adaptive tube-based nonlinear mpc for ecological

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Adaptive Tube-based Nonlinear MPCfor Ecological Autonomous CruiseControl of Plug-in Hybrid ElectricVehicles

bijan sakhdari, nasser lashgarian azadDate Submitted: 19 February 2018Date Published: 20 February 2018

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Subject Classification Vehicular Technology

Keywords Advanced driver assistant systems; ecological autonomouscruise controller; adaptive tube-based model predictive control,real-time control and plug-in hybrid electric vehicles;;

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY,VOL.NUMBER, NO.NUMBER,MONTHYEAR 1

Adaptive Tube-based Nonlinear MPC for EcologicalAutonomous Cruise Control of Plug-in Hybrid

Electric VehiclesBijan Sakhdari, Nasser L. Azad

Abstract—This paper proposes an adaptive tube-based nonlin-ear model predictive control (AT-NMPC) approach to the designof autonomous cruise control (ACC) systems. The proposedmethod utilizes two separate models to define the constrainedreceding horizon optimal control problem. A fixed nominal modelis used to handle the problem constraints based on a robust tube-based approach. A separate adaptive model is used to define theobjective function, which utilizes least square online parameterestimators for adaption. By having two separate models, thismethod takes into account uncertainties, modeling errors anddelayed data in the design of the controller and guaranties robustconstraint handling, while adapting to them to improve controlperformance. Furthermore, to be able implement the designedAT-NMPC in real-time, a Newton/GMRES fast solver is employedto solve the optimization problem. Simulations performed ona high-fidelity model of the baseline vehicle, the Toyota plug-in Prius, which is a plug-in hybrid electric vehicle (PHEV),show that the proposed controller is able to handle the definedconstraints in the presence of uncertainty, while improving theenergy cost of the trip. Moreover, the result of the hardware-in-loop experiment demonstrates the performance of the proposedcontroller in real time application.

Index Terms—Advanced driver assistant systems; ecologicalautonomous cruise controller; adaptive tube-based model predic-tive control, real-time control and plug-in hybrid electric vehicles.

I. INTRODUCTION

The current number of vehicles in the world is approxi-mately 1 billion, a number that with the existing high demandfor personal transportation, is expected to double over the nextfew decades. [1]. The rapid growth in the number of vehicleshas resulted in high air pollution, increased energy demand,higher traffic intensity, longer travel times and increased risk ofaccidents. In fact, the annual cost of traffic injuries worldwideis estimated at $518 billion [2]. Approximately 1.3 millionpeople die in car accidents each year; this number is expectedto increase to 1.9 million annually, based on the current growthrate in vehicle ownership [3]. At the same time, air pollutioncaused by the transportation sector is escalating in many partsof the world. Greenhouse gas emissions produced by the

Manuscript received Month DAY, YEAR; revised Month Day, Year. Thiswork is a part of a project funded by the Natural Sciences and EngineeringResearch Council (NSERC) of Canada, Toyota Technical Center and theOntario Centers of Excellence. The Associate Editor for this paper was —-.

The authors are with the Department of Systems Design Engineering,University of Waterloo, Waterloo, ON N2L-3G1, Canada (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are availableon-line at http://ieeexplore.ieee.org.

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transportation sector has doubled in the past few decades, andis currently responsible for approximately 22% of the totalanthropogenic global greenhouse gas emissions[4]. Issues suchas these have provided the basis for many researchers to pursuethe development of safe and efficient transportation systemsusing modern technology.

In the recent years, advancements in embedded digitalcomputing and communication networks have enabled thedevelopment of automated driving systems, with AdvanceDriver Assistance Systems (ADAS) being one of the mostimportant of these developments. ADAS technologies assistdrivers according to the changes in the environment based onsensing the vehicles surroundings [1]. ADAS can reduce theeffect of error in human judgment in emergency situations toimprove driving safety and performance by making optimaldecisions to enhance the autonomous interaction with theenvironment. Autonomous Cruise Control (ACC) is a topicof interest among several types of ADAS, spurring manyscientific studies in the field. ACC is an advanced version ofthe cruise control system that has the ability to automaticallyaccelerate or decelerate, without any additional input fromthe driver, when the preceding vehicle is speeding up orslowing down. This system is beneficial in many ways, as itcan improve traffic flow, reduce the possibility of accidentsand provide a comfortable driving setting through a semi-autonomous driving experience. [5].

In order to maintain a safe inter-vehicular distance, the mostcommonly developed ACC system utilizes linear controllers.Authors of [6], in their study of adaptive cruise control-basedconcept, developed a single-lane ACC with intelligent rampmetering by enabling vehicles to move in short inter-vehiculardistances to increase highway capacity. In [7], a more complexACC system was created with the ability to adapt itself todiffering driving/road conditions. The researchers applied analgorithm to automatically detect traffic conditions based onsurrounding information, which then subsequently changed theparameters of the ACC system in accordance with each trafficsituation. In [8], authors developed a PID-based ACC aimingto perform and behave as a human driver would. Other authorsemployed linear methods in their design [9], [10], resulting inlow-complexity tracking performance. These methods couldprovide a satisfactory tracking performance with low com-plexity. Also, PID controllers offer tuning parameters that canbe easily adjusted to match different situations and systems.

To improve the benefits of ACC systems, it is possibleto consider fuel efficiency in its design. Considering future

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prediction of traffic motion and environment of the vehiclecould be very useful in this direction [11], [12]. Utilizing ACCto improve fuel efficiency of the vehicles has been widelyinvestigated in previous research. In [13], the authors devel-oped an Ecological ACC (Eco-ACC) system based on ModelPredictive Control. They assumed a stationary condition (zeroacceleration) on surrounding vehicles in order to perform theirfinite horizon optimization. Their result shows that utilizingfuture prediction of the preceding vehicles trajectory yieldsbetter fuel economy. In [11], a smooth acceleration degrada-tion in the prediction horizon was employed to predict thepreceding vehicles trajectory and a jamming wave predictionwas presented to prevent jamming waves while maintaining asafe inter-vehicular distance. The authors used MPC to addressthis problem and showed that their method improves trafficflow and driver comfort, as well as fuel efficiency. In [14],[12]a higher energy efficiency has been achieved via exploitingnonlinear MPC in designing Eco-ACC for Plug-in HybridElectric Vehicles (PHEV).

