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Robust H∞ filtering of fuzzy dynamic systems
ROBUST H[Infinity] FILTERING OF FUZZY DYNAMIC SYSTEMS...
14th World Congress ofIFAC
Copyright CO 1999 IFAC 14th Triennial World Congress~ Beijing, P.Ra China
K-3e-12-1
Robust H~ Filtering of Fuzzy Dynamic Systems G. Feng*~ J. Ma+~ Z.
x. Han* and N. Zbang*
*&~'cho()1 (?{ Electrical EnJ~jneeringr (injversi~v (?{ lVelV ~~'outh rVales. &';':v(ine~v, N.r.,TV 2052. ~4u.,. .t ralia.
Tel: (61 2) 9385 5374. Fax: (61 2) 9385 5993. Elllai/:
(;.Fel1g(~I(Un.\"tJ.edu.au
+Departluenl (~rIn. (0r11Joti 0/1 ~~:v.,·ten1.", (~i~v rrniver."i(v (?( Hong Konl~1 Ho}]..!? Kong
Abstract This paper addresses robust H oo filtering of cOlnplex nonlinear systelns which can be represented by a fuzzy dynalnic model. Based on a continuous Lyapunov function and (I piecc\vise cOlltinuous Lyapunov function respectively~ two kinds of new filtering design tuethods are proposed using quadratic stability theory and Linear Matrix Inequalities (LMI). Copyright
,e, 1999IFAC
KeY'\'ords: Fuzzy sysfeIlls, RV-' filtering~ LMt quadratic stability.
1.
observers for fuzzy dynalnic systelns in (Feng~ et al. 1995). but there is~ to the authors· best knowledge~ very rev.- result available on design offilters for fuZZ}f dynaluic SystCITIS.
Introduction
Since the first paper on fuzzy sets (Zadeh~ 1973) "vas published~ fuzzy logic control (FLC) has recently proved to be a successful control approach for cOlnplex nonlinc3r systclns_ During the last few
years4 \ve have proposed a nUlnber of ne\v Jnethods for the systelnatic analysis and design of fuzzy logic controllers based on t he so-called fuzzy dynaluic tnodeI ~ \vhich consists of Cl f~unily of local linear Inodels smoothly connected through fuzzy Inelnbership functions (Cao_ et al. 1996~ 1997a, 1997b, 1997c~ 1997d: Fcng~ et al. 1994a 1995~ 1996~ 1997). This lllodel can be recognised as an extension of Takagi-Sugenols Inodel (TakagL and Sugeno, 1985) in the sense that Takagi-Sugeno's InodeJ is lnorc function 3pproxinlation oriented \vhiJe our Inode) is Inore systelns and control oriented. The basic idea of the Inethods is: (i) to represent the fuzzy systelu as a faluily of local state space linear 11lodeJs: (ii) to design a state feedback controller for each local state space tnodel: (Hi) to construct a global controller frollI those local controllers. In all those luetbodSa the controller are al\vays based on full state feedback. Howevec not all the states are available in practice, and thus observers or filters are needed. There is SOlue discussion on design of
In this paper. we ,vill address filtering design of filzzy dynatuic systeuls. Two design tnethods \vill be developed based on H.-r. theory. quadr
that the approaches proposed in this paper could provide a ne,"' way for filtering design of a class of nonIinear systems. The paper is organised as follows. Section 2 discusses the fuzzy dynamic tnodel for a class of nonlinear systems. Based on a continuous Lyapunov function and a piecewisc continuous Lyapunov function respecth.-ely, section 3 and section 4 discuss two kinds of H X' filter design luethods using qnadratic stability theory and LMI techniques. A nUlnerical example is presented in section 5. Section 6 concludes some brief rClnarks. All proofs are olnitted due to space Iilnit.
2..
Fuzz~'
s)'stcm modelling
Many physicaJ systelns are very cOlnplex in practice so that rigorous Inatheluatical lnodels can be very difficult to obtain if not inlpossible. HoweveL many of these systems can be expressed in some fOTln of
5445
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress of IFAC
ROBUST H[Infinity] FILTERING OF FUZZY DYNAMIC SYSTEMS...
luathctuatical InodcI locally. or as an aggregation of a set of Inatheln,Ulcal ruodels. Here ",re consider using the follo\ving fuzz)'" dynalnic Inodel (0 represent a cOluplcx Inulti-input Inulti-output systelll. RI: IF \"1 is Fl' anli ... Vs is
F.!
Then
:K( 1 )
=:
.A, x( t) + RI
)jlr
z(t)
==
Lx(t)
,vhere RI denotes the i-th fuzzy inference rule. 111 the nUlnber of inference rules. j == 1. 2, ~ .. ,,'0 ) aTC
FJ (
3. Robust model
of the inferred fuzzy set P \",here
n
FT(
(2.2a)
I
and m
L:
PI
= 1.
"00 filter
design based on a nominal
In this section. we present an H oo filter design fOT the system described in eqn. (2.1) using Linear Matrix Inequalities (LMI). It can be seen from eqn. (2.3) that the global systelll is a nonlinear tDodel in nature. However~ if we introduce a nOlninal systelu (./-10. BO, C'O~ DO. L) and ~se eqn. (2~2) the systetn (2.3) can be rc\vritten in the following fonn x(t) == (Af) + ll.4( J1 »)x( t) + ( B o + M3( jl )ll~( t ) y(t) = ((~o + L\C~( I-l))r(t) +( Do + MJ( f.l))w(l) z( ( ) == L x( r )
Let 11/(Z(t)) be the nOftnalized luembership function
j~l
ReJuork 2.1: The assutnption 2.1 (a) is required to guarantee the unifonn asymptotic stability of the estiInation error dynamics: and the (b) is a·standard assulnption to guarantee nonsingular estitnation.
z
fuzzy sets. (.-1/1 BI, (), DJ) is the I-th subsysteJu of the systeln (2.1)" L is a kno,",n tnatrix representing the infonnation to be estitnated:- x(tJ E ..9 '1 are the state variables. ).1; E9iQ are the noise variables of the system which belong to L2[O, cX~. YI(I) E~l1r are the output variables of the l-th subsysteln. z(t) is a linear colnbination of the state variables to be estilnated~ and l'(t) := {VI (f). 1'2(1), ... , vs{t}} are SOlne Incasurable variables of the systetn. The final output of the systeln is obtained by taking the weighted average of all subsystems,
Ft ;:::
(2.6 )
The objective of this paper is to design a suitable filter for the systems (2.1) or (2.3) for estilnation of variable z \vith a guaranteed perforlnance in the H oo sense. that is_ given a prescribed level of noise attenuation )'>0. find a suitable filter such that the induced L~-nonn of the operator froIn IF to the filtering error. z - ~ is less than y under zero intial conditions I! Z - zlb
(2.1)
1. 2,···,111
~
I == 1, 2, .. ·111.
1)
y,(l)=("JX(f)+D,\1'(t)
I
r > (),
(b) [)J[)
(2.2b)
(3.1)
/:::1
nI
and AI is defined as the set of the IT\Clnbership functions satis1)ring (2.2b).
L1Afp(t)) =
L.
111
fit L\Ar Llli(p(t)) =
'::::;;:1
rn
L1C'(J1(t))
By using a standard fuzzy inference lllethod~ that is. using a singleton fuzzjfier. product fuzzy inference and centre-average defuzzifier. the foUo\ving global fuzzy dynalnic lnoocl can be obtained (Cao. et 011997a~
1996.
I997b_ 1997c. 1997d: Feng.
