online semi-active control system of a magnetorheological...

International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:03 130

192203-4848-IJMME-IJENS © June 2019 IJENS I J E N S

Online Semi-Active Control System of a

Magnetorheological Fluid Damper using LQR

Algorithm Mohammed A. Al-Shujairi, Mohammed JawadAubad,SalwanObaidWaheedKhafaji*, Mustafa Turki Hussein

University of Babylon, Collage of Engineering, Department of Mechanical Engineering, Iraq

*Corresponding author, E-mail: [email protected]

Abstract-- In this paper, passive and semi- active suspensions

system of a magnetorheological fluid damper for an

automobile application under different excitations (step,

sinusoidal, and random) as road profiles is studied. This work

is presented in two parts, the dynamic responses of the MR

fluid damper under the expiation are presented in the first part

and developing a simple and efficient controller to control the

damper behavior is presented in the second one. Bingham

plastic model is adopted for the mathematical modeling and

analyze the hysteretic behavior of the MR fluid damper. LQR

algorithm is used for the control processes. The results showed

that the input current to the magnetic circuit plays an

important role regarding the dynamic response and damping

force for the excitation, however, current effect is different for

the excitations. Magnetic saturation is noticed in the dynamic

response. Damping force can be controlled by controlling the

input current to reduce the overshoot, steady-state error, and

steady state time response. In addition, the linear quadratic

regulator (LQR) has successfully stabilized the system and removes the vibration without any abnormal behavior.

Index Term-- Online control, semi-active control,

Magnetorheological fluid damper, LQR algorithm.

1- INTRODUCTION One of the important requirements for any dynamic system

is an effective control process due to vibration, noise, and

instability problems [1]. The fluctuation of system vibration

close to system natural frequencies may cause mechanical

failure due to resonance domination and that may increase

the requirements of maintenance and total operation cost. In

the light of these problems, people have found a way to

partially solve these issues by design isolations systems [1–

8] and developing active and passive control methods [9].

Very good literature about control can be reviewed referring

to.A good general introduction to about method of vibration and isolation systems and its application can be found in [10

and 11].One of the promising ways in the direction of

counteracts and dissipates the vibration energies is the idea

of using semi-active control. The key parameters used to

introduce semi-active control are low power, interphase

simplicity between the mechanical and electrical parts, and

the good performance. Semi-active control has been

developed as a combination of the expensive, high

performance active vibration control and cheap-low

performance passive vibration control [12]. The respected

reader is referred to [13-18] for more information about

semi-active control mainly used. People have also worked

on application semi-active vibration control for vehicle

suspension system [20]. Using magnetorheological fluid

damper (MRF damper) is very good example of semi-active

control of dynamical system vibration [21].MRF damper

can be defined as a semi-active controlled device that achieves a wide range of controllable damping fore with

high performance compared to the conventional hydraulic

passive damper. Low power requirements, high

performance, safe to fail, fast response, and high level of

reliability are the main advantages of using MRF damper.

The MRF damper is composed of the same mechanical parts

of the conventional hydraulic damper filled with a specific

MR fluid. MR fluid consists of base fluid and magnetic-

nanoparticles. In the presence of magnetic field, the

nanoparticles align themselves in the direction of the

developed magnetic lines producing chain- like structure. Stiffness and strength of the developed chains depend on the

magnetic field strength and electromagnetic properties of

the nanoparticles. [19, 20].The time of the MR fluid is in the

order of milliseconds.

However, the mathematical model of semi-active MRF

damper is complex and complexity level is high because of

the nonlinear dynamic behavior[22].Regarding the

mathematical model complexity, F. AliadoptedBouc-Wen model for the mathematical modelling taking input current

and frequency amplitude into the considerations. He showed

that the dynamic response of the MRF damper is well

described. Vincenzo Paciello [23]developed a new

approach for characterization of MRF damper for different

mathematical models.M. Khusyaieet, al[24]Investigated

different parameters of MRF damper and explained how

these parameters variations vary the accuracy of the studied

models of the MRF damper. He used “nonlinear least square

fitting method” to achieve his study.DavidCaseet, al [25]

characterize the dynamic response for a small-scale MRF damper for tremor-suppression orthosis applications. Third

order transfer function is used to model both input current of



the coil and nanoparticles. He showed that the developed

mathematical model demonstrates good results compared to

the experimental one. In his research, ZekeriyaParlaket,

al.[26]figured out an optimal configuration design of MRF

damper using design of experiments (DOA) This study

deals with the optimal configuration of an MR damper using the Taguchi experimental design approach. The dynamic

force and damping force were the design parameters need to

deal with. Sulaymon L. Eshkabilov[27]presented very

good review for the mathematical models used to formulate

the MRF damper for a quarter-car model for different

excitations.He shows that accuracy of the simulations

results varies for each model depending on the way the

model used. In his study, Mohammad Saadet, al. [28]used

PID controller along with Bouc-Wen model for semi-active

control of MRF damper. MATLAB and Simulink were used

for the simulation.The results showed the desired design can

be achieved. Geoffrey Geldhof 2013[29]used MRF damper for semi-active vibration control for atwo-cart system.

