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Robust Sampled-Data Control for Fuzzy Uncertain Systems.

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Dev. Chem Eng. Mineral Process., 9(IR),pp.175-182, 2001.
Robust Sampled-Data Control for Fuzzy
Uncertain Systems
L.-S. Hu', H.-H. Shao, Y.-X. Sun'
Department of Automation, Shanghai Jiaotong University, Shanghai
200030, P. R. China
#National Laboratory of Industrial Control Technology, Zhejiang
University, Hangzhou 310027, P. R. China
Sampled-data control which is capable of stabilizing general nonlinear systems
with the intersample behavior taken into account axe of great current interest.
In this paper, the Thkagi-Sugeno fuzzy models were considered for representing
the nonlinear plants. For the Thkagi-Sugeno fuzzy model and the Takagi-Sugeno
uncertain fuzzy model, the fuzzy sampled-data control and the robust sampleddata control were considered in this paper. The robust sampled-data control
for the TakagiSugeno uncertain fuzzy model with the constraints on the states
was also developed. The results were described as the matrix inequalities, which
could be solved by LMITool. Finally, a numerical example based on the missile autopilot model was considered, the result showed the effectiveness of the
proposed procedures.
Introduction
Mathematical models of the system under consideration are the basis of the
control design methodologies. In most cases, the industrial process presented is
highly nonlinear, we usually need to linearize the system on the nominal operating point so that we are able to proceed with the control design using linear
system theory. However, for systems with shifting or multiple operating conditions, the traditional mathematical model may not be sufficient to represent
the dynamic behavior. As each local model is valid only for a certain range of
operating conditions, so these results can only guarantee the local stability of
nonlinear systems. Recently, a nonlocal approach, the so-called Takagi-Sugeno
(TS)type fuzzy model, which is conceptually simple and straightforward is proposed for nonlinear systems design via fuzzy control [l,21. The Takagi-Sugeno
fuzzy model, proposed by Takagi and Sugeno [l]is based on a fuzzy partition
of input-output space. In each fuzzy subspace a linear input-output relation is
formed. The output of fuzzy reasoning is given by the aggregation of the values
inferred by some implications that were applied to an input. There are a number of works aimed to analyze and synthesize fuzzy control systems based on the
Takagi-Sugeno fuzzy model (see [3, 41 and refer therein).
*Author for correspondence (email: lshu@maill.sjt u .edu. cn).
175
L-S. Hu, H.-H. Shao and Y.-X. Sun
We purpose to study the design procedure of sampled-data control for the
Takagi-Sugeno fuzzy model in this paper. Sampled-data control is a direct design procedure of a digital controller for the continuous time systems, which
takes account of the effects of the intersample behavior of the system and has
no degrade of closed-loop performance. Although there is much research on H2
and H , sampled-data control for linear systems [2], a few results of sampleddata control for the nonlinear systems are obtained. Hagiwara et al. [5] and
Okuyama and Takemori [6] dealt with stability analysis of sampled-data systems with a sector nonlinearity. Rui et al. [7] dealt with sampled-data control
for a class of nonlinear systems. In this paper, the sampled-data control for the
nonlinear systems described as Takagi-Sugenofuzzy models was considered. For
the Takagi-Sugeno fuzzy model and uncertain fuzzy model, the fuzzy sampleddata control and the fuzzy robust sampled-data control were considered. The
results were described as the matrix inequalities. The fuzzy robust sampled-data
control for the Takagi-Sugeno uncertain fuzzy model with the hard constraints
on the states was also developed. The result was also described as the matrix
inequality, which could be solved by LMITool. Finally, to demonstrate the design procedure, we applied the design procedure to a nonlinear missile autopilot
problem [3]. Simulation result shows the effectiveness of the proposed approach.
This paper is organized as follows. First, we present some preliminaries on the
Takagi-Sugenofuzzy model. Then, we present the main results of fuzzy sampleddata control and robust sampled-data control and constrained robust sampleddata control for the Takagi-Sugenofuzzy model. After that, a numerical example
based on the missile autopilot model was considered to show the effectiveness of
the proposed procedures. Finally, we give our conclusions.
