APPLICATION OF DIGITAL CELLULAR RADIO FOR MOBILE LOCATION ESTIMATION


IIUM Engineering Journal, Vol. 24, No. 1, 2023 Zainuddin et al. 
https://doi.org/10.31436/iiumej.v24i1.2341 

PERFORMANCE ANALYSIS OF PREDICTIVE 

FUNCTIONAL CONTROL FOR AUTOMOBILE ADAPTIVE 

CRUISE CONTROL SYSTEM 

MOHAMED AL-SIDEQUE ZAINUDDIN1,2, MUHAMMAD ABDULLAH1*,

SALMIAH AHMAD1,  MOHD SUHAIMI UZAIR3  

AND ZAID MUJAIYID PUTRA AHMAD BAIDOWI4

 
1
Department of Mechanical and Aerospace Engineering,  

International Islamic University Malaysia, Jalan Gombak, 53100, Kuala Lumpur, Malaysia 
2
Department of Automotive Engineering Technology, Kolej Kemahiran Tinggi MARA,  

Masjid Tanah, Melaka, Malaysia     
3
Testing & Development, Engine Development, Powertrain R&D, Powertrain Division, 

Proton Holdings Berhad, Shah Alam, Malaysia  
4Centre of Foundation Studies, Universiti Teknologi MARA, Cawangan Selangor,  

Kampus Dengkil, 43800 Dengkil, Selangor, Malaysia 

*
Corresponding author: mohd_abdl@iium.edu.my 

(Received: 28th February 2022; Accepted: 21st June 2022; Published on-line: 4th January 2023) 

ABSTRACT: This paper presents the performance analysis of Predictive Functional 

Control (PFC) for Adaptive Cruise Control (ACC) application. To cope with multiple 

driving objectives of modern ACC systems such as passenger comfort, safe distancing, 

and fast time response, an advanced optimal controller such as Model Predictive Control 

(MPC) is often used. Nevertheless, MPC requires a high computation load due to its 

complex formulation and may overload the processing power of a microcontroller. Thus, 

the prime objective of this work is to propose a PFC algorithm as an alternative 

controller, while providing a formal comparison between MPC and the traditional 

Proportional Integral (PI) controller. A standard kinematic model for vehicle longitudinal 

dynamics was modelled and used to derive the control law of PFC. Since the open-loop 

dynamic of the derived transfer function is not stable, the second objective is to propose 

a pre-stabilized loop or cascade PFC structure for the system. A complete tuning 

procedure and analysis were presented. The simulation result shows that although MPC 

performance is the best for the ACC application with Root Mean Square Error (RMSE) 

of 1.4873, PFC has shown a promising response with RMSE of 1.5501, which is better 

compared to the PI controller with RMSE of 1.6219. All the imposed driving constraints 

such as maximum acceleration, maximum deceleration and safe distance were satisfied 

in the car following application. Thus, the findings from this work can become a good 

initial motivation to further explore the capability of the PFC algorithm for future ACC 

development.   

ABSTRAK: Kajian ini adalah berkenaan analisis prestasi Kawalan Fungsi Ramalan 

(PFC) aplikasi Kawalan Mudah Suai (ACC). Bagi memenuhi pelbagai keperluan objektif 

sistem pemanduan moden ACC seperti keselesaan penumpang, penjarakan selamat dan 

tindak balas pantas, kawalan optimum terbaru seperti Model Kawalan Ramalan (MPC) 

sering digunakan. Walau bagaimanapun, MPC memerlukan beban pengiraan tinggi 

kerana rumusnya yang kompleks dan mungkin mengakibatkan beban berlebihan kuasa 

pemprosesan mikrokawalan. Oleh itu, matlamat utama kajian ini adalah bagi 

mencadangkan algoritma PFC yang mempunyai pengiraan mudah sebagai kawalan 

alternatif, sementara menyediakan perbandingan formal antara MPC dan kawalan 

213



IIUM Engineering Journal, Vol. 24, No. 1, 2023 Zainuddin et al. 
https://doi.org/10.31436/iiumej.v24i1.2341 

 

 

tradisional Berkadar Keseluruhan (PI). Oleh kerana model ini tidak stabil, objektif kedua 

adalah mencadangkan penggunaan struktur PFC berlapis bagi menstabilkan sistem 

terlebih dahulu sebelum algorithma kawalan digunakan atau dengan menggunakan 

struktur PFC secara berturut pada sistem. Prosedur lengkap dan terperinci untuk analisis 

