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Performance Analysis of Quarter Car Active Suspension System: LQR and Roo Control

Strategies. M. P. Nagarkar, Dr. G.J. Vikhe Patil

Abstract- This paper presents a time domain performance

analysis of LQR and Hoo control strategy implemented to a

quarter car suspension model The performance analysis is

carried out on the basis of body trave� body acceleration and

suspension deflection criteria. The simulation results show

that LQR controller performs better to improve vehicle ride

comfort.

Keywords - LQR, Hoo, Performance Analysis.

I. INTRODUCTION

Suspension system has been widely applied, to the vehicles from the horse drawn carriages with flexible leaf springs, to the modern automobiles with complex control algorithms. The main functions of an automotive suspension system are to isolate the vehicle from the road unevenness and to provide vehicle support, stability and directional control during handling manoeuvres. The passive suspension systems are trade-off between ride comfort and performance. A case with a nice cushy ride usually wallows through the corners whereas a car with high performance suspension, like Fl cars, will hang on tight through the corners but will make the passengers feel very every little dip and bump in the road. The intent of the active suspension system is to replace the classical passive elements by a controlled system, active suspension system, which can supply force to the system. [22]

In a typical active suspension system, acceleration sensors are attached at parts of chassis above each wheel and at wheel axles respectively to measure each wheel behavior for independent control. Outputs from these sensors determine various system variables like sprung mass (body) velocity , unsprung mass (wheel) velocity, suspension deflection and wheel and body acceleration. In an active suspension, the passive components are augmented by actuators that supply additional forces. These additional forces are determined by a feedback control law using data from sensors attached to the vehicle. [4, 22]

Researchers have investigated active suspensions using number of different vehicle models. Hall and Gill have summarized the advantages of active suspension as follows:

1. Low natural frequencies resulting in improved ride while maintaining small static deflections.

2. Low dynamic deflections under transient excitations.

M. P. Nagarkar is PhD student at SCoE, Pune (MS), India. (phone: +91-0241-2423887; fax: +91-0241-2568383; e-mail: maheshnagarkar@rediffinai1.com). Dr. G. J. Vikhe Patil is Principal at AVCoE, Sangamner, Dist-Ahmednagar (MS), India.

2.Suspension characteristics maintained irrespective of loading. 3.Flexibility in the choice of dynamic characteristics. And disadvantage as appreciable power requirements to operate the actuators. [10]

Literature describes several techniques to control a car using an active suspension. Various active suspension system for vehicles such as LQG (Linear Quadratic Gaussian) control [1] , LQ and Sliding mode LQ controller with pneumatic actuators [3] , optimal state feedback [11] , backsteeping method [13] , optimal state feedback [[2] and proportional­integral sliding mode control are implemented using various vehicle models. During simulations, the vehicle models are subjected to irregular excitations from road profile [1] , road profile estimated by using minimum observer based on linear system [3] , random and deterministic road profile [5] , sinusoidal road profile [7] and typical bumps [8] . The simulation results indicate that the proposed active suspension system proves to be effective in the vibration isolation of the suspension system. The ride comfort is improved by means of the reduction of the body acceleration caused by the car body when road disturbances from uneven road surfaces, pavement points etc. act on the tires of running cars.

This paper presents a performance study of LQR and Hoo control strategy. LQR and Hoo controller is used with passive suspension system to improve the vehicle ride comfort. The vehicle is subjected to the disturbances like step, bump and ISO road profIles with various vehicle speeds. The performance analysis is carried on the basis of body travel, body acceleration and suspension deflection criteria for LQR, Hoo control schemes and passive suspension system. MATLAB/Simulink environment is used to carry the simulations.

II. CAR MODEL

Researchers have utilized various car models in their work. Christophe Lauwerys, Jan Swevers, Paul Sas [4] and Daniel Fischer, Rolf [sermann [6] presented the design of a robust linear controller for an active suspension mounted in a quarter car test-rig, which does not require a (nonlinear) physical model of either the car or the shock-absorber. Chuanyin Tang, Tiaoxia Zhang [I] and E. Foo, R. M. Goodall [9] implemented a four degree of freedom half car model. S. [kenaga, F. L. Lewis, J. Campos and L. Davis [2] used the full-vehicle suspension system is represented as a linearized seven degree-of-freedom (DOF) system.

