Closed-loop design for scalable performance of vehicular formations

TL;DR

The paper tackles scalability challenges in vehicular formations facing instability and poor transient behavior under conventional second-order consensus, especially in directed or large-scale networks. It introduces serial consensus, modeling the closed-loop as two first-order consensus systems in series via the transfer $(sI+L_2)(sI+L_1)X(s)=U_{\mathrm{ref}}(s)$ and a controller $u=u_{\mathrm{ref}}-(L_2+L_1)\dot{x}-L_2L_1x$, ensuring stability when each Laplacian has a connected spanning tree. The main results establish a closed-form bound for scalable performance, $\alpha=\dfrac{p_1+p_2+\max\{2,2p_1p_2\}}{|p_1-p_2|}$, under $u=-(p_1+p_2)L\dot{x}-p_1p_2L^2x$ with $p_1,p_2>0$, and show that the poles are the union of the eigenvalues of $-L_1$ and $-L_2$; the paper also details practical implementations via message passing or extended local measurements to realize the required feedback. Through numerical examples, it demonstrates scalable stability and performance for directed graphs and discusses extensions to 2-hop signaling and estimator-based approaches. Overall, the work offers a scalable, decentralized framework for robust vehicular platoons applicable to large, possibly directed, networked systems.

Abstract

This paper presents a novel control design for vehicular formations, which is an alternative to the conventional second-order consensus protocol. The design is motivated by the closed-loop system, which we construct as first-order systems connected in series, and is therefore called serial consensus. The serial consensus design will guarantee stability of the closed-loop system under the minimum requirement of the underlying communication graphs each containing a connected spanning tree -- something that is not true in general for the conventional consensus protocols. Here, we show that the serial consensus design also gives guarantees on the worst-case transient behavior of the formation, which are independent of the number of vehicles and the underlying graph structure. In particular this shows that the serial consensus design can be used to guarantee string stability of the formation, and is therefore suitable for directed formations. We show that it can be implemented through message passing or measurements to neighbors at most two hops away. The results are illustrated through numerical examples.

Figures (3)

• Figure 1: Vehicle platoon with directed measurements and message passing to implement serial consensus.
• Figure 2: Simulation of the initial value response to $x=0$, $\dot{x}_{i\neq1}(0)=0$, and $\dot{x}_1(0)=1$. For the serial consensus, $p_1=2$ and $p_2=0.5$ was used and for the conventional $r_1=2.5$ and $r_0=1$ was used. Different graph structures and number of vehicles $N$ was tested. For each plot the inter-vehicle distances $e_p(t)=L_\mathrm{ahead-path}x(t)$ are shown. The conventional consensus system can be seen to degrade with increasing number of vehicles $N$, while the serial consensus displays scalable stability and performance.
• Figure 3: Simulation of the initial value response to $x=0$, $\dot{x}_{i\neq1}(0)=0$, and $\dot{x}_1(0)=1$. The conventional consensus system \ref{['eq:undir_conv']} is considered with $r_1=2.5$ and $r_0=1$, while for the serial consensus system \ref{['eq:undir_serial']}, $p_1=2$ and $p_2=0.5$ are used. The results illustrate that for some choices of graph Laplacians the serial and conventional consensus can have comparable performance.

Theorems & Definitions (18)

• Definition 1: Second-order consensus
• Remark 1
• Definition 2: $q$-step implementable relative feedback
• Example 1
• Remark 2
• Definition 3: Second-order serial consensus system
• Theorem 1
• Proposition 1
• Definition 4: Scalable performance
• Theorem 2