Bouc-Wen model wasadoptedfor the mathematical

modeling.MATLAB and LabView were used for the

numerical implementation. The results showed the semi-

active control can be improved to gain 20% performance

than the passive control.S. Talatahari [30] proposed a

comprehensive optimization procedure to obtain the

optimum design of MRF damper based on the parameters

used in his mathematical model and dynamic hysteresis. The

optimized model tested experimentally and the results

showed good results. In this paper, passive and semi- active suspensions system

of a magnetorheological fluid damper for an automobile

application under different excitations (step, sinusoidal, and

random) as road profiles is studied. This work is presented

in two parts, the dynamic responses of the MR fluid damper

under the expiation are presented in the first part and

developing a simple and efficient controller to control the

damper behavior is presented in the second one. Bingham

plastic model is adopted for the mathematical modeling and

analyze the hysteretic behavior of the MR fluid damper. LQR algorithm is used for the control processes.The results

showed that the input current to the magnetic circuit plays

an important role regarding the dynamic response and

damping force for the excitation, however, current effect is

different for the excitations. Magnetic saturation is noticed

in the dynamic response. Damping force can be controlled

by controlling the input current to reduce the overshoot,

steady-state error, and steady state time response. In

addition, the linear quadratic regulator (LQR) has

successfully stabilized the system and removes the vibration

without any abnormal behaviour.

2- MATHEMATICAL FORMULATION OF A QUARTER-CAR

MODEL In order to test the performance of the magnetorheological

fluid damper under several dynamic excitations, an MR

fluid damper attached to a real physical model represented

by a quarter-car is studied. To simplify the mathematical

model, the car and all loads are evenly distributed across all

tires androughness of terrain isthe same for all the wheels of

the car. Moreover, it is assumed that the damping effect of

the tire is smaller than that corresponding of the attached

MR fluid damper so the tire damping effect can be neglected. Figure (1) shows a quarter- car model for both

active and semi-active systems. Newton’s law is simply

used to derive the mathematical model for both cases in Eq.

(1) and Eq. (2), respectively.

a) Passive design b) Semi-active design

Fig. 1. Physical model of a quarter-car model for both passive and semi-active

Suspension designs.

sc

sm

um

sk

uk uc



0s s s s u s s u

s s s s u s s u

u u s u u u

m z c z z k z z

m z c z z k z z

c z k z k r c r

(1)

and

s s s s u s s u c

s s s s u s s u

u u u u c u u

m z c z z k z z U

m z c z z k z z

c z k z U k r c r

(2)

where 𝑚𝑢 and 𝑚𝑠are the half mass of axle and the wheel

and quarter of the body mass, 𝑍𝑢 , �̇�𝑢 , and 𝑍�̈� are the

displacement, velocity, and acceleration of half mass of

the axel, , 𝑍𝑠, �̇�𝑠, and 𝑍�̈�are the displacement, velocity,

and acceleration of half mass of the quarter-car, 𝑐𝑢, and

𝑐𝑠 , are the damper coefficients of the corresponding

masses, 𝑘𝑠 , and 𝑘𝑧 , are the stiffnesses of the

corresponding masses, 𝑟, �̇�, are displacement and velocity of the external excitation on the systems, which depends

on the road roughness. 𝑈𝑐 is the generated force of the

designed controller with respect to the external excitation

(disturbance due to road roughness) and the vehicle speed.

It is worth to mention that one of the objectives in this

work is to design a controller to control the MR fluid

damper to achieve best ride and lower dynamic load

factor which may harm the system due to impact load

exerted on the tire by different road conditions. Several

models were proposed to describe the relationship between shear stress and shear strain in the presence of

the magnetic field. In this work, Bingham plastic model is

used and will be explained in the next section.