L.14/ L1('j
y(l) = (~(f.1(J)) x(t) Z(t) =
et of
m
L
PI ~1/, B(fA(tJ)
m
==
("(p(t)) =
~
L
PI B,. (2.4)
l~l
PI
(~J
ji(l):=-= j.1(p(tJ):
co_
r
(f
[)(ft(t)) =
i
t1D/~
= C~f
-
=
B/- BO~
C'O~
H
+ [)()1(t)) H'(t)
l=i
L: PI I=l
"uncertainties t ' in all matrices except L. However~ it should be noted that the "uncertainties of the fuzzy dynalnic systems are quite different from those oftJlC naflnal uncertain linear systelns~ in fact the "unccrtainties u in the fuzzy dynalnic lnodel are not real uncertainties because they are knov..:n a priori. The fundalnental idea involves constructing a quadratic form \vhich serves as an upper bound for the Lyapunov derivative corresponding to the closed loop fit ter S)rstelll. Thus we need to define the foUo\\'ing upper bounds of the systetn (3,1).
L X(O
A(fA(t)) =
m
Jll !lC',. dD(p(t)) =
.41 - .AO~ &31
:= •
/-it &3/~
LJDf = DI - DO. ! = 1 ~ 2. ... ~ nl. (3 .2) This is a linear systelll \vith time-varying nonlinear
(2.3)
11'(t)
L 1:::::1
199..f.~
1995. 1996. 1997). x( I ) = .il. (p(!)) x(tJ + B(f/(t))
=:
L
1::::=1
Jil JJ/.
I::.::.i
ff.il(v(t)). JI]{v(tJ) . .. ~, Jinl(V(t))}.
In this paper. \ve consider RJ"J filtering and adopt the following assulnption (Xie. and Souza. 1995). .A.,"sunJption 2.1: (a) The systcnl (2.1) or (2.3) is quadraticaIly stable. that is. there exist a positive definite luatrix P such that A( J-l ) T P + l~ 4. ( J.l) < 0 V j..J E }t. f : (2.5 )
d4{p)L1A r.u)T :5 EAE.~ ~ dB(j.L)AR(.u;Ts ERE~.
VIA EA!..
(J.3)
\lpEAf~
(].4)
~( f.l )&(~( J.1)T ~ EcE~.
'V pEAl _
(3.5)
MJ(p)MJ(p)T.s;E[JEL,
V'tJEAf.
(3.6)
{j,(
5446
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ROBUST H[Infinity] FILTERING OF FUZZY DYNAMIC SYSTEMS...
14th World Congress of IFAC
There arc several ways to obtain the upper bounds in (3.1 )-(3 .6). See (Cao. et al. 1996. 1997b) for details.
rr
R
"'1
i.",· ~J. v:! is
i( 1 )
Then
=:: JiJ e
F; '···1
x( f ) + A
(! •
J1( t
1 ••• ,
)
(3.7)
1 -
n,.
\",here >1" and K(, are Inatrices to be detcnnined. The global filter equation thus can be described as ;( t ) = .4 e x"( t ) + K e .v( t ) (3 .8)
z(1 ) ::; £J.l / 14 x( t )
=:
L x( t ) _
(1.9)
1=1
The error dynatnics equation of the global filter systelu can be obtained by cOlnbining eqn.(3.1) and (3.8) as fol1o~TS.
;. J Lr~40 + ·.4~Je +- Re.A X- i r
Re
e -
=:
+
Re .040 + AA - Re
Kf!(Dn +MJ)
]
f
8 0 + /)J1- K e ( Do + 1VJ)
J[
X ] X -
X
_ ( . .A e Ke(fO 4 4 I ~ 0 - .' r! ~ .l\. e
-
Bc
=
Let
+.
P"
rlB
Kef)o
('
-0
4
=[
heM)
A(!('O
~ i (I -
~
I (. .fi.. e . 0
l
We will show later on that if there exist a matrix c; ~ a positive definite Inatrix P and some constant c> () such that the follo\ving Inatrix inequality holds,! then the robust filter problelu will be solved. (~4 + BGe JP + A + B(;C~ )T + 2iEA EJ.
pr
1 - - -T- T +-pp +2eF(]IJEc E'c (J.;"GH) {;...
+2r~
-
-
-~
T
~~
(By'.' +BGDw)(B w +BGD w
)
1-r +4y- ERER
~t:
. ~
]
r:::
(3.11)
l]r=[I
-I' -
('0
l' It
B:
~
(
0
ER
]~ Rc: == lEe()
0 Ec
l~
(3.15)
(3.16)
!Jr T
B
x-x
L];-.
1=[0 °JB=[I-1
()
E~.l
F:=[~ ~IJ.H:=[~ ~lR=[~J
6B~J.:.eAD
~·lo'
J
\\'ith
Recently~ an approach of Linear Matrix Inequalities (LMI) has been developed for various controLLer designs (Skelton. and IwasakL 1995). It has been 5ho\\'0 (Skelton_ and I\\'asaki~ (995) that lllany controller design probleJns including HuJ~ H:;.~ Robust control. etc. can be tral1SfOrlned as an LMI probleln \vllich can be solved efficiently by convex progralnluing. Here the LMI techniques will be used for the filter design. III order to separate the fi Iter paralneters froln the plant Inatrices. \\le define the follo\\ling augluented luatrices.
~·lo
T
;
(3.13)
rGA+(l'GA)T+(~)
=l x ,. j
~ = (~4c + A-4~. )( + ( Bc: + !lEe )lV
~.
l1
It can be easily shown by using the Schur cOlupletuent formula a few tilDes that the above inequality is equivalent to the standard form of the luatrix inequality discussed in (Skelton~ and I\vasaki. 1995) as
eqn. (3.10) can be rc\\'ritten as e ~ [0
-~
+ B(;D + Mc)
+y-lpP'[p
El : =[ 0 -' E"A
J
o - KcD o K A(~ ] K /!,.(~ t...1 c = [ t:J..-l_eK,.~C M-eA-eliC 6B. (
~
)( EH
(3.14)
where Re=Ke(C'o-1 6(·) and e is the estilnation error defined as e ~ z ~ Z . .'
~-.~
+Y- (B}i + BC;D w + ABc
,,,here (3.10)
~4 c
G
+4y-1FC;REDEbrFGRJT +y-lpl![p <0
w
Ll[x:xJ
e==[O
L].
I
Bo
and a positive definite tnatrix P such that the following equation holds. (A + BGC + A.4c )P+ P(~4 + BCr(· + Mc)T
z( t) ~L;(t)
I:::: L2
1
Ae
Then the H~ filtering problcln is to find 3 Inatrix
F./
is
Vs
.
1
(3.12)
The proposed filter can be described as J ::
0] B)1.'- -= [0lJ' D.... - ~ [0 ]L =[0 Do
(-.~ = [~~eo
0J
0
0
()
D~,
-T
,AT =
0
T
F
EcH
F
E~RT
tI)
pr
LP
Sw
1.J
()
()
-rJ
0
0
()
()
B~,
0
_L]
0
0
()
2 0
0
!)
-61
()
()
0
0
[)
0
I --1
()
()
0
0
()
0
_L]
-r
(.:.) =
()
-T
26
4 (3.17) \\!here <}):::::
AP + PAT + 2I£AE.~
+4y~) EeEJ.
Then the solution of the filter design can be obtained by solving the LMI as discussed in (Skcltoll~ and
(·0'
5447
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ROBUST H[Infinity] FILTERING OF FUZZY DYNAMIC SYSTEMS...
14th World Congress of IFAC
Ilvasaki~ 1995}. Therefore. "ve Jla,'c tJJe fOnoll1ing other tnain result of this section.