Preliminary Discussion
In this section, the nonlinear plant is represented as the following Takagi-Sugeno
fuzzy model, and its ith Plant Rule is
I f zl(t) is
Fjl
and - . . and z p ( t ) i s Fip Then k ( t ) = Aiz(t) + B+(t). . (1)
i = 1,2, . . .,T , where F,j is the fuzzy set and T is the number of If-Then rules.
zj(t), j = 1 , 2 , . - -, p are the premise variables. Given a pair of (z(t),u(t)),
the
final output of the fuzzy systems is inferred as follows
+
k ( t )=
. . . (2)
Xi(z(t))(Aiz(t) BiU(t)),
where ~ ( t=)( z l ( t ) ,za(t),. . . ,z p ( t ) ) .Xi(z(t)) =
xi==,
*), hi
*
where haMt)) =
*=I
II;=lF,j(zj(t)) for all t , satisfy
X i ( z ( t ) ) = 1, X i ( z ( t ) ) 20, i = 1,2,...,r.
F i j ( z j ( t ) )is the grade of membership of z j ( t ) in Fij. Traditionally, the parallel
distributed compensation can be employed to design each local fuzzy control
rule so as to compensate each local rule of a fuzzy system (as in many reports).
Except for the discreteness of the fuzzy sampled-data controller, its ith Control
Rule (3) has the same structure as Plant rule i (1)of the fuzzy model
If
176
zl(tk)
is
Fil
and
.
and
zp(tk)
is
FiP Then G i [ t k ] = Fi[tk]z(tk). .. (3)
Robust Sampled-DataControlfor Fuuy Uncertain Systems
for i = 1,2,. .. ,T . Then the fuzzy sampled-data controller is inferred as follows
= EL1Xi(z(tk))Gi[tk],
with Sa[tk]= F i [ t k ] ~ ( t kLet
) . Z ( t ) = ( Z T ( t ) ,uT(t))T,
and use the zero order holder, u ( t ) = ii[tk),for t E (tk,&+I],k = 0,1,2,-.-,we
obtain the following fuzzy hybrid system
i ( t )=
c;=,
Ai(.z(t>)Ai5(t),
t E (tk,tk+l],
c;=lAi(Z(tk))(A+ BFi[tk])qtk),
qt;> =
[
Ai = A* Bi
1,
[
I 0
[
. . . (4)
1,
o], B =
pi[tk[tkl= [ & [ t k ] 0 1 .
The fuzzy sampled-data controller design is to determine the local sampleddata feedback gains Fi[tk]in the consequent parts. The feedback gains kftk]
are determined by an LMI-based design technique presented in Theorem 1 in
the next section. However, as the reason presented in the following, the main
purpose of this paper is to present the sampled-data control, with the form
where
A =
u(t>= f i [ t k ] , f i [ t k ] = F[tk]Z(tk),
t E (tk, t k + l ] ,
for the fuzzy uncertain systems and the ones with state constraints.
. .. ( 5 )
f i z z y Sampled-Data Control
In this section, we first present fuzzy sampled-data control for the system of
equation (1),then consider sampled-data control for the fuzzy uncertain system
and the one with state constraints.
-
Theorem 1 For given positive scalars pk 5 1, k = 0,1,2 - -,and the sampling
period T,,the system (1) is stabilizable by the sampled-data fuzzy controller (3),
if there exist symmetric piecewise continuous matrix functions P[t]= P [ t i ]+
(t - t b ) Y [ t ~
>]0, &I [t]= Ql[tt]
+ (t- tk)Z[tk]2 0 , for t E ( t k , t k + 1 ] , the semipositive symmetric matrices Q&], and the controller gain Fi[tk],i = 1,2, ' - .,r,
satisfying the following matrix inequalities
Y[tk]f P[t]A¶+ Apqt]
fori < j ,
2Y[tk]4- P[t](Ai4-Aj)
for t
+
(T
- l)Ql[t] 5 0,
+ (Ai + Aj)TP[t]- 2Q1[t]5 0,
E ( t k , tk+l], and
Proof. From Corollary 1 of [8] and Theorem 1 of [9],the proof is trivial.
I77
L-S. Hu. H.-H. S ~ L Wand
Y.-X.Sun
Along the same lines of Theorem 1, robust fuzzy sampled-data control for the
uncertain fuzzy model can be obtained. However, in the following, we presented
robust sarnpled-data control and constrained robust sampled-data control with
the form ( 5 ) for the uncertain fuzzy systems to improve the precision of control.