PFC dibentangkan. Dapatan simulasi kajian menunjukkan walaupun prestasi MPC 

adalah baik bagi aplikasi ACC dengan Ralat Punca Min Kuasa Dua (RMSE) bernilai 

1.4873, namun PFC menunjukkan tindak balas baik dengan RMSE bernilai 1.5501 

berbanding kawalan PI yang mempunyai RMSE sebanyak 1.6219. Kesemua kekangan 

seperti pecutan dan nyahpecutan maksima, dan penjarakan selamat bertepatan dengan 

aplikasi kenderaan ini. Dengan itu, penemuan ini adalah motivasi awal yang baik bagi 

meneroka lebih jauh keupayaan algoritma PFC bagi membangun ACC pada masa 

hadapan. 

KEY WORDS: predictive functional control; model predictive control; PID; adaptive 

cruise control  

1. INTRODUCTION  

Adaptive Cruise Control (ACC) is one of the basic features for an Advanced Driving 

Assistant System (ADAS), where its function is to regulate the speed of a vehicle while 

retaining a safe following distance. Compared to the conventional Cruise Control system, 

ACC has two modes of operation: speed control and space control. With the advancement 

of sensor and microprocessor technologies, an energy efficient ACC system which can 

satisfy multiple driving objectives such as passengers’ comfort, fuel efficiency, and safe 

distancing with acceptable time response are currently being developed. This type of ACC 

system requires a more sophisticated control algorithm and structure as each of the driving 

objective functions needs to be optimized to get the optimum control action. It is also well 

noted that the vehicle longitudinal dynamics for full range operation is highly nonlinear 

and thus a hierarchal control structure is required where the upper-level system is 

controlled to provide a desired signal for the lower-level system to track [1]. At the same 

time the switching between space and speed control also needs to be considered to ensure 

the safety and comfort of the passenger.  

With a traditional PI controller, a special switching algorithm is used to ensure a 

smooth transition between the two modes to avoid jerking [2]. This operation can be done 

with the help of a look up table. Nevertheless, the control performance is not robust since 

the decision making is solely based on the current measurement. To improve the 

robustness, a proper tuning strategy can be implemented, as demonstrated by [3]. 

However, the control structure has become more complicated and there is no systematic 

tuning approach for this type of modification leaving only trial and error or use of some 

optimal algorithm tuning scheme. Several works also have proposed the use of Fuzzy 

Logic Controller (FLC) for an ACC system. Using a common logic rule, the switching 

strategy is developed based on the relative distance with the lead vehicle. The strategy is 

quite straightforward and implantable, yet as reported by [4], in the presence of 

unmeasured disturbance, the control performance will be deteriorated. Indeed, there are 

several options to overcome these issues such as improving the accuracy of the fuzzy rule 

by using a different logic function or combining it with other advanced controllers [5]. 

However, there is no systematic rule that can be implemented as there are many 

parameters that need to be tuned and most of the modifications are system dependent.     

A predictive controller is another suitable option to control the upper level of an ACC 

system. A representative kinematic mathematical model is used to predict the future 

214



IIUM Engineering Journal, Vol. 24, No. 1, 2023 Zainuddin et al. 
https://doi.org/10.31436/iiumej.v24i1.2341 

 

 

velocity and the input acceleration will be optimized to get the desired response while 

satisfying other driving constraints such as passenger’s comfort, fuel efficiency, and safe 

distancing. As reported in the literature, various authors have managed to prove that this 

algorithm can effectively control the vehicle speed in satisfying the multiple driving 

objectives either in simulation or real-time implementation [6-8].  The main reason behind 

this is the prediction capability where the controller can anticipate what will happen in the 

future and try to provide the best solution. Hence, no special switching strategy is required 

and it can be replaced by a more systematic optimization problem. Indeed, there are also 

several limitations such as a proper selection of the tuning parameters of prediction and 

control horizons based on certain applications. However, even with suitable parameters, 

the computation burden will still be heavy to implement in existing vehicle 

microprocessors. In a short sampling time i.e., 0.1 s, the algorithm needs to compute 

several heavy mathematical operations such as prediction, optimization, and constraints 

handling using a standard quadratic cost function. Of course, there are several options to 

reduce the computation such as offline implementation [9] and changing the prediction 

structures using a special function [7,10], yet still the basic microprocessor in existing 

vehicles needs to be upgraded. This requirement will increase the cost and, in the future, 

full implementation of autonomous vehicles would further lead to a greater computation 

demands.  

