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Research Papers

Compensation of Time-Varying Input and State Delays for Nonlinear Systems

[+] Author and Article Information
Nikolaos Bekiaris-Liberis1

Department of Mechanical and Aerospace Engineering,  University of California, San Diego, CA 92093-0411nbekiari@ucsd.edu

Miroslav Krstic

Department of Mechanical and Aerospace Engineering,  University of California, San Diego, CA 92093-0411krstic@ucsd.edu

1

Corresponding author.

J. Dyn. Sys., Meas., Control 134(1), 011009 (Dec 02, 2011) (14 pages) doi:10.1115/1.4005278 History: Received March 21, 2011; Revised August 10, 2011; Accepted August 22, 2011; Published December 02, 2011; Online December 02, 2011

We consider general nonlinear systems with time-varying input and state delays for which we design predictor-based feedback controllers. Based on a time-varying infinite-dimensional backstepping transformation that we introduce, our controller achieves global asymptotic stability in the presence of a time-varying input delay, which is proved with the aid of a strict Lyapunov function that we construct. Then, we “backstep” one time-varying integrator and we design a globally stabilizing controller for nonlinear strict-feedback systems with time-varying delays on the virtual inputs. The main challenge in this case is the construction of the backstepping transformations since the predictors for different states use different prediction windows. Our designs are illustrated by three numerical examples, including unicycle stabilization.

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Figures

Grahic Jump Location
Figure 1

System’s response for Example 1. Dot-lines: System with φ(t) = t and the controller 210. Dash-lines: system with φ(t) as in Eq. 211,11 and the uncompensated controller 210. Solid-lines: System with φ(t) as in Eq. 211,11 and the delay-compensating controller 214.

Grahic Jump Location
Figure 2

The trajectory of the robot and the heading θ(t) with the compensated controller 222,223,224,225, 227,228,229 (solid line), the uncompensated controller 222,223,224,225, 226 (dashed line) and the controller 222,223,224,225, 226 for the delay-free system (dotted line) with initial conditions x(0) = y(0) = θ (0) = 1 and ω(s) = v(s) = 0 for all φ(0) ≤ s ≤ 0

Grahic Jump Location
Figure 3

The control efforts v(t) and ω(t) with the controller 222,223,224,225, 227,228,229 (solid line), the controller 222,223,224,225, 226 (dashed line) and the controller 222,223,224,225, 226 for the delay-free system (dotted line) with initial conditions x(0) = y(0) = θ (0) = 1 and ω(s) = v(s) = 0 for all φ(0) ≤ s ≤ 0

Grahic Jump Location
Figure 4

System’s response for Example 3. Dot-lines: System with ϕ(t) = t and the uncompensated controller. Dash-lines: System with φ(t) as in Eq. 211,11 and the uncompensated controller. Solid-lines: System with φ(t) as in Eq. 211,11 and the delay-compensating controller.

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