Robust Sampled-data Control
Instead of the system (l),we consider the following uncertain fuzzy model
) is Fil and ... and zp (t )i s FipThen
{ I5 (f tz)=l( t(Ai
+ A A i )z(t )+ (Bi + ABi)u(t),
. , . (6)
i = 1,2,. . - ,T , where Ai, Bi are matrices with appropriate dimensions, AAi
and ABi are real valued continuous uncertain matrix functions specified as
[ AAi ABi ] = DiAi [ Eia Eib 3, where Di, Ei, and Eib are real matrices with appropriate dimensions, Aj are Lebesgue measurable unknown matrix
functions satisfying the following bounds ATAi 5 I. Similar to ( 2 ) , we obtain
k ( t ) = C61Xi(z(t))((Ai+ A A i ) z ( t )+ (Bi+ ABi)u(t)).
. . . (7)
If usmg the sampled-data controller ( 5 ) to stabiliie the plant (7), the closedloop system is described as
k ( z ( t ) ) ( A+
i AAi)z(t),tE
tk+1],
{ i ( t )=-C&1
(A +
k = 0,1,2, - . ,
(tk,
+
BP[tk])E(tk),
. . . (8)
Theorem 2 For given positive scalars / l k 5 1, k = 0,1,2. ., and the sampling
period TB,the system (6) is stabilizable by the sampled-data controller (S), if
there exists symmetric piecewise continuous matrix function P[t]= P [ t i ]+ (t t k ) Y [ t k ] > 0, 91[t]= & l [ t ; ] + ( t - t k ) 2 [ t k ] 2 0, f o r t E (tk,tk+l]t =0 ,1 ,2 ,'.-,
positive scalars ( i , ( l i , <2i, C k , and the controller gain F[tk] satisfying the followang matrix inequalities
. . . (9)
5 0,i< j ,
. . . (10)
. .. (11)
178
Robust Sampled-Data Controlfor Fuuy Uncertain Systems
Proof. The proof is trivial. We present the outline of the proof. It is easy to
know that the system ( 8 ) is equivalent to
i ( t )= fr CI=1C,'=iAi(t(t))Aj(Z(t))(Ai
+A& + Aj
+ AAj)Z(t),
for t E (h,tk+i],and
z(t;) = ( A+ BF[tk])Z(tk),
k = 0,1,2,.. ..
Choose the Lyapunov function candidate as V ( t ,5 ) = ZT(t)P[t]Z(t),
which
is right continuous on (tk,tk+l],we obtain
c;==,
V ( t , Z )= ZT(t){Y[tk]
+ c;=,
Xi(Z(t))Aj(t(t))
(P[t](Ai
+ A& + Aj + AAj) + (Ai + AAi + Aj + AAj)TP[t])}Z(t)
= xi==,
A?(t(t))ZT(t)(Y[tk]
P[t](Ai AA,)
+(Ai + AAi)TP[t])Z(t)2
&(t(t))Aj(t(t))ZT(t)(Y[tk]
+iP[t](Ai AAi Aj + AAj) + $(Ai+ AAi + Aj + AAj)TP[tJ)Z(t).
3
+
&j
+
+
+
+ z:<j
If there exist the semipositive symmetric matrix Ql[t],
positive scalars tli,
and ck, such that (10)holds for i < j,then we obtain from Corollary 1 of [8]
V(t,Z)5
+(T
Ci=1A : ( ~ ( t ) ) Z ~ ( t ) ( Y+[P[t](A,
t k ] + AAi) + (Ai + AAi)TP[t]
- 1)Qi[t])Z(t)
- 2ck
x:<j
Ai(.Z(t))Aj(Z(t))ZT(t)Z(t),
for t E (tk,tk+lI.
It is easy to know if the condition (9) holds, then V ( t , E ) 5 -Ck
for
t E (tk,tk+l].For the given positive scalars pk, we obtain, if the condition (11)
holds, then V ( t ; , Z ( t t ) )- p k V ( t k , Z ( t k ) ) 5 0. Furthermore from Theorem 1 of
191, we know the Theorem 2 holds.
