Research Article Nonlinear Optimal Control of Continuously … · 2019. 7. 31. · Optimal control theory has rarely been used in the previous control approaches applied to such systems

Research ArticleNonlinear Optimal Control of Continuously VariableTransmission Powertrain

Reza Kazemi, Mohsen Raf’at, and Amir Reza noruzi

Deptartement of Mechanical Engineering, K. N. Toosi University of Technology, Tehran P.O. Box 19395-1999, Iran

Correspondence should be addressed to Reza Kazemi; [email protected]

Received 31 May 2013; Accepted 14 July 2013; Published 1 January 2014

Academic Editors: A. Lidozzi and D. Sanders

Copyright © 2014 Reza Kazemi et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Optimization of gear ratio with the objectives of fuel consumption reduction and vehicle longitudinal performance improvementhas been the subject of many studies for years. Finding a strategy for changing gears with specific control objectives, especially inthe design of vehicles equipped with Continuously Variable Transition system (CVT), which has advantage of arbitrary selectionof gear ratio, has been the aim of some recent researches. Optimal control theory has rarely been used in the previous controlapproaches applied to such systems due to the limitations in the use of fast computational systems. The aim of this study is todesign the aforementioned gear ratio change strategy and related control rules on the basis of optimal control. A driver model isalso designed for the simulation of driving cycle using MATLAB Simulink Toolbar. Results of implementing optimal control rulesin vehicle longitudinal movement simulation with the aim of fuel consumption reduction are finally represented. The presentedmethod has the remarkable advantage of considerable fuel consumption reduction in comparison to other proposed approachesfor gear ratio change strategies.

1. Introduction

Continuously variable transmission (CVT) is an attractivetechnology and has become more practical with recentimprovements in the technology. CVT is effective in achiev-ing continuously smooth shifting and enables the engine tooperate in itsmost efficient state. CVThas now reached a levelthat permits a large-scale usage of these devices even in a full-size passenger car class. Maximum torques of approximately300Nm can be handled with push belts (cf. Gesenhaus,2000, or Goppelt, 2000). Other systems (e.g., toroidal drives),which cover even larger torque ranges, have been proposed aswell.

Although the efficiencies of CVT are inherently lowerthan those of cog wheel gear boxes, a more efficient total-system behavior can be obtained by shifting the engineoperating points for a certain demanded power towardshigher loads and lower speeds.

A reliable method of control system design is usuallytrial and error in which different iterative methods are usedfor determining design parameters of an acceptable system.

Appropriate performance of the system is usually intro-duced by some characteristics such as rising time, settlingtime, and overshoot or with some frequency characteristicssuch as phase margin, gain margin, and band width. Withthis method for multi-input multioutput systems, differentcriteria or performance characteristics for achieving thetechnological requirements are needed. For example, planeposition plan control which minimizes fuel consumptionis not possible with usual techniques. A direct method forintroducing these complex systems is called optimal controland it has been a practical solution using digital computers.The objective of optimal control is to determine the controlsignals which satisfy mechanical constrains and maximize orminimize performance or special criteria.

2. Complete Detailed CVTPowertrain Modeling

The powertrain is modeled in its longitudinal behavior withminimum simplification being taken into account.

This model is appropriate to test control unit operationafter applying on vehicle longitudinal dynamic.

Hindawi Publishing CorporationISRN Automotive EngineeringVolume 2014, Article ID 479590, 11 pageshttp://dx.doi.org/10.1155/2014/479590

2 ISRN Automotive Engineering

1

24

5

67

8

3

𝜔w

𝜔e

Figure 1: Schematic of CVT powertrain subsystems.

Powertrain subsystems are the following as shown inFigure 1:

(1) internal combustion engine,(2) torque converter,(3) DNR,(4) CVT,(5) final reduction,(6) differential,(7) drive shaft,(8) tires.

2.1. Internal Combustion Engine. The mean-value model ofan engine describes the engine behavior in a cycle aver-aged sense. Though it does not contain individual cylindertransient dynamics, the average of engine dynamics overseveral cycles provides adequate low-frequency dynamicinformation and it is suitable for many control problems.Figure 2 depicts a schematic of the mean-value enginemodel including throttle airflow dynamics, intake manifolddynamics, fuel film dynamics, engine torque production, andcrankshaft dynamics.

The time domain mean-value engine model, introducedin the next section, assumes that exhaust gas recirculation(EGR) depends internally on variable valve timing (VVT)and spark advance (SA) remains constant in certain operatingconditions. Air fuel ratio (AFR) is also well maintained atstoichiometric. Therefore, this engine model is broken downinto four subsystems: electronically controlled throttle body,intake manifold, combustion, and crank shaft Dynamicsneglecting fuel dynamics [1].

Intake Manifold. The intake manifold is the plenum betweenthe ETB and the engine cylinders. Equation describes amean-value filling and emptying intake model based on thecontinuity principle and the ideal gas law [2].The total air thatgoes into the cylinders is expressed in an empirical equation.

