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Multi Model Adaptive Control for CACC applicationsFrancisco Navas, Vicente Milanés, Carlos Flores, Fawzi Nashashibi

To cite this version:Francisco Navas, Vicente Milanés, Carlos Flores, Fawzi Nashashibi. Multi Model Adaptive Controlfor CACC applications. IEEE Transactions on Intelligent Transportation Systems, IEEE, In press,�10.1109/TITS.2020.2964320�. �hal-02470639�

Multi Model Adaptive Control for CACC applicationsFrancisco Navas, Vicente Milanes, Carlos Flores and Fawzi Nashashibi

Abstract—This paper proposes a multi-model adaptive control(MMAC) algorithm based on Youla-Kucera (YK) theory to dealwith heterogeneity in cooperative adaptive cruise control (CACC)systems. The main idea of MMAC is to choose the plant ina predefined set that best approximates the system dynamics,applying the corresponding predesigned controller. A set oflinear plants describing different vehicle dynamics is defined.Different CACC controllers are designed depending on theselinear plants. Simulation and experimental results prove howMMAC determines the closest plant in the set, choosing theCACC system able to ensure string stability.

Index Terms—Heterogeneous string, Homogeneous string, Co-operative Adaptive Cruise Control, Youla-Kucera parameteriza-tion, Multi-Model Adaptive Control.

I. INTRODUCTION

Traffic demand has increased significantly over recent years.According to the European Commission, road transport consti-tutes 82.4% of the whole passengers’ transport in the EuropeanUnion (EU) [1]. This leads to traffic congestion problems,which if not solved, could have a cost up to 213 billion eurosin the next 10 years [2].

To overcome the problem of this huge rising traffic volume,intelligent transportation systems (ITS) technology providesefficient solutions without additional infrastructure cost. Anexample of such a system is the commercially available adap-tive cruise control (ACC) [3], that can maintain a minimumsafety time gap with the vehicle ahead. Recently, researchfocuses on the cooperative version of this system, so-calledcooperative ACC (CACC) [4]. Vehicle-to-vehicle (V2V) com-munication is added to ACC, providing a lower minimumsafety time gap, which could increment highway capacity inalmost a 100 % [5].

The main criteria to evaluate a CACC system is the min-imum time gap achievable that guarantees string stability.The latter is defined as the attenuation of disturbances alongthe string of vehicles [6]. As it needs to be ensured toavoid collisions, improve highway capacity and reduce fuelconsumption, string stability analysis for a string of vehicleshas been a hot topic in research.

When vehicles within the string have identical dynamics(i.e. homogeneous string), disturbances attenuates/amplifiesuniformly along the string. String stability for homogeneousstring of vehicles has been widely studied [6], being ensuredfor any positive time gap value if no delay is present inthe communication link between vehicles. This statement isvalidated through experimental results [7] [8].

On the contrary, when vehicles in the string have notidentical dynamics, disturbances do not attenuate/amplify in

F. Navas is with AKKA Research Department (France); butwork was developed with RITS Team at INRIA Paris, France.{francisco-martin.navas-matos@akka.eu}.

the same way downstream; e.g. a vehicle with slower dynamicswill have difficulties to follow a vehicle with faster dynamics.This was first noticed during the GCDC 2011, where ATeamremarked how string stability is explicitly affected by preced-ing vehicle’s dynamics–the minimum time gap achievable goesfrom 0.6s to 1.5s [9]. Guidelines are given in [10] [11] abouthow to change variables of a CACC controller depending onwhether slower or faster dynamics vehicle is in front of theego-vehicle.

String stability for a heterogeneous string of vehicles hasbeen mainly addressed in two different ways: Robust control–a fixed controller able to work under bounded uncertaintiesin the plant. Dynamics difference between ego and precedingvehicles is seen as uncertainty in the platoon model; Adaptivecontrol– controller changes with the unknown uncertainties inthe system, e.g. control is modified depending on ego andpreceding dynamics. Adaptive control is needed when robustcontrol is not able to offer decent behaviour due to all theparameters changes that can occur in the system.

Related to robust control, [12] added some local condi-tions for heterogeneity to a robust controller designed for ahomogeneous string of vehicles. Dynamics of the differentagents within the strings need to be known to validate ifthe required local conditions are fulfilled. [13] proposed acontroller design based on linear matrix inequality (LMI)to ensure individual stability, while an H∞ control algorithmconsiders disturbances that affect string stability. Again someconditions based on the components of the string shouldbe validated. A H∞ robust control design was presented in[14] to overcome uncertainty between model connections. Therange of model parameters needs to be known apriori. [15]presented a H∞ control method which can guarantee bothindividual and string stabilities to parameters uncertainties andcommunication delay. Heterogeneity is reflected by differentvehicle mass and time lag of powertrain dynamics. Wind gustand road slopes are also considered as external disturbances.

Related to adaptive control, almost all the existing worktakes as baseline controller a CACC feedforward/feedbackproportional-derivative (PD) controller. This type of controllerhas been extensively used due its simplicity and performance.[16] designed a new feedforward controller for a heteroge-neous string of vehicles to minimize spacing error and makeindependent string stability from preceding vehicles dynamics.Models of ego and preceding vehicles are included in thefeedforward, so string stability remains as in a homogeneousstring of vehicles. However, there is not mention of howto obtain the dynamics of both vehicles in order to adaptthe controller. More recently, one-vehicle look-ahead topol-ogy with model reference adaptive control is considered in[17]. They augmented a normal feedforward/feedback CACCsystem working in the homogeneous case with an adaptive

term compensating unknown driveline vehicle dynamics. Theidea is the same, but dynamics estimation for subsequentadaptation is included. Preceding vehicle dynamics estimationis also developed in [18] with a particle filter algorithm. [19]proposes an adaptive optimal control solution able to learnthe optimal feedback loop based on online data. However,vehicle dynamics are considered as a first-order function withdifferences of 0.04s in their response time, which seems unrealto simulate heterogeneity. Finally, an adaptive robust controlsolution is proposed in [20]. It deals with different informationdelays, communication topologies, ego-vehicle uncertainty andexternal disturbances. However, dynamics differences betweenvehicles are expressed as a maximum speed, distance anddifferent controller parameters, leading to simplified vehiclemodels that again could not represent the real difference in aheterogeneous string of vehicles.

