Abstract-- In this paper, adaptive speed control of a light

weighted all-electric vehicle (EV) is presented for strict feedback form with unknown time delay. The EV is driven by DC motor. The unknown time delay has been compensated through the appropriately chosen Lyapunov–Krasovskii functional. In present work, state feedback control is presented via backstepping, applied to a strict feedback form of the EV system. The controller singularity problem has been resolved by relaxing our control objective of convergence to a bounded region rather than the origin and assembling unknown parameters together. The proposed adaptive backstepping design has been proved that the output of the system to converge to an arbitrarily small neighborhood of the origin. The new European driving cycle (NEDC) test is performed to test the control performance. Through simulation results the effectiveness of the proposed control scheme are shown.

Index Terms—Adaptive control, backstepping control, DC motor, .electric vehicle, new European driving cycle (NEDC), time delay.

I. INTRODUCTION N recent years, electric vehicles (EVs) are developing fast due to stricter emission standards and the shortage of energy

sources. EV has many advantages over the conventional internal combustion engine (ICE) vehicle such as high efficiency, quiet and smooth operation. There have been tremendous efforts by researchers and the automobile industry to develop zero-polluting EV for energy conservation and environmental protection. EV has come out to be a promising alternative to improve fuel economy while meeting the tightened emission standards [1]. The propulsion system and its control method are the key technologies of electric vehicle. Researches and investigations on the power propulsion system of EVs and its control have drawn significant attention among academician and in the automobile industry as well [2].

EVs are powered by electric motors through transmission unit which includes tyre and differential gears. The dynamic behavior of the electric motors with input-output state feedback controller via adaptive backstepping observer of a typical series hybrid EV is addressed in [3].

V. Sharma is with the Department of Electrical Engineering, Motilal

Nehru National Institute of Technology, Allahabad-211004, India (e-mail: vikassharma.ecb@gmail.com).

S. Purwar is with the Department of Electrical Engineering, Motilal Nehru National Institute of Technology, Allahabad-211004, India (e-mail: spurwar@rediffmail.com).

978-1-4799-4939-7/14/$31.00 ©2014 IEEE

A robust adaptive backstepping controller for speed control of permanent magnet synchronous motor used in EVs is discussed in [4]. Battery supplies DC power, therefore, EVs driven by DC motor is a favorable selection. The control of DC motor is simple and it can provide comparatively larger startup torque. Nonlinear control methods are reported for series wound DC motor speed control in [5], [6]. Additionally the primary function of propulsion to the EV, the DC motor can also be used adequately as the braking device because of its capability of regeneration and fast torque response characteristics [7], [8].

Nonlinear optimal and robust speed control of series wound DC motor driven EV has been intensively investigated in [9]. Nonlinear backstepping speed control is investigated for EV system in strict feedback form in [10].

Robust adaptive backstepping control for nonlinear systems in which uncertainties are not only from unknown nonlinear functions as well as parametric has been presented in [11]. Robust adaptive control of nonlinear systems with unknown time delays and adaptive neural network control of nonlinear systems with unknown time delays have been addressed in [12], [13]. For a class of nonlinear time-delay systems, a robust stabilizing controller design based on the Lyapunov–Krasovskii functional was reported in [14].

It is not possible to model the dynamics of an EV system precisely as some parameters may vary with time and some parameters may vary with conditions. For example, the resistance of the armature winding in the DC motor changes during EV running condition as the temperature varies. In this paper, the resistance in the armature winding ( aR ) of the DC motor and the mass of the passengers ( MΔ ) are considered to be varying with time resulting in unknown parameters. The aim of this paper is to design nonlinear adaptive controller for EV and test the performance of the system on new European driving cycle (NEDC).

The paper is organized as follows. In Section II, the EV system description in strict feedback form will be presented. The problem statement is introduced in the Section III. The design of adaptive controller is described in Section IV. Section V, validates the performance of the proposed controller through simulations followed by conclusion.

