Nonlinear Stability of a Delayed Feedback Controlled Container Crane

Nonlinear Stability of a Delayed Feedback Controlled Container Crane THOMAS ERNEUX Université Libre de Bruxelles Optique Nonlinéaire Théorique Campus Plaine, C.P. 231 1050 Bruxelles, Belgium (terneux@ulb.ac.be) TAMÁS KALMÁR-NAGY Department of Aerospace Engineering, Texas A&M University, College Station, TX 77845, USA (Received 15 December 2005 accepted25july2006) Abstract: A simplified model of a container crane subject to a delayed feedback is investigated. The conditions for a Hopf bifurcation to stable/unstable limit-cycle solutions are determined. It is shown that a subcritical Hopf bifurcation to unstable oscillations cannot be ruled out and the undesired coexistence of stable large amplitude oscillations and a stable equilibrium endangers the robustness of time-delay control strategies. The bifurcationisanalyzedbothanalyticallyand numerically using a continuation method. Keywords: Sway reduction, Hopf bifurcation, stable oscillations 1. INTRODUCTION Gantry cranes are used for moving objects within shipyards, ports, railyards, factories, and warehouses (Figure 1). These cranes can lift several hundred tons and can have spans of well over 50 meters. For fabrication and freight-transfer applications, it is important for the crane to move payloads rapidly and smoothly. If the gantry moves too fast the payload may start to sway, and the crane operator might lose control of the payload. During the last four decades, different strategies for controlling payload pendulations without including the operator in the control loop have been investigated. Henry (1999), Henry et al. (2001), Masoud and Nayfeh (2003) and Masoud et al.(2003, 2004, 2005) developed a control strategy based on a time-delayed position feedback of the payload cable angles. The main objective of this type of control strategy is to add damping to the system. It has been known since the work of Minorsky (1942, 1962) that time delay may produce significant damping of the oscillations. The efficiency of this technique has been investigated by numerical simulations of detailed mathematical models and by experiment (Henry, 1999 Henry et al., 2001 Nayfeh 2002 Nayfeh et al., 2005). In these papers, the stability of the delayed position feedback control was examined close to the equilibrium position. At a recent ASME meeting (ASME, 2005), the question was raised whether the linear stability analysis is sufficient to ascertain the Journal of Vibration and Control, 13(5): 603 616, 2007 DOI: 10.1177/1077546307074245 2007 SAGE Publications Figures 4 7 appear in color online: http://jvc.sagepub.com

604 T. ERNEUX and T. KALMÁR-NAGY Figure 1. Rail-mounted gantry cranes are used as yard cranes. Except for the immediate loading or unloading tasks, all operations must be done automatically with an efficient anti-sway control technique. robustness of the proposed delayed-position feedback. In particular, nonlinear systems can have unstable limit cycles close to the stable steady-state, thus even small perturbations of the system could lead to a catastrophic loss of stability. The main objective of this paper is to emphasize the need for nonlinear studies. Two specific bifurcation phenomena associated with a delay differential equation deserve attention. First, it is known from studies of mechanical and electronical systems that the delay can lead to a subcritical Hopf bifurcation, allowing an overlap of a branch of stable limit-cycles and a branch of stable steady states (Kalmar-Nagy et al., 1999, 2001 Kalmar-Nagy, 2002 Kevorkiann and Cole, 1996, Landry et al., 2005 Larger et al., 2004 Gilsinn, 2002). Second, it is also known from problems in nonlinear optics (Alsing et al., 1996 Pieroux et al., 1994, 2000) and from analytical studies of simple oscillator problems (Erneux, 2005) that several isolated branches of periodic solutions may coexist with a stable steady state if the delay is sufficiently large. In this paper, we examine the direction of the Hopf bifurcation appearing in a simplified model of the crane system. We determine conditions on the delay and the gain of the controller so that the resulting Hopf bifurcation is supercritical and leads to stable oscillations. The plan of the paper is as follows. Section 2 is devoted to the mathematical formulation of the model. Section 3 discusses the linear stability properties of the basic equilibrium. Section 4 and Section 5 concentrate on the local analytical and global numerical bifurcation diagrams, respectively. Section 6 summarizes our main results.