The majority of studies in this area to date assume thatradar measurements are reliable with little to no imperfectionor uncertainty. However, this assumption is not realistic asradar and Lidar performance are highly dependent on weatherconditions and weather elements, such as snow, rain, or fog,which can substantially reduce their accuracy [15], [16]. Thesensors precision is also contingent on the number of objectswithin range and the type of background [17]. In order tomaintain its stability and performance, a reliable ACC mustbe resilient to data uncertainty and modeling error. With windand road disturbances as their main consideration, the authorsin [18], developed an H2/H∞ control method-based model-based controller, which lead to improved tracking performancedemonstrated in simulations. In [19], the authors assumed alinear model for their vehicle with disturbances on the states todevelop a robust ACC system. This led to improved trackingperformance, but also resulted in higher computational effortthrough the utilization of min-max robust MPC method, whichmakes this method not suitable for real-time applications.Another version of robust MPC is tube-based MPC (T-MPC),which works based on a tube resulted from bounded uncer-tainties in the system [20], [21]. In order to ensure the tightbondage of the real states inside the demarcated restraints, T-MPC maintains the nominal system inside a tighter area. Thecomputational demand of T-MPC is only slightly higher thanregular MPC, since the requisite tube can be calculated offline,making it suitable for real-time applications. In [22] and [23],the authors used T-MPC to design a semi-autonomous groundvehicle. They considered the uncertainties and nonlinearities asan additive disturbance, then calculated a tube for the disturbedstates. Their MPC used the resulted tube to gain robustnessagainst system uncertainties. In [24], the authors developed arobust ACC controller using linear T-MPC. Their simulationon a high-fidelity vehicle model showed that this methodcan ensure robustness against delayed data, uncertainty andmodeling errors in a car following scenario.

Robust control methods can guarantee safety and stability;however, they are usually conservative and can deteriorateperformance of the controlled system. Therefore, many re-

searchers prefer adaptive control approaches that estimatechanges in parameters and respectively adapt to them tomaintain performance and stability of the system. Moreover,due to changing conditions and aging of the vehicle, the actualparameters of a car might change and, therefore, an on-lineoptimization algorithm with a fixed model might not be able tofind the actual optimal control decisions. To consider this mat-ter, in [25], the authors designed a hierarchical cruise controlfor connected vehicles and used a gradient-based parameterestimation to estimate the changes in vehicle parameters.They fed the estimated parameters to a low-level sliding-modecontroller to regulate axle torque so that the desired statesare followed. In [26], authors used a recursive least squareparameter estimator and adaptive nonlinear MPC to design acruise controller with fuel optimization. They used parameterestimation to improve the control-oriented model of their MPCand, by performing vehicle experiments, showed that theirmethod can achieve a 2.4% improvement in fuel economy,compared to a production cruise controller. Similar adaptivecontrol approaches can be found in [27],[28],[29],[30]. Al-though these methods can capture the changes in the modeland act accordingly, in the event of a sudden change in pa-rameters or wrong estimation, they may loose performance andstability. Specially, for close car following, the controller mustbe able to guarantee safety of the system while improving theperformance. Therefore, a method is needed that can adapt tochanges while being robust to uncertainties, disturbances andmodel errors. To combine robustness and model adaptation,in [31] and [32] authors used a type of adaptive MPC thatthey called learning-based model predictive control. In theirmethod, a linear controller generates optimal inputs based ona learned linear model, while a separate model checks if theconstraints will be satisfied. Nonlinear learning based MPCwas used in [33] and [34] for path tracking control of a mobilerobot in outdoor and off-road environment. They used a simpleknown model and a Gaussian process disturbance model thatcan be learned based on trial experience. Their experimentalresults on different robot platforms show that their controlleris able to reduce path tracking error by learning and improvingdisturbance model through experience.

In this paper, an adaptive tube-based nonlinear MPC (ATN-MPC) controller is presented that can improve the performanceof Eco-ACC by adapting to the changes in system, whilemaintaining robustness against uncertainties, disturbances andmodeling errors. This method decouples performance from ro-bustness and, therefore, is able to maintain stability and safetywhile adapting to changes in the system and the environment.First, nonlinear T-MPC method is used to design a robustcontroller that can handle the uncertainties in estimation ofthe drag coefficient, gravitational forces due to uncertainty inroads grade estimation, uncertainty in the preceding vehiclesacceleration, and also delay in the data gathered from theon-board vehicle radar. The designed controllers optimizesvehicles motion in finite horizon to improve the consumedenergy cost of the vehicle, while handling the defined con-straints in the pretense of uncertainty and disturbances. Then,to capture changes in the system and enhancing the control-oriented model, an on-line parameter estimation algorithm is



used that estimates new parameter values based on minimizingthe error between estimated and actual output of the system.This way the on-line optimization will find the actual optimalpoint based on the adapted model while constraints are handlesbased on the original nominal model.

