Best suited suspension system for luxury SUVs

Analysis of Active and Passive Suspension Systems in Luxury SUVs
EGME 476A-52 Dynamic Systems and Controls Laboratory
California State University Fullerton
Kristopher Kerames
Due: 7/3/2021
Submitted to: Yong Seok Park, Ph.D.

EGME 476A-52 Analysis of Active and Passive Suspension Systems
Kristopher Kerames Page 2 of 19
Abstract
This experiment explores how proportional (P), derivative (D), and integral (I) controllers can be combined to
deliver the best driving experience in a luxury, performance sport utility vehicle (SUV). The combinations of
controllers used were PD, PI, and PID controllers. These were also compared to a passive suspension system
without any form of PID control. Governing equations of motion were developed for a quarter-SUV going
over a single bump, and the passive suspension system – without a PID controller – was simulated in
SIMULINK. The Zeigler-Nichols Method was used to tune the PID, and simulate all combinations of the
PID-controlled system in SIMULINK. In each case, the SUV settled close to a steady-state value of 0 m. The
PID controller allowed for the fastest rise time, the lowest peak amplitude of 4.1 mm, and the fastest 2%
settling time of 3.85 sec. For comparison, the system with the passive suspension had a peak amplitude of
32.6 mm which was also the largest of all systems tested. Having such a low peak amplitude results in the PID
controller offering a much more comfortable ride to the passengers. PID controllers are more expensive than
the other options considered in this experiment, but considering the context, they are the best option. In the
luxury SUV class, components that are integral to the driving experience are expected by consumers to be
factored into the cost. Therefore, cost is likely to be met with demand when including components like the
PID controller.

Table of Contents
Abstract................................................................................................................................................. 2
List of Symbols and Units..................................................................................................................... 4
Introduction........................................................................................................................................... 5
Theory................................................................................................................................................... 6
Test Setup and Procedure...................................................................................................................... 8
Sample Calculations............................................................................................................................ 10
Error Analysis ..................................................................................................................................... 11
Results................................................................................................................................................. 12
Discussion and Conclusion................................................................................................................. 14
References........................................................................................................................................... 15
Appendix A......................................................................................................................................... 16
Appendix B......................................................................................................................................... 18

List of Symbols and Units
Name of Factor Symbol Unit
Damping Coefficient c 𝑁 ∙ 𝑠/𝑚
Spring Stiffness k 𝑁/𝑚
Ground Height g 𝑚
Initial Height 𝑦𝑖 m
Marginal Stability 𝑘∗
N/m
Period 𝑇∗
seconds
Integral Controller 𝑘𝑖 𝑁/𝑚
Derivative Controller 𝑘𝑑 𝑁/𝑚
Proportional Controller 𝑘𝑝 𝑁/𝑚
Force 𝐹 𝑁
Acceleration 𝑎 𝑚/𝑠2

Introduction
One of the key considerations to make when designing a car is the suspension system. This system
affects important characteristics of a car including its handling, level of comfort, and durability. Components
that make up the suspension system are the shock absorbers, springs, and mechanical linkages connecting
them to the rest of the car. Parameters that can be changed in this system include the damping coefficient,
spring constant, and mass of the components it consists of. These are called passive parameters and cannot be
actively changed while driving. When left to their own devices, passive components alone cannot always
achieve the desired driving characteristics. In the case of a luxury vehicle, a comfortable ride is desirable
without compromising on handling. Reducing the spring constant can create a more comfortable ride, but will
hinder the car’s handling. In cases like this, an active suspension system can be used to adjust the
characteristics of the suspension according to certain inputs. Active suspension systems can use proportional
integral derivative controllers (PIDs) which allow stiffer springs to be used for improving handling while the
PID controller actively adjusts the suspension response making the ride more comfortable. PIDs apply
corrective adjustments to a system in a feedback loop and can mitigate oscillations in the suspension system
caused by bumps in the road. P, I, and D controllers can be combined in any order. The P-controller linearly
relates the controller output and the system error in order to reduce rise times. The I-controller removes
deviations in the output in order to reduce the error in the system’s steady-state value. The D-controller
measures the change in error and can allow the PID to control the response rate, minimizing the percent
overshoot. This experiment will focus on separately using a PD, PI, and PID controller to optimize the
suspension system in a luxury, high-performance (SUV). Luxury SUVs have been increasing in production,
and a relatively small amount of research has been dedicated, specifically, to cars in this high-demand class.
Simulations in this study will focus on luxury SUVs, as research dedicated to improving these vehicles would
benefit many auto manufacturers. The methods used are based on an article on modeling car suspension
systems in SIMULINK [1]. However, the article covers different types of controllers and parameters than this
experiment does, and it generalizes the simulation to model a typical car. The main objective of this
experiment is to compare the three different PID configurations and a passive suspension system in a quarter-
car-model to determine which is most effective in the context of a luxury, performance SUV.