DELAYED FEEDBACK CONTROLLED CONTAINER CRANE 605 Figure 2. Simple pendulum model of a container crane. 2. FORMULATION A realistic model of a quay-side container crane which includes a spreader bar to lift the container has been considered in Masoud and Nayfeh (2003) and Masoud et al. (2005). A simplified version of this model was then used to obtain guidelines on the stability of the controller. While the bifurcation analysis of the full nonlinear system has not yet been presented, it is important to understand the local nature of the possible bifurcations and complement the analytical studies with numerical methods including branch following. Therefore we consider the simpler pendulum model of a container crane (Figure 2) to demonstrate the need for this kind of analysis. We assume that the cable is inextensible or its length is slowly varying compared to the time scale of the payload oscillations. Using the Lagrangian approach (Omar Hanafy, 2003) with Rayleigh dissipation (Masoud et al., 2005), we obtain the following equations of motion m Mu cu ml cos ml 2 sin cml cos F t (1) l c g sin cosu cu 0 (2) where c is the friction coefficient. The force F t acting on the cart is provided by an ideal motor. This control input was specified in Masoud et al. (2005) and Nayfeh et al. (2005) as F t K u u 0 H T (3) where u 0 is the desired final position of the crane, K is a proportional gain, T t T is the delayed value of the angle (where T is the delay), and H is a judiciously chosen function of T and satisfying H0 0 0. The parameters can be tuned for optimal performance of the control system in order, for example, to introduce as much damping

606 T. ERNEUX and T. KALMÁR-NAGY as possible. To focus on nonlinear phenomena associated with the delayed feedback force F t H T, we neglect the operator input (as in Nayfeh et al., 2005). For moderate oscillations of the system ( 2) the horizontal dynamics of the system can be expressed as a damped oscillator whose driving is provided by the payload oscillations. From (2), we find u cu l c cos g tan (4) Introducing (4) into equation (1), we obtain an equation for only, which is given by l M m sin 2 c m M g tan ml 2 sin H cos T (5) At this point, we introduce the dimensionless time s t, where is defined as the frequency of the linearized trolley-payload system (i.e. the system Ml c m Mg 0) given by M m Ml g (6) Equation (5), then becomes 1 sin 2 2 tan 2 sin h cos (7) Prime now means differentiation with respect to the nondimensional time s, c 1 2is the dimensionless friction coefficient, and s with T being the dimensionless time delay. The parameter mm is the ratio of the payload and trolley masses. The function in the right-hand side of equation (7) is defined as h H Ml and represents the nondimensional control force. The shape of the control force determines not only the linear stability of the system but also its bifurcation structure, therefore its choice is crucial from the control point of view. A simple delayed feedback of the form h k (8) was used by Pyragas (1992) to control a chaotic system, while Nayfeh et al. (2005) used h k sin in the context of gantry crane control. For both of these control functions the only equilibrium state 0 is the same as that of the the free crane system and this will be our basic reference state. In this paper, we investigate in detail Pyragas control strategy (8). With this, equation (7) can be rewritten as the following second-order delay differential equation 2 cos tan 2 sin k 1 sin 2 0 (9)

DELAYED FEEDBACK CONTROLLED CONTAINER CRANE 607 3. LINEAR STABILITY Linearizing equation (9) leads to 2 k 0 (10) The linear stability boundaries are found by introducing expis into equation (10). From the real and imaginary parts, we obtain 2 1 kcos 1 0 (11) 2 k sin 0 (12) The solution for k k can be determined analytically (see Kalmár-Nagy et al., 2001). If we wish to avoid the inverse trigonometric functions, we may obtain the solution in parametric form using x 2 0 as parameter. Eliminating k in equations (11) and (12), and inserting 2x, we obtain a quadratic equation for It always admits a positive real root given by 2 x tanx x 2 tan 2 x 1 (13) Having x we determine k using (12) with 2x 1 : k 4x (14) sin2x By continuously increasing x from zero, the successive Hopf bifurcation curves are generated by (13) and (14) (full lines in Figure 3). The friction coefficient is generally small and if 0, the expressions for the Hopf bifurcation lines considerably simplify. From equations (11) and (12), we find the following three cases: k 0 0 and 0 1 (15) 0 2n and 0 1 (16) k 0 1 2n 1 2 2n 1 1 and 0 (17) 2 where n 0 1 2,... The horizontal line k 0 the vertical line defined by (16) with n 1, and the lines defined by (17) with n 0and1areshownbybrokenlinesinFigure3. It can be shown (Kalmár-Nagy et al., 2001 Stépán, 1989) that the crosshatched domain in Figure 3 corresponds to a stable steady-state of equation (10), and a transversal (i.e. with nonzero velocity) root crossing occurs on the curves separating this region from the unstable one. We thus expect the emergence of a Hopf bifurcation at the stability boundaries. In the next section, we propose a nonlinear bifurcation analysis of equation (9) and discuss the direction of the Hopf bifurcation.