Fuel dynamics are notmodeled here since engine air fuel ratio(AFR) is always well maintained at stoichiometric operatingconditions [3].

Fuel consumption is thus calculated as the total air massflow rate entering the cylinders divided by

��

𝑎= ��

𝑎𝑖− ��

𝑎𝑜,

��

𝑎𝑖= MAX ⋅ TC ⋅ PRI,

TC = {

1 − cos (1.14459𝛼 − 1.06) 𝛼 ≤ 79

∘

,

1 𝛼 ≥ 79

∘

,

PRI = 1 − (exp(9

𝑃

𝑚

𝑃atm) − 1) ,

𝑉

𝑚

𝑑𝑃

𝑚

𝑑𝑡

= ��

𝑎𝑅𝑇

𝑚,

��

∞= 𝐶

1⋅ 𝜂VOl ⋅ 𝑚𝑎 ⋅ 𝜔𝑒,

𝐶

1=

𝑉

𝑒

4𝜋𝑉

𝑚

.

(1)

Combustion. Engine combustion takes air and fuel as inputsand produces torque and exhausts with losses. Torque pro-duction from combustion is usually estimated by a regressionmodel that takes air flow, SA, AFR, and engine speed intoaccount. Since AFR is assumed to be constant in this model,its effect on produced torque is combined into 𝑇

0term. The

engine torque therefore becomes 𝑇

𝑒. 𝑇𝑒in equation is the

engine brake torque which considers both engine productionand friction torques. Air in this equation is delayed by tdwhich varies in the time domain due to varying.

Engine speed. The engine torque is bounded by wide openthrottle and minimum throttle torques according to:

𝜂VOl = (24.5𝜔

𝑒− 3.10 × 10

4

)𝑚

2

𝑎

+ (−0.167𝜔

𝑒+ 222)𝑚

𝑎

+ (8.1 × 10

−4

𝜔

𝑒+ 0.352) , 𝑇

𝑖

= 𝐶

𝑇×

��

∞(𝑡 − Δ𝑡

𝑖𝑡)

𝜔

𝑒(𝑡 − Δ𝑡

𝑖𝑡)

× AFI (𝑡 − Δ𝑡

𝑖𝑡)

× SI (𝑡 − Δ𝑡

𝑠𝑡) ,AFI

= cos(7.3834 (

𝐴

𝐹

− 13.5)) , SI

= (cos (SA −MBT))2.875.

(2)

Crank Shaft. Crank shaft speed dynamics are intrinsicallybased onNewton’s second law for a rotational object;𝑇

𝑝in (2)

represents the load torque from the torque converter pump.Idle and redline are the physical speed constraints for theengine. These relationships are represented as [1]

𝐼

𝑒��

𝑒= 𝑇

𝑖− 𝑇

𝑝. (3)

2.2. Torque Converter (TC). The primary functions of thetorque converter (TC) include torque multiplication to

ISRN Automotive Engineering 3

EGR valve

ExhaustvalveSpark

plug

Intakevalve

FuelManifoldpressuresensorIntake manifold

Ide bypass valve

ThrottleAir

MAFsensor

EGOsensorTPS

sensor

Fuelinjector

Canister purge valve

Air fuelmixture

Figure 2: Internal combustion engine subsystems [4].

provide sufficient torque during vehicle launch and fluiddamping to smooth torque fluctuations in the powertrain. Afluid-filled three-element TC has two phases: torque multi-plication phase and fluid coupling phase. The TC impeller(also referred to as the TC pump) is driven by the engineand the turbine is attached to the transmission input shaft.The turbine and the stator that are connected to the TChousing via a one-way clutch are initially at rest duringvehicle launch. The turbine speed begins to increase underthe angular momentum of the impeller that is transmittedthrough circulating fluid inside the TC.When the ratio of theturbine speed to the pump speed is low, the stator remainsat rest and it redirects the fluid flow in the same directionas the pump torque such that the resulting output torque ofthe TC is amplified. This is called the torque multiplicationor torque amplification phase. At higher turbine speed, thestator rotates freely in the same direction of the pump and itis considered to consume no torque. Therefore, the turbinetorque in this torque coupling phase is the same as the pumptorque [4].

TC act as hydraulic dampers to interrupt vibration prop-agation originated from either engines or road bumps andto provide torque multiplication during vehicle launch [5].Since TC is essentially a damper, losses are not negligible.However, these losses can be reduced by employing a TCbypass clutch, which mechanically connects the TC pumpand the turbine when the clutch is engaged. This connectionimproves TC efficiency at the price of losing the capabilityto absorb oscillations in the powertrain. A compromisingsolution is proposed by Kazemi et al., allowing 1 to 2% ofclutch slip to achieve similar results as the TC is working asa damper [4]. Obviously, people desire to minimize this slipfor efficiency consideration.This type of bypass clutch is a so-called minimal slip-type TC clutch.

