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DESIGN OF ADAPTIVE FUZZY CONTROLLER WITH NONLINEAR SLIDING MODE
m
Copyright 02002 IFAC
ELS E V I ER
15th Triennial World Congress, Barcelona, Spain
PUBLICATIONS
www.elsevier.com/locate/ifac
DESIGN OF ADAPTIVE FUZZY CONTROLLER WITH NONLINEAR SLIDING MODE Dae-Sik Leea and Kwang Y. Leeb a
School of Computer and Communication Engineering, The Daegu Universig,712714, Korea Department of Electrical Engineering, The Pennsylvania State University, Universig Park, PA 16802, U.S.A
Abstract: Although the general sliding mode control has the robust property, bounds on the disturbances and parameter variations should be known a priori to the designer of the control system. Fuzzy logic provides an effective way to design a controller of the system with disturbances and parameter variations. Therefore, combination of the best feature of the fuzzy logic control and the sliding mode control is considered. The adaptive fuzzy variable structure controller developed in t h ~ spaper employs fuzzy controller with nonlinear sliding mode. Therefore, the resulting system can resolve the conflict between the opposing requirements of static and dynamic accuracy that are encountered when designing the linear sliding mode. A variable length pendulum system is used to demonstrate the availability of the proposed approach. Copyright 0 2002 IFAC Keywords: fuzzy control, adaptive control, sliding mode control, nonlinear systems 1. INTRODUCTION
response of the error, during sliding mode, can be prescribed in advance and the system is completely robust and insensitive to parameter variations and external disturbances.
Automatic control with the concept of variable structure system (VSS) is a special class of nonlinear controls. This control differs from other control systems mainly in that its structure is not fixed but is varied during the control process.
The fuzzy set theory, established by Zadeh [4], has been developed for over 20 years. Its most prominent application is in the area of control engineering. In general, if a system is ill defined or too complex to have its mathematical model, fuzzy control provides an effective way to design a control. Based on fuzzy logic, the fuzzy controller converts a linguistic control strategy into an automatic control strategy [5]. The most advantageous and controversial property is that fuzzy controller does not rely on the mathematical model of the process. Control rules are constructed by using the knowledge of experts or operators' experience. In other words, the fuzzy controller is human dependent. Different expertise will give different rules for the same performance. There have been many studies investigating the relationship between sliding mode control and fuzzy control. Kawaji and Matsunaga[6] proposed a method of generating fuzzy rules for servo motors based on the VSS control. Yager and Filev[7] determined the fuzzy rules according to the sliding mode condition. Glower and Munighan[S] presented fuzzy sliding mode controller and assured stability for each of this fuzzy controllers. Lu and Chen[9] combined the best features of self-organizing fuzzy control and sliding mode control to achieve rapid and accurate tracking control of a class of nonlinear systems. The fuzzy rule base is used to approximate the VSC through
The structure of a control system may change either accidentally or intentionally in accordance with a definite rule. The VSS considered in t h ~ spaper fall into the category of systems where the structure is intentionally changed during the transient in accordance with the present structured control law. The variable structure control (VSC) law developed, based on the theory of VSS, usually provide for changes in the structure of the controlled system whenever the state crosses a certain surface called the switching surface. If the system is forced to remain on the predetermined switching surface, it results in a dynamic behavior that is largely determined by design parameters and equations defining the switching surface. Consequently, new properties that were not present in the original system can be obtained for the controlled system. Therefore the system is robust and insensitive to external disturbances and parameter variations. The state trajectory undergoes a small-amplitude highfrequency chattering motion along the surface; which is known as the sliding mode or the sliding regime. The designed surface is then referred to as the sliding manifold or the switching surface [1]-[3]. The advantage of t h ~ sapproach is that the transient
151
self-organization and the VSC effort is used to compensate for the approximation error and to provide exponential convergence of the sliding variable. Lin and Chiu [lo] combined the merits of the sliding mode control, the fuzzy inference mechanism and adaptive algorithm. The VSS can be prescribed with an arbitrary transient response. The adaptive fuzzy variable structure controller used in this paper employs fuzzy controller with nonlinear sliding mode [ 111, while the previous methods used the fuzzy controller with linear sliding mode. This can resolve the conflict between the opposing requirements of static and dynamic accuracy that are encountered when designing the linear sliding mode. Thus, the speed of response is improved without overshoot and a good transient response has been obtained. 2. VARIABLE STRUCTURE CONTROL
WITH NONLINEAR SLIDING MODE 0 0
Consider a single-input n -th order nonlinear system =f(X,t)+g(X,t)u(t)
a = ; .:
(1)
0
(2)
u+ u
where s ( 2 , t ) = O
s ( 2 , t )> 0 s ( 2 , t )< o
2 dt
:
:
0
"'
1
.