608 T. ERNEUX and T. KALMÁR-NAGY Figure 3. Successive Hopf bifurcation lines (solid) in the k vs parameter plane for (8) and 0025. Thebrokenlinescorrespondtothelimitingcaseofnofriction( 0 and are shown for the first four Hopf bifurcations. The crosshatched domain corresponds to a stable steady state. 4. NONLINEAR ANALYSIS In this section, we determine the periodic solutions near their Hopf bifurcation points. Two cases need to be examined depending on the choice of the control parameter. 4.1. Bifurcation Parameter k We first examine the Hopf bifurcation using k as the bifurcation parameter and assuming small positive values of (the case of arbitrary can be similarly analyzed). Specifically, we apply the Poincaré Lindstedt method (Drazin, 1992 Kevorkian and Cole, 1996 Casal and Freedman, 1980) and seek a 2-periodic solution of the form 1 S 2 3 S (18) where S s and the small parameter is defined by We also expand the frequency and the bifurcation parameter k as 0 2 2 and k k 0 2 k 2 (19) where k 0 and 0 are defined by (17). The power series in 2 in (18) comes from the fact that the Taylor expansion of the nonlinear function in (7) only generates odd powers of

DELAYED FEEDBACK CONTROLLED CONTAINER CRANE 609 The scaling 2 and the fact that the first non-zero corrections are O 2 in (19) are required by the solvability conditions. We also need the expansion S S 0 2 2 S 0 (20) where prime now means differentiation with respect to time S. After introducing (18) (20) into equation (9), we equate the coefficients of each power of to zero. The problems for the unknown functions 1, 3 are given by L 1 2 0 1 1 k 0 1 S 0 1 0 (21) 1 1 L 3 2 0 1 3 1 2 0 6 2 1 1 k 0 2 1 S 0 2 2 0 2 1 k 0 2 1 S 0 k 2 1 S 0 1 (22) The solution of equation (21) is 1 A expis cc (23) where A is an unknown amplitude and cc stands for complex conjugate terms. An equation for A is determined by applying a solvability condition (to remove secular terms) on the right-hand side of equation (22). Using expi 0 1 this condition is given by 2 A2 0 ik 0 A2i 0 N A 2 2k 2 0 (24) where N is defined as N 3 2 2 0 1 (25) 2 Assuming A 0 the real and imaginary parts of equation (24) lead to two conditions for k 2 and 2 After eliminating 2 2 0 k 0 we find A 2 2k 2 k 2H N 0 (26) where k 2H 22n 12 2 3 k 0 k 0 0 (27) is the critical value of k 2 at the Hopf bifurcation point. The inequality in equation (25) defines the direction of bifurcation, i.e. the possible values of the deviation k 2 k 2H where a limit cycle exists. It clearly depends on the sign of N, which may be positive or negative depending on the value of If k 0 0 we note from Figure (3) the basic equilibrium solution 0 is unstable if k 2 k 2H 0 The inequality in (25) then implies that the bifurcation is super

610 T. ERNEUX and T. KALMÁR-NAGY Table1.Rowsofthetablerepresentthesignrequirementsforthetermsinthefirstrowin order for the bifurcations to be supercritical (Case (17), is fixed, and k is the bifurcation parameter). k 0 k 2 k 2H N critical if N 0 1. Similarly, if k 0 0 we note from Figure (3) that 0 is unstable if k 2 k 2H 0 The inequality in (25) then implies that the bifurcation is supercritical if N 0 The different possibilities for a supercritical Hopf bifurcation are listed in Table 1. If one of the two conditions in Table 1 is violated, the bifurcation is subcritical and a branch of unstable periodic solutions will overlap the stable equilibrium 0 It is instructive to discuss the sign of N in terms of From (25), we find that N 0if c 1 2 3 which is impossible if the delay is too small since mm 2 2n1 2 2 2 must be positive. In other words, a supercritical Hopf bifurcation for k 0 0 requires the 3 necessary condition 2n 1 Thus, contrary to what we naively might expect, a 2 small delay does not guarantee a supercritical Hopf bifurcation. Another point that is worth stressing is the dramatic change of sign of N in Table 1 as soon as k 0 changes sign. At the critical point k 0 0 equals 1 regardless of n From (25), we then note that N 0 This means that in the vicinity of k 0 0 the bifurcation is supercritical only in the domain where N 0, i.e. when k 0 0 4.2. Bifurcation Parameter In the case of (17) with as the control parameter and k fixed, the analysis is quite similar and we summarize only the main result. We introduce into equation (9). 0 and 0 are defined by 0 2 2 and 0 2 2 (28) 0 2n 1 2n 1 and 0 (29) 1 2k 0 where n 0 1. andk 12. The limit-cycle solution is given by (23) to a first approximation and the amplitude A satisfies A 2 2 0 0 2 2H 0 (30) where 2H 2 0 k and N 3 2 1 2k 1 (31) 2

