Proceedings of

‘Second International Conference on Emerging Research in Computing, Information,

Communication and Applications’

(ERCICA-14)

243

Comparative Analysis of Multiple Controllers for Semi-Active Suspension System

a*Prashantkumar R., Surag R. Kulkarni, Deepak Sharma, Rakesh Sambrani, Varun V. Desai

M. Tech II Semester, Department of ECE, S.D.M.College of Engineering and Technology, Dharwad-580002, India

Abstract

The primary objective of this study is to model and simulate multiple controllers for vehicle suspension system and optimize the response of the

system for different driving modes and road profiles. The fundamental goals of a suspension system are to support the vehicle mass, to isolate

the vehicle body from road disturbances and to maintain the traction force between the tire and the road surface. The approach employed in this

work is to realize and simulate the mathematical model of the single wheel semi-active suspension system, with two degrees of freedom (2-

DOF). Multiple controller models are designed and are tuned to obtain optimal response for the system. Performances of PID, Skyhook and

Groundhook controllers are then analyzed in depth for different road profiles. Matlab Simulink provides a very good platform for modeling,

analyzing and optimizing the 2-DOF single wheel suspension system.

Keywords: Driving modes; Road profiles; 2-DOF; PID controller; Skyhook controller; Groundhook controller; Matlab Simulink.

1. Introduction

Basically, suspension system comprises of a coil spring, a damper and linkages that connects a vehicle to its wheels [1]. The purpose of suspension system in automobiles is to improve ride comfort and road holding. Ride comfort literally indicates the easiness experienced by the rider even on uneven terrains. Road holding points to the traction maintained between the tire and the road surface. In normal road condition, ride comfort is the main concern but on sports track, road holding concerns the most. Unfortunately, optimizing both parameters simultaneously has not been possible. Hence while designing the suspension systems focus is on choosing the spring and damper coefficients, such that a perfect trade-off is achieved. With the recent advances in ride-by-wire concept and damper technology, it is possible for the rider to switch between different modes, like comfort, city or sports mode. Suspension system can be broadly categorized into three types: Passive suspension system, Active suspension system and Semi-active suspension system. Semi-active system is particularly grabbing the attention of researchers for its sheer advantages over the other two systems. Literature survey justifies that semi-active systems have less mechanical failure rate, less complexity, and have much low power requirements compared to active systems [3]-[5]. Furthermore, as pointed out in [6] and [7] semi-active systems have a very competitive performance and are good choice for overcoming system performance conflict. Controller is the inevitable part of the semi-active system. Various controlling schemes have been proposed employing different principles in [8]-[10]. In this paper few of the popular controller systems like PID controller, Skyhook controller [9] and Groundhook controller are designed and compared.

2. Single Wheel Suspension Model

The model shown in Fig.1 represents a 2-DOF single wheel suspension model. The dynamic model, which can describe the

relationship between the input and output, enables one to understand the behaviour of the system. Equations of motion (1) and (2)

are used in building mathematical model of the system. As we have considered mass movements only on the vertical axis,

ignoring the other movements of the vehicle, this system is said to have 2-DOF. Since the distance (x1–w) is hard to measure and

the deformation of the tire (x2–w) is negligible, the displacement (x1–x2) is used to analyze the behaviour of the suspension system.

* Corresponding author. Tel.:+918123868668;

E-mail address:prashantkumar21@outlook.com.

Prashantkumar R. .et.al.

244

Equations of motion:

𝑚1𝑥1" + 𝑏1 𝑥1

′ − 𝑥2′ + 𝑘1 𝑥1−𝑥2 = 0 (1)

𝑚2𝑥2" + 𝑏2 𝑥2

′ − 𝑥1′ + 𝑘1 𝑥2−𝑥1 + 𝑏2 𝑥2

′ − 𝑤 ′ + 𝑘2 𝑥2 − 𝑤 = 0 (2)

Fig. 1. 2-DOF Single Wheel Suspension Model.

Here: m1 - Sprung mass; m2 - Unsprung mass; k1 - Spring stiffness coefficient of the suspension; k2 - Spring stiffness coefficient of the tire; b1 - Damping coefficient of the suspension; b2 - Damping coefficient of the tire; x1 - Vertical displacement of sprung mass; x2 - Vertical displacement of unsprung mass; w - Road excitation.