Due to the above reasons, this work intended to propose an alternative low 

computation predictive controller known as Predictive Functional Control (PFC), which is 

novel in the ACC application. In general, the algorithm principle is the same, but the 

optimization process is simplified without the use of the quadratic cost function. In return, 

the computation time can be reduced significantly [11]. The PFC algorithm has also been 

implemented in other engineering fields ranging from chemical, mechanical, and electrical 

industries [12,13]. However, there is a tradeoff in using PFC, where the optimality of the 

control solution will be reduced. This is a well-known tradeoff but for a Single Input 

Single Output (SISO) application such as an ACC system, it may become a good 

alternative strategy. Besides, there are also a number of works that have improved the 

existing algorithm such as [14,15]. Thus, the main objective of this work is to investigate 

and provide a formal performance comparison of PFC with MPC and a traditional PI 

controller to assess its capability in a vehicle following application.  

Another important issue that needs to be highlighted is the application of PFC for 

marginally stable processes or type 1 processes where there will be an independent 

integrator in the transfer function’s denominator. Since the input to the ACC system is 

acceleration and the output is speed, if a step input is given, the output will not converge to 

a steady state value. Thus, when a conventional PFC algorithm is implemented, the control 

solution will be ill-posed, where the prediction and actual control performance will differ 

[16]. To cope with this issue, a cascade PFC with feedback compensator will be used 

where the prediction structure is pre-stabilized before implementing the conventional 

algorithm [17]. A proper analysis on the tuning process will be discussed and denoted as 

one of the contributions of this works. 

2. METHODOLOGY  

2.1  Vehicle Longitudinal Dynamics Modelling 

The mathematical model is the most important part of predictive control formulation. 

In general, the inputs to the system will be the pedal pressure from the throttle and brake, 

while the output of interest will be velocity. In a traditional cruise control system, the 

215



IIUM Engineering Journal, Vol. 24, No. 1, 2023 Zainuddin et al. 
https://doi.org/10.31436/iiumej.v24i1.2341 

 

 

braking effect is not considered, thus the standard kinetic vehicle longitudinal model can 

be used straight away to derive the control law of PFC as presented in [18]. However, the 

ACC system needs braking input as well to keep the safe distancing with a lead vehicle. If 

a full kinetic modelling is used to derive the control law as in [18], the constraint 

implementation will be quite complicated especially if it is designed for full range 

dynamic operation where the nonlinearities will be very high due to the changing gear 

ratio and coefficient of friction between road and tire [2]. Thus, a hierarchal control 

strategy is often proposed in the literature [5,6] as shown in Fig. 1.  

 
Fig. 1: Hierarchical structure for vehicle longitudinal dynamics. 

In the hierarchal structure, the throttle and brake actuator correspond to powertrain 

and brake line dynamics respectively. As for the lower-level dynamic, it is responsible for 

the relationship between traction force 𝐹𝑡 and acceleration, 𝑎, by Newton’s second law. 
The details of these models are available in many references including academic textbooks 

[2]. Since the lower-level model is nonlinear in nature, the PFC controller will be derived 

based on the upper-level system that represents the kinematic dynamic between 

acceleration and velocity where the transfer function is given as: 

𝑣(𝑠) =
1

𝑠(𝜏𝑠+1)
𝑎(𝑠) (1) 

The additional time constant 𝜏 in Eq. (1) corresponds to the estimation of the first 
order lag of the lower-level controller [2]. In this case, it is expected that the car will track 

the velocity imperfectly and thus the nominal time constant 𝜏 is approximately equal to 0.5 
s [2]. In this work, it is assumed that the lower-level controller can retain the nominal 

value in every operating speed, although in reality, it will be changed due to nonlinearities, 

especially for low-speed operation where the gear ratio is not constant. The investigation 

of the impact of nonlinearities on the overall control performance of an ACC system and 

the possibility of using other robust control methods will denote future work.   

2.2  Cascade PFC Prediction  

Since the PFC algorithm works in the digital domain, the transfer function in Eq. (1) 

needs to be discretized a with sampling time of 0.1 s [6]. From here on, a standard symbol 

for input 𝑢 will be used for acceleration 𝑎 and output 𝑦 for velocity 𝑣. The discrete 
transfer function can be represented as: 

𝐺(𝑧) =
𝑦(𝑧)

𝑢(𝑧)
=

0.009365𝑧 −1+0.008762𝑧−2

1−1.819𝑧−1+0.819𝑧−2
 (2) 

Another important part that needs to be considered is the nature of the transfer 

function in Eq. (1) and Eq. (2), where it is a type 1 system with independent integrator in 

the denominator. Since the output response will not converge when given a bounded input, 

it needs to be pre-stabilized before deriving the control law. If not, it will lead to an ill-

posed solution as reported in the literature [14]. There are two ways to overcome this issue 

[14,17], one is by splitting Eq. (2) into two separate transfer functions that run in parallel, 

one using partial fractions and the other using a proportional gain, 𝐾, to stabilize the 

216



IIUM Engineering Journal, Vol. 24, No. 1, 2023 Zainuddin et al. 
https://doi.org/10.31436/iiumej.v24i1.2341 

 

 

system as shown in Fig. 2. This work utilizes the second option since only a minor 

modification in the computed input is required when implementing the control law.  