Adaptive control based on identification covers a greaterdynamic range of vehicle dynamics than robust control. Withthe idea of avoiding an identification process that could slowdown the control loop, a set of linear plants that describes arange of vehicle dynamics can be defined. Plants should begeneral enough to cover a wide range of possible dynamics,properly emulating the intended string heterogeneity. Like-wise, a string stability analysis should be carried out in orderto define application limits, which is missing in the literature.

Controller reconfiguration depending on a set of linearplants it is called in the literature Multi Model Adaptive Con-trol (MMAC) [21]. Different applications of this methodologyare found in the literature: clinical stroke rehabilitation [22],attitude tracking control for a spacecraft [23] or large-scalephotovoltaic plants [24]. The idea of MMAC is a supervisorwho chooses the proper controller among pre-designed candi-dates controllers once more information is known about theplant. Pre-designed controllers are conceived with the set oflinear plants. MMAC is able to determine the closest plant inthe set, switching to its corresponding controller to maintainthe desired performance. [25] proposed an indirect adaptivecontrol based on the identification of linear plants by using thev-gap metric. As the metric is difficult to obtain in real time,[26] proposed a similar approach but using model unfalsifica-tion: If a model and a controller are unable to reproduce theobserved behaviour in closed-loop, then the set plant-controlleris not the correct representation of the system. However, noisecorrelation problems are affecting system performance. Thenoise correlation problem is later solved in [27] by using thedual Youla-Kucera (YK) parameterization.

Here the MMAC algorithm based on YK in [27] is usedtogether with the widely used CACC feedforward/feedbackcontrol structure. A set of linear plants describing differentvehicle dynamics is defined representing heterogeneous stringsfor MMAC validation. Stability analysis is provided, clearlydefining application limits. MMAC should estimate the closestplant in this set to ego and preceding vehicles, so the pre-designed CACC controller able to ensure string stability ischosen. An outline of this algorithm is in Fig. 1. Encouragingresults are obtained both in simulation and in real tests.

In brief, main contributions in this paper are the following:• Adaptive control solution for CACC heterogeneous/ho-

Fig. 1. Outline MMAC for CACC applications.

mogeneous string of vehicles without need of identifica-tion algorithm.

• First and second order models are employed with widerparameter variations in order to properly emulate stringheterogeneity.

• Application limits are well-known, and applicable to awider range just by setting more plants in the set.

• String stability is ensured for a a specific dynamics rangein both ego and preceding vehicles.

• YK-based MMAC approach has been adapted to CACCapplications and applied for the first time experimentally.

• Dynamics estimation results faster than other estima-tion/identification processes in the literature [17].

• Experimental results belong to the very first heteroge-neous CACC algorithm results in the literature togetherwith [28].

The paper is structured as follows. Section 2 gives somebackground for a good understanding of the paper. Section3 describes MMAC for a general set of linear plants andcontrollers. The same is modified in Section 4 for its appli-cation to CACC systems. Simulation and experimental resultsare analysed in Sections 5 and 6. Finally, some concludingremarks are given in Section 7.

II. PRELIMINARIES

This section describes some basic notation, which willbe used extensively in the sequel. MMAC relies on theYK parameterization [29] for controller reconfiguration andsupervision.

A general description of a set of linear plants and its cor-responding predesigned controllers is given, to later apply thedoubly coprime factorization used by the YK parameterization.

A. System description and assumptions

Let’s consider a set of nominal plants represented as {G}={G0...,Gi...,Gn}, describing different dynamics of a system.Gi denotes the ith linear, time-invariant, discrete plant mappinginput signals ui ∈ Rm in output signals yi ∈ Rp. For each ofthese plants a discrete feedback controller Ki mapping errorsignals ei in input signals ui is designed such that the closed-loop (CL) behavior of the system is the desired one. Thus, aset of candidate controllers is defined relying on the nominalset of plants {K}= {K0...,Ki, ...,Kn}. Plants and controllersare represented in state space as:

Gi =

[Ai BiCi Di

];Ki =

[Ac

i Bci

Cci Dc

i

](1)

Time variations in the real plant Greal are considered slowcompared to input-output dynamics. The longitudinal low level

of the vehicle is considered to be properly designed, keepingthe same LTI response with uncertainties as slope, wind gust,.... Nominal plants are defined, such that once a candidatecontroller is selected, it remains unchanged–i.e. the variationsin Greal are smaller than those needed to change from onenominal plant to other within the set.

B. Doubly coprime factorization

[Gi,Ki] needs to be factorized through the doubly coprimefactorization [30] to apply later the basis of MMAC, that is,the YK parameterization.[

Mi Ni][Xl,i

Yl,i

]= MiXl,i + NiYl,i = I (2)

These coprime factors should be such that Gi and Ki are:

Gi = NiMi−1 = Mi

−1Ni

Ki = UiVi−1 = Vi

−1Ui(3)

At the same time, these coprime factors Ui ∈ RHmxp∞ , Ui ∈

RHmxp∞ , Vi ∈ RH pxp

∞ , Vi ∈ RH pxp∞ , Ni ∈ RH pxm

∞ , Ni ∈ RH pxm∞ ,

Mi ∈ RH pxm∞ and Mi ∈ RH pxm

∞ should satisfy the doubleBezout’s identity [31].

[32] allows to obtain coprime factors for [Gi,Ki], when Giand Ki are described in state-space form, Eq. (1). These factorssatisfy (3) and the double Bezout’s identity.

III. MULTI MODEL ADAPTIVE CONTROL

Multi-model adaptive control (MMAC) is a supervisorswitching among prespecified candidate controllers as newdata about the plant is known, until one of these controllersfinally remains unchanged.

