Observers for takagi-sugeno fuzzy systems - Systems, Man and ...
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Observers for Takagi–Sugeno Fuzzy Systems P. Bergsten, R. Palm, and D. Driankov
Abstract—We focus on the analysis and design of two different sliding mode observers for dynamic Takagi–Sugeno (TS) fuzzy systems. A nonlinear system of this class is composed of multiple affine local linear models that are smoothly interpolated by weighting functions resulting from a fuzzy partitioning of the state space of a given nonlinear system subject to observation. The Takagi–Sugeno fuzzy system is then an accurate approximation of the original nonlinear system. Our approach to the analysis and design of observers for Takagi–Sugeno fuzzy systems is based on extending sliding mode observer schemes to the case of interpolated multiple local affine linear models. Thus, our main contribution is nonlinear observers analysis and design methods that can effectively deal with model/plant mismatches. Furthermore, we consider the difficult case when the weighting functions in the Takagi–Sugeno fuzzy system depend on the estimated state. Index Terms—Dominant linear models, fuzzy sliding mode observers, interpolation of multiple linear local models, Luenberger fuzzy observers, Takagi–Sugeno (TS) fuzzy systems.
I. INTRODUCTION This paper describes the analysis and design of two sliding mode observers for a class of nonlinear systems modeled as Takagi–Sugeno (TS) fuzzy systems. A dynamic TS [1] fuzzy system is composed of multiple local affine dynamic linear models. These local models, for the purpose of this paper are related to local linearizations, via Taylor series expansion, of the original nonlinear system at off-equilibrium points [2]. Thus, the local models have no equilibrium point within their regions of validity, i.e., they are off-equilibrium local models (for detailed exposition see [3]). As shown in [4], incorporating off-equilibrium linearization in the model can significantly improve the transient response in closed-loop control. In [3] it is proved that a TS fuzzy system where the local affine dynamic models are off-equilibrium local linearizations leads to an arbitrary close approximation of the linear time varying (LTV) dynamic system resulting from dynamic linearization of the original nonlinear system about an arbitrary trajectory (see also [5] for related results). Thus, the results concerning observers for TS fuzzy systems are also relevant to systems such as linear parameter varying (LPV) systems, piecewise linear systems, and conventional gain-scheduled systems. This paper focuses on analysis and design aspects of observers for TS fuzzy systems of the type introduced above, i.e, we allow possible affine TS systems, which is not normally considered in the literature, e.g., [6] and [7]. Our main contribution is analysis and design methods in the case of model/plant mismatches represented as matched and unmatched uncertainties. By matched uncertainties for observers, we mean that the uncertainties appear in the output channels only. Unmatched uncertainties do not appear in the output channels. We also explicitly consider the case when the weighting functions in the TS fuzzy system depend on the estimated state. This case is considered to be difficult [7] and the usual assumption is that the weights depend only Manuscript received July 24, 2001. This paper was recommended by Associate Editor C. T. Lin. P. Bergsten and D. Driankov are with the Center for Applied Autonomous Sensor Systems, Department of Technology, Örebro University, Örebro, Sweden (e-mail: [email protected]; [email protected]). R. Palm is with Siemens AG Corporate Technology, Otto-Hahn-Ring, Munich, Germany (e-mail: [email protected]). Publisher Item Identifier S 1083-4419(02)00027-4.
on measurable parameters (see [6] and [8]). Two observer types are proposed: 1) the extension of a so called Fuzzy–Thau Luenberger observer [9] with a switching term to account for matched uncertainties and 2) the affine TS fuzzy system is rewritten as a dominant linear system in order to treat it as a linear system subjected to known and unknown uncertainty. The latter approach is inspired from a similar idea used for controller design [10]. Using this approach, design methods for sliding mode observers for linear systems can be applied. Lastly, the analysis and design methods proposed utilize available Matlab software which facilitates reproducing the results reported. Section II introduces the TS fuzzy system subject to observation and defines a Luenberger type of observer for it. The latter is realized in terms of a parallel distributed compensation scheme incorporating an interpolation between local Luenberger observers. This is the type of nonlinear observer that has received most attention in the fuzzy control literature, but all results reported assume no matched/unmatched uncertainties. In Section III-A, we take into account matched uncertainties only and present a sliding mode fuzzy observer that is related to the so called min–max observer described by Zak and Walkot [11]. This observer utilizes interpolation between local observer gains which has one major negative effect: large number of local models will give a large number of linear matrix inequalities (LMI) in the stability analysis and design, which may prohibit the use of existing LMI tools. For this reason, in Section III-B, we reduce the TS fuzzy system to a dominant linear one, that is one local model is chosen and the effect of the rest are incorporated in it in terms of known deviations interpreted as uncertainties. This avoids the use of LMIs for analysis and design and instead a direct sliding mode observer design is possible. Furthermore, model mismatches are taken into account as known upper bounds of matched and unmatched uncertainties. In Section IV the performance of the two different types of sliding mode observers is illustrated on a model of a cart with a pendulum. These results are also compared to the ones produced by a fuzzy Thau–Luenberger observer for the same system. Details about the latter one can be found in [9]. II. TS FUZZY SYSTEM Consider a nonlinear system of the form
x_ =f (x; u; ) y =g(x; )
(1)
where
f : R n 2 Rm 2 Rs ! Rn
nonlinear function that satisfies the Lipschitz condition; g : R n 2 R s ! Rp measurement function; x state vector; u control input vector; vector of possibly time varying parameters; y 2 Rp output vector. In the following we assume that (1) can be represented or approximated sufficiently well by a TS fuzzy system (this assumption need not to be true for an arbitrary system of the form (1), however). The TS system consists of a fuzzy rule base, where each rule i is of the form Rule i:
1083-4419/02$17.00 © 2002 IEEE
IF 1 is i1 ( 1 ) and . . . and r is ir ( r ) THEN
x_ = Ai x + Bi u + ai y = Ci x + ci
(2)
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where the vector of premise variables 2 Rr is a subset of x, u, and and ij : R ! [0; 1]. The function ij ( j ) denotes the j th membership function in the ith rule which applies to the j th premise variable. The Cartesian product i1 ( 1 ) 21 1 12 ir ( r ) defines a fuzzy region in Rr . The system matrices Ai and Bi may be obtained by a linearizing transformation [12] or through Taylor expansion in some point (xi ; ui ; i ) corresponding to i in the fuzzy region described by each rule i
Bi = @f and Ai = @f @x (x ;u ; ) @u (x ;u ; ) Ci = @g : (3) @x (x ; ) Note that (xi ; ui ; i ) does not have to be an equilibrium point which means that constant or affine terms in each subsystem i may be obtained as
ai =f (xi ; ui ; i ) 0 Ai xi 0 Bi ui ci =g(xi ; i ) 0 Ci xi : The global TS fuzzy system described by (2) is then written as
x_ = y=
l i=1 l i=1
wi ( )(Ai x + Bi u + ai ) + fum;x wi ( )(Ci x + ci ) + fum;y
(4)
where l is the number of rules and
wi ( ) =
r hi ( ) h ( ) = ij ( j ) i l h ( ) k k=1 j =1
with l i=1
wi ( ) = 1:
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orem for a fuzzy controller and observer. In this paper, we consider the latter case which is more difficult than the former one—see [9] where a fuzzy Thau–Luenberger observer is introduced based on ideas from [13]. We will also constrain ourselves to the continuous time case although there exist stability results for the discrete time case (see [6], [7], and [14]). Furthermore, the above type of observer does not take into account approximation errors or uncertainties in the fuzzy model; that is, it assumes that the TS system (4) is the nominal one. In Section III-A, we will consider the case y = Cx which covers a large class of real technical systems. In Section III-B, we extend the class of systems to the case y = li=1 wi ( )Ci x. III. SLIDING MODE OBSERVERS FOR TS FUZZY SYSTEMS As already mentioned, the observer from (5) does not consider modeling errors, i.e., it is assumed that a perfect nonlinear model of the plant is available. However, in the presence of modeling errors or unmodeled system uncertainties the previous approach may lead to degraded performance and, in some cases, even to instability. Modeling errors or unmodeled system uncertainties can be represented as matched and unmatched uncertainties. In this section we show how sliding mode techniques can be used in observer design to cope both with matched and unmatched uncertainties. An extensive representation of these techniques can be found in [15] and [16]. In [17], fuzzy sliding mode observers were first introduced and conditions were given under which matched uncertainties are eliminated. These types of observers are related to the so-called min–max observer described by Zak and Walcott [11]. In Section III-A, we show how a fuzzy Luenberger observer can be extended to a sliding mode observer whose analysis and design is based on LMIs. Then in Section III-B, we present a technique for observer design applicable to a particular form of a TS fuzzy system called dominant linear TS fuzzy system. The latter enables us to design a sliding mode observer in a direct way [15] without using LMIs.
