MEV


 

 

Journal of Mechatronics, Electrical Power, and Vehicular Technology 9 (2018) 89–100 

 

Journal of Mechatronics, Electrical Power, 

and Vehicular Technology 
 

e-ISSN: 2088-6985  

p-ISSN: 2087-3379  

 

 

www.mevjournal.com 
 

 

 

doi: https://dx.doi.org/10.14203/j.mev.2018.v9.89-100 

2088-6985 / 2087-3379 ©2018 Research Centre for Electrical Power and Mechatronics - Indonesian Institute of Sciences (RCEPM LIPI).  

This is an open access article under the CC BY-NC-SA license (https://creativecommons.org/licenses/by-nc-sa/4.0/).  

Accreditation Number: (LIPI) 633/AU/P2MI-LIPI/03/2015 and (RISTEKDIKTI) 1/E/KPT/2015. 

Designing optimal speed control with observer using  

integrated battery-electric vehicle (IBEV) model for energy efficiency 
 

Rina Ristiana a, b, *, Arief Syaichu Rohman a, Estiko Rijanto c,  

Agus Purwadi a, Egi Hidayat a, Carmadi Machbub a 
a School of Electrical Engineering and Informatics, Institut Teknologi Bandung 

Jl. Ganesha No. 10, Bandung, West Java, Indonesia 
b Instrumentation Development Unit, Indonesian Institute of Sciences 

Komplek LIPI Jl. Sangkuriang, Bandung, West Java, Indonesia  
c Research Centre for Electrical Power and Mechatronics, Indonesian Institute of Sciences 

Komplek LIPI Jl. Sangkuriang, Bandung, West Java, Indonesia  

 

Received 23 October 2018; received in revised form 19 December 2018; accepted 20 December 2018 

Published online 30 December 2018 

 

 

Abstract 

This paper develops an optimal speed control using a linear quadratic integral (LQI) control standard with/without an 

observer in the system based on an integrated battery-electric vehicle (IBEV) model. The IBEV model includes the dynamics 

of the electric motor, longitudinal vehicle, inverter, and battery. The IBEV model has one state variable of indirectly measured 

and unobservable, but the system is detectable. The objectives of this study were: (a) to create a speed control that gets the exact 

solution for a system with one indirect measurement and unobservable state variable; and (b) to create a speed control that has 

the potential to make a more efficient energy system. A full state feedback LQI controller without an observer is used as a 

benchmark. Two output feedback LQI controllers are designed; including one controller uses an order-4 observer and the other 

uses an order-5 observer. The order-4 observer does not include the battery state of charge as an observer state whereas the 

order-5 observer is designed by making all the state variable as the observer state and using the battery state of charge as an 

additional system output. An electric passenger minibus for public transport with 1500 kg weight was used as the vehicle model. 

Simulations were performed when the vehicle moves in a flat surface with the increased speed from stationary to 60 km/h and 

moves according to standard NEDC driving profile. The simulation results showed that both the output feedback LQI controllers 

provided similar speed performance as compared to the full state feedback LQI controller. However, the output feedback LQI 

controller with the order-5 observer consumed less energy than with the order-4 observer, which is about 10% for NEDC driving 

profile and 12% for a flat surface. It can be concluded that the LQI controller with order-5 observer gives better energy efficiency 

than the LQI controller with order-4 observer. 

©2018 Research Centre for Electrical Power and Mechatronics - Indonesian Institute of Sciences. This is an open access 

article under the CC BY-NC-SA license (https://creativecommons.org/licenses/by-nc-sa/4.0/).  

Keywords: integrated battery-electric vehicle (IBEV) model; speed control; electric vehicle; linear quadratic integral; observer 

system; energy efficient. 

 

 

I. Introduction 

In the future, electric vehicles will be more widely 

used for mass transportation, implemented in special 

lines empowered by automatic systems such as 

driverless systems, assisted drive systems, self-driving 

systems and so on. This prospect has opened up new 

research areas for innovation in technology based on 

automation of specifically controlled systems. One of 

the limitations of electric vehicles is the limited amount 

of energy they can carry, which is mainly stored in its 

battery [1]. Assuming that this limited capacity is 

because of existing battery technology, the problem 

should be solved using an energy-efficient strategy [2].  

Energy-efficient strategies for electric vehicles are 

one of several types of strategies that involve control 

design of the vehicle. The control design of an electric 

vehicle is implemented with vehicle/motor speed 

control [3] and torque control [4][5].  

 

 

* Corresponding Author. Tel: +62 813 7991 7553 

E-mail address: rina005@lipi.go.id 

https://dx.doi.org/10.14203/j.mev.2018.v9.89-100
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R. Ristiana et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 9 (2018) 89–100 

 

90 

An important factor in designing such a control 

system is the electric vehicle model. In [3] and [4] an 

electric vehicle model with battery dynamics integrated 

into the system was presented. The use of an integrated 

model in electric vehicle control design (speed or 

torque) has been shown to have potential in achieving 

a more energy-efficient system. Although the 

integrated model has one unobservable state variable, 

the system is still detectable. 

Ideally, all state variables should be available for 

feedback in the system, but not all state variables are 

available for feedback. Therefore, it needs to estimate 

unavailable state variables. Estimation of unavailable 

state variables is called state observer. A state observer 

estimates the state variables based on the measurements 

of the output and control variables. The observers 

consist of: a full-order observer that is used to estimate 

all the state variables of the system that are considered 

available for direct measurement [6]. 