 

Fig. 2: Proportional feedback loop for plant pre-stabilization. 

The inner loop 𝑇(𝑧) will be used as a prediction model instead of 𝐺(𝑧) as in Eq. (2) to 
get a stable response and it is computed as: 

𝑇(𝑧) =
𝑦(𝑧)

𝑥(𝑧)
=

𝐺(𝑧)𝐾

1+𝐺(𝑧)𝐾
 (3) 

where, 𝑥(𝑧) is the modified controlled input. A suitable value of proportional gain 𝐾 can 
be selected based on trial and error or by using a more systematic approach as presented in 

[17]. As in this work, the MATLAB PID tuner is utilized. Using the function 

pidtune(sys,type), it provides the proportional value of 𝐾 = 1.147. According to the 
function description in MATLAB help desk, the given value is computed based on the 

balance performance between response time and robustness.  

A linear prediction structure based on the superimposed principle for transfer function 

in Eq. (3) can be formed, and since the derivation is standard [23-26], only the final form 

is presented. The 𝑖-step ahead prediction at the k sampling is presented as:   

𝑦(𝑘 + 𝑖|𝑘) = 𝐻𝑋 + 𝑃𝑋0 + 𝑄𝑌  + 𝑑 (4) 

The dimension of matrix 𝐻, 𝑃, and 𝑄 depends on the model parameters, and for a standard 
second order transfer function, the parameters 𝑋, 𝑋0 and 𝑌 are: 

𝑋 = [

𝑥(𝑘)

𝑥(𝑘 + 1)
⋮

𝑥(𝑘 + 𝑖)

] , 𝑋0 = 𝑥(𝑘 − 1), 𝑌 = [
𝑦(𝑘)

𝑦(𝑘 − 1)
] 

The term 𝑑 in Eq. (4) corresponds the correction term to get an unbiased prediction using 
an independent model structure as shown in Fig. 3. Technically, it is the difference 

between the actual measured output form a plant 𝑦𝑝 and the calculated output 𝑦 from 𝑇(𝑧) 

in Fig. 2.  

𝑑 = 𝑦𝑝 − 𝑦 (5) 

 

Fig. 3: Independent model structure for PFC prediction. 

2.3  Cascade PFC Control Law  

217



IIUM Engineering Journal, Vol. 24, No. 1, 2023 Zainuddin et al. 
https://doi.org/10.31436/iiumej.v24i1.2341 

 

 

To compute the control input, the prediction in Eq. (4) at n-step ahead is forced to 

coincide with a first order setpoint trajectory r, which is given as [11]: 

𝑟(𝑘 + 𝑛|𝑘) = (1 − 𝜆𝑛)𝑅 + 𝜆𝑛𝑦𝑝(𝑘) (6) 

where R is the desired set point speed. In this stage, there are two tuning parameters that 

need to be specified. The first one is the coincidence horizon 𝑛, where the prediction in 
Eq. (4) is forced to match with the target trajectory in Eq. (6). The second one is 𝜆, which 
corresponds to the desired Closed-loop Time Response (time taken to reach 95 % form the 

steady state value): 

𝜆 = 𝑒−3𝑇𝑠/𝐶𝐿𝑇𝑅 (7) 

The term 𝑇𝑠 in Eq. (7) denotes the sampling time. For a second order system, the 
selection of n has a general tuning guideline as presented in [17]. It should be selected 

between 40% and 80% of the response to avoid prediction mismatch. Once all the tuning 

parameters have been selected, the general control law of PFC can be derived. Using its 

standard assumption that the future modified control input will be constant i.e., 𝑥(𝑘) =
𝑥(𝑘 + 1) = ⋯ . 𝑥(𝑘 + 𝑛) [25], the n-th row of matrix 𝐻𝑛 will reduce to a single value in 
the form of: 

ℎ𝑛 = 𝐻𝑛 [

1
1
⋮
1

] (8) 

Extracting the n-th prediction form each matrix and equating the predictions in Eq. (4) and 