Theorem 3.1: Given a SystCIll as described in (2.1). and et filter of the fonD (3.7) or (3 .R)~ if the inequality (3.14) or equivaJentJy (3. J6) has a positive definite luatrix solution p~ then (i) the error dynanlics of the global filter systeln is as}rluptotically stable and the nann of the operator frOlll \1· to the filter error is less than y. (ii) MoreoveL the filter paralneters are given b}f
c; = _pr T <1)AT ,-·P
(3.18)
\vhere the scalar p is the free paraznetcr c]):=(rr.
T
1,-1 ~~(...» ) >0
su~ject
P and the Inatrix JP is defined by '¥: = (A
r
Al, /J,.
E
A-:J.Jj
~
!:J( ~Ii
1·1, -
;t:.
~-4{;!1Bli ~
;::: (',. - (
f == 1,2
'\11' E ~\'I
O.
·r •M.Jh
(3.20)
It is noted that one essential assuluption about the systeln for the above filter design is the existence of a nOluinallnodel. This is Cl very restrictive assulnption and only a very lilnited class of nonlinear systelns. can satisfy such assumption. Then the following new design luethod is needed.
Ri - Br '
= 1)1 -
\\-'here p, "",P",I are Ill! luelnbership functions \\'hich are not equal to zero when the I-Ih subsystem plays a dOlninant role~ that is. v( f ) E 5;, . Siluilar to the upper bounds discussed in Section J. \ve define the follo\\ring upper bounds for the above Imcertainties_
AB,J1BT S
ER!E~!,
(4.6) 8.(~J6.("T ~EcIE~J; MJ,MJT ~ED1EL{. I == 1,2.·· ·.n} . Then the idea of the robust filter design is to find a robust filter on each subspacc following the approach discussed in Section 3. The proposed filter on each subspace can be described as J R :: ff VI is ~J. "1 is F! ..... " s is Jt:! ;( t ) ::: .44"1
Then
x( t ) + K
eJ
J/( t )
(4.7)
zf t) = L i( 1) v(t) E5'~I' I
= t2,···.",.
\vhere . A",] and Kef, I == t2.·· ·/11 4. Robust Bd) filter design based on piecewise Lyapunov function
(4.5)
L)[.
111
t .. q
/!A[M,T S EA1E~/'
to
(3.19)
1
lif
_
are matrices to be
determined.
Let r; i == t2.··· . T be T tilue instants at which the variable v(t) of the systeln (2.3) lueets the boundary of one of the subspaces ~)'" I = 1.2., -·111. Then \vhen the systetn comes to the boundary. its filter paralneters are chosen as follows~ 1
In tl1is section. ""e lviJ] present one Ine1hod of piecewise Lyapunov function based approach_ In order to facilitate the filter design~ \ve decolupose the tllodel (2.3) into 111 independent subsystelns. First. \ve define In subspaces of the systeln~ 5', ~ (vl.,u,(V) > l'l(V) i = 1,2, ,_., 111,i '* 1./ (4. I) 1=1,2
1 •••
,/1,
==
{I
v( t
)
o
E
.'J', or v{ f -
a-,~/ ~
) E
olherliJ;.";e
A' ==
i=J
_ {K er J.:..(! ==.
x(t) ;::: ..4 ,(jl)X(tj+B/(jt)l1J(f)
e
r(
11 ) x( t )
) ~
MJ( f.1 ))1-1'( t
llA., ( f.-l) ~
L 71 j Li4 1i , /illJ ( 1-') =
j=',,:Li ffJlf
8,(·1 (J.1)::::;
nit
~
),
+[
p; liB".
e~[O
i=l,iot=c.{
m,
L. Jii~(\·' AD, (J')::;; L JtiMJJj. i""'J.I~r
..-'le]
J..:. ej
z(l)=Lx(I), nJ;(
() _. kaT
If v( r f
)
E ,\, • I
= ri
.
<
\
,V( t) E'<"1.' t *_T i (r i ) E ~S J • t - r
v( T
_ j
'.
• i
E ~\ j ' t ~ i i
)
(4.9) '
[X~~]=[A'~~4:~:r .4~~~rJ[x:x]
+ Dr 1-' ) H'( t )
+ A( Y/-l ))x(l ) + (D{ +
l· - -,
el
0
The error dynalnics equation of the global system can be obtained by cOlubining (4.4) and (4.9) as follows.
(4.4)
== ( .4., + LiA., ( It ))x( f ) + ( E, + tM3J ( /1 )))""'( I
==
{Ad
f 'It ( t ) == 1
be expressed by
~ ((~,
l [A
Therefore., the filter paralueters are detenllined by the following equations.
(4.3) Then. on every subspace the fuzzy systeln (2.3) can
.v( I ) =
0~
J.:.. e
(4.8)
The boundaf)' of S'/ is denoted by ~~ 1 = {pi PT ( }) ) = j.J i ( v) j = 1.2., -. , 111. i ;c J} ( 4. 2 ) The characteristic function of ~~.( is defined by I} I (t )
.4<:. [ ()
II
~ KeD '"-' B-KeD
li-
l'{ J) E
~r;.,~(
Ll[x:xl (4.10)
r=l.i::t:1
,,,,here e is the estillliition error defined as
e == z -
z.
5448
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ROBUST H[Infinity] FILTERING OF FUZZY DYNAMIC SYSTEMS...
(=[
Let
~-Jd
=[
cl
Kel~-~I
~4t:'J - h
...:J.J -
H
i ,,]_ X~X
...-4('1,+
::::= [
Ad f),
Ke/("J]
_,
~41 - Kl!I(~J ·
eJ ( I
].
B r -J..: e1 D 1
•
Kel~(T( - he[ Ii.f
~~/= [ c
14th World Congress of IFAC
il.4 1
Ke/A(~{ AiI f
T
-
K ci~( ~I
Renlork ..j, J: If we use a fixed P=P(t), '\It the condition (4. 15) is the standard condition of quadratic stability ,vith a disturbance attenuation y for linear continuous-time systems. Howevec if P(t) is a piecewise differentiable lnatrix {unc(ion~ we should expect boundary conditions (4.14).
1 I
MJ~~I~~J
Md == [
eqn. (4.10) can be re\:vritten on each subspace .~fll
I=
as
AcJ + l~r(.~1 Jt; + ( Bc! + ABcl
: ; -::=; (
e
t2~'~~'11
== (0
L); ~
with the same disturbance attenuation y.
v( t) E If.,T r .
HO"'eVeL it should be noted that even if all tIle filters on the subspaces are stable the global filler syslelll cannot be guaranteed to be stable unless some other conditions are satisfied. We will use the follo''''ing piecewise differentiable quadratic (PDQ) Lyapunov function for stability analysis~
=x
f'''(t)
T
r71
p
m
With the following
- _[0.A, ..4,0J
~'1[ - ,
(~."
~
I
-=-_[1
B -
.
[ I (~J
notations~
0 ] C','
-
'~J
l[°1J D-
B"'I = B
-1
W1
JJ .
-1
[Aoe/
- = G -r
T
P(t)x = L,}/(f H'! = Lll,(t)x P'X ':;;::1
Consider the dynamic fuzzy system
1:
(4.13) .. if it is quadraticlly stable with a disturbance attenuation Y1 then~ it is also asylnptoticaHy stable
(4.11)
)lfJ
Len,nlQ ..r
()
0 K el
1~
(4.17a)
lOJ L- "" l0
= D
•
IJ( ( ) ~ 'h (t )~ + '''J.J t JP"! +.... H+l1 m( t )Pn; (4.12) \vhece (PE, I';!. .... Pm) is a set of nxn fixed positivedefinite symtnetric matrices and P(!) is a symmetric piece\\'ise differentiable Inatrix function.