This paper describes how to design an optimal 

speed control using the LQI control standard 

with/without an observer in the system. The goals of 

this research were to create a control design: (a) that 

gets the exact solution for one state variable in the 

system which is unobservable and can only be 

measured indirectly, and (b) has the potential to be 

more energy efficient. The LQI control systems have 

been built in three cases, i.e. LQI control without 

observer (assumption that all variables are available for 

feedback), LQI control with an order-4 observer 

(ignoring one state variable of the system during 

designing the observer), and LQI control with an order-

5 observer (adding one state variable in the output of 

the system), which were compared to find the best 

response characteristics and to increase energy 

efficiency. 

II. Materials and methods 

A. Integrated battery-electric vehicle (IBEV) model 

The battery-electric vehicle (BEV) model was built 

as an integrated model. This means that it is a model 

with battery dynamics involved in the system (Figure 

1). It includes an electric motor [7], an inverter [8], a 

longitudinal vehicle [9], and battery dynamics [10][11].  

The integrated model is a linearized model derived 

from a nonlinear model. It is assumed that only the 

battery supplies the electric motor of the vehicle, hence 

the current of the battery are the same as the motor 

current. The gear trains have no backlash; they are rigid 

bodies. The shaft stiffness and each gear ratio are 

proportional to the radius of the gear [9]. The 

longitudinal dynamic equations were influenced by 

traction, acceleration, and total resistance forces as load 

(see Figure 1). The total resistance forces included drag 

force, gradient force, rolling resistance force, and 

curvature resistance force [12].  

According to [4], differential equations of the motor 

speed (1), the motor current (2), the first (3) and the 

second (4) capacitor voltage of the battery, and the 

charge extracted from the battery (5) respectively can 

be written as: 

𝑑𝜔𝑚(𝑡)

𝑑𝑡
= −

𝑏𝑚

𝑛𝐽𝑡𝑜𝑡
𝜔𝑚(𝑡) +

𝑘𝑡

𝑛𝐽𝑡𝑜𝑡
𝑖𝑚(𝑡) −

𝑛2𝐾𝑑𝑟𝑤
3

2𝐽𝑡𝑜𝑡
𝜔𝑚

2(𝑡) +
𝑚𝑣𝑟𝑤𝑔

𝐽𝑡𝑜𝑡
(sin 𝜃 +

𝐶𝑅𝑥 cos 𝜃 +
𝑘𝑡𝑘

𝑅
)  (1) 

𝑑𝑖𝑚(𝑡)

𝑑𝑡
= −

𝑘𝑒

𝐿𝑚
𝜔𝑚(𝑡) −

𝑅𝑚

𝐿𝑚
𝑖𝑚(𝑡) +

𝐾𝑐

𝐿𝑚
(−𝑅𝑑𝑖𝑚(𝑡) −

𝑉𝑐1(𝑡) − 𝑉𝑐2(𝑡) + 2𝑎1𝑆𝑂𝐶𝑛(𝑡) + 2𝑎1 +
2𝑎0)𝑢𝑐(𝑡) (2) 

𝑑𝑉𝑐1(𝑡)

𝑑𝑡
= −

1

𝑅𝑡1𝐶𝑡1
𝑉𝑐1(𝑡) +

1

𝐶𝑡1
𝑖𝑏(𝑡)  (3) 

𝑑𝑉𝑐2(𝑡)

𝑑𝑡
= −

1

𝑅𝑡2𝐶𝑡2
𝑉𝑐2(𝑡) +

1

𝑖𝑏(𝑡) 𝐶𝑡2⁄
𝑖𝑏(𝑡)  (4) 

𝑑𝑆𝑂𝐶𝑛(𝑡)

𝑑𝑡
= −

1

𝑄𝑛
𝑖𝑏(𝑡)  (5) 

The battery voltage can be represented as: 

𝑉𝑏(𝑡) = 𝑉𝑂𝐶(𝑡) − 𝑅𝑑𝑖𝑏(𝑡) − 𝑉𝑐1(𝑡) − 𝑉𝑐2(𝑡) (6) 

The open-circuit voltage (two batteries) is 𝑉𝑂𝐶(𝑡) =

2𝑎1𝑆𝑂𝐶(𝑡) + 2𝑎0  and the state of charge is 𝑆𝑂𝐶(𝑡) =

(𝑆𝑂𝐶0(𝑡) + 𝑆𝑂𝐶𝑛(𝑡)) with𝑆𝑂𝐶0(𝑡) = 𝑄0 𝑄𝑛⁄ = 1, where 

𝑅𝑑 , 𝑖𝑏 , 𝑅𝑡1 , 𝐶𝑡1 , 𝑅𝑡2 , 𝐶𝑡2 , 𝑎1 , 𝑎0 , 𝑄0  and 𝑄𝑛  are 
suitable constants [4][11]. 

The state variables are defined as 𝑥1(𝑡) = 𝜔𝑚(𝑡), 
𝑥2(𝑡) = 𝑖𝑚(𝑡) , 𝑥3(𝑡) = 𝑉𝑐1(𝑡) , 𝑥4(𝑡) = 𝑉𝑐2(𝑡)  and 
𝑥5(𝑡) = 𝑆𝑂𝐶𝑛(𝑡)  and the output variable as 𝑦(𝑡) =
𝜔𝑚(𝑡) = 𝑥1(𝑡). 

