Antifragile Control Systems The case of an oscillator-based network model of urban road trac dynamics Cristian Axenieab Margherita Grossia

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Antifragile Control Systems: The case of an oscillator-based network model

of urban road traﬃc dynamics

Cristian Axeniea,b,∗, Margherita Grossia

aIntelligent Cloud Technologies Laboratory, Huawei Munich Research Center, Riesstraße 25, 80992 Munich, Germany

bAudi Konfuzius-Institut Ingolstadt Laboratory, Technische Hochschule Ingolstadt, Esplanade 10, 85049 Ingolstadt, Germany

Abstract

Urban road traﬃc is a highly nonlinear process which continuously evolves under uncertainty. Short-term

sporadic events can determine large changes in traﬃc ﬂow which propagate in both space and time across

an urban road network. Existing traﬃc control systems only possess a local perspective over the multiple

scales of traﬃc evolution, namely the intersection-level, the corridor-level, and the region-level respectively.

Aggregating such local views of traﬃc evolution over a large region is what current control systems try by

using traﬃc models. Yet, capturing uncertainty under such complex spatio-temporal interactions is a very

diﬃcult problem and we often experience how fragile such systems are in reality. Fortunately, despite its

complex mechanics, traﬃc is described by various periodic phenomena. Workday ﬂow distributions in the

morning and evening commute times can be exploited to make traﬃc adaptive and robust to disruptions.

Additionally, controlling traﬃc is also based on a periodic process, choosing the phase of green time to

allocate to opposite directions right of pass and complementary red time phase for adjacent directions. In our

work, we harmonize this perspective and consider a novel system for road traﬃc control based on a network of

interacting oscillators. Such a model has the advantage of capturing both temporal and spatial interactions

of traﬃc light phasing, as well as the network-level evolution of the traﬃc macroscopic features (i.e. ﬂow,

density). We demonstrate that periodicity exploiting closed-loop control systems, which are capturable by

such a model, have the potential to tame periodic dynamics and are robust to the inherent uncertainty in

traﬃc evolution. In this study, we propose a new realization of the antifragile control framework to control

a network of interacting oscillator-based traﬃc light models to achieve region-level ﬂow optimization. We

demonstrate that antifragile control can capture the volatility of the urban road environment and the

uncertainty about the distribution of the disruptions that can occur. We complement our control-theoretic

design and analysis with experiments on a real-world setup to comparatively discuss the beneﬁts of an

antifragile design for traﬃc control.

Keywords: Antifragile Control; Traﬃc Optimization; Oscillator-based Models; Network Models;

1. Introduction

Urban road traﬃc congestion is still resistant to straightforward solutions, given the strong impact it

has on infrastructure as well as the economic and social dimensions of life in large cities. Traﬃc modelling,

control, and optimization remain among the hardest problems across disciplines where only ”incremental”

research pushes a slow progress, as emphasized in the review of van Wageningen-Kessels et al. (2015).

Considering actual technology instantiations, such as SCOOT from Hunt et al. (1982), SCATS from Lowrie

(1990), PRODYN from Henry et al. (1984), OPAC from Gartner (1983), RHODES from Mirchandani &

Head (1998), or LISA from GmbH (2021), adaptive traﬃc signal control systems use a simple model of traﬃc

∗Corresponding author.

Email address: cristian.axenie@huawei.com; cristian.axenie@gmail.com (Cristian Axenie)

Preprint submitted to / to be submitted January 25, 2023

arXiv:2210.10460v3 [eess.SY] 24 Jan 2023

Figure 1: Traﬃc control signaling. The fundamental dynamics in a pair of signalized crosses is described

by the space-time diagram of the platoon of cars passing from South ⇔North for the duration of the green

light signal. The periodic signal behavior ensures that adjacent directions (i.e. West ⇔East transit) are

allowed to pass through (not shown in space-time). The control algorithm should compute the green and

red time signals while taking into account an oﬀset to enable safe sequencing of driving directions.

dynamics and feed it with sensor-based vehicles detection in order to optimize signal timings. Subsequently,

detections from multiple crosses, are aggregated into a central system, which models the ﬂow of traﬃc in

the area. Finally, the underlying traﬃc model is used to adapt the phasing of the traﬃc light signals in

accordance with the ﬂow of traﬃc, thus minimizing unnecessary green phases and allowing the traﬃc to

ﬂow most eﬃciently. For a basic depiction of the space-time dynamics of signalized road traﬃc, we refer the

reader to Figure 1. The focus of our work is to provide a control framework that can handle uncertainty

which prevails in urban traﬃc dynamics and inherent anomalies over regular traﬃc patterns.

