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Safe Delay-Adaptive Control of Strict-Feedback Nonlinear Systems with Application in Vehicle Platooning

Zhenxu Zhao, Ji Wang

Abstract

This paper presents a safe delay-adaptive control for a strict-feedback nonlinear ODE with a delayed actuator, whose dynamic is also a strict-feedback nonlinear ODE and the delay length is unknown. By formulating the delay as a transport PDE, the plant becomes a sandwich configuration consisting of nonlinear ODE-transport PDE-nonlinear ODE, where the transport speed in the PDE is unknown. We propose a predictor-based nonovershooting backstepping transformation to build the nominal safe delay-compensated control, guaranteeing that the output of the distal ODE safely tracks the target trajectory from one side without undershooting. To address the uncertainty in the delay, we incorporate recent delay-adaptive and safe adaptive technologies to build a safe adaptive-delay controller. The adaptive closed-loop system ensures 1) the exact identification of the unknown delay in finite time; 2) the output state stays in the safe region all the time, especially in the original safe region, instead of a subset, after a finite time; 3) all states are bounded, and moreover, they will converge to zero if the target trajectory is identically zero. In the simulation, the proposed control design is verified in the application of safe vehicle platooning. It regulates the spacing between adjacent vehicles to converge to a small distance and avoids collisions by ensuring they do not breach the safe distance at any time, even in the presence of large unknown delays and at a relatively high speed.

Safe Delay-Adaptive Control of Strict-Feedback Nonlinear Systems with Application in Vehicle Platooning

Abstract

This paper presents a safe delay-adaptive control for a strict-feedback nonlinear ODE with a delayed actuator, whose dynamic is also a strict-feedback nonlinear ODE and the delay length is unknown. By formulating the delay as a transport PDE, the plant becomes a sandwich configuration consisting of nonlinear ODE-transport PDE-nonlinear ODE, where the transport speed in the PDE is unknown. We propose a predictor-based nonovershooting backstepping transformation to build the nominal safe delay-compensated control, guaranteeing that the output of the distal ODE safely tracks the target trajectory from one side without undershooting. To address the uncertainty in the delay, we incorporate recent delay-adaptive and safe adaptive technologies to build a safe adaptive-delay controller. The adaptive closed-loop system ensures 1) the exact identification of the unknown delay in finite time; 2) the output state stays in the safe region all the time, especially in the original safe region, instead of a subset, after a finite time; 3) all states are bounded, and moreover, they will converge to zero if the target trajectory is identically zero. In the simulation, the proposed control design is verified in the application of safe vehicle platooning. It regulates the spacing between adjacent vehicles to converge to a small distance and avoids collisions by ensuring they do not breach the safe distance at any time, even in the presence of large unknown delays and at a relatively high speed.
Paper Structure (29 sections, 8 theorems, 71 equations, 5 figures, 1 table)

This paper contains 29 sections, 8 theorems, 71 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Delay-adaptive control
  3. CBF safe control
  4. Contributions
  5. Organization
  6. Notation
  7. Problem Formulation
  8. Safe Delay-Compensated Controller
  9. Nonundershooting backstepping transformation for the $Y$ part
  10. Predictor-based nonundershooting backstepping transformation for the $X$ part
  11. Inverse transformations
  12. Selection of Design Parameters
  13. Result with the nominal safe delay-compensated control
  14. Safe Delay-Adaptive Controller
  15. Delay-adaptive control design
  16. ...and 14 more sections

Key Result

Proposition 1

The inverse transformations of zi--hi are where the functions $\bar{h}_i(\underline{z}_i,\underline{s}^{(i-1)})$, $\bar{\psi}_i(\underline{z}_i,\underline{s}^{(i-1)})=\psi_i(\underline{y}_i)$ are continuously differentiable in all their arguments and satisfy $\bar{h}_i(0,0)=0$, $\bar{\psi}_i(0,0)=0$. The inverse of the transformation ker is where $\delta(x,t)$ is given by inverpred. The inverse t

Figures (5)

  • Figure 1: The transformation between the $Y,u,X$-original and $Z,w,R$-target systems with the predictors $p,\delta$.
  • Figure 2: The diagram of the safe delay-adaptive control.
  • Figure 3: Vehicle platooning with leader $E_0$, and the followers $E_i,i=1,2$, where the safe distances to be maintained are $d_{oi}$.
  • Figure 4: Results for the distance between vehicle $E_0$ and $E_1$, i.e., $d_1(t)=l_0(t)+y_{11}(t)$.
  • Figure 7: Results for the output force of the actuator $F_i=x_{i1}(t)$ and the input voltage $\mathcal{V}_i(t)=\frac{rLU_i(t)}{k_t}$.

Theorems & Definitions (20)

  • Remark 1
  • Proposition 1
  • proof
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 10 more