From equation (1) to (5), the state equation may be 

described as: 

�̇�𝑣(𝑡) = 𝑓(𝑥𝑣(𝑡)) + 𝑔(𝑥𝑣(𝑡))𝑢𝑐(𝑡) + 𝐻𝑑𝐿  

𝑦𝑣(𝑡) = 𝐶𝑣𝑥𝑣(𝑡)  (7) 

Its matrices are given by: 

 

Figure 1. Integrated battery-electric vehicle (IBEV) model [4] 



R. Ristiana et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 9 (2018) 89–100 91 

𝑓(𝑥𝑣(𝑡)) =

[
 
 
 
 
𝑎11 + 𝑎𝑁𝐿 𝑎12 0 0 0

𝑎21 𝑎22 0 0 0
0 𝑎32 𝑎33 0 0
0 𝑎42 0 𝑎44 0
0 𝑎52 0 0 0]

 
 
 
 

 , 

𝑔(𝑥𝑣(𝑡)) = [0 𝑔2 0 0 0]
𝑇, 

𝐻 = [1 0 0 0 0]𝑇, 

𝐶𝑣 = [1 0 0 0 0], 

where: 

𝑎11 = − 𝑏𝑚 𝑛𝐽𝑡𝑜𝑡⁄  , 𝑎𝑁𝐿 = 𝑛
2𝐾𝑑𝑟𝑤

3𝑥1
2(𝑡) 2⁄ , 

𝑎12 = 𝑘𝑡 𝑛𝐽𝑡𝑜𝑡⁄ , 𝑎21 = − 𝑘𝑒 𝐿𝑚⁄  ,  

𝑎22 = − 𝑅𝑚 𝐿𝑚⁄  , 𝑎32 = 1 𝐶𝑡1⁄  ,  

𝑎33 = 1 𝑅𝑡1𝐶𝑡1⁄  , 𝑎42 = 1 𝐶𝑡2⁄  , 𝑎44 = 1 𝑅𝑡2𝐶𝑡2⁄ , 

𝑎52 = 1 𝑄𝑛⁄  , 𝐽𝑡𝑜𝑡 =  (𝑚𝑣𝑟𝑤𝑛 + 𝐽𝑒𝑞)𝑟𝑤, 

𝐽𝑒𝑞 = 𝐽𝑚 + (𝐽𝑡 𝑛𝑔
2⁄ ) + (𝐽𝑤 𝑛𝑔

2𝑛𝑡
2⁄ ), and 

𝑔2 = −(𝑅𝑑𝑥2(𝑡) + 𝑥3(𝑡) + 𝑥4(t) − 2𝑎1𝑥5(𝑡) −
2(𝑎0 + 𝑎1))𝐾𝑐/𝐿𝑚. 

With 𝐾𝑑 = 𝜌𝐶𝑑𝐴𝑓 ; 𝑚𝑣 , 𝑟𝑤 , 𝜌, 𝐶𝑑 , 𝐴𝑓 , 𝐶𝑅𝑥 , 𝑔 , 𝜃 , 

𝑘𝑡𝑘 , 𝑅, 𝑅𝑑 , 𝑖𝑏 , 𝑅𝑡1 , 𝐶𝑡1, 𝑅𝑡2 , 𝐶𝑡2, 𝑎1 , 𝑎0 , 𝑄0 , 𝑄𝑛 , 𝐿𝑚 , 
𝑅𝑚, ke, , n=1/nggntt; g and t are suitable constants 
[4]. 

B. Control system design 

The speed control system was designed using the 

linear control integral (LQI) method. The LQI 

computes an optimal state feedback control law for the 

tracking loop with the assumption that all state 

variables are available for feedback in the system. In 

this paper, three LQI controllers are designed, i.e. a 

state feedback LQI controller and two output feedback 

LQI controllers with observer systems such as order-4 

observer and order-5 observer. The state feedback LQI 

controller is used as a benchmark for comparison study. 

Luenberger observer is used in each output feedback 

LQI controller [13]. 

The first purpose of the LQI controller design is that 

the control design can answer in a proper way if there 

is a state variable in a system that is indirectly 

measurable and unobservable. The second purpose is to 

get one control design that has the potential to be more 

energy efficient.  

1) LQI control  

The LQI control used is as shown in Figure 2. Based 

on (7), by ignoring 𝑑𝐿, a linearized plant can be derived 
as follows:  

�̇�𝑣(𝑡) = 𝐴𝑣𝑥𝑣(𝑡) + 𝐵𝑣𝑢𝑐(𝑡)  

𝑦𝑣(𝑡) = 𝐶𝑣𝑥𝑣(𝑡)  (8) 

The set point tracking is given by: 

�̇�𝑖(𝑡) = 𝑟(𝑡) − 𝐶𝑣𝑥𝑣(𝑡)  (9) 

The full state feedback control is: 

𝑢𝑐(𝑡) = −𝑘𝑣𝑥𝑣(𝑡) − 𝑘𝑖𝑥𝑖(𝑡) = −𝐾𝑧𝑥𝑧(𝑡)  (10) 

The augmented state equation is obtained from [13] is: 

�̇�𝑧(𝑡) = 𝐴𝑧𝑥𝑧(𝑡) + 𝐵𝑧𝑢𝑐(𝑡) + 𝐺𝑧𝑟(𝑡)  (11) 

where  

𝐴𝑧 = [
𝐴𝑣 0
−𝐶𝑣 0

], 

𝐵𝑧 = [
𝐵𝑣
0

],  

𝐺𝑧 = [
0
1
], and  

𝑥𝑧(𝑡) = [𝑥𝑣(𝑡) 𝑥𝑖(𝑡)]
𝑇. 