Eq. (6) gives: 

ℎ𝑛 𝑥(𝑘) + 𝑃𝑛 𝑋0 + 𝑄𝑛𝑌  + 𝑑 = (1 − 𝜆
𝑛)𝑅 + 𝜆𝑛𝑦𝑝(𝑘) (9) 

The compensated input 𝑥 can be computed as: 

𝑥(𝑘) = ℎ𝑛
−1[(1 − 𝜆𝑛)𝑅 + 𝜆𝑛𝑦𝑝(𝑘) − 𝑃𝑛 𝑋0 − 𝑄𝑛𝑌  − 𝑑] (10) 

It should be noted that since a cascade structure is used to pre-stabilized the response 

as explained in Section 2.2. By referring to Fig. 4, the actual input that will be sent to the 

plant is computed as:  

𝑢(𝑘) = 𝐾[𝑥(𝑘) − 𝑦𝑝(𝑘)] (11) 

 

 

Fig. 4: Cascade PFC structure. 

2.4  Constraint Handling for Passenger Comfort and Safe Distancing 

For the predictive controller framework, the conventional switching strategy from 

speed mode to space mode can be implemented by formulating a constraint control 

problem. Fig. 5 shows the schematic diagram for vehicle following application and how 

the ACC controller is implemented.   

218



IIUM Engineering Journal, Vol. 24, No. 1, 2023 Zainuddin et al. 
https://doi.org/10.31436/iiumej.v24i1.2341 

 

 

 

Fig. 5: ACC schematic diagram for vehicle following application. 

With the help of a distance sensor such as LIDAR, a relative distance 𝐷𝑟  or relative 
velocity 𝑣𝑟  with a lead vehicle can be measured. These two parameters can be used to 
regulate the acceleration to retain a safe following distance if necessary. A standard safe 

following distance equation as recommended in [2] is used, where:  

𝐷𝑠𝑎𝑓𝑒 (𝑘) = 𝐷𝑑𝑒𝑓𝑎𝑢𝑙𝑡 + 𝑇𝑔𝑎𝑝𝑣(𝑘) (12) 

The default distance 𝐷𝑑𝑒𝑓𝑎𝑢𝑙𝑡 is set to 10 m, it means that if the velocity of vehicle is 0 

m/s (both lead and ego vehicle are not moving or approaching a complete stop), the 

relative distance should not be less than 𝐷𝑑𝑒𝑓𝑎𝑢𝑙𝑡. The safe time gap 𝑇𝑔𝑎𝑝 between the 

vehicle is set to 1.4 s. Based on Eq. (12), the higher the velocity of a vehicle, the larger the 

safe distance needed to be maintained because it will take more time to slow down the car. 

Based on the output velocity prediction in Eq. (4), the future relative position between the 

car 𝐷𝑟  can be estimated by assuming the future velocity of the lead car is constant at the 
instantaneous sampling  

𝐷𝑟 (𝑘 + 1) = [𝑣𝑙   −  𝑣(𝑘)]𝑇𝑠 + 𝐷𝑟 (𝑘) (13) 

Using superposition, at each sampling time, the maximum velocity to keep the safe 

distance can be formed as: 

𝑣𝑚𝑎𝑥 = (𝑣𝑙 𝑇𝑠 + 𝐷𝑟 (𝑘)– 𝐷𝑑𝑒𝑓𝑎𝑢𝑙𝑡 )/(𝑇𝑔𝑎𝑝 + 𝑇𝑠) (14) 

Then a normal PFC output constraints formulation can be implemented by selecting a 

suitable validation horizon 𝑛𝑖 . The algorithm is given as below:  

Algorithm (A). At each sample time: 

(1) Compute the unconstrained compensated input 𝑥 as in Eq. (10). 

(2) A simple ‘for’ loop is used to check the constraint violation while updating Eq. 
(13) and Eq. (14) 

(3) If 𝑣 > 𝑣𝑚𝑎𝑥 , then: 

𝑥(𝑘) = 𝐻𝐿𝑖
−1[𝑣𝑚𝑎𝑥 (𝑖) − 𝑃𝑖 𝑈0 − 𝑄𝑖 𝑌  − 𝑑] (15) 

(4) Else, the value of 𝑥 is retained.   

