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Research Papers

# Infinite-Dimensional Pole-Optimization Control Design for Flexible Structures Using the Transfer Matrix Method

[+] Author and Article Information
Ryan W. Krauss

Associate Professor
Mechanical and Industrial Engineering,
Southern Illinois University Edwardsville,
Edwardsville, IL 62026
e-mail: rkrauss@siue.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 29, 2012; final manuscript received August 30, 2013; published online October 4, 2013. Assoc. Editor: Hiroyuki Sugiyama.

J. Comput. Nonlinear Dynam 9(1), 011004 (Oct 04, 2013) (12 pages) Paper No: CND-12-1230; doi: 10.1115/1.4025352 History: Received December 29, 2012; Revised August 30, 2013

## Abstract

This paper presents an approach to control design for flexible structures based on the transfer matrix method (TMM). The approach optimizes the closed-loop pole locations while working directly on the infinite-dimensional TMM model. The approach avoids spatial discretization, eliminating the possibility of modal spillover. The design strategy is based on an iterative process of optimizing the closed-loop pole locations using a Nelder-Mead simplex algorithm and then performing hardware-in-the-loop experiments to see how the pole locations are affecting the closed-loop step response. The evolution of the cost function used to optimized the pole locations is discussed. Contour plots (three dimensional Bode plots) in the complex s-plane are used to visualize the pole locations. A computationally efficient methodology for finding the closed-loop pole locations during the optimization is presented. The technique is applied to a single-flexible-link robot and experimental results show that the optimization procedure improves upon an initial, Bode-based compensator design, leading to a lower settling time.

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## Figures

Fig. 1

Schematic of a three link robot used to illustrate TMM analysis of flexible multibody systems

Fig. 2

A cantilever beam driven by an external torque at the end of the beam. This system is used to illustrate how the TMM models continuous elements.

Fig. 3

Bode plots of the cantilever beam system from Fig. 2 generated using the TMM and an ROM model

Fig. 4

Robot with two flexible links used to illustrate TMM modeling and some of the risks of reduced-order control design

Fig. 5

Bode plots from the infinite-dimensional TMM model and the reduced-order state-space model (ROM) for the system in Fig. 4

Fig. 6

Flow chart of the infinite-dimensional control optimization algorithm

Fig. 7

The single-flexible-link robot (SFLR) used as the experimental test bed for this paper

Fig. 8

Block diagram for the closed-loop system with both joint angle θ and tip acceleration x··tip feedback

Fig. 9

A schematic of the SFLR shown in Fig. 7, showing all of the elements necessary for accurate modeling of the open-loop response of the system

Fig. 10

Closed-loop Bode plot for the joint angle θ/θd, the desired joint angle, for the system with joint angle feedback, but without acceleration feedback

Fig. 11

Closed-loop Bode plot for the acceleration near the tip of the beam x··tip/θd, the desired joint angle, for the system with joint angle feedback, but without acceleration feedback

Fig. 12

Block diagram of the closed-loop system with joint angle θ feedback, but without acceleration feedback

Fig. 13

Step response for the system with joint angle θ feedback, but without acceleration feedback

Fig. 14

Closed-loop Bode plot for the acceleration near the tip of the beam x··tip/θd for the system with joint angle and tip acceleration feedback. Comparing this plot to the one from Fig. 11, the acceleration feedback has significantly reduced the height of the peak near 2.5 Hz.

Fig. 15

Step response for the system with both joint angle and tip acceleration feedback. The 2% setting time for this system is 0.87 s and the overshoot is 23%. The acceleration feedback compensator Ga(s) was designed using classical Bode techniques.

Fig. 16

Bode plot for x··tip/θd for the ROM system with the LQG controller converted to a feedback compensator. The LQG controller has reduced the height of both the major peaks.

Fig. 17

Contour (3D Bode) plot for the infinite-dimensional TMM model with the LQG vibration suppression compensator. While the damping coefficients for two of the major poles have been significantly increased, the pole for the third mode has been driven unstable.

Fig. 18

Contour plot for the system with θ and x··tip feedback. Ga(s) for this contour plot is the result of the Bode-based design discussed in Sec. 5.2.1 and shown in Figs. 14 and 15. This design is the starting point for contour-based optimization.

Fig. 19

Zooming in on the portion of the contour plot of Fig. 18 that is near the origin

Fig. 20

The portion of the contour plot near the origin for a system without acceleration feedback

Fig. 21

Contour plot for Case 1, zoomed in near the origin

Fig. 22

Step response for Case 1

Fig. 23

Contour plot for Case 2, zoomed in near the origin

Fig. 24

Step response for Case 2

Fig. 25

Contour plot for Case 3, zoomed in near the origin

Fig. 26

Step response for Case 3

Fig. 27

Contour plot for Case 4, zoomed in near the origin

Fig. 28

Step response overlay showing the improvement in the settling time from the initial, Bode-based design to the final result of the optimization process (Case 4)

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