.
.
(8)
... h,_
P
(3)
i = 1,2,. ..,q , and q = { p ( p + l)(p + 2)) / 6
j=l
Step 3: From (5), (6) and (9), the nonlinear switching function can be written as follows:
i=l
(4)
j=1
The nonlinear parts used to improve the performance in the transient state are given in the second term. Differentiating s ( 2 , t ) with respect to time yields
where y is positive real constant [l]. Define a nonlinear sliding surface s ( 2 , t ) in the state space R" as follows:
s ( 2 ,t ) = SL (2, t )+ SNL(2, t)
:
where di is a specified constant value, x p u = 3 ,
is the switching surface and
s2 =sS<-ys2
0 0
(9)
2 = X - X , . The condition for the sliding mode to exist on the switching surface is _I _d
"'
"'
regarded as a nonlinear feedback system with the state vector z and the control sNL Then, from [ 111, the following form can be obtained
Both f ( X , t ) and g ( X , t ) are assumed to be continuous in X . The control problem is to find u ( t ) such that X will track the desired trajectory X , = [x,, X,,. ..,x,(~-~)]' . The control input u(t) has the following form: u(t>=
0
0 1
-4 h, 4
where u ( t ) is the control input, X = [x,X,...,x(~-')]' is the state vector, function f ( X , t ) is unknown but bounded by a known continuous function F ( X , t ) , and the control gain g ( X , t ) is unknown but limited in a certain fixed range as follows: 0 < Gmi, < g ( X , t ) < G,,
1 0
(5)
152
where
r is a number of nonlinear terms given by the n
derivative of (10) and cq,= 3 , j
r
KS >-> Gln,
= 1,2,. .. , r . From
i=l
r>o
(23)
then the resulting system satisfies the sliding mode existence condition in (4), and the error trajectoly move toward the switching. Therefore, the dynamics of the system are always stable and the error 2 will slide into the origin.
(1) and (111,
(12) In the sliding mode, the equivalent control ueq is
3. ADAPTIVE FUZZY VS S
obtained as follows:
In general, the sliding mode control assumes that the uncertainties are bounded and their bounds are available to the designer. However, bounds on the uncertainties may not be easily obtained because of the complexity of the system structure involving uncertainties. Moreover, the magnitude of external disturbance cannot be easily estimated. Thus the cost for implementing the sliding mode control is rather high for a system having large upper bound of uncertainties. On the other hand, the fuzzy inference mechanism can be constructed to estimate the system uncertainties. In t h ~ spaper a fuzzy sliding mode control is developed in which a fuzzy inference mechanism is used to estimate the upper bound of the system uncertainties.
(13) However, since f ( X , t ) and g ( X , t ) are unknown, the following equation is used which is estimated as precisely as possible: ieq
=
i - l
(P.7
+',own)
(14)
7
and i are estimated value of f and g , where respectively, such that
Replacing KN by K f in (19), the following equation
and uhown is an independent term for system parameters and external disturbance, i.e.,
can be obtained:
u(t) = Zi, +us - K , sgn(s) where
Now, consider the following state feedback control law : u = ueq+u, +us
estimated by
fuzzy
inference
satisfy (4) is
(18)
K i =g-'f
-i-'?+(i-' -g-')uhOwn
(25)
But the ideal value K i cannot be obtained exactly
(19)
owing to the parameter variations and external disturbances. Therefore, it is assumed that there exists a specific K, that achieves minimal control
(20)
effort, i.e., satisfying the sliding mode existence condition (4) and
And us is defined by us =-K,s
is
mechanism. An ideal optimal value of K , that will
where uN is used to eliminate the influence of uncertain terms of (15) and (16) as follows: uN = -KN sgn(s)
K,
(24)
which determines the dynamic response of s ( 2 , t ) .
K, - K i l ~ l s l
Inserting (18) into (12), the derivative of s ( 2 , t ) is given by
(26)
where E is a small positive real constant. In general, the fuzzy control rules are a set of fuzzy implementations of linguistic term. Consider a singleoutput fuzzy controller with 2 inputs, s and S , and m rules in its rule base, represented by R,, R2,...,R, , respectively. The general form of the .j -th rule is
S= f -gi-lj
+(&' - l ) u b o w n -gKN sgn(s>-gKSs (21) If control gains are selected as
R,
:If(s is
where
153
A:) A(S is A:), then K+ = a, (27)
AND intersection operation,
A
and let the adjustable parameter error vector be
fuzzy set characterizing the variable A& of the premise, vp , in the j -th rule, output form the
-aj
consequent parameter.