Theory
The suspension system can be modeled in a quarter-car going over a bump as in Figure 1,
Figure 1. Diagram of a quarter-car going over a bump defined by 𝑔.
𝑚2 is the mass of the car’s frame, and 𝑚1 is the mass of the suspension components, including the wheel. The
height of the bump, 𝑔, is defined by the piecewise function,
𝑔 = {
0, 0 ≤ 𝑡 < 1
5𝑡 − 5, 1 ≤ 𝑡 ≤ 1.25
0, 1.25 < 𝑡
(1)
𝑢 is the input of the PID controller. 𝑘2 is the spring stiffness of the suspension system, and 𝑘1 is the spring
stiffness. 𝑐2 is the damping coefficient of the dampers in the suspension system, and 𝑐1 is the damping
coefficient of the tire.
The equations of motion (EOM) can be found from the free body diagrams. The free body diagram of 𝑚2is in
Figure 2, and the free body diagram of 𝑚1 is in Figure 3.
Figure 2. Free body diagram of the car’s mass.

Figure 3. Free body diagram of the suspension system’s mass.
Using Newton’s 2nd
Law of motion,
Σ 𝐹
⃗ = 𝑚𝑎
⃗ (2)
The EOM for 𝑚1 becomes,
𝑚1𝑦̈1 = − 𝑐2(𝑦̇2 − 𝑦̇1) − 𝑐1(𝑦̇1 − 𝑔̇ ) + 𝑘2(𝑦2 − 𝑦1) − 𝑘1(𝑦1 − 𝑔) − 𝑢
⇒ 𝑦̈1 =
1
𝑚1
[− 𝑐2(𝑦̇2 − 𝑦̇1) − 𝑐1(𝑦̇1 − 𝑔̇ ) + 𝑘2(𝑦2 − 𝑦1) − 𝑘1(𝑦1 − 𝑔) − 𝑢]
(3)
The EOM for 𝑚2 is,
𝑚2𝑦̈2 = −𝑐2(𝑦̇2 − 𝑦̇1) − 𝑘2(𝑦2 − 𝑦1) + 𝑢
⇒ 𝑦̈2 =
1
𝑚2
[−𝑐2(𝑦̇2 − 𝑦̇1) − 𝑘2(𝑦2 − 𝑦1) + 𝑢]
(4)

Test Setup and Procedure
Define all constants in MATLAB that are used in the SIMULINK models.
Run all simulations using a 0.001 sec fixed time-step.
1. Open SIMULINK to set up a system representing the passive suspension components that uses the
EOMs defined in Equations (3) and (4). Set 𝑢 = 0.
2. Find 𝑘∗
by adding a feedback loop to the system in step 1 that uses 𝑘𝑝 as a gain value. Adjust the gain
value until the response is marginally stable. At that point, 𝑘∗
= 𝑘𝑝. An example of the marginally
stable response can be seen in Figure
3. Find 𝑇∗
by using the measuring tool to measure the period of the marginally stable system. The graph
with the measuring tool should look as it does in Figure 4.
4. Create a PD-controlled system by adding 𝑘𝑝 and 𝑘𝑑 gains to the loop to form a PD controller.
5. Create a PI-controlled system by adding 𝑘𝑝 and 𝑘𝑖 gains to the loop to form a PI controller.
6. Create a PID-controlled system by adding a built-in PID controller from SIMULINK.
7. Tune the PID controller using the parameters in the “tune” window. These values can be seen in the
bottom-right of Figure 5.
8. For all simulations, determine the values of peak response, peak time, percent overshoot, settling
time, and steady-state error.
The complete set of SIMULINK block diagrams can be found in Appendix A.

Figure 4. Marginally stable system with measuring tool in use.
Figure 5. Control parameters are in the bottom-right of this window.