DELAYED FEEDBACK CONTROLLED CONTAINER CRANE 611 Table2.Rowsofthetablerepresentthesignrequirementsforthetermsinthefirstrowin order for the bifurcations to be supercritical (Case (17), k is fixed and is the bifurcation parameter). k 2 2H N Table3.Rowsofthetablerepresentthesignrequirementsforthetermsinthefirstrowin order for the bifurcations to be supercritical (Case (16), k is fixed and is the bifurcation parameter). k 2 2k 1 The different possibilities for a supercritical Hopf bifurcation are listed in Table 2. Note from (31) that if k 0, N is always positive meaning that the bifurcation is always subcritical. In the case of (16), we are forced to use as our control parameter because the leading approximation of the Hopf bifurcation point is independent of k. With (28) where 0 and 0 are defined by (16), the sequence of problems to analyze now is L 1 1 1 k 1 S 0 1 0 (32) L 3 1 1 2 1 3 1 2 1 6 1 k 2 1 S 0 2 2 2 1 k 2 0 1 S 0 k 2 1 S 0 (33) Equation (32) admits (23) as solution. Solvability of equation (33) now requires where 2 A2 ik 0 ik 2 A A2i N A 2 0 (34) N 1 2 0 (35) 2 Eliminating 2 from the real and imaginary parts leads to the following expression for A 2 : A 2 2 N 0 2 2k 1 (36) If k 0, the steady state is unstable if 2 2k 1 and since N 0, the bifurcation is supercritical and stable. On the other hand if k 0 the steady state is unstable if 2 2k 1 But since N 0, the bifurcation is subcritical and unstable. Thus the only possibility of observing a supercritical Hopf bifurcation is indicated in Table 3.

612 T. ERNEUX and T. KALMÁR-NAGY Figure 4. Bifurcation diagram using the delay as the control parameter and for a fixed gain (k 01). From the outer to the inner closed Hopf bifurcation branches, the damping coefficient is increased from 001 to 005 5. NUMERICAL RESULTS To numerically verify our analytical predictions, we used the MATLAB toolbox DDE-BifTool written by Koen Engelborghs et al. (2001, 2002), to find the Hopf bifurcation branches of equation (9). We also used a numerical integration routine written in Mathematica to verify these results at several parameter values. Figures 4 7 appear in different colors online. The colors online correspond to different values of the friction coefficient (001 Blue, 002 Red,003 Green, 004 Magenta, 005 Black). Figure 4 represents the bifurcation diagram of the periodic solutions (the oscillation amplitude A as a function of the bifurcation parameter ) for a fixed positive value of k. The figure shows that the Hopf bifurcations near 0and 2 are supercritical while the criticality of bifurcations near and 3 depend on the value of the damping. For small, these bifurcations are subcritical. This agrees with our analysis since, evaluating (31) with 01 andk 01, we find N 0 implying a subcritical bifurcation (Table 2). Similarly, the supercritical Hopf bifurcations near 0and 2 are predicted by our analysis (Table 3). For fixed negative k, we find the same super-subcritical structure for each Hopf bifurcation branches (Figure 5). The supercritical bifurcation near k and k 3 and the subcritical bifurcation near k 2 are predicted by our analysis (Table 2 and Table 3, respectively). We also considered k as our bifurcation parameter keeping fixed. Figures 6 and 7 show the Hopf bifurcation branches at two different values of. In Figure 6, 1 and the bifurcation is supercritical for all values of the damping rate Decreasing, the Hopf bifur-