The main contribution of this paper is in combining ro-bustness against uncertainties with parameter adaptation in thedesign of controller. The TA-NMPC approach is proposed thatcan robustly handle defined constraints while adapting to thechanges and improving performance of the Eco-ACC systemfor PHEVs. Moreover, to be able to execute the designed op-timal controller in real-time, a Newton/GMRES fast solver isadapted to solve the AT-NMPC optimal control problem. Therest of the paper is structured as follows: Section II presentspreliminary definitions; In section III, modeling procedure isexplained and a model for car-following is presented that canbe used for the design of ACC controllers, also uncertaintybounds are defined based on the presented model as an additivedisturbance. In section IV, the controller design is explainedby taking advantage of the AT-NMPC method to achieverobustness and high performance in ACC design. Section V isdedicated to control evaluation. In this section, the proposedmethod has been simulated on a high-fidelity vehicle model ina car-following scenario and in a simulation environment withinjected uncertainties. Moreover, HIL experiments has beenpresented in this section for the proposed controller. Finally,conclusions are presented in Section VI.

II. PRELIMINARIES

In this paper, ⊕ and indicate the Minkowski sum and thePontryagin’s set difference. If X and Y are sets, then: Y ⊕X ={x+y : x ∈ X, y ∈ Y } and Y X = {v ∈ Rn : v⊕X ⊆ Y }.Also, X⊕(Y⊕V ) = {x+y+v : x ∈ X, y ∈ Y, v ∈ V }, AX ={Ax : x ∈ X}, AX +BY = {Ax+By : x ∈ X, y ∈ Y } and⊕m

i=nYi = Yn + Yn+1 + · · ·+ Ym.

III. MODELING

This section explains the models that have been used fordesign and evaluation of the proposed controller. Control eval-uation has been done on a high-fidelity model of the base-linevehicle, which is a Toyota Plug-in Prius. It consists of complexhigh-fidelity models and mappings of all the components inthe vehicle that can affect longitudinal motion and energyconsumption. The high accuracy of this model makes it areliable tool for evaluation of the designed controllers. Forcontrol design, however, a simple model is needed that has lowcomputational demand but is descriptive enough to capture thegeneral behavior of the system. Here, different control-orientedmodels are presented that represent longitudinal motion andenergy consumption of the vehicle.

A. High-fidelity power-train model

Fig. 1 illustrates the power-train of a power-split ToyotaPlug-in Prius. It has an internal combustion engine and twoelectric motors that combined with each other and throughplanetary gears, power the vehicle. To evaluate performance ofthe designed controllers, a high-fidelity model of the vehicle

Fig. 1. Schematic of Toyota power-split power-train

is needed. Therefore, a model of the base-line vehicle wasdeveloped in Autonomie which is a new generation of PSATsoftware developed by Argon National Lab. Autonomie has alibrary of vehicle models and components that can be selectedto generate a high-fidelity model of the whole vehicle. Itallows the selection of a two wheel vehicle with hybrid power-split power-train and modification of its components andcharacteristics to simulate the base-line PHEV. The proceduretaken in development of the high-fidelity model and testingits validity has been presented in previous publications by theauthors research group [35],[36]. This model has been used forevaluation of the proposed controllers in terms of longitudinalmotion control and trip energy cost.

B. Car following control-oriented model

To develop a model for car-following problem shown inthe Fig. 2, it is necessary to define a safe car following rule.Among different spacing policies in literature, constant timeheadway rule was chosen in this paper: d = d0 + hvh, whered is the desired distance, d0 is the minimum distance at standstill, vh is the host vehicle’s velocity and h is the constantheadway time. This spacing policy requires increasing distancewith respect to velocity so it takes a specific constant amountof time for the host vehicle to reach to its preceding. Basedon the chosen gap policy, the sate equations of the system canbe written as follows:

x = Ax+Bu(t− τa) +Bpap +BgFr

x =

epevvhTw

, A =

0 0 1 − h

mrw0 0 0 − 1

mrw0 0 0 1

mrw0 0 0 − 1

η

,

B =

000Ka

η

, Bp =

0100

, Bg =

−h−110

,

Fr = −1

2ρaAcCd(vh + vw)

2 − (µr0 + µrvvh)mg cos(φr)

−mg sin(φr),(1)

where ep = pp − ph − hvh, ev = vp − vh and ah, vh and phare the acceleration, velocity and position of the host vehicle,pp, vp and ap are the velocity, position and acceleration ofthe preceding vehicle, Fr represents sum of all the resistance



Fig. 2. Car-following with autonomous cruise control

forces, ρa is the air density, Ac is the frontal area of the car, Cd

is the drag coefficient, µr0 and µrv are the rolling resistancecoefficient, m is the vehicle’s mass, vw is the head wind speedand φr is road grade. Tw is the wheel torque and rw isthe wheel radius. (1) represents a nonlinear control-orientedmodel that is used for calculating the disturbance sets. Thismodel includes drag forces by vehicle motion and wind, rollingresistance force and gravity force due to grade change. Todefine a cost function based on energy economy improvement,a control-oriented model for the consumed energy is needed.Because our base-line vehicle is a PHEV, we have to considerboth fuel and electricity costs. Therefore, instead of fuel rateand electrical current, we define the cost function based oncombined energy cost of the two sources

Ecost = −Cfmf

vh− Ce

˙SOC

vh, (2)

where Ecost is the cost of energy, Cf and Ce are the costof gasoline and electricity, respectively, and SOC is the stateof charge of the battery. The energy cost has been dividedby the velocity to eliminate the effect of traveled distance. Atlower velocities, the energy cost will be assumed constant toavoid singularities. An energy management algorithm decidesthe distribution of energy between the energy sources whilethe vehicle is running to keep the power-train near its optimalworking point. Therefore, it can be assumed that the engine isalways working in its optimum working point and approximatefuel consumption with the following equation [37]:

mfi = α1 + α2Pe + α3P2e + α4vh, (3)

where mfi is the fuel rate, Pei is the engine power and α1, α2,α3 and α4 are constant coefficients. To estimate the electricityrate, we used the following equation:

˙SOC = γ1 + γ2Pm + γ3P2m, (4)

where Pm is the motors’ or generators’ power and γ1, γ2and γ3 are constant coefficients. The squared electric powerhas been included in the model to represent ohmic losses.As mentioned, energy management decides the power ratiobetween electricity and gasoline. Therefore, based on powerratio, the energy cost can alternate for different total powerdemands. Based on power ratio the power demand from eachsource can be calculated:

PR =Pe

Ptotal,

Pe = PR ∗ vh ∗ u ∗m,

Pm = (1− PR) ∗ vh ∗ u ∗m,

(5)

where PR is power ratio and Ptotal is the total power demand.