To stabilize the system of (11), a state feedback 

controller can be designed using 𝐾𝑧 = −𝑅
−1𝐵𝑧

𝑇𝑃, by 
assuming R > 0 and Q ≥ 0, P is the solution of the 

following algebraic Ricatti equation:  

𝑄 + 𝐴𝑧
𝑇𝑃 + 𝑃𝐴𝑧 − 𝑃𝐵𝑧𝑅

−1𝐵𝑧
𝑇𝑃 = 0  (12) 

Such a feedback controller minimizes the following 

performance index: 

𝐽 = ∫ (𝑥𝑧(𝑡)
𝑇𝑄𝑥𝑧(𝑡) + 𝑢𝑐(𝑡)

𝑇𝑅𝑢𝑐(𝑡))
∞

0
𝑑𝑡  (13) 

The closed-loop system using LQI control with 

reference input is described by the augmented state 

equation that is obtained from:  

[
�̇�𝑣
�̇�𝑖

] = [
𝐴𝑣 − 𝐵𝑣𝐾𝑧 0

−𝐶𝑣 0
] [

𝑥𝑣
𝑥𝑖

]  (14) 

2) LQI control with order-4 observer 

The LQI control with an order-4 observer is 

designed with the assumption that it has one state 

variable which can be directly measured (𝑥1(𝑡)) and 
three state variables, (𝑥2(𝑡), 𝑥3(𝑡) and 𝑥4(𝑡)), are not 

 

 

Figure 2. The LQI control design [13] 

 



R. Ristiana et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 9 (2018) 89–100 

 

92 

directly measurable. Figure 3 shows the LQI control 

system with an order-4 observer. In (7) the state 

variable 𝑥5(𝑡)  is dependent on the state variable 
𝑥2(𝑡). Therefore, the state variable 𝑥5(𝑡)  is ignored 
during observer design. Equation (7) can be expressed 

as follows. 

�̇�𝑎(𝑡) = 𝐴𝑎𝑥𝑎(𝑡) + 𝐹𝑎𝑥5(𝑡) + 𝐵𝑎𝑢𝑐(𝑡)   

𝑦𝑎(𝑡) = 𝐶𝑎𝑥𝑎(𝑡) (15) 

where 𝑥𝑎(𝑡) = [𝑥1(𝑡) 𝑥2(𝑡) 𝑥3(𝑡) 𝑥4(𝑡)]
𝑇, 

𝐴𝑎 = [

𝑎11 𝑎12 0 0
𝑎21 𝑎22 𝑎23 𝑎24
0 𝑎32 𝑎33 0
0 𝑎42 0 𝑎44

] , 𝐹𝑎 = [

0
𝑎25
0
0

] ,  

𝐵𝑎 = [0 𝑏2 0 0]
𝑇, and  

𝐶𝑎 = [𝑐1 0 0 0]. 

The state space equation for state variable 𝑥5(𝑡)  is 
given by (16). 

�̇�5(𝑡) = 𝐴5𝑎𝑥𝑎(𝑡) + 𝐴5𝑏𝑥5(𝑡) + 𝑏5𝑢𝑐(𝑡)  (16) 

where:  

𝐴5𝑎 = [0 𝑎52 0 0],  

𝐴5𝑏 = [0], and  

𝑏5 = 0.  

State space equation of the order-4 observer is given 

by (17).  

�̇̃�𝑎(𝑡) = (𝐴𝑎 − 𝐿𝑎𝐶𝑎)�̃�𝑎(𝑡) + 𝐹𝑎𝑥5(𝑡) + 𝐵𝑎𝑢𝑐(𝑡) +
𝐿𝑎𝑦𝑎(𝑡)  (17) 

State estimation error is given by (18).  

𝑒𝑎(𝑡) = 𝑥𝑎(𝑡) − �̂�𝑎(𝑡)  (18) 

Therefore, the following equation holds. 

�̇�𝑎(𝑡) = �̇�𝑎(𝑡) − �̇̂�𝑎(𝑡)  (19) 

By substituting (16) and (17) into (19), the following 

equation is obtained.  

�̇�𝑎(𝑡) = (𝐴𝑎 − 𝐿𝑎𝐶𝑎)𝑒𝑎(𝑡)  (20) 

The state feedback control based on the observed 

state �̂�𝑎(𝑡) is: 

𝑢𝑐(𝑡) = −𝑘𝑎�̂�𝑎(𝑡) − 𝑘𝑏𝑥5 − 𝑘𝑖𝑥𝑖(𝑡)  (21) 

By substituting (20) into (16), the following equation is 

obtained. 