(3 1)
&=u-&
j -th implication,
Ki
(30)
K, = & W T
s, S inputvariables,
Choose the Lyapunov function candidate as
There are m rules to define the rule base of the fuzzy controller and each rule can give a distinct value of the output, the weighted average of the individual output, K+, j = 1,2,...,m , is used to obtain the
where y is a positive constant. Differentiating V with respect to time, the following equation is obtained:
output, K , . The weighting assigned to each K+ is
(33)
the firing strength of the j -th rule, designed as uJ. The firing strength of the j -th rule is determined from U J = min (P:
(4)
(s), P:
Substituting (12), (18)-(20) and (25)-(26) into (33) gives
(28)
where p&(vp) is the value of the membership function at vp in the fuzzy set A,;. The membership functions of A, are shown in Fig. 1, where the ordinary triangle shape function is used for convenient. Fuzzy output K , can be calculated by the center of
=
.( f +g 4 ,
gKss - U h o w x )
f + gi,,
Y
uhowx)- gKJs - gKss2
-
+g(KJ - K f )I s I +LabT I s(
m
f + gi,,
Y
,=l
uhowx)- gKJs - gKss2
-
+
[
Gm,WIsI ---a
Y
If adaptive law is defined as
where a = [al,a,,..., a,] is a adjustable parameter vector; -aJ is the center of the membership function of K+ , and
-
Y
-gKJ s + gKJs - gKf I s I +LabT I s(
area (COA) defuzzification as follows:
,=l
1-: +--a-a~
i,, - K~ sgn(s) - K ~ S -)
.I-
-aT -Gm,~s2 (34)
bT = Y G,, Is1 W T
w = [al,w,, .,W, ] I C W , is a firing strength
(35)
then (35) becomes
..
,=I
V 5 -G,,E
vector.
sz - G,,,K, sZ 5 -7 sz < 0
which requires that
... “1
... “2
i‘
>L+%E . G,i”
G,”
4. SIMULATION RESULTS
I
“2
A: A,’
“1
...
K,
(36)
In order to demonstrate the validity of the proposed controller, the variable length pendulum in Fig. 2 is used as an example of the control [12], where B is the angle, l is the length g is the acceleration of gravity, and m is the mass. This model was used to identify the gymnast’s center of mass as the kip is performed. The dynamic equation of the system can be written as
Asm-’ A,”
...
I
Fig. 1. The membership functions of
A,
Define & as the optimal vector that achieves the minimal effort of sliding mode control as follows:
154
4
Rule 4: if s is Z and S is P then K , is NS
b
Rule 5: if s is Z and S is Z then K , is ZE Rule 6: if s is Z and S is N then K , is PS Rule 7: if s is N and S is P then K , is PM Rule 8: if s is N and S is Z then K , is PB Rule 9: if s is N and S is N then K , is PH Rule 1 implies that the switching variable s is far away from the switching surface and it is keep increasing. Therefore, a large K , is required in order to reach the slidmg-mode condition
g / t o = l O , and m l d = l
!,IC,=fl.S.
Fig. 2. Variable length pendulum.
B = 0.5sin~(1+0.5cos8)8’/A(Q) -10 sin8(1+ cos 8)/ A ( 8 ) +u / A ( 8 ) + v(t)cos 8 where A(8)
= 0.25 (2
+ cos 8)’
(37)
and the disturbance is
v(t) = 2cos(3t) . Following Lin and Chiu [ l l ] , the membership functions for the fuzzy sets corresponding to switching variable s , S and upper bound of the K , is defined in Fig. 3, where the associated fuzzy sets involved in the fuzzy control rules are defined as follows: N : negative P : positive NB:negative big NS negative small PS: positive small PB: positive big
(20) is selected as K ,
a‘
G,,
Z : zero NH:negative huge NM: negative medium ZE: zero PM: positive medium PH: positive huge
“6
“7
“8
“9
initial value of states is
5. CONCLUSIONS An adaptive fuzzy sliding mode controller is designed using the advantages of the sliding mode control, the fuzzy inference mechanism and the adaptive algorithm in this paper. The VSS can be designed with an arbitrary transient response and is theoretically robust on disturbances and parameter variations. Moreover, the VSS with nonlinear sliding mode can resolve the conflict between the opposing requirements of static and dynamic accuracy that are encountered with the linear sliding mode control. The fuzzy inference mechanism is used to estimate the upper bound of the disturbances and parameter variations. In addition, the adaptive fuzzy inference mechanism estimate the optimal bound of the
K,
Fig. 3. Membership functions of fuzzy sets. Rule 1: if s is P and S is P then K , is NH Rule 2: if s is P and S is Z then K , is
= 4 . And the
The responses of linear fuzzy adaptive VSS (LFAVSS) and nonlinear fuzzy adaptive VSS (NFAVSS) are shown in Fig. 4, and as can be seen, the transient response is greatly improved. The Fig. 5 gives the responses for the 0.1Hz square wave input with x(0) = [1,0] . The system response for the first square wave input is plot (a) in the Fig. 5. The plot (b) is the response of the second square wave input. The responses over the 4-th square input are the plot (d). Fig. 5 shows the adaptation process of the nonlinear fuzzy VSS with adaptive law, where the adaptive law improves the system response.