II. EV SYSTEM DESCRIPTION An EV system description broadly constitutes of two parts:

the vehicle dynamics and dynamics of the motor system, as shown in Fig. 1. Motor system dynamics is associated to the vehicle system dynamics through transmission unit, which

Adaptive Speed Control of a Light Weighted All Electric Vehicle with Unknown Time Delay

Vikas Sharma, Student Member, IEEE, and Shubhi Purwar, Member, IEEE

I

includes the gearing system. In this paper, we considering the Ev system in strict feedback form discussed in [10].

Fig. 1. Electric vehicle system.

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( )

1 2

2 21 3 7 1 3 82 2 1 2

2 2

1 51 42 1 2

2 2

1 3 92

2

1

2 2

2

2

z t z tK K K K K K

z t x t z t x tm K m K

K KK Kz t z t z t

m K m KK K K

x t u tm K

y t z t

=

= − −+ +

− −+ +

++

= .

(1)

where, ( )1z t and ( )2z t are the state variable, ( )u t is the

system input, ( )y t is the output and ( ) ( ) ( )( )2 1 2,x t f z t z t=

is the state transformation variable [10]. System constants 1K to 9K are as given below

2 21K G r= , ( )2 2

2K G r J= , 3 afK L= , 4K B= ,

( ) ( )3 35 1 2 dK AC r Gρ= , ( )6 rrK r G gμ= ,

( ) ( )( )7 a f a fieldK R R L L= + + , ( )8 af a fieldK L L L= + , and

( )9 1 a fieldK L L= + .

where, rrμ is the rolling resistance coefficient, m is the mass of the EV, g is the gravity acceleration, ρ is the air density, A is the frontal area of the vehicle, dC is the drag coefficient, r is the tyre radius of the EV, G the gearing ratio, J is the inertia of the motor, including the gearing system and the tyres; , , ,a a field fL R L R are the armature inductance, armature resistance, field winding inductance, and field winding resistance respectively, B is the viscous coefficient, afL is the mutual inductance between the armature winding and the field winding, generally nonlinear because of saturation.

In [1] m is a constant which is a very rigorous assumption. In the proposed work m includes the mass of vehicle M and the mass of passengers MΔ i. e., m M M= + Δ . Thus m is varying with time and not a constant, and the resistance in the armature winding ( aR ) of the DC motor is considered to be varying as the armature winding resistance of the DC motor changes as the temperature varies. The parameters used on a lightweight all-electric vehicle are specified in Table I.

TABLE I PARAMETERS OF THE EV SYSTEM [9]

Motor Vehicle

a fieldL L+ (mH) 6.008 A (m2) 1.8

a fR R+ (Ω) 0.12 ρ (kg/m3) 1.25

B (N.M.s.) 0.0002 dC 0.3

J (kg m2) 0.05 rrμ 0.015

afL (mH) 1.766 r (m) 0.25

u (V) 0 ~ 48 G 11 M (kg)=800, and MΔ =100 (kg) for 0< t <=200, MΔ =220 for 201< t <=500, MΔ =300 (kg) for 501< t <800 and MΔ =150 (kg) for 800<= t <=1180

III. PROBLEM STATEMENT AND BACKGROUND Consider (1) as a class of nonlinear time delay EV system

and rewritten as

( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( )( )

( ) ( )

1 2

21 3 9 1 3 82 2 1 2

2 2

2 221 3 7 1 51 42

22 2 2

1

2 2

2 2

z t z tK K K K K K

z t x t u t z t x tm K m K

z tK K K K KK Kx t

z tm K m K m K

y t z t

ττ

=

= −+ +

⎡ − ⎤⎡ ⎤− − ⎢ ⎥⎢ ⎥ −+ + +⎣ ⎦ ⎣ ⎦=

(2)

where, [ ]1 2, Tz z z= and 2τ is the unknown time delay. To make precise problem statement, the system in (2) is

represented as follows:

( ) ( )( ) ( ) ( )( )

( )( ) ( )( )( ) ( )

