Using Fuzzy logic to control

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					     FUZZY CONTROL OF MECHANICAL VIBRATING SYSTEMS

        Kateřina HYNIOVÁ and Antonín STŘÍBRSKÝ and Jaroslav HONCŮ
                           Department of Control Engineering,
              Faculty of Electrical Engineering, Czech Technical University
     Karlovo náměstí. 13, 121 35 Prague, Czech Republic, honcu@electra.felk.cvut.cz


Abstract: The main role of a car suspension system is to improve the ride comfort and to
better the handling property. It usually consists of a spring and a damper. To improve the
properties of suspension systems, many studies based on adaptive control methods, preview
control, sliding mode control, H control etc. have been done.
     In this paper, fuzzy logic is used to control active hydropneumatic suspension. It is one
of the most active research and development areas of artificial intelligence at the present
time, particularly in the automobile industry because fuzzy logic can improve vehicle ride
comfort and road handling performance. 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.

Key words: vehicle, suspension, one-quarter-car model, fuzzy logic control


1.  Quarter-car model
    In this paper, we are considering a quarter car model with two degrees of freedom.
This model uses a unit to create the control force between body mass and wheel mass.
The motion equations of the car body and the wheel are as follows:
                  m b  b  f a  k 1 (z b  z w )  c s (z b  z w )
                      z                                         

                  m w  w  f a  k 1 (z b  z w )  k 2 (z w  z r )
                      z
with the following constants and variables which respect the static equilibrium position:
          mb ….. body mass (one quarter of the total body mass)          250 kg
          mw ….. wheel mass                                                35 kg
          k1 ….. spring constant (stiffness) of the body              16 000 N/m
          k2 ….. spring constant (stiffness) of the wheel            160 000 N/m
          fa ….. desired force by the cylinder
          cs ….. damping ratio of the damper                             980 Ns/m
          zr .…. road displacements
          zb ….. body displacement
                   zw ….. wheel displacements

To model the road input let us assume that the
vehicle is moving with a constant forward
speed. Then the vertical velocity can be taken
as a white noise process which is
approximately true for most of real roadways.
To transform the motion equations of the
quarter car model into a space state model, the
following state variables are considered:

                           x=[x1, x2, x3, x4]T
where:
     x1= zb-zw -                   body displacement
     x2= zw-zr -                   wheel displacement
     x3= z b
              -                   absolute velocity
                                   of the body
        x4= z w
                          -       absolute velocity
                                   of the wheel
                                                                       Fig.1. One-quarter-car model

     Then the motion equations of the quarter car model for the active suspension can be
written in state space form as follows:
                                                     x  A.x  B. f a  F .z r
                                                                          
with
            0             0       1      1       0 
            0                                               0
            k             0       0       1  
                                                    0 
                                                              1
                                    c      cs       1 
       A  
                1
                           0       s           B      F 
            mb                     mb    mb       mb     0
            k1                                               
                           k2      cs       cs     1      0
                                                   
            mw            mw      mw      mw       mw 



2.     Fuzzy logic controller

       The fuzzy logic controller used in the active suspension has three inputs : body
acceleration        z
                    b ,       body velocity   zb , body deflection velocity zb  zw and one output :
                                                                             
desired actuator force fa. The control system itself consists of three stages : fuzzification,
fuzzy inference machine and defuzzification.
     The fuzzification stage converts real-number (crisp) input values into fuzzy values
while the fuzzy inference machine processes the input data and computes the controller
outputs in cope with the rule base and data base. These outputs, which are fuzzy values, are
converted into real-numbers by the defuzzification stage.
     A possible choice of the membership functions for the four mentioned variables of the
active suspension system represented by a fuzzy set is as follows:


     for body deflection velocity z b  z w
                                                for body velocity z b
                                                                    




    Fig. 2.: Membership function for body             Fig. 3.: Membership function for body
             deflection velocity                               velocity


     for body acceleration   z
                             b         for desired actuator force   fa




    Fig. 4.: Membership function for body              Fig. 5.: Membership function for
             acceleration                                       desired actuator force


The abbreviations used correspond to:

    NV …        Negative Very Big                      PS    …     Positive Small
    NB …        Negative Big                           PM    …     Positive Medium
    NM …        Negative Medium                        PB    …     Positive Big
    NS …        Negative Small                         PV    …     Positive Very Big
    ZE  …       Zero