;””:
4 5
values of The boundary
q o ) T = [1,0] .
-0 1
“4
initial = 20 .
4 and 9
** “3
adaptive parameter
values of control gain g ( X , t ) are Gmi,= -
mode control is determined by the following fuzzy inference rules:
“2
= 10. And
are updated with a(0)= [5,10,20,40] and y
Since only three fuzzy subsets, N , Z and P , are defined for s and S , the fuzzy inference mechanism only contains nine rules. The K , for the sliding
“1
Let X T = [X1,X2]= [8-8,,8-8,] . From (lo), the nonlinear switching variable is obtained as s(X) = X2 + 42, + 52; . The estimated equivalent control input of (14) is determined as Zi, = -42’ -15X;Xz . The value of gain constant of
NB
Rule 3: if s is P and J. is N then K , is NM
155
disturbances and parameter variations. A variable length pendulum system is used to demonstrate the applicability of the proposed controller. Simulation has shown that the transient response of the system is greatly improved with the nonlinear fuzzy adaptive control.
a : LFAVSS b : NFAVSS
REFERENCES 0
Itkis, U. (1976). Control Systems of Variable Structure, John Wiley & Sons. Utlun, V.I. (1983). Variable Structure Systems: Present and Future. Autom. Remote. Control, V01.12, pp. 1105-1 120. Decarlo, R.A., Zak, S.H. andMatthews, G.P. (1988). Variable Structure Control of Nonlinear Multivariable Systems: a Tutorial. Proc. IEEE, V01.76, No.3, pp.212-232. Zadeh, L.A. (1965). Fuzzy Sets. Inform. Control, V01.8, pp. 338-3 53. Park, Y.-M, Moon, U. and Lee, K.Y. (1995). A Self Organizing Fuzzy Logic Controller for Dynamic Systems Using a Fuzzy Auto-Regressive Moving Average (FARMA) Model. IEEE Trans. On Fuzzy System, Vo1.3, No.1, pp.75-82. Kawaji, S. and Matsunaga , N. (1990). Generation of Fuzzy Rules for Servomotor. in Proc. IEEE Inter. Wkshp. Intelligent Motion Contr., Istanbul, Turkey, pp.77-82. Yager, R.R. and Filev, D.P. (1994). Essentials of Fuzzy Modelling and Control, Wiley, New York. Glower, J.S. and Munighan, J. (1997). Designing Fuzzy Controllers from a Variable Structures Stand Point. IEEE Trans. On Fuzzy Systems, Vo1.5, No.1. Lu, Y.S. and Chen, J.S. (1994). A Self-organizing Fuzzy Sliding Mode Controller Design for a Class of Nonlinear Servo Systems. IEEE Trans. Indust. Elect., Vo1.41, pp.492-502. Lin, F.J. and Chiu S.L. (1998). Adaptive Fuzzy Sliding mode Control for PM Synchronous Servo Motor Drives. IEE Proc.-Control Theory Appl., Vo1.145, No.1. Lee, D.S., Kim, M.G., Kim, H.K. and Youn, M.J. (1991). Controller Design of Multivariable Variable Structure System with Nonlinear Switching Surfaces. IEE Proc. -Control Theory Appl., Vo1.138, No.5. Nakawaki, D.E., Joo, S. and Miyazaki, F. (1999). Identification of Motor Skill Characteristics Using a Reduced Model Approach. IEEE Systems, Man, and Cybernetics '99 Conference Proc., Vo1.4.
156
1
2
4
6
10
8
Time[sec]
Fig. 4. Comparison of responses.