Sample Calculations
The following constants have values which would typically be seen in a luxury vehicle. These constants are
based off of parameters mentioned in a study on ride comfort [2].
• 𝑚1 = 20 𝑘𝑔
• 𝑚2 = 1800 𝑘𝑔
• 𝑘1 = 500000 𝑁/𝑚
• 𝑘2 = 16812
𝑁
𝑚
• 𝑐1 = 2500 𝑁 ∙ 𝑠/𝑚
• 𝑐2 = 1000 𝑁 ∙ 𝑠/𝑚
The following parameters are for the active suspension system responses using the Zeigler-Nichols Method.
The value, 𝑘∗
, that makes the P-controlled loop marginally stable is,
𝑘∗
= 7.75 × 103
𝑁/𝑚
The period, 𝑇∗
, is obtained from the plot of the marginally stable system and has a value of,
𝑇∗
= 1.877 𝑠𝑒𝑐
This period was found by using the measuring device in SIMULINK to measure the difference in heights
between peaks in the plot of the marginally stable response (Fig. 4).
For the system using the PD controller, the proportional response is,
𝑘𝑝 = 0.45𝑘∗
= 0.45(7.75 × 103
𝑁/𝑚) = 3487.5 𝑁/𝑚
The derivative gain for this system is,
𝑘𝑑 = 0.125 ∗ 𝑘∗
∗ 𝑇∗
= 0.125 (7.75 × 103 𝑁
𝑚
)(1.877 𝑠𝑒𝑐 ) = 1818.34 𝑁/𝑚
For the PI controlled system, the proportional stiffness is the same as it is for the PD controlled system,
𝑘𝑝 = 3487.5 𝑁/𝑚
And the integral gain is,
𝑘𝑖 =
1.2 𝑘∗
𝑇∗
=
1.2 (7.75 × 103 𝑁
𝑚
)
1.877 𝑠𝑒𝑐
= 4954.71 𝑁/𝑚
MATLAB script used to calculate the responses can be found in the Appendix B.

Error Analysis
Steady state errors were calculated using 2% settling values. Results obtained in this error analysis can also be
found in tables 1 and 2.
A. Error analysis of passive system
𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝑂𝑣𝑒𝑟𝑠ℎ𝑜𝑜𝑡 = 100
𝑦𝑚𝑎𝑥 − 𝑦𝑖
𝑦𝑖
= 100
0.0326 − 0
0
= 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
𝑆𝑡𝑒𝑎𝑑𝑦 𝑆𝑡𝑎𝑡𝑒 𝐸𝑟𝑟𝑜𝑟 = |𝑦𝑜𝑢𝑡 − 𝑦𝑖| = | − 6.45 × 10−4
− 0| = 6.45 × 10−4
𝑚
𝑆𝑡𝑒𝑎𝑑𝑦 𝑆𝑡𝑎𝑡𝑒 𝐸𝑟𝑟𝑜𝑟 𝑃𝑒𝑟𝑐𝑒𝑛𝑡 = 100
|𝑦𝑜𝑢𝑡 − 𝑦𝑖|
𝑦𝑖
= 100
|6.45 × 10−4
− 0|
0
B. Error analysis of active system
PD-controller:
0.0245 − 0
0
𝑆𝑡𝑒𝑎𝑑𝑦 𝑆𝑡𝑎𝑡𝑒 𝐸𝑟𝑟𝑜𝑟 = | − 4.90 × 10−4
− 0| = 4.90 × 10−4
𝑚
|5.0983 − 0|
0
PI-controller:
0.0293 − 0
0
− 0| = 1.14 × 10−4
𝑚
|1.14 × 10−4
− 0|
0
PID-controller:
0.0041 − 0
0
− 0| = 8.24 × 10−5
𝑚
|8.24 × 10−5
− 0|
0

Results
The passive response was stable, but had the highest peak value at 0.0326 (Fig. 6 and Table 1). The
PID-controlled response had the lowest peak value by a significant margin (Fig. 6). The PI-controlled system
had the longest 2% settling time when compared to all other systems while the PID-controlled system had the
smallest 2% settling time (Table 1). The plot of the marginally stable response has a constant amplitude, so it
does not reach a steady-state value (Fig. 7). All other curves approached a steady-state value of 0 m.
Figure 6. Plot of all responses to passive and active suspension systems.
0 5 10 15 20 25 30
Time (s)
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
Height
(m)
All Passive and Active Responses
Passive Response
PD Controlled Response
PI Controlled Response
PID Controlled Response

Figure 7. Plot of the marginally stable system operating at k*
.
Table 1. Closed Loop Step Response Values
Passive PD Controller PI Controller PID Controller
Rise Time (sec) 0.0022 5.53 × 10−10 0.001 0
Peak Response (m) 0.0326 0.0245 0.0293 0.0041
Peak Time (sec) 1.65 1.58 1.6 1.31
Percent Overshoot (%) 475810 1.4 × 1012
2.18 × 104
5.155 × 1019
Settling Time (sec) 14.90 6.37 25.67 3.8456
Steady-State Error (m) 6.45 × 10−4
4.90 × 10−4
1.14 × 10−4
8.24 × 10−5
Steady-State Error Percentage (%) undefined undefined undefined undefined
Note: The percentages on the order of ~104
or higher were calculated in MATLAB, but are in reality
undefined values.
0 10 20 30 40 50 60 70 80 90 100
Time (s)
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
Height
(m)
Marginally Stable Response

