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Research Papers

Filtered Dynamic Inversion for Vibration Control of Structures With Uncertainty

[+] Author and Article Information
T. M. Seigler

Assistant Professor
e-mail: seigler@engr.uky.edu

Jesse B. Hoagg

Assistant Professor
e-mail: jhoagg@engr.uky.edu
Department of Mechanical Engineering,
University of Kentucky,
281 Ralph G. Anderson Building,
Lexington, KY 40506-0503

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received May 6, 2012; final manuscript received January 30, 2013; published online May 23, 2013. Assoc. Editor: YangQuan Chen.

J. Dyn. Sys., Meas., Control 135(4), 041017 (May 23, 2013) (16 pages) Paper No: DS-12-1130; doi: 10.1115/1.4023670 History: Received May 06, 2012; Revised January 30, 2013

This paper presents a controller for uncertain structures that are minimum phase and potentially subject to unknown-and-unmeasured disturbances. The controller combines dynamic inversion with a low-pass filter to yield a single-parameter high-gain-stabilizing controller. Filtered dynamic inversion requires limited model information, is independent of system order, requires only output feedback, and makes the average power of the command following error arbitrarily small despite the presence of unknown disturbances. The controller is applied to structures modeled by finite-dimensional vector second-order systems with unknown and arbitrarily large order. We also present an adaptive filtered-dynamic-inversion controller, which uses a high-gain adaptive law to increase the controller parameter until the desired performance is achieved. Finally, the controller is extended to vector second-order nonlinear systems, in which case full-state feedback may be required. Examples are given to demonstrate the application and performance of the controller.

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References

Figures

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Fig. 1

Schematic diagram of the filtered-dynamic-inversion controller in feedback with the plant

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Fig. 2

Closed-loop poles of G˜yr and G˜yw as k is increased from 0 to infinity. As k tends to infinity, two closed-loop poles tend to the poles of Gm, and the real parts of the two remaining poles tend to negative infinity.

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Fig. 3

Bode plot of G˜yw for different values of k

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Fig. 4

Closed-loop disturbance rejection of unknown band limited white noise for increasing values of k. Note that the control is turned on at 5 s.

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Fig. 5

Bode plot of G˜yr for increasing values of k. As k is increased, the Bode plot of G˜yr tends to the Bode plot of Gm over any finite frequency range.

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Fig. 6

Closed-loop step response with unknown band limited white-noise disturbance. As k is increased, y approaches ym and u approaches u*.

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Fig. 7

Closed-loop step response with unknown band limited white-noise disturbance and uncertainty in b0. As k is increased, y approaches ym and u approaches u*.

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Fig. 8

Closed-loop poles of G˜yr and G˜yw as k is increased from 0 to infinity. As k tends to infinity, one closed-loop pole tends to the pole of Gm, one pole tends to the open-loop zero, and the real parts of the two remaining poles tend to negative infinity. The closed-loop poles are in the open-left-half-plane for all k > 0.

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Fig. 9

Closed-loop disturbance rejection of unknown band limited white noise for increasing values of k. Note that the control is turned on at 5 s.

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Fig. 10

Closed-loop step response with unknown band limited white-noise disturbance. As k is increased, y approaches ym and u approaches u*.

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Fig. 11

Mass-spring-damper structure

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Fig. 12

Controlled response with k = 200 and k = 500 of the four mass-spring-damper system shown in Fig. 11, where w1 = sin(4πt) N and w2 = sin(2πt + π/4) N. Note that the vertical axes for the k = 200 and k = 500 results are different that the k = 0 case.

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Fig. 13

Controlled response with k = 200 and k = 500 and with uncertainty in Hd

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Fig. 14

Uncontrolled response (k = 0), controlled response with k = 50 and k = 300, and control input for a cantilever beam subject to an unknown point disturbance w(t) = sin 10t + 0.5 sin (65t + π/6) N

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Fig. 15

The minimum stabilizing gain increases with the number of modes used to model the cantilever beam

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Fig. 16

Closed-loop response for varying p and n. The performance depends on p but not on n.

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Fig. 17

Controlled response of modes 1 and 2 with k = 200 and k = 500

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Fig. 18

Response of a three-mass structure with an unknown band limited Gaussian white-noise disturbance. The system is uncontrolled for 15 s and then the adaptive filtered-dynamic-inversion controller (Eqs. (51)–(56)) is turned on. The adaptive parameter k(t) converges to approximately 589 and the motion y of mass m1 is reduced by over an order of magnitude compared to the open-loop response.

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Fig. 19

Two-link rotating arm with payload mass

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Fig. 20

Controlled response of the two-link arm for k = 0, k = 25, and k = 200

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