\lie have the follo\\-'ing results.
Because \ve use the piecewise differentiable matrix function. the folJo\",ing notations are used in this paper. For a lnatrix function IVt),
Theorem -1.1: Given a systetn as described in (2.1), and a filter of the fonn (4.7) and (4. 9)~ if the following conditions are satisfied.
p( f
):~» -4 ~H nxn .
the left -liluit of
[J(r)
at
t
91 is defined as
( E ~11
T
<
(4.17b)
E8/:""
EAT'
(4.18)
cc,
(2)
there exist a set of positive B I • I :=: 1.2.··· 111 such that the inequality
P( t- }:;::::; lint 1)( t - E:) £---+0 .
if the lilnit exists, and the right limit of
1
--- lO 0J - lOl EH1l
E A /:= EAr
(I)
E
T
P(t)
at
(AI
+BG,rr)P, +F/( . 4 . r +BCi;0)T +C,EA1E;;,
+~F(J-} +2BIF~HEc,E{.I( f~(r?l-{)T
is defined as J)( I ~ ): -:=. /hll Pr t + & ) H-4()
E,
1 -
---
+2y - (B",/ + BC]! D wl
jfthe lituit exists.
constants
)(
-
-~-
B wl + BCT, D W1
+4y-l Es1EJn )
T
+4y -I f~G,HYi[)lE:;l(FG,H)T + y-l p'rrI~ < 0
We \-vill consider the quadratic stability_ De.{inUiol1 -I. 1: Given a dyn(:unic fuzzy closed loop
systelu x( t ) = ..4", ( !J ) x( t ) +
z( t )
Bc ( !-I }t-e( ( )
== Lx( f )
where (4.13 )
the systetD (4-.13) is said to be quadratically stable with a disturbance attenuation y if there exist a set of fixed positive-defini le SYIUluetric Inatrices ( ~ . p;. .. . ~ P",) and a constant et such that the
following conditions are (I)
T
I == 1. 2•... In
satisfied~
(4.14)
F:=
_ [E
E c :=
i'
o
0]
Ec
( 4~
19)
;
[~~ ~IJ H." = [~ ~J
v.,hich can be re\"-ritten as rc.',A, + (rc;:.A,)T + (...)/ < 0 ~ I:=: 1,2;"'111 with
R ""
[~l (4.20)
5449
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ROBUST H[Infinity] FILTERING OF FUZZY DYNAMIC SYSTEMS...
B
p,(7r
0
n D
0
]'=
.A~ =
()
References
:, 1
0
Cao.
S.G.. N.W. Rees_ and G. Feng (1996). Quadratic stability analysis and design of continuous tilne fuzzy control systems u 1nl. J . .\:y.\~te"g..· ~':tcience ~ vot 27_ pp. 19J -20) . Cao_ S.G.. N.W, Rees~ and G. Feng (1997a). ~-Analysis and design for a class of cOlnplex control systelns~ Part I: fuzzy modeling and identification ~- ~ .04 ut0I11at;ca~ vo!. 3 J ~ no.6.
]{r;, HT
F
il
E~/RT
J1~
:>
(1)1
P,T!
Bwl
P,
0
0
LI',
-yl
[)
()
0
0
~:'I
()
_L]
()
0
0
!}
0
0
-Et!
0
0
()
()
0
0
__1_ 1 28 1
0
0
0
0
_LJ
,
0), ==
14th World Congress of IFAC
()
2
0
pp.iO 17-1028.
4
(4.21) -
-T
= ,Al~ + IJA/
-
-T
-1-
~T
+ 2c/E Af E A { + 4y EA1ErH has a set of positive definite Inatrix solutions 11, I = 1,2.·· ~111 ~ then (i) the error dynarnics of the global filter systeln is aSylnptotically stable \vith a disturbance attenuation ¥, (ii) MoreoveL the filter parameters are given by T (;, == -Prr a)/A~\¥, ~ J;:: 1.2.···n' (4.22) where the scalar Pt is the free parallleter subject to l
l1>/:==(rr T
1. -1 --(""')1)
>0,
4 23) ('.-
PI
and the Inatrix lf1 is defined by q/,:== (A/cJ),A~ )-r.
(4.24)
ReJllork -+.!: It should be noted that the above
stability results of filter design are based on one boundary condition T < 00, which is not easy to test. For sOlne Inore easily checkable conditions~ sce (Cao~ et al. 1997d).
5. An example of simulation We consider the probleln of balancing an inverted pendululn on a cart. The silllulation results are olnitted due to space liInit.
6. Conclusions
In this paper~ a nUluber of robust HOC) filter design lucthods are proposed for fuzzy dynamic systelllS based on quadratic stability theol)' and LMI techniques. It has been shown that the resulting robust H oo filter S}~steln is Quadratically stable ~rith disturbance attenuation jf a silnple linear Inatrix inequality has a positive definite symJuetric Inatrix solution or if a set of simple linear Inatrix inequalities has a set of of positive definite sylUll1etric lnatrix solutions depending on t\\'O kinds of design strategies.
Cao. S.G.. N.W. Recs, and G. Feng (1997b). ~·Analysis and design for a class of complex control systems~ Part 11: fuzzy controller design·· ~ ..4utonlatica_ vo1.33~ pp. 1029-1039. Cao. S.G.~ N.W. Rees, and G. Feng (1997c).•1fL., control of nonlinear continuous-time systelns based on dynamic fuzzy luodels u • Int. J .
pp.821-830. N.W. Rees.. and G. Feng (1997d). ~~Further results about quadratic stability of continuous time fuzzy control systeJnsH~ Int..1. S:vstems l';;cience. 'l0l.28, no.4~ pp.391-404. Feng~ G.~ S.G. Cao~ N.W. Rees and C.K. Chak (1994). 11 An approach to controller design for a class of fuzzy systems u .. Proc. AJVZIfS.. Brisbane~ pp.229-233. Feng~ G.~ S.G. Cao.. N.W. Rees and C.K. Chak (1995). "Design of fuzzy control systems ba'sed on state feedback".. J. Intelligent de FuzZ)-' l~v.\;te111 ...-,·, vo1.3~ pp.295-304. Feng. G.. S.G. C30~ and N.W. Rees (1996). ·'An approach to Hu') control of a class of nonlinear systelns", Auto/natica~ vo1.32. pp. 1469-1474. Feng. G... S.G. Cao~ N.W. Rees and C.K. Chak (1997). "Design of fuzzy control systems with guaranteed stability" ~ Fuzz.v /\eL\" and .'-,)lslen,s~ vo1.85. pp.1-l0. Skelton. R.E. and T. Iwasaki (1995). "'Increased roles of linear algebra in control education·~~ IEEE C"ontrol .S:v.,·tents A/ag... vol. 15. pp.76-90. Takagi T. and M. Sugeno (1985). "Fuzz~/ identification of systems and its application to luodelling and control" _ lEEE Trans. .)~v.,,·tenl.\· A/an C:vbernelics, vol. 15 ~ pp, 1 16-132. Xie~ L. and C.E. de Souza (1995). .tO n robust filtering for linear systems paralueter uncertainty. u Pr()c.3.Jth IEEE (LJ(T, Zadeh. L.A. (1973). "Outline of a ne," approach to the analysis of cOlnplex systelns and decision processes ". 1l:!.~E Tran.\'. ~t;.,~v.\·teI11"'; Alan C:vhernetics. vo1.3. pp.28-44.