During the analysis, a linear quarter car model utilized to compare the essential performance criteria of an active system with well-designed passive system. This model consists of a sprung mass and unsprung mass supported by

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springs and dampers [3, 5, 7, 8] . A linear quarter car model is considered as it is simple to model, yet we can observe the basic elements of the suspension system such as sprung and unsprung masses, sprung mass deflection, sprung mass acceleration, tire deflection, rattle/suspension space. Modeling and analysis of a quarter car suspension system is done in the vertical plane. Longitudinal or transverse deflections of the suspension components are considered negligible in comparison to vertical deflections. The complete vehicle mass is divided into two masses i.e. the sprung mass and the unsprung mass respectively. Springs and dampers are connected between the sprung and unsprung masses and unsprung mass and ground respectively as shown in Fig. I. The force U, in kN, applied between the sprung and unsprung masses, is controlled by feedback control law. The force U represents the active component of the suspension system.

ms __ J Xs sprung mass

Fig l. Linear Quarter Car Model.

Let us ignore the dynamics of the actuator, and assuming that the control signal is the force U.

Let us defme the states as -Xl = X"X2=X·" X3 = Xus and X4=X·us

The quarter car dynamics in the state-space form can be described as follows -

Xl =X2 -I X2 =-[kJxl -X3)+C,(X2 -X4)-U] m,

} (])

X2 =_l_ [k, (XI - X3 )+ C, (X2 -x4)-k,(x3 -Xr)-U] mus

The above equation (1) can be arranged in state space form as,

x (3= Ax (j- Bu (: y(j=Cx(}Du(: } (2)

III. CONTROLLER DESIGN

A. LQR CONTROL The design of a stable control system based on quadratic

performance indexes. Let us consider the system is expressed as Eq. (3).

x = Ax+Bu

where x = state vector (n-vector); u = control vector(r-vector); A = n x n constant matrix

B = n x r constant matrix

(3)

In designing control systems, one often interested in choosing the control vector u(t) such that a given performance index is minimized. The quadratic performance index is expressed as Eq. (4).

(4)

where matrix Q is positive-defmite (or positive-semi defmite) Hermitian or real symmetric matrix, R is positive defmite Hermitian or real symmetric matrix. and L(x,u) is a quadratic function or Hermitian function of x and u, leads to a linear control law.

(5)

where KrQR is a LQR gain matrix, The block diagram of LQR controller is shown in figure 2.

u x r-+ x=Ax+Bu

I -k I I I

Figure 2: Block Diagram of LQR control scheme.

We obtain the value of K, again matrix, which minimizes JLQRas Eq. (4):

(6)

where P is positive defmite matrix. Matrix P must satisty the matrix Riccati equation.

The detailed description and design procedure of the LQR control theory can be found in detail in [14, 15, 16, 17] .

B. Hoo CONTROL: There is a wide range of optimal controllers derived for a

quarter car model. The approach used is based on Hoo framework. The standard H-infmity control scheme is plotted in the Fig. 3. When open loop transfer matrix from ul to yl is

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denoted Ty[ul , then a standard optimal H-infinity controller problem is to find all admissible controllers K(s) such that infinity normllTylu [II is minimal.

An Hoo controller is computed as Tylu [ norm minimization. So it is possible to shape a characteristic in an open loop to improve performance of the whole system. For active suspension system the performance and robustness output should be weighted. The performance weighting has to include the passenger comfort and car stability (suspension displacement, actuator force).

'U1 .. Suspension Yl ... .... ....

r* System

!-

u Y '"---- Controller f+-

Fig. 3: Hoo control configuration

Here u I is the disturbance vector, u is the control vector, y I is the controlled output vector, y is the measurements vector. The detailed description of the H-infinity control theory can be found in [18, 19, 20] .

