Proceedings of the 3Sm Conference on Decision & Control Phoenix, Arizona USA December 1999 WeA08 09:lO Closed-Loop Input Shaping for Flexible Structures using Time-Delay Control Vikram Kapila, Anthony Tzes, and Qiguo Yan Department of Mechanical, Aerospace, and Manufacturing Engineering Polytechnic University, Brooklyn, NY 11201 Abstract Input shaping techniques reduce the residual vibra- tion in flexible structures by convolving the command input with a sequence of impulses. The exact cancella- tion of the residual structural vibration via input shaping is dependent on the amplitudes and instances of impulse application. A majority of the current input shaping schemes are inherently open-loop where impulse ap li cation at inaccurate instances can lead to system perkr: rnance degradation. In this paper, we develop a closed- loop control design framework for input shaped systems. This framework is based on the realization that the dy- namics of input shaped systems give rise to time delays in the input. Thus, we exploit the feedback control theory of time dela systems for the closed-loop control of input shaped flexiile structures. A Riccati equation-based and a linear matrix inequality-based frameworks are devel- oped for the stabilization of systems with uncertain, mul- tiple input delays. Next, the aforementioned framework is applied to an input shaped flexible structure system. This framework guarantees closed-loop system stability and performance when the impulse train is applied at inaccurate instances. A simulation study demonstrates that the closed-loop system with the proposed time de- lay controller outperforms the open-loop, input shaped system and the standard linear quadratic regulator when the impulse is applied at an inexact instance. 1. Introduction Active vibration control of flexible structures, such as flexible robotic manipulator systems, has experienced rapid growth in recent years. Flexible structure dynam- ics consist of underdamped poles and zeros resulting in a lightly damped impulse response (Tzes and Yurkovich 1993 . In recent years, many researchers have focused on t ; e development of precompensation schemes to re- duce the residual vibration that result in the structure when control is applied (Bodson (1998), Magee et al. (1997), Pao and Lau (1999), Singer and Seering (1990), Singhose et al. (1996, 1997)). Amongst these schemes, the input shaping technique relies on the development of an impulse train, with appropriate magnitudes and time-intervals, that convolves in real-time with the ref- erence input (Singer and Seering (1990), Swigert (1980), Tallrnan and Smith (1958)). The input shaping filter es- sentially attempts to add zeros to the system at the exact locations of the system poles. Thus, the amplitudes and instances of application of the impulses in input shaping technique de end on the damping factors and natural frequencies o f t h e system poles. Several techniques have recently been proposed that enhance the performance of input shaping filter to ac- Research supported in part by the Air Force Office of Scientific Research under Grant F49620-93-0063, Air Force Research Laboratory/VAAD, WPAFB, OH, under IPA: Vis- iting Faculty Grant, and the NASA/New York Space Grant Consortium under Grant 32310-5891. 0-7803-5250-5/99/$10.00 0 1999 IEEE 1561 count for uncertainties in the damping factors and natural frequencies of the flexible structure (Pa0 and Lau (1999), Singh and Vadali (1993), Singhose et al. (1996, 1997)). These schemes typically lengthen the du- ration of the impulse sequence and result in a slower system response. Despite the advantages offered by the input shaping framework (simplicity, ease of implemen- tation, saturation avoidance, etc.) compared to other precompensators (Tzes and Yurkovich 1989), closed-loop control for input shaped system has received only scant at tention. The dynamics of a flexible structure coupled with an input shaper give rise to time delays in the control input. Since the feedback control theory for time delay systems had not sufficiently advanced until recently (Dugard and Verriest 1997) , the aforementioned closed-loop con- trol problem for input shaped system did not have a tractable solution. However, with the recent advance- ments in the feedback control theory for time delay sys- terns (Dugard and Verriest (1997), Haddad et al. (1997), Kapila et al. (1998), Kim et al. (1996), Moheimani and Petersen (1995), Mori et al. (1983), Niculescu et al. (1994), Niculescu et al. (1997), Shen et al. (1991), Ver- riest and Ivanov (1994)), it is now possible to account for the delay dynamics that arise in the input shaped flexible structures. In addition, since the time delay con- trol theory can account for uncertainty in the amount of input delay (Dugard and Verriest (1997), Kapila et al. (1998), Kim et al. (1996), Niculescu et al. (1997 ), one structures that are robust to inaccuracies in the applica- tion of instances of impulses. Note that inaccuracies in the application of instances of impulses can arise in real- time control, e.g., due to the sampled-data controller im- plementation which restricts the command to be applied at sampling instances. This paper is organized as follows. The input shaped flexible structure dynamic modeling is reviewed in Sec- tion 2. A full-state feedback control design framework for linear systems with multiple, arbitrary, input time delays is presented in Section 3. A Riccati equation- based and a linear matrix inequality (LMI) based delay- independent stabilization techniques for time-invariant dynamical systems with multiple input delays are devel- oped in this section. The proposed framework also ac- counts for degradation in a quadratic performance func- tional in the presence of input delays. An illustrative nu- merical example is given in Section 4 which demonstrates the efficacy of the proposed approach for closed-loop in- put shaping. Finally, concluding remarks are provided in Section 5. can design feedback controllers for input shaped B exible B, Brx s, Br - IrjOr - Y,Pr - 21 < 2, - Nomenclature real numbers, r x s real matrices, Rrxl r x r identity matrix, r x T zero matrix r x r symmetric, positive-definite matrices 2, - 21 E Pr; Zl,Z, E 9'