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Research Papers

Supervisory Control of Dynamical Systems With Uncertain Time Delays

[+] Author and Article Information
Bo Song

School of Engineering, University of California, 5200 North Lake Road, Merced, CA 95343bosong1979@gmail.com

Jian-Qiao Sun1

School of Engineering, University of California, 5200 North Lake Road, Merced, CA 95343jqsun@ucmerced.edu

1

Corresponding author.

J. Vib. Acoust 132(6), 061003 (Sep 14, 2010) (6 pages) doi:10.1115/1.4001846 History: Received October 20, 2009; Revised March 23, 2010; Published September 14, 2010; Online September 14, 2010

A study of controlling dynamical systems with uncertain and varying time delays is presented in this paper. The uncertain time delay is assumed to fall in a range with known upper and lower bounds. We apply the supervisory control algorithm to deal with uncertainties in the time delay. An index is defined for each of the predetermined controls for a discrete set of time delays sampled from the range. Based on this index, a hysteretic switching rule selects a control from the predetermined controls with optimal feedback gains. Each predetermined control must be stable for any time delay in the range. Two control design methods are discussed, namely, the mapping method and a higher order approach. Examples of linear systems are used to demonstrate the theoretical work.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

Boundaries of stability domains (lines) in the gain space and the optimal feedback gains (○) for the LTI systems with four sampled time delays τi(i=1,2,3,4)

Grahic Jump Location
Figure 2

The closed-loop response of the LTI system under feedback controls designed for the sampled time delays when the system true time delay is τ4 and is assumed to be unknown. When the feedback gain is K2, K3, or K4 for the time delay close to the actual time delay, the control performance is quite good. When the mismatch gap is large, or when K1 designed for τ1 is implemented for the system with the true time delay τ4, the performance deteriorates.

Grahic Jump Location
Figure 3

The closed-loop response of the system under the switching control when the initial gain of the control is K1 designed for τ1 while the system true time delay is τ4. This is to be compared with the upper-left subfigure in Fig. 2.

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Figure 4

The switching signal (bottom) and the control index (top) of the hysteretic switching algorithm for the closed-loop response of the LTI system under the lower order control in Fig. 3

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Figure 5

The closed-loop response of the LTI system under higher order controls designed for the sampled time delays when the system true time delay is τmax=9τ/30 and is assumed to be unknown. When the feedback gain is K7 or K8 for the time delay close to the actual one, the control performance is quite good.

Grahic Jump Location
Figure 6

The closed-loop response of the LTI system under higher order controls designed for the sampled time delays when the system true time delay is τmax=9τ/30 and is assumed to be unknown. When the feedback gain K=K9 or K10 for the time delay close to the actual one, the control performance is quite good.

Grahic Jump Location
Figure 7

The closed-loop response of the LTI system under the switching higher order control when the initial gain of the control is K1 designed for τ1 while the system true time delay is τ10 (bottom), as compared with the case when the gain is fixed at K1 (top)

Grahic Jump Location
Figure 8

The switching signal (bottom) and the control index (top) of the hysteretic switching algorithm for the closed-loop response of the LTI system under the higher order control in Fig. 7

Grahic Jump Location
Figure 9

The boundaries of stability domains (lines) in the gain space and the optimal feedback gains (○) for the periodic system with five different time delays τi(i=1,2,3,4,5). The stability boundaries move down along kd axis as the time delay increases.

Grahic Jump Location
Figure 10

The closed-loop response of the periodic system under PD feedback controls designed for the sampled time delays when the system true time delay is τ1 and is assumed to be unknown

Grahic Jump Location
Figure 11

The closed-loop response of the periodic system under the switching PD control when the initial gain of the control is K4 designed for τ4 while the system true time delay is τ1 (bottom), as compared with the case when the gain is fixed at K4 (top)

Grahic Jump Location
Figure 12

The switching signal (bottom) and the control index (top) of the hysteretic switching algorithm for the closed-loop response of the periodic system under the lower order control in Fig. 1

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