Cao.
S.G.~
5450
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress ofIFAC
Copyright CO 1999 IFAC 14th Triennial World Congress~ Beijing, P.Ra China
K-3e-12-1
Robust H~ Filtering of Fuzzy Dynamic Systems G. Feng*~ J. Ma+~ Z.
x. Han* and N. Zbang*
*&~'cho()1 (?{ Electrical EnJ~jneeringr (injversi~v (?{ lVelV ~~'outh rVales. &';':v(ine~v, N.r.,TV 2052. ~4u.,. .t ralia.
Tel: (61 2) 9385 5374. Fax: (61 2) 9385 5993. Elllai/:
(;.Fel1g(~I(Un.\"tJ.edu.au
+Departluenl (~rIn. (0r11Joti 0/1 ~~:v.,·ten1.", (~i~v rrniver."i(v (?( Hong Konl~1 Ho}]..!? Kong
Abstract This paper addresses robust H oo filtering of cOlnplex nonlinear systelns which can be represented by a fuzzy dynalnic model. Based on a continuous Lyapunov function and (I piecc\vise cOlltinuous Lyapunov function respectively~ two kinds of new filtering design tuethods are proposed using quadratic stability theory and Linear Matrix Inequalities (LMI). Copyright
,e, 1999IFAC
KeY'\'ords: Fuzzy sysfeIlls, RV-' filtering~ LMt quadratic stability.
1.
observers for fuzzy dynalnic systelns in (Feng~ et al. 1995). but there is~ to the authors· best knowledge~ very rev.- result available on design offilters for fuZZ}f dynaluic SystCITIS.
Introduction
Since the first paper on fuzzy sets (Zadeh~ 1973) "vas published~ fuzzy logic control (FLC) has recently proved to be a successful control approach for cOlnplex nonlinc3r systclns_ During the last few
years4 \ve have proposed a nUlnber of ne\v Jnethods for the systelnatic analysis and design of fuzzy logic controllers based on t he so-called fuzzy dynaluic tnodeI ~ \vhich consists of Cl f~unily of local linear Inodels smoothly connected through fuzzy Inelnbership functions (Cao_ et al. 1996~ 1997a, 1997b, 1997c~ 1997d: Fcng~ et al. 1994a 1995~ 1996~ 1997). This lllodel can be recognised as an extension of Takagi-Sugenols Inodel (TakagL and Sugeno, 1985) in the sense that Takagi-Sugeno's InodeJ is lnorc function 3pproxinlation oriented \vhiJe our Inode) is Inore systelns and control oriented. The basic idea of the Inethods is: (i) to represent the fuzzy systelu as a faluily of local state space linear 11lodeJs: (ii) to design a state feedback controller for each local state space tnodel: (Hi) to construct a global controller frollI those local controllers. In all those luetbodSa the controller are al\vays based on full state feedback. Howevec not all the states are available in practice, and thus observers or filters are needed. There is SOlue discussion on design of
In this paper. we ,vill address filtering design of filzzy dynatuic systeuls. Two design tnethods \vill be developed based on H.-r. theory. quadr
that the approaches proposed in this paper could provide a ne,"' way for filtering design of a class of nonIinear systems. The paper is organised as follows. Section 2 discusses the fuzzy dynamic tnodel for a class of nonlinear systems. Based on a continuous Lyapunov function and a piecewisc continuous Lyapunov function respecth.-ely, section 3 and section 4 discuss two kinds of H X' filter design luethods using qnadratic stability theory and LMI techniques. A nUlnerical example is presented in section 5. Section 6 concludes some brief rClnarks. All proofs are olnitted due to space Iilnit.
2..
Fuzz~'
s)'stcm modelling
Many physicaJ systelns are very cOlnplex in practice so that rigorous Inatheluatical lnodels can be very difficult to obtain if not inlpossible. HoweveL many of these systems can be expressed in some fOTln of
5445
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress of IFAC
ROBUST H[Infinity] FILTERING OF FUZZY DYNAMIC SYSTEMS...
luathctuatical InodcI locally. or as an aggregation of a set of Inatheln,Ulcal ruodels. Here ",re consider using the follo\ving fuzz)'" dynalnic Inodel (0 represent a cOluplcx Inulti-input Inulti-output systelll. RI: IF \"1 is Fl' anli ... Vs is
F.!
Then
:K( 1 )
=:
.A, x( t) + RI
)jlr
z(t)
==
Lx(t)
,vhere RI denotes the i-th fuzzy inference rule. 111 the nUlnber of inference rules. j == 1. 2, ~ .. ,,'0 ) aTC
FJ (
3. Robust model
of the inferred fuzzy set P \",here
n
FT(
(2.2a)
I
and m
L:
PI
= 1.
"00 filter
design based on a nominal
In this section. we present an H oo filter design fOT the system described in eqn. (2.1) using Linear Matrix Inequalities (LMI). It can be seen from eqn. (2.3) that the global systelll is a nonlinear tDodel in nature. However~ if we introduce a nOlninal systelu (./-10. BO, C'O~ DO. L) and ~se eqn. (2~2) the systetn (2.3) can be rc\vritten in the following fonn x(t) == (Af) + ll.4( J1 »)x( t) + ( B o + M3( jl )ll~( t ) y(t) = ((~o + L\C~( I-l))r(t) +( Do + MJ( f.l))w(l) z( ( ) == L x( r )
Let 11/(Z(t)) be the nOftnalized luembership function
j~l
ReJuork 2.1: The assutnption 2.1 (a) is required to guarantee the unifonn asymptotic stability of the estiInation error dynamics: and the (b) is a·standard assulnption to guarantee nonsingular estitnation.
z
fuzzy sets. (.-1/1 BI, (), DJ) is the I-th subsysteJu of the systeln (2.1)" L is a kno,",n tnatrix representing the infonnation to be estitnated:- x(tJ E ..9 '1 are the state variables. ).1; E9iQ are the noise variables of the system which belong to L2[O, cX~. YI(I) E~l1r are the output variables of the l-th subsysteln. z(t) is a linear colnbination of the state variables to be estilnated~ and l'(t) := {VI (f). 1'2(1), ... , vs{t}} are SOlne Incasurable variables of the systetn. The final output of the systeln is obtained by taking the weighted average of all subsystems,
Ft ;:::
(2.6 )
The objective of this paper is to design a suitable filter for the systems (2.1) or (2.3) for estilnation of variable z \vith a guaranteed perforlnance in the H oo sense. that is_ given a prescribed level of noise attenuation )'>0. find a suitable filter such that the induced L~-nonn of the operator froIn IF to the filtering error. z - ~ is less than y under zero intial conditions I! Z - zlb
(2.1)
1. 2,···,111
~
I == 1, 2, .. ·111.
1)
y,(l)=("JX(f)+D,\1'(t)
I
r > (),
(b) [)J[)
(2.2b)
(3.1)
/:::1
nI
and AI is defined as the set of the IT\Clnbership functions satis1)ring (2.2b).
L1Afp(t)) =
L.
111
fit L\Ar Llli(p(t)) =
'::::;;:1
rn
L1C'(J1(t))
By using a standard fuzzy inference lllethod~ that is. using a singleton fuzzjfier. product fuzzy inference and centre-average defuzzifier. the foUo\ving global fuzzy dynalnic lnoocl can be obtained (Cao. et 011997a~
1996.
I997b_ 1997c. 1997d: Feng.
