Michael Kenison Development Engineer, Schlumberger, 14910 Airline Road, Rosharon, TX 77583 William Singhose Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332 Concurrent Design of Input Shaping and Proportional Plus Derivative Feedback Control Systems that exhibit flexible dynamics are widespread and present a very challenging control problem when their performance is pushed to the limit. If there is some knowledge of the flexible modes, then command signals can be generated to negate the detrimental dynamics. These vibration-reducing commands are dependent on the feedback controller gains because the gains influence the flexible modes. This paper presents a method for concurrently designing a PD feedback controller and a command generator so that performance is optimized. The design method takes into account limits on allow- able overshoot, residual vibration, and actuator effort. Furthermore, the structure of the method allows a wide range of performance requirements, such as disturbance rejection, to be integrated into the design. Results demonstrate that a PD controller cannot achieve the same performance as a PD controller augmented with a command generator. DOI: 10.1115/1.1486009 1 Introduction The vibration of flexible systems often limits speed and accu- racy. If the system dynamics are known with some confidence, then there are several techniques for generating commands that will negate the system’s flexible modes 1–3. Input shaping is one type of command generation scheme that is implemented by convolving a sequence of impulses with the command signal, as shown in Fig. 1 4. In general, the input shaper can contain any number of impulses and the reference command can take on any shape. Input shaping can be used in conjunction with any type of feed- back controller. A block diagram for the case with an input shaper outside the feedback loop and a Proportional-Derivative PD feedback controller is shown in Fig. 2. A method is presented in this paper for formulating residual vibration and auxiliary perfor- mance constraints to simultaneously calculate the input shaper impulses and the PD feedback gains. Input shaping has been implemented on a variety of systems. The performance of long-reach manipulators 5,6, cranes 7–9, and coordinate measuring machines 10–12 was improved with input shaping. In particular, input shaping has shown great prom- ise when used in conjunction with feedback control on flexible robot arms. Hillsley and Yurkovich used input shaping for large- angle movements of a two-link robot, then switched to feedback control when near the desired position 13. A combined input shaper and feedback controller was successfully implemented on a five-bar linkage manipulator in 14. Magee and Book used input shaping in conjunction with feedback control to reduce the vibra- tion of a small articulated robot mounted on the end of a long, slender beam 6. Designing input shapers in the z-domain has proven to be easy and effective 15–17. In the literature discussed above, all of the combined input shaping and feedback controllers have one thing in common: the feedback control gains were calculated first, and an input shaper was then designed for the resulting frequencies of the overall closed-loop system. However, this procedure fails to maximize the system performance. The pole locations of the closed-loop system depend on the feedback controller, and the input shaper depends on these pole locations. Improved performance can be achieved if the feedback gains and input shaper are determined simultaneously. The proposed approach has the following benefits: 1. Limits on maximum overshoot and residual vibration can be satisfied while minimizing the settling time. 2. Performance requirements can be met over a range of oper- ating parameters to account for modeling inaccuracies. 3. Without saturating the actuators, a control scheme can be devised that provides a faster vibration-free response than is possible with PD control alone. 4. Higher levels of performance can be achieved than with the traditional method of sequentially designing feedback gains and then designing the input shaper. 2 Command-Enhanced Feedback The approach used to simultaneously determine the input shaper parameters and feedback gains is called the Command- Enhanced Feedback CEF design method. The position control of a single mass is analyzed in order to demonstrate the CEF method. The mass position is maintained with PD control and friction to ground is neglected. It is assumed that the command Y d ( s ) in Fig. 2 will consist of step changes in position of varying magnitude. The mass is considered settled when the position remains within 5 percent of the desired step size. The plant transfer function is: G P  s  = 1 ms 2 = Y  s  U s  (1) where m is the value of the mass. Ignoring the effect of the dis- turbances, the controller and overall closed-loop transfer functions are given by: G PD  s  =K D s +K P = U s  E  s  (2) and, G CL  s  = K D m s + K P m s 2 + K D m s + K P m = Y  s  R  s  (3) Contributed by the Dynamic Systems and Control Division for publication in the JOURNAL OF DYNAMIC SYSTEMS,MEASUREMENT, AND CONTROL. Manuscript received by the Dynamic Systems and Control Division September 2000. Associate Editor: C. Rahn. 398 Õ Vol. 124, SEPTEMBER 2002 Copyright © 2002 by ASME Transactions of the ASME