The torque converter is expressedwith a regressionmodelbased onKotwicki’s research [5]. In thismodel, there are threemodes in the forward drive case (power is flowing from the

engine to the wheels) and two modes in the backward drivecase (overrun case), and they are shown in (4) to (7).

Forward (𝜔𝑝> 𝜔

𝑡).

(1) Torque multiplication mode (𝑇𝑡> 𝑇

𝑝):

𝑇

𝑝= 𝑏

1𝜔

2

𝑝+ 𝑏

2𝜔

𝑝𝜔

𝑡+ 𝑏

3𝜔

2

𝑡,

𝑇

𝑡= 𝑐

1𝜔

2

𝑝+ 𝑐

2𝜔

𝑝𝜔

𝑡+ 𝑐

3𝜔

2

𝑡.

(4)

(2) Torque coupling mode (𝑇𝑡= 𝑇

𝑝):

𝑇

𝑝= 𝑇

𝑡= 𝑎

1𝜔

2

𝑝+ 𝑎

2𝜔

𝑝𝜔

𝑡+ 𝑎

3𝜔

2

𝑡. (5)

(3) Lockup mode (𝜔𝑝= 𝜔

𝑡):

𝑇

𝑡≤ 𝑇clutch max. (6)

Backward (overrun) (𝜔𝑡> 𝜔

𝑝).

(1) Torque coupling mode: (𝑇𝑡= 𝑇

𝑝)

𝑇

𝑃= 𝑇

𝑡= 𝑑

1𝜔

2

𝑃+ 𝑑

2𝜔

𝑃𝜔

𝑡+ 𝑑

3𝜔

2

𝑡. (7)

(2) Lockup mode: the same as in the forward drive case.

As shown in Figure 3, this TC has the maximum torqueratio (turbine torque over pump torque) of about 1.65 and thecoupling point at the speed ratio (turbine speed over pumpspeed) of 0.88. Its efficiency before the coupling point is lowerthan 90% and that in the lockup mode is around 99%.

2.3. Continuously Variable Transmission (CVT). A CVT is acontinuous speed reduction device with infinite number oftransmission ratios between two limits. Comparing the three


Torque converter map

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.2

0.4

0.6

0.8

1.2

1

1.4

1.6

Torq

ue ra

tio effi

cien

cy

Speed ratio (𝜔t/𝜔p)

Figure 3: Torque converter operating diagram.

10.95

0.90.85

0.80.75

0.70.65

CVT

emcle

ncy

CVT input torque (N·m) CVT output speed (mm)

10080

6040

200 0

10002000

30004000

5000

r1

r2r4

r5

CVT efficiency map

Figure 4: CVT efficiency map [4].

types of the CVTs used in automobiles, that is, mechani-cal, hydraulic, and electrical, the mechanical CVT is moreattractive due to its better performance on efficiency, noiselevel, size, weight, and cost [6]. Among themechanical CVTs,the variable pulley CVT is more commercialized than thevariable stroke CVT and the traction drive CVT.The variablepulley could be rubber belt, chain, or push-belt and the latteraccounts for the largest share of the market.

The CVT ratio can be varied by changing the radii ofthese twopulleyswith a hydraulic control system. Shifting of apush-belt CVT neglecting the dynamics, the CVT ismodeledwith an efficiency map with speed, torque, and transmissionratio as the arguments [7].

In Figure 4, the CVT has higher efficiency at lower speed,lower CVT ratio, and medium torque. 𝑟1 to 𝑟5 representdistributed CVT ratios 0.5 to 2.5. The efficiency at any ratioin between is obtained by using linear interpolation. The

efficiency of a steel-belt CVT with special oil containingrubber molecules to lock the belt with the pulley can reachup to 97%, similar to that of a manual transmission [4].

Compared to a powertrain equipped with a stepped-geartransmission, the one with a CVT has better overall efficiencyand drivability. A variable pulley type CVT with a metal 𝑉belt is introduced here [8].The input-output speed and torqueare expressed as functions of the efficiency and the CVT ratio:

𝜏CVT𝑑𝑟CVT𝑑𝑡

+ 𝑟CVT = 𝑟CVT req,

𝐽

2

𝑑𝜔

𝑡

𝑑𝑡

= 𝑇

𝑡− 𝑇CVT 𝑝,

𝐽

3

𝑑𝜔CVT 𝑠𝑑𝑡

= 𝑇CVT 𝑝 − 𝑇

𝑓𝑑,

𝑟CVT =

𝜔CVT 𝑝

𝜔CVT 𝑠,

𝑇CVT 𝑠 = 𝜂CVT ⋅ 𝑟CVT ⋅ 𝑇CVT 𝑝.