[v. RESULTS AND DISCUSSIONS

A mathematical model of the suspension system is described in state space form (equations I and 2) and the proposed LQR and Hoo controller in section III. Here the LQR and Hoo controller is compared with the passive suspension system. The simulations are carried out in Matlab/SIMULINK® [23] environment. The input data to the MATLAB model are as follows:

m,= 250 kg, k,= 16000 N/m, C,= 1000 Ns/m mus = 50 kg, k, = 190000 N/m,

While simulation, it is assumed that the vehicle hits a Pot Hole (step) of O.15m. and a bump such that, road disturbance xr(t):

xr(t) = a(1 - cos871l) for 0 <:: t <:: 0.25s (7) xr(t) = 0 otherwise.

where a=0.05 corresponds to a road bump of peak magn itude 0.1 m.

For the analysis of control strategies, three parameters are observed during the simulations. They are displacement, acceleration of sprung mass and the rattle space or the suspension space deflections i.e. xs(t)- Xl�(t). Following figures (4 and 5) shows the performance of both the control strategies with the passive suspension system to bump and step input. Figure 4 and 5 shows that the active suspension system using controllers shows better performance by

effectively absorbing and quickly damping the vehicle vibrations.

Road Profile

r'l Ci 0.1

DO 0 : r---:--�-------:--�---:--:---l o 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

I 02

.. � 0.1

Body Travel

Time (sec)

1- Hinf Control

- LQR Control

- - - Passive System

Fig 4: Step Input Response

Road Profile

: : : : : : : : : 1 0.5 1.5 2.5 3.5 4.5 5

Body Tra\lel o 15 ,-----�-�-��-�-�___,-,_______:c:_-____, 1 - Hinf Control

01

.- \

- LQR Control

- - - PaSSil'8 System I 005 1\ o ���/_-�'��=----------------�

�/ -0 05

0'--------:0'"=.5-

---'---------:I.L.5-

-"-------=-2.L5 -

-'--------=-'3.5=------'--

-::-'4.5,--------.J

l;r . "OOk:'_"" • • • •

o 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 r���:··''"",.":' • • • • o 05 1 1 5 25 35 4 5 5

Time (sec)

Fig 5: Bump Response

Here from table 1, the active suspension with LQR controller is having better performance with minimum body travel, minimum body accelerations and minimum rattle space requirements.

The performance analysis of the control strategies is also compared by simulating the vehicle model over an ISO road profile with various speeds.

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Table: I Comparative performance analysis of Step and B I ump nput.

LQR

Roo

PASSIVE

LQR

Hoo

PASSIVE

Xl Max (m)

0.1972

0.1727

0.2373

Xl Max (m)

0.0348

0.0785

0.0726

STEP RESPONSE

Acceleration Rattle (m/s2) Space (m)

25.7305 0.1767

30.7913 0.1978

27.2769 0.2054

BUMP RESPONSE

Acceleration Rattle (m/s2) Space (m)

5.5631 0.0580

13.5428 0.0801

8.0359 0.0857

The road profIle can be represented by a PSD function. The power spectral densities of roads show a characteristic drop in magnitude with the wave number. To determine the power spectral density function, or PSD, it is necessary to measure the surface profIle with respect to a reference plane. Random road profIles can be approximated by a PSD in the form of

Sq(Q) = Sq(QO{�o f" OR Sq(n) = sq(no{ � r (8)

where, Sq(Q) is the spatial power spectral density, Sq(Qo) is the coefficient of road roughness, Q is the spatial angular frequency, and Qo is the reference

spatial angular frequency =O.I(cycles/m). 0) is called waviness and indicates whether the road has

more long wavelengths ( w is large) or short wavelengths (0) is small). The waviness 0) is found within the range l.75 :S O):S 2.25, generally 0) = 2.

n is spatial frequency(l/m) and Sq(no) is called as coefficient of road roughness, and no = 1I2IT. [21]