        The rule base used in the active suspension system can be represented by the
following table with fuzzy terms derived by modelling the designer’s knowledge and
experience.
 Table 1.: Rule base
  zb  zw z b
                      z
                          b        fa         zb  zw
                                                            zb
                                                                      z
                                                                        b       fa
     PM         PM              ZE        ZE      PM              PM     P or N        NS
     PS         PM              ZE        NS      PS              PM     P or N        NM
     ZE         PM              ZE        NM      ZE              PM     P or N        NB
     NS         PM              ZE        NM      NS              PM     P or N        NB
     NM         PM              ZE        NB      NM              PM     P or N        NV
     PM         PS              ZE        ZE      PM              PS     P or N        NS
     PS         PS              ZE        NS      PS              PS     P or N        NM
     ZE         PS              ZE        NS      ZE              PS     P or N        NM
     NS         PS              ZE        NM      NS              PS     P or N        NB
     NM         PS              ZE        NM      NM              PS     P or N        NB
     PM         ZE              ZE        PS      PM              ZE     P or N        PM
     PS         ZE              ZE        ZE      PS              ZE     P or N        PS
     ZE         ZE              ZE        ZE      ZE              ZE     P or A        ZE
     NS         ZE              ZE        ZE      NS              ZE     P or N        NS
     NM         ZE              ZE        NS      NM              ZE     P or N        NM
     PM         NS              ZE        PM      PM              NS     P or N        PB
     PS         NS              ZE        PM      PS              NS     P or N        PB
     ZE         NS              ZE        PS      ZE              NS     P or N        PM
     NS         NS              ZE        PS      NS              NS     P or N        PM
     NM         NS              ZE        ZE      NM              NS     P or N        PS
     PM         NM              ZE        PB      PM              NM     P or N        PV
     PS         NM              ZE        PM      PS              NM     P or N        PB
     ZE         NM              ZE        PM      ZE              NM     P or N        PB
     NS         NM              ZE        PS      NS              NM     P or N        PM
     NM         NM              ZE        ZE      NM              NM     P or N        PS


    The linguistic control rules of the fuzzy logic controller obtained from the table above
used in such case are as follows:
                                                                   z
                  R1: IF ( z b  z w =PM) AND ( z b =PM) AND (  b =ZE) THEN (fa=ZE)
                                                

                                                               z
                 R75: IF ( z b  z w =NM) AND ( z b =NM) AND (  b =P) THEN (fa=PS)
                                              

Thus the rules of the controller have the general form of:

                                                                 z
                  Ri: IF ( z b  z w =Ai) AND ( z b = Bi ) AND (  b = Ci ) THEN (fa= Di)
                                              

     where Ai, Bi, Ci and Di are labels of fuzzy sets representing the linguistic values of
                  z
z b  z w , z b ,  b and fa respectively, which are characterised by their membership
          
functions.
         The output of the fuzzy controller is a fuzzy set of control. As a process usually
requires a non-fuzzy value of control, a method of defuzzification called “center of gravity
method” is used here:


                                           f *
                                          Fa
                                                        D   ( f ).df
                                   fa 
                                               
                                               Fa
                                                    D   ( f ).df

     where D(f) is corresponding membership function.

3.   Simulation results
     In this section, the controller was tested to compare the results of the designed fuzzy
logic controller with a traditional pasive suspension system.

     First, a step response (step is 10 cm height) was simulated:




                  Fig.6.: Active and passive suspension system – step response

The step response of the active suspension build with fuzzy command is clearly more
efficient than the step response of the passive suspension.
         Next, the response of the body displacement of active suspension with fuzzy logic
controller when white noise road input is used is compared to the response of the passive
suspension (Fig. 7).
         The result of the active suspension is four times better than the result of the passive
suspension, the body displacement is less than 0.1mm for the input of more than 15cm.




              Fig.7.: Active and passive suspension system – white noice response

4.   Conclusion

     In this paper, the new active suspension control system is proposed to achieve both ride
comfort and good handling. This aim was achieved with respect to the results of the
simulation; the results of the active suspension system based on the fuzzy logic controller
also show the improved stability of the one-quarter-car model.

References

[1] Cai, B. - Konik, D.: Intelligent Vehicle Active Suspension Control Using Fuzzy Logic,
         IFAC World Congress, Sydney, Vol.2, pp.231-236, 1993
[2] Rouieh, S. - Titli, A.: Design of Active and Semiactive Automotive Suspension Using
         Fuzzy Logic, IFAC World Congress, Sydney, Vol.2, pp. 253-257, 1993