(8)

2.4. Final Drive (differential). A final drive is represented as agear set. The ratio is defined as the final drive speed over thedriveshaft speed. Efficiency of the final drive is simplified bytaking a constant value:

𝑟

𝑓𝑑=

𝜔CVT 𝑠𝜔

𝑑𝑠

,

𝑇

𝑓𝑑= 𝜂

𝑓𝑑⋅ 𝑟

𝑓𝑑⋅ 𝑇CVT 𝑠.

(9)

2.5. Driveshaft. Shaft flexibility is modeled as lumped com-pliance, which is helpful in absorbing oscillations in thepowertrain. The nonlinear damper is characterized as afunction of driveshaft speed:

𝑑𝑇

𝑤ℎ

𝑑𝑡

= 𝐾

𝑑𝑠(𝜔

𝑑𝑠− 𝜔

𝑤ℎ) ,

𝑇

𝑓𝑑= 𝑇

𝑤ℎ+ 𝐷

1𝜔

𝑑𝑠+ 𝐷

2𝜔

2

𝑑𝑠.

(10)

2.6. Vehicle Longitudinal Dynamics. Resistance forces includ-ing aerodynamic drag, rolling resistance, and gravity forcesare expressed as follows [9, 10]:

𝐹

𝑡𝑓− 𝐹

𝑡𝑟− 𝐹

𝑔− 𝐹

𝑎= 𝑀

𝑉,

𝐹

𝑎=

1

2

𝜌air𝐶𝑑𝐴𝑓𝑉2

,

𝐹

𝑡𝑟= 𝑀𝑔𝐶

𝑑cos (𝜃) ,

𝐹

𝑔= 𝑀𝑔 sin (𝜃) .

(11)

3. Control-Based Modeling of CVT Powertrain

Some simplifications in model are taken into account forcontrol design purposes. To have feasible nonlinear optimalcontrol design, these simplifications are necessary.Therefore,


only nonzero vehicle velocities are assumed; that is, thevehicle launch (which needs a clutch or a torque converter)is not analyzed.

First subsystem taken into account is the engine. A staticmodel of engine is derived to have a map of engine operationand its efficient performance in all regions of throttle openingand engine rotational speed [11].

Next supposition is the state of torque converter opera-tion and it is assumed to be in locked state. Driveshaft is sup-posed to be rigid and input torque is equal to output torquein this part. Other subsystems are similar as mentioned inSection 2 [12].

4. Optimal Control Theory Overview

The objective of optimal control is to determine control sig-nals that will cause the system tominimize ormaximize someperformance criteria while satisfying the physical constraints[13]. Performance criteria, in general, are expressed by thecost function 𝐽 = 0 for a system of 𝑥 = 𝑓(𝑥, 𝑢, 𝑡). Thevariables 𝑡

0and 𝑡

𝑓are the initial and final time, ℎ and 𝑔 are

scalar functions and 𝑡

𝑓may be fixed or free depending on the

problem statement.Starting from the initial value 𝑥(𝑡

0) = 𝑥

0and applying the

optimal control 𝑢∗(𝑡) for 𝑡 ∈ [𝑡

0, 𝑡

𝑓], the system will follow

some state trajectory with minimum cost. (The superscriptasterisk on 𝑥, 𝜆 and 𝑢 represents the optimal trajectories forstate, costate, and control variables.) [14].

4.1. Application of the Steeping Descends Method to OptimalControl Problems. Suppose that the nominal control graph𝑢

(𝑖)

(𝑡), 𝑡 ∈ [𝑡

0, 𝑡

𝑓] was developed for the solution of the

flowing differential equations:

��

(𝑖)

(𝑡) = 𝑎 (𝑥

(𝑖)

(𝑡) , 𝑢

(𝑖)

(𝑡) , 𝑡) ,

𝑝

(𝑖)

(𝑡) = −

𝜕𝐻

𝜕𝑥

(𝑥

(𝑖)

(𝑡) , 𝑢

(𝑖)

(𝑡) , 𝑝

(𝑖)

(𝑡) , 𝑡) ,

(12)

so that the state-costate equation 𝑥

(𝑖)

, 𝑝

(𝑖) would meet thefollowing boundary conditions:

𝑥

(𝑖)

(𝑡

0) = 𝑥

0,

𝑝

(𝑖)

(𝑡

𝑓) =

𝜕ℎ

𝜕𝑥

(𝑥

(𝑖)

(𝑡

𝑓)) .

(13)

It should also be noticed that 𝑥

(𝑖)

(𝑡), 𝑝

(𝑖)

(𝑡), and 𝑢

(𝑖)

(𝑡)

will be externals, if the nominal control graph satisfies thefollowing equation:

𝜕𝐻

𝜕𝑢

(𝑥

(𝑖)

, 𝑢

(𝑖)

, 𝑝

(𝑖)

, 𝑡) = 0 𝑡 ∈ [𝑡

0, 𝑡

𝑓] .