L.14/ L1('j
y(l) = (~(f.1(J)) x(t) Z(t) =
et of
m
L
PI ~1/, B(fA(tJ)
m
==
("(p(t)) =
~
L
PI B,. (2.4)
l~l
PI
(~J
ji(l):=-= j.1(p(tJ):
co_
r
(f
[)(ft(t)) =
i
t1D/~
= C~f
-
=
B/- BO~
C'O~
H
+ [)()1(t)) H'(t)
l=i
L: PI I=l
"uncertainties t ' in all matrices except L. However~ it should be noted that the "uncertainties of the fuzzy dynalnic systems are quite different from those oftJlC naflnal uncertain linear systelns~ in fact the "unccrtainties u in the fuzzy dynalnic lnodel are not real uncertainties because they are knov..:n a priori. The fundalnental idea involves constructing a quadratic form \vhich serves as an upper bound for the Lyapunov derivative corresponding to the closed loop fit ter S)rstelll. Thus we need to define the foUo\\'ing upper bounds of the systetn (3,1).
L X(O
A(fA(t)) =
m
Jll !lC',. dD(p(t)) =
.41 - .AO~ &31
:= •
/-it &3/~
LJDf = DI - DO. ! = 1 ~ 2. ... ~ nl. (3 .2) This is a linear systelll \vith time-varying nonlinear
(2.3)
11'(t)
L 1:::::1
199..f.~
1995. 1996. 1997). x( I ) = .il. (p(!)) x(tJ + B(f/(t))
=:
L
1::::=1
Jil JJ/.
I::.::.i
ff.il(v(t)). JI]{v(tJ) . .. ~, Jinl(V(t))}.
In this paper. \ve consider RJ"J filtering and adopt the following assulnption (Xie. and Souza. 1995). .A.,"sunJption 2.1: (a) The systcnl (2.1) or (2.3) is quadraticaIly stable. that is. there exist a positive definite luatrix P such that A( J-l ) T P + l~ 4. ( J.l) < 0 V j..J E }t. f : (2.5 )
d4{p)L1A r.u)T :5 EAE.~ ~ dB(j.L)AR(.u;Ts ERE~.
VIA EA!..
(J.3)
\lpEAf~
(].4)
~( f.l )&(~( J.1)T ~ EcE~.
'V pEAl _
(3.5)
MJ(p)MJ(p)T.s;E[JEL,
V'tJEAf.
(3.6)
{j,(
5446
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ROBUST H[Infinity] FILTERING OF FUZZY DYNAMIC SYSTEMS...
14th World Congress of IFAC
There arc several ways to obtain the upper bounds in (3.1 )-(3 .6). See (Cao. et al. 1996. 1997b) for details.
rr
R
"'1
i.",· ~J. v:! is
i( 1 )
Then
=:: JiJ e
F; '···1
x( f ) + A
(! •
J1( t
1 ••• ,
)
(3.7)
1 -
n,.
\",here >1" and K(, are Inatrices to be detcnnined. The global filter equation thus can be described as ;( t ) = .4 e x"( t ) + K e .v( t ) (3 .8)
z(1 ) ::; £J.l / 14 x( t )
=:
L x( t ) _
(1.9)
1=1
The error dynatnics equation of the global filter systelu can be obtained by cOlnbining eqn.(3.1) and (3.8) as fol1o~TS.
;. J Lr~40 + ·.4~Je +- Re.A X- i r
Re
e -
=:
+
Re .040 + AA - Re
Kf!(Dn +MJ)
]
f
8 0 + /)J1- K e ( Do + 1VJ)
J[
X ] X -
X
_ ( . .A e Ke(fO 4 4 I ~ 0 - .' r! ~ .l\. e
-
Bc
=
Let
+.
P"
rlB
Kef)o
('
-0
4
=[
heM)
A(!('O
~ i (I -
~
I (. .fi.. e . 0
l
We will show later on that if there exist a matrix c; ~ a positive definite Inatrix P and some constant c> () such that the follo\ving Inatrix inequality holds,! then the robust filter problelu will be solved. (~4 + BGe JP + A + B(;C~ )T + 2iEA EJ.
pr
1 - - -T- T +-pp +2eF(]IJEc E'c (J.;"GH) {;...
+2r~
-
-
-~
T
~~
(By'.' +BGDw)(B w +BGD w
)
1-r +4y- ERER
~t:
. ~
]
r:::
(3.11)
l]r=[I
-I' -
('0
l' It
B:
~
(
0
ER
]~ Rc: == lEe()
0 Ec
l~
(3.15)
(3.16)
!Jr T
B
x-x
L];-.
1=[0 °JB=[I-1
()
E~.l
F:=[~ ~IJ.H:=[~ ~lR=[~J
6B~J.:.eAD
~·lo'
J
\\'ith
Recently~ an approach of Linear Matrix Inequalities (LMI) has been developed for various controLLer designs (Skelton. and IwasakL 1995). It has been 5ho\\'0 (Skelton_ and I\\'asaki~ (995) that lllany controller design probleJns including HuJ~ H:;.~ Robust control. etc. can be tral1SfOrlned as an LMI probleln \vllich can be solved efficiently by convex progralnluing. Here the LMI techniques will be used for the filter design. III order to separate the fi Iter paralneters froln the plant Inatrices. \\le define the follo\\ling augluented luatrices.
~·lo
T
;
(3.13)
rGA+(l'GA)T+(~)
=l x ,. j
~ = (~4c + A-4~. )( + ( Bc: + !lEe )lV
~.
l1
It can be easily shown by using the Schur cOlupletuent formula a few tilDes that the above inequality is equivalent to the standard form of the luatrix inequality discussed in (Skelton~ and I\vasaki. 1995) as
eqn. (3.10) can be rc\\'ritten as e ~ [0
-~
+ B(;D + Mc)
+y-lpP'[p
El : =[ 0 -' E"A
J
o - KcD o K A(~ ] K /!,.(~ t...1 c = [ t:J..-l_eK,.~C M-eA-eliC 6B. (
~
)( EH
(3.14)
where Re=Ke(C'o-1 6(·) and e is the estilnation error defined as e ~ z ~ Z . .'
~-.~
+Y- (B}i + BC;D w + ABc
,,,here (3.10)
~4 c
G
+4y-1FC;REDEbrFGRJT +y-lpl![p <0
w
Ll[x:xJ
e==[O
L].
I
Bo
and a positive definite tnatrix P such that the following equation holds. (A + BGC + A.4c )P+ P(~4 + BCr(· + Mc)T
z( t) ~L;(t)
I:::: L2
1
Ae
Then the H~ filtering problcln is to find 3 Inatrix
F./
is
Vs
.
1
(3.12)
The proposed filter can be described as J ::
0] B)1.'- -= [0lJ' D.... - ~ [0 ]L =[0 Do
(-.~ = [~~eo
0J
0
0
()
D~,
-T
,AT =
0
T
F
EcH
F
E~RT
tI)
pr
LP
Sw
1.J
()
()
-rJ
0
0
()
()
B~,
0
_L]
0
0
()
2 0
0
!)
-61
()
()
0
0
[)
0
I --1
()
()
0
0
()
0
_L]
-r
(.:.) =
()
-T
26
4 (3.17) \\!here <}):::::
AP + PAT + 2I£AE.~
+4y~) EeEJ.
Then the solution of the filter design can be obtained by solving the LMI as discussed in (Skcltoll~ and
(·0'
5447
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ROBUST H[Infinity] FILTERING OF FUZZY DYNAMIC SYSTEMS...
14th World Congress of IFAC
Ilvasaki~ 1995}. Therefore. "ve Jla,'c tJJe fOnoll1ing other tnain result of this section.