1 1 2

2 2 2 2

2 2 2 2

1

Tf

Tf h

z t g z t

z t g u t F z t

z t H z t

y t z t

φ

δ φ τ

=

= +

+ + −

=

(3)

where, 1 1g = , ( )1 3 92 2

2

2K K Kg x t

m K=

+, 1 3 8

22

2f

K K Km K

φ = −+

,

1 51 42

2 2

2h

K KK Km K m K

φ⎡ ⎤

= − −⎢ ⎥+ +⎣ ⎦, ( )2F ⋅ and ( )2H ⋅ are known

smooth function vectors. 2fφ and 2hφ are unknown parameter

vectors, ( )2fδ ⋅ is unknown smooth function that satisfy the following bound condition [12]

( )( ) ( )( )2 2 2f fz t c z tδ ϕ≤ (4)

where, 2fc is unknown constant parameter, and ( )2ϕ ⋅ is known nonnegative smooth function. The aim of this paper is to design and test the performance of an adaptive controller for EV system, which forces the plant output ( )y t to track a

specified reference trajectory ( )dy t in the presence of time varying mass ‘ m ’ and varying armature winding resistance

( aR ), while all the signals in closed loop system are bounded.

The desired trajectory is defined as [ ]2 , Td d dz y y= .

Assumption 1: The functions ( )ig ⋅ , where 1, 2i = are assumed to be bounded away from zero and there exists two constants , 0i ig g > such that ( )i i ig g g≤ ⋅ ≤ . Without the

loss of generality, in this paper, we have considered 0.ig >

Assumption 2: The unknown time delay is bounded by a known constant, such that 2 maxτ τ≤ .

Assumption 3: The desired trajectory dy and its derivatives up to second order are bounded.

Lemma 1[12]: For 1 0ε > and any Rυ ∈ , the following inequality holds

( )1 10 tanh kυ υ υ ε ε≤ − ≤

where, k is a constant that satisfies ( )1kk e− += , i.e., k =0.2785.

IV. ADAPTIVE CONTROLLER DESIGN

In the backstepping controller design, we treat ( )2z t as fictitious control signal. In this stage, we design the fictitious controller ( )1 tα . Second, we design an actual controller for

( )u k to force the error between ( )1z t and ( )dy t as small as possible. The error dynamics are defined as 1 1 de z y= − and

2 2 1e z α= − .

A. Fictitious Controller Design Firstly consider the error dynamics of 1e subsystem

( ) ( ) ( )1 1 de t z t y t= − (5)

Substituting ( )1z t from (3), we have

( ) ( ) ( )( ) ( )( ) ( )

1 1 2

1 2 1

d

d

e t g z t y t

g e t t y tα

= −

= + − (6)

Now, we define the scalar function

( ) ( ) ( )21 1 11 2eV t g e t= (7)

The time derivate of ( )1eV t is given by

( ) ( ) ( ) ( )( ) ( ) ( ) ( )1 1 2 1 1 11e dV t e t e t t g e t y tα= + − (8)

As there is no time delay term in this subsystem, therefore we can choose the fictitious controller ( )1 tα , as follows

( ) ( ) ( ) ( ) ( )( )( ) ( )( )

1 1 2 1 1 1 1

1 10 1 1 1

Te

T

V t e t e t e t t F

t k e t F

φ

φ

α φ

α φ

= + +

= − − (9)

where, 10k >0 is the design parameter, ( )1T

dF y tφ = −⎡ ⎤⎣ ⎦ and

( )1 11T

gφ = ⎡ ⎤⎣ ⎦ .

B. Actual Controller Design Now, we consider the 2e subsystem

( ) ( ) ( )2 2 1e t z t tα= − (10)

Substituting ( )2z t from (3), we have

( ) ( ) ( )( )( )( ) ( )( ) ( )

2 2 2 2

2 2 2 2 1

Tf

Tf h

e t g u t F z t

z t H z t t

φ

δ φ τ α

= +

+ + − − (11)

Now, we define the scalar function

( ) ( ) ( )22 2 21 2eV t g e t= (12)

The time derivate of ( )2eV t is given by

( ) ( ) ( )( )( ) ( )( )

( )( ) ( )2 2 2

2 22 2 2 2 1

1Tf f

e Th

F z t z tV t e t u t

g H z t t

φ δ

φ τ α

⎧ ⎫⎡ ⎤+⎪ ⎪⎢ ⎥= +⎨ ⎬⎢ ⎥+ − −⎪ ⎪⎣ ⎦⎩ ⎭

(13)