Table 2' Road roughness values classified by ISO

Degree of roughness Sq(no)(1 0-6m2/( cycle/m)) where no = 0.1 cycle/m

Road Class Lower Geometric Limit Mean

A (Very Good) - 16

B (Good) 32 64

C (Average) 128 256

D (Poor) 512 1024

Upper Limit

32

128

512

2024

x 10.3 Road Profile

Body Travel X 10.3

4�--------------------------�--�--, --- Hinf Control --- LQR Control - - - Passive S,"stem

2 4 5 6 10 Body Acceleration

04 ,--,---,---,--,---.---,--,--�--�--,

'k 0.2

� 0

� -0.2 I

5 Time (sec)

Fig.6: Random Road Response (Vehicle Travelling with lOKmph on Class C Road)

-2

0.5

-0.5

X 10.3 Road Profile

10

10

-��--�--�--�--�--��--�--��

� -2 a:

-4 o

Suspension Deflection

Time (sec)

Fig.7: Random Road Response (Vehicle Travelling with 20Kmph on Class C Road)

10

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Road Profile

Body Acceleration

0.5

i 0 �r"\m,VI ! ,�run�IIV'i!iWJ1n 1i :t. -0.5

Time (sec)

Fig.8: Random Road Response (Vehicle Travelling with 30Kmph on Class C Road)

1 ih II

�� ° W\� WI" -1 y! v '

'"

-2 -2

� -2 '"

Body Travel

4 5 6 Body Acceleration

10

10

10

-40L--'----------:'2,----------:------'------:-----:------:------:------:':-----:'10 Time (sec)

Fig.9: Random Road Response (Vehicle Travelling with 40Kmph on Class C Road)

Road Profile

4 5 6 Body Acceleration

10

10

�O 10 Time (sec)

Fig.IO: Random Road Response (Vehicle Travelling with 50Kmph on Class C Road)

Road Profile

E I

� _: �;�'�V�,!f��1\\,��V}{��\'�1r(1'��I�MF��1���1AVAj�!�;f��VfM�j\���I�V�� t2 I

0.5 "t: 1 0 1i � -05

10

Body Acceleration

-40·L------:'------:-----:------'-------:':-----:------=---:---::-----:'10 Time (sec)

Fig.ll: Random Road Response (Vehicle Travelling with 60Kmph on Class C Road)

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x

X 10.3 Performance Comparision of LQR, Hinf and Passive Suspension System

4,-----,-----------------------------------, I + LQR Control Hin! Control + Passive System

+ + + +

+

� 0L-------2� 0--------3LO------�40--------5LO------�60 0.8 ,-------,----------,----------,--------,----------,

;[ 07 + +

10.6 c

"i 0.5

] 0.4 :c

10

+

20

+

30 40 50 60 X 10.3 5,-------,----------,----------,--------,----------,

+ + + +

\0L-------2=-'=0------ ------::3LO ------�40:------ ------:5,c-0------ �60 Speed (Kmph)

Fig.12: Performance Comparison of LQR, Roo and passive suspension system at various vehicle speed.

From above figures 6 to 11 it is observed that LQR controller shows better performance over Hoo and passive suspension system by keeping body travel, body acceleration and rattle space requirements minimum. Also as the speed goes on increasing, the body travel goes on decreasing whereas the body acceleration and rattle space requirement goes on increasing [Refer fig 11] .

V. CONCLUSION

The main objective this paper is to demonstrate performance analysis of the active control strategy - LQR and Roo controller with a passive suspension system. The mathematical model of a suspension system is determined in State Space form. A passive suspension system without any controller and an active suspension system with LQR and Hoo controller were modeled and simulated using Matlab/Simulink environment. The simulation result shows that the active suspension system using LQR and Hoo controller can improve the ride quality by minimizing the body travel and body acceleration than the passive suspension system. Also rattle space requirements are minimum in LQR and Hoo controller.

Also, LQR and Hoo controller with passive system are tested against ISO road profiles with various vehicle speeds. The results show that the LQR control shows a better performance over the Roo control strategy.

REFERENCES

[1] Chuanyin Tang and Tiaoxia Zhang, "The research on control algorithms of vehicle active suspension system", IEEE International Conference on Vehicular Electronics and Safety (2005),320-325.