�̇�𝑎 = (𝐴𝑎 − 𝐵𝑎𝑘𝑎)𝑥𝑎 + (𝐹𝑎 − 𝐵𝑎𝑘𝑏)𝑥5 +
𝐵𝑎𝑘𝑎𝑒𝑎(𝑡) − 𝐵𝑎𝑘𝑖𝑥𝑖(𝑡)  (22) 

From (9), (20), and (22), the system using the LQI 

control with the order-4 observer and using the 

assumption that the system has a reference input, can 

be described by the following augmented state equation. 

[

�̇�𝑎
�̇�𝑎
�̇�𝑖

] = [
𝐴𝑎 − 𝐵𝑎𝑘𝑎 𝐵𝑎𝑘𝑎 𝐵𝑎𝑘𝑖

0 𝐴𝑎 − 𝐿𝑎𝐶𝑎 0
−𝐶𝑎 0 0

] [

𝑥𝑎
𝑒𝑎
𝑥𝑖

] +

[
𝐹𝑎 − 𝐵𝑎𝑘𝑏

0
0

] 𝑥5  (23) 

where �̂�𝑎(𝑡) is the observer state variable, 𝐶1�̂�𝑎(𝑡) is 
estimated output, 𝑦𝑎(𝑡) is the system output, 𝑢𝑐(𝑡) is 
control variable, and 𝐿𝑎  is the Luenberger observer 
gain matrix. 

 

Figure 3. LQI control with order-4 observer 



R. Ristiana et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 9 (2018) 89–100 93 

3) LQI control with order-5 observer 

The LQI control with an order-5 observer is 

designed with the assumption that it has one state 

variable which can be directly measured (𝑥1(𝑡)), three 
state variables, ( 𝑥2(𝑡) , 𝑥3(𝑡)  and  𝑥4(𝑡) ), are not 
directly measurable, and one state variable 𝑥5(𝑡)  is 
unobservable. Figure 4 shows the LQI control system 

with an order-5 observer. In (7) the state variable 𝑥5(𝑡) 
is an integral of state variable 𝑥2(𝑡). Therefore, in order 
to make the system be observable, 𝑥5(𝑡)is used as an 
additional output. Equation (7) can be expressed as 

follows. 

�̇�𝑣(𝑡) = 𝐴𝑣𝑥𝑣(𝑡) + 𝐵𝑣𝑢𝑐(𝑡)  

𝑦𝑏(𝑡) = 𝐶𝑏𝑥𝑣(𝑡)  (24) 

where:  

𝑥𝑣(𝑡) = [𝑥1 𝑥2 𝑥3 𝑥4 𝑥5]
𝑇,  

𝑦𝑏(𝑡) = [𝑦𝑣 𝑦𝑤]
𝑇,  

𝑥𝑏(𝑡) = [𝑥𝑣 𝑥5]
𝑇, 

𝐴𝑣 =

[
 
 
 
 
𝑎11 𝑎12 0 0 0
𝑎21 𝑎22 𝑎23 𝑎24 𝑎25
0 𝑎32 𝑎33 0 0
0 𝑎42 0 𝑎44 0
0 𝑎52 0 0 0 ]

 
 
 
 

 ,  

𝐵𝑣 =

[
 
 
 
 
0
𝑏2
0
0
0 ]

 
 
 
 

 , 

𝐶𝑏 = [𝐶𝑣 𝐶𝑤]
𝑇,  

𝐶𝑣 = [𝑐1 0 0 0 0], and  

𝐶𝑤 = [0 0 0 0 1]. 

 

State space equation of the order-5 observer is given 

by: 

�̇̃�𝑣(𝑡) = (𝐴𝑣 − 𝐿𝑣(𝐶𝑣 + 𝐶𝑤))�̃�𝑣(𝑡) + 𝐵𝑣𝑢𝑐(𝑡) +

𝐿𝑣(𝑦𝑣(𝑡) + 𝑦𝑤(𝑡))  (25) 

State estimation error is given by (26). 

𝑒𝑣(𝑡) = 𝑥𝑣(𝑡) − �̂�𝑣(𝑡)  (26) 

Thus, the following equation holds.  

�̇�𝑣(𝑡) = �̇�𝑣(𝑡) − �̇̂�𝑣(𝑡)  (27) 

By substituting (24) and (25) into (27), the 

following equation is obtained. 

�̇�𝑣(𝑡) = (𝐴𝑣 − 𝐿𝑣(𝐶𝑣 + 𝐶𝑤))𝑒𝑣(𝑡)  (28) 

The state feedback control based on the observed 

state �̃�𝑣(𝑡) is: 

𝑢𝑐(𝑡) = −𝑘𝑤�̃�𝑣(𝑡) − 𝑘𝑖𝑥𝑖(𝑡)  (29) 

By substituting (29) into (24), the following equation is 

obtained. 

�̇�𝑣(𝑡) = (𝐴𝑣 − 𝐵𝑣𝑘𝑤)𝑥𝑣(𝑡) − 𝐵𝑣𝑘𝑤𝑒𝑣(𝑡) −
𝐵𝑣𝑘𝑖𝑥𝑖(𝑡) (30) 

From (9), (28), and (30), the system using the LQI 

control with the order-5 observer, and using the 

assumption that the system has a reference input, can 

be described by the following augmented state  

equation. 