fum;x and fum;y are modeling/approximation errors which have to be
A. Fuzzy Luenberger Sliding Mode Observer
taken into consideration in the observer design. Note that other matched and unmatched uncertainties are lumped together with these errors. Note also that (4) is suited to be obtained by nonlinear system identification [1], due to its particular form. The most well known parallel distributed (PD) type of observer proposed in [7] is an extension of the classical Luenberger observer to the above TS fuzzy system and is obtained as follows. Consider the following observer structure:
The observer developed in this section is based on the ideas found in [17] and [18]. Assume that the fuzzy approximation of a nonlinear system reads
x^_ = y^ =
l i=1 l i=1
wi ( ) (Ai x^ + Bi u + Li (y 0 y^) + ai ) wi ( ) (Cix^ + ci ) :
(5)
Note that the affine terms are not considered in [6] and [7]. Assume that only contains measurable parameters, i.e., does not depend on the estimated states and define the error between the estimated states x^ and the real states x in (4) as e = x 0 x^, then we get the following error differential equation
e_ =
l
l
i=1 j =1
wi ( )wj ( ) [(Ai 0 Li Cj )e 0 Li fum;y ] + fum;x :
(6)
Here the effect of the modeling/approximation errors is clearly seen, and the main issue is to ensure asymptotic stability of (6). As pointed out in [7], one can distinguish between two cases: 1) does not depend on some of the unmeasurable states and 2) does indeed depend on the unmeasurable states, i.e., := ^ . The authors in [6] considered the former case and extended the results in [7] with a separation the-
x_ =
l i=1
wi ( )(Ai x + Bi u + ai ) + D
(7) y =Cx where 2 Rq , D 2 Rn2q , and C 2 Rp2n with p q . The term D
represents the matched modeling/approximation errors. That is, we assume that the present uncertainties can be transformed to the output channels (see Assumption 1). Furthermore, as can be seen in Theorem 1, the observer scheme also gives some robustness to unmatched growth bounded uncertainties. This fact is an inherent feature stemming from the attempt to deal with the case := ^ discussed above. The following assumptions are made. Assumption 1: There exists a linear change of coordinates z = T x such that 0 (8) D = T D = and C = CT 01 = [ 0 Ip ] D2
2 with D
2 Rp2q , T 2 Rn2n , and z 2 Rn . Ai = T Ai T 01 ; Bi = T Bi ;
and a i = T ai :
(9)
Using Assumption 1 we may simply define matched uncertainty as uncertainty that is structured so that the linear coordinate transformation T —with the desired properties—exists. Hence, Assumption 1 effectively constrains the class of uncertainty. Here it can be seen that the
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case with a varying C matrix will give rise to a nonlinear coordinate transformation which may be difficult to obtain. In this section we will therefore only consider the case with constant C . Assumption 2: There exists a scalar > such that k k . Assumption 2 effectively bounds the norm of the uncertainty by a positive constant. If one knows that k k depends on some measurable parameters # it may be possible to state a varying bound like k # k # . In any case, the a-priori knowledge that is needed is an estimate of the norm of the uncertainty . One case when there may be a problem to obtain such an estimate is when (7) is unstable and k k is an increasing function of kxk. Using the transformed state vector z T x we have that
0
()
()
=
l
z_ =T x_ = T
=
l
i
i=1
i
i
i
i
+ T D (10)
and
y = CT 01 z:
(11)
Hence, we get the transformed system
z_ =
l
w ( ) A z + B u + a i
i=1
y =Cz:
i
i
+ D
i
(12)
]
z_1 y_
=
l
A11 A21
w ( ) i
i=1
+ y =[ 0 I
p
]
;i
;i
z1 : y
a1 a2
A12 A22
z1 y
;i
;i
+ D02
;i ;i
+
z^_ =
B1 B2
;i ;i
i
i=1
y^ =C z^
i
i
i
+ L (y 0 y^) + D i
P e
y
y
then becomes
e_ =
l
+ D + 1( ; ^; z; u) w (^ ) A 0 L C e 0 D i
i=1
i
i
p
i
p
n
i
i
p
p
i
V_ (e) = eT
l
w (^ ) (A i
i=1
+P (A 0 L C ) i
i
0 L C )TP i
+ eT 2PD + 21T Pe : e 0 eT 2P D
(22)
P
=
we have that eT P D
V_ (e) 0 eT Qe + 21T P e T T 0 2 e P2DDT2PD2e P2 e + 2eTP2 D 2 2 2 0 eT Qe + 21T P e 0 2 D 2T P2 e + 2k k D 2T P2 e : y
y
y
y
y
(23)
y
Obviously, the maximum positive part of the uncertainty D is canceled in the Lyapunov function derivative by the switching term D . Now, there always exist a > such that
0 V_ (e) 0eT Qe + eT P P e + 01 k1k2 02 D 2T P2 e + 2k k D 2T P2 e y
T = k (15) k ; if D2 P2 e 6= 0 0; else where e = y 0 y^ and P2 is a symmetric positive definite matrix (see Theorem 1). Note also that the general case with := ^ is considered. The fact that may contain the nontransformed state does not alter the presentation. The error differential equation, for the error e = z 0 z^, P e
p
with the matrices Li being local observer gain matrices and D
p
D
p n
Due to the structural constraint (18) on eyT P2 D2 which gives
(14)
D
D
u
(13)
w (^ ) A z^ + B u + a
2T (if N = ; then where N is a basis of the null space of D 4 (=00)22( ).0 If) there exists a scalar > 0 and matrices P1 = P1T 2 , P2 = P2T 2 R 2 , P3 2 R( 0 )2 , and Q = QT > R 0 such that P P4 P = T 1 T 3 >0 (18) 4 P3 P2 T (19) A 0 L C P + P A 0 L C 0 Q Q 0 2 P >0 (20) P I k1( ; ^; z; u)k kek (21) for i = 1; 2; . . . ; l, then (16) is asymptotically stable. Proof: Consider the Lyapunov function candidate V (e) = eT P e = eT P e with > 0. Taking the derivative of V (e) along the
i
Also note that both the transformed state vector z and the nontransformed state vector x (incorporated in ) may be present in (12). This does not, however, pose any problem in the observer design. Consider the following observer for (12) l
(17)
trajectories gives
From the properties of T in Assumption 1, the state vector z is partitioned as z z1T yT T and (12) now has the uncertainty in the output channels. For clarity, (12) can be written in partitioned form as
=[
T 4 = N0 D
n
i
w ( ) T A T 01 z + T B u + T a i
i=1
i
4
p
w ( ) (A x + B u + a ) + D
(^)) + + 0 1( ^ ) 0
1( ^ ) = ( () 1( ^ ) 0 1( ^ )
l ; ; z; u wi 0 wi Ai z Bi u ai . where i=1 Note that ; ; z; u ! when e ! . That is, ; ; z; u acts like an unstructured vanishing perturbation that is supposed to be growth bounded as k ; ; z; u k kek for some > . However, assuming no particular structure of such perturbations may be quite conservative, as pointed out in [19]. Sufficient stability conditions for (16) are given by the following theorem. Theorem 1: Let 2 Rp2p be structured as
y
= we obtain V_ (e) 0 eT Qe + eT P P e + 2 kek2 0 2 D 2T P2 e + 2k k D 2T P2 e 0 min (Q 0 P P )kek2 + 2 kek2 0 2 D 2T P2 e + 2k k D 2T P2 e
:
(24)
If we choose
y
y
y
y
(25)
which is negative definite if (16)
min (Q 0 P P ) > 2 :
(26)
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By the Schur complement [20], (26) is equivalent to
Q0 P
P >0 I
2
(27)
and the theorem follows. Remark 1: If k ; ; z; u k and using the arguments in [11], one can conclude that sliding motion takes place on the surface S e j D2T P2 Ce . Remark 2: Note that if D2 2 Rp2p with full rank the sliding surface is S fe j Ce g, that is, similar to the observer proposed in [18]. Now, we can state the design of this observer as an eigenvalue ; ; ; l so problem (EVP) in , Q, P1 , P2 , P3 , and Mi for i that we maximize the robustness measure and solving for the gain matrices at the same time
1( ^ ) = 0 =0 = =0
=
= 1 2 ...
Q = QT > 0 P~1 + P~2 > 0 AT P~1 + P~1 A + AT P~2 + P~2 A 0 C TM T 0 M C 0Q Q 0 P~1 + P~2 ~P1 + P~2T I > 0
maximize subject to
i
i
where
P~1 = and
P~2 =
0 0 4T
I
P1
0
0
P2
0
P3
(28)
i
w (^ )(A x^ + B u + a i
i=1
i
i
i
i
j
0 0 4
:
i
(31)
1 =
l
j
j
j
1C =
w ( )((A i
i=1 l
j
i
w ( )(C i
i=1
+1 +f j
um;x
(34)
um;y
0 A )x + (B 0 B )u + (a 0 a )) j
i
and defined as in (15). The concept of LMI regions for placing the eigenvalues of the local systems may be used to get a good dynamic behavior of the convergence of the observer. Such region constraints on the eigenvalues are easily incorporated in the design and will show up as additional linear matrix inequalities (LMI) constraints in the EVP above. See, for example, [12] and [21] for more details on using LMI regions in the context of fuzzy observers and controllers. B. Sliding Mode Observer for a Dominant Linear Fuzzy System In the design and analysis method described in Section II we used an LMI-based approach. However, when the number of LMIs is too large this approach may have computational difficulties. In the following the nonlinear plant is approximated with an affine TS fuzzy system. Further, one of the resulting subsystems is selected as to be dominant linear for the whole plant which can be decided in a practical application by the designer. An analytical way of how to obtain an appropriate dominant linear system for the controller design is reported in [10]. For the
i
j
0 C )x:
i
(35)
j
1
1
z=T x=
z1 y~
j
(32)
j
Notice that the “uncertainties” j and Cj are not real ones because they are completely known. As in Section II we assume that there exists a nonsingular transformation Tj such that
with
Li = T 01 L i for i = 1; 2; . . . ; l
(33)
where (30)
+ L (y 0 y^)) + D
y =C x^
um;x
um;y
j
j
I
i
w ( )C x + f i
i=1
i
x_ =A x + B u + a y =C x + 1C + f
(29)
coordinate system as l
l
i
where fum;x and fum;y are the vectors of matched and unmatched uncertainties. Furthermore, let the j th subsystem in (33) dominate the whole system [10]. In this case we call the system “dominant linear.” Then, (33) can be rewritten as
i
x^_ =
w ( )(A x + B u + a ) + f i
i=1
y=
0 p The robustness measure is obtained as = and the gain matrices as L = (P~1 + P~2 )01 M . The observer is then stated in the original P3T
l
x_ =
i
i
observer design there are many options to choose a dominant linear subsystem. One particular option is to choose a subsystem such that the remaining errors are minimal. Another option is to choose that subsystem which is believed to be occupied most of the operating time. A further option is to choose that subsystem in which the unforced system comes to rest (the equilibrium point). The final result is a linear system with known modeling errors and further uncertainties whose upper bounds are supposed to be known. The next step is to transform the so obtained linear system into a canonical form which separates the nonmeasured states from the measured outputs. This transformation is always possible if the system meets certain rank conditions [15]. In the next design step, a linear and a nonlinear observer in canonical form are computed. Finally the inverse transformation leads to both the linear and nonlinear observer gain for the original observer equation. We are concerned with the following fuzzy system:
i
i
117
where z1 2 Rn0p , Tj 2 Rn2n , and y (34) is then transformed into
(36)
~ = y 0 1C 0 f
z_ = T A T 01 z + T B u + T j
j
j
j
j
j
j
1 +T f j
j
. Equation
um;y
um;x
+T a j
(37)
j
that can be written as
z_1 =A11 z1 + A12 y~ + B1 u + f y~_ =A21 z1 + A22 y~ + B2 u + f ;j
;j
;j
z ;j
;j
;j
;j
y ;j
+f +f
+ a1 + a2
(38)
+f +f
+ a1 + a2
(39)
z ;j y ;j
;j ;j
or
z_1 =A11 z1 + A12 y + B1 u + f~ y_ =A21 z1 + A22 y + B2 u + f~ ;j
;j
;j
z ;j
;j
;j
;j
y ;j
z ;j y ;j
where
f~ f~
z ;j
y ;j
=f =f
z ;j y ;j
0 A12 1C + 1_ C 0 A22 1C ;j j
j
;j
j
;j ;j
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= =
=
f
z ;j
l
w ( ) (A11
0 A11 )z1 + (A12 0 A12 ) 1 (y 0 1C ) + (B1 0 B1 )u + (a1 0 a1 )
i
i=1
;i
;j
;i
j
;i
;i
f
=
y ;j
l
;j
;j
;i
j
;i
;i
= 0 Q2 Q^ =Q1 + AT21 P2 Q201 P2 A21 ^ P1 + P1 A11 = 0 Q:
AT11;j
~
Let fz
;j
;j
~
, fy
;j
, fz
=T a j
;j
z ;j
y ;j
and
j
:= h(f
z ;j
um;x
y ;j
;f
; f_
um;y
um;y
z ;j
)
and the matrix A11;j is assumed to be stable. Consider a corresponding observer of the following form
z^_ 1 =A11 z^1 + A2 y^ + B1 u + f^~ y^_ =A21 z^1 + A22 y^ + B2 u + f^~ + (A22 0 A )e 0 : 2 R 2 is a stable design matrix and ;j
;j
;j
s;j
p
s;j
p
z ;j
;j
+ A12 + a2
;j
e
y
p
0;
e
s;j
s;j
z ;j
y ;j
;j
y ;j
j
z ;j
=
w (^ ) (A11 i
i=1
;i
= 0Q2
(42)
V_
z
f^
y ;j
=
l
w (^ ) (A21 i
i=1
;i
=
^
z
y
;j
z
;j
z
z ;j
s;j
z
z
z ;j
z ;j
y ;j
y ;j
E
;j
(49)
z1
y
(50)
y
;j
s;j
s;j
z
e
y
y
y ;j
y ;j
z ;j
z
:
y ;j
y ;j
y
y
;j
;j
y
z ;j
z
y
y ;j
y
y ;j
z ;j
z
0 Q201P2 A21 e 1 T Q2 e 0 Q201P2 A21 e 1 = eT Q2 e 0 eT AT21 P2 e 0 eT P2 A21 e ;j
y
z
y
z
;j
y
;j
;j
y
z
:
T
z
P2 Q201 P2 A21 e 1 : With e~ = e 0 Q201 P2 A21 e 1 and (53) we finally obtain V_ = 0 eT Q1 e 0 e~T Q2 e~ + 2eT P2 + 2eT P1 f + 2eT P2 (f + f ) + 2eT P1 f y
;j
;j
;j
y
;j
z
z
(53)
z
y
y
z ;j
z
z
y
y ;j
y
y ;j
z ;j
z
:
(54) Substituting (41) and (47) into (54) yields
V_
(43)
. In order to say something about the stability of (43) we state the following theorem. z ;j
z
y
y
z
z ;j
y ;j
+ kE6
y
y
y
z ;j
y
= y 0 y^, we obtain the following
y
y
;j
+ eT1 AT21
;j
+ f~ + f + A e + + f~ + f where f~ = f~ 0 f^~ and f~ = f~ 0 f^~ e_ 1 =A11 e e_ =A21 e
z
z
y
;j
For the errors ez z1 0 z1 and ey error differential equation
;j
^ 0 eT Q2 e = 0 eT Qe + eT AT21 P2 e + eT P2 A21 e + 2eTP2 + 2eT P1 f + 2eT P2 (f + f ) + 2eT P1 f e
1 (^y 0 1^ C ) + (B2 0 B2 )u + (a2 0 a2 ) : ;i
k2
Equation (52) can be simplified by
;j
;i
;i
z
(52)
0 A21 )^z1 + (A22 0 A22 ) j
;j
;j
z
;j
;j
k E 1 and (48)
y
+ kkPP12 kk
k3
;j
z
;j
1 (^y 0 1^ C ) + (B1 0 B1 )u + (a1 0 a1 ) ;i
z1
=eT AT11 P1 + P1 A11 e + eT AT P2 + P2 A + eT AT21 P2 e + eT P2 A21 e + 2eTP2 + 2eT P1 f + 2eT P2 (f + f ) + 2eT P1 f z
V_
;i
;i
> 0 such that ke
Substituting (44) and (46) into (51), we obtain
j
0 A11 )^z1 + (A12 0 A12 ) j
(47)
(51)
j
;j
y
The derivative of V with respect to time is given by
z
f^
;j
= eT P1 e + eT P2 e :
V
(41)
and l
z
holds, the error system (43) is asymptotically stable. Proof: Consider the Lyapunov function candidate
6= 0
;j
;j
;j
+
=f^ 0 A12 1^ C =f^ + 1^_ C 0 A22 1^ C z ;j
y
;j
and Q2 2 Rp2p is a symmetric positive definite design matrix. Furthermore, we have
f^~ f^~
;j
;j
;j
is the solution of the Lyapunov equation
AT P2 + P2 A
z
;j
(40)
if ey else.
e
;j
min (Q1 ) >2k1 kP1 k >k5 + k4 E
;j
y
0kP2 kP201 k k ;
k k1 ke 1 k + k2 ke k k k3 ke 1 k + k4 ke k k k5 k k6
and assume there are scalars Ez1 ; Ey y y . Then, if
;j
y ;j
be bounded by
;j
ke k E
+ a1
p
= P2 2 R 2
;j
;j
;j
A
and fy
;j
kf~ kf~ kf kf
;j
;j
f f
(46)
;j
with ;j
(45)
;j
;j
y ;j
a1 a2
(44)
s;j
s;j
0 A21 )z1 + (A22 0 A22 ) 1 (y 0 1C ) + (B2 0 B2 )u + (a2 0 a2 ) ;i
>
AT P2 + P2 A
;j
;j
w ( ) (A21 i
i=1
0 and Q2 = Q2T > 0 be design 0 and P2 = P2T > 0 are solutions to
Q1T Theorem 2: Let Q1 matrices such that P1 P1T > the following matrix equations
and
0 min (Q1 )ke k2 0 e~T Q2e~ + 2k1 kP1 kke 1 k2 + 2ke kke 1 k (kP1 kk2 + kP2 kk3 ) + 2ke k2 kP2 kk4 + 2ke kkP2 k(0 + k5 ) + 2ke kkP1 kk6 z
y
y
y
z
;j
y
;j
z
;j
;j
z
;j
y
which is negative definite by the conditions.
;j
(55)
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119
Remark 3: In [6] it is shown that an ideal sliding motion occurs on ^. the hyperplane S = fe j Ce = 0g where e = x 0 x The observer can now be written in the nontransformed form
x^_ =
l i=1
wi ( ^) (Ai x^ + Bi u + ai ) + Gl (y 0 y^) 0 Gnl
y^ =C x^
(56)
Fig. 1.
Cart with a noninverted pendulum.
where
Gl = Tj01
A12;j A22;j 0 As;j
and Gnl = Tj01
0
(57)
Ip
and defined as in (41). Remark 4: If the nonlinear process f (x; u; ) is known, the observer can be formulated in terms of the nonlinear structure
x^_ =f (^x; u; ) + Gl (y 0 y^) 0 Gnl y^ =C x^:
(58)
IV. SIMULATION EXAMPLE This section illustrates the performance of the observers considered in theorems 1 and 2 on a simulated model of a cart with a pendulum. In the following, we refer to the observers in theorems 1 and 2 as fuzzy sliding mode (FSM) and dominant linear fuzzy sliding mode (DLSM), respectively. The results are compared to those of a fuzzy Thau–Luenberger observer that is referred to as FTL. The FTL observer is basically a FSM observer without switching term, and the design of the observer gains may be less conservative due to the absence of the structural constraint on P (see [9]). A. Mathematical Model The model considered here is a revised version of the model of a cart with an inverted pendulum considered in [22]. The difference is that we consider the cart with a noninverted pendulum (see Fig. 1). Let x1 = , x2 = _ , x3 = x, and x4 = x_ .
x_ 1 x_ 2 x_ 3 x_ 4
=
x2 0g sin(x )0 0mla cos x4 +
0ma cos
w1 (x1 ) =
0mla cos 0
(x )
(u
0 fc )
x_ = (w1 (x1 )A1 + w2 (x1 )A2 ) x w1 (x1 )B1 + w2 (x1 )B2 ) 2 2
0
A1 =
017:31
1
0
0
0
0
0
1:7312
0
0
1
0
0
0
0
C=
014:32
1
0
0
0
0
0
0:716
0
0
1
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
1
(60)
0
B1 =
00:1765 0
0:1176 0
B2 =
00:1147 0
0:1081
:
B. Design of DLSM-Observer For the design of this observer, the functions obsfor.m and disobs.m of the SMC Matlab Control Toolbox by Edwards and Spurgeon [23] were used. According to option 3 mentioned in the beginning of Section III-B subsystem A1 , B1 around the equilibrium point x1 = 0 is chosen as the dominant linear system. In obsfor.m the desired stable sliding mode pole is chosen as p = 03. This gives the transformation matrix T1 and the transformed system matrices 3
sin(x1 )
1 + exp
1
014(x 0 )) 014(x1 + 8 )
1+exp(
The system (59) was driven by the control input u = 10 sin(5t). Fig. 2 shows the response of the real system and its fuzzy approximation.
(x )
where g = 9:81 [m/s2 ], m = 2 [kg], M = 8 [kg], a = 1=(m + M ), l = 0:5 [m], and fc = c sgn(x4 ) is the friction of the cart where = 0:005. The system is linearized around x1 = 0 and x1 = 6=4 and a fuzzy model is obtained from [22] as
1 (u 0 fc + mlx
0
and with A2 = A3 and B2 = B3 the matrices Ai ; Bi (i = 1 . . . 3) are
(x )
0
1
w2 (x1 ) =1 0 w1 (x1 )
A2 =
0ma cos
y =Cx
where the fuzzy weights are defined as
0
(x )
+
+
Fig. 2. Comparison between the real (above) and the fuzzy (below) system (x —solid, x —dotted, x —dash-dotted, and x —dashed).
+ fum;x
T1 = (59)
1
01
0
01:5
0
0
0
0
1
0 0
0
0
0
1
120
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T1 A1 T 01 = 1
03 23:71 0 04:5 01 3 0 01:5 0 0
0 0 1:73 0
0 0 T1 B1 = 0 0:1176
1 0
and CT101
0 1 0 0 = 0 0 1 0 0 0 0 1
:
According to (56) and (57) the observer gains laws Gl and Gnl are calculated by the function disobs.m of the SMC toolbox. Three output estimation poles p = [01; 02; 04] are selected which, according to (40), gives the stable design matrix As;1 . With the design matrix Q2 = I3 , see (42), the following matrices are obtained: 01 0 0 0:5 0 0 As;1 = 0 02 0 P2 = 0 0:25 0 0 0 04 0 0 0:125 4 0 01:5 014:31 0 06 Gl =
Gnl
0 2 1 1:73 0 4 0:1176 0 0 0 :3529 0 0 0 :1765 = 0 0:1176 0 0 0 0:1176
Fig. 3. Error signals for FTL, FSM, and DLSM observer (upper, middle, lower), respectively, applied to the fuzzy system with no uncertainty.
:
Now the conditions (48) and (49) have to be validated. Experiments with different initial values have shown that k1;1 = 0:16, k2;1 = 0:51, k3;1 = 0:14, k4;1 = 0:47, k5;1 = 0:6, k6;1 = 0:6, Ez1 = 1:5, and Ey = 0:5. Choosing Q1 = 1 we obtain with (45) kP1 k = 0:208. According to (49) and with kP2 k = 0:5 we obtain min (Q1 ) = 1 > 2 1 0:16 1 0:208 = 0:067. The other condition is obtained by > 0:6 + 0:47 1 0:5 + (0:14 + 0:208=0:5 1 (0:51 + 0:6=0:5))11:5 = 2:46. Thus we define = 3 > 2:46. C. Design of the FSM-Observer When the observer is applied to the original system we will have uncertainties both in x_ 2 (unmatched) and x_ 4 (matched). Choosing D = B1 may give some tolerance against the unmatched uncertainties. The transformation matrix was chosen as 0 01 0 01:5
T which gives
2 = D
= 10 0
0 0 0:1176
0 0 0
0 1 0
0 0 1
together with
P2 and
01 0 0 and 4 = 0 01 0 0
Fig. 4. Error signals for FTL, FSM, and DLSM observer, respectively, applied to the fuzzy system with matched uncertainty.
0
0
4:619 4:429E 0 4 0:1018 = 4:429E 0 4 4:404 03:159E 0 5 0:1018 03:159E 0 5 4:403
= 3 as in the design for the DLSM observer.
D. Simulation Results
:
The LMI control toolbox for MATLAB [24] was used for solving the LMIs in the design of this observer. An LMI region was chosen to be in the intersection of the strip [01; 03] and the cone with an inner angle of 80. Maximizing the robustness measure by solving an EVP gave the robustness measure FSM = 0:77 and gain matrices (transformed to original coordinates) 5:440 3:492E 0 4 01:665 0 6 :517 5:679 04:587 L1 = 01:331E 0 4 2:773 0:1 1:794 05:605E 0 6 2:775 5:428 3:583E 0 4 01:663 0 3 :507 5:787E 0 4 04:583 L2 = 01:297E 0 4 2:773 0:1 0:7791 05:510E 0 6 2:775
Three experiments were performed, all with the following initial conditions: x(0) = [0:5; 0; 0:1; 0]T and x ^(0) = [0; 0; 0; 0]T . In the first experiment the observers are applied to the fuzzy system without matched uncertainties (see Fig. 3). As expected, all three observers are able to reconstruct the states. The second experiment incorporates matched uncertainties in the fuzzy model of the form B1 3 sin(2t); the result is shown in Fig. 4. It can be seen that the FTL observer has some difficulty tracking the state accurately, although it is not unstable. In the third experiment, the observers are applied to the original nonlinear model as shown in Fig. 5. The observers are based on the fuzzy approximation, and the approximation errors give rise to both matched and unmatched uncertainties and all observers have some difficulty to track the states accurately. The FSM observer is a little bit “calmer” than the FTL observer. We further notice that the DLSM observer gives the best tracking. This may be explained by looking at the computation of the switching terms in the FSM and DLSM cases. In the FSM case,
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Fig. 5. Error signals for FTL, FSM, and LSM observer, respectively, applied to the original system.
2T is involved in the computation. D 2T has in this example the matrix D a fairly large null space which eliminates some information from ey . In the case of DLSM all information from ey is used in the computation of the observer’s estimate. This shows that the DLSM observer very well may be used for weakly nonlinear systems. Note also that the chattering that may occur in the states of a sliding mode observer can be smoothed out by defining such that 0 as ey 0.
!
!
V. CONCLUSION This paper essentially presented two types of fuzzy sliding mode observers. A Thau-Luenberger fuzzy observer was extended to a sliding mode fuzzy observer which is able to reject matched uncertainty effectively. This type of observer can be effectively designed using LMI tools. The second observer presented is based on the assumption that a multiple model system can be represented by a selected subsystem that is dominant linear over the other subsystems. Explicit design procedure was also given for that observer. The performance of the observers was also illustrated on model of cart with a pendulum. One topic for future research may be in the use of parameter dependent Lyapunov functions for proving stability. Another topic dealing with the fuzzy sliding mode observer may be to consider a varying disturbance distribution matrix.
REFERENCES [1] R. Babuˇska, “Fuzzy modeling for control,” in International Series in Intelligent Technologies. Boston, MA: Kluwer, 1998. [2] T. A. Johansen, K. J. Hunt, and P. J. Gawthrop, “Transient performance, robustness and off-equilibrium linearization in fuzzy gain scheduled control,” in Advances in Fuzzy Control, D. Driankov and R. Palm, Eds, Heidelberg, Germany: Physica-Verlag, 1998, vol. 16, Studies in Fuzziness and Soft Computing, pp. 357–375. [3] T. A. Johansen, R. Smith, and R. Murray-Smith, “On the interpretation and identification of dynamic Takagi–Sugeno models,” IEEE Trans. Fuzzy Syst., vol. 8, pp. 297–313, June 2000. [4] K. J. Hunt et al., “Speed control design for an experimental vehicle using a generalized gain scheduling approach,” IEEE Trans. Control Syst. Technol., vol. 8, pp. 381–395, May 2000.
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[5] D. J. Leith and W. J. Rugh, “Analytic framework for blended multiple model systems using local linear models,” Int. J. Control, vol. 72, pp. 605–619, 1999. [6] X.-J. Ma, Z.-Q. Sun, and Y.-Y. He, “Analyis and design of fuzzy controller and observer,” IEEE Trans. Fuzzy Syst., vol. 6, pp. 41–51, Feb. 1998. [7] K. Tanaka, T. Ikeda, and H. Wang, “Fuzzy regulators and fuzzy observers: Relaxed stability conditions and LMI-based design,” IEEE Trans. Fuzzy Syst., vol. 6, pp. 250–265, May 1998. [8] A. Jadbabaie et al., “Observer-based controller synthesis for model-based fuzzy systems via linear matrix inequalities,” in Fuzzy Control: Synthesis and Analysis, S. S. Farinwata, D. Filev, and R. Langari, Eds. New York: Wiley, 2000, ch. 11, pp. 239–252. [9] P. Bergsten and R. Palm, “Thau-Luenberger observers for TS fuzzy systems,” in Proc. IEEE Ninth Int. Conf. Fuzzy Syst., vol. 2, San Antonio, TX, May 7–10, 2000, pp. 671–676. [10] S. H. Zak, “Stabilizing fuzzy models using linear controllers,” IEEE Trans. Fuzzy Syst., vol. 7, pp. 236–240, Apr. 1999. [11] S. H. Zak and B. L. Walcott, “State observation on nonlinear control systems via the method of Lyapunov,” in Deterministic Control of Uncertain Systems, A. S. I. Zinober, Ed. London, U.K.: Peter Peregrinus, 1990, ch. 16, pp. 333–350. [12] P. Korba, A Gain-Scheduling Approach to Model-Based Fuzzy Control. Düsseldorf, Germany: VDI Verlag, 2000. [13] F. E. Thau, “Observing the state of nonlinear systems,” Int. J. Control, vol. 17, pp. 471–479, 1973. [14] J. Joh, Y.-H. Chen, and R. Langari, “On the stability issues of linear Takagi–Sugeno fuzzy models,” IEEE Trans. Fuzzy Syst., vol. 6, pp. 402–410, Aug. 1998. [15] C. Ewards and S. Spurgeon, Sliding Mode Control: Theory and Applications. London, U.K.: Taylor & Francis, 1998. [16] A. S. I. Zinober, Ed., Deterministic Control of Uncertain Systems. London, U.K.: Peter Peregrinus, 1990. [17] R. Palm and D. Driankov, “Toward a systematic analysis of fuzzy observers,” in Proc. 18th Int. Conf. North American Fuzzy Inform. Processing Soc. NAFIPS, New York, NY, June 10–12, 1999, pp. 179–183. [18] C. P. Tan and C. Edwards, “An LMI approach for designing sliding mode observers,” in UKACC Int. Conf. Control, Cambridge, U.K., Sept. 2000. [19] H. K. Khalil, Nonlinear Systems, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996. [20] S. Boyd et al., “Linear matrix inequalities in system and control theory,” in SIAM Studies in Applied Mathematics. Philadelphia: SIAM, 1994. [21] J. Joh and R. Langari, “A framework for the synthesis of PDC-type Takagi–Sugeno fuzzy control systems: An LMI approach,” in Fuzzy Control: Synthesis and Analysis, S. S. Farinwata, D. Filev, and R. Langari, Eds, New York: Wiley, 2000, ch. 13, pp. 267–282. [22] M. C. M. Teixeira and S. H. Zak, “Stabilizing controller design for uncertain nonlinear systems using fuzzy models,” IEEE Trans. Fuzzy Syst., vol. 7, pp. 133–142, Apr. 1999. [23] C. Edwards and S. Spurgeon. (1998) A Sliding Mode Control Toolbox, Tech. Rep. 98-9. Control Systems Research, Department of Engineering, University of Leicester, U.K. [Online]. Available: http://www.le.ac.uk/eg/ce14/vscbook/vscbook.html [24] P. Gahinet et al., Control Toolbox: For Use With MATLAB. Natick, MA: Mathworks, Inc., 1995.
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Observers for Takagi–Sugeno Fuzzy Systems P. Bergsten, R. Palm, and D. Driankov
Abstract—We focus on the analysis and design of two different sliding mode observers for dynamic Takagi–Sugeno (TS) fuzzy systems. A nonlinear system of this class is composed of multiple affine local linear models that are smoothly interpolated by weighting functions resulting from a fuzzy partitioning of the state space of a given nonlinear system subject to observation. The Takagi–Sugeno fuzzy system is then an accurate approximation of the original nonlinear system. Our approach to the analysis and design of observers for Takagi–Sugeno fuzzy systems is based on extending sliding mode observer schemes to the case of interpolated multiple local affine linear models. Thus, our main contribution is nonlinear observers analysis and design methods that can effectively deal with model/plant mismatches. Furthermore, we consider the difficult case when the weighting functions in the Takagi–Sugeno fuzzy system depend on the estimated state. Index Terms—Dominant linear models, fuzzy sliding mode observers, interpolation of multiple linear local models, Luenberger fuzzy observers, Takagi–Sugeno (TS) fuzzy systems.
I. INTRODUCTION This paper describes the analysis and design of two sliding mode observers for a class of nonlinear systems modeled as Takagi–Sugeno (TS) fuzzy systems. A dynamic TS [1] fuzzy system is composed of multiple local affine dynamic linear models. These local models, for the purpose of this paper are related to local linearizations, via Taylor series expansion, of the original nonlinear system at off-equilibrium points [2]. Thus, the local models have no equilibrium point within their regions of validity, i.e., they are off-equilibrium local models (for detailed exposition see [3]). As shown in [4], incorporating off-equilibrium linearization in the model can significantly improve the transient response in closed-loop control. In [3] it is proved that a TS fuzzy system where the local affine dynamic models are off-equilibrium local linearizations leads to an arbitrary close approximation of the linear time varying (LTV) dynamic system resulting from dynamic linearization of the original nonlinear system about an arbitrary trajectory (see also [5] for related results). Thus, the results concerning observers for TS fuzzy systems are also relevant to systems such as linear parameter varying (LPV) systems, piecewise linear systems, and conventional gain-scheduled systems. This paper focuses on analysis and design aspects of observers for TS fuzzy systems of the type introduced above, i.e, we allow possible affine TS systems, which is not normally considered in the literature, e.g., [6] and [7]. Our main contribution is analysis and design methods in the case of model/plant mismatches represented as matched and unmatched uncertainties. By matched uncertainties for observers, we mean that the uncertainties appear in the output channels only. Unmatched uncertainties do not appear in the output channels. We also explicitly consider the case when the weighting functions in the TS fuzzy system depend on the estimated state. This case is considered to be difficult [7] and the usual assumption is that the weights depend only Manuscript received July 24, 2001. This paper was recommended by Associate Editor C. T. Lin. P. Bergsten and D. Driankov are with the Center for Applied Autonomous Sensor Systems, Department of Technology, Örebro University, Örebro, Sweden (e-mail: [email protected]; [email protected]). R. Palm is with Siemens AG Corporate Technology, Otto-Hahn-Ring, Munich, Germany (e-mail: [email protected]). Publisher Item Identifier S 1083-4419(02)00027-4.
on measurable parameters (see [6] and [8]). Two observer types are proposed: 1) the extension of a so called Fuzzy–Thau Luenberger observer [9] with a switching term to account for matched uncertainties and 2) the affine TS fuzzy system is rewritten as a dominant linear system in order to treat it as a linear system subjected to known and unknown uncertainty. The latter approach is inspired from a similar idea used for controller design [10]. Using this approach, design methods for sliding mode observers for linear systems can be applied. Lastly, the analysis and design methods proposed utilize available Matlab software which facilitates reproducing the results reported. Section II introduces the TS fuzzy system subject to observation and defines a Luenberger type of observer for it. The latter is realized in terms of a parallel distributed compensation scheme incorporating an interpolation between local Luenberger observers. This is the type of nonlinear observer that has received most attention in the fuzzy control literature, but all results reported assume no matched/unmatched uncertainties. In Section III-A, we take into account matched uncertainties only and present a sliding mode fuzzy observer that is related to the so called min–max observer described by Zak and Walkot [11]. This observer utilizes interpolation between local observer gains which has one major negative effect: large number of local models will give a large number of linear matrix inequalities (LMI) in the stability analysis and design, which may prohibit the use of existing LMI tools. For this reason, in Section III-B, we reduce the TS fuzzy system to a dominant linear one, that is one local model is chosen and the effect of the rest are incorporated in it in terms of known deviations interpreted as uncertainties. This avoids the use of LMIs for analysis and design and instead a direct sliding mode observer design is possible. Furthermore, model mismatches are taken into account as known upper bounds of matched and unmatched uncertainties. In Section IV the performance of the two different types of sliding mode observers is illustrated on a model of a cart with a pendulum. These results are also compared to the ones produced by a fuzzy Thau–Luenberger observer for the same system. Details about the latter one can be found in [9]. II. TS FUZZY SYSTEM Consider a nonlinear system of the form
x_ =f (x; u; ) y =g(x; )
(1)
where
f : R n 2 Rm 2 Rs ! Rn
nonlinear function that satisfies the Lipschitz condition; g : R n 2 R s ! Rp measurement function; x state vector; u control input vector; vector of possibly time varying parameters; y 2 Rp output vector. In the following we assume that (1) can be represented or approximated sufficiently well by a TS fuzzy system (this assumption need not to be true for an arbitrary system of the form (1), however). The TS system consists of a fuzzy rule base, where each rule i is of the form Rule i:
1083-4419/02$17.00 © 2002 IEEE
IF 1 is i1 ( 1 ) and . . . and r is ir ( r ) THEN
x_ = Ai x + Bi u + ai y = Ci x + ci
(2)
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 32, NO. 1, FEBRUARY 2002
where the vector of premise variables 2 Rr is a subset of x, u, and and ij : R ! [0; 1]. The function ij ( j ) denotes the j th membership function in the ith rule which applies to the j th premise variable. The Cartesian product i1 ( 1 ) 21 1 12 ir ( r ) defines a fuzzy region in Rr . The system matrices Ai and Bi may be obtained by a linearizing transformation [12] or through Taylor expansion in some point (xi ; ui ; i ) corresponding to i in the fuzzy region described by each rule i
Bi = @f and Ai = @f @x (x ;u ; ) @u (x ;u ; ) Ci = @g : (3) @x (x ; ) Note that (xi ; ui ; i ) does not have to be an equilibrium point which means that constant or affine terms in each subsystem i may be obtained as
ai =f (xi ; ui ; i ) 0 Ai xi 0 Bi ui ci =g(xi ; i ) 0 Ci xi : The global TS fuzzy system described by (2) is then written as
x_ = y=
l i=1 l i=1
wi ( )(Ai x + Bi u + ai ) + fum;x wi ( )(Ci x + ci ) + fum;y
(4)
where l is the number of rules and
wi ( ) =
r hi ( ) h ( ) = ij ( j ) i l h ( ) k k=1 j =1
with l i=1
wi ( ) = 1:
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orem for a fuzzy controller and observer. In this paper, we consider the latter case which is more difficult than the former one—see [9] where a fuzzy Thau–Luenberger observer is introduced based on ideas from [13]. We will also constrain ourselves to the continuous time case although there exist stability results for the discrete time case (see [6], [7], and [14]). Furthermore, the above type of observer does not take into account approximation errors or uncertainties in the fuzzy model; that is, it assumes that the TS system (4) is the nominal one. In Section III-A, we will consider the case y = Cx which covers a large class of real technical systems. In Section III-B, we extend the class of systems to the case y = li=1 wi ( )Ci x. III. SLIDING MODE OBSERVERS FOR TS FUZZY SYSTEMS As already mentioned, the observer from (5) does not consider modeling errors, i.e., it is assumed that a perfect nonlinear model of the plant is available. However, in the presence of modeling errors or unmodeled system uncertainties the previous approach may lead to degraded performance and, in some cases, even to instability. Modeling errors or unmodeled system uncertainties can be represented as matched and unmatched uncertainties. In this section we show how sliding mode techniques can be used in observer design to cope both with matched and unmatched uncertainties. An extensive representation of these techniques can be found in [15] and [16]. In [17], fuzzy sliding mode observers were first introduced and conditions were given under which matched uncertainties are eliminated. These types of observers are related to the so-called min–max observer described by Zak and Walcott [11]. In Section III-A, we show how a fuzzy Luenberger observer can be extended to a sliding mode observer whose analysis and design is based on LMIs. Then in Section III-B, we present a technique for observer design applicable to a particular form of a TS fuzzy system called dominant linear TS fuzzy system. The latter enables us to design a sliding mode observer in a direct way [15] without using LMIs.
fum;x and fum;y are modeling/approximation errors which have to be
A. Fuzzy Luenberger Sliding Mode Observer
taken into consideration in the observer design. Note that other matched and unmatched uncertainties are lumped together with these errors. Note also that (4) is suited to be obtained by nonlinear system identification [1], due to its particular form. The most well known parallel distributed (PD) type of observer proposed in [7] is an extension of the classical Luenberger observer to the above TS fuzzy system and is obtained as follows. Consider the following observer structure:
The observer developed in this section is based on the ideas found in [17] and [18]. Assume that the fuzzy approximation of a nonlinear system reads
x^_ = y^ =
l i=1 l i=1
wi ( ) (Ai x^ + Bi u + Li (y 0 y^) + ai ) wi ( ) (Cix^ + ci ) :
(5)
Note that the affine terms are not considered in [6] and [7]. Assume that only contains measurable parameters, i.e., does not depend on the estimated states and define the error between the estimated states x^ and the real states x in (4) as e = x 0 x^, then we get the following error differential equation
e_ =
l
l
i=1 j =1
wi ( )wj ( ) [(Ai 0 Li Cj )e 0 Li fum;y ] + fum;x :
(6)
Here the effect of the modeling/approximation errors is clearly seen, and the main issue is to ensure asymptotic stability of (6). As pointed out in [7], one can distinguish between two cases: 1) does not depend on some of the unmeasurable states and 2) does indeed depend on the unmeasurable states, i.e., := ^ . The authors in [6] considered the former case and extended the results in [7] with a separation the-
x_ =
l i=1
wi ( )(Ai x + Bi u + ai ) + D
(7) y =Cx where 2 Rq , D 2 Rn2q , and C 2 Rp2n with p q . The term D
represents the matched modeling/approximation errors. That is, we assume that the present uncertainties can be transformed to the output channels (see Assumption 1). Furthermore, as can be seen in Theorem 1, the observer scheme also gives some robustness to unmatched growth bounded uncertainties. This fact is an inherent feature stemming from the attempt to deal with the case := ^ discussed above. The following assumptions are made. Assumption 1: There exists a linear change of coordinates z = T x such that 0 (8) D = T D = and C = CT 01 = [ 0 Ip ] D2
2 with D
2 Rp2q , T 2 Rn2n , and z 2 Rn . Ai = T Ai T 01 ; Bi = T Bi ;
and a i = T ai :
(9)
Using Assumption 1 we may simply define matched uncertainty as uncertainty that is structured so that the linear coordinate transformation T —with the desired properties—exists. Hence, Assumption 1 effectively constrains the class of uncertainty. Here it can be seen that the
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case with a varying C matrix will give rise to a nonlinear coordinate transformation which may be difficult to obtain. In this section we will therefore only consider the case with constant C . Assumption 2: There exists a scalar > such that k k . Assumption 2 effectively bounds the norm of the uncertainty by a positive constant. If one knows that k k depends on some measurable parameters # it may be possible to state a varying bound like k # k # . In any case, the a-priori knowledge that is needed is an estimate of the norm of the uncertainty . One case when there may be a problem to obtain such an estimate is when (7) is unstable and k k is an increasing function of kxk. Using the transformed state vector z T x we have that
0
()
()
=
l
z_ =T x_ = T
=
l
i
i=1
i
i
i
i
+ T D (10)
and
y = CT 01 z:
(11)
Hence, we get the transformed system
z_ =
l
w ( ) A z + B u + a i
i=1
y =Cz:
i
i
+ D
i
(12)
]
z_1 y_
=
l
A11 A21
w ( ) i
i=1
+ y =[ 0 I
p
]
;i
;i
z1 : y
a1 a2
A12 A22
z1 y
;i
;i
+ D02
;i ;i
+
z^_ =
B1 B2
;i ;i
i
i=1
y^ =C z^
i
i
i
+ L (y 0 y^) + D i
P e
y
y
then becomes
e_ =
l
+ D + 1( ; ^; z; u) w (^ ) A 0 L C e 0 D i
i=1
i
i
p
i
p
n
i
i
p
p
i
V_ (e) = eT
l
w (^ ) (A i
i=1
+P (A 0 L C ) i
i
0 L C )TP i
+ eT 2PD + 21T Pe : e 0 eT 2P D
(22)
P
=
we have that eT P D
V_ (e) 0 eT Qe + 21T P e T T 0 2 e P2DDT2PD2e P2 e + 2eTP2 D 2 2 2 0 eT Qe + 21T P e 0 2 D 2T P2 e + 2k k D 2T P2 e : y
y
y
y
y
(23)
y
Obviously, the maximum positive part of the uncertainty D is canceled in the Lyapunov function derivative by the switching term D . Now, there always exist a > such that
0 V_ (e) 0eT Qe + eT P P e + 01 k1k2 02 D 2T P2 e + 2k k D 2T P2 e y
T = k (15) k ; if D2 P2 e 6= 0 0; else where e = y 0 y^ and P2 is a symmetric positive definite matrix (see Theorem 1). Note also that the general case with := ^ is considered. The fact that may contain the nontransformed state does not alter the presentation. The error differential equation, for the error e = z 0 z^, P e
p
with the matrices Li being local observer gain matrices and D
p
D
p n
Due to the structural constraint (18) on eyT P2 D2 which gives
(14)
D
D
u
(13)
w (^ ) A z^ + B u + a
2T (if N = ; then where N is a basis of the null space of D 4 (=00)22( ).0 If) there exists a scalar > 0 and matrices P1 = P1T 2 , P2 = P2T 2 R 2 , P3 2 R( 0 )2 , and Q = QT > R 0 such that P P4 P = T 1 T 3 >0 (18) 4 P3 P2 T (19) A 0 L C P + P A 0 L C 0 Q Q 0 2 P >0 (20) P I k1( ; ^; z; u)k kek (21) for i = 1; 2; . . . ; l, then (16) is asymptotically stable. Proof: Consider the Lyapunov function candidate V (e) = eT P e = eT P e with > 0. Taking the derivative of V (e) along the
i
Also note that both the transformed state vector z and the nontransformed state vector x (incorporated in ) may be present in (12). This does not, however, pose any problem in the observer design. Consider the following observer for (12) l
(17)
trajectories gives
From the properties of T in Assumption 1, the state vector z is partitioned as z z1T yT T and (12) now has the uncertainty in the output channels. For clarity, (12) can be written in partitioned form as
=[
T 4 = N0 D
n
i
w ( ) T A T 01 z + T B u + T a i
i=1
i
4
p
w ( ) (A x + B u + a ) + D
(^)) + + 0 1( ^ ) 0
1( ^ ) = ( () 1( ^ ) 0 1( ^ )
l ; ; z; u wi 0 wi Ai z Bi u ai . where i=1 Note that ; ; z; u ! when e ! . That is, ; ; z; u acts like an unstructured vanishing perturbation that is supposed to be growth bounded as k ; ; z; u k kek for some > . However, assuming no particular structure of such perturbations may be quite conservative, as pointed out in [19]. Sufficient stability conditions for (16) are given by the following theorem. Theorem 1: Let 2 Rp2p be structured as
y
= we obtain V_ (e) 0 eT Qe + eT P P e + 2 kek2 0 2 D 2T P2 e + 2k k D 2T P2 e 0 min (Q 0 P P )kek2 + 2 kek2 0 2 D 2T P2 e + 2k k D 2T P2 e
:
(24)
If we choose
y
y
y
y
(25)
which is negative definite if (16)
min (Q 0 P P ) > 2 :
(26)
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By the Schur complement [20], (26) is equivalent to
Q0 P
P >0 I
2
(27)
and the theorem follows. Remark 1: If k ; ; z; u k and using the arguments in [11], one can conclude that sliding motion takes place on the surface S e j D2T P2 Ce . Remark 2: Note that if D2 2 Rp2p with full rank the sliding surface is S fe j Ce g, that is, similar to the observer proposed in [18]. Now, we can state the design of this observer as an eigenvalue ; ; ; l so problem (EVP) in , Q, P1 , P2 , P3 , and Mi for i that we maximize the robustness measure and solving for the gain matrices at the same time
1( ^ ) = 0 =0 = =0
=
= 1 2 ...
Q = QT > 0 P~1 + P~2 > 0 AT P~1 + P~1 A + AT P~2 + P~2 A 0 C TM T 0 M C 0Q Q 0 P~1 + P~2 ~P1 + P~2T I > 0
maximize subject to
i
i
where
P~1 = and
P~2 =
0 0 4T
I
P1
0
0
P2
0
P3
(28)
i
w (^ )(A x^ + B u + a i
i=1
i
i
i
i
j
0 0 4
:
i
(31)
1 =
l
j
j
j
1C =
w ( )((A i
i=1 l
j
i
w ( )(C i
i=1
+1 +f j
um;x
(34)
um;y
0 A )x + (B 0 B )u + (a 0 a )) j
i
and defined as in (15). The concept of LMI regions for placing the eigenvalues of the local systems may be used to get a good dynamic behavior of the convergence of the observer. Such region constraints on the eigenvalues are easily incorporated in the design and will show up as additional linear matrix inequalities (LMI) constraints in the EVP above. See, for example, [12] and [21] for more details on using LMI regions in the context of fuzzy observers and controllers. B. Sliding Mode Observer for a Dominant Linear Fuzzy System In the design and analysis method described in Section II we used an LMI-based approach. However, when the number of LMIs is too large this approach may have computational difficulties. In the following the nonlinear plant is approximated with an affine TS fuzzy system. Further, one of the resulting subsystems is selected as to be dominant linear for the whole plant which can be decided in a practical application by the designer. An analytical way of how to obtain an appropriate dominant linear system for the controller design is reported in [10]. For the
i
j
0 C )x:
i
(35)
j
1
1
z=T x=
z1 y~
j
(32)
j
Notice that the “uncertainties” j and Cj are not real ones because they are completely known. As in Section II we assume that there exists a nonsingular transformation Tj such that
with
Li = T 01 L i for i = 1; 2; . . . ; l
(33)
where (30)
+ L (y 0 y^)) + D
y =C x^
um;x
um;y
j
j
I
i
w ( )C x + f i
i=1
i
x_ =A x + B u + a y =C x + 1C + f
(29)
coordinate system as l
l
i
where fum;x and fum;y are the vectors of matched and unmatched uncertainties. Furthermore, let the j th subsystem in (33) dominate the whole system [10]. In this case we call the system “dominant linear.” Then, (33) can be rewritten as
i
x^_ =
w ( )(A x + B u + a ) + f i
i=1
y=
0 p The robustness measure is obtained as = and the gain matrices as L = (P~1 + P~2 )01 M . The observer is then stated in the original P3T
l
x_ =
i
i
observer design there are many options to choose a dominant linear subsystem. One particular option is to choose a subsystem such that the remaining errors are minimal. Another option is to choose that subsystem which is believed to be occupied most of the operating time. A further option is to choose that subsystem in which the unforced system comes to rest (the equilibrium point). The final result is a linear system with known modeling errors and further uncertainties whose upper bounds are supposed to be known. The next step is to transform the so obtained linear system into a canonical form which separates the nonmeasured states from the measured outputs. This transformation is always possible if the system meets certain rank conditions [15]. In the next design step, a linear and a nonlinear observer in canonical form are computed. Finally the inverse transformation leads to both the linear and nonlinear observer gain for the original observer equation. We are concerned with the following fuzzy system:
i
i
117
where z1 2 Rn0p , Tj 2 Rn2n , and y (34) is then transformed into
(36)
~ = y 0 1C 0 f
z_ = T A T 01 z + T B u + T j
j
j
j
j
j
j
1 +T f j
j
. Equation
um;y
um;x
+T a j
(37)
j
that can be written as
z_1 =A11 z1 + A12 y~ + B1 u + f y~_ =A21 z1 + A22 y~ + B2 u + f ;j
;j
;j
z ;j
;j
;j
;j
y ;j
+f +f
+ a1 + a2
(38)
+f +f
+ a1 + a2
(39)
z ;j y ;j
;j ;j
or
z_1 =A11 z1 + A12 y + B1 u + f~ y_ =A21 z1 + A22 y + B2 u + f~ ;j
;j
;j
z ;j
;j
;j
;j
y ;j
z ;j y ;j
where
f~ f~
z ;j
y ;j
=f =f
z ;j y ;j
0 A12 1C + 1_ C 0 A22 1C ;j j
j
;j
j
;j ;j
118
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= =
=
f
z ;j
l
w ( ) (A11
0 A11 )z1 + (A12 0 A12 ) 1 (y 0 1C ) + (B1 0 B1 )u + (a1 0 a1 )
i
i=1
;i
;j
;i
j
;i
;i
f
=
y ;j
l
;j
;j
;i
j
;i
;i
= 0 Q2 Q^ =Q1 + AT21 P2 Q201 P2 A21 ^ P1 + P1 A11 = 0 Q:
AT11;j
~
Let fz
;j
;j
~
, fy
;j
, fz
=T a j
;j
z ;j
y ;j
and
j
:= h(f
z ;j
um;x
y ;j
;f
; f_
um;y
um;y
z ;j
)
and the matrix A11;j is assumed to be stable. Consider a corresponding observer of the following form
z^_ 1 =A11 z^1 + A2 y^ + B1 u + f^~ y^_ =A21 z^1 + A22 y^ + B2 u + f^~ + (A22 0 A )e 0 : 2 R 2 is a stable design matrix and ;j
;j
;j
s;j
p
s;j
p
z ;j
;j
+ A12 + a2
;j
e
y
p
0;
e
s;j
s;j
z ;j
y ;j
;j
y ;j
j
z ;j
=
w (^ ) (A11 i
i=1
;i
= 0Q2
(42)
V_
z
f^
y ;j
=
l
w (^ ) (A21 i
i=1
;i
=
^
z
y
;j
z
;j
z
z ;j
s;j
z
z
z ;j
z ;j
y ;j
y ;j
E
;j
(49)
z1
y
(50)
y
;j
s;j
s;j
z
e
y
y
y ;j
y ;j
z ;j
z
:
y ;j
y ;j
y
y
;j
;j
y
z ;j
z
y
y ;j
y
y ;j
z ;j
z
0 Q201P2 A21 e 1 T Q2 e 0 Q201P2 A21 e 1 = eT Q2 e 0 eT AT21 P2 e 0 eT P2 A21 e ;j
y
z
y
z
;j
y
;j
;j
y
z
:
T
z
P2 Q201 P2 A21 e 1 : With e~ = e 0 Q201 P2 A21 e 1 and (53) we finally obtain V_ = 0 eT Q1 e 0 e~T Q2 e~ + 2eT P2 + 2eT P1 f + 2eT P2 (f + f ) + 2eT P1 f y
;j
;j
;j
y
;j
z
z
(53)
z
y
y
z ;j
z
z
y
y ;j
y
y ;j
z ;j
z
:
(54) Substituting (41) and (47) into (54) yields
V_
(43)
. In order to say something about the stability of (43) we state the following theorem. z ;j
z
y
y
z
z ;j
y ;j
+ kE6
y
y
y
z ;j
y
= y 0 y^, we obtain the following
y
y
;j
+ eT1 AT21
;j
+ f~ + f + A e + + f~ + f where f~ = f~ 0 f^~ and f~ = f~ 0 f^~ e_ 1 =A11 e e_ =A21 e
z
z
y
;j
For the errors ez z1 0 z1 and ey error differential equation
;j
^ 0 eT Q2 e = 0 eT Qe + eT AT21 P2 e + eT P2 A21 e + 2eTP2 + 2eT P1 f + 2eT P2 (f + f ) + 2eT P1 f e
1 (^y 0 1^ C ) + (B2 0 B2 )u + (a2 0 a2 ) : ;i
k2
Equation (52) can be simplified by
;j
;i
;i
z
(52)
0 A21 )^z1 + (A22 0 A22 ) j
;j
;j
z
;j
;j
k E 1 and (48)
y
+ kkPP12 kk
k3
;j
z
;j
1 (^y 0 1^ C ) + (B1 0 B1 )u + (a1 0 a1 ) ;i
z1
=eT AT11 P1 + P1 A11 e + eT AT P2 + P2 A + eT AT21 P2 e + eT P2 A21 e + 2eTP2 + 2eT P1 f + 2eT P2 (f + f ) + 2eT P1 f z
V_
;i
;i
> 0 such that ke
Substituting (44) and (46) into (51), we obtain
j
0 A11 )^z1 + (A12 0 A12 ) j
(47)
(51)
j
;j
y
The derivative of V with respect to time is given by
z
f^
;j
= eT P1 e + eT P2 e :
V
(41)
and l
z
holds, the error system (43) is asymptotically stable. Proof: Consider the Lyapunov function candidate
6= 0
;j
;j
;j
+
=f^ 0 A12 1^ C =f^ + 1^_ C 0 A22 1^ C z ;j
y
;j
and Q2 2 Rp2p is a symmetric positive definite design matrix. Furthermore, we have
f^~ f^~
;j
;j
;j
is the solution of the Lyapunov equation
AT P2 + P2 A
z
;j
(40)
if ey else.
e
;j
min (Q1 ) >2k1 kP1 k >k5 + k4 E
;j
y
0kP2 kP201 k k ;
k k1 ke 1 k + k2 ke k k k3 ke 1 k + k4 ke k k k5 k k6
and assume there are scalars Ez1 ; Ey y y . Then, if
;j
y ;j
be bounded by
;j
ke k E
+ a1
p
= P2 2 R 2
;j
;j
;j
A
and fy
;j
kf~ kf~ kf kf
;j
;j
f f
(46)
;j
with ;j
(45)
;j
;j
y ;j
a1 a2
(44)
s;j
s;j
0 A21 )z1 + (A22 0 A22 ) 1 (y 0 1C ) + (B2 0 B2 )u + (a2 0 a2 ) ;i
>
AT P2 + P2 A
;j
;j
w ( ) (A21 i
i=1
0 and Q2 = Q2T > 0 be design 0 and P2 = P2T > 0 are solutions to
Q1T Theorem 2: Let Q1 matrices such that P1 P1T > the following matrix equations
and
0 min (Q1 )ke k2 0 e~T Q2e~ + 2k1 kP1 kke 1 k2 + 2ke kke 1 k (kP1 kk2 + kP2 kk3 ) + 2ke k2 kP2 kk4 + 2ke kkP2 k(0 + k5 ) + 2ke kkP1 kk6 z
y
y
y
z
;j
y
;j
z
;j
;j
z
;j
y
which is negative definite by the conditions.
;j
(55)
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119
Remark 3: In [6] it is shown that an ideal sliding motion occurs on ^. the hyperplane S = fe j Ce = 0g where e = x 0 x The observer can now be written in the nontransformed form
x^_ =
l i=1
wi ( ^) (Ai x^ + Bi u + ai ) + Gl (y 0 y^) 0 Gnl
y^ =C x^
(56)
Fig. 1.
Cart with a noninverted pendulum.
where
Gl = Tj01
A12;j A22;j 0 As;j
and Gnl = Tj01
0
(57)
Ip
and defined as in (41). Remark 4: If the nonlinear process f (x; u; ) is known, the observer can be formulated in terms of the nonlinear structure
x^_ =f (^x; u; ) + Gl (y 0 y^) 0 Gnl y^ =C x^:
(58)
IV. SIMULATION EXAMPLE This section illustrates the performance of the observers considered in theorems 1 and 2 on a simulated model of a cart with a pendulum. In the following, we refer to the observers in theorems 1 and 2 as fuzzy sliding mode (FSM) and dominant linear fuzzy sliding mode (DLSM), respectively. The results are compared to those of a fuzzy Thau–Luenberger observer that is referred to as FTL. The FTL observer is basically a FSM observer without switching term, and the design of the observer gains may be less conservative due to the absence of the structural constraint on P (see [9]). A. Mathematical Model The model considered here is a revised version of the model of a cart with an inverted pendulum considered in [22]. The difference is that we consider the cart with a noninverted pendulum (see Fig. 1). Let x1 = , x2 = _ , x3 = x, and x4 = x_ .
x_ 1 x_ 2 x_ 3 x_ 4
=
x2 0g sin(x )0 0mla cos x4 +
0ma cos
w1 (x1 ) =
0mla cos 0
(x )
(u
0 fc )
x_ = (w1 (x1 )A1 + w2 (x1 )A2 ) x w1 (x1 )B1 + w2 (x1 )B2 ) 2 2
0
A1 =
017:31
1
0
0
0
0
0
1:7312
0
0
1
0
0
0
0
C=
014:32
1
0
0
0
0
0
0:716
0
0
1
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
1
(60)
0
B1 =
00:1765 0
0:1176 0
B2 =
00:1147 0
0:1081
:
B. Design of DLSM-Observer For the design of this observer, the functions obsfor.m and disobs.m of the SMC Matlab Control Toolbox by Edwards and Spurgeon [23] were used. According to option 3 mentioned in the beginning of Section III-B subsystem A1 , B1 around the equilibrium point x1 = 0 is chosen as the dominant linear system. In obsfor.m the desired stable sliding mode pole is chosen as p = 03. This gives the transformation matrix T1 and the transformed system matrices 3
sin(x1 )
1 + exp
1
014(x 0 )) 014(x1 + 8 )
1+exp(
The system (59) was driven by the control input u = 10 sin(5t). Fig. 2 shows the response of the real system and its fuzzy approximation.
(x )
where g = 9:81 [m/s2 ], m = 2 [kg], M = 8 [kg], a = 1=(m + M ), l = 0:5 [m], and fc = c sgn(x4 ) is the friction of the cart where = 0:005. The system is linearized around x1 = 0 and x1 = 6=4 and a fuzzy model is obtained from [22] as
1 (u 0 fc + mlx
0
and with A2 = A3 and B2 = B3 the matrices Ai ; Bi (i = 1 . . . 3) are
(x )
0
1
w2 (x1 ) =1 0 w1 (x1 )
A2 =
0ma cos
y =Cx
where the fuzzy weights are defined as
0
(x )
+
+
Fig. 2. Comparison between the real (above) and the fuzzy (below) system (x —solid, x —dotted, x —dash-dotted, and x —dashed).
+ fum;x
T1 = (59)
1
01
0
01:5
0
0
0
0
1
0 0
0
0
0
1
120
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T1 A1 T 01 = 1
03 23:71 0 04:5 01 3 0 01:5 0 0
0 0 1:73 0
0 0 T1 B1 = 0 0:1176
1 0
and CT101
0 1 0 0 = 0 0 1 0 0 0 0 1
:
According to (56) and (57) the observer gains laws Gl and Gnl are calculated by the function disobs.m of the SMC toolbox. Three output estimation poles p = [01; 02; 04] are selected which, according to (40), gives the stable design matrix As;1 . With the design matrix Q2 = I3 , see (42), the following matrices are obtained: 01 0 0 0:5 0 0 As;1 = 0 02 0 P2 = 0 0:25 0 0 0 04 0 0 0:125 4 0 01:5 014:31 0 06 Gl =
Gnl
0 2 1 1:73 0 4 0:1176 0 0 0 :3529 0 0 0 :1765 = 0 0:1176 0 0 0 0:1176
Fig. 3. Error signals for FTL, FSM, and DLSM observer (upper, middle, lower), respectively, applied to the fuzzy system with no uncertainty.
:
Now the conditions (48) and (49) have to be validated. Experiments with different initial values have shown that k1;1 = 0:16, k2;1 = 0:51, k3;1 = 0:14, k4;1 = 0:47, k5;1 = 0:6, k6;1 = 0:6, Ez1 = 1:5, and Ey = 0:5. Choosing Q1 = 1 we obtain with (45) kP1 k = 0:208. According to (49) and with kP2 k = 0:5 we obtain min (Q1 ) = 1 > 2 1 0:16 1 0:208 = 0:067. The other condition is obtained by > 0:6 + 0:47 1 0:5 + (0:14 + 0:208=0:5 1 (0:51 + 0:6=0:5))11:5 = 2:46. Thus we define = 3 > 2:46. C. Design of the FSM-Observer When the observer is applied to the original system we will have uncertainties both in x_ 2 (unmatched) and x_ 4 (matched). Choosing D = B1 may give some tolerance against the unmatched uncertainties. The transformation matrix was chosen as 0 01 0 01:5
T which gives
2 = D
= 10 0
0 0 0:1176
0 0 0
0 1 0
0 0 1
together with
P2 and
01 0 0 and 4 = 0 01 0 0
Fig. 4. Error signals for FTL, FSM, and DLSM observer, respectively, applied to the fuzzy system with matched uncertainty.
0
0
4:619 4:429E 0 4 0:1018 = 4:429E 0 4 4:404 03:159E 0 5 0:1018 03:159E 0 5 4:403
= 3 as in the design for the DLSM observer.
D. Simulation Results
:
The LMI control toolbox for MATLAB [24] was used for solving the LMIs in the design of this observer. An LMI region was chosen to be in the intersection of the strip [01; 03] and the cone with an inner angle of 80. Maximizing the robustness measure by solving an EVP gave the robustness measure FSM = 0:77 and gain matrices (transformed to original coordinates) 5:440 3:492E 0 4 01:665 0 6 :517 5:679 04:587 L1 = 01:331E 0 4 2:773 0:1 1:794 05:605E 0 6 2:775 5:428 3:583E 0 4 01:663 0 3 :507 5:787E 0 4 04:583 L2 = 01:297E 0 4 2:773 0:1 0:7791 05:510E 0 6 2:775
Three experiments were performed, all with the following initial conditions: x(0) = [0:5; 0; 0:1; 0]T and x ^(0) = [0; 0; 0; 0]T . In the first experiment the observers are applied to the fuzzy system without matched uncertainties (see Fig. 3). As expected, all three observers are able to reconstruct the states. The second experiment incorporates matched uncertainties in the fuzzy model of the form B1 3 sin(2t); the result is shown in Fig. 4. It can be seen that the FTL observer has some difficulty tracking the state accurately, although it is not unstable. In the third experiment, the observers are applied to the original nonlinear model as shown in Fig. 5. The observers are based on the fuzzy approximation, and the approximation errors give rise to both matched and unmatched uncertainties and all observers have some difficulty to track the states accurately. The FSM observer is a little bit “calmer” than the FTL observer. We further notice that the DLSM observer gives the best tracking. This may be explained by looking at the computation of the switching terms in the FSM and DLSM cases. In the FSM case,
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 32, NO. 1, FEBRUARY 2002
Fig. 5. Error signals for FTL, FSM, and LSM observer, respectively, applied to the original system.
2T is involved in the computation. D 2T has in this example the matrix D a fairly large null space which eliminates some information from ey . In the case of DLSM all information from ey is used in the computation of the observer’s estimate. This shows that the DLSM observer very well may be used for weakly nonlinear systems. Note also that the chattering that may occur in the states of a sliding mode observer can be smoothed out by defining such that 0 as ey 0.
!
!
V. CONCLUSION This paper essentially presented two types of fuzzy sliding mode observers. A Thau-Luenberger fuzzy observer was extended to a sliding mode fuzzy observer which is able to reject matched uncertainty effectively. This type of observer can be effectively designed using LMI tools. The second observer presented is based on the assumption that a multiple model system can be represented by a selected subsystem that is dominant linear over the other subsystems. Explicit design procedure was also given for that observer. The performance of the observers was also illustrated on model of cart with a pendulum. One topic for future research may be in the use of parameter dependent Lyapunov functions for proving stability. Another topic dealing with the fuzzy sliding mode observer may be to consider a varying disturbance distribution matrix.
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