C. Reduced model

The control-oriented model needs to be updated based onthe on-line measurements in the system. In this paper, adapta-tion has been done based on a recursive Least-square method,which requires a parametric model. The following formulationhas been used for longitudinal dynamic’s parametric model:

ah =Rgηprwm

Tcom − ρaAcCd

2m(vh + vw)

2

− g(µr0 + µrv )cos(φr)− gsin(φr),(6)

where Rg is gear ratio, ηp is power-train efficiency, Tcom isthe commanded torque and other parameters are as definedbefore. This formulation can be rearranged as follows:

svh =Rgηprwm

Tcom − ρaAcCd

2m(vh)

2

− (ρaAcCd

mvw + gµrv )(vh)− g(φr)

− ρaAcCd

2mvw

2 − gµr0 .

(7)

Finally, by using stable filtering this model can be reduced tothe following model:

s

s+ λvh =θ1

Tcom

s+ λ− θ2

vh2

s+ λ− θ3

vhs+ λ

− θ4φr

s+ λ− θ5

1

s+ λ,

(8)

where λ is the stable filter’s time constant. Therefore thereduced parametric model is:

ah = ΘTΦd, (9)

where ah is the estimated acceleration and:

Θ =[θ1 θ2 θ3 θ4 θ5

]T, (10)

which Θ is the vector of the estimated parameters and:

Φd =1

s+ λ

[Tcom −vh

2 −vh φr 1]T

, (11)

where Φd is the regressors’ vector for longitudinal dynamicestimator. Equations 3 and 4 are already in the parametric formof mf = AΦe and ˙SOC = ΓΦm with:

Φe =[1 Pe Pe

2 vh]T

,

Φm =[1 Pm Pm

2]T

,

A =[α1 α2 α3 α4

]T,

Γ =[γ1 γ2 γ3

]T.

(12)

The hat shows the estimated value of a parameter. Presentedmodels in this section will be used for control design andevaluation in the following sections.

IV. CONTROL DESIGN

This section is devoted to the control design procedure.First, the effective disturbances and uncertainties are ana-lyzed and an additive disturbance term that captures themis presented. A linear feedback controller (Kc) is designedthat stabilizes the system and bounds the effect of additive



Fig. 3. AT-NMPC controller architecture

disturbances on the system’s sates. Then, nonlinear T-MPCdesign procedure is explained, which is able to handle the de-fined constraints in the presence of bounded uncertainties anddisturbances. The final control input to the system is generatedby combining the designed linear controller with the outputof MPC. Finally, an on-line least square parameter estimatoris presented, which estimates the uncertain parameters of thesystem in real-time. The estimated parameters are used insidethe MPC controller to improve its performance in case of achange in the parameter values. This way the final systemwill be robust to changes in the uncertain parameters thatcan also adapt to them to improve the control performance.Therefore, robustness and performance will be decoupled andachieved simultaneously. Fig. 3 illustrates the proposed Eco-ACC architecture.

A. Disturbance set

To design a robust MPC, a bound must be established onthe states’ error caused by disturbances or a robust positiveinvariant set defined below.

Definition 1: For an autonomous system x[k+1] = Ax[k]+Bw[k] with bounded disturbance w[k] ∈ W, robust positiveinvariant set Φ is the set of all x[k] ∈ Φ such that for allw[k] ∈ W and i > 0, x[k + i] ∈ Φ [38].

Suppose a nonlinear system in the following format:

x[k + 1] = Ax[k] +Bu[k] + g(x[k]) + w[k],

subject tox[k] ∈ X,u[k] ∈ U,w[k] ∈ Wm,

(13)

where x is the state of the system, u is input, w is an additivedisturbance, X, U and Wm are bounds on state, input anddisturbance in the system and g(x) is the nonlinear part ofthe system. Now suppose that the input to the system has thefollowing form:

uk = −Kx[k − τd] + c0 (14)

where K is a linear stabilizing controller, c0 is the inputgenerated by the model predictive controller and τd is thetotal delay in radar and actuation. We ignore τd in the rest ofcalculations and model it as part of uncertainty in Proposition1. By considering this input, (13) can be rewritten as

x[k + 1] = Acx[k] +Bc0 + g(x[k]) + w[k], (15)

where Ac = A−BK. On the other hand, without consideringthe disturbance term, the nominal system can be written as

x[k + 1] = Acx[k] +Bc0 + g(x[k]), (16)

where x is the nominal state. By reducing (16) from (15) andconsidering state error as e = x − x, error dynamic can bedefined.

e[k + 1] = Acek + (g(x[k])− g(x[k])) + w[k] (17)

An RPI set of this system is equivalent to the maximum errorcaused by the additive disturbance. To be able to find RPI setof this system, we need to handle the error in the nonlinearterm. Authors of [23] showed that if the nonlinear term g(x) isLipschitz continues, ‖g(x)− g(x)‖2 can be bounded. If g(x)is Lipschitz in the region x ∈ X then:

‖g(x)− g(x)‖2 ≤ L ‖x− x‖2 , ∀x1, x2 ∈ X, (18)

where the smallest L satisfying this condition is the Lipschitzconstant. Now if L(X) is the Lipschitz constant over X thenit can be obtained from (18) that

∀x, x ∈ X & e ∈ E, ‖g(x)− g(x)‖∞≤ L(X) max

e∈E‖e‖2 ,

(19)

where E is a subset of X which includes the origin. Thisinequality defines a boxed shaped set that bounds the errorin the nonlinear term.