(14)

If the last equation is not satisfied, the augmented func-tional change, 𝐽

𝑎, in the state-costate equation will be defined

as [15]

𝛿𝐽

𝑎= [

𝜕ℎ

𝜕𝑥

(𝑥

(𝑖)

(𝑡

𝑓)) − 𝑝

(𝑖)

(𝑡

𝑓)]

𝑇

𝛿𝑥 (𝑡

𝑓)

+ ∫

𝑡𝑓

𝑡0

{[

𝑝

(𝑖)

(𝑡) +

𝜕𝐻

𝜕𝑥

(𝑥

(𝑖)

(𝑡) , 𝑢

(𝑖)

(𝑡) , 𝑝

(𝑖)

(𝑡) , 𝑡) ]

𝑇

𝛿𝑥 (𝑡)

+ [

𝜕𝐻

𝜕𝑢

(𝑥

(𝑖)

(𝑡) , 𝑢

(𝑖)

(𝑡) , 𝑝

(𝑖)

(𝑡) , 𝑡)]

𝑇

𝛿𝑢 (𝑡)

+ [𝑎 (𝑥

(𝑖)

(𝑡) , 𝑢

(𝑖)

(𝑡) , 𝑡) − ��

(𝑖)

(𝑡)]

𝑇

𝛿𝑝 (𝑡) } 𝑑𝑡,

(15)

in which

𝑥 (𝑡) ≜ 𝑥

(𝑖+1)

(𝑡) − 𝑥

(𝑖)

(𝑡) ,

𝛿𝑢 (𝑡) ≜ 𝑢

(𝑖+1)

(𝑡) − 𝑢

(𝑖)

(𝑡) ,

𝛿𝑝 (𝑡) ≜ 𝑝

(𝑖+1)

(𝑡) − 𝑝

(𝑖)

(𝑡) .

(16)

If the above equations are met, then

𝛿𝐽

𝑎= ∫

𝑡𝑓

𝑡0

[

𝜕𝐻

𝜕𝑢

(𝑥

(𝑖)

(𝑡) , 𝑢

(𝑖)

(𝑡) , 𝑝

(𝑖)

(𝑡) , 𝑡)]

𝑇

𝛿𝑢 (𝑡) 𝑑𝑡,(17)

when 𝐽

𝑎is the linear part of the increment Δ𝐽

𝑎≜ 𝐽(𝑢

(𝑖+1)

) −

𝐽(𝑢

(𝑖)

), and the norm ‖𝑢

(𝑖+1)

−𝑢

(𝑖)

‖ is as 𝛿𝐽𝑎.With the objective

of 𝐽𝑎minimization, the term Δ𝐽

𝑎should be negative.

If the change in variable 𝑢 is defined as

𝑢 (𝑡) = 𝑢

(𝑖+1)

(𝑡) − 𝑢

(𝑖)

(𝑡) = −𝜏

𝜕𝐻

(𝑖)

𝜕𝑢

(𝑡) ,

𝑡 ∈ [𝑡

0, 𝑡

𝑓] ,

(18)

for 𝑢 > 0, then,

𝛿𝐽

𝑎= −∫

𝑡𝑓

𝑡0

[

𝜕𝐻

(𝑖)

𝜕𝑢

(𝑡)]

𝑇

[

𝜕𝐻

(𝑖)

𝜕𝑢

(𝑡)] 𝑑𝑡 ≤ 0.

(19)

The integrand is nonnegative for all 𝑡 ∈ [𝑡

0, 𝑡

𝑓].

The equation will be met, if and only if the followingequation is satisfied:

𝜕𝐻

(𝑖)

𝜕𝑢

(𝑡) = 0, 𝑡 ∈ [𝑡

0, 𝑡

𝑓] .

(20)

Determining 𝛿𝑢 with infinitesimal ‖𝛿𝑢‖ will assure us ofthe fact that each least value function will be as small as theprevious value. Finally, when 𝐽

𝑎reachs the relative least value,

the vector 𝜕𝐻/𝜕𝑢 will be equal to zero in the whole timeinterval of [𝑡

0, 𝑡

𝑓].


4.2. Optimal Control Approach Algorithm

4.2.1. Solving State Equations. Consider the following:

��

∗

(𝑡) =

𝜕𝐻

𝜕𝑝

= 𝑎 (𝑥

∗

(𝑡) , 𝑢

∗

(𝑡) , 𝑡) ,

𝑥

∗

(𝑡

0) = 𝑥

0.

(21)

For solving equations represented above, and determin-ing state variables in whole simulation time when vehicleworks on all three state of torque convertor performance, wehave used simulation model which is the result of softwareequation simulation in this software.

4.2.2. Solving Costate Equation. Consider the following:

𝑝 (𝑡) = −

𝜕𝐻

𝜕𝑥

= [

𝜕𝑎 (𝑥

∗

(𝑡) , 𝑢

∗

(𝑡) , 𝑡)

𝜕𝑥

]

𝑇

𝑝

∗

(𝑡)

−

𝜕𝑔 (𝑥

∗

(𝑡) , 𝑢

∗

(𝑡) , 𝑡)

𝜕𝑥

𝑝

∗

(𝑡

𝑓) =

𝜕ℎ

𝜕𝑥

(𝑥

∗

(𝑡

𝑓)) .