It considers that the real plant Greal belongs to a set of nomi-nal plants {G}= {G0...,Gi...,Gn}, or at least is close to one ofthem. Each of the nominal plants is associated to a controllerto give a desired performance {K}= {K0...,Ki, ...,Kn}.

The supervisor is at a higher level, specifying which isthe switching sequence γ that makes the system converge tothe best controller for the unknown real plant Greal. If Grealcoincides with one of the nominal plants in the set {G}, agood candidate controller Ki is straightforward. Otherwise, theclosest nominal plant in the set should be chosen, switchingto the corresponding controller. Both supervisor and switchingare related to Youla-Kucera (YK) parameterization.

YK parameterization is divided in two: Parameterizationof all the controllers that stabilize a given plant; useful forperforming switching between the candidate controllers [33].And its dual formulation (also called Hansen scheme), allthe plants stabilized by a given controller, which recasts CLidentification into an open-loop-like problem [34]; employedwithin the supervisor to determine the switching sequence γ

based on signals u and y.Both theorems are briefly explained in their corresponding

sections. Once the basis is known, subsection MMAC algo-rithm explains how they are used together, examining the setplant-controller [Gi,Ki] and choosing the controller that is bestable to fulfill performance requirements.

yu

r

Qi

V0-1 U0

Mi- ~

~

Ni~

~

G

γi

(a)

Vi Ui

Si

N0 M0-1

y

Ki

u

r

ζ0

-

zi

~ ~

~ ~

(b)

Fig. 2. (a) YK parameterization for controller reconfiguration and (b) DualYK parameterization for CL identification.

A. YK parameterization. Switching

YK parameterization provides all stabilizing controllers fora given plant Gi within the set {G} by interconnecting aninitial controller K0 with Qi, called YK parameter. The initialcontroller could be any in the set {K}. Doubly coprime factorsshould be obtained, so the corresponding Qi can be obtainedas in [35].

Thus, different controllers Ki can be implemented just bygetting the YK parameter Qi. Different Qi’s are obtained foreach controller in the set {K}; so the YK parameter set is{Q} = {Q0, ...,Qi, ...,Qp}. As the initial controller is K0, itscorresponding Q0 = 0. The control structure able to switchfrom an initial controller K0 to any in the set {K} is depictedin Fig. 2 (a).

When doing controller transitions, the scalar factor γi playsa key role. It regulates the different level of activation of theYK parameter Qi. γi may vary from 0 to 1, being 0 a 100%contribution of K0 and 1 a 100% contribution of Ki. If the setof controllers is greater than 2, n > 2, a linear combination ofall the controllers could be implemented as:

Q =n

∑i=1

γiQi withn

∑i=1

γi = 1 (4)

In the present work, just one of the candidate controllers isactivated at the same time. Controller reconfiguration dependson the supervisor.

B. Dual YK parameterization. Supervisor

Dual YK parameterization is used to recast CL identificationinto an OL-like problem [36]. It provides all the plants stabi-lized by a given controller. Again, once double coprime factorsare obtained for an initial plant G0 and a fixed controller Kiin the set {K}, all the plants stabilized by the controller canbe represented by the dual YK parameter Si.

From the general description of any plant stabilized by Ki,it is possible to identify the real plant Greal connected to thecontroller, just by identifying the dual YK parameter Si. Thenthe set of all plants stabilized by Ki is represented by thescheme in Fig. 2 (b). To identify Si, output ζ0 and input zineed to be obtained from measurable data:

zi = Viu− Uiy (5)

ζ0 = M0y− N0u (6)

Even when u and y are measured in CL, the identificationof Si is OL-like by employing the filtered signals ζ0 and zi.Identification algorithms as [37] [38] [39] are suitable for Siidentification.

Si can be interpreted as the difference between real Grealand initial plant G0. Its output, ζ0, gives an insight of thisdifference. When Greal = G0 the value of ζ0 is zero. MMACalgorithm uses this property, so an identification algorithm isno longer needed to figure out the closest plant in the set {G}.

C. MMAC algorithm

Let {K} = {K0...,Ki...,Kn} be a set of predesigned con-trollers for different plants {G} = {G0...,Gi...,Gn}. γ is theswitching sequence: {γ}= {γ0...,γi...,γn}. It determines whichcontroller Ki is activated by setting the corresponding γi to 1at the YK reconfiguration controller structure shown in Fig. 2(a). Only one γi can be set to 1 at the same time.

The switching sequence γ is specified by the supervisor. Thegoal is to figure out which plant in the set {G} is the closest tothe real plant Greal. As outlined above, Si does not need to bedirectly identified to know the closest plant in the set [27]. IfGreal coincides with the initial plant G0, ζ0 should be zero forany value of u and y. By choosing different coprime factors Miand Ni for every nominal plant in the set {G}, ζi = Miy− Niugives the closeness to these plants. The smallest truncated 2-norm Ji = (‖ζi‖2)

2 will activate the signal γi corresponding tothe controller Ki able to fulfill performance requirements.

The MMAC algorithm for a real plant Greal with a setof stabilizing controllers {K} designed for a set of nominalplants {G} is described in Algo. 1. h should be positive, andexpresses the mandatory difference between two norms forcontroller change.

Algorithm 1 Multi Model Adaptive Control1. Initializationγ[n] = [0] . Switching sequence initializationK[n] = [K0, ...,Ki, ...,Kn] . Candidate controllersM[n] = [M0, ...,Mi, ...,Mn] . Left coprime factor M for GiN[n] = [N0, ..., Ni, ..., Nn] . Left coprime factor N for Giζ [n] = [0] . Si output initializationJ[n] = [0] . Truncated 2-norm initializationloop2. YK Controller reconfigurationUpdateController(γ) . Apply controller Ki, with i = γ

Get(u,y) . Obtain measurements u and y3. Supervisor3.1 Closeness to plants in setfor i in (0,n) do

ζ [i] = M[i]y− N[i]u . Output of SiJ[i] = (norm2(ζ [i]))2 . Compute truncated 2-norm/ Closeness

to nominal plantsimin = argmini{i ∈ n | J[i]} . The smallest norm corresponds to

the closest plant3.2 Evaluate switching sequenceif (J[γ[i] == 1]≤ J[imin]+h) then

γ = γ . Previous controller remainselse

γ[imin] = 1 . Controller changesγ[∀i except i = imin] = 0

end loop

IV. MMAC FOR CACC APPLICATIONS

This section describes how MMAC can be used togetherwith a CACC feedforward/feedback structure to address theproblem of heterogeneity in CACC string of vehicles.