Wg = {ζ ∈ Rn| ‖ζ‖∞ ≤ L(X)maxe∈E

‖e‖2}, (20)

which can be added to Wm to make W = Wg ⊕ Wm.Basically, if a bound can be defined on the nonlinear termin the constraints region, then we can consider it as part ofthe additive disturbance. The next step is to find a bound forWm. To be able to use this method, all sources of uncertaintymust be combined into a single additive disturbance. On majorcause of uncertainty on this system is the delay in feedbackloop due to τd. This uncertainty can be bounded by findingthe maximum state change that can happen in the maximumdelay time. The following proposition explains the calculationof this bound.

Proposition 1: Let x[k] ∈ X,u[k] ∈ U, ap[k] ∈ Ap, w[k] ∈W where all of the sets X,U,Ap,W are bounded. Further-more, assume that the radar and actuator delay is upper-bounded by Td, i.e., 0 6 τd 6 Td. Then wτ , the uncertaintycaused by delay, will be bounded by the set:

Wτ = TdBdKc × {AX ⊕ (BU ⊕ (EAp ⊕W ))} .

Proof: : In order to prove this proposition, we use thefact that the difference between x[k] and x[k − τd] is givenby the rate of changes of x in τd-duration multiplied by τd(assuming that τd is small). Rigorously

x[k − τd] = x[k]− τdx[k].

Next, note that according to (1), the set of all possible statechange rates ∆X can be given by:

∆X = {x | x = Ax+Bu+Bpap + w,

∀x ∈ X, ∀u ∈ U,∀ap ∈ Ap,∀w ∈ W},Downloaded from engine.lib.uwaterloo.ca on 25 November 2021 Page 5 of 12


which, looking back at the preliminary definitions, is equiva-lent to the following Minkowski sum:

∆X = {AX ⊕ (BU ⊕ (BpAp ⊕W ))} .

Therefore, by equation (7), the total amount of uncertainty thatdelay produces in the system is given by

Wτ = TdBdKc × {AX ⊕ (BU ⊕ (BpAp ⊕W ))},

which is what we aimed to show.Another source of uncertainty is the acceleration of the

preceding vehicle. In (1) preceding vehicle’s acceleration aphas been modeled as an additive disturbance. Therefore, wap

which is the uncertainty caused by ap can be bounded byknowing a bound for maximum possible acceleration for thepreceding vehicle.

wap ∈ Wa = BpAp. (21)

Unknown model parameters can also increase model uncer-tainty. Uncertainty in vehicle mass, tire radius, drag coeffi-cient, rolling resistance coefficients, road grade, wind speedand power-train efficiency must be considered in controldesign. If a bound for each of these uncertain parametersis available, equation (6) can be used to find maximummodel error that the uncertain parameters can cause. SupposeΥ =

[m rw Cd µrv µr0 φr vw ηp

]as the vector

of uncertain parameters in (6) with Υ as the vector theirnominal value and Υmax and Υmin as the vector of their max-imum and minimum values. Then the following optimizationproblem will find the maximum model error.

eamin= min

Υ,vh,Tcom

ah(Υ, vh, Tcom)− ah(Υ, vh, Tcom),

eamax= max

Υ,vh,Tcom

ah(Υ, vh, Tcom)− ah(Υ, vh, Tcom),

subject toΥmin ≤ Υ ≤ Υmax,

0 ≤ vh ≤ vhmax ,

Tmin ≤ Tcom ≤ Tmax.

where eaminand eamax

are the minimum and maximum errorcaused by parameter uncertainty which based on them the setof all possible acceleration errors due to parameter uncertaintycan be defined as: ea ∈ Ea and bounded additive disturbancedue to the parameter uncertainty can be calculated

wah∈ Wh = BgEa (22)

where Bg is as defined in (1). Combination of all the uncer-tainty sources will be the bounded additive disturbance term.

W = Wg ⊕Wτ ⊕Wa ⊕Wh (23)

This disturbance set will be used for the design of the tube-based controller. (17) can be rewritten as

e[k + 1] = Ace[k] + w[k] w ∈ W. (24)

Using this stable model with a bounded additive disturbanceand Minkovski sum, the finite reachable set for the error canbe calculated.

Φn = ⊕ni=0Ac

iW (25)

where Φn is the finite reachable error set and it’s infinity limitΦ∞ is called the robust positive invariant set [39]. In thispaper, our T-MPC is similar to [40] which use finite invariantset instead of infinity RPI set with fixed current state.

B. Model adaptationA model adaptation method is employed to adapt to changes

in the system and environment in order to maintain theperformance of designed controllers. In this paper, we usea least square parameter adaption method with forgettingfactor similar to [41] with same notation. This method usespreviously presented parametric models, to estimate the valueof each effective parameter. It works based on minimizing thesquared error between the estimated and measured output ofthe system by minimizing the following cost function.

J(θ) =1

2

∫ t

0

e−β(t−τ)(z(τ)− θT (t)φ(τ))2

m2s(τ)

dτ

+1

2e−βt(θ(t)− θ0)

TQ0(θ(t)− θ0),

(26)

where θ is the estimated parameters vector, θ0 is the initialestimated parameter, φ is the measured input signal, z is themeasured output signal, β is a forgetting factor, Q0 is a weight-ing matrix, P is covariance matrix and m2

s is a normalizingterm that can be chosen as: m2

s = 1+αφTφ, α ≥ 0. By min-imizing this cost function, the algorithm can find an estimationof the parameters. The first term penalizes the estimationerror and the second term penalizes the convergence rate witha decaying factor in order to increase estimation robustnessagainst disturbances. Forgetting factor gives a higher weight tothe new measurements, so that in case of change in a parameterthe algorithm can adapt to it. Based on this cost function arecursive least square algorithm is defined as follows:

˙θ(t) = P (t)ε(t)φ(t),

P (t) = βP (t)− P (t)φ(t)φT (t)

m2s(t)

P (t),

ε(t) =z(t)− θT (t)φ(t)

m2s(t)

.

