Robust neuro-fuzzy control of multivariable systems by tuning
consequent membership functions
Wei-Song Lin
a;∗, Chih-Hsin Tsai
a, Jing-Sin Liu
baDepartment of Electrical Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Taipei 106, Taiwan, ROC bInstitute of Information Science, Academia Sinica, Nankang, Taipei, Taiwan 115, ROC
Received 10 August 1998; received in revised form 8 October 1999; accepted 6 April 2000
Abstract
A robust neuro-fuzzy controller with tuning mechanism of membership functions and neural weights to achieve the tracking control ofcomposite multivariable systems is proposed. The control strategy is developed to facilitate robust property by self-tuning the consequent membership functions of the fuzzy controllers. By an on-line tuning mechanism, the fuzzy system can e5ectively deal with the equivalent uncertainties that may appear in the subsystems due to plant uncertainty, function approximation error, or external disturbance. By using Lyapunov stability theory, the overall system with the proposed controller has been proved to be uniform ultimate bounded. Simulation results of a two-link robot control demonstrate the e5ectiveness and robustness ofthe design. c 2001 Elsevier Science B.V. All rights reserved.
Keywords: Neuro-fuzzy control; Consequence membership function; Multivariable systems
1. Introduction
The fuzzy control method has been demonstrated to have advantage of robustness through industrial appli-cations [9,11] and theoretical analysis [4,6,19]. In [4], a robustness measurement, which gives the bound on allowable uncertainties or nonlinearity, and robust stability offuzzy control systems have been studied through the Popov–Lyapunov approach. However, the control system to be analyzed should be able to transform into a perturbed Lur’e system. Yi and Chung [19] presented control theoretic analysis ofa fuzzy control system in the sense ofLyapunov based on the similarity between prevalent fuzzy logic controllers and the variable structure controller. In [6], Johansen used fuzzy sets and fuzzy inference to construct a nonlinear model of plant and provided an analysis ofstability, robustness and performance ofthe control loop. The controller is a discrete-time nonlinear decoupler but nonfuzzy model based.
∗Corresponding author. Tel.: +886-2-2363-5251 ext. 413; fax: +886-2-2363-8247.
E-mail address: [email protected] (W.-S. Lin).
0165-0114/01/$ - see front matter c 2001 Elsevier Science B.V. All rights reserved. PII: S 0165-0114(00)00119-6
Recently, using the fuzzy basis function expansion (FBFE) to represent the unknown nonlinearity of plants, several researchers have proposed adaptive fuzzy control methods [7,13,17,20]. These methods take the ad-vantages of fuzzy basis function and stable parameter adaptation schemes and are derived by using Lyapunov theory. However, as indicated in [8,16], the adaptive fuzzy controllers which will change their parameters of membership function of fuzzy rules used in fuzzy inference may result in ine5ectualness of the robust property of fuzzy control. In [2], by adding a robust stability factor to the controller, an adaptive law is obtained to be robust with respect to the modeling error resulting from the fuzzy approximators. Here we propose a robust neuro-fuzzy controller with self-tuning on the consequent membership function to encounter the equivalent uncertainty resulted by function approximation error, external disturbances, and measurement noise. But no additional robust stability factor as in [2] is added in the design since it will make the entire controller not being in a context offuzzy logic control system. The robustness ofthe neuro-fuzzy control system against system uncertainties is analyzed; this analysis gives an account ofthe relationship between control performance and the design parameters of the neuro-fuzzy controller, which was previously obscure in the theory offuzzy=neural control. Comparing with the results in [12,14,17], our e5ort has been on the extension to composite multivariable systems and the robust parameter adaptation schemes. Two weight adap-tation schemes are proposed and compared. One is the gradient weight scheme that is widely used in adaptive fuzzy=neural control law [10,13,17]. Another is a robust adaptation scheme with self-tuning on the consequent membership functions of the fuzzy part. The result of the former approach can only be proved when the parameters are initialized not too far from their optimal values (local stability) [10]. The latter we proposed is shown to release the above assumptions and be able to considerably reduce the tracking error residual set and obtain robustness in the sense that the self-tuning mechanism can automatically adapt the controller to react to the e5ect ofequivalent uncertainties for unknown plant dynamics. The overall system with the adaptation schemes has been proved to be able to guarantee uniform ultimate bounded.