[2] S. Ikenaga, F. L Lewis, 1. Campos, L Davis, "Active Suspension Control of Ground Vehicle based on a Full-Vehicle Model", Proceedings of the American Control Conference - Chicago, (June 2000), 4019-4024.

[3] T. Yoshimura, A Kume, M. Kurimoto And 1. Hino, "Construction of an active suspension system of a quarter car model using the concept of sliding mode control", Journal of Sound and Vibration 239(2) (2001), 187-199.

[4] Christopher Lauwerys, Jan Swevers, Paul Sas, "Robust linear control of an active suspension on a quarter car test-rig", Control Engineering Practice 13 (2005), 577-586.

[5] Yuping He and John McPhee, "Multidisciplinary design optimization of mechatronic vehicles with active suspensions", Journal of Sound and Vibration 283 (2005), 217-241.

[6] Daniel Fischer and Rolf Isermann, "Mechatronic semi-active and active vehicle suspensions", Control Engineering Practice 12 (2004), 1353-1367.

[7] Nizar AI-Holou, Tarek Lahdhiri, Dae Sung Joo, Jonathan Weaver and Faysal AI-Abbas, "Sliding Mode Neural Network Inference Fuzzy Logic Control for Active Suspension Systems", IEEE Transactions on Fuzzy Systems 10(2), (APRIL 2002),234-246.

[8] Yahaya Md. Sam, Johari B. S. Osman, M Ruddin A Ghani, "A class of proportional-integral sliding mode control with application to active suspension system", Systems & Control Letters 51 (2004) 217 - 223.

[9] E. Foo and R M. Goodall, "Active suspension control of flexible­bodied railway vehicles using electro-hydraulic and electro-magnetic actuators", Control Engineering Practice 8 (2000) 507-518.

[10] Hall, B. B. and K. F. Gill. , "Performance evaluation of motor vehicle active suspension systems", IME-Proceedings of the Institution of Mechanical Engineers- Transportation 20 I (02) (1987), 135-148.

[11] A Alleyne, JK Hedrick, Nonlinear adaptive control of active suspensions, IEEE Trans. Control System TechnoL 3, 1997,94-101.

[12] E Esmailzadeh, H.D. Taghirad, Active vehicle suspensions with optimal state-feedback control, 1. Mech. Sci. 200, 1996, 1-18.

[13] J. S. Lin, I. Kanellakopoulos, Nonlinear design of active suspension, IEEE Control System Mag. 17, 1997, 45-59.

[14] MP. Nagarkar, G.J Vikhe, KR Borole and VM Nandedkar, "Active Control of Quarter-Car Suspension System Using Linear Quadratic Regulator", International Journal of Automotive and Mechanical Engineering (HAME), Volume 3, 2011, 364-372.

[15] Ogata, K. , Modern Control Engineering. Fourth Edition, Prentice Hall, India. 2002

[16] Anderson, B. and Moore, 1. "Optimal Control - Linear Quadratic Methods" Prentice Hall, N1. 1990.

[17] Hrovat, D., "Survey of advanced suspension developments and related optimal control applications", Automatica, 33(10), 1997 1781-l787.

[18] K. Zhou and J. C Doyle, Essentials of Robust Control. Prentice Hall, 1998.

[19] Hong Chen, and Kong-Hui Guo, "Constrained Hoo Control of Active Suspensions: An LMl Approach" IEEE Transactions on Control Systems Technology, Vol. 13, No. 3, May 2005,412-421.

[20] Ramin Amirifar and Nasser Sadati, "Low-Order Hoo Controller Design for an Active Suspension System via LMIs", IEEE Transactions on Industrial Electronics, Vol. 53, No. 2, April 2006, 554-560.

[21] L-J. ZHANG, C-M LEE and Y S. WANG, "A Study on Non­stationary Random Vibration of A Vehicle in Time and Frequency Domains", International Journal of Automotive Technology, Vol. 3, No. 3,2002, 101-109.

[22] CR. Fuller, SJ. Elliott and PA Nelson, "Active Control of Vibrations", Academic Press, London, 1996.

[23] MATLAB/Simulink Heip Library.

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