3- BINGHAM PLASTIC MODEL

Bingham plastic model is one of the models that used to

express the dynamic behavior of the MR fluid in the

presence of magnetic field [19]. Fig. (2) Presents the

mechanical discrete model of the Bingham plastic model [33]. The mathematical modelling of this system can be

described by a first order differential equation as,

0 0 0sgnmr cF F z c z k z F (3)

Where, 𝑧 is the same as 𝑍𝑠in the quarter-car model which

represents the displacement of the damper piston 𝐹𝑐 is the

controlled frictional force, 𝐹0 is a constant force of the

damper, 𝑐𝑜 and 𝑘𝑜 is the damping and stiffness

coefficients of the damper, respectively. The sgn function

is introduced to eliminate direction of motion dependency

for the controlled frictional forces against the relative

velocity�̇�, Due to presence of the sign function in Eq. 3,

the numerical representation for𝐹𝑚𝑟, is not easy to use in this form because of the involved discontinuity resulted

by sign function. However, another model can be used to

overcome this problem by introducing inverse tan

function as shown in Eq. 4 below [33]:

1

0 0 0

2 tan .c

mr

F d zF c z K z F

(4)

where a new term d, introduced in eq.(4), is a form factor.

By increasing the value of d, better performances can be

attained in terms of damped vibration [33]. The respected

author is referred to [33] for more information about that.

The updated mathematical representation of Eq. (4) can

be represented by Fig.(2).

Fig. 2.Bingham plastic model proposed

4- RESULTS AND DISCUSSION

4-1 Damping characteristics

This section is divided into two parts, dynamic response

of the MR fluid damper under different excitations (step, ramp, and random) are presented in the first part and

developing a simple and efficient controller to control the

damper behavior is presented in the second one. The

numerical values of the parameters presented in Eq. (1)

through Eq. (4) are selected to provide better ride and

presented in Table I.



Table I

Numerical values of the quarter-car model [34].

Parameter Value

Sprung mass (ms) 450 kg

Un-sprung mass (mu) 68 kg

Suspension stiffness (ks) 28500 [N/m]

Tire stiffness (ku) 293.9[KN/m]

Damping coeffs. of spring mass (cs) 2700 [N.s/m]

Damping coefficients of the tire (cu) 0

For the Bingham plastic model, some values such as𝐹𝑐, 𝑐𝑜

and 𝐹0 are adopted from the experimental work. 𝐹0 is

assumed to equal 40N while the other two parameters are

computed dependently on the input current to the

magnetic circuit of the MR fluid damper. Their values are

given as a function of the current as [33]:

3 2

4 3 2

0

910.09 986.49

663.56 52.19

48.74 106.39 66.00

1.43 0.53

cf I I I

I

c I I I I

I

(5)

Figure (3) presents effect of input current on the damping

force along with damper relative velocity. Effect of input

current is more dominant at the higher values of the

current even for the smaller values of the velocities and

this make the MR fluid damper more efficient than the

conventional one. Figure (4) presents acceleration

response of the body car and force damper of system for a

sinusoidal excitation of 3.75cm amplitude and 7.77rad/sec

frequency. Fig (5) and Fig (6) presents acceleration response and damping force due to random and step input.

For the three figures above, it is noted that the

acceleration and the body responseare inversely

proportional to the input current while the damping force

is linearly proportional. The acceleration and body

response decreases with the input current while damping

force increases. However, the rate of decreasing and

increasing of response and damping force decreases with

higher values of the input current due to magnetic field

saturation, the critical value of the current after which no

further increasing in MR fluid viscosity and thereby damping force. It is noted that MR fluid damper can cover

a wide range of damping characteristics for different road

configurations and input current and this encourages

people to build a simple and efficient controller to control

damping force. This will be presented in the next section.

Fig. 3. Effect of input current on the damping force



Fig. 4. The acceleration, response of the body car and force damper of system for sinusoidal excitation with amplitude 0.0375m and 7.77 rad/sec



Fig. 5. The acceleration, response of the body car and force damper of system for random excitation



Fig. 6. The acceleration, response of the body car and force damper of system for step bump excitation with amplitude 0.01m



4-2- State Feedback Controller Design

In this section the state feedback control of the car suspension system is addressed. The main controller

objective in this section is stabilizing the states of closed-

loop system and removes the body vibration as fast as

possible. Online algorithm is adopted for the applications

require very fast time response for the automobile

applications [36, 37, 38, 39]. Online algorithm is adopted

in this work. The controller of the system is designed

based on the linear model derived from the nonlinear

model described in Eq. (3 and 4). The linear model is

generated automatically from the Simulink model. The

dynamic equations of the system under study are

rearranged as:

�̇�(𝑡) = 𝐴𝑥(𝑡) + 𝐵 [𝑢(𝑡)𝑟(𝑡)

] (6)

𝑦(𝑡) = 𝐶𝑥(𝑡) + 𝐷 [𝑢(𝑡)𝑟(𝑡)

] (7)

Here 𝐴, 𝐵, 𝐶, and 𝐷 are the state space model matrices,

𝑥(𝑡) = [𝑧𝑢 𝑧𝑠 �̇�𝑢 �̇�𝑠], 𝑢(𝑡) is the control input, 𝑟(𝑡)

is the disturbance of the road. The derived state space

model is checked for the controllability and observability.