(22)

4.2.3. Control Input Modification. After determining stateand costate variables related to simulation time for

𝜕𝐻

(𝑖)

𝜕𝑢

(𝑡

𝑘) = [

𝜕𝑎 (𝑥

(𝑖)

(𝑡

𝑘) , 𝑢

(𝑖)

(𝑡

𝑘) , 𝑡

𝑘)

𝜕𝑢

]

𝑇

𝑝

(𝑖)

(𝑡

𝑘)

+

𝜕𝑔 (𝑥

(𝑖)

(𝑡

𝑘) , 𝑢

(𝑖)

(𝑡

𝑘) , 𝑡

𝑘)

𝜕𝑢

,

(23)

modification of control input which is gear ratio, accord-ing to equation below we must determine the amount of(𝜕𝐻

(𝑖)

/𝜕𝑢)(𝑡

𝑘) function in all time of simulation, and then we

will have

𝑢

(𝑖+1)

(𝑡

𝑘) = 𝑢

(𝑖)

(𝑡

𝑘) − 𝜏

𝜕𝐻

(𝑖)

𝜕𝑢

(𝑡

𝑘) ,

𝑡 ∈ [𝑡

𝑘𝑡

𝑘+1] , 𝑘 = 0, 1, 2 . . . , 𝑁.

(24)

And easily its step value of 𝑢(𝑖+1)(𝑡𝑘), which is new input

value at next step, is determined.

4.3. Optimal Control of the Automotive Longitudinal Dynam-ics. As it was mentioned before, the aim of this study isto control the longitudinal motion of the locomotive sothat the fuel consumption is significantly reduced in thecondition ofmaintaining performance characteristics, retain-ing driving abilities, or even changing longitudinal behaviorparameters if possible. In the first step, the governing statespace equations of the dynamic system must be derived. Theindex performance and the Hamiltonian function are thendefined according to the desirable control input. Finally, thederivative of the Hamiltonian function relative to the controlinput is determined and then optimized [16].

0 1 2 3 4 5 6 7 8

1

1.2

1.4

1.6

1.8

2.2

2

Time (s)

0-top speed, acceleration test

CVT

ratio

in fi

rst i

tera

tion

Figure 5: CVT ratio in first iteration.

0 1 2 3 4 5 6 7 8

78.9489

78.9489

78.9489

78.9489

78.9489

78.9489

78.9489

78.9489

78.9489

Thro

ttle o

peni

ng in

firs

t ite

ratio

n


Time (s)

Figure 6: Throttle opening in first iteration.

0 1 2 3 4 5 6 7 8550

560

570

580

590

600

610

620

630

640

650

Time (s)


DesireDatum

𝜔e

(rpm

)

Figure 7: Engine speed.


0 1 2 3 4 5 6 7 8 90

100

200

300

400

500

600

Time (s)

P1

Costate variable in 0-top speed, acceleration test

Figure 8: Costate parameter 1.

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8

9

P2

×104

Time (s)

Costate variable in 0-top speed, acceleration test

Figure 9: Costate parameter 2.

0 1 2 3 4 5 6 7 8

1

1.2

1.4

1.6

1.8

2

2.2

2.40-top speed, acceleration test

CVT

ratio

opt

imiz

atio

n

Time (s)

Figure 10: CVT ratio optimization.

0 10 20 30 40 50 600

50100150200250300350400450500

Iteration step number

Ja


Figure 11: Performance index variations.

0 10 20 30 40 50 600

5

10

15

Iteration step number

Diff

eren

tial H

amilt

onia

n


Figure 12: Hamiltonian differential variations.

4.3.1. Extraction of Governing Equations. Torque convertor islocked in this state, governing variables are engine speed andCVT ratio and control input is desired CVT ratio that mustbe followed by CVT first-order system.

State EquationsConsider the following:

𝑈CVT =

𝑈desire − 𝑈CVT𝜏CVT

= 𝐹

1,

��

𝑒= (𝑇

𝑡∗ [𝐽

3⋅ 𝑁

𝑓𝑑⋅ 𝐾

2+ 𝐽

4⋅

𝐾

2

𝑁

𝑓𝑑

+ 𝑀 ⋅ 𝑟

2

⋅

𝐾

2

𝑁

𝑓𝑑

−

𝑟

3

⋅ 0.5 ⋅ 𝑅

𝑜⋅ 𝐶

𝑑⋅ 𝐴 + 0.00000004 ⋅ 𝑔 ⋅ 𝑀 ⋅ 𝜔

2

𝑒

𝑁

2

𝑓𝑑⋅ 𝑈

2

CVT

+ 𝐶

𝑟⋅ 𝑟 ⋅ 𝑔 ⋅ 𝑀] × (𝑁

𝑓𝑑⋅ 𝑈CVT)

−1

)


0 1 2 3 4 5 6 7 81015202530354045505560

Time (s)

Vehi

cle sp

eed

(M/s

)


Figure 13: Vehicle speed.