(27)

This algorithm updates the covariance and estimated param-eters on-line when the vehicle is running. To prevent wrongestimation, it is necessary to limit the estimated parameters.Therefore, parameter projection was used to put constraintson estimations. Moreover, to avoid the covariance matrix frombecoming very large, it is also necessary to put a constrainton its maximum value. Assuming the desired constraint on theparameters is defined by: S = {θ ∈ Rn|g(θ) ≤ 0}, where g isa smooth function and R0 as an upper bound for P , projectioncan be defined as follows:

θ =

Pεφ if θ ∈ So or

θ ∈ δ(S) & (Pεφ)T∇g ≤ 0

Pεφ− P ∇g∇gT

∇gTP∇gPεφ otherwise

P =

βP − P φφT

m2sP if ‖P‖ ≤ R0 &{θ ∈ So or

θ ∈ δ(S) & (Pεφ)T∇g ≤ 0}0 otherwise

(28)Downloaded from engine.lib.uwaterloo.ca on 25 November 2021 Page 6 of 12


Projection ensures that the estimation will not go out of theconstraint region and will move along the border when itreaches to its limits. This adaptation algorithm is used estimatefuel consumption, electricity consumption and longitudinaldynamics parameters based on the reduced model presentedin the modeling section. The estimated parameters will beused in the control-oriented model of AT-NMPC so that theoptimization problem will find the updated optimal point ofthe system.

C. Adaptive robust controllerUsing the disturbance set and the parameter adaption

method above, it is possible now to define our adaptive robustcontrol problem. This controller includes a linear controllerthat stabilizes the system, and an NMPC that controls thesystem based on its nominal control-oriented model withoutconsidering the uncertainty and disturbances. NMPC keeps thenominal state of the system in a tighter region to ensure that theactual system states will remain inside the defined constraints.Moreover, parameter adaptation updates the control-orientedmodel that is used in the definition of the cost function tomaintain system performance. Therefore, two control-orientedmodels are used here, one for updating the cost function andperforming a future prediction and another one for handlingof the constraints.

minco

{ Np∑i=1

(ω1e

2p(x, u) + ω2e

2v(x, u) + ω3u

2

+ ω4Ecost(x, u, PR, A, Γ))}

,

subject tox[n] = x[n], x[n] = x[n],

x[n+ i+ 1] = fn(x[n+ i], c0[n+ i]),

x[n+ i+ 1] = fn(x[n+ i], c0[n+ i]),

u[n+ i] = −Kcx[n+ i] + c0[n+ i],

x[n+ i+ 1] ∈ X Φ[i],

c0[n+ i+ 1] ∈ U (−KcΦ[i]),

(29)

where f and f are the nominal and estimated nonlinear longi-tudinal dynamic model of the vehicle, Np is the predictionhorizon’s length, ω1, ω2, ω3 and ω4 are weights on eachterm and other parameters are as defined before. The controlproblem finds a vector c0 that minimizes the cost function inthe prediction horizon while the actual input to the system iscombination of c0 and the linear controller.

Remark 1: In this control problem, the current state of thesystem is not a decision variable and x[n] has a fixed value.Both nominal and adapted control-oriented models start fromthe same initial point but perform future predictions based ontheir own parameters.

Remark 2: The tighter constraints on the states ensures thatthe system states would remain inside X , the defined stateconstraint, for any amount of disturbance that satisfies w ∈W. Moreover, the tighter constraints on the input reserves apart of available actuation for the linear controller to maintainsystem’s robustness.

Remark 3: Having two separate control-oriented modelsensures that constraints are always satisfied based on thefixed nominal control-oriented model. Therefore even if theadapted control-oriented model has low accuracy, the system’srobustness will be maintained. This is specially important inthe case that a sudden change in model parameters occur.Because the parameter estimator may not be able to recognizethe change in the model immediately, it is necessary to makesure that the system will remain safe and robust while themodel is getting adjusted which is achievable by using separatemodels as has been done here.

D. Fast optimizerTo implement the proposed robust Eco-ACC on a vehicle

control system, the AT-NMPC problem must be solved inreal-time. Therefore a fast solver is required that can solvethe nonlinear optimization problem with little computationaldemand. To this end, in this paper, Newton/GMRES methodhas been used to solve the control optimization problem. Thismethod is claimed to be very fast as it solves the differentialequation once at each time step [42]. In the current study, theauthors use an automatic multi-solver NMPC code generator,called MPSee, to generate the NMPC code based on the New-ton/GMRES algorithm which has been previously developedand tested in the authors’ research group [43],[44]. MPSee isa MATLAB-based mathematical program that enables usersto develop GMRES-based NMPC codes for different optimalcontrol problems and carry out simulations in Simulink. Todefine the Newton-GMRES solver, field vector and constraintshave been defined as follows:

f(x, u) =d

dt([ph vh Tw pp vp ap ph vh Tw

]T)

=

vhTw

mrwηp− {

12ρaCaAw

m v2h + gφr + µvgvh + gµ0}− Tw

τa+

u−K[ep ev Tw]τa

vpap

−σapvh

Θ1Tw −Θ2v2h −Θ3φr −Θ4vh −Θ5

− Th

τa+

u−K[ep ev Th]τa

(30)

C(x, u) = Hx[i]

ep[i]ev[i]

u[i]−K[ep[i] ev[i] Tw[i]]

− hx[i]

(31)where Hx[i] and hx[i] are used to define the polytopicconstraints in each step equivalent to X Φ[i] in (29). Thetwo separate control-oriented models have been implementedin here in the field vector and constraints where barredvariables show the nominal values and hatted variables are theestimated ones. σ is the decaying factor for preceding vehicle’sacceleration as defined in [12] .