Time[sec]
Fig. 5. The system responses for the 0.1Hz square wave.
Copyright 02002 IFAC
ELS E V I ER
15th Triennial World Congress, Barcelona, Spain
PUBLICATIONS
www.elsevier.com/locate/ifac
DESIGN OF ADAPTIVE FUZZY CONTROLLER WITH NONLINEAR SLIDING MODE Dae-Sik Leea and Kwang Y. Leeb a
School of Computer and Communication Engineering, The Daegu Universig,712714, Korea Department of Electrical Engineering, The Pennsylvania State University, Universig Park, PA 16802, U.S.A
Abstract: Although the general sliding mode control has the robust property, bounds on the disturbances and parameter variations should be known a priori to the designer of the control system. Fuzzy logic provides an effective way to design a controller of the system with disturbances and parameter variations. Therefore, combination of the best feature of the fuzzy logic control and the sliding mode control is considered. The adaptive fuzzy variable structure controller developed in t h ~ spaper employs fuzzy controller with nonlinear sliding mode. Therefore, the resulting system can resolve the conflict between the opposing requirements of static and dynamic accuracy that are encountered when designing the linear sliding mode. A variable length pendulum system is used to demonstrate the availability of the proposed approach. Copyright 0 2002 IFAC Keywords: fuzzy control, adaptive control, sliding mode control, nonlinear systems 1. INTRODUCTION
response of the error, during sliding mode, can be prescribed in advance and the system is completely robust and insensitive to parameter variations and external disturbances.
Automatic control with the concept of variable structure system (VSS) is a special class of nonlinear controls. This control differs from other control systems mainly in that its structure is not fixed but is varied during the control process.
The fuzzy set theory, established by Zadeh [4], has been developed for over 20 years. Its most prominent application is in the area of control engineering. In general, if a system is ill defined or too complex to have its mathematical model, fuzzy control provides an effective way to design a control. Based on fuzzy logic, the fuzzy controller converts a linguistic control strategy into an automatic control strategy [5]. The most advantageous and controversial property is that fuzzy controller does not rely on the mathematical model of the process. Control rules are constructed by using the knowledge of experts or operators' experience. In other words, the fuzzy controller is human dependent. Different expertise will give different rules for the same performance. There have been many studies investigating the relationship between sliding mode control and fuzzy control. Kawaji and Matsunaga[6] proposed a method of generating fuzzy rules for servo motors based on the VSS control. Yager and Filev[7] determined the fuzzy rules according to the sliding mode condition. Glower and Munighan[S] presented fuzzy sliding mode controller and assured stability for each of this fuzzy controllers. Lu and Chen[9] combined the best features of self-organizing fuzzy control and sliding mode control to achieve rapid and accurate tracking control of a class of nonlinear systems. The fuzzy rule base is used to approximate the VSC through
The structure of a control system may change either accidentally or intentionally in accordance with a definite rule. The VSS considered in t h ~ spaper fall into the category of systems where the structure is intentionally changed during the transient in accordance with the present structured control law. The variable structure control (VSC) law developed, based on the theory of VSS, usually provide for changes in the structure of the controlled system whenever the state crosses a certain surface called the switching surface. If the system is forced to remain on the predetermined switching surface, it results in a dynamic behavior that is largely determined by design parameters and equations defining the switching surface. Consequently, new properties that were not present in the original system can be obtained for the controlled system. Therefore the system is robust and insensitive to external disturbances and parameter variations. The state trajectory undergoes a small-amplitude highfrequency chattering motion along the surface; which is known as the sliding mode or the sliding regime. The designed surface is then referred to as the sliding manifold or the switching surface [1]-[3]. The advantage of t h ~ sapproach is that the transient
151
self-organization and the VSC effort is used to compensate for the approximation error and to provide exponential convergence of the sliding variable. Lin and Chiu [lo] combined the merits of the sliding mode control, the fuzzy inference mechanism and adaptive algorithm. The VSS can be prescribed with an arbitrary transient response. The adaptive fuzzy variable structure controller used in this paper employs fuzzy controller with nonlinear sliding mode [ 111, while the previous methods used the fuzzy controller with linear sliding mode. This can resolve the conflict between the opposing requirements of static and dynamic accuracy that are encountered when designing the linear sliding mode. Thus, the speed of response is improved without overshoot and a good transient response has been obtained. 2. VARIABLE STRUCTURE CONTROL
WITH NONLINEAR SLIDING MODE 0 0
Consider a single-input n -th order nonlinear system =f(X,t)+g(X,t)u(t)
a = ; .:
(1)
0
(2)
u+ u
where s ( 2 , t ) = O
s ( 2 , t )> 0 s ( 2 , t )< o
2 dt
:
:
0
"'
1
.
