Dan Martinec

Nonzero Bound on Fiedler Eigenvalue Causes Exponential Growth of H-Infinity Norm of Vehicular Platoon

We consider platoons composed of identical vehicles and controlled in a distributed way, that is, each vehicle has its own onboard controller. The regulation errors in spacing to the immediately preceeding and following vehicles are weighted differently by the onboard controller, which thus implements an asymmetric bidirectional control scheme. The weights can vary along the platoon. We prove that such platoons have a nonzero uniform bound on the second smallest eigenvalue of the graph Laplacian matrix-the Fiedler eigenvalue. Furthermore, it is shown that existence of this bound always signals undesirable scaling properties of the platoon. Namely, the H-infinity norm of the transfer function of the platoon grows exponentially with the number of vehicles regardless of the controllers used. Hence the benefits of a uniform gap in the spectrum of a Laplacian with an asymetric distributed controller are paid for by poor scaling as the number of vehicles grows.

Wave-absorbing vehicular platoon controller

Abstract The paper tailors the so-called wave-based control, popular in the field of flexible mechanical structures, to the field of distributed control of vehicular platoons. The proposed solution augments the symmetric bidirectional control algorithm with a wave-absorbing controller implemented on the leader, and/or on the rear-end vehicle. The wave-absorbing controller actively absorbs an incoming wave of positional changes in the platoon and thus prevents oscillations of inter-vehicle distances. The proposed controller significantly improves the performance of platoon manoeuvrers such as acceleration/deceleration or changing the distances between vehicles without making the platoon string unstable. Numerical simulations show that the wave-absorbing controller performs efficiently even for platoons with a large number of vehicles, for which other platooning algorithms are inefficient or require wireless communication between vehicles.

Transients of Platoons with Asymmetric and Different Laplacians

Abstract We consider an asymmetric control of platoons of identical vehicles with nearest-neighbor interaction. Recent results show that if the vehicle uses different asymmetries for position and velocity errors, the platoon has a short transient and low overshoots. In this paper we investigate the properties of vehicles with friction. To achieve consensus, an integral part is added to the controller, making the vehicle a third-order system. We show that the parameters can be chosen so that the platoon behaves as a wave equation with different wave velocities. Simulations suggest that our system has a better performance than other nearest-neighbor scenarios. Moreover, an optimization-based procedure is used to find the controller properties.

Zeros of transfer functions in networked control with higher-order dynamics

Abstract This paper presents some results regarding location of transfer function zeros in general network control systems with dynamics of arbitrary order. The numerator polynomial of the transfer function is derived as a function of single agent dynamics and a Laplacian matrix. The results already known in literature are extended from single integrator and bidirectional formations to general dynamics and general interconnection structures. The location of zeros is related to poles of a slightly modified structure. Therefore, in some cases the zeros must follow the same root-locus-like rules as the poles do and they interlace.

PDdE-based analysis of vehicular platoons with spatio-temporal decoupling

Abstract The paper provides exact analytical solutions of wave-like PDdE-based (partial differential-difference equation) models of several popular vehicular platooning problems with bidirectional distributed control. Efficient evaluation of eigenvalues, eigenfunctions and wave velocities is proposed for a finite vehicular platoon without an approximation by a continuum model. Thanks to the decoupling of spatial and temporal dynamics, closed-loop stability can be analyzed by looking at a characteristic polynomial using a root-locus-like graphical technique. Although the paper only contains discussions for static state feedback controllers (using measurements of inter-vehicular distances and relative velocities), extensiona to higher-order (dynamic) controllers is straighforward and will be included in the extended version of the paper.

Harmonic instability of asymmetric bidirectional control of a vehicular platoon

This paper deals with harmonic instability of asymmetric control of vehicular platoons. The considered platoons are composed of identical linear models of arbitrary order but there must be at least two integrators in the open loop. Each vehicle only responds to the movement of its immediate neighbors-the predecessor and the follower-and these two couplings are nonidentical. We show that for arbitrary asymmetry of intervehicular coupling, there is an exponential growth of the peak in the platoons magnitude frequency responses as the number of vehicles grows-a phenomenon called harmonic instability in the literature. Whereas it has been already shown for some particular cases that the beneficial effects of having the Laplacian eigenvalues lower-bounded by a nonzero constant come at the cost of harmonic instability, here we generalize this result to any linear controllers.

Scaling in Bidirectional Platoons With Dynamic Controllers and Proportional Asymmetry

We consider platoons composed of identical vehicles with an asymmetric nearest-neighbor interaction. We restrict ourselves to intervehicular coupling realized with dynamic arbitrary-order onboard controllers such that the coupling to the immediately preceding vehicle is proportional to the coupling to the immediately following vehicle. Each vehicle is modeled using a transfer function and we impose no restriction on the order of the vehicle. The only requirement on the controller and vehicle model is that the platoon is stable for any number of vehicles. The platoon is described by a transfer function in a convenient product form. We investigate how the H-infinity norm and the steady-state gain of the platoon scale with the number of vehicles. We conclude that if the open-loop transfer function of the vehicle contains two or more integrators and the second smallest eigenvalue of the graph Laplacian is uniformly bounded from below, the norm scales exponentially with the growing distance in the graph. If there is just one integrator in the open loop, we give a condition under which the norm of the transfer function is bounded by its steady-state gain—the platoon is string-stable. Moreover, we argue that in this case it is always possible to design a controller for the extreme asymmetry—the predecessor following strategy.

Vehicular platooning experiments with racing slot cars

The paper reports on an affordable platform for indoor experimental verification of algorithms for distributed control of platoons of vehicles. The platform is based on commercially available racing slot cars (by Carrera) equipped with an on-board 32-bit microcontroller-based control system. The assembled PCB board was provided by Freescale Semiconductor, Rožnov pod Radhoštěm, Czech Republic, within their Freescale Race Challenge 2012 competition of individual autonomous slot cars organized for student teams in Czech and Slovak republics. The documentation is freely available online. Some extra components such as an infrared range sensor and a wireless communication module have been added to the original system. The capabilities and potentials of the proposed platform are demonstrated on platooning experiments using two of the most popular platoon control schemes: predecessor following and leader following. Some technical parameters of the platform as well as design experience are shared in the report. It is hoped to encourage other teams to build a similar setup and possibly expose their new distributed control algorithms to an experimental competition.

On the Necessity of Symmetric Positional Coupling for String Stability

We consider a distributed system with identical agents, constant-spacing policy, and asymmetric bidirectional control, where the asymmetry is due to different controllers, which we describe by transfer functions. By applying the wave-transfer function approach, it is shown that if there are two integrators in the dynamics of agents, then the positional coupling must be symmetric; otherwise, the system is string unstable. The main advantage of the transfer function approach is that it allows us to analyze the bidirectional control with an arbitrary complex asymmetry in the controllers, for instance, the control with symmetric positional but asymmetric velocity couplings.

Vehicular platooning experiments with LEGO MINDSTORMS NXT

The paper reports on experimenting with a popular introductory robotics toolkit LEGO® MINDSTORMS® NXT that was aimed at demonstrating a few algorithms for distributed control of a long platoon of autonomous vehicles. The configuration used in this report measures the distance to the vehicle ahead and the velocity of the vehicle itself. The acquired technical experience with this platform is shared in the report, including recommendation towards improvements.