For taking care of passenger comfort and fuel efficiency, the input acceleration is 

constrained in between 𝑢𝑚𝑖𝑛= -3 to 𝑢𝑚𝑎𝑥 = 2 m/s
2. As discussed in many PFC papers [16-

17] a simple clipping strategy is enough to cater for input constraint such that if 𝑥 <
𝑦(𝑘) + 𝑢𝑚𝑖𝑛 /𝐾, then:  

𝑥(𝑘) = 𝑦(𝑘) + 𝑢𝑚𝑖𝑛/𝐾 (16) 

Similarly, if 𝑥 > 𝑦(𝑘) + 𝑢𝑚𝑎𝑥 /𝐾, then:  

𝑥(𝑘) = 𝑦(𝑘) + 𝑢𝑚𝑎𝑥 /𝐾 (17) 

219



IIUM Engineering Journal, Vol. 24, No. 1, 2023 Zainuddin et al. 
https://doi.org/10.31436/iiumej.v24i1.2341 

 

 

3. SIMULATION RESULT  

3.1  Analysis of Tuning Parameter of PFC  

The first step in tuning the PFC is to select a suitable compensator gain to stabilize the 

open-loop behavior of the vehicle dynamics. Since the open-loop transfer function is type 

1 (with one integrator in the denominator), a single proportional gain is enough to stabilize 

the system. As discussed in Section 2.2, an autotune PI controller in MATLAB is used to 

find an optimum proportional gain 𝐾 = 1.147 that will provide a balanced performance 
between response time and robustness [17]. Figure 5 shows the open-loop response of the 

pre-stabilized system output 𝑦 against the desired output trajectory 𝑅 with CLTR of 5s (𝜆 
= 0.9418).  

 

Fig. 5: Pre-stabilized open-loop response compared to the desired trajectory. 

The objective of the PFC is to force the system to match the desired target trajectory 

at a specific coincidence horizon, 𝑛. In this stage, a proper coincidence horizon needs to be 
selected carefully. In general, a smaller coincidence horizon will provide more aggressive 

response and input, while higher coincidence horizon will be less aggressive and slow. 

The effect of coincidence horizon may be varied for different systems since the dynamics 

are different. Several coincidence horizons are simulated to track a desired velocity of 5 

m/s with 0.1 s sampling time, as shown in Fig. 6. 

 

  

Fig. 6: Closed-loop response with varying coincidence horizon (n). 

It can be observed that a smaller horizon will lead to slower response and a larger 

value will lead to faster response than the desired one (black dashed line). Based on the 

tuning guide proposed by [17], a suitable value should be selected between 40% and 80% 

rise of step response to its steady state value. In this case 𝑛=8 provides the closest 
response compared to the desired trajectory. A similar effect can be seen in the control 

220



IIUM Engineering Journal, Vol. 24, No. 1, 2023 Zainuddin et al. 
https://doi.org/10.31436/iiumej.v24i1.2341 

 

 

effort where large horizon will need larger over actuation compared to the smaller horizon 

value. Nevertheless, all the control efforts violate the maximum allowable input 

acceleration, which is 2 m/s2. 

To satisfy the driving constraint, the algorithm (A) in section 2.4 is implemented. As 

expected, Fig. 7 shows the response where the implied constraint is satisfied 

systematically, yet with a slower response in tracking the desired target trajectory.  

  
Fig. 7: Closed-loop response when input constraint is activated. 

 

Fig. 8: Closed-loop response for vehicle following application. 

221



IIUM Engineering Journal, Vol. 24, No. 1, 2023 Zainuddin et al. 
https://doi.org/10.31436/iiumej.v24i1.2341 

 

 

For the car following application, the output velocity also needs to be constrained to 

ensure that the safe following distance is always respected. With the prediction capability 

of PFC, it can anticipate the future relative distance and change the input accordingly. 

Nevertheless, a suitable validation horizon (how far ahead the constraint is implemented) 

needs to be selected. Figure 8 overlays different selections of validation horizons when 

following a lead vehicle with a constant velocity of 3 m/s and initial relative distance of 21 

m. As can be observed, all three responses managed to prevent the relative distance from 

going lower than the safe following distance and respecting all the input constraints. 

Nevertheless, there is a clear tradeoff in the selection of validation horizon, if it is too 

short (𝑛𝑖 =2 green solid line), an aggressive control effort is needed. Nevertheless, if it is 
too large (𝑛𝑖 =18, red dash dotted line), the response may be too conservative, and the 
computation burden of the controller will increase as it needs to compute more 

mathematical operation. For this case, a validation of 𝑛𝑖 =8 (blue dotted line), which is 
equal to the coincidence horizon, is selected since it respects all the constraints with 

optimum control effort. This feature is very important to ensure the safety and comfort of 

the passengers. 