This work proposes a feedforward/feedback fractional-order(FO) PD controller [40], able to keep performance but in-creasing flexibility design by moving from integer to fractionalorder differentiation. This CACC FOPD controller is extendedwith the feedforward filter in [16] to ensure string stabilityeven if different dynamics are present in the string. A model ofeach of the vehicles in the string would be necessary. Furtherdetails are in subsection CACC control structure.

In order to avoid any identification process that could slowdown the control loop, MMAC is used as a supervisor ableto choose the proper controller among a pre-designed CACCcontroller set {K}. The task of the supervisor will be toestimate the closest plant to the ego and preceding vehicles, sothe controller ensuring string stability will be chosen. Detailshow Algo. 1 is modified for a CACC application are insubsection Supervisor and switching.

A. CACC control structure

A string of z vehicles driving in the same lane is considered.j determines the order of a vehicle inside the string, j ∈ [1,z].Vehicle j denotes ego-vehicle and vehicle j− 1 precedingvehicle. The solution here focuses on ego-vehicle situationdepending on preceding’s dynamics. From now on, Greal isequivalent to ego-vehicle Gj.

Vehicle j tracks its preceding vehicle, regulating the dis-tance with respect to the desired inter-vehicle distance pro-vided by a constant time gap spacing policy [6]. As seen inEq. (7) the desired intervehicle distance dre f , j is the additionof two terms, a fixed standstill distance s j and a variableterm composed by the vehicle’s velocity v j multiplied by theconstant time gap h j.

dre f , j = s j +h jv j (7)

This desired distance is compared with the real one d j, soa distance error needs to be regulated to zero: ed, j = d j −dre f , j. A FOPD controller (see Eq. (8)) is designed followingguidelines in [41].

Kj = Kp, j +Kd, jsα j =

[Ac

j Bcj

Ccj Dc

j

](8)

String stability is defined as the attenuation of speed distur-bances along the string of vehicles. A sufficient condition ispresented in [42]: The absolute position of each vehicle cannotbe amplified as it propagates along the string:

SS = ‖x j/x j−1‖∞ ≤ 1 f or i > 1 (9)

When ideal communication is considered (no delays arepresent in the communication link), the choice of a feedfor-ward filter as Fj = 1/(h js + 1) = [Af

j Bfj;C

fj Df

j] ensuresstring stability for any positive value of h j [43]. The statementabove is no longer valid when a heterogeneous string of

Kj Gj 1/s

hj

-

ej

-

vj-1

uj

xj xj-1

sj

Fj

v j

dj uj,ff

uj,c

Fig. 3. CACC SISO structure.

[Kj Fj] [0 Gj]Tvj-1

1/s

hj

-

ej

-

[

[vj-1 uj [

[vj 0

xj

xj-1

sj

dj

Fig. 4. CACC MISO structure.

vehicles is considered. [16] included dynamics of ego andpreceding vehicles in Fj (see Eq. 10), proving string stabilityeven when dynamics are not the same.

Fj =1

h js+1G j−1

G j=

[Af

j Bfj

Cfj Df

j

](10)

Both ego Gj and preceding vehicle Gj−1 need to be knownto ensure string stability. When Gj = Gj−1, the feedforwardcontroller reduces to the homogeneous case.

Figure 3 shows the classical SISO structure for CACCsystems. Notice how the control signal u j is the addition ofu j, f f and u j,c. Here the controller structure is modified toMIMO, so FOPD controller Kj and feedforward filter Fj canbe changed at once by using the YK parameterization. Themodified MIMO structure is shown in Fig. 4.

Notice that there are as many controllers as there arepossible dynamics combinations between ego and precedingvehicles.

B. Supervisor and switching

Let’s consider n number of nominal plants in the set {G}={G0...,Gi...,Gn}. Preceding Gj−1 and ego Gj vehicles canbe described by any of these nominal plants; or have a closebehavior to one of them. Depending on ego and precedingdynamics combination, a set of (n + 1)2 CACC controllersis created {K} = {Kxr}, where x is the closest plant in {G}to the preceding vehicle Gj−1, and r the closest to the ego-vehicle Gj. Similarly, γ is the switching sequence {γ}= {γxr},specifying which controller in {K} is activated by modifyingto 1 the corresponding γxr at the YK controller reconfigurationstructure. Only one γxr can be set to 1 at the same time.

The switching sequence γ is specified by the supervisor.The goal is to determine which of the plants in the set {G}is the closest to the real one Gj. As already outlined, thisis done through the signal ζi. The smaller its truncated 2-norm Ji the closer to the plant Gi. The same kind of systemis installed in the preceding vehicle, sending to ego-vehiclewhich is the closest plant to Gj−1. Once x and r are known,the corresponding switching signal γxr is activated, so stringstability is ensured.

The general MMAC algorithm in Algo. 1 is modifiedfor CACC applications. It remains the same, except for aswitching sequence γ that depends on two different indexes, acommunication link added to get the value of x when obtainingmeasurements u and y, and an extra condition when ego-vehicle is exactly between two plants in the set {G}. Evenif unlikely, the fastest model is chosen in order to provide themost string stable controller in the most critical situation. Themandatory difference between two norms h is set to 0.4.