As ( )2fδ ⋅ is bounded, therefore from (4), we have

( ) ( ) ( )

( ) ( )( )( ) ( )( )

( ) ( )( )( ) ( )

2 2 2

2 2 22 2

2 2 2 2 2

2 1

1

Tf

fe T

h

e t F z t

e t c z tV t e t u t

g e t H z t

e t t

φ

ϕ

φ τ

α

⎡ ⎤⎢ ⎥⎢ ⎥+

≤ + ⎢ ⎥+ −⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦

(14)

In (14), the unknown parameter 2Thφ and unknown time

delay 2τ are jumbled together, which make more complexity to tackle the problem. Using young’s Inequality [15], we can treated them separately as

( ) ( )

( ) ( )( )( ) ( )( )

( )

( )( ) ( )( )( ) ( )

2 2 2

2 2 2

22 2 2 2 2

2

2 2 2 2

2 1

1 1212

Tf

f

Te h h

T

e t F z t

e t c z t

V t e u t e tg

H z t H z t

e t t

φ

ϕ

φ φ

τ τ

α

⎡ ⎤⎢ ⎥⎢ ⎥+⎢ ⎥⎢ ⎥⎢ ⎥≤ + +⎢ ⎥⎢ ⎥+ − −⎢ ⎥⎢ ⎥⎢ ⎥−⎣ ⎦

(15)

The following Lyapunov-Krasovskii functional is considered to prevail over the design problem from the unknown time delay 2τ

( ) ( ) ( )( )2

2 2 21 2t

U tV t g U z d

ττ τ

−= ∫ (16)

where, ( )( )2U z τ is a positive definite function given as

( )( ) ( )( ) ( )( )2 2 2TU z H z t H z tτ = (17)

The time derivate of ( )2UV t is given by

( ) ( )( )( ) ( )( )

( )( ) ( )( )2 2

2 22 2 2 2

1 2T

U T

H z t H z tV t g

H z t H z tτ τ

⎡ ⎤⎢ ⎥=⎢ ⎥− − −⎣ ⎦

(18)

Therefore, we obtain ( ) ( )2 2e UV t V t+ so that the unknown time delay term has been completely removed. Now, for clarity we will exclude the time variable, yields

( )2 2 2 2 2 20 2 20T

e UV V e u F eφφ φ ϕ+ ≤ + + (19)

where, ( )20 2 2fc gφ = , 20 2ϕ ϕ= ,

( ) ( ) ( ) ( )2 2 2 2 2 2 1 2 2, , , 1TT T

f h hg g g g gφ φ φ φ⎡ ⎤= ⎣ ⎦ , and

( ) ( )2 2 2 10 2 2 2 2 10, 1 2 , , 1 2TT T

dF F e k z e H H k yφ ⎡ ⎤= +⎣ ⎦ . Therefore, from (19) the desired control signal is chosen as

( )2 2 2 2 2 2 2

2 2

ˆ ,0,

Te

e

k t e F e cu

e cφφ β⎧− − − ≥⎪= ⎨ <⎪⎩

(20)

where, 2φ is the estimate of 2φ and 2 20 2ˆβ φ ξ= with

( )2 20 2 20 2tanh eξ ϕ ϕ ε= , where 20φ is the estimate of 20φ . The adaptive control law can be considered as follows

( )20 2 2 2 20 20ˆ ˆeφ γ ξ σ φ= − , (21)

( )2 2 2 2 2 2ˆ ˆF eθφ σ φ= Γ − , (22)

( ) ( ) ( )( )max

22 20 2 21

t

tk t k e U z d

ττ τ

−= + ∫ (23)

where, 20k >0 is a design parameter, 2ε >0 is a positive small constant, 2γ >0 is constant, 1

2 2−Γ = Γ >0, 20σ and 2σ are small

constants.

C. Stability Analysis Theorem 1: Consider the EV system (3) along with

Assumptions 1-3 holds. If the control law is chosen as (20) and the adaptive law is updating according to (21), (22), then for any finite initial conditions, all the signals in the closed loop system are bounded, and the tracking error [ ]1 2, Te e e= will converge to the compact set defined by

2e ee R e cΩ = ∈ ≤ .