Theorem 3.1: Given a SystCIll as described in (2.1). and et filter of the fonD (3.7) or (3 .R)~ if the inequality (3.14) or equivaJentJy (3. J6) has a positive definite luatrix solution p~ then (i) the error dynanlics of the global filter systeln is as}rluptotically stable and the nann of the operator frOlll \1· to the filter error is less than y. (ii) MoreoveL the filter paralneters are given b}f
c; = _pr T <1)AT ,-·P
(3.18)
\vhere the scalar p is the free paraznetcr c]):=(rr.
T
1,-1 ~~(...» ) >0
su~ject
P and the Inatrix JP is defined by '¥: = (A
r
Al, /J,.
E
A-:J.Jj
~
!:J( ~Ii
1·1, -
;t:.
~-4{;!1Bli ~
;::: (',. - (
f == 1,2
'\11' E ~\'I
O.
·r •M.Jh
(3.20)
It is noted that one essential assuluption about the systeln for the above filter design is the existence of a nOluinallnodel. This is Cl very restrictive assulnption and only a very lilnited class of nonlinear systelns. can satisfy such assumption. Then the following new design luethod is needed.
Ri - Br '
= 1)1 -
\\-'here p, "",P",I are Ill! luelnbership functions \\'hich are not equal to zero when the I-Ih subsystem plays a dOlninant role~ that is. v( f ) E 5;, . Siluilar to the upper bounds discussed in Section J. \ve define the follo\\ring upper bounds for the above Imcertainties_
AB,J1BT S
ER!E~!,
(4.6) 8.(~J6.("T ~EcIE~J; MJ,MJT ~ED1EL{. I == 1,2.·· ·.n} . Then the idea of the robust filter design is to find a robust filter on each subspacc following the approach discussed in Section 3. The proposed filter on each subspace can be described as J R :: ff VI is ~J. "1 is F! ..... " s is Jt:! ;( t ) ::: .44"1
Then
x( t ) + K
eJ
J/( t )
(4.7)
zf t) = L i( 1) v(t) E5'~I' I
= t2,···.",.
\vhere . A",] and Kef, I == t2.·· ·/11 4. Robust Bd) filter design based on piecewise Lyapunov function
(4.5)
L)[.
111
t .. q
/!A[M,T S EA1E~/'
to
(3.19)
1
lif
_
are matrices to be
determined.
Let r; i == t2.··· . T be T tilue instants at which the variable v(t) of the systeln (2.3) lueets the boundary of one of the subspaces ~)'" I = 1.2., -·111. Then \vhen the systetn comes to the boundary. its filter paralneters are chosen as follows~ 1
In tl1is section. ""e lviJ] present one Ine1hod of piecewise Lyapunov function based approach_ In order to facilitate the filter design~ \ve decolupose the tllodel (2.3) into 111 independent subsystelns. First. \ve define In subspaces of the systeln~ 5', ~ (vl.,u,(V) > l'l(V) i = 1,2, ,_., 111,i '* 1./ (4. I) 1=1,2
1 •••
,/1,
==
{I
v( t
)
o
E
.'J', or v{ f -
a-,~/ ~
) E
olherliJ;.";e
A' ==
i=J
_ {K er J.:..(! ==.
x(t) ;::: ..4 ,(jl)X(tj+B/(jt)l1J(f)
e
r(
11 ) x( t )
) ~
MJ( f.1 ))1-1'( t
llA., ( f.-l) ~
L 71 j Li4 1i , /illJ ( 1-') =
j=',,:Li ffJlf
8,(·1 (J.1)::::;
nit
~
),
+[
p; liB".
e~[O
i=l,iot=c.{
m,
L. Jii~(\·' AD, (J')::;; L JtiMJJj. i""'J.I~r
..-'le]
J..:. ej
z(l)=Lx(I), nJ;(
() _. kaT
If v( r f
)
E ,\, • I
= ri
.
<
\
,V( t) E'<"1.' t *_T i (r i ) E ~S J • t - r
v( T
_ j
'.
• i
E ~\ j ' t ~ i i
)
(4.9) '
[X~~]=[A'~~4:~:r .4~~~rJ[x:x]
+ Dr 1-' ) H'( t )
+ A( Y/-l ))x(l ) + (D{ +
l· - -,
el
0
The error dynalnics equation of the global system can be obtained by cOlubining (4.4) and (4.9) as follows.
(4.4)
== ( .4., + LiA., ( It ))x( f ) + ( E, + tM3J ( /1 )))""'( I
==
{Ad
f 'It ( t ) == 1
be expressed by
~ ((~,
l [A
Therefore., the filter paralueters are detenllined by the following equations.
(4.3) Then. on every subspace the fuzzy systeln (2.3) can
.v( I ) =
0~
J.:.. e
(4.8)
The boundaf)' of S'/ is denoted by ~~ 1 = {pi PT ( }) ) = j.J i ( v) j = 1.2., -. , 111. i ;c J} ( 4. 2 ) The characteristic function of ~~.( is defined by I} I (t )
.4<:. [ ()
II
~ KeD '"-' B-KeD
li-
l'{ J) E
~r;.,~(
Ll[x:xl (4.10)
r=l.i::t:1
,,,,here e is the estillliition error defined as
e == z -
z.
5448
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ROBUST H[Infinity] FILTERING OF FUZZY DYNAMIC SYSTEMS...
(=[
Let
~-Jd
=[
cl
Kel~-~I
~4t:'J - h
...:J.J -
H
i ,,]_ X~X
...-4('1,+
::::= [
Ad f),
Ke/("J]
_,
~41 - Kl!I(~J ·
eJ ( I
].
B r -J..: e1 D 1
•
Kel~(T( - he[ Ii.f
~~/= [ c
14th World Congress of IFAC
il.4 1
Ke/A(~{ AiI f
T
-
K ci~( ~I
Renlork ..j, J: If we use a fixed P=P(t), '\It the condition (4. 15) is the standard condition of quadratic stability ,vith a disturbance attenuation y for linear continuous-time systems. Howevec if P(t) is a piecewise differentiable lnatrix {unc(ion~ we should expect boundary conditions (4.14).
1 I
MJ~~I~~J
Md == [
eqn. (4.10) can be re\:vritten on each subspace .~fll
I=
as
AcJ + l~r(.~1 Jt; + ( Bc! + ABcl
: ; -::=; (
e
t2~'~~'11
== (0
L); ~
with the same disturbance attenuation y.
v( t) E If.,T r .
HO"'eVeL it should be noted that even if all tIle filters on the subspaces are stable the global filler syslelll cannot be guaranteed to be stable unless some other conditions are satisfied. We will use the follo''''ing piecewise differentiable quadratic (PDQ) Lyapunov function for stability analysis~
=x
f'''(t)
T
r71
p
m
With the following
- _[0.A, ..4,0J
~'1[ - ,
(~."
~
I
-=-_[1
B -
.
[ I (~J
notations~
0 ] C','
-
'~J
l[°1J D-
B"'I = B
-1
W1
JJ .
-1
[Aoe/
- = G -r
T
P(t)x = L,}/(f H'! = Lll,(t)x P'X ':;;::1
Consider the dynamic fuzzy system
1:
(4.13) .. if it is quadraticlly stable with a disturbance attenuation Y1 then~ it is also asylnptoticaHy stable
(4.11)
)lfJ
Len,nlQ ..r
()
0 K el
1~
(4.17a)
lOJ L- "" l0
= D
•
IJ( ( ) ~ 'h (t )~ + '''J.J t JP"! +.... H+l1 m( t )Pn; (4.12) \vhece (PE, I';!. .... Pm) is a set of nxn fixed positivedefinite symtnetric matrices and P(!) is a symmetric piece\\'ise differentiable Inatrix function.