.
.
(8)
... h,_
P
(3)
i = 1,2,. ..,q , and q = { p ( p + l)(p + 2)) / 6
j=l
Step 3: From (5), (6) and (9), the nonlinear switching function can be written as follows:
i=l
(4)
j=1
The nonlinear parts used to improve the performance in the transient state are given in the second term. Differentiating s ( 2 , t ) with respect to time yields
where y is positive real constant [l]. Define a nonlinear sliding surface s ( 2 , t ) in the state space R" as follows:
s ( 2 ,t ) = SL (2, t )+ SNL(2, t)
:
where di is a specified constant value, x p u = 3 ,
is the switching surface and
s2 =sS<-ys2
0 0
(9)
2 = X - X , . The condition for the sliding mode to exist on the switching surface is _I _d
"'
"'
regarded as a nonlinear feedback system with the state vector z and the control sNL Then, from [ 111, the following form can be obtained
Both f ( X , t ) and g ( X , t ) are assumed to be continuous in X . The control problem is to find u ( t ) such that X will track the desired trajectory X , = [x,, X,,. ..,x,(~-~)]' . The control input u(t) has the following form: u(t>=
0
0 1
-4 h, 4
where u ( t ) is the control input, X = [x,X,...,x(~-')]' is the state vector, function f ( X , t ) is unknown but bounded by a known continuous function F ( X , t ) , and the control gain g ( X , t ) is unknown but limited in a certain fixed range as follows: 0 < Gmi, < g ( X , t ) < G,,
1 0
(5)
152
where
r is a number of nonlinear terms given by the n
derivative of (10) and cq,= 3 , j
r
KS >-> Gln,
= 1,2,. .. , r . From
i=l
r>o
(23)
then the resulting system satisfies the sliding mode existence condition in (4), and the error trajectoly move toward the switching. Therefore, the dynamics of the system are always stable and the error 2 will slide into the origin.
(1) and (111,
(12) In the sliding mode, the equivalent control ueq is
3. ADAPTIVE FUZZY VS S
obtained as follows:
In general, the sliding mode control assumes that the uncertainties are bounded and their bounds are available to the designer. However, bounds on the uncertainties may not be easily obtained because of the complexity of the system structure involving uncertainties. Moreover, the magnitude of external disturbance cannot be easily estimated. Thus the cost for implementing the sliding mode control is rather high for a system having large upper bound of uncertainties. On the other hand, the fuzzy inference mechanism can be constructed to estimate the system uncertainties. In t h ~ spaper a fuzzy sliding mode control is developed in which a fuzzy inference mechanism is used to estimate the upper bound of the system uncertainties.
(13) However, since f ( X , t ) and g ( X , t ) are unknown, the following equation is used which is estimated as precisely as possible: ieq
=
i - l
(P.7
+',own)
(14)
7
and i are estimated value of f and g , where respectively, such that
Replacing KN by K f in (19), the following equation
and uhown is an independent term for system parameters and external disturbance, i.e.,
can be obtained:
u(t) = Zi, +us - K , sgn(s) where
Now, consider the following state feedback control law : u = ueq+u, +us
estimated by
fuzzy
inference
satisfy (4) is
(18)
K i =g-'f
-i-'?+(i-' -g-')uhOwn
(25)
But the ideal value K i cannot be obtained exactly
(19)
owing to the parameter variations and external disturbances. Therefore, it is assumed that there exists a specific K, that achieves minimal control
(20)
effort, i.e., satisfying the sliding mode existence condition (4) and
And us is defined by us =-K,s
is
mechanism. An ideal optimal value of K , that will
where uN is used to eliminate the influence of uncertain terms of (15) and (16) as follows: uN = -KN sgn(s)
K,
(24)
which determines the dynamic response of s ( 2 , t ) .
K, - K i l ~ l s l
Inserting (18) into (12), the derivative of s ( 2 , t ) is given by
(26)
where E is a small positive real constant. In general, the fuzzy control rules are a set of fuzzy implementations of linguistic term. Consider a singleoutput fuzzy controller with 2 inputs, s and S , and m rules in its rule base, represented by R,, R2,...,R, , respectively. The general form of the .j -th rule is
S= f -gi-lj
+(&' - l ) u b o w n -gKN sgn(s>-gKSs (21) If control gains are selected as
R,
:If(s is
where
153
A:) A(S is A:), then K+ = a, (27)
AND intersection operation,
A
and let the adjustable parameter error vector be
fuzzy set characterizing the variable A& of the premise, vp , in the j -th rule, output form the
-aj
consequent parameter.