V. CONTROL EVALUATION

This section presents the evaluation of the proposed Eco-ACC in terms of estimation, robustness and ecological im-



Fig. 4. grade and wind profile injected in the simulation environment

Fig. 5. Longitudinal dynamic parameter estimator

provement. A high-fidelity model of the base-line vehicle,developed in Autonomie, is used for evaluation tests. First, theperformance of the three estimators is presented. Second, ro-bustness of the proposed controller is tested using high-fidelitymodel of the base-line vehicle. Third, ecological improvementcaused by the the proposed Eco-ACC is discussed. Finally, theresult of HIL experimet is presented that shows the real-timecapability of the proposed controller.

A. Parameter estimation

We used three least-square parameter estimators to improvethe control-oriented model of our predictive controller. Thefirst estimator gets velocity, road grade and propulsion orbraking torque, and then finds the parameters of the longi-tudinal model based on (8). Fig. 5 (a-e) illustrate the resultof on-line parameter estimation for this estimator and Fig. 5(f) shows the estimated acceleration by the estimated modelcompared to the measured acceleration. It is worth noting

Fig. 6. Fuel consumption parameter estimator

Fig. 7. Electricity rate parameter estimator

that the estimated parameters may not converge to their realvalue which, for adaptive control use, is acceptable as long asthe estimated output matches the real value [41]. This modelpredicts the motion of the vehicle inside the prediction horizonand adjusts to changes, so that the prediction would be moreaccurate. The two other estimators are shown in Fig. 6 for fuelconsumption and in Fig. 7 for electricity. As it can be seenboth models are able to estimate the desired output closelyand adapt to changes in the parameters. The initial guess forthe estimators was chosen arbitrary, which is the reason forinitial oscillations in the parameter estimations. To improve theestimators’ performance, forgetting factor has higher value inthe beginning and decreases gradually afterward to a lowervalue. The main objective of using estimators is for makingsure that the optimizing the defined cost function wouldoptimized the actual system. Therefore, the result of theseestimators will be used in the cost function of the optimizationproblem.



B. Robust constraint handling

Other than on-line parameter adaptation, the proposed AT-NMPC based Eco-ACC is able to handle bounded uncertain-ties. To show the validity of this statement, we conductedsimulations by utilizing a high-fidelity model of the base-line vehicle and then added a variable wind speed and roadgrade to the simulation environment, as shown in Fig. 4.During the simulation, host vehicle follows a preceding vehiclein a drive cycle by receiving inter-vehicular distance andvelocity from the radar. To simulate the effect of delay inthe radar’s data, 400ms transport delay was injected in theinter-vehicular distance and velocity during the simulation.Moreover, parametric errors were considered in the control-oriented model by 20% error in the vehicle mass, 50% error inthe drag coefficient, and ignoring the rolling resistance forces.Then based on the method presented in section IV, disturbancesets were calculated and used for defining the constraints of theAT-NMPC problem. Fig. 8 shows the result of the simulationin a standard FTP-75 drive cycle. The preceding vehiclefollows the given drive cycle and the host vehicle follows thepreceding vehicle during the simulation using a regular MPCbased ACC and proposed AT-NMPC Eco-ACC. As shown inFig. 8 (a) both controllers have acceptable velocity trackingperformance while following the preceding vehicle. However,the NMPC based ACC has harsher accelerations compared tothe proposed Eco-ACC. Harsher accelerations are due the factthat non-robust NMPC can not handle the defined constraintsdue to the existed uncertainties in the system. Therefore, theseuncertainties can push the system out of the defined constraintsand NMPC has to perform harsh braking and acceleration to goback into the constraints. However, AT-NMPC is robust againstthese uncertainties and therefore it has less harsh accelerations.Fig. 8 (c) compares the position tracking error of the twocontrollers. NMPC is not able to handle the defined constraintsand it has higher position error than the defined limits. On theother hand, AT-NMPC handles the constraint perfectly becausethe effect of uncertainties has been considered in its designbased on the optimization problem in (29).

C. Ecological Improvement

The objective function of the proposed controller in definedto minimize the cost of energy in a drive cycle. Basedon the given dynamic model of the vehicle and availableroad elevation, AT-NMPC predicts the future host vehiclestrajectory and based on that calculates the expected powerdemand in the prediction horizon. Then, based on the powerratio of energy management system, it calculates the powerdemand of each energy source and also energy cost duringthe prediction horizon. Parameter estimators make sure that theoptimal point of the cost function is the actual optimal point ofthe system. Fig. 9 compares the trip energy cost of a trackingNMPC, ecological NMPC and the proposed AT-NMPC inthree consecutive FTP-75 drive cycles. A longer drive cycleis chosen to minimize the effect of energy management in theachieved results. Tracking NMPC has higher weightings onposition and velocity tracking term to increase the trackingperformance. Therefore, it mostly sacrifices energy cost for

Fig. 8. (a) velocity, (b) acceleration and (c) position error in car-followingsimulation

Fig. 9. Energy consumption in three consecutive FTP-75 drive cycles

better tracking and has the highest energy cost in this drivercycle by $2.56. The Eco-NMPC has a higher weighting onenergy cost term, which means that it sacrifices tracking, insidethe defined constraints, to have better energy cost. As such, ithas an energy cost of $2.4, which is about 6.2% lower thantracking NMPC. However, in this case, NMPC controller isnot able to handle the defined constraints and uncertaintiespush the system out of the defined constraints set. To be ableto move back into the constraints, NMPC has to do harshbraking and accelerations, which increase the energy cost. AT-NMPC, on the other hand, handles the defined constraints andalso minimizes a cost function that adapts to the actual vehiclebehavior. It has an energy cost of $2.29, which is about 10.5%lower compared to tracking NMPC.