Constrained Robust Sampled-Data Control
In this section, we consider the robust sampled-data control for the system (6)
with the hard constraints on the states. For given positive scalars 6 and 0 with
6 < 0,the state constraints are described as Ix(t)l < 0,t 2 0, for 1x01 < 6. Let
.=[; ;].
Theorem 3 For given time-domain performance indexes 6 and 0 with 6 < 0,
positive scalars pk 5 1, C(k 5 1, k = 0 , 1 , 2 , - . - ,and the sampling period T,, the
system (6) with 1x01 < 6 , is stabilizable by the sampled-data controller (S), and
satisfies Ix(t)l < r, t 2 0 , if there exist symmetric piecewise continuous matrix
function ~ 1 [ =
t ]Pl[t;] ( t - tk)Yl[tk]> 0, P2[t]= Pz[t;] + ( t - tk)Y2[tk]> 0,
Q[t]=
+ ( t - tk)Z[tk]2 0, Qi[t]= Qi[t;] + ( t - tk)Zi[tkI 2 0, for
t E (tk,tk+l],k = 0 , 1 , 2 , . . . , let P[t]= Pl[t]+ P2[t],Y[tk]= Yl[tk]+ Y2[tk],
positive scalars cli, Q , &it &j, J l l i , E 2 1 j , Ck, the positive scalars a l , a2 and b
satisfying
Q[tt]
a1J2
+
+ a2b2 < bu2,
179
L.-S. Hu, H.-H. Shao and Y.-X. Sun
and the controller gain F[tk]satisfying the following matrix inequalities
. . . (13)
P2[t]2 bCTC,
. . . (17)
Proof. Choose the Lyapunov function candidate as V ( t ,Z) = Z*(t>P[t]Z(t),
Vl(t,Z)= ZT(t)P1[t]Z(t).
If there exist the semipositive symmetric matrix Q[t],
Q l [ t ]positive
,
scalars <I,, ci, &i, &,
& j , Ck such that the conditions (13)
hold for a < j , and the conditions (12) hold, then V ( t , Z ) 5 0, and Vl(t,Z)5
-Ck li512, for t E ( t k , t k + l ] . For the given positive scalars p k 5 1, p i 5 1, if the
conditions (14) hold, then V(tl,Z(tkf))- pkV(tk,Z(tk)) 5 0, K(tl,Z(tk+))
pf’V1(tk,S(tk))5 0. From Theorem 1 of [9],we h o w the Theorem 3 holds.
Remark 1 The conditions listed in Theorem 1, Theorem 2 and Theorem 3 are
not the linear matrix inequalities exactly, even if we specified Fi[tk] or F[tk].
180
Robust Sampled-Data Controlfor Fuuy Uncertain Systems
However PI and P2, jkom their definitions, are piecewise linear continuous functions. So we only need to check the conditions on two points of the sampling
intervals ( t k , &+I], for k = 0,1,2,.-. In such cases, these conditions are the
linear matrix inequalities if we specified F,[tk] or F [ t k ] . After specifying all the
parameters, such as pk, Ck etc., then using Matlab LMITool, optimizing Fi[tk] or
F[tk] at each sampling intervals by other optimal procedures, we can solve these
mat* inequalities.