0 1 2 3 4 5 6 7540

560

580

600

620

640

660

Time (s)

DesireDatumOptimized

𝜔e

(rpm

)


Figure 14: Engine speed.

× (𝐽

2+

𝐽

3

𝑈

2

CVT+

𝐽

4

𝑁

2

𝑓𝑑⋅ 𝑈

2

CVT+ 𝑀⋅

𝑟

2

𝑁

2

𝑓𝑑⋅ 𝑈

2

CVT)

−1

=𝐹

2,

𝐹

2=

𝑈CVT (

𝜔

𝑒

𝑈

2

CVT) .

(25)

Performance Index & Hamiltonian Equation

𝐽

𝑎= ∫

𝑡𝑓

𝑡𝑜

(𝜔ool − 𝜔

𝑒)

2

+ 𝑃

1∗ (

𝑈CVT − 𝐹

1)

+ 𝑃

2∗ (��

𝑒− 𝐹

2) 𝑑𝑡,

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

Time (s)

MPG

(mile

/gal

lon)

OptimizedDatum


Figure 15: Specified parameter of fuel consumption.

0 20 40 60 80 1000

5

10

15

20

25

Time (s)

Vehi

cle sp

eed

(M/s

)

Federal highway driving scheduale (FHDS)

DesireDatumOptimized

Figure 16: Federal highway driving cycle.

𝐻locked = (𝜔ool − 𝜔

𝑒)

2

+ 𝑃

1∗

𝑈CVT

+ 𝑃

2∗ ��

𝑒.

(26)

Costate Equations and Hamiltonian Differential

𝑃

1= −

𝜕𝐻locked𝜕𝑈CVT

,


0 20 40 60 80 100−0.5

0

0.5

1

1.5

2

2.5

3Federal highway driving schedule (FHDS)

Time (s)

DatumOptimized

Vehi

cle ac

cele

ratio

n (M

/s2)

Figure 17: Acceleration Vehicle, (dashed), optimized (line).

0 10 20 30 40 50 60 70 80 90 1000

200400600800

10001200140016001800

Federal highway driving schedule (FHDS)

Dist

ance

(M)

Time (s)

DatumOptimized

Figure 18: Distance, datum (dashed), optimized (line).

𝑃

2= −

𝜕𝐻locked𝜕𝜔

𝑒

,

𝐻minlocked =

𝜕𝐻locked𝜕𝑈CVT

.

(27)

Initial & Final Conditions

𝑥

∗

(𝑡

0) = 𝑥

0,

𝑈

∗

CVT (𝑡

0) = 𝑈

0,

𝑝

∗

1(𝑡

𝑓) =

𝜕ℎ

𝜕𝑥

(𝑥

∗

(𝑡

𝑓)) = 0,

𝑝

∗

2(𝑡

𝑓) =

𝜕ℎ

𝜕𝑥

(𝑥

∗

(𝑡

𝑓)) = 0.

(28)

0 20 40 60 80 100

0

5

10

15

20


Time (s)

Thro

ttle o

peni

ng (d

eg)

Figure 19: Throttle opening input.

0 20 40 60 80 100

Fit 1Fit 2

−3

−2.5

−2

−1.5

−1

−0.5

0CV

T ra

tio d

ot (1

/s)

Time (s)


Figure 20: CVT ratio variation, datum (dashed), optimized (line).

Mechanical Constrains

700 < 𝜔

𝑒< 6000 rpm,

0.5 < 𝑈CVT < 2.5.

(29)

5. Optimal Control Derivation

A maneuver that is selected for longitudinal vehicledynamic simulation is maximum acceleration testwhich is proposed to determine optimal control input.See Figures 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15.


0 20 40 60 80 100

DatumOptimized

0.5

1

1.5

2

2.5

CVT

ratio

Time (s)


Figure 21: CVT Ratio, datum (dashed), optimized (line).

0 20 40 60 80 100

0

50

100

150

200

DatumOptimized

−50

Torq

ue (N

·m)

Time (s)


Figure 22: Engine torque, datum (dashed), optimized (line).

6. Optimal Control Approach Test in FederalHighway Driving Cycle

See Figures 16, 17, 18, 19, 20, 21, 22, 23, 24, and 25.

7. Conclusion

There exists indeed a need to investigate the fuel-optimalcontrol solution and how much fuel consumption can beimproved, because the classical control approaches are basedon heuristic deliberations only. Thus in this paper, havebeen introduced an appropriate system model for wholepowertrain and its subsystems. Optimal control is incapable

0 20 40 60 80 1000

50

100

150

200

250

300

DatumOptimizedDesire

Time (s)


𝜔e

(rpm

)Figure 23: Engine Speed, datum (dashed), optimized (thin Line),desired (thick line).