DELAYED FEEDBACK CONTROLLED CONTAINER CRANE 613 Figure 5. Bifurcation diagram using the delay as the control parameter and fixed gain (k 01). From the outer to the inner closed Hopf bifurcation branches the damping rate is gradually inceased from 001 to 004 Figure 6. Bifurcation diagram using k as the control parameter and for a fixed delay 1. supercritical Hopf bifurcation becomes progressively more vertical as we decrease The

614 T. ERNEUX and T. KALMÁR-NAGY Figure 7. Bifurcation diagram using k as the control parameter and for a fixed delay 7. The bifurcation for negative k and the one near k 0 are supercritical and stable. The bifurcation at k 04 is unstable because it emerges from an unstable equilibrium. cation point comes closer to k 0 and the branch becomes sharper. This particular behavior is not captured by our analysis, which only considered Case (16) and Case (17) characterized by non-zero values of k if 0 The analysis of Case (15) with k 0 0 requires a higher order analysis and shows the progressive transition to a vertical Hopf bifurcation as 0 This interesting case (small k) will be described in detail elsewhere. Figure 7 shows the bifurcations at 7 and indicates a supercritical Hopf bifurcation for both positive and negative values of k. The supercritical Hopf bifurcation for negative k agrees with our analysis: evaluating (25) with 01 7 0 3 leads to N 0 which implies a supercritical bifurcation (Table 1). As in Figure 6, the Hopf bifurcation near k 0 becomes sharper as 0 We also note a second primary Hopf bifurcation appearing at k 04, which corresponds to the Hopf bifurcation point defined by Case (17) with n 0. If is decreased from 7 to 2the two Hopf bifurcation points coalesces at a double Hopf bifurcation near k 38 0375 In the vicinity of this point, we found numerically limit-cycles and tori suggesting secondary bifurcation phenomena, in agreement with Campbell et al. (1995). 6. DISCUSSION We have analyzed a simple crane model with nonlinear delayed feedback (Pyragas control). We showed analytically and numerically the existence of both supercritical and subcritical Hopf bifurcations. These observations, which are also found for other second-order oscillator

DELAYED FEEDBACK CONTROLLED CONTAINER CRANE 615 problems with delayed feedback, indicate the necessity for nonlinear studies. The existence of unstable limit cycles in the linearly stable regime deteriorates the robustness of the control algorithm. The numerical bifurcation diagrams indicate the presence of stable high amplitude limit cycles coexisting with the stable equilibrium. Whether or not the motion of the crane payload system can become accidentally attracted by these stable limit-cycles needs to be analyzed in detail by simulations of the time dependent problem. In our analysis, we also found several cases of double Hopf bifurcation points, which we shall examine in the near future. Pyragas control is purely linear and the question whether we may design a nonlinear control for better stability properties remains open. NOTE 1. A supercritical (subcritical) Hopf bifurcation overlaps the unstable steady state (the stable steady state). Acknowledgements. The authors would like to acknowledge the help of Kevin Daugherty in producing some of the numerical results with DDE-BifTool. A Mathematica package written by Didier Pieroux has been very useful to translate the equations into MATLAB format. TE acknowledges the support of the Fond National de la Recherche Scientifique (Belgium). REFERENCES Alsing, P. M., Kovanis, V., Gavrielides, A., and Erneux, T., 1996, Lang and Kobayashi phase equation, Physical Revue A 53, 4429 4434. ASME, 2005, 5th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Long Beach, CA, September. Campbell, S. A., Bélair, J., Ohira, T., and Milton, J., 1995, Limit cycles, tori and complex dynamics in a secondorder differential equation with delayed negative feedback, Journal of Dynamics and Differential Equations 7, 213 236. Casal, A. and Freedman, M., 1980, A Poincaré Lindstedt approach to bifurcation problems for differential-delay equations, IEEE Transactions on Automatic Control 25(5), 967 973. Drazin, P. G., 1992, Nonlinear Systems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge. Engelborghs, K., Luzyanina, T., and Roose, D., 2002, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Transactions on Mathematics and Software 28(1), 1 21. Engelborghs, K., Luzyanina, T., and Samaey, G., 2001, DDE-BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations. Technical Report TW-330, Department of Computer Science, K.U.Leuven, Leuven, Belgium. Erneux, T., 2005, Multiple time scale analysis of delay differential equations modeling mechanical systems, Proceedings of IDETC/CIE 2005, ASME 2005 International Design Engineering Technical Conferences, & Computers and Information in Engineering Conference, September, Long Beach, CA, USA. Gilsinn, D. E., 2002, Estimating critical Hopf bifurcation parameters for a second-order delay differential equation with application to machine tool chatter, Nonlinear Dynamics 30(2), 103 154. Henry, R. J., 1999, Cargo pendulation reduction on ship-mounted cranes, Master s Thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA. Henry, R. J., Masoud, Z. N., Nayfeh, A. H., and Mook, D. T., 2001, Cargo pendulation reduction on ship-mounted cranes via boom-lu angle actuation, Journal of Vibration and Control 7, 1253 1264. Kalmár-Nagy, T., Pratt, J. R., Davies, M. A., and Kennedy, M. D., 1999, Experimental and analytical investigation of the subcritical instability in metal cutting, in Proceedings of DETC 99 17th ASME Biennial Conferemce on Mechanical Vibration and Noise.