V. SIMULATION RESULTS

This section presents the MMAC CACC performance whenthe string of vehicles is heterogeneous. In the following sub-sections, the sets of nominal plants and corresponding CACCcontrollers are introduced. These sets are chosen accordingto the requirements of convergence and stability. Differentsimulations have been carried out. The first of them (sub-section Matching case), dynamics of both vehicles, precedingand ego, coincide exactly with one of the plants in the set{G}. Subsection Non-matching plant considers that precedingand/or ego-vehicle have dynamics close to one of the plantsin the set.

A. Plant and controller sets

Convergence and stability for MMAC algorithms are as-sured by assuming that the true plant is sufficiently close tothe identified model in the set [44] [45]. Thus, determinationof the correct set of nominal plants ensures convergence andstability [25]. The idea is to determine the stability criterionfor each pair plant/model, analyzing the maximum uncertaintyaround the plant without affecting the performance/stability ofthe system. In that way, one can define the required numberof plants for the desired application range.

Vehicles with fast or slow dynamics, and with a certainovershoot are considered through second-order transfer func-tions: They are LTI plants mapping control velocity signals inoutput velocity signals:

Gi =w2

n,i

s2 +2 fdamping,iwn,is+w2n,i

=

[Ai BiCi Di

](11)

where fdamping,i is the damping factor and wn,i is the naturalfrequency in rad/s. Application range is defined by lim-its in fdamping,i and wn,i: fdamping,i ∈ [0.55,0.65] and wn,i ∈[0.9524,6.667]. Notice that these limits could be extendedin order to deal with every possible vehicle in the market.These limits are in consistency with real vehicle models inthe literature as [46], [47] and [48], validating the selectedapplication range.

Once the application range is defined, string stability inEq. (9) is extended in order to get the minimum number ofnominal plants in {G} able to cover the whole applicationrange with string stability guarantee. For doing so, uncer-tainties in damping factor and natural frequency of both egoand preceding vehicles are included into the string stabilitycondition (see Eqs. (12) and (13)) as ∆ fdamping, j, ∆ fdamping, j−1,∆wn, j and ∆wn, j−1. MMAC uses the same LTI plant model fora given vehicle operational range. When doing so, there are

G1 G2G0

0.15 0.3 0.45 0.6 0.75 0.9 1.05

0.65

0.55

f da

mpi

ng

1/ωn(s)

Fig. 5. Uncertainty range string stable for pairs (G0,K0), (G1,K1) and(G2,K2).

inaccuracies between the associated model and the real model.This can be seen as heterogeneous string stability problemwithin the given operational range. String stability must beguaranteed in the worst-case scenario, providing the numberof nominal plants for the MMAC design.

For the specific range defined above, three models in the set{G} are enough to ensure string stability in the whole applica-tion range. String stable areas for pairs (G0,K0), (G1,K1) and(G2,K2) are depicted in Fig. 5. Small overlapping betweenthem is carried out in order to have a safer solution.

X j

X j−1(∆ fdamping, j,∆ fdamping, j−1,∆wn, j,∆wn, j−1) =

=

G j(∆ fdamping, j ,∆wn, j)FjG j−1(∆ fdamping, j−1,∆wn, j−1)

− G j(∆ fdamping, j ,∆wn, j)K js

1− (G j(∆ fdamping, j ,∆wn, j)K j

s +G j(∆ fdamping, j,∆w j)K jh j)(12)

Gi(∆ fdamping,i,∆wn,i) =

=(wn,i +∆wn,i)

2

s2 +2( fdamping,i +∆ fdamping,i)(wn,i +∆wn,i)s+(wn,i +∆wn,i)2

with i = j, j−1(13)

Values for nominal plants in the set are in Table I. G0, G1and G2 go from faster to slower dynamics. Plants Gx1, Gx2and Gx3 are also present in Table I, and they will be the non-matching cases in the corresponding subsection. Notice howGx3 have first order dynamics corresponding to the productionvehicle in [47]. The latter allows to test how the algorithmbehaves in a high-speed scenario and with a different ordermodel. τi refers to the time constant of the first order transferfunction.

TABLE IPLANTS AND PARAMETERS.

G0 fdamping,0 = 0.6 wn,0 = 3.3333

G1 fdamping,1 = 0.6 wn,1 = 1.6667

G2 fdamping,2 = 0.6 wn,2 = 1.1111

Gx1 fdamping,2 = 0.65 wn,2 = 6.6667

Gx2 fdamping,2 = 0.55 wn,2 = 0.9524

Gx3 τ3 = 0.2

For every plant in the set {G} = {G0,G1,G2} a FOPDcontroller has been designed with the objective of beingrobust to uncertainties. Guidelines in [41] are followed, re-sulting in a controller set {K} = {K0,K1,K2}. Controller

TABLE IIFOPD CONTROLLER PARAMETERS

Kp Kp α h

K0 Kp,0 = 0.35 Kd,0 = 0.15 α0 = 0.3847 h0 = 1s

K1 Kp,1 = 0.5 Kd,1 = 0.225 α1 = 0.3847 h1 = 1s

K2 Kp,2 = 0.6 Kd,2 = 0.3 α2 = 0.3847 h2 = 1s

parameters are shown in Table II. This set is extended in-cluding the feedforward filter; Eq. (10) depends on possi-ble combinations between plants in {G}, yielding {Kxr} ={K00,K01,K02,K10,K11,K12,K20,K21,K22}, where x is theplant in the preceding vehicle, and r the plant in the ego-vehicle.

Once nominal plants and candidate controllers are defined,Theorem 1 is applied to get the coprime factors needed inTheorem 3, so the set of YK parameters is obtained as {Qxr}={Q00,Q01,Q02,Q10,Q11,Q12,Q20,Q21,Q22}. Each of thempermits controller reconfiguration from an initial controller toa controller in the set {Kxr}. This set is reduced by choosingan initial controller K00. Thus, the new set of YK parameters is{Qxr} = {Q01,Q02,Q10,Q11,Q12,Q20,Q21,Q22}. Controllertransition is done through the scalar factor γxr associated toQxr. In the switching sequence γ only one γxr can be setto one at the same time. If none of them is activated, theinital controller K00 will be applied. Table III gathers theinformation related to the controller set and the correspondingYK parameter.