3.2  Performance Comparison with PI and MPC  

To analyze the control performance of PFC, its closed-loop response is compared 

with two benchmark controllers: PI and MPC. The reason the PI controller is used instead 

of the PID is because it is a type 1 system. Figure 9 shows the comparison of the input 

acceleration, output velocity and relative distance of the three controllers.  The lead 

vehicle (black-dashed line) is assumed to operate with sinusoidal acceleration signal with 

amplitude of 0.35 m/s2 and frequency of 0.1 rad/s. The ego vehicle (the controlled vehicle) 

is set to track a desired velocity of 30 m/s while respecting the safe following distance. For 

predictive controllers (PFC and MPC), the safe following distance is treated as a 

constraint, while for PI controller, a simple switching strategy is used to change between 

speed and space mode. The gains are selected based on an auto tune function in MATLAB 

with P = 0.8, I = 0.001. At the same time, input acceleration is constrained between -3 m/s2 

to 2 m/s2 for passenger comfort.  

As expected, it can be observed that from the result in Fig. 9, MPC (green dashed-

dotted line) provides the most optimal response with less aggressive input acceleration 

followed by PFC (blue dotted line) and PI (red solid line). The difference in output 

velocity is not quite significant except that a large spike response is generated by the PI 

controller at 10 s when the ego car is trying to adjust its speed to match the leading 

vehicle. As for the safe distancing, both PFC and MPC managed to retain it, while PI 

needs to violate the safe distancing to satisfy input acceleration constraints at 60 s, 120 s 

and 180 s.  

Besides the qualitative results, some quantitative values are shown in Table 1 for each 

controller to numerically demonstrate their output velocity performance in terms of rise 

time τr, settling time τs and Root Mean Square Error (RMSE). These quantitative values 
are measured over the duration of 40 s for tracking 30 m/s from rest and will be the 

performance indices of the controllers. It can be observed from Fig. 9 and Table 1 that the 

PI controller for the constrained closed-loop response requires longer settling times 

(199.5386 s) as compared to others although it has the fastest rise time (2.7213 s). Another 

observation is that all the controllers do not produce an overshoot. However, the MPC and 

PFC provide the best fit response curve to the desired speed of 30 m/s with lower RMSE 

of (1.5501) and (1.4873) respectively compared to PI controller RMSE of (1.6219).  

 

222



IIUM Engineering Journal, Vol. 24, No. 1, 2023 Zainuddin et al. 
https://doi.org/10.31436/iiumej.v24i1.2341 

 

 

 

Fig. 9: Closed-loop performance comparison between PI, PFC, and MPC  

for vehicle following application. 

Table 1: Performance indices for all controllers. 

Performance Criteria PI PFC MPC 

Rise time, τr 2.7213 2.7348 2.9551 

Settling time, τs 199.5386 199.5348 199.5295 

RMSE 1.6219 1.5501 1.4873 

 Based on the analysis, it can be explained that the prediction capability is very 

important in an ACC system to provide optimal response. By estimating a future behavior, 

a better control input can be generated while satisfying the constraints systematically.  

Nevertheless, there is an obvious difference between PFC and MPC because the 

optimization algorithm is simplified to reduce the computation burden.  However, it worth 

pointing out that PFC performance is better compared to PID. The main reason is because 

the PI computes the control action based on the current measurement rather than 

prediction. Indeed, one may argue that if it is tuned properly or a suitable special switching 

strategy is used, a better performance can be obtained. Nevertheless, it also should be 

noted that the tuning procedure is not as straightforward as expected since many trial-and-

error procedures need to be implemented.     

223



IIUM Engineering Journal, Vol. 24, No. 1, 2023 Zainuddin et al. 
https://doi.org/10.31436/iiumej.v24i1.2341 

 

 

4. CONCLUSION  

In summary, this paper describes a design that proposes the use of PFC in controlling 

the ACC system of a vehicle. Based on the simulation results, it is found that although the 

PFC performance with RMSE of 1.4873 is not comparable with the more advanced MPC 

algorithm with RMSE of 1.5501, it gives a better performance compared to the PI 

controller with RMSE of 1.6219. Besides, the PFC algorithm is also simple and 

straightforward to implement and requires less calculation compared to the MPC while 

retaining its predictive advantage compared to other traditional controllers. The tuning 

process is also intuitive, being based on the desired time constant and coincidence horizon 

selection. Indeed, there is a lot of room for improvement and future work will provide a 

more systematic analysis of the computation requirement of both MPC and PFC 

controllers and their robustness properties. Hence, it can be concluded that PFC can 

become a good alternative to MPC and PI by trading off some of the optimality properties 

for a lower computation burden.  

