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Internal and String Stability of an Observer-based Controller for Vehicle Platooning under the MPF Topology

Wei Jiang, Elham Abolfazli, Themistoklis Charalambous

Abstract

In this paper, we study the internal stability and string stability of a vehicle platoon under the constant time headway spacing (CTHS) policy and the multiple-predecessor-following (MPF) vehicle-to-vehicle information flow topology. More specifically, we depart from the conventional Proportional-Integral-Derivative (PID) controller design for such systems and we propose the design of an observer-based controller. For designing our observer-based controller, we first design a distributed observer, with which each follower estimates their position, speed and acceleration error with respect to the leader. The observer is designed by means of constructing an observer matrix whose parameters should be determined. Next, we simplify the design of the matrix of the observer in such a way that the design boils down to choosing a single scalar value; this design further simplifies the structure of the controller, whose simplicity enables the derivation of string stability conditions by means of a frequency response method. Subsequently, the string stability conditions for a given time headway, are transformed to conditions for the controller parameters. We obtain controller parameters that satisfy the stability conditions by designing a novel heuristic search algorithm. Furthermore, we extend the search algorithm by incorporating a bisection-like algorithm, which allows to obtain (within some deviation tolerance) the minimum available value of the time headway. Finally, we provide insights about how to finalize the observer-based controller parameters from above algorithms to avoid the peaking phenomenon. The performance of the proposed observer-based controller is demonstrated via illustrative examples. Additionally, a comparison with a widely-used PID controller for MPF topology shows that our proposed observer-based controller has better convergence performance.

Internal and String Stability of an Observer-based Controller for Vehicle Platooning under the MPF Topology

Abstract

In this paper, we study the internal stability and string stability of a vehicle platoon under the constant time headway spacing (CTHS) policy and the multiple-predecessor-following (MPF) vehicle-to-vehicle information flow topology. More specifically, we depart from the conventional Proportional-Integral-Derivative (PID) controller design for such systems and we propose the design of an observer-based controller. For designing our observer-based controller, we first design a distributed observer, with which each follower estimates their position, speed and acceleration error with respect to the leader. The observer is designed by means of constructing an observer matrix whose parameters should be determined. Next, we simplify the design of the matrix of the observer in such a way that the design boils down to choosing a single scalar value; this design further simplifies the structure of the controller, whose simplicity enables the derivation of string stability conditions by means of a frequency response method. Subsequently, the string stability conditions for a given time headway, are transformed to conditions for the controller parameters. We obtain controller parameters that satisfy the stability conditions by designing a novel heuristic search algorithm. Furthermore, we extend the search algorithm by incorporating a bisection-like algorithm, which allows to obtain (within some deviation tolerance) the minimum available value of the time headway. Finally, we provide insights about how to finalize the observer-based controller parameters from above algorithms to avoid the peaking phenomenon. The performance of the proposed observer-based controller is demonstrated via illustrative examples. Additionally, a comparison with a widely-used PID controller for MPF topology shows that our proposed observer-based controller has better convergence performance.

Paper Structure

This paper contains 31 sections, 2 theorems, 54 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem formulation
  4. Longitudinal vehicle platooning dynamics
  5. Inter-vehicle distance using constant time headway
  6. Inter-vehicle communication structure
  7. Control objectives
  8. Distributed observer-based controller
  9. Leader-following platooning tracking error model
  10. Distributed observer design
  11. Parameter matrix $L$ design
  12. Platoon safety
  13. String stability
  14. Conditions for string stability
  15. Parameters $k_1, k_2, k_3$ and $\alpha$ for string stability
  16. ...and 16 more sections

Key Result

Lemma 1

Under Assumption assump_mpf with $r_i$ being the number of predecessors of follower vehicle $i$, the predecessor-follower platooning tracking error will converge to zero asymptotically, i.e., $\lim_{t\rightarrow \infty} \bar{x}_{i}(t) = 0$, by designing parameter matrices $K$ and $L$ such that $A-BK

Figures (4)

  • Figure 1: One example of MPF IFT with seven follower vehicles among which position $p_i$, velocity $v_i$, acceleration $a_i$ and observer $\hat{x}_i, i \in \textbf{I}_1^7$ (see more details in the proposed observer \ref{['observer_hatx']}) are communicated via V2V communication. The leader sends only its $p_0, v_0$ and $a_0$ to its three followers. Based on Assumption \ref{['assump_mpf']}, we have $r_1 = 1, r_2 = 2, r_i = r = 3, i \in \textbf{I}_3^7$.
  • Figure 2: Performance comparison of followers tracking leader's speed in (i-iv) and comparison of platooning string stability performance with the predecessor-follower platooning spacing error $\bar{e}_i$\ref{['bar_e_i']} in (v-vi).
  • Figure 3: Heuristic search algorithm validation for our controller parameters with fixed time headway $h=0.198s$ in Sec. \ref{['exampleA1']}.
  • Figure 4: Feasible region for Sec. \ref{['exampleA1']} and Algorithm \ref{['alg']} for controller parameters with time headway minimization in Sec. \ref{['exampleB']}.

Theorems & Definitions (15)

  • Remark 1
  • Lemma 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem \oldthetheorem
  • proof
  • Remark 5
  • Remark 6
  • ...and 5 more