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Research Papers

Time-Delayed Control of SISO Systems for Improved Stability Margins

[+] Author and Article Information
A. Galip Ulsoy

ASME Fellow
Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109-2125
e-mail: ulsoy@umich.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received March 27, 2014; final manuscript received August 29, 2014; published online November 7, 2014. Assoc. Editor: M. Porfiri.

J. Dyn. Sys., Meas., Control 137(4), 041014 (Apr 01, 2015) (12 pages) Paper No: DS-14-1143; doi: 10.1115/1.4028528 History: Received March 27, 2014; Revised August 29, 2014

While time delays typically lead to poor control performance, and even instability, previous research shows that time delays can, in some cases, be beneficial. This paper presents a new benefit of time-delayed control (TDC) for single-input single-output (SISO) linear time invariant (LTI) systems: it can be used to improve robustness. Time delays can be used to approximate state derivative feedback (SSD), which together with state feedback (SF) can reduce sensitivity and improve stability margins. Additional sensors are not required since the state derivatives are approximated using available measurements and time delays. A systematic design approach, based on solution of delay differential equations (DDEs) using the Lambert W method, is presented using a scalar example. The method is then applied to both single- and two-degree of freedom (DOF) mechanical systems. The simulation results demonstrate excellent performance with improved stability margins.

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Figures

Grahic Jump Location
Fig. 1

Closed-loop response to initial condition x(t) = 1.0 for −h < t< 0 for the scalar system in Ex. 3 with SF control (solid red line) and with TDC (dashed blue line) when g = 1 and h = 0.1 s

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Fig. 2

Block diagram of a SISO unity feedback control system with controller C(s) and plant G(s)

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Fig. 3

Bode plot of C(s)G(s) for SF control. GM = ∞ and PM = 91.5 deg.

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Fig. 4

Bode plot of C(s)G(s) for SSD control with g = 1. GM = ∞ and PM = 180 deg (an improvement in PM of 88.5 deg over SF control)

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Fig. 5

Bode plot of C(s)G(s) for TDC (i.e., approximate SSD control) with g = 1 and h = 0.1 s. GM = ∞ and PM = 115.3 deg (an improvement in PM of 23.8 deg over SF control).

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Fig. 6

Schematic of a single-DOF mass-spring-damper system

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Fig. 7

Response x(t) of single-DOF system with SF control (K1 = 3 and K2 = 3.9) to initial condition x(0) = [1 0]T. Red solid line is x1(t) and dashed blue line is x2(t).

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Fig. 8

Loop transfer function Bode plot for SF control (K1 = 3 and K2 = 3.9) of single–DOF system. GM = ∞ and PM = 81.1 deg.

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Fig. 9

Response of the single-DOF system with approximate PD control (TDC, h = 0.1, Kp = 42, and Kd = −39) to initial condition x(0) = [1 0]T. Red solid line is x1(t) and dashed blue line is x2(t).

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Fig. 10

Loop transfer function Bode plot for approximate PD control (TDC, h = 0.1, Kp = 42, and Kd = −39) of single–DOF system. GM = 21.7 dB and PM = 69.4 deg.

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Fig. 11

Loop transfer function Bode plot for SSD control (F1 = 7, F2 = 7.9, G1 = 0, and G2 = 1) of a single–DOF system. GM = ∞ and PM = 180 deg.

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Fig. 12

Response of the single–DOF system with TDC (approximate SSD control and h = 0.1) to initial condition x(0) = [1 0]T. Red solid line is x1(t) and dashed blue line is x2(t).

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Fig. 13

Loop transfer function Bode plot for TDC (approximate SSD control and h = 0.1) of a single–DOF system. GM = ∞ and PM = 98 deg (an improvement in PM of 16.9 deg over SF control).

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Fig. 14

Schematic of a two–DOF mass–spring–damper system

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Fig. 15

Response x(t) of two–DOF system with SF control (K1 = 23, K2 = 1, K3 = 8.7, and K4 = 33.7) to initial condition x(0) = [0 1 0 0]T. Solid red line: x1 and dashed blue line: x2.

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Fig. 16

Loop transfer function Bode plot for SF control (K1 = 23, K2 = 1, K3 = 8.7, and K4 = 37.7) of a two–DOF system. GM = ∞ and PM = 71.8 deg.

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Fig. 17

Loop transfer function Bode plot for SSD control of a two–DOF system. GM = ∞ and PM = 180 deg (an improvement in PM of 108.2 deg over SF control).

Grahic Jump Location
Fig. 18

Response x(t) of two–DOF system with approximate SSD feedback (h = 0.1) to initial condition x(0) = [0 1 0 0]T. Solid red line: x1 and dashed blue line: x2.

Grahic Jump Location
Fig. 19

Loop transfer function Bode plot for TDC (approximate SSD control and h = 0.1) of a two–DOF system. GM = ∞ and PM = 80 deg (an improvement in PM of 8.2 deg over SF control).

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