ACKNOWLEDGEMENT  

The authors would like to acknowledge the Ministry of Higher Education, Malaysia for 

funding this work under the Fundamental Research Grant Scheme 

FRGS/1/2021/TK02/UIAM/02/2 (FRGS21-240-0849). The first author would like to 

acknowledge Majlis Amanah Rakyat (MARA) for his tuition fee waiver. 

REFERENCES  

[1] Jiang Y, Deng W, He R, Yang S, Wang S, Bian N. (2017) Hierarchical framework for 
adaptive cruise control with model predictive control method (No. 2017-01-1963). SAE 

Technical Paper. 

[2] Rajamani R. (2011) Vehicle dynamics and control. Springer Science & Business Media. 
[3] Haroon Z, Khan B, Farid U, Ali SM, Mehmood CA. (2019). Switching control paradigms 

for adaptive cruise control system with stop-and-go scenario. Arabian Journal for Science 

and Engineering, 44(3): 2103-2113. 

[4] Alomari K, Mendoza RC, Sundermann S, Goehring D, Rojas R. (2020). Fuzzy Logic-based 
Adaptive Cruise Control for Autonomous Model Car. In ROBOVIS (pp. 121-130). 

[5] Phan D, Amani AM, Mola M, Rezaei AA, Fayyazi M, Jalili M., ...  Khayyam H. (2021). 
Cascade Adaptive MPC with Type 2 Fuzzy System for Safety and Energy Management in 

Autonomous Vehicles: A Sustainable Approach for Future of Transportation. Sustainability, 

13(18): 10113. 

[6] Takahama T, Akasaka D. (2018). Model predictive control approach to design practical 
adaptive cruise control for traffic jam. International Journal of Automotive Engineering, 

9(3): 99-104. 

[7] Li SE, Jia Z, Li K, Cheng B. (2014). Fast online computation of a model predictive 
controller and its application to fuel economy–oriented adaptive cruise control. IEEE 

Transactions on Intelligent Transportation Systems, 16(3): 1199-1209. 

[8] Guo L, Ge P, Sun D, Qiao Y. (2020). Adaptive cruise control based on model predictive 
control with constraints softening. Applied Sciences, 10(5): 1635. 

[9] Borek J, Groelke B, Earnhardt C, Vermillion C. (2019, July). Optimal control of heavy-duty 
trucks in urban environments through fused model predictive control and adaptive cruise 

control. In 2019 American Control Conference (ACC) (pp. 4602-4607). IEEE. 

[10] Awad N, Lasheen A, Elnggar M, Kamel A. (2022). Model predictive control with fuzzy 
logic switching for path tracking of autonomous vehicles. ISA transactions, 129: 193-205. 

[11] Rossiter JA. (2018). A first course in predictive control. CRC press. 

224



IIUM Engineering Journal, Vol. 24, No. 1, 2023 Zainuddin et al. 
https://doi.org/10.31436/iiumej.v24i1.2341 

 

 

[12] Nasiri Soloklo H. (2018). Predictive Functional Control for Tracking of Core Power 
Variations in Pressurized Water Reactor based on Laguerre functions and Reduced-Order 

Model. Modares Mechanical Engineering, 18(1): 299-306. 

[13] Li MY, Lu KD, Dai YX, Zeng GQ. (2020). Fractional-Order Predictive Functional Control 
of Industrial Processes with Partial Actuator Failures. Hindawi Complexity, 2020: 1-25. 

[14] Abdullah M, Rossiter JA. (2018). Input shaping predictive functional control for different 
types of challenging dynamics processes. Processes, 6(8): 118. 

[15] Abdullah M, Rossiter JA, Ghaffar AFA. (2021). Improved constraint handling approach for 
predictive functional control using an implied closed-loop prediction. IIUM Engineering 

Journal, 22(1): 323-338. 

[16] Abdullah M, Rossiter JA. (2021). Using Laguerre functions to improve the tuning and 
performance of predictive functional control. International Journal of Control, 94(1): 202-

214. 

[17] Rossiter JA, Aftab MS. (2021). A Comparison of Tuning Methods for Predictive Functional 
Control. Processes, 9(7): 1140. 

[18] Zainuddin MAS, Abdullah M, Ahmad S, Tofrowaih KA. (2022). Performance Comparison 
Between Predictive Functional Control and PID Algorithms for Automobile Cruise Control 

System. International Journal of Automotive and Mechanical Engineering, 19(1): 9460-

9468. 

225