3. Types of Suspension System

3.1 Passive Suspension System

In Passive Suspension system the characteristics of the suspension elements are constant, in other words it has fixed spring and

damping coefficient. In the design of these systems, there is an inherent compromise between good ride comfort and road holding.

The 2-DOF single wheel passive suspension system is shown in Fig 1. This system is modelled in Simulink, as shown in Fig 2.

Fig. 2. 2-DOF Passive suspension model in Simulink.

Fig. 3. 2-DOF Semi-active suspension model in Simulink.

3.2 Active Suspension System

Active suspension system retains the basic structure of passive system, but the damper is replaced by an actuator and the force it exerts is determined by a controller with closed loop control. The controller uses sensory information as feedback and provides the actuating signal. Actuator generates appropriate force depending on the instant condition of the suspension motion. Actuators required to produce such force require a large power source and this very reason has made automotive industry to hunt for alternate system. Fig 4(a) shows the 2-DOF single wheel active suspension system.

3.3 Semi-Active Suspension System

The Semi-active suspension system offers the same response as active suspension. The dampers that are required to realize it

consume less energy and are comparatively less expensive. In semi-active suspension, the actuator is replaced with an adjustable

damper. Fig 4(b) shows the 2-DOF single wheel semi-active suspension system. The input to this new damper block is the control

m2

m1

k2

m1

m1

k1

m1

m1

b2

m1

m1

b1

m1

m1

w

m1

x2

m1

m1

x1

m1

m1

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signal from the controller. Controllers with different controlling schemes are discussed in the next section. Fig 4 shows the

Simulink model of semi-active suspension system.

Fig. 4. (a) 2-DOF Single wheel active suspension system. (b) 2-DOF Single wheel semi-active suspension system.

4. Controller Schemes

The response of active suspension system or semi-active suspension system depends on the type of the controller used. We

have designed three commonly used controller systems namely: PID, Skyhook and Groundhook.

4.1 PID Controller

The mathematical equation of the PID controller is:

𝑐 𝑡 = 𝑘𝑝𝑒(𝑡) + 𝑘𝑖 𝑒 𝑡 𝑑𝑡 + 𝑘𝑑𝑑𝑒 (𝑡)

𝑑𝑡 (3)

Fig. 5. PID controller Simulink model.

Where, c(t) is the control signal, kp is proportional coefficient, ki is integral coefficient, kd is derivative coefficient, e(t) is the

error signal (e=u-uref), u is the measured process variable and uref is the reference variable. The reference variable uref is set to 0.

The control signal c(t) is the input signal to the adjustable damper. The Simulink model of PID controller system is shown in Fig.

5. The input to the PID controller could be either x1 (vertical displacement of sprung mass) or x2 (vertical displacement of

unsprung mass)

4.2 Skyhook Controller

Fig. 6. (a) 2-DOF Ideal Skyhook suspension system. (b) 2-DOF Ideal Groundhook suspension system.

m2

m1

k2

m1

m1

k1

m1

m1

fa

m1

m1

w

m1

x2

m1

m1

x1

m1

m1

Controller

x1’

m1

m1

x2’

m1

m1

b2

m1

m1

A

m2

m1

k2

m1

m1

k1

m1

m1

fb

m1

m1

w

m1

x2

m1

m1

x1

m1

m1

Controller

x1’

m1

m1

x2’

m1

m1

b2

m1

m1

(a) (b)

c(t)

c(t)

m2

m1

k2

m1

m1

k1

m1

m1

w

m1

x2

m1

m1

x1

m1

m1

b2

m1

m1

csky

m1

m1

m2

m1

k2

m1

m1

k1

m1

m1

w

m1

x2

m1

m1

x1

m1

m1

b2

m1

Cgnd

m1

m1

(a) (b)

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Force of the ideal Skyhook damper can be defined mathematically as:

𝐹𝑠𝑘𝑦 = 𝑐𝑠𝑘𝑦 𝑥1′

However in ideal Skyhook system, the damper connection to the fixed point can be performed only in a stationary system. In mobile systems like vehicle suspension, a force actuator between sprung and unsprung masses is used to simulate the Skyhook damper force. In a semi-active suspension system, a controllable damper is used as the force actuator.