1.1. Traﬃc control

In principle, closed-loop road traﬃc control could scale to city-level, in an ideal scenario, when all the

temporal and spatial interactions among the crosses are known and precisely modelled. But, this is never

the case and additional dimensions emerge. For instance uncertainty in the weekly patterns of traﬃc load

on ﬁxed capacity arteries (see work of Chen (2010)), volatility of hourly capacity in multi-lane streets in

the city center or during sport activities (see work of Ossenbruggen et al. (2012)), and variability of the

external traﬃc entering the city or heavy rain during late autumn (see work of Ko et al. (2006)) are clear

components of traﬃc dynamics. Such components make traﬃc contexts fragile towards degenerating into

congestion. The problem lies in the fact that such phenomena are hard to model, predict, and control.

Technically, the core diﬀerentiating aspect among the existing systems is their underlying traﬃc model,

in other words, the dynamics of traﬃc they capture and how this model handles the inherent uncertainty,

volatility, and variability of the captured variables. For instance, based on large amounts of high-resolution

ﬁeld traﬃc data, the work of Hu & Liu (2013) used the conditional distribution of the green signal times

and traﬃc demand to improve urban ﬂow. However, such data were expensive to acquire and the statistical

method couldn’t handle long-tail events (e.g. daily traﬃc patterns ﬂuctuations on weekends vs. week days).

Using a relatively simple model to predict arrivals at coordinated signalled crosses, the work of Day & Bullock

(2020) assumes nearest-neighbor interactions between signals and uses a linear superposition of distributions

to optimize traﬃc lights phase duration. Despite ﬁnding the optimal coordination, the algorithm couldn’t

handle unpredictable changes to platoon shapes (i.e. occasionally caused by platoon splitting and merging)

or unpredictably saturated conditions (i.e. traﬃc jams, accidents). Such limited adaptation capabilities in

the face of uncertainty and disruptions, which can propagate in time and space throughout the system and

lead to heavy congestion, make the system fragile.

The main goal of this study is to introduce the application of antifragile control to traﬃc control and

optimization under uncertainty, volatility, and variability. As coined in the book of Taleb Taleb (2012),

antifragility is a property of a system to gain from uncertainty, randomness, and volatility, opposite to

what fragility would incur. An antifragile system’s response to external perturbations is beyond robust,

such that small stressors can strengthen the future response of the system by adding a strong anticipation

component. For the closed-loop system we employ the model introduced by Axenie et al. (2021) that uses

oscillator-based dynamics modelling. Such an approach has the advantage of capturing both the periodicity

in daily/seasonal traﬃc patterns and the periodicity of the local signal timings, and as shown in Axenie

et al. (2021) is robust and eﬃcient at scale. We now propose an alternative control mechanism, based on

the antifragile control framework introduced by Axenie et al. (2022) and built on top of the principles in

the seminal work of Taleb & Douady (2013).

1.2. Oscillator-based network modelling and interaction dynamics

Oscillator-based modelling and control is an approach emerging from physics that proves to be a plausi-

ble application in traﬃc control. In a very good review and perspective, the study of Chedjou & Kyamakya

(2018) introduced the formalism of oscillator-based traﬃc modelling and control. Despite the strong math-

ematical grounding, the proposed approach was static, in that it removed all convergence and dynamics

of the oscillator-based model by replacing it with the steady-state solution. Such an approach proves its

beneﬁts at the single cross level, as the authors claim, but will fail in large-scale heterogeneous road networks

(i.e. non-uniform road geometry, disrupted traﬃc patterns, volatility of traﬃc load, uncertainty of weather

conditions). The approach we take in the current study is to design a closed-loop antifragile control system

in which a novel variable structure sliding mode controller (see Utkin (2008)) is designed for a nonlinearly-

coupled oscillators model based on the model of Strogatz (2000). We demonstrate through our experiments

that the oscillator-based model with antifragile control can stabilize in a plausible solution of signal timing

under dynamical demand changes based on measurement of local traﬃc data. A similar oscillator-based

model approach was used in the work of Nishikawa & Kuroe (2004) and later in the work of Fang et al.

(2013) as area-wide signal control of an urban traﬃc network. Yet, due to their complex-valued dynamics

and optimization, the systems could not capture both the spatial and temporal correlations under a realistic

computational cost for real-world deployment.

1.3. Fragility-robustness-antifragility continuum in traﬃc control

Traﬃc dynamics is highly nonlinear and sensitive to multiple sources of uncertainty. Here we go beyond

uncertainty in capturing the real dynamics of traﬃc through a model (i.e. structured/parametric uncertain-

ties) and consider the changes that disruptions, such as weather, accidents, social events, and infrastructure

availability (i.e. unstructured uncertainties, or un-modelled dynamics), induce in the overall ﬂow of cars.

The uncertainty, volatility, and variability inherent in such disruptions are described by a stochastic evo-

lution in the space-time-intensity reference system. The compound eﬀect of such disruptions (typically

additive in nature) reﬂects itself in computed measures of quality of traﬃc, for instance travel time for

cars over a certain itinerary. Such unstructured uncertainties determine traﬃc signal re-computations that,

subsequently, alter the shape of the travel time distribution and, hence, the overall travel time distribution

– as depicted in Figure 2.