[

�̇�𝑣
�̇�𝑣
�̇�𝑖

] = [

𝐴𝑣 − 𝐵𝑣𝑘𝑤 𝐵𝑣𝑘𝑤 𝐵𝑣𝑘𝑖
0 𝐴𝑣 − 𝐿𝑣(𝐶𝑣 + 𝐶𝑤) 0

−(𝐶𝑣 + 𝐶𝑤) 0 0
] [

𝑥𝑣
𝑒𝑣
𝑥𝑖

] 

 (31) 

where �̂�𝑣(𝑡) is the observer state variable, 𝐶𝑏�̂�𝑣(𝑡) is 
estimated output, 𝑦𝑏(𝑡) is the system output, 𝑢𝑐(𝑡) is 
control variable, and 𝐿𝑣  is the Luenberger observer 
gain matrix. 

 

 

Figure 4. LQI control with order-5 observer  



R. Ristiana et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 9 (2018) 89–100 

 

94 

III. Results and discussions 

A. Model parameter Molina 

The model parameters were taken from an 

experimental electric vehicle called Molina ITB Type-

3 where the specifications can be seen in Table 1. This 

vehicle was designed as a passenger minibus for public 

transport with 1500 kg weight and a wheel diameter of 

58 cm. The used electric motor is a brushless DC 

(BLDC) electric motor with an input voltage of 48 V, 

10 kW of power, 3500 rpm of motor speed rate, and 

120 A of motor current. Meanwhile, the used power 

supply consisted of two 24 V lithium-ion batteries 

installed in series. Each battery had a normal capacity 

of 100 Ah. 

B. Linearized integrated model 

For 24 V input voltage, a linearized integrated 

model was obtained at operating point xT = [m im Vc1 

Vc2 SOCn]
T = [1721 147.4 0.15 0.15 99.96]T. By 

ignoring 𝑑𝐿 in (7), the linearized integrated model (8) 
is in the following form: 

𝐴 =

[
 
 
 
 
−0.402 1603.77 0 0 0
−0.019 −3.941 −0.003 −0.003 −0.0002

0 294.118 −0.291 0 0
0 294.118 0 −0.291 0
0 294.118 0 0 0 ]

 
 
 
 

, 

𝐵 = [0 0.9871 0 0 0]𝑇, 

𝐶 = [1.5305 0 0 0 0], and  

𝐷 = [0]. 

From these matrices, the poles of the open-loop 

system are given by -2.1710+5.3327i, -2.1710-5.3327i, 

-0.0001, -0.2912, -0.2907. The poles of the open-loop 

system can be placed at any desired location, which 

means that the system of the plant is stable. The system 

of the open-loop system is fully controllable (Av, Bv ) 
but it is not fully observable (Av, Cv ), where the system 
has an observability rank of four. It means that the 

system has one state variable that is not observable, i.e. 

SOCn, but the system is detectable. 

C. Cases of control design 

The various cases of the LQI control design were as 

follows: 

1) Case 1: LQI control 

The LQI control system is based on (9), the 

augmented state equation is given by (11), the 

performance index is using (13), the gain full state 

feedback is given by Kv = [0.0234 5.6992 0.0008 

0.0008 0.0015], and the gain integral is expressed in Ki 

= [-0.0316]. The weighting matrices of the LQI are 

chosen based on trial and error approach. In order to 

obtain the optimum state feedback control gains, the 

weighting matrices were selected as follows:  

Q = diag[0.1], and R = 100. 

A gain of state feedback that is defined by the 

eigenvalues of the system is necessarily needed to solve 

the problem. The eigenvalues of the closed-loop system 

in (14) are given as −4.884 + 7.007𝑖 , −4.884 −
7.007𝑖 , 0.224 + 0.104𝑖 , −0.224 − 0.104𝑖  , −0.044, 
and 0.291.  

Table 1. 

Parameter of Molina ITB Type-3 

Specifications Symbol Value Units 

Motor BLDC    

Resistance Rm 12.4  m 

Inductance Lm 34  uH 

Torque constant Kt 0.1082  Nm/A 

Inertia Jm 48×10-6  kgm2 

Stiffness bm 79×10-4  Nms/rad 

Bmf constant Ke 0.0128  Vs/rad 

Lithium-ion battery    

Inner resistance  2  m 

Terminal resistance,  Rt 1.72  m 

Terminal capacitance,  Ct 2000  F 

n-capacity,  Qn 100  Ah 

Vehicle    

Mass,  mv 1500  kg 

Wheel radius,  rw 0.29  m 

Wheel inertia,  Jw 12×10-6  kgm2 

Transmission Inertia,  Jt 53×10-6  kgm2 

Air density,   1.25  kg/m
3 

Drag coefficient,  Cd 0.417  Ns2/kgm 

Frontal area,  Af 1.581  m2 

Rolling coefficient,  Crx 0.015  

Gravity coefficient,  g 9.8  m/s2 

 

 



R. Ristiana et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 9 (2018) 89–100 95 

2) Case 2: LQI control with order-4 observer 

To provide a solution for Case 2, the partition state 

variables can be obtained using (15). The matrices are 

given as: 

𝐴𝑎 = [

−0.402 1603.77 0 0
−0.019 −3.941 −0.003 −0.003

0 294.118 −0.291 0
0 294.118 0 −0.291

],  

𝐹𝑎 = [

0
−0.0002

0
0

] , 

𝐵𝑎 = [

0
0.987

0
0

], 

𝐶𝑎 = [1.531 0 0 0]
𝑇, 

𝑘𝑎 = [0.023 5.699 0.0008 0.0008] ,  

𝑘5 = [−0.0316], 

𝑘𝑖 = [−0.0316], and 

𝐿𝑎 = [−0.4180 −0.0126 −0.330 −0.330]
𝑇.  