𝐹𝑑 = 𝑐1(𝑥1′ − 𝑥2

′ )

But, Fd must be equal to Fsky

∴ 𝑐1 =𝑐𝑠𝑘𝑦 𝑥1

′

𝑥1′ − 𝑥2

′

Since the semi active damping force cannot possibly be applied in the same direction as the Skyhook damping force, the best that can be achieved is to minimize the damping force.

∴ 𝑐1 =

𝑐𝑠𝑘𝑦 𝑥1′

𝑥1′ −𝑥2

′ , 𝑥1

′ 𝑥1′ − 𝑥2

′ ≥ 0

0, 𝑥1′ 𝑥1

′ − 𝑥2′ < 0

(4)

The Simulink model of practical Skyhook controller is realized using (4) in a user defined Matlab function block. In the

Skyhook construction, the damper is connected to the sprung mass and the vibration is damped, so better ride comfort is achieved.

Whereas the unsprung mass vibration remains without damping force. To overcome this limitation we have designed a

Groundhook controller which provides better road holding.

4.3 Groundhook Controller

Force of the ideal Groundhook damper can be defined mathematically as: 𝐹𝑔𝑛𝑑 = 𝑐𝑔𝑛𝑑 𝑥2′

Similar to Skyhook controller: 𝐹𝑑 = 𝑐2(𝑥1′ − 𝑥2

′ )

But, Fd must be equal to Fgnd

∴ 𝑐2 =𝑐𝑔𝑛𝑑 𝑥2

′

𝑥1′ − 𝑥2

′

Since the semi-active damping force cannot possibly be applied in the same direction as the Groundhook damping force, the best that can be achieved is to minimize the damping force.

∴ 𝑐2 =

𝑐𝑔𝑛𝑑 𝑥2′

𝑥1′ −𝑥2

′ , −𝑥2

′ 𝑥1′ − 𝑥2

′ ≥ 0

0, −𝑥2′ 𝑥1

′ − 𝑥2′ < 0

(5)

The Simulink model of practical Groundhook controller system is realized using (5) in a user defined Matlab function block.

Groundhook controller enhances the traction force between tire and road (better road holding). However, ride comfort degrades

considerably.

5. Optimizing Controllers

5.1 Saturation Limit

It is also important to note that practical dampers [2], [4], can only accept input in a specified range (depending on manufacturer). As a result, a more limited range for the high and low level of damping should be defined as:

𝑏𝑎 = 𝑏𝑚𝑎𝑥 𝑐 > 𝑏𝑚𝑎𝑥

𝑏𝑚𝑖𝑛 𝑐 < 𝑏𝑚𝑖𝑛

𝑐 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

Here, bmax = 3000; bmin = -3000;

5.2 Tuning Controller Parameters

Tabulation I and II shows the coefficient of PID, Skyhook and Groundhook controllers, tuned to obtain the optimum response

from the system for respective modes.

247

Table I: Model Coefficients for PID

Coefficients of optimized PID controller for different modes

Mode/Coeff kp ki kd

Comfort 0.552 5.52 0.5

Sports 50 5.52 0.5

City 50 100 -0.5

Table II: Model Coefficients for Skyhook and Groundhook

Coefficients of optimized Skyhook and Groundhook controller

Mode/Coeff bsky= csky bgnd= cgnd

Comfort 2450 -

Sports - 100,000

6. Road Profiles

6.1 Road Profile 1

This road profile represents edges of the roads. The simulation of the same is shown in Fig. 7. It is generated using a pulse

generator from Simulink library with following parameters: Amplitude = 0.1m; Period = 40s; Pulse width = 10s

Fig. 7. Simulation of Road Profile 1

Fig. 8. Simulation of Road Profile 2

6.2 Road Profile 2

This road profile represents irregular road surface [3]. Fig. 8 shows the simulation of the road profile 2. It is generated by

realizing (6) in a user defined Matlab function block included in Simulink library.