TABLE IIICONTROLLER SET AND YK PARAMETER

Kxr Gj−1 Gj Kj Fj Qxr γxr

K00 G0 G0 K01

hs+1 – –

K01 G0 G1 K11

hs+1G0G1

Q01 γ01

K02 G0 G2 K21

hs+1G0G2

Q02 γ02

K10 G1 G0 K01

hs+1G1G0

Q10 γ10

K11 G1 G1 K11

hs+1 Q11 γ11

K12 G1 G2 K21

hs+1G1G2

Q12 γ12

K20 G2 G0 K01

hs+1G2G0

Q20 γ20

K21 G2 G1 K11

hs+1G2G1

Q21 γ21

K22 G2 G2 K21

hs+1 Q22 γ22

B. Matching case

The matching case considers that ego and preceding vehicleshave dynamics that coincide exactly with one of the plantsin the set {G}. Specifically, preceding vehicle has dynamicscorresponding to plant Gj−1 =G0, while ego-vehicle has fasterdynamics like Gj = G2.

The dual YK parameterization is used in the supervisor, soidentification algorithms are not needed to determine which isthe closest plant in the set. ζi is used instead. By obtaining thesignal value for every plant in the set, one can easily determinecloseness. The top graph in Fig. 6 shows the evolution of thesesignals through time for the ego-vehicle Gj, while the bottomgraph depicts the truncated 2-norm Ji related to ζi. Different

system activation times are considered to see how the systemreacts to different initial conditions. Solid-line indicates thecase where MMAC is activated at 0s, while dotted line showsthe situation where MMAC is activated at 29s. In both cases,ζ2 and J2 are always zero, as Gj = G2. This plant index r isused together with the received index x to specify the switchingsequence γ .

Figure 7 depicts the performance of MMAC CACC algo-rithm when preceding vehicle is Gj−1 = G0 and ego-vehicleGj = G2. A comparison is made between an ego-vehicle withan erroneous controller and with the controller that makesthe system string stable, observing the transition from oneto another when using MMAC. The initial and erroneouscontroller in the YK controller reconfiguration is K00. The topgraph plots vehicles’ speeds when using K00 (dotted red line),K02 (dotted blue line), MMAC activated at 0s (solid greenline) and MMAC activated at 29s (solid pink line). Precedingvehicle speed is also shown (black solid line). The secondgraph plots the switching sequence γ given by the supervisor.The switching sequence is determined by the signals in Fig.6. Dot-line indicates the case where MMAC is activated at29s. Only γ02 is shown to have a lighter graph; the rest arezero even when MMAC is activated at 29s. Finally, the bottomgraph represents the distance error for each of the controllers.

At second 0 Ego-incorrect, Ego-MMAC 1 and Ego-MMAC2 are overlapping; it is not until second 1.8 that the correspond-ing γ02 is activated, making a transition of Ego-MMAC 1 fromEgo-Incorrect to Ego-Correct. From second 1.5 to 32 Ego-MMAC 2 matches with Ego-Incorrect as the system has beenactivated at second 29, and the correct γ02 is not identified untilsecond 32. It is then when Ego-MMAC 2 starts its transitionfrom Ego-Incorrect to Ego-Correct.

From these results, one observes the importance of usingdifferent controllers depending on vehicle dynamics. Whenusing K00 the string of vehicles results unstable. Ego-vehiclespeed is amplified in comparison with the preceding one,and distance error tracking is large. No matter the activationtime, MMAC is able to specify in a few seconds the correctswitching sequence that makes the string stable. Thus, distanceerror tends to zero and the transition between controllers isreally soft. To the best of authors’ knowledge, the only workmixing controller reconfiguration and dynamics estimationfor an heterogeneous string of vehicles is the one in [17].Parameters estimation takes 31s, which is much slower thanthe presented work.

C. Non-matching case

The non-matching case considers that ego-vehicle and/orpreceding vehicle are not one of the plants in the set {G},but within the application range: Gx1, Gx2 and Gx3. TheVinnicombe ν-gap in [49] is employed to figure out whichis the closest plant in the set, so later can be verified ifthe proposed MMAC algorithm is properly working. ν-gapgoes from 0 to 1, and expresses the difference between twoplants; the closer to zero the more the two plants look alike.In the case of Gx1, the lower value of ν-gap is for G0 (ν-gap(Gx1,G0)= 0.5336), so it is the closest plant to Gx1 in

0 10 20 30 40 50-4

-2

0

2

ζ0 1 ζ

1 1 ζ

2 1 ζ

0 2 ζ

1 2 ζ

2 2

Time(s)0 10 20 30 40 50

0

50

100

150

J0 1 J

1 1 J

2 1 J

0 2 J

1 2 J

2 2

Fig. 6. ζi and Ji comparison. Gj−1 = G0 and Gj = G2. Solid-line indicatesthe case where the system is activated at 0s, while dotted line shows the casewhere the system is activated at 29s.

0 10 20 30 40 50

Vel

. (m

/s)

0

2

4

PrecedingEgo - CorrectEgo - IncorrectEgo-MMAC 1Ego-MMAC 2

0 10 20 30 40 50

γ

-0.5

0

0.5

1

1.5γ

01γ

02 1 γ

02 2 γ

10γ

11γ

12γ

20γ

21γ

22

Time(s)0 10 20 30 40 50

Dis

tanc

e er

ror

(m)

-1

0

1

Fig. 7. Simulation results MMAC CACC. Gj−1 = G0 and Gj = G2 .

the set. In the case of Gx2, the closest plant results G2 withν-gap(Gx2,G2)= 0.1449. Finally, in the case of the first orderplant Gx3 the closest results G0 with ν-gap(Gx3,G0)= 0.5722.