Based on (23), the eigenvalues of the closed-loop 

system are given as -1.804+9.754i, -1.804-9.754i,  

-0.782, -0.289, -0.291, -2.168+5.391, -2.168-5.391,  

-0.297, and -0.291. 

3) Case 3: LQI control with order-5 observer 

To provide a solution for Case 3, the partition state 

variables can be obtained using (24). The matrices are 

given by as: 

𝐴𝑣 =

[
 
 
 
 
−0.402 1603.77 0 0 0
−0.019 −3.941 −0.003 −0.003 −0.002

0 294.118 −0.291 0 0
0 294.118 0 −0.291 0
0 294.118 0 0 0 ]

 
 
 
 

, 

𝐵𝑣 = [0 0.9871 0 0 0]
𝑇, 

𝐶𝑣 = [1.5305 0 0 0 0], 

𝐶𝑤 = [0 0 0 0 1], 

𝐾𝑤 = [0.0234 5.699 0.0008 0.0008 0.0015], 

𝐾𝑖 = [−0.0423], and 

𝐿𝑤 = [−0.008 −0.003 −0.007 −0.007 −0.003]
𝑇. 

Based on (36), the poles or eigenvalues of the 

closed-loop system are given as -4.944+6.941i, -4.944-

6.941i, -0.0393+0.042i, -0.0.393-0.042i, -0.292,  

-2.164+5.265i, -2.164-5.265i, -0.002, -0.292, -0.291 

and -0.291. 

All the eigenvalues of the closed-loop system and 

the observers must be negative. Theoretically, these 

eigenvalues can be arbitrarily moved to minus infinity 

to achieve extremely fast convergence. The problem of 

selecting good eigenvalues is not easily solved. 

However, the observer may be slightly faster than the 

rest of the closed-loop system.  

Generally, the formula is defined with 2 to 6 times 

larger poles for the observer than for the closed-loop 

systems’ poles. This can increase the noise on the 

observer side. In this case, the poles were set 5 times 

larger for the observer than for the closed-loop system. 

This means that the observer may be slightly faster than 

the closed-loop system and the observation error 

decays shortly to zero.  

Initial condition values influence the state variables 

values forward through time. In other words, the state 

variables are a function of time and the initial condition 

values. The initial state variables values were selected 

as x(0) = [1 0 0 0 0]T.  

Based on Figure 5, in which the response to state 

variables versus time is shown, all state variables were 

defined. The state variables were: 𝑥1 = 𝜔𝑚, 𝑥2 = 𝑖𝑚, 
𝑥3 = 𝑉𝑐1, 𝑥4 = 𝑉𝑐2, 𝑥5 = 𝑆𝑂𝐶𝑛 , and 𝑥𝑖  is the integral 
state. For all cases of the control design, it can be seen 

that the motor speed response (𝑥1)  and the motor 
current response (𝑥2) were the same, whereas 𝑥3, 𝑥4, 
𝑥5 and 𝑥𝑖 had a different response. It can be seen that 
𝑥3 and 𝑥4 had the same response in Case 1 (red line) 
and Case 3 (black line), and reached steady state after 

3 seconds, so that Case 2 (green line) reached steady 

state after 4 seconds.  

Also, 𝑥𝑖  was the same in Case 1 and Case 2, and 
reached steady state after 6 seconds. This was also the 

case in Case 3, reaching a steady state after 1 seconds, 

which means faster than Case 1 and Case 2 by around 

5 seconds. However, for 𝑥5 , Case 2 had undershoot, 
while it reached steady state in the same time as Case 2, 

i.e., after 6 seconds. Case 3 had the best response, 

reaching a steady state after 2.6 seconds. This means 

that Case 3 had unexploited battery energy.  

To obtain the response of the observer error vector 

to the following initial observer error e(0) = [1 0 0 0]T. 

The response to state estimate versus time with the 

initial observer error is shown in Figure 6. The error 

was happened just for Case 2 and Case 3, while there is 

no error for Case 1 because Case 1 is designed without 

any observer. The state estimate in Case 2 (red line) was 

𝑒1 = �̃�𝑚, 𝑒2 = 𝑖̃𝑚, 𝑒3 = �̃�𝑐1, and 𝑒4 = �̃�𝑐2. In Case 3 
(blue line) it was 𝑒1 = �̃�𝑚 , 𝑒2 = 𝑖̃𝑚 , 𝑒3 = �̃�𝑐1 , and 
𝑒4 = �̃�𝑐2 and 𝑒5 = 𝑆𝑂�̃�𝑛 . 

The response of Case 3 is the fastest, which means 

that the observer has the same structure as the system, 

with a feedback driving term where the observation 

error decays shortly to zero. This means that Case 3 had 

the best observer error response. 

D. Energy consumption 

The purpose of this simulation was to see how the 

use of a BEV model combined with the observer in the 

speed control design influences the energy 

consumption of the electric vehicle. An electric vehicle 

was simulated using a small-scale simulator, and the 

energy usage for a certain driving profile was presented 

in [14].  