616 T. ERNEUX and T. KALMÁR-NAGY Kalmár-Nagy, T., Stépán, G., and Moon, F. C., 2001, Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations, Nonlinear Dynamics 26, 121 142. Kalmár-Nagy, T., 2002, Delay-differential models of cutting tool dynamics with nonlinear and mode-coupling effects. PhD Thesis, Cornell University. Kevorkian, J. and Cole, J. D., 1981, Perturbation Methods in Applied Mathematics, Applied Mathematical Sciences 34, Springer, New York. Kevorkian, J. and Cole, J. D., 1996, Multiple Scale and Singular Perturbation Methods, Applied Mathematical Sciences 114, Springer, New York. Landry, M., Campell, S. A., Morris, K., and Aguilar, C. O., 2005, Dynamics of an inverted pendulum with delayed feedback control, SIAM Journal on Applied Dynamical Systems 4(2), 333 351. http://www.math.uwaterloo. ca/~sacampbe/reprints/invpend.pdf Larger, L., Goedgebuer, J.-P., and Erneux, T., 2004, Subcritical Hopf bifurcation in dynamical systems described by a scalar nonlinear delay differential equation, Physical Revue E 69, 036210. Masoud, Z. N., Nayfeh, A. H., and Al-Mousa, A., 2003, Delayed position-feedback controller for the reduction of payload pendulations of rotary cranes, Journal of Vibration and Control 9, 257 277. Masoud, Z. N. and Nayfeh, A. H., 2003, Sway reduction on container cranes using delayed feedback controller, Nonlinear Dynamics 34, 347 358. Masoud, Z. N., Nayfeh, A. H., and Mook, D. T., 2004, Cargo pendulation reduction of ship-mounted cranes, Nonlinear Dynamics 35, 299 311. Masoud, Z. N., Nayfeh, A. H., and Nayfeh, N. A., 2005, Sway reduction on quay-side container cranes using delayed feedback controller: simulations and experiments, Journal of Vibration and Control 11, 1103 1122. Minorsky, N., 1942, Self-excited oscillations in dynamical systems possesing retarded actions, Journal of Applied Physics 9, A65 A71 Minorsky, N., 1962, Nonlinear Oscillations D. Van Nostrand, Princeton, NJ, Chapter 21. Nayfeh, N. A., 2002, Adaptation of delayed position feedback on the reduction of sway of container cranes. Master s thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA. Nayfeh, N. A., Masoud, Z., and Baumann, W., 2005, A comparison of three feedback controllers for container cranes, in Proceedings of IDETC/CIE 2005, ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, September, Long Beach, CA, USA. Omar, H. M., 2003, Control of gantry and tower cranes. Master s thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA. Pieroux, D., Erneux, T., and Otsuka, K., 1994, Minimal model of a class-b laser with delayed feedback: cascading branching of periodic solutions and period-doubling bifurcation, Physical Revue A 50, 1822 1829. Pieroux, D., Erneux, T., Gavrielides, A., and Kovanis, V., 2000, Hopf bifurcation subject to a large delay in a laser system, SIAM Journal of Applied Mathematics 61, 966 982) reprinted in SIAM Revue 45, 523 540 (2003). Pyragas, K., 1992, Continuous control of chaos by self-controlling feedback, Physical Revue Letters A 170, 421. Stépán, G., 1989, Retarded Dynamical Sytems: Stability and Characteristic Functions,Wiley.