Proof: Let the Lyapunov function candidate has been considered as defined in (24) and follows the same line as discussed in [12], considering all the three cases. For brevity the whole proof is not presented here.

( ) ( )1 2

1 2 12 2 20 2 2 21 2 1 2 T

e e UV V V V γ φ φ φ− −= + + + + Γ (24)

where, 20 20 20ˆφ φ φ= − and 2 2 2

ˆφ φ φ= − .

V. SIMULATION RESULTS The currently used drive cycle tests for light weighted EVs

are New European Driving Cycle (NEDC), Federal Test Procedure (FTP-75), and Japanese Cycle (JC08). The NEDC

is used in Europe, the low powered EV version of this cycle is used in India. The FTP 75 cycle is used in the USA and the JC08 in Japan.

In order to show the validity of the proposed Adaptive controller for EV system, the NEDC is used for testing the performance. The NEDC is a driving cycle consisting of four repeated ECE-15 driving cycles and an extra-urban driving cycle (EUDC) [1]. The maximum speed of NEDC is 120 km/h but it is scaled to 50 km/h when applied in this paper [9].

For simulation purpose to test the performance of designed adaptive controller for mass variation is considered as given in (25). Passengers mass is increased / decreased at different points of time, in the driving cycle. The variation in armature winding resistance of the DC motor due to temperature changes is considered as give in (26) and the unknown time delay 2τ is considered as 1 sec.

900 0 2001 020 201 5001100 501 800950 800 1180

tt

mt

t

< ≤⎧⎪ < ≤⎪= ⎨ < <⎪⎪ ≤ ≤⎩

(25)

0.0867 0 5000.0947 500 1180a

tR

t< ≤⎧

= ⎨ < ≤⎩ (26)

The design parameters chosen for simulations are as

follows: The initial conditions ( ) ( )1 20 , 0T

z z⎡ ⎤⎣ ⎦ = [ ]0.1, 0.1 T ,

2γ =1, 2Γ = [ ]1 0;0 1 , 20σ =0.1, 2σ =0.5, 020φ =0, 0

2φ =0,

10k =0.6, 20k =0.8, 2ε =0.1 and 2ec =0.1. For simulations on light weight EVs, four batteries of 12

volts each are taken into account and this restricts the control signal in the range of 0 ~ 48 V (see Table I). The drive cycle test performance and the tracking error for proposed adaptive controller for EV system is shown in Figs. 2-3 respectively. Figs. 4-6 shows the boundedness of the control signal, the parameters estimates 20φ and 2φ , respectively.

0 200 400 600 800 1000 1200

0

10

20

30

40

50

60

Time (s)

Spe

ed (

km/h

r)

Adaptive Controller

NEDC Standard

Fig. 2. Performance of adaptive controller.

0 200 400 600 800 1000 1200-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Time (s)

Err

or in

spe

ed (

km/h

r)

Fig. 3. Tracking Error.

0 200 400 600 800 1000 12000

10

20

30

40

50

Time (s)

Con

trol

Eff

ort

(vol

ts)

Fig. 4. Control effort.

0 200 400 600 800 1000 12000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (s)

Fig. 5. Boundedness of 20φ .

0 200 400 600 800 1000 12000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (s)

Fig. 6. Boundedness of 2φ .

From the simulation results, it is clear that the proposed adaptive controller for EV system can effectively perform on NEDC drive cycle test. Note also that after some initial time that is required for adaptation to implement the control action, the controller tracks the NEDC Standards closely and effectively.

VI. CONCLUSION An adaptive speed control of a light weighted EV has been

discussed for strict feedback form with unknown time delay. The unknown time delay has been compensated through the appropriately chosen Lyapunov–Krasovskii functional. The output of the system has been proved to converge to an arbitrarily small neighborhood of the origin. The NEDC test is performed to test the control performance. It is shown that the tracking performance of the proposed adaptive backstepping controller designed in this paper is satisfactory. The real time implementation of the proposed controller will be carried out in near future.

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20φ

2φ

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