(3 1)
&=u-&
j -th implication,
Ki
(30)
K, = & W T
s, S inputvariables,
Choose the Lyapunov function candidate as
There are m rules to define the rule base of the fuzzy controller and each rule can give a distinct value of the output, the weighted average of the individual output, K+, j = 1,2,...,m , is used to obtain the
where y is a positive constant. Differentiating V with respect to time, the following equation is obtained:
output, K , . The weighting assigned to each K+ is
(33)
the firing strength of the j -th rule, designed as uJ. The firing strength of the j -th rule is determined from U J = min (P:
(4)
(s), P:
Substituting (12), (18)-(20) and (25)-(26) into (33) gives
(28)
where p&(vp) is the value of the membership function at vp in the fuzzy set A,;. The membership functions of A, are shown in Fig. 1, where the ordinary triangle shape function is used for convenient. Fuzzy output K , can be calculated by the center of
=
.( f +g 4 ,
gKss - U h o w x )
f + gi,,
Y
uhowx)- gKJs - gKss2
-
+g(KJ - K f )I s I +LabT I s(
m
f + gi,,
Y
,=l
uhowx)- gKJs - gKss2
-
+
[
Gm,WIsI ---a
Y
If adaptive law is defined as
where a = [al,a,,..., a,] is a adjustable parameter vector; -aJ is the center of the membership function of K+ , and
-
Y
-gKJ s + gKJs - gKf I s I +LabT I s(
area (COA) defuzzification as follows:
,=l
1-: +--a-a~
i,, - K~ sgn(s) - K ~ S -)
.I-
-aT -Gm,~s2 (34)
bT = Y G,, Is1 W T
w = [al,w,, .,W, ] I C W , is a firing strength
(35)
then (35) becomes
..
,=I
V 5 -G,,E
vector.
sz - G,,,K, sZ 5 -7 sz < 0
which requires that
... “1
... “2
i‘
>L+%E . G,i”
G,”
4. SIMULATION RESULTS
I
“2
A: A,’
“1
...
K,
(36)
In order to demonstrate the validity of the proposed controller, the variable length pendulum in Fig. 2 is used as an example of the control [12], where B is the angle, l is the length g is the acceleration of gravity, and m is the mass. This model was used to identify the gymnast’s center of mass as the kip is performed. The dynamic equation of the system can be written as
Asm-’ A,”
...
I
Fig. 1. The membership functions of
A,
Define & as the optimal vector that achieves the minimal effort of sliding mode control as follows:
154
4
Rule 4: if s is Z and S is P then K , is NS
b
Rule 5: if s is Z and S is Z then K , is ZE Rule 6: if s is Z and S is N then K , is PS Rule 7: if s is N and S is P then K , is PM Rule 8: if s is N and S is Z then K , is PB Rule 9: if s is N and S is N then K , is PH Rule 1 implies that the switching variable s is far away from the switching surface and it is keep increasing. Therefore, a large K , is required in order to reach the slidmg-mode condition
g / t o = l O , and m l d = l
!,IC,=fl.S.
Fig. 2. Variable length pendulum.
B = 0.5sin~(1+0.5cos8)8’/A(Q) -10 sin8(1+ cos 8)/ A ( 8 ) +u / A ( 8 ) + v(t)cos 8 where A(8)
= 0.25 (2
+ cos 8)’
(37)
and the disturbance is
v(t) = 2cos(3t) . Following Lin and Chiu [ l l ] , the membership functions for the fuzzy sets corresponding to switching variable s , S and upper bound of the K , is defined in Fig. 3, where the associated fuzzy sets involved in the fuzzy control rules are defined as follows: N : negative P : positive NB:negative big NS negative small PS: positive small PB: positive big
(20) is selected as K ,
a‘
G,,
Z : zero NH:negative huge NM: negative medium ZE: zero PM: positive medium PH: positive huge
“6
“7
“8
“9
initial value of states is
5. CONCLUSIONS An adaptive fuzzy sliding mode controller is designed using the advantages of the sliding mode control, the fuzzy inference mechanism and the adaptive algorithm in this paper. The VSS can be designed with an arbitrary transient response and is theoretically robust on disturbances and parameter variations. Moreover, the VSS with nonlinear sliding mode can resolve the conflict between the opposing requirements of static and dynamic accuracy that are encountered with the linear sliding mode control. The fuzzy inference mechanism is used to estimate the upper bound of the disturbances and parameter variations. In addition, the adaptive fuzzy inference mechanism estimate the optimal bound of the
K,
Fig. 3. Membership functions of fuzzy sets. Rule 1: if s is P and S is P then K , is NH Rule 2: if s is P and S is Z then K , is
= 4 . And the
The responses of linear fuzzy adaptive VSS (LFAVSS) and nonlinear fuzzy adaptive VSS (NFAVSS) are shown in Fig. 4, and as can be seen, the transient response is greatly improved. The Fig. 5 gives the responses for the 0.1Hz square wave input with x(0) = [1,0] . The system response for the first square wave input is plot (a) in the Fig. 5. The plot (b) is the response of the second square wave input. The responses over the 4-th square input are the plot (d). Fig. 5 shows the adaptation process of the nonlinear fuzzy VSS with adaptive law, where the adaptive law improves the system response.