In this part of work, the energy consumption can be 

observed in two ways. First, the vehicle moves on a flat 

surface with a constant vehicle speed of 60 km/h in the 

 



R. Ristiana et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 9 (2018) 89–100 

 

96 

simulation, and second, a simulation was performed 

according to the standard NEDC (a new European 

driving cycle) driving profile. The NEDC is a test 

procedure as long as the vehicle moves at a speed 

profile. The speed profile has a major impact on the 

resulting energy consumption [15]. 

The formulation of the various performance index 

to observe the energy consumption was based on the 

following characteristics: 

 

• Control energy  

𝐸1 = ∫ 𝑉𝑚(𝑡)
2∞

0
𝑑𝑡 or 𝐽1 = ∫ 𝑢𝑐

2∞

0
𝑑𝑡 

• Mechanical energy  

𝐸2 = ∫ 𝑇𝑚(𝑡)𝜔𝑚(𝑡)
∞

0
𝑑𝑡 or 𝐽2 = ∫ 𝑥2𝑥1

∞

0
𝑑𝑡  

• Motor energy input  

𝐸3 = ∫ 𝑉𝑚(𝑡)𝐼𝑚(𝑡)
∞

0
𝑑𝑡 or 𝐽3 = ∫ 𝑢𝑐𝑥2

∞

0
𝑑𝑡 

 

 

 

 

 

 

Figure 5. Initial condition response (state variable versus time) 



R. Ristiana et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 9 (2018) 89–100 97 

1) Constant vehicle speed 

In this simulation, the vehicle was moving on a flat 

surface with a constant speed at 60 km/h for 15 seconds 

duration. In Figure 7, it was shown that the motor speed 

reached 3000 rpm, and control signal about 41 V with 

the same response for all cases. However, it was also 

shown that all three cases had different time settling. In 

Case 1, it was a faster settling time, while in Case 2, it 

was a slower settling time. The response of the motor 

current showed the same transient response. This 

means that if the motor current has different values for 

reaching 3000 rpm or 60 km/h, it has an effect on 

energy consumption. The energy consumption was 

presented by J1, J2, and J3.  

 

 

 

 

 

Figure 6. The error of observer response (state observer versus time) 

Table 2. 

Energy consumption 

State  

feedback 

Energy consumption (Watt-hour) 

J1 J2 J3 

Constant Vehicle Speed at 60 km/h (during 15 seconds) 

Case 1 0.798×103 2.205×10
3 2.796×103 

Case 2 0.701×103 1.944×103 2.465×103 

Case 3 0.626×103 1.732×103 2.196×103 

NEDC Profile (during 1200 seconds) 

Case 1 1.223×103 5.025×10
3 6.369×103 

Case 2 1.061×103 4.396×10
3 5.528×103 

Case 3 0.964×103 3.964×10
3 5.020×103 

 



R. Ristiana et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 9 (2018) 89–100 

 

98 

In Table 2, it was shown that the energy 

consumption in Case 3 was 27.45% (J1), 27.27% (J2), 

and 27.34% (J3) better than in Case 2. The energy 

consumption in Case 3 also showed 12.04%, 12.21% 

and 12.24%, for J1, J2, and J3 respectively, which were 

better than in Case 1. This result means that the energy 

consumption in Case 3 was the most efficient out of 

these three cases. 

2) NEDC driving profile 

A simulation was performed on the moving vehicle 

according to the NEDC driving profile for 1200 

seconds. The simulation result can be seen in Table 2 

where the energy consumption for the vehicle using 

NEDC profile in Case 3 was 21.17% (J1), 21.12% (J2) 

and 21.18% (J3) better than in Case 2. The energy 

consumption in Case 3 also showed 10.04%, 10.09% 

 

 

(a) 

 

 

(b) 

 

 

(c) 

 

Figure 7. Response system when the vehicle moved; (a) motor speed response; (b) control signal response; (c) current response 



R. Ristiana et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 9 (2018) 89–100 99 

and 10.12% better than in Case 1 for J1, J2, and J3 

respectively. This result means that the energy 

consumption in Case 3 is the most efficient out of these 

three cases.  

IV. Conclusion 

 Optimal speed control with observer applied to an 

integrated battery-electric vehicle (IBEV) model was 

presented. An LQI control design was used for the 

feedback control design, and a Luenberger observer 

was used to design the observer. In the design of the 

observer, it was assumed that there was one indirectly 

measurable and unobservable state variable in the 

system that was used to build the LQI control with 

order-5 observer. For comparison, an LQI control only 

and an LQI control with order-4 observer were also 

designed. All control design cases simulated a vehicle 

moving on a flat surface and moving according to the 

NEDC driving profile. The LQI control with order-5 

observer (Case 3) provided the highest energy 

efficiency. Moreover, the transient response in Case 3 

was  slightly faster than in Case 2. An optimal speed 

control design with observer was shown to have the 

potential to provide higher energy efficiency for 

integrated battery-electric vehicles. Its application is 

currently under further research. 

Acknowledgement 

The authors would like to acknowledge the support 

of LPNK scholarship given to the first author by the 

Indonesian Ministry of Research, Technology, and 

Higher Education. The valuable comments from the 

reviewers and also from the scholars of the School of 

Electrical Engineering and Informatics, Institut 

Teknologi Bandung, Indonesia are also very much 

appreciated. 

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