Fuzzy Sliding Mode Control Application to Large time Varying Systems

Copyright @IFAClnlelligcnt Components IIId Instruments for Control Applications. Budapest, Hungary. 1994

FUZZY SLIDING MODE CONTROL APPLICATION TO LARGE TIME VARYING SYSTEMS

P. Cerf, S. Boverie, F. Vemieres SIEMENS AUTOMOTWE SA.lAboraloire MIRGAS. averwe du Mirail. BP 1149.31036 TouloMSe CedeJ:. FRANCE

Abstract- This paper presents an application of fuzzy control techniques for large time varying systems. This study is based on an analogy between fuzzy control and sliding mode control techniques. First, special attention will be payed 10 the analysis of sliding mode control properties. next a generalization of sliding mode control structure to large time varying systems is presented and finally this concept will be extended to fuzzy controllers. Some application results are shown in the last section. Keywords: Fuzzy controllers. sliding mode. large time varying systems.process control.

We

1. INTRODUCTION

define

in

S=E+AE=

Fuzzy logic which appeared back in the 1960's. has just recently shown signs of slrong growth. One of its prefered fields of application is process conlrol. Fuzzy conlrol sometimes presents an interesting alternative when the processes to be controlled are non-linear, difficult to model or are time varying. It also has the advantage of presenting a good trade~ff between cost and performance.Different authors like Mac Vicar-Whelan, have proposed some formalization of the fuzzy conlrOl SlJ"Uctures. We present in this paper an extension of the Mac Vicar-Whelan SlrUCture in order to take into account large time varying systems. This study is based on the analogy between fuzzy conlrol and non-linear sliding mode conlrol. The first section is dedicated to the sliding mode conlrOl. The second one presents the new control SlrUCture. Finally, we present some results obtained on an application in the last section.

the

phase

plane

a

o called the sliding line.

line

= 0

Fig.2 Phase plane analysis

When S>O ( S
2. SLIDING MODE CONTROL

2.1. SLIDING MODE CONTROL The sliding mode control stems from the studies on relay based analogic conlrol (Fig.l). The principles of this robust non-linear control method are quite simple and can be analyzed in the phase plane ( E ,E ) (Fig 2). e

Fig.1 Sliding mode control structure

115

open loop system behavior when we apply the maximum admissible control value.

subsystem 1

E

~---...r---'lf .

t

,_,

+f1u

..

----------~-~r-----------~ ,, , .

t

,

- f1u

',Fast system

Fig.6 Representation of the sliding line! in the phase plme

For the first sub-system we can define: S (S :: E + A. E :: 0 ) with If S > then ~Ul :: KI

Fig.3lnfluence of the proces! dynamic respome

2.2. GENERALIZED SLIDING

MODE CONTROL

If S < - then

~UI

=-

KI

Sliding mode control is not without problems. A rapidly varying, large amplitude control effort ± f1u applied at each S sign changing is not always imaginable (non-linearity problems, oscillations, hard constraints on the controlled process on the actuators,

If -- < S < then ~UI = KI S and for the second sub-system: s (s = OE = 0) with If s > then ~U2 = K2<1>

...).

If s < - then ~U2

With the development of digital control, it was possible to extend this concept and to consider that area around the sliding line corresponding to a smooth control. This control could be chosen proportional to the distance from the sliding line. Beyond this area. the control values are saturated (Fig4).

If -- < s < then ~U2

K2<1>

=

K1s

Combining the output of each sub-system, in the non-saturated area we have: f1ul = KI [E + A. E) f1U2 = K2 0 E

i

~u = ~UI + ~U2:: Kt [E + (A. + K2 0 Kt

+t.u max

.

S =E+ A£=O

S =E+ ).£=-4> Fig.4 rotation of the sliding line for a time variant system

The control law can be expressed by (Fig 5): ~u

= ~UIIW< ~u

If

S > 4> then

If

S < - 4> then

If

-4> < S < 4> then

::

~u

E

+ A. ' E = 0 with

A' = A + K2 0 KI

In the transient areas, when only one of the two systems are saturated we have (Fig. 7): ~u = KI[E + A. E) ± K2 or ~u = K2 0 E ± KI4>

~UIIW< ::

=

It is possible to rotate the sliding line around the point (E = i = 0 ) by modifying the values of the gains Ki. and the equivalent dynamic response defined by the sliding line could be easily adapted to the change of the process . When both the subsystems are saturated we can write: iaul = IKI4> ± K24>1

. .. . '.

K.S

+6.

e

*~ I

)t ]

Then a new sliding line is defined: S'

-6u max

=-

I

Fig.S Genera1i.zcd sliding mode controller strucwre

2.3. APPuCATION OF 1lIE GENERALIZED SLIDING MODE CONTROL TO LARGE TIME VARYING PROCESSES

If we consider a physical system with a time varying dynamic response, we have seen in section 2.1 that this can imply some constraints for the choice of the parameter A. A new control structure is proposed Fig-

ure6.

Fig.? Rotation of the sliding line

116

The control action can be determined as a non-linear = F(S(£. ,i If we examine the control surface, it is possible to define for each modal point ( £j ,i j ) a gain K ij (in the non saturated area) like ~u = I(;j . S(£., i ) . S being an image of the distance to the sliding surface. Between these modal points the control values are calculated with a non-linear or a linear interpolation (Raymond el aI, 93) (depending on the fuzzy operators: min-max,.max) and ~u can be expressed as: du = K(E, £ )S( E, 'E.) .As for the conventional sliding modes, this sliding curve defines the behavior of the system. The control value ~u applied to the process tries to bring the state vector ( £ ,i ) on this curve. If we consider a process with time varying response this involves some constraints in the choice of the sliding curves. In order to take into account the different dynamics, it is necessary to rotate the sliding curve around the point (£. = i = 0 ). To do that it is possible to consider the same structure as for the conventional sliding modes (Fig 11).