;””:
4 5
values of The boundary
q o ) T = [1,0] .
-0 1
“4
initial = 20 .
4 and 9
** “3
adaptive parameter
values of control gain g ( X , t ) are Gmi,= -
mode control is determined by the following fuzzy inference rules:
“2
= 10. And
are updated with a(0)= [5,10,20,40] and y
Since only three fuzzy subsets, N , Z and P , are defined for s and S , the fuzzy inference mechanism only contains nine rules. The K , for the sliding
“1
Let X T = [X1,X2]= [8-8,,8-8,] . From (lo), the nonlinear switching variable is obtained as s(X) = X2 + 42, + 52; . The estimated equivalent control input of (14) is determined as Zi, = -42’ -15X;Xz . The value of gain constant of
NB
Rule 3: if s is P and J. is N then K , is NM
155
disturbances and parameter variations. A variable length pendulum system is used to demonstrate the applicability of the proposed controller. Simulation has shown that the transient response of the system is greatly improved with the nonlinear fuzzy adaptive control.
a : LFAVSS b : NFAVSS
REFERENCES 0
Itkis, U. (1976). Control Systems of Variable Structure, John Wiley & Sons. Utlun, V.I. (1983). Variable Structure Systems: Present and Future. Autom. Remote. Control, V01.12, pp. 1105-1 120. Decarlo, R.A., Zak, S.H. andMatthews, G.P. (1988). Variable Structure Control of Nonlinear Multivariable Systems: a Tutorial. Proc. IEEE, V01.76, No.3, pp.212-232. Zadeh, L.A. (1965). Fuzzy Sets. Inform. Control, V01.8, pp. 338-3 53. Park, Y.-M, Moon, U. and Lee, K.Y. (1995). A Self Organizing Fuzzy Logic Controller for Dynamic Systems Using a Fuzzy Auto-Regressive Moving Average (FARMA) Model. IEEE Trans. On Fuzzy System, Vo1.3, No.1, pp.75-82. Kawaji, S. and Matsunaga , N. (1990). Generation of Fuzzy Rules for Servomotor. in Proc. IEEE Inter. Wkshp. Intelligent Motion Contr., Istanbul, Turkey, pp.77-82. Yager, R.R. and Filev, D.P. (1994). Essentials of Fuzzy Modelling and Control, Wiley, New York. Glower, J.S. and Munighan, J. (1997). Designing Fuzzy Controllers from a Variable Structures Stand Point. IEEE Trans. On Fuzzy Systems, Vo1.5, No.1. Lu, Y.S. and Chen, J.S. (1994). A Self-organizing Fuzzy Sliding Mode Controller Design for a Class of Nonlinear Servo Systems. IEEE Trans. Indust. Elect., Vo1.41, pp.492-502. Lin, F.J. and Chiu S.L. (1998). Adaptive Fuzzy Sliding mode Control for PM Synchronous Servo Motor Drives. IEE Proc.-Control Theory Appl., Vo1.145, No.1. Lee, D.S., Kim, M.G., Kim, H.K. and Youn, M.J. (1991). Controller Design of Multivariable Variable Structure System with Nonlinear Switching Surfaces. IEE Proc. -Control Theory Appl., Vo1.138, No.5. Nakawaki, D.E., Joo, S. and Miyazaki, F. (1999). Identification of Motor Skill Characteristics Using a Reduced Model Approach. IEEE Systems, Man, and Cybernetics '99 Conference Proc., Vo1.4.
156
1
2
4
6
10
8
Time[sec]
Fig. 4. Comparison of responses.
Time[sec]
Fig. 5. The system responses for the 0.1Hz square wave.