The remainder ofthis paper is organized as follows: The output-tracking problem ofcomposite multivariable nonlinear systems is formulated in Section 2. The robust neuro-fuzzy controller for multivariable systems is proposed and formulated in Section 3. The gradient adaptation and the robust adaptation schemes are proposed and compared in Section 4. In Section 5, the control ofa two-link robot carrying a heavy load is simulated to illustrate the e5ectiveness and robustness ofthe proposed control system. Section 6 is the conclusion.
2. Problem formulation
Consider a composite multivariable nonlinear plant governed by
y(r)= f (x) + G(x)u + d(x; t); (1)
where y = [y1; : : : ; ym]T and y(r)≡ [y1(r1); : : : ; y(rmm)]T denote the output vector and its derivative, respectively,
r = [r1; : : : ; rm] with mi=1ri = n is deHned as the system relative degree, u = [u1; : : : ; um]T is the system
in-put, x = [x1; : : : ; xn]T= [y1; : : : ; y(r11−1); : : : ; ym; : : : ; ym(rm−1)]T is the state vector, f (x) = [f1(x); : : : ; fm(x)]T; G(x)
= [+1(x); : : : ; +m(x)]; fi(x) and +i(x) = [gi1(x); : : : ; gim(x)]T are unknown smooth functions satisfying the mild
assumption |gij(x)|¿0; ∀x; and d(x; t) = [d1(x; t); : : : ; dm(x; t)] is the disturbance with the properties
ofstan-dard smoothness and boundedness. Then given a desired trajectory yd(t), the purpose ofdesign is to Hnd
a controller for the composite multivariable plant described by (1) so that the tracking error represented by
e = [eT
1; : : : ; eTm]T with ei= [yi− yid; ˙y − ˙yid; : : : ; yi(ri−1)− y(ridi−1)]T will be attenuated to an arbitrarily small
Fig. 1. ConHguration ofa robust neuro-fuzzy control system. Fig. 2. The synaptic connection topology ofthe ith multi-layer fuzzy system.
3. Design of the robust neuro-fuzzy controller
The control ofan MIMO nonlinear system poses diKculties mainly in three aspects. Firstly, the interac-tions among subsystems usually cause the input applied to one subsystem undesirably a5ecting some other subsystems. Secondly, the functions f and G or parameters ofthe system are being unknown or diKcult to measure. The Hnal one is the presence of equivalent uncertainties which resulted from the function approxi-mation error and the disturbances. To overcome the above diKculties and shrink the tracking error residual set, the proposed robust neuro-fuzzy controller is composed of the following three parts: a multi-layer fuzzy system with rule credit assignment, a self-tuning mechanism on the consequent membership functions, and a decoupling network. The conHguration ofthe robust neuro-fuzzy control system in the case ofcontrolling two-input=two-output nonlinear system is shown in Fig. 1. The multi-layer fuzzy system and the decoupling network are nominal designs based on an on-line approximation ofthe unknown nonlinear functions ofthe plant. The self-tuning mechanism is designed to encounter the equivalent uncertainty which resulted from the plant uncertainty, the function approximation error, or the external disturbances.
3.1. The multi-layer fuzzy system
Fig. 2 shows the proposed synaptic connection topology ofthe ith multi-layer fuzzy system for the ith subsystem ofthe controlled plant. Considering the request ofnumerical input and output ofthe fuzzy sys-tem, a particular class of fuzzy system with the singleton fuzziHed, algebraic product T-norm, the sup star compositional operator [17] and the local mean-of-maximum method [1] are used. The basic components of the multi-layer fuzzy system and its self-tuning mechanism will be discussed and formulated in the following paragraph.