Numerical Examples
Example 1. The missile autopilot model [3] is the complex nonlinear one. We
propose the robust control design based on the Takagi-Sugenofuzzy model as an
alternative approach to the missile guidance problem. This model represents a
missile traveling at Mach 3 at an altitude of 20,000ft..The nonliiear dynamic
model is as follows
where a angle of attack, q pitch rate, W weight, V speed, IvD pitch moment of
inertia, Q dynamic pressure, S reference area, d reference diameter, Z = C,QS
normal force, M = C,QSd pitch moment. The normal force and pitch moment aerodynamic coefficients C, and Cm are approximated by C, = +=(a)
+
b,d, C, = 4m(a) b,6, where 6 is fin deflection. b, = -0.034(1 + Ab,),
b, = -0.206(1+ Abm), 4z(a) = 0.000103~~~
- 0-00945ala1 - 0.170(1+ Adz)al)
#,(a) = 0.000215cU3- 0.00195ala1 0.051(1+ A4,)a. It is noted that there
exist uncertain parameter variations which are denoted by Ab,, Ab,, Ad, and
A&,. These parameter variations may be due to modeling error, system uncertainty, and external disturbance. We assume that these variations are within
30%, i.e., lAb,(t)l 5 0.3, lAb,(t)l 5 0.3, lA$,(t)J5 0.3,and lA#,(t)l 5 0.3,
V t 2 0. We linearize the original system around the following angle of attack, a
= 0, 30°,-30’. Let x1 denote the angle of attack, 5 2 denote the pitch rate q and
u denote the fin deflection 6. Let x = ( x I , x ~ then
) ~ , the system was described
as the following 2-rule Takagi-Sugeno fuzzy model
+
+
I”
51
is Small Then k ( t )= Alx(t)+ Blu(t),
..- (18)
If 5 2 is Large Then z(t)= Azs(t)+ BZu(t),
-O.SOS(l + Arb,) 1
-0.121(1+ Ab,)
[
32.404(1+ A&) 0 ] ’ B1 = [ -130.888(1+
-1.112(1+ A$z)
-0.105(1+ Ab,)
[ -216.3464(1+
A&) 0 1, B2 = [ -130.888(1+ Ab,) 1,
where
A1
=
Abm)
1
1,
A2
=
and lAbz(t)l 5
0.3,lAb,(t)l 5 0.3,lA+,(t)l 5 0.3, lA$m(t)l5 0.3,V t 2 0. Triangular membership functions are adopted in this model. It was noted that the 2-rule TakagiSugeno fuzzy model has lower natural frequency than the original missile system.
The robust sampled-data controller was designed to make the missiles to follow
certain trajectories with the same structure as (18). Choose the sampling pe181
L-S. Hu. H.-H. Shao and Y.-X. Sun
1,
riod T, = 5ms and F[O] = [ 10 0.5
pk = 0.95, for simplicity. We apply
the robust sampled-data controller, designed using Theorem 2, to the original
missile system. The angle of attack follows the desired trajectory after 1.5 seconds, which is superior to that of [3] (4 seconds). These show the effectiveness
of proposed method.
Conclusions
In this paper, the Takagi-Sugeno fuzzy models are considered for representing
nonlinear plants. For the Takagi-Sugenofuzzy model and uncertain fuzzy model,
the fuzzy sampled-data control and the robust sampled-data control were considered in the paper. The results were described as the matrix inequalities. The
robust sampled-data control for the Takagi-Sugeno uncertain fuzzy model with
the constraints on the states was also developed. The result was also described
as the matrix inequalities, which could be solved by LMITool. Finally, a numerical example was considered, the result shows the effectiveness of the proposed
procedures.
References
1. Takagi, T., and Sugeno, M., 1985. Fuzzy identification of systems and its applications t o
modeling and control. IEEE Trans. Sys., Man, Cybern, 15, 116-132.
2. Wang, H. O., Tanaka, K., and Griffin, M. F., 1996. An approach to fuzzy control of
nonlinear systems: stability and design issues. IEEE Trans. Fuzzy Sys., 4(1), 1423.
3. Lee, T. S., Chen, Y.H., and Chuang, J . , 1998. Fuzzy modeling and uncertainty-based
control for nonlinear systems. Proc. of American Control Conference, Philadelphia.
4. Chen, T., and Francis, B., 1995. Optimal Sampled-data Control Systems. Springer.
5. Hagiwara, T., Kuroda, G., and Araki, M., 1996. Stability of sampled-data systems with a
timeinvariant memoryless sector nonlinearity. Proc. of the 35th Conference Decision and
Control, 1264-1265.
6. Okuyama, Y., and Takemori, F., 1996. Robust stability evalution of sampled-data control
systems with a sector nonlinearity. IFAC, 13th Triennial World Conference, 41-46.
7. Rui, C., Kolmanovsky, I., and McClamroch, N. H., 1997. Hybrid control for stabilization
of class of cascade nonlinear systems. Proc. of American Control Conference.
8. Tanaka, K., Ikeda, T., and Wang, H., 1996. Design of fuzzy control systems based on
relaxed LMI stability conditions. Proc. of the 35th Conference on Decision and Control,
598-603.
9. Hu, L., Shao, H., and Sun, Y.,1999. Constrained robust sampled-data control for a class
of nonlinear uncertain systems. IFAC, 14th Triennial World Congress, 521-526.
Received: 28 October 1999; Accepted after revision: 12 May 2000.
182
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