This structure allows a global rotation of the control surface around the point (£. = i = 0 ). Some distortions of this control surface will appear in the transient Nevertheless it is possible to choose c)l, A and 0 in order to minimize their influences.

function:~u

areas.

3. EXTENSION TO THE FUZZY CONTROL The structure of a conventional fuzzy controller uses as inputs the error £ and the derivative of the error i . Mac Vicar-Whelan has suggested asymmetrical control structure (Fig.8 ) :The fuzzy control value is equal to zero on the diagonal line and positive ( negative) above ( below) this line; the value of this fuzzy control increases progressively to the distance from this line.

::' pi( ZE NS

PS ZE

NM

NS

PM PS ZE

NB

NM

NS

: NS ; NB

NB

NM

: : NM

NB

NB

,::>~::~;";5 PM

::~;-~;o~

,d

pr'

':;:.:f<~ : ZE

::;;. < ~. <~. ;: ;;.<~, <;;.

'
NB

PB PM

PB PB

PB PB

PS

PB

ZE NS NM

PM PS ZE NS

PM PS ZE

:, "

PB PB PB

».

PB PM PS ZE

Fig.8 11nIC1UJ'e of fuzzy controller proposed by Mac Vicar-Whelan

Fi,. ll Fuzzy sliding mode controller I1nIcture

We have in the non-saturated area: dUI = KI[ K(E,'E) S( E,'E } ] du2 du = ~UI + ~U2

It is possible to represent the equivalent control surface in a three dimensional space (..1", £. , i ). We can see that the control surface is non linear, depending on the choice of the modal points of £ and i (Fig 9).

du

=

= K2

k(£)1

KI K(E,'E ) [S(E,'E ) +

K2k(£) .£] KIK(E,£ ) ~u = KIK(E,£) S'( E,'E ,)and S' is the new sliding curve. In the saturated areas we have: lau ... 1 IKI ± K2 1 ~U..,. with ~UI..... - ~Ulm In some cases it should be suitable to choose Kl and K2 as K 1+ K:z=ete. This allows to avoid any modification of the maximum value of the control variable. Some distortions will appear between saturated and non-saturated areas, nevertheless the errors are relatively small and have no influence on the closed loop control behavior. As for conventional sliding modes, by modifying the values of the gains Ki this structure allows a global rotation of the control surface around the point (£. = i = 0 ). lE

- I -4j

Fig.9 Fuzzy control surface in a three dimensionnai space

4. APPLICATION

The intersection between this surface and the plane ( £. , i ) gives the zero control curve S(£., i ) = 0/..1" = 0 (Fig 10) which is equivalent to a sliding curve (Palm, 92).

This concept has been tested in real time on an automotive 4 stroke engine for controlling idle speed. Idle speed control is always considered as a difficult problem.lt must be efficient every time and assuming a compromize between conflictual goals: - for comfort, it must ensure a good stability of the engine speed and a good robustness with respect to load disturbances (power steering, lights, air conditionner, ...), - for consumption and pollutant emissions the regulation set point must be as low as possible. On the other hand the features of an automotive engine are very different according to its utilization (stationary and rolling vehicle). In the first case the

1:

fast bevahiour Fi,.IO ZlCIO control alIVe in the phase plane

117

enq!ne apeed rpm

process to be controlled is only the engine (engine inertia) in the second case, the process to be controlled is the overall vehicle through the mechanical coupling (vehicle inertia), which corresponds to a tremendous change of parameter values. The sensors located on the engine allow the detection of these different phases, and allow one to supervise the gains in order to rotate the sliding curve. The results obtained show the quality of the control : the decrease of the engine speed must be smooth, without irregularities or undershoot

400

300

I I

200

J""-

100 0

'-

"-

I

I o

0.5

r-..... 1.5

2

2.5

3 •• c .

Fig.IS Reaching idle lpeed set point with load (power Iteering+ventilator) engine apeed rPM

enq!ne apeed rpa

1700 '50

1500 I~

1300

~

900

~JI.I

I..

1100 900

850 .lec .

22

20

26

24

28

2

OfT

~

l"'lJ1I!

800

., 'Y"'",. ....,.. ••• IIdJNI . " 1

,.. " I

15

10

20

... e

Fig.13 Idle speed regulation with load (power steering)

1000

5

~

7

8

9

aec .

REFERENCES

engine apeed rpm

1500

...

C. Raymond. S.Boverie, I.M. Le Quellec, "Practical realization of fuzzy controUer- comparison with conventionn" methods", Proceedings of !he EUFIT symposium in A.chen, SeplI993. R. Palm, " Sliding mode fuzzy control" FUZZ-IEEE 92', San Diego, March 92

2500 2000

IAA

In this study, we have developed an original control structure based on an analogy between a non-linear control technique like sliding mode and the fuzzy control technique. This structure allows to take into account large dynamic variations of the processes. This algorithm has been tested succesfully in real time on an idle speed controller.

n ON

700

--

5. CONO-USION

I I

al.•

900

'"

Fig. 16 Reaching idle speed set point with load and with nmning vehicle

~

100 0

"""

700

30

Fig. 12 Idle speed regulation without 10lld

engine apeed rpa

"

"\.

\

". .e

500 3

Fig.14 Reaching idle speed set point wi!hout load

118