Fuzzy rule base: A multivariable system can be controlled by the following linguistic rules
Rj: IF x
1 is Aj1 AND · · · AND xn is Ajn
THEN u1 is B1j· · · um is Bjm; j = 1; : : : ; N + 1;
where N + 1 is the number offuzzy rules, the antecedent part, Ajk, is deHned as the following Gaussian type:
and the consequent membership function of the consequent part is deHned as Bji(ui) = (1 + ((cij− ui)=aLi)2)−1 if ui6cji; (1 + ((ui− cji)=aRi)2)−1 if ui¿ cji; (3)
where {akj; mjk} and {aLi; aRi; cij} are referred to the premise and consequence parameters, respectively.
Rule credit assignment: The basic idea ofthe rule credit assignment is to reward good rules by increasing
the conHdence of the consequent fuzzy sets and the recommendation fuzzy output of this rule. Denote !iij¿1
(or !iij¡1) as a reward (or a punishment) o5ered to the jth rule in the ith knowledge rule base, then the
consequent membership function (3) can be reshaped into ˜Bj i(ui) = (1 + (!jii(cji − ui)=aLi)2)−1 if ui6cji; (1 + (!jii(ui− cji)=aRi)2)−1 if ui¿ cji (4) and the recommendation fuzzy output of each rule is determined in singleton form as follows:
!jiiI(Aj(x0); ˜Bj i(ui)) = !jiiAj(x0) f or u i= ˜cji; 0 otherwise; (5) where x0= (x0
1; : : : ; xn0); Aj(x0) = A1j(x10)A2j(x02) · · · Anj(x0n) denotes the given input and the matching degree,
respectively, ˜cij denotes the location ofthe singleton implication fuzzy set and is deHned as (see Fig. 5 [1])
˜cij= the centroid of the set {ui: ˜Bji(ui)¿Aj(x0)}: (6)
Using (4) and (6) can be resolved into (see Appendix A)
˜cji = cij−aLRi !jii 1 Aj − 1 (7) where aLRi= (aLi− aRi)=2.
The study ofassigning rule credit assignment may be complicated, where, the modiHcation ofcontrol rules is achieved by giving a credit or reward value to individual rules engaged in the problem solving process. In [15,18], the credit value is obtained from a fuzzy algorithm which deHnes the desired performance linguisti-cally. However, as addressed in [3,21], the control rules may often be improperly modiHed when the set-point changes. Generally, for a fuzzy=neural system, these parameters are updated according to the output value and the associated teacher signals. But for the control problems under consideration, the teacher signals are not available and only the error information between the plant and the desired trajectory can be used. Therefore, in this paper, the entire problem is approached in the context ofLyapunov-based adaptive systems theory to
provide on-line tuning rules for !iij as shown in the next section.
Self-tuning mechanism: Physically, the parameter aLRi represents the left–right spread di5erence of the
consequent membership functions. In traditional fuzzy logic control system, aLi is set to be equivalent to aRi
or the consequent membership is just in singleton form [13]. In this paper, this term is employed as a robust control component and a robust adaptive law for it is proposed in the next section.
Analytical formulation of the multi-layer fuzzy system: Using the center average defuzziHcation, the output response ofthe fuzzy controller is
u0i = Fi(Aj; !jii) = N+1 j=1 !jiiAj˜cji N+1 j=1 !jiiAj : (8)
In the rule base, the (N + 1)th rule is chosen to be ofTakagi–Sugeno type and its consequent membership
function BN+1
i is singleton with support represented as the form of the synthesis input
c
i= ydi(ri)− $Tiei (9)
where i= [$i1; : : : ; $iri]T being positive constants chosen such that pn+$iripri−1+· · ·+$i1is a stable (Hurwitz)
polynomial. The curvature control parameter ofits antecedent membership function, aN+1
k , is assumed to
approach to inHnity so that this rule will be Hred whatever x0 is. The credit assignment takes place in rules
Rj; j = 1; : : : ; N but assigned to be 1 for RN+1. Accordingly, using (7) and (9), the analytical formulation of
the multi-layer fuzzy system in Eq. (8) resolves into
u0 = ˆD−1(− ˆf + c− aLRT fˆLR); (10)
where ˆD = Block diag[!T
11ˆ+!; : : : ; !Tmmˆ+!]; ˆf = [T1 ˆf&; : : : ; Tm ˆf&]T with !ii and ˆ+! are (N + 1) × 1 column
vectors composed of !iij and Aj, respectively,
i and ˆf& are N × 1 column vectors composed of !iijcij and
− Aj, respectively, c= [c 1; : : : ; cm]T; aLR= [aLR1; : : : ; aLRm]T and ˆfLR= N j=1 Aj 1=Aj− 1.