Fig. 10. Schematic of the HIL experiment setup

D. Real-time Implementation

To further examine the performance of the proposed con-troller, hardware-in-the-loop (HIL) experiments have beenconducted to study the potential and capability of AT-NMPCfor real-time implementation in a vehicle control system.HIL tests take into account the computational limits andcommunication issues, and their result is considered morepractical than software-in-the-loop simulations. Because phys-ical prototyping in the early stages of development of vehiclecontrol systems could be very expensive, HIL experiments,which are less expensive and also faster and safer, usuallycarried out before manufacturing the prototype vehicle [45].In this paper, dSPACE Micro-Autobox II control prototypinghardware was used for HIL tests. This setup is one of thewidely used instruments for calibration and testing of ECUsspecifically for automotive applications. As shown in Fig.10,the HIL setup has three main components: 1) a prototypeECU (MicroAutoBox II), which is an independent processingmodule that runs the uploaded controller; 2) a real-timesimulator (DS1006 processor board), which is responsible forrunning the complex high-fidelity model of the vehicle in real-time fashion; and 3) a personal computer (PC) that serves asthe human-machine interface and is used for programmingthe real-time machine and prototype ECU, as well as forrecording the desired test signals. All the communicationsbetween the prototype ECU and the real-time simulator areperformed through a Controller Area Network (CAN).

For HIL tests, a C code was generated of the designedcontroller by using the dSPACE Real-Time Workshop codegenerator and uploaded to the prototype ECU. With a sim-ilar procedure, the high-fidelity model was uploaded to therealtime simulator using the human-machine interface. Fig.11illustrates the turnaround time of the controller in FTP-75 drivecycle simulation. The turnaround time for this controller isbetween 400µs and 700µs at all times for a prediction horizonof Np = 10. The maximum inner and outer iterations set tobe less than 5 to enable real-time implementation.

VI. CONCLUSIONS

This paper proposed an adaptive tube-based nonlinear modelpredictive controller for the design of autonomous cruise

Fig. 11. Turn-around time of the controller in HIL experiment

control systems. This method ensures the robust satisfaction ofthe defined constraints in the presence of uncertainty, and alsoimproves the systems performance by adapting to the changesin the vehicle control-oriented model. Therefore, in a way,this method decouples performance and robustness by usingseparate models one for constraint handling and another onefor defining the objective function.

In the modeling step, a nonlinear control oriented model waspresented for a vehicle that performs car-following. This modelwas used for evaluation of the safe sets in the presence ofadditive disturbance. Moreover, models for fuel consumptionand electricity rate were presented to estimate the cost ofenergy in the prediction horizon and based on them, reducedmodels for parameter estimation were generated. A high-fidelity model of the base-line PHEV, Toyota plug-in Prius,was used to evaluate the controller.

In the control design step, first, a linear controller wasused to stabilize the system. Then by analyzing the existeduncertainties, they were translated into an additive disturbanceterm to define a bound for maximum uncertainty. Next, designof the three least square parameter estimators were explainedthat estimate the parameters of the control-oriented model.Then, AT-NMPC control problem was defined that uses twoseparate models: one for defining the objective function andanother one for constraint handling. Using separate modelsensures that constraints are handled based on the fixed nominalcontrol-oriented model, while the objective function is definedbased on an adapted control-oriented model to ensure that theoptimal point of the cost function and the actual system areequivalent. The objective function of the controller was definedto minimize the energy cost while following a precedingvehicle.

In the control evaluation step, simulations on high-fidelitymodel were performed by injecting uncertainties and delayinto the simulation environment. Control evaluations showedthat the proposed AT-NMPC is able to handle the definedconstraints in the presence of uncertainty while improvingthe trip energy cost by 10% compared to a tracking NMPC.Finally, an HIL experiment was conducted to show the real-time capability of the proposed controller, which showed thatAT-NMPC has low computation costs while running on aprototype ECU.

VII. ACKNOWLEDGMENT

Authors would like to gratefully express their appreciationto Toyota, NSERC and Ontario Centers of Excellence for



financial support of this study.

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Bijan Sakhdari graduated with B.Sc. and M.Sc.degrees in Mechanical Engineering from Sharif Uni-versity of Technology, Tehran, Iran. He is currentlya Ph.D. candidate in the Department of SystemsDesign Engineering (SYDE) at the University ofWaterloo (UW) in Waterloo, Ontario, Canada. Hewas previously a Research Associate with the Sur-gical Robotic Laboratory, Emam hospital, Tehran,Iran where he was responsible for developing controlsystems for medical robots especially for the SINAtele-surgical system. He is a member of the Smart

Hybrid and Electric Vehicle Systems (SHEVS) Laboratory, SYDE, UW. Hisresearch interests include intelligent vehicle control systems, robust controlof connected vehicles and control of autonomous vehicles.

Dr. Nasser L. Azad is currently an Associate Pro-fessor in the Department of Systems Design Engi-neering, University of Waterloo, and the Director ofHybrid and Electric Vehicle Systems (SHEVS) Lab-oratory. He was a Postdoctoral Fellow with the Ve-hicle Dynamics and Control Laboratory, Departmentof Mechanical Engineering, University of California,Berkeley, CA, USA. Dr. Azads primary researchinterests lie in control of connected hybrid andelectric vehicles, autonomous cars, and unmannedaerial vehicle quadrotors. He is also interested in ap-

plications of Artificial Intelligence for solving different engineering problems.Due to his outstanding work, Dr. Azad received an Early Researcher Awardin 2015 from the Ministry of Research and Innovation, Ontario, Canada.


adaptive tube-based nonlinear mpc for ecological

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