Discussion and Conclusion
Each configuration for the suspension system offers different strengths and weaknesses. The
passive suspension system reached its steady-state value in a similar amount of time as the more
expensive PI and PD controlled systems. Although it reached steady-state slower than the PID
controlled system did, it can be implemented at a lower cost than the PID can. A key drawback of
the passive suspension system is that it has the highest peak value. This means it would cause the
passengers of the SUV to move a greater distance up and down as the car goes over bumps. This
would not be ideal for the passengers. The PI controller resulted in the longest settling time, and only
had the third lowest peak amplitude of the four options. Its best performance measure is its steady-
state error value (Table 1), and that was only second best behind the PIDs steady-state error value.
Given that it is relatively expensive compared to a passive suspension system, but offers no
significant benefit, it is one of the worst solutions for a luxury SUV. The PD controlled system has
the second smallest 2% settling time, and the second lowest peak value. Given that it costs less to
use in the SUV than the PID does, it is an alternative that strikes a balance between cost and benefit.
Nonetheless, the PID controller remains the best option among the four for this application. In the
market for luxury, performance SUVs, consumers expect costs to be increased by features that
materially improve the driving experience. The simulations in this experiment support that the PID
controller offers the best driving experience by a large margin. Given that luxury SUVs are in
increasingly high demand, higher costs of using the PID are likely to meet the demand of consumers
looking for this class of vehicle.
There is a large amount of research simulating quarter-car models, but few focus on the
luxury SUV market. As this is an emerging market, auto manufacturers would benefit from this
research, as well as future research that improves upon the simulations in this experiment. One area
in which there is room for improvement would be the input from the road. The bump in this
experiment, expressed by 𝑔, has a triangular shape. As there are many different types of
imperfections in road surfaces, future research would benefit from exploring additional road features
and how they affect the system.

References
[1] “Suspension: Simulink Controller Design,” Control Tutorials for MATLAB and Simulink - Suspension:
Simulink Control, (2021). [Online]. Available:
https://ctms.engin.umich.edu/CTMS/index.php?example=Suspension§ion=SimulinkControl. [Accessed: 01-
Jul-2021].
[2] A. Tiwari, S. More, and A. Shahane, (2018). “Study of Road Holding and Ride Comfort Analysis With
the Help of Quarter Car Model”. International Journal of Scientific Development
t and Research. Vol. 3. Available: https://www.ijsdr.org/papers/IJSDR1812050.pdf

Appendix A
SIMULINK Models

Appendix B
MATLAB Script
%EGME 476A Final Lab
%Author: Kit Kerames
clc; clear; close all
%Constants
m1=20;
m2=1800;
k1=500000;
k2=16812;
c1=2500;
c2=1000;
% Makes system margnally stable
KpStar=7.75e3;
Tstar=1.877;
KdMS=0;
% Makes system PD controlled
Kp=0.45*KpStar;
Kd=0.125*KpStar*Tstar;
% Makes system PI controlled
Ki=KpStar*Tstar/1.2;
%Runs simulation
sim('EGME476AFinalLab')
%Plots passive system response
figure(1)
plot(ans.y2);
hold on
plot(ans.y2PD);
plot(ans.y2PI);
plot(ans.y2PID);
legend('Passive Response', 'PD Controlled Response',...
'PI Controlled Response', 'PID Controlled Response')
grid on
title('All Passive and Active Responses');
xlabel('Time (s)')
ylabel('Height (m)');
%Plots marginally stable response
figure(2)
plot(ans.y2MS);
grid on
title('Marginally Stable Response');
xlabel('Time (s)')
ylabel('Height (m)');
%Table information
PASsystem=stepinfo(ans.y2.data,ans.y2.time);
PDsystem=stepinfo(ans.y2PD.data,ans.y2PD.time);

PIsystem=stepinfo(ans.y2PI.data,ans.y2PI.time);
PIDsystem=stepinfo(ans.y2PID.data,ans.y2PID.time);
%Calculating steady-state values at 2% settling
PASss=ans.y2.data(find(ans.y2.time==14.9));
PDss=ans.y2PD.data(find(ans.y2PD.time==6.37));
PIss=ans.y2PI.data(find(ans.y2PI.time==29.97));
PIDss=ans.y2PID.data(find(ans.y2PID.time==3.85));

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