3.2. The decoupling neural network and the overall control law
Since the multi-layer fuzzy system does not take the interconnection among subsystems into consider-ation, the decoupling network is required to eliminate the interaction. The construction ofthe decoupling
neural network is conceptually explained as below. Let D(x) = Block diag[g11; : : : ; gmm] and assuming that
the subsystems are not interconnected (i.e., gij= 0; ∀i = j), and the nonlinear function f (x) ofthe system is
known exactly, then the desired control input u∗
0 for each individual subsystem without disturbance can be
theoretically computed from
u∗
0 = D−1(− f + c); (11)
where i; i = 1; : : : ; m are chosen to obtain exponential decay ofthe tracking error. When the subsystems
are interconnected (i.e. gij = 0), to ensure e → 0 as t → ∞, the desired control input needs to counteract the
interactions as u∗= G−1(− f + c) = (D + C)−1(− f + c) (12) or using (11) to obtain u∗= (D + C)−1Du∗ 0 = u∗ 0+ Mu∗0; (13)
where C = G − D and M = − [Im+ C−1D]−1 with Im denoting an m × m identity matrix.
When the system is disturbed and has unmodeled dynamics, (13) cannot be practically applied. Accordingly this paper proposes the robust neuro-fuzzy controller that uses the multi-layer fuzzy controller (10) as an alternative of(11) and invokes a decoupling neural network to counteract the interaction by learning as
follows:
u = u0+ ˆMu0; (14)
where the matrix ˆM is chosen as
ˆ M = − (Im+ ˆC−1D)ˆ −1 (15) and ˆ C = 0 !T 12ˆ+! · · · !T1mˆ+! !T 21ˆ+! 0 · · · !T2mˆ+! ... ... ... ... !T m1ˆ+! !Tm2ˆ+! · · · 0 (16)
Fig. 1 illustrates the concept ofthe robust neuro-fuzzy controller with the decoupling neural network. Using (10), (14), (15) and the matrix inversion lemma [5]
(A + BCD)−1= A−1− A−1B(DA−1B + C−1)−1DA−1 (17)
the robust neuro-fuzzy controller resolves into
u = (Im− (Im+ ˆC−1D)ˆ −1) ˆD−1(− ˆf + c− aTLRfˆLR)
= ˆG−1(− ˆf + c− aT
LRfˆLR); (18)
where ˆG = ˆC + ˆD:
4. Learning algorithms and performance analysis
Let i= [iT; !Ti1; : : : ; !Tim]T being bounded by M&i= {i: |i|6i; Max}, and deHne the parameters ofthe best
function approximation to be
∗
i ≡ arg min&
i∈M&i[sup |fi− T
i ˆf&|];
!∗
ij≡ arg min!ij∈M
&i[sup |gij− !
T
ijˆ+!|]: (19)
Applying (18) to (1), then subtractingmj=1!T
ijˆ+!uj and adding − iTˆf&+ ci− aTLRifˆLR to the right-hand side,
the ith component is obtained as y(ri) i = ydi(ri)− Tiei+ (∗Ti − Ti ) ˆf&+ m j=1 (!∗T ij − !Tij) ˆ+!uj+ *i− aLRifˆLR (20) or ˙ei= Aiei− bi˜Tiw + bi(*i− aLRifˆLR); (21)
where *i= (fi− ∗Ti ˆf&) +mj=1(+ij− !∗Tij ˆ+)uj+ di, ˜i= i− ∗i denotes the parameter estimation error, and Ai= 0 1 0 · · · 0 0 0 1 · · · 0 ... ... ... ... ... 0 0 0 · · · 1 − $i1 − $i2 − $i3 · · · − $iri ; bