Proceedings of the 38*

Conference on Decision & Control

Phoenix, Arizona USA December 1999
FrA12 09: Io
A Robust Stabilizing Controller For a Class
of Fuzzy Systems
Shehu. S. Farinwata,Member,IEEE
Ford Motor Company, Research Laboratory
Dearborn, MI 48121-2053, USA
Email: sfarinwa@ford. com
Abstract
This paper discusses the construction of Q robust sta-
bilizing controller f or Q Takagi-Sugeno class of fuzzy sys-
tems. It should be clear at the outset that The fuzzy sys-
t em is inherently uncertain due t o the vagueness of the
associated linguistic terms, even though crisp and use-
ful results are obtained via inferencing and defuzzifica-
tion. This vagueness uncertainty is the essential fuzzi-
ness of the system ( I S introduced via the membership func-
tions. However, the fuzzy system considered here is un-
certain f or two other reasons. One, the parameterized
membership functions are not completely known but the
bounds of the characterizing parameters are known. Two,
the mathematical model itself is uncertain via parameters
whose bounds are known. The problem is cast within Q ro-
bust, bounded parameter design which then allows analy-
sis within that framework. The parameters of the mem-
bership functions considered are those that determine its
spread, and therefore the fuzziness, i n the universe of dis-
course. It is shown that by knowing the bounds of the
parametric uncertainties in both the system model and
the membership functions, and using Q matching condi-
tion, the local feedback controller gains can be selected so
that the overall control law stabilizes the system. Further-
more, i t is shown that the resulting Lyapunov equation
that needs to be solved f or global stability is indeed depen-
dent on the fuzzy firing order which has not been the case
with previous results.
1 Introduction
The Takagi-Sugeno (TS) fuzzy system continues to re-
ceive attention from researchers in the control community
who are involved with fuzzy control system design and
analysis.The great appeal may be attributed to the stan-
dard state space form for linear systems for which there
is an abundant design and analysis tools. The works of
Tanaka and his colleagues have been formidable in ascer-
taining and exploiting such tools as pole-placement de-
sign, LMI design and observer designs, and has there-
fore set certain trends along these lines. Other efforts
in this area include the works of Langari et al. [8],
[19], J adbabaie et. al., [18], Calcev et. al., [23] and
Filev [20], to mention just a few. The overall TS con-
troller design may be viewed as being similar to design of
gain-scheduling control system where local controllers are
sought for a linear or linearized subsystem, given the de-
sired operating points. However, the efficacy and sophis-
tication of the design may be better appreciated when
viewed within the framework of large-scale system de-
sign, say along the works of Siljak [13], J amshidi [12],
Lunze [14], and others where a complex system is de-
composed into finite number of smaller sub-systems for
control. There also, sub-systems play a major role, via
strengths of interconnections, in the overall system be-
havior. The same may be observed in the case of the TS
scheme,via the local contributions (degree of fulfilment)
of the fuzzy rules as these perpetuate all through the ex-
pression of the overall control law. This is perhaps an im-
portant facet, which differentiates these designs from the
ordinary gain-scheduling.From theoretical point of view,
the TS approximation of systems suffers similar limita-
tions as linear systems, in terms of design and analysis
tools, when a highly complex, non-linear system is en-
countered. However, the practical relevance seems to be
very broad as may be ascertained from [21, 221.
In this paper, an uncertain TS model is considered whose
uncertainty stems from two sources: the parameters of the
membership functions that can vary in a known ranges
and parameters in the model of the plant which can also
vary in some known ranges. On one hand,the parame-
ters in the plants model basically give rise to a family
of plants with a fixed, known structure. On the other
hand, a given membership function is parametrized via
the spread and location parameters.These give the mem-
bership function its shape [l]. When these parameters
vary in specified ranges, a parameter-family of member-
ship functions results. The task here is to construct a
controller that will stabilize such families of plants and
membership functions, given the bounded sets of the pa-
rameters.
0-7803-5250-5/99/$10.00 0 1999 IEEE 4355
2 The Nominal System
Consider the continuous-time T-S fuzzy system below:
Plant Rule:R.:
i sL;, THEN
If zl (t) is Li l and z z ( t ) is Lia and ,and z, (t)
k ( t ) = A. z( t ) +B, u( t ) , Z(0) =20 ( 1 )
Y( t ) = Cz ( t )
for 2 =I,. . . , r, where z E lR", U E EI, and the matrices are of
appropriate dimensions; r is the number of rules, L,, are the lingulstic
labels which are fuzzy sets, p , >, in the universe of discourse, X , for z ( t ) .
The fuzzy system is inferred as:
with
u ( t ) = - 2 h . ( z ( t ) ) K . z ( t )
,=1
and with the normalized degree of fulfillment, h , , given as:
with
We will denote the overall system's nominal matrices by:
3 Quasi Local Dynamical Systems
(3)
One of the ways to view the T-S fuzzy system, which
is shared here, is as local linearizations of a general
nonlinear dynamical system about several equilibrium
points. The origin of such a nice, closed-form nonlinear
map involving the system states and inputs including
(4) parameters is an important concern addressable in
nonlinear identification and is not the subject of this
paper.
(5) Suppose we start out with the general nonlinear system
expressed as:
w, ( z) is the degree of fulfillment of the a-th rule. C , w , #0 is
necessary for the existence of solution of the defuzzification scheme. I t
is clear from (4) that:
i =f(+), u ( 4, a) (10)
where z ( t ) E W is the vector of the states, u( t ) E ",the
input vector and a E nt' is vector of real, bounded para-
meters,with in this space as the nominal value of the
g h . ( z ) = 1 , h , ( z ) > O, i =l , . . . , r
( 6 )
.=I
I t should also be clear that the uncertainty or essential
fuzziness is embedded in hi(.) via pij. Let uj E R
be the parameter vector that specifies ,u; j in X , for
j =1, . . . , p , the cardinality of the fuzzy set. ujo will
be used to denote the component-wise nominal value
of the parameter vector. Similarly, A; and Bi will be
considered parametrized as Ai((. ) and Bi (a), with a
E a and a nominal value, ajo. We will consider p; j and
a to be set at their nominal values, so that wehave the
representation below of the nominal system under fuzzy
control.
at which the nonlinear system will be linearized where
R; =( x ; *, ui *, a~) for i =1, ..., s. These linear systems
for each rule i will constitute the subsystem models for
the T-S fuzzy system and are given below:
The feasible or admissible universes of discourse which
are all subsets of their corresponding spaces as already
specified, will be denoted by X and U for the states and
inputs respectively.
Conversely, in the neighborhood of Ri, we expect, and it
follows, that f is well-behaved. Again, observe that the
parameter a results from the model itself, whereby we
: (6 R.; ( A. ( ao ) , B. ( ao ) , C, p , , ( o , o ) , u ( t ) )
,=I
That is:
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actually introduce the parameter, Uj , by the use of fuzzy
membership functions for the rules. Thus, we may con-
sider a independent of uj. Furthermore, wemay associate
AAT(Y) = BT k M, ( y ) , ABT(Y) =BT 9 E.(r)
(17)
with each subsystem ( Ai , B;, C) at Ri , a local dynamical
=I =I
system of the form: where Mi and E, are continuous in y. This consideration will be
shown t o render (15) stabilizable for all bounded variations in ai and a.
Now, weassume that f and h; depend continuously on a
and uj respectively. Thus, the solution of (11) is assumed
to depend continuously on these parameters. It follows
that Fi, being a product of continuous functions, is con-
tinuous on (x, (YO, U ~ O ) so that it has a unique solution.
Now, as long as we stay in some neighborhoods of these
parameters, ie., IIa - a011 <car and I[u - ujo[l <cor we
may rewrite (11) simply as:
i j ( t ) =l q x i , U i , U, a), $ ( t o ) =20, (12)
The choice of rn and e should carefully be made over
the cardinality of the rules linguistic term sets. This is
why wehave opted to generate the matrices from rule to
rule since the term-set for each rule may have different
cardinality. In fact, this also allows for each term in a
term-set to have different shape.
The following theorem addresses the issue of robust
stabilization under constant feedback gain of the overall
fuzzy system.
Equivalently, wewrite above as:
Theorem 1 (Fuzzy Robust Stabi l i zati on) Suppose (Ai, B, ) IS
completely controllable, i =1 , . . . , r .
Let the matching conditions (16) and the bounds in (18) hold.
Xi(t) = hi(0) { Aj ( a) $i ( t ) +Bi (a)Ui } and if e <1,
(a) The uncertain system (is) can be robustly stabilized at a stable
= ~ %( Y ) x i +E) i ( Y) %, =[(. a] ( l 3) A , by:
with
4 The Perturbed System
As a result of the parameter variations, let us consider
a A( . ) variation in the nominal matrices of (9). Further-
more, let us restrict the point of entry of the variation via
the input distribution matrix, and propagate through the
system via only the state equation. The local dynamics,
neglecting the subscript on the states for convenience, is
given below:
i ( t ) = (Ai +AAi(Y))S(t) +(hi +Ahi ( y) ) . i ( t [ l 5)
y ( t ) = Cc( t ) x( 0) =20.
(b) where the overall feedback gain i s given by:
where P >0 and Q >0 solve a lyapunou equation, i =1,. . . r
Proof
Lemma 1 :Any completely controllable linear time-invariant (LTI)
system is stabilizable 1.241.
. Thus A, is stabilizable by feedback. Furthermore, it is straight
forward to show that the controllability of ( A, , B; ) implies that of
( i , , E i ) , h, >0.
This consideration allows us to impose the following local
matchzng conditzons:
Controller Rule c, If zl (t) IS L, , and z z ( t ) IS L , ~ , THEN
u . ( t ) = - K, x( t ) , t =1, , r (21)
A&(y) = Bi Mi ( y) , Agj(y) =BiEi(y) (16) Above is used t o place the poles at out stable, chosen eigenvalues, for
al l i =l , . , . , r,
we intend to generate the matching matrices locally, that
is, from rule to rule and extrapolate the result to the
overall system as will be further explained shortly. So
the overall system the perturbations are matched as:
Now, from our choice of stable x for the overall system, the overall
system IS asymptotlcally stable The next lemma follows immedlately
Lemma z . Any asymptotically stable system zs stabilizable [24].
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I t follows that the overall system ( x i hi ( z) A, , Ci hi ( z) Bi ) can be
stabilized by feedback. The proposed form is given in (3) above.
Since AT isasymptotically stable 3 a scalar function, V( z ) >0, given
by V( z) =zTPz, P >0, as a Lyapunov function candidate.
Let us denote by V,(z), the open-loop Lyapunov function so that:
For the closed-loop system, we have:
where
21 =(- h, ( z) K. z( t ) )
,=1
Upon using the matching conditions from
one obtains:
(17) and greatly reducing,
Rewriting above using the right-hand side of (19), and using V( z) =
zTPz and expanding, we have:
and
With these bounds in place, (26) becomes:
By completing the square and using e <1, above is rewritten as:
V 5 v,+- ma 11. 11
4 ~ ( 1 - e)
nom e =- zTQz, Q > 0 and using the fact from the so-called
Rayleighs theorem [25], i.e.,
Am,,,(Q) 5 ITQZ +~2 5 Ama=( Q) , T82 =Rayleigh quotient, (30)
X T 2
werewrite (28) as:
Above will be negative for all z #0 if:
ma
-Am*n(Q) +- < 0, or
4 ( 1 - e)
ma
4 (1 - e)Am*n(Q)
P >
Consider (19) from the theorem reproduced below, that is,
Upon squaring both sides of above and applying Cauchys inequality
to the left-hand side, wehave:
A second application of a version of Cauchys inequality on the left-
hand side namely,
,=I 2 1 / r ( k K , > 1=1 (35)
We caution that the left-hand sides of the last three equatlons above
are only, mathematically convenient intermediate steps and are not put
into any computational use. From (34) and (35), we have:
This and (32) establish part (b) of the theorem I
To establish part (a) of the theorem, one only needs to show that
the proposed feedback control law (3) does preserve the uncertainty
bound e and the matchings in the perturbed system. This is achieved
without much difficulty and will not be carried out here for lack of space.
Remark 1:
The significance of above derivation is such that the number of rules
and the activation order are taken into account in the selection of the
local feedback gains. This gives an added degree of freedom in tweaking
the controller gain for the i - t h rule.
5 The matching conditions and
matrices
In the following, the perturbations AAi and AB, will be determined
by construction. From these, the matching matrices, E; and Mi are
determined. Observe that the uncertainties in AT and BT are due to
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ii t
Figure 1: Parameterized Membership Function with p
labels
Figure 2: Variation of pjj(x) with parameter, uj
hi in (4) through the membership function parameter, uj as shown in
figure 1, and the systema parameter, a.
It is assumed that the membership functions for the linguistic labels
have the same shape. Furthermore, they are sufficiently convex and
normal. Sufficient convexity, excluding the trivially convex case, sin-
gleton membership functions, ensures that the if-then rule set, U, Ri
IS essentially fuzzy. Figure 2 is used to illustrate the situation under
consideration.
A membership function pi ; for the i-th linguistic label of the i-th
rule is parametrized by U and has a nominal parameter U =(10. The
parameter U in this case determines the spread and therefore the
fuzziness of the label. Three cases are illustrated in Figure 2 where
o equivalently affects the concentration of the label. In terms of
fuzziness, the nominal label with uo is the least fuzzy , while the
label with uz is the most fuzzy of the three. This may further be
illustrated by considering some so E X, the memberships are such
that in general, p z( s 0) 2 pl (r o) 2 po( s 0) in the same linguistic label
having the membership function, pi;. This is seen to be the effect of
U changing from U =oo to U =U Z . Thus, a fuzzy controller that has
performed optimally when designed with the membership functions
parametrized at their nominal values may actually exhibit a less than
optimal performance, if the parameters were to change inadvertently
to some other values. This may happen due to measurement or sensor
error in gathering the data for the membership function or due to
other non-stationarity effects. If the aggregate effect, which manifests
in the defuzzification scheme via the overall degree of fulfilment, is
such that the closed-loop fuzzy control system is destabilized, then the
fuzzy controller lacks robustness. We caution that mere change in the
aggregate firing order due to parametric variation by itself does not
constitute a robustness problem. It may only point at a sensitivity issue
which may not cripple the closed-loop system per sea. Sensitivity is-
sues for a class of T-S fuzzy control system have been addressed in [lo].
In the construction of the matching matrices, Ap will be used
directly rather than A6 as the former is explicit in the defuzzification
scheme. The effect of Ao is obviously captured implicitly. A practical
conventional wisdom, so to say, in robust parameter design is followed
here, where it might be more practicable and fruitful to deal with the
effect of the changing parameter on observable quantities rather the
changing parameter itself. Another way put, we will attempt to deal
with the effect of the variability rather than eliminate its source which
may not even be known.
Definition: maximum membership change of the j - t h linguistic term in
a linguistic term set of cardinality p , due to a change in a characteristic
parameter, U as:
Apju =maz l puo( z ) - p0(z)I
such that the Support, Supp(puo) #0, Supp(pui) #0 , Vs E X , where
uo is the nominal value of the parameter and uJ, uo E [uJm.,,,uJmaz]
So for a term-set of cardinality p for i-th rule, we denote the set of
maximum deviations as:
APi ={ A c I , A P ~ , . . . , A P ~ }
where the additional subscript U is dropped in Apjs for convenience.
Consider the local perturbation, Ad;. The expression for this is con-
structed approximately below based on the form of the inferred overall
system matrices in (9) and the local ones in (13). Thus, weapproximate
the local perturbation for rule i in ti; as:
For the overall system we have:
for Bi, wehave:
similarly:
Equations (39) and (42) are the matching conditions that have been
proposed in (16). Next, M, ( a) and Ei ( a) are determined.
nom (37), wehave:
Observe that E, above is a real quantity for our case, and is a square
matrix for the multiple input case. Also, realize that the quantities,
M; and E, must be determined for each rule. Once this is done, we
determine the respective norm bounds, m and e on the sum of these
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matrices for all the rules as proposed in (18)
Remark 2:
This remark addresses the determination of the overall P. Since wehave
chosen the eigenvalues of the overall system, E, h,(z)A, at A, and that
the Subsystems, A,, are stabilizable, wecan either find the common P
by inspection - a formidable task, or solve the linear matrix inequality
( LMI ) system below as a generalized eigenvalue problem (GEVP):
where y >0, is the decay rate, and select Q =yP, for i =1,. . . , r .
Above equation does not seemto be computationally appealing with the
firing order included in the expression. Moreover, one does not have to
solve a Lyapunov equation with cross subsystems (211 if the the a good
mapping principle is observed in the membership functions. However,
the Lyapunov equation with cross-subsystems, though stringent, can be
solved to determine P and Q. This is given below:
( A, - B, K, ) TP - P( A, - B, K, ) - yP < 0 (48)
for w, w, #0, and i , j =l ;..,r.
More on this equatlon and alternate forms may be found in (211.
6 Conclusion
The theorem basically allows one to pick the closed form
constant feedback of the individual sub-system of the i-th
rule, from a closed-form expression of the overall control
law. The formulation deliberately makes use of the rules
degree of fulfillment and the number of rules, so as to
provide more adjusting freedom for the local controller
gains. It also provides an upper bound for these gains. It
still remains to determine P and Q. As already stated,
the intent of this paper is not to dwell on the determi-
nation of P and Q which has been handled very well in
[21, 221. However, if weemploy the converse view that the
subsystems have been stabilized by fuzzy control, then P
and Q are easily obtained in a slightly different manner
by solving a generalized eigenvalue LMI problem. We
have proceeded further to show that under some struc-
tural constraints for the uncertainty, and certain bounds,
the overall control law stabilizes the fuzzy system for the
entire set of parameter variation, which is a new approach
in this area. We then give the closed-form expression of
the robustly stabilization controller.
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