0 10 20 30 40 50 60 70 80 90 1000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

OptimizedDatum

Fuel

cons

umpt

ion

(kg)

Time (s)


Figure 24: Fuel consumption, datum (dashed), and optimized(line).

of solving the CVT-based powertrains problem in transientconditions and it is defined in this paper; then the problemsolved via nonlinear optimal control theory.

Regarding fuel consumption optimization, the strategy offollowing optimal performance curve of the engine is consid-ered; thus, if the required power for the driver is supplied,there will be no decrease in the automotive performance aswell as some improvements in the longitudinal performance.


0 20 40 60 80 1000

5

10

15

20

25

30

35

40

45

50

Datum

Time (s)

Optimized


MPG

(mile

/gal

lon)

Figure 25: Specified parameters of fuel consumption Datum(dashed), optimized (line).

As it is shown in Figure 24, the fuel consumption index has aremarkable increase from40.5 to 46.5. Also, decreasing clutchengagement and optimal control strategy has improved fuelconsumption about 11%.

The improvements in fuel consumption are obtained bychanging the controller software. Since no new powertraincomponents are needed, the costs associated with thesechanges are expected to be reasonably small.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this article.

References

[1] R. Kazemi, R. Marzooghi, and S. M. Ansari Movahhed, “A toolfor dynamic design of power train via modularity patterning,”in Proceedings of the International Conference of Iranian Societyof Mechanical Engineering (ISME ’02), 2002.

[2] M. R. Peterson and J.M. Starly, “NonlinearVehicle PerformanceSimulation with Test Correction and Sensitivity Analysis,” SAE960521.

[3] F. Xie, J. Wang, and Y. Wang, “Modelling and co-simulationbased on AMESim and Simulink for light passenger car withdual state CVT,” in Proceedings of the International Workshopon Automobile, Power and Energy Engineering (APEE ’11), pp.363–368, April 2011.

[4] R. Kazemi, M. Kabganiyan, and M. Rajayi, “Nonlinear robustcontrol of engine torque to the vehicle slip control,” in Pro-ceedings of the International Conference of Iranian Society ofMechanical Engineering (ISME ’02), 2002.

[5] M. Zhou, X. Wang, and Y. Zhou, “Modeling and simulation ofcontinuously variable transmission for passenger car,” in Pro-ceedings of the 1st International Forum On Strategic Technology(IFOST ’06), pp. 100–103, October 2006.

[6] R. Pfiffner, Optimal Operation of CT-Based Powertrains [Ph.D.Dissertation], Swiss Federal Institute of Technology, Zurich,Switzerland, 2001.

[7] L. Kong and R. G. Parker, “Steady mechanics of layered, multi-band belt drives used in continuously variable transmissions(CVT),”Mechanism and MachineTheory, vol. 43, no. 2, pp. 171–185, 2008.

[8] R. Krick, J. Tolstrup, A. Appelles, S. Henke, and M. Thumm,“The relevance of the phosphatidylinositolphosphat-bindingmotif FRRGT of Atg18 and Atg21 for the CVT pathway andautophagy,” FEBS Letters, vol. 580, no. 19, pp. 4632–4638, 2006.

[9] R. Krick, J. Tolstrup, A. Appelles, S. Henke, and M. Thumm,“The relevance of the phosphatidylinositolphosphat-bindingmotif FRRGT of Atg18 and Atg21 for the CVT pathway andautophagy,” FEBS Letters, vol. 580, no. 19, pp. 4632–4638, 2006.

[10] F. Xie, J. Wang, and Y. Wang, “Modelling and co-simulationbased on AMESim and Simulink for light passenger car withdual state CVT,” Procedia Engineering, vol. 16, pp. 363–368, 2011.

[11] G. Mantriota, “Performances of a parallel infinitely variabletransmissions with a type II power flow,” Mechanism andMachine Theory, vol. 37, no. 6, pp. 555–578, 2002.

[12] N. Srivastava and I. Haque, “Transient dynamics of metal V-belt CVT: effects of band pack slip and friction characteristic,”Mechanism and Machine Theory, vol. 43, no. 4, pp. 459–479,2008.

[13] N. Srivastava and I. Haque, “A review on belt and chain con-tinuously variable transmissions (CVT): dynamics and control,”Mechanism and Machine Theory, vol. 44, no. 1, pp. 19–41, 2009.

[14] K. Hyunso and L. Jaeshin, “Analysis of belt behavior andslip characteristics for a metal V-belt CVT,” Mechanism andMachine Theory, vol. 29, no. 6, pp. 865–876, 1994.

[15] D. Kirk,Optimal ControlTheory: An Introduction, Prentice-Hill,Englewood Cliffs, NJ, USA, 1974.

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Research Article Nonlinear Optimal Control of Continuously … · 2019. 7. 31. · Optimal control theory has rarely been used in the previous control approaches applied to such systems