Figure 8 shows the evolution through time of ζi and Ji whenGj−1 = G1 and ego-vehicle Gj = Gx3 (Citroen C4 model usedin [47]). Again different system activation times are consid-ered, 10s and 35s in solid and dotted lines respectively. In bothcases, one can observe how the minimum value correspondsto ζ0. Even if Gx3 is a production vehicle model and not acandidate plant in the set {G}, dual YK parameterization givesa good insight of the closest plant. Notice that ζ0 is not zero

10 15 20 25 30 35 40 45 50-2

-1

0

1

2 0 1

1 1

2 1

0 2

1 2

2 2

10 15 20 25 30 35 40 45 50

Time(s)

0

100

200

300

400

500J

0 1 J

1 1 J

2 1 J

0 1 J

1 2 J

2 2

Fig. 8. ζi and Ji comparison. Gj−1 = G1 and Gj = Gx1. Solid-line indicatesthe case where the system is activated at 0s, while dotted line shows the casewhere the system is activated at 20s.

10 15 20 25 30 35 40 45 5022

24

26

28

Vel

. (m

/s)

Preceding

Ego - Correct

Ego - Incorrect

Ego-MMAC 1

Ego-MMAC 2

10 15 20 25 30 35 40 45 50-0.5

0

0.5

1

1.5 01 02 10 1

10 2

11 12 20 21 22

10 15 20 25 30 35 40 45 50

Time(s)

-0.5

0

0.5

Dis

tance

err

or

(m)

Fig. 9. Simulation results MMAC CACC. Gj−1 = G1 and Gj = Gx3.

in these cases, but it is much more smaller than ζ1 and ζ2.Figure 9 depicts the performance of MMAC CACC algo-

rithm when preceding vehicle is Gj−1 = G1 and ego-vehicle isthe production vehicle Gj = Gx3. Again incorrect (K00, dottedred line) and correct controller (K10, dotted blue line) arecompared with MMAC activated at 10s (solid green line) andMMAC at 35s (solid pink line). Preceding vehicle (black solidline) follows a speed profile that goes between 22 and 28m/s.Indexes are identified in preceding and ego vehicles, so thecorresponding γ10 is activated in few seconds no matter the

0 20 40 60 80 100-5

-3

-1

1

3

5 ζ0

ζ1

ζ2

Time(s)0 20 40 60 80 100

0

200

400

600 J0

J1

J2

Fig. 10. ζi and Ji comparison. Gj−1 = Gx1 and Gj = Gx2. System shutdownat 71.5s.

0 20 40 60 80 100

Vel

. (m

/s)

0

2

4

PrecedingEgo - CorrectEgo - IncorrectEgo-MMAC

0 20 40 60 80 100

γ

-0.5

0

0.5

1

1.5γ

01γ

02γ

10 1 γ

10 2 γ

11γ

12γ

20γ

21

Time(s)0 20 40 60 80 100

Dis

tanc

e er

ror

(m)

-1

0

1

Fig. 11. Simulation results MMAC CACC. Gj−1 =Gx1 and Gj =Gx2. Systemshutdown at 71.5s.

activation time, making the distance error zero in a smoothway.

At second 10 Ego-incorrect, Ego-MMAC 1 and Ego-MMAC 2 are overlapping; it is not until second 12.8 that thecorresponding γ10 is activated, making Ego-MMAC 1 goesfrom Ego-Incorrect to Ego-Correct. It is not until second 36that Ego-MMAC 2 starts its transition from Ego-Incorrect tocorrect as the system has been activated at second 35, and thecorrect γ10 is identified at second 36.

Finally, a case with two non-matching models in precedingand ego vehicles is considered: Gj−1 = Gx1 and ego-vehicleGj =Gx2. This is the most critical case possible, as it considers

the fastest preceding vehicle together with the slowest ego-vehicle, and different damping factors in the limits of theapplication range. Evolution through time of ζi and Ji for ego-vehicle is in Fig. 10. The closest plant is G2 as ζ2 is the lowerone. Differences between the unstable controller and stable oneare remarkable in the distance error tracking.

Figure 11 depicts the performance of MMAC CACC al-gorithm when preceding vehicle is Gj−1 = Gx1 and ego-vehicle Gj = Gx2. Incorrect (K00, dotted red line) and correctcontrollers (K02, dotted blue line) are compared with MMACsolution (solid green line). Indexes are correctly identifiedin preceding and ego vehicles, so the corresponding γ02 isactivated, preserving string stability. MMAC shut down isalso analysed in the same figure; the system shuts off at71.5s passing smoothly from a string stable behaviour to anunstable one. Notice how in the unstable case, the distanceerror tracking gets bigger than in previous cases. This distanceerror is directly associated with an ego-vehicle speed that isamplified.

At second 0 Ego-incorrect and Ego-MMAC are overlapping;it is at second 3 that the corresponding γ02 is activated, makingEgo-MMAC goes from Ego-Incorrect to Correct. At second71.5 MMAC system is switched off, making Ego-MMACcomes back to Ego-Incorrect.

It has been proved the correct behaviour of MMAC forCACC applications even if the real vehicle Gj is not exactlyone of the candidate plants in the set {G}. Different orders andspeed scenarios have been used. Real vehicle dynamics shouldbe within the application range. A larger set of models andcontrollers would be needed in practice to cover the differentdynamics in a fleet of vehicles.

VI. EXPERIMENTAL RESULTS

A string of two cycabs is used as the experimental testplatform. Cycabs are mobile platforms used in several researchlabs conceived for urban applications [50]. The vehicle iscontrolled by a commanded velocity. The velocity is limitedup to 4m/s, because it is mainly designed for crowded areaswhere higher speeds can lead to unsafe situations. MMACCACC algorithm is implemented in C++ using RTMaps1

prototyping software.Experimental results serve as convergence validation of a

real system. This is a non-matching case. Since both vehiclesin the string have similar dynamics, the string will always behomogeneous, not being required the activation of any γxr inthe switching sequence γ . The initial controller is modified toK01; in that way, a modification of γ will be mandatory toobtain the best performance possible.