\lie have the follo\\-'ing results.
Because \ve use the piecewise differentiable matrix function. the folJo\",ing notations are used in this paper. For a lnatrix function IVt),
Theorem -1.1: Given a systetn as described in (2.1), and a filter of the fonn (4.7) and (4. 9)~ if the following conditions are satisfied.
p( f
):~» -4 ~H nxn .
the left -liluit of
[J(r)
at
t
91 is defined as
( E ~11
T
<
(4.17b)
E8/:""
EAT'
(4.18)
cc,
(2)
there exist a set of positive B I • I :=: 1.2.··· 111 such that the inequality
P( t- }:;::::; lint 1)( t - E:) £---+0 .
if the lilnit exists, and the right limit of
1
--- lO 0J - lOl EH1l
E A /:= EAr
(I)
E
T
P(t)
at
(AI
+BG,rr)P, +F/( . 4 . r +BCi;0)T +C,EA1E;;,
+~F(J-} +2BIF~HEc,E{.I( f~(r?l-{)T
is defined as J)( I ~ ): -:=. /hll Pr t + & ) H-4()
E,
1 -
---
+2y - (B",/ + BC]! D wl
jfthe lituit exists.
constants
)(
-
-~-
B wl + BCT, D W1
+4y-l Es1EJn )
T
+4y -I f~G,HYi[)lE:;l(FG,H)T + y-l p'rrI~ < 0
We \-vill consider the quadratic stability_ De.{inUiol1 -I. 1: Given a dyn(:unic fuzzy closed loop
systelu x( t ) = ..4", ( !J ) x( t ) +
z( t )
Bc ( !-I }t-e( ( )
== Lx( f )
where (4.13 )
the systetD (4-.13) is said to be quadratically stable with a disturbance attenuation y if there exist a set of fixed positive-defini le SYIUluetric Inatrices ( ~ . p;. .. . ~ P",) and a constant et such that the
following conditions are (I)
T
I == 1. 2•... In
satisfied~
(4.14)
F:=
_ [E
E c :=
i'
o
0]
Ec
( 4~
19)
;
[~~ ~IJ H." = [~ ~J
v.,hich can be re\"-ritten as rc.',A, + (rc;:.A,)T + (...)/ < 0 ~ I:=: 1,2;"'111 with
R ""
[~l (4.20)
5449
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ROBUST H[Infinity] FILTERING OF FUZZY DYNAMIC SYSTEMS...
B
p,(7r
0
n D
0
]'=
.A~ =
()
References
:, 1
0
Cao.
S.G.. N.W. Rees_ and G. Feng (1996). Quadratic stability analysis and design of continuous tilne fuzzy control systems u 1nl. J . .\:y.\~te"g..· ~':tcience ~ vot 27_ pp. 19J -20) . Cao_ S.G.. N.W, Rees~ and G. Feng (1997a). ~-Analysis and design for a class of cOlnplex control systelns~ Part I: fuzzy modeling and identification ~- ~ .04 ut0I11at;ca~ vo!. 3 J ~ no.6.
]{r;, HT
F
il
E~/RT
J1~
:>
(1)1
P,T!
Bwl
P,
0
0
LI',
-yl
[)
()
0
0
~:'I
()
_L]
()
0
0
!}
0
0
-Et!
0
0
()
()
0
0
__1_ 1 28 1
0
0
0
0
_LJ
,
0), ==
14th World Congress of IFAC
()
2
0
pp.iO 17-1028.
4
(4.21) -
-T
= ,Al~ + IJA/
-
-T
-1-
~T
+ 2c/E Af E A { + 4y EA1ErH has a set of positive definite Inatrix solutions 11, I = 1,2.·· ~111 ~ then (i) the error dynarnics of the global filter systeln is aSylnptotically stable \vith a disturbance attenuation ¥, (ii) MoreoveL the filter parameters are given by T (;, == -Prr a)/A~\¥, ~ J;:: 1.2.···n' (4.22) where the scalar Pt is the free parallleter subject to l
l1>/:==(rr T
1. -1 --(""')1)
>0,
4 23) ('.-
PI
and the Inatrix lf1 is defined by q/,:== (A/cJ),A~ )-r.
(4.24)
ReJllork -+.!: It should be noted that the above
stability results of filter design are based on one boundary condition T < 00, which is not easy to test. For sOlne Inore easily checkable conditions~ sce (Cao~ et al. 1997d).
5. An example of simulation We consider the probleln of balancing an inverted pendululn on a cart. The silllulation results are olnitted due to space liInit.
6. Conclusions
In this paper~ a nUluber of robust HOC) filter design lucthods are proposed for fuzzy dynamic systelllS based on quadratic stability theol)' and LMI techniques. It has been shown that the resulting robust H oo filter S}~steln is Quadratically stable ~rith disturbance attenuation jf a silnple linear Inatrix inequality has a positive definite symJuetric Inatrix solution or if a set of simple linear Inatrix inequalities has a set of of positive definite sylUll1etric lnatrix solutions depending on t\\'O kinds of design strategies.
Cao. S.G.. N.W. Recs, and G. Feng (1997b). ~·Analysis and design for a class of complex control systems~ Part 11: fuzzy controller design·· ~ ..4utonlatica_ vo1.33~ pp. 1029-1039. Cao. S.G.~ N.W. Rees, and G. Feng (1997c).•1fL., control of nonlinear continuous-time systelns based on dynamic fuzzy luodels u • Int. J .
pp.821-830. N.W. Rees.. and G. Feng (1997d). ~~Further results about quadratic stability of continuous time fuzzy control systeJnsH~ Int..1. S:vstems l';;cience. 'l0l.28, no.4~ pp.391-404. Feng~ G.~ S.G. Cao~ N.W. Rees and C.K. Chak (1994). 11 An approach to controller design for a class of fuzzy systems u .. Proc. AJVZIfS.. Brisbane~ pp.229-233. Feng~ G.~ S.G. Cao.. N.W. Rees and C.K. Chak (1995). "Design of fuzzy control systems ba'sed on state feedback".. J. Intelligent de FuzZ)-' l~v.\;te111 ...-,·, vo1.3~ pp.295-304. Feng. G.. S.G. C30~ and N.W. Rees (1996). ·'An approach to Hu') control of a class of nonlinear systelns", Auto/natica~ vo1.32. pp. 1469-1474. Feng. G... S.G. Cao~ N.W. Rees and C.K. Chak (1997). "Design of fuzzy control systems with guaranteed stability" ~ Fuzz.v /\eL\" and .'-,)lslen,s~ vo1.85. pp.1-l0. Skelton. R.E. and T. Iwasaki (1995). "'Increased roles of linear algebra in control education·~~ IEEE C"ontrol .S:v.,·tents A/ag... vol. 15. pp.76-90. Takagi T. and M. Sugeno (1985). "Fuzz~/ identification of systems and its application to luodelling and control" _ lEEE Trans. .)~v.,,·tenl.\· A/an C:vbernelics, vol. 15 ~ pp, 1 16-132. Xie~ L. and C.E. de Souza (1995). .tO n robust filtering for linear systems paralueter uncertainty. u Pr()c.3.Jth IEEE (LJ(T, Zadeh. L.A. (1973). "Outline of a ne," approach to the analysis of cOlnplex systelns and decision processes ". 1l:!.~E Tran.\'. ~t;.,~v.\·teI11"'; Alan C:vhernetics. vo1.3. pp.28-44.
Cao.
S.G.~
5450
Copyright 1999 IFAC
ISBN: 0 08 043248 4