the system 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 optimization is 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 two masses approach the final position without any overshoot when the second-order compensator is used compared to the case in which a first-order compensator is used. Figure 6 illustrates that the control reaches its limit rapidly, unlike the first-order compensator, which never reaches the saturation value of the control input. and
449 Figure 4. Evolution of the position of the first mass. Figure 5. Evolution of the position of the second mass.
.. 450 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 the problem of controlling an overhead crane performing rest-to-rest maneuvers. We consider two cases: the first models the cable and the suspended mass as a rigid link that corresponds to the case when the suspended mass is much larger than the cable mass, and the second case models 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 the cable, which carries a suspended mass m2. The kinetic and potential energy of the system when 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 due to gravity, and I is the length of the cable. The Euler-Lagrange equations of motion, assuming small angular displacement of the cable, are _
451 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
452 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 differentiator to form a reactive system since the complex poles will always have a magnitude greater than the zeros, leading to an alternating pattern of poles and zeros. Since the plant is passive, any SPR compensator will stabilize the system in a IC2 sense. We select a second-order feedback compensator and optimize for the dynamics of the compensator to minimize the ITAE with the constraint that the transfer function of the compensator 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 transfer function is
453 Figure 9. Overhead crane with a flexible rope. Figures 8a and 8b illustrate the evolution of the position of the platform, which reaches its final position with no overshoot, and the tip sway of the pendulum, which indicates smallamplitude oscillations. Figure 8c indicates that the control force does not violate the control constraints 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 cable or smaller than the cable mass, the dynamics of the cable cannot be ignored. We now need to model the dynamics of the cable by the wave equation. The equations of motion of the overhead 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
454 where p is unit mass per length for rope, T is tension of the cable, and 0 and w are the rigid and flexible displacements, respectively. (:) indicates the derivative of (.) with respect time, 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 to the The Euler-Lagrange equations can be shown to be where u(t) is the control force, and Assuming C2 If, = (35), we arrive at where.1 c is the wave speed, and substituting this definition into equation 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
455 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 tip deflection 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, we assume 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 control input 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 inputoutput transfer function. Since the plant is passive, any SPR compensator will asymptotically
456 Figure 10. Time response of the rigid body stabilize the system. We optimize for the gains of a second-order SPR compensator, resulting in 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 simulations revealed that any model greater than a 12th-order model did not significantly change the system 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 position with no overshoot. It can also be seen from Figure 11 that the flexible deflection of the tip of the cable is only 4 cms, providing us with an excellent sway-minimizing controller. The evolution of the control (see Figure 12) shows that the control input reaches its maximum values quickly to minimize the maneuver time. To gauge the performance of the proposed passivity-based noncollocated controller, a simple Lyapunov-based collocated control strategy is studied. This controller only requires sensors to measure the states of the actuator mass. Optimizing for the collocated controller resulted in a passive controller whose parameters are nearly identical to those of the noncollocated case. Plotting the time response of the flexible tip deflection in Figure 13, it can be seen that although the maximum cable deflection is slightly smaller than the noncollocated case, the settling time has been significantly increased.
457 Figure 11. Cable tip sway. Figure 12. Control input.
458 Figure 13. Comparison of flexible tip displacement for the collocated and noncollocated sensor-actuator cases. Figure 14. Time responses with uncertain system parameters.
459 Finally, the robustness of the passivity-based controller is studied by perturbing the suspended mass by 50%. Figure 14 illustrates that the system response does not change significantly 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 a system with alternating poles and zeros is passive is exploited to determine an output variable that forces the input-output transfer function to be passive. Following the establishment of an output, passive controllers of different orders are optimized to minimize the integral of the time absolute error. Numerical simulations illustrate the improvement of second-order controllers over a first-order passive controller for the benchmark problem. The same approach is followed to design passive controllers for an overhead crane. The first simulation assumes that the suspension cable is rigid, followed by a model in which the wave equations are used to model the 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, a collocated controller and a noncollocated controller are compared based on the sway of the payload, and it is shown that the passive noncollocated controller outperforms the collocated controller. Numerical simulations are also used to illustrate the robustness of the passive controller. 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 positivity concepts," 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 an hybrid PDE-ODE system," in Proceedings of the 1991 European Control Conference, Grenoble, France, July 2-5, pp. 2244-2248. Ducourau, L., Singh, T., and Mayne, R. W, 1996, "Automated parameter optimization for structural and controller design," 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-space model," m Proceedings of IEEE International Conference on Decision and Control, Brighton, England, December, 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 mechanical systems," 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 suspended payloads to advanced integrated maintenance systems," in Proceedings of the 38th Conference on Remote Systems 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 Research 11(6), 572-578.