Figure 12 depicts the evolution through time of ζi and Jiduring the experimental test. MMAC algorithm is activatedat 0s. The minimum value corresponds to ζ0, so G0 is theclosest plant to the real dynamics of a cycab. It is not until2.5s that the differences between J0, J1 and J2 are big enoughto determine the switching sequence γ .

Figure 13 depicts the performance of MMAC CACC algo-rithm when applied to a string of two cycabs. A comparison is

1https://intempora.com/

0 5 10 15 20 25 30-1

0

1

2

3ζ

0

ζ1

ζ2

Time(s)0 5 10 15 20 25 30

0

100

200

J0

J1

J2

Fig. 12. ζi and Ji comparison. Experimental results. Homogeneous string oftwo cycabs.

0 5 10 15 20 25 30

Vel

. (m

/s)

0

1

2

3

4

PrecedingEgo-CorrectEgo-IncorrectEgo-MMAC

0 5 10 15 20 25 30

γ

0

1

2 γ00

γ02

γ10

γ11

γ12

γ20

γ21

γ22

Time (s)0 5 10 15 20 25 30

Dis

t. er

ror

(m)

-1.5

-1

-0.5

0

0.5

Fig. 13. Experimental results MMAC CACC. Homogeneous string of twocycabs.

made between the second cycab in the string with an erroneouscontroller and with the controller that makes the systemstring stable, showing the transition between both when usingMMAC. The top graph plots cycabs’ speeds when using K10(dotted red line), K00 (dotted blue line) and MMAC activatedat 0s (solid green line). First cycab’s speed is also shown (blacksolid line). The middle graph shows the switching sequence γ

determined by signals in Fig. 12. The bottom graph plots thecorresponding distance errors when using correct, erroneousand MMAC controllers. Notice that initial distance errors arenot the same for the 3 tests; a maximum error of 20cm wasconsidered to start each of the tests.

From these results, one can see how important it is totake into account vehicle dynamics to ensure string stability.Inappropriate performance can be clearly appreciated on the

distance error tracking. Even if this is less remarkable in thespeed response, distance error is longer when the velocitystep is big enough. The difference in the second step is lessremarkable, as the speed step is smaller, but oscillations arepresent in the distance error. For the MMAC algorithm, ittakes 2.5s to determine that both vehicles are closer to G0,switching to the homogeneous controller K00 through γ00. Thetransition between both controllers is smooth. The resultingtransient is a combination of the initial distance error andthe controller reconfiguration induced by γ00. These resultsvalidate the simulation results seen in the previous section.

Experimental results serve as a convergence proof of areal vehicle with some of the plants in {G}. As vehiclesare homogeneous, the most important task is to detect thatboth vehicles have similar dynamics, in order to switch to theclassical homogeneous CACC controller.

VII. CONCLUSIONS

This paper explores the use of MMAC for CACC applica-tions in heterogeneous/homogeneous string of vehicles.

Youla-Kucera theory is explained as the basis of a generalMMAC algorithm based on a set of nominal plants and pre-defined controllers. MMAC acts as a supervisor determiningthe closest plant to the real system in the set, switching tothe predesigned controller that will give the best performancepossible. No identification algorithm is needed, the dual YKparameterization is used instead.

The general MMAC algorithm is later applied to a feed-forward/feedback FOPD CACC system, so a structure able tohandle different dynamics in a string without the need of anidentification algorithm is obtained.

Performance of MMAC CACC algorithm is analysedthrough simulation results. A set of 3 linear plants is con-sidered for validation purposes. Real vehicles models areconsidered within the application range. Dynamics matchingand non-matching (even with different model order) caseswith a plant in the set are considered, verifying how thesupervisor is able to provide the closest plant in the set,activating the controller that ensures string stability. Dynamicsestimation results much faster than other estimation processesin the literature. A test setup of two cycabs is used, showingthat MMAC CACC design is not only theoretically, but alsopractically feasible.

Although the set {G} is composed by 3 plants, this couldbe extended to n plants, so a larger dynamic range can becovered in heterogeneous CACC applications.

Finally, as communication delays have a direct impact inthe string stability of the system, the developed YK controllerstructure will be extended in order to deal with such delaysand different communication topologies.

VIII. ACKNOWLEDGMENTS

Authors express their gratitude to the french project VALET(ANR-15-CE22-0013) and the RITS Team for its support inthe development of this work.

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Francisco Navas received his PhD degree insidethe RITS team at INRIA (Paris) and MINES Paris-Tech, PSL Research University, in November 2018.Since 2018 he is with the Research Department atAKKA Technologies. His research interests includeautonomous vehicles, Youla-Kucera parameteriza-tion, switching controllers, and vehicle-infrastructurecooperation.

Vicente Milanes received his Ph.D. degree inelectronic engineering from University of Alcala,Madrid, Spain, in 2010. He was with the AUTOPIAprogram at the Center for Automation and Robotics(UPM-CSIC, Spain) from 2006 to 2011. Then, hewas awarded with a two-years Fulbright fellowshipat California PATH, UC Berkeley. In 2014, he joinedthe RITS team at INRIA, France. Since 2016, he iswith the Research Department at Renault, France.Dr. Milanes has been awarded with the Best PaperAward in three conferences and his PhD has received

three major awards. His research interests include autonomous vehicles,vehicle dynamic control, intelligent traffic and transport infrastructures, andvehicle-infrastructure cooperation.

Carlos Flores received his Ph.D. degree in the RITSTeam, INRIA (Paris, France) and MINES ParisTech,PSL Research University (France). He is currentlyworking with PATH at University of California,Berkeley. His research interests include automatedvehicles, cooperative driving, system modelling andcontrol, and fractional-order calculus.

Fawzi Nashashibi is a senior researcher and theprogram manager of RITS team of INRIA (France)and a senior researcher and program manager inthe robotics centre of MINES ParisTech. He has aPh.D. degree in robotics from Toulouse Universityprepared in LAAS/CNRSlaboratory. His current in-terest focuses on advanced urban mobility throughthe design and development of highly AutomatedTransportation Systems.