Going further with our travel time example, the change in the shape of the distribution of the travel time

can be described through the actual type of response (i.e. the shape) to the space-time-intensity character-

istics of the uncertainty. We can then describe the distribution of such responses to uncertainty-triggered

signal re-computation with respect to the travel time itself, as shown in Figure 3. As also show in the excel-

lent work of Taleb & West (2022), if we combine the reaction to uncertainties, or the uncertainty response

(i.e. signal re-computation) S(T ravelT ime), with the distribution of the travel time P(T ravelT ime), we

can describe the probability distribution of the signal re-computation P(S(T ravelT ime)). The core idea

is that we can vary the parameters of the signal re-computation such that the shape of S(T ravelT ime)

changes to handle the changes in P(T ravelT ime) given uncertainty.

Figure 2: Uncertainty, volatility and variability in traﬃc control, when considering travel time over an

itinerary. Space-time-intensity of various sources of uncertainty in traﬃc (e.g. weather, accidents, social

events, infrastructure availability) and the traﬃc signal re-computation that alters the distribution of travel

time. The overall (whole itinerary) travel time distribution change as response to signal re-computation.

Figure 3: Mapping from uncertainty eﬀects on travel time P(T ravelT ime) to the fragile-robust-antifragile

spectrum under signal re-computation. Signal re-computation response of travel time changes shape

S(T ravelT ime) can vary based on the closed-loop control variables and push the system towards a diﬀerent

region of the response to signal re-computation distribution P(S(T ravelT ime)). Due to the periodic nature

of the traﬃc light signalling the shape of S(T ravelT ime) can be convex (i.e. antifragile behavior), concave

(i.e. fragile behavior), linear (i.e. robust behavior), or even mixed convex–concave (i.e. non-stationary

behavior).

The unique conﬁguration of space-time-intensity of the uncertainty actually determines the critical points

where the traﬃc dynamics would not be able to compensate for uncertainty, and become fragile. In traﬃc

engineering, the spectrum of macroscopic behaviors is captured through macroscopic fundamental diagrams

(MFD). As systematically introduced in the work of Li & Zhang (2011), MFD describe bivariate equilibrium

relationships of traﬃc ﬂow, density, and speed that can provide a versatile mapping to the fragile-robust-

antifragile spectrum. The alignment among the two spectra is depicted in Figure 4. Here, we consider

the theoretical MFD, the deﬁnition of the three possible equilibrium states–free ﬂow, bound ﬂow, and

congestion–and their placement in the velocity–density, ﬂow–velocity, and ﬂow–density characteristics (see

Figure 4 a).

Figure 4: Mapping macroscopic fundamental diagrams (MFD) to the fragile–robust–antifragile spectrum in

road traﬃc control under uncertainty. a) Analytic form of the MFD and the traﬃc regimes characteristics.

b) Real MFD extracted from a real-world dataset on a highway segment with three lanes. c) The fragile–

robust–antifragile distributions shapes based on the mapping of MFD on the real data curves. The shapes

of the distribution in the loss and the gain regions (on the MFD) are then matched using Taleb’s heuristic

(see Taleb & Douady (2013)) to the fragile–robust–antifragile continuum.

As one can see in Figure 4 b, the real MFD indicate that both capacity drop and concave-–convex MFD

shapes abound in practice. In this example, the detector data used to compute the MFD in Figure 4 b

is considering a highway scenario from the open-source SUMMER-MUSTARD dataset of Axenie (2021)

available on Zenodo1. Important to note that the analysis is based on a highway scenario, so there’s no

traﬃc signal control. We chose to ﬁrst describe the behavior mapping when traﬃc dynamics evolve without

control. More precisely, we plot the MFD for a three lane road segment (excluding the high-occupancy

lane) in China over a day. We can map the MFD regions, and implicitly traﬃc dynamics, to the fragile–

robust–antifragile spectrum, as shown in Figure 4 c. We use the taxonomy provided Taleb (2012) and the

mathematical identiﬁcation heuristic in Taleb & Douady (2013) to map the loss and gain domains in terms

of the distribution of points in the road traﬃc MFD in Figure 4 a, b.

Considering the ﬂow–velocity characteristic, the captured shape from the data follows the analytic shape

and we can, hence, identify a direct mapping. Here the free ﬂow region corresponds to a fat tail in the gain

domain (i.e. increasing velocity with sub-linearly increasing ﬂow) and a thin tail in loss domain (i.e. high

velocity and high ﬂow), practically corresponding to congestion region of the MFD. The robust behavior

1Data available at: https://zenodo.org/record/5025264

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AntifragileControlSystems:Thecaseofanoscillator-basednetworkmodelofurbanroadtracdynamicsCristianAxeniea,b,,MargheritaGrossiaaIntelligentCloudTechnologiesLaboratory,HuaweiMunichResearchCenter,Riesstrae25,80992Munich,GermanybAudiKonfuzius-InstitutIngolstadtLaboratory,TechnischeHochschuleIngolstadt,...

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Antifragile Control Systems The case of an oscillator-based network model of urban road trac dynamics Cristian Axenieab Margherita Grossia

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