𝑤(𝑡) = 𝑢1(𝑡) 0.25 < 𝑡 ≤ 0.5

𝑢1(𝑡) 0.75 < 𝑡 ≤ 0.90 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

(6)

Where, 𝑢2 𝑡 = 0.05 1 − cos 8𝜋𝑡

𝑢1 𝑡 = 0.025 1 − cos 8𝜋𝑡

7. Simulation Results

Once the suspension system, controllers and road profiles are modelled in Simulink using necessary built-in and user defined

models from the library, they are subjected to simulation. Necessary parameters and coefficients have been listed in previous

sections. Note that the y-axis represents amplitude (in meter) and the x-axis represents time (in second). The simulation results for

all types of suspension system and controller schemes are presented in this section.

Fig. 9. Body displacement: Passive, Semi active (PID & Skyhook).

Fig. 10. Wheel displacement: Passive, Semi active (PID & Skyhook).

Prashantkumar R. .et.al.

248

Fig. 11. Body displacement (Road Profile 1): PID & Skyhook.

Fig. 12. Wheel displacement (Road Profile 1): PID & Skyhook.

Fig. 13. Body displacement (Road Profile 2): PID & Skyhook.

Fig. 14. Wheel displacement (Road Profile 2): PID & Skyhook.

Fig. 15. Body displacement (Road Profile 1): PID & Groundhook.

Fig. 16. Wheel displacement (Road Profile 1): PID & Groundhook.

Fig. 17. Body displacement (Road Profile 2): PID & Groundhook.

Fig. 18. Wheel displacement (Road Profile 2): PID & Groundhook.

Fig. 19. City mode (Road Profile 1): PID controller

Fig. 20. City mode (Road Profile 2): PID controller

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Fig. 9 and Fig. 10 are the simulation results of body and wheel displacement for passive and semi-active system (PID and

Skyhook controller). Passive system has overshoot and it requires larger settling time. It neither provides good ride comfort nor road holding. For body displacement, the system with PID settles down quickly and for system with Skyhook, response is smoother. For wheel displacement, system with PID has less overshoot than the system with Skyhook.

Fig. 11 - Fig. 14 are the simulation results of body and wheel displacement for semi-active system with PID (with x1 as reference) and Skyhook controller with two different road profiles. Road profile 1: In terms of body displacement, PID system has higher overshoot than Skyhook. In terms of body displacement Skyhook system has higher overshoot than PID system. Road profile 2: Both in terms of body and wheel displacement, Skyhook system offers slightly better performance. Hence Skyhook controller is ideal for Comfort mode.

Fig. 15 - Fig. 18 are the simulation results of body and wheel displacement for semi-active system with PID (with x2 as reference) and Groundhook controller with two different road profiles. Road profile 1: In terms of body displacement, PID system has lesser overshoot and requires lesser settling time than the Groundhook. In terms of wheel displacement, Groundhook offers quicker road contact than PID but it oscillates for a while before settling. Road profile 2: Both in terms of body and wheel displacement, PID system have lesser overshoot and require lesser settling time than the Groundhook system. Hence Groundhook controller is ideal for Sports mode.

It is evident from the above simulation results that the Skyhook controller is ideal for comfort mode, whereas Groundhook controller is ideal for sports mode. The PID controller indeed offers reasonable performance for both comfort mode and sports mode. This very reason has encouraged us to employ PID controller for city mode by taking the average displacement value of sprung mass and unsprung mass of the vehicle. Fig. 19 - Fig. 20 are the simulation results of body and wheel displacement for semi-active system (PID controller) with two different road profiles for City mode. PID controller is tuned to obtain perfect trade-off between ride comfort and road holding.

8. Conclusion

This paper has outlined the improvements in performance of semi-active suspension system over passive suspension system.

We have presented the response of different controllers for various road profiles and driving modes. It is evident from the

simulation results that no control scheme offers best response for all conditions. Skyhook controller offers better performance for

Comfort mode, where better ride quality is desired. Groundhook controller offers better performance for Sports mode, where

better road holding is desired. Lastly, PID controller provides optimal response for city mode where both better ride quality and

road holding are desired.

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