The control input is generated using the feedback gain

matrix by linear quadratic regulator (LQR). This method

has an advantage over the pole placement method because

it allows the feedback gain matrix to be determined that will result in the minimum amount of energy being

required to stabilize the system [Franklin et. al. 2018].

The proposed controller block diagram is shown in

Fig.(7), where 𝐿 is the observer gain vector. The state

feedback control law is given by:

𝑢(𝑡) = −𝐾𝑐𝑥(𝑡) (8)

In this study, the observer is designed in order to estimate

the immeasurable derivative of the states in the system.

There are many different design techniques used to design

the observer gain matrices. In this work, the gains of the

controller and the full state observer are found by

minimizing a linear quadratic performance index 𝐽 =

∫ (𝑥𝑇𝑄𝑠 𝑥 + 𝑢𝑇𝑅 𝑢) 𝑑𝑡∞

0, which leads to solving the

algebraic Riccati equation:

𝐴𝑇𝑃𝑠 + 𝑃𝑠𝐴 + 𝑄𝑠 − 𝑃𝑠𝐵𝑅−1𝐵𝑇𝑃𝑠 = 0. (9)

The derived dynamic model in conjunction with state

feedback controller is simulated and the results are

addressed in this section. The controller gains are

designed to be𝐾𝑐 = [0 240 600 90]. Different disturbance signals are used to check the

effectiveness of the designed controller. Step signal is first

used as 𝑟(𝑡) = 0.1 ℎ(𝑡) , where ℎ(𝑡) is a unit step

function. The uncontrolled and the controlled positions of

the car are shown in Fig.(8), the controlled displacement

in Fig.(8-b) of the car is reduced drastically due to controller action. A square disturbance is tested in

simulation, the disturbance signal shown in Fig.(9). The

behavior of the system is shown in Fig.(10-a) and (b). It

can be noticed from the behavior that the system can

overcome the effect of the disturbance very effectively.

Another test is performed in simulation, the signal in Fig.

(11-a), is performed as disturbance. The response of the

open and closed loop system are presented in fig. (11-b

and c). Fig. (12), demonstrates the trajectory of the states

( 𝑧𝑠 , �̇�𝑠 ) for the cases of square disturbance and the

disturbance in Fig. (11-a), the figures reveal that the controlled system is globally asymptotically stable

according to Lyapunov. Generally, the linear quadratic

regulator has successfully stabilized the system and

removes the vibration without any abnormal behaviour.

Fig. 8. Dynamic

response due to Input𝑢(𝑡)

−𝐾𝑐

𝐿 +

−

𝐶 Desired

States + −

Disturbance 𝑟(𝑡)

Estimated States �̂�

Fig. 7. State feedback and observer diagram.



Fig. 8. Dynamic response due to step disturbance: (a) Uncontrolled (b)

Controlled

Fig. 10. Response due to square disturbance: (a) Uncontrolled (b)

Controlled

Fig. 9. Square disturbance signal

Fig. 11. Disturbance signal and response: (a) Signal

Time [𝑚]

𝑧𝑠 [𝑚]

(a)

Time [𝑚]

𝑧𝑠 [𝑚]

(b)

Time [𝑚]

𝑟(𝑡) [𝑚]

𝑧𝑠 [𝑚]

Time [𝑚]

(a)

Time [𝑚]

𝑧𝑠 [𝑚]

(b)

𝑟(𝑡) [𝑚]

Time [𝑚]

(a)



Fig. 11. Disturbance signal and response: (b) Uncontrolled (c)

Controlled

Fig. 12. System states trajectory of [𝑧𝑠, 𝑧�̇�]: (a) square disturbance, (b)

Disturbance signal described

5- CONCLUSIONS Several conclusion remarks can be drawn from the

analysis and results of the work. The conclusion can be

listed as:

1- That the Bingham plastic model can be used for

the analysis and modeling of the magnetorheological fluid damper.

2- That the magnetorheological fluid damper offers

a wide range of damping force and damping

characteristics for different road excitation

(sinusoidal, step, and random). However, the

passive response of the damper is not enough to

get a good performance.

3- Damping force and characteristics can be

controlled by control the input current to the

magnetic circuit. However, the magnetic

saturation problem is the main reason to limit the damper performance. In addition, current effect

is not the same for the external excitations.

4- The linear quadratic regulator (LQR) has

successfully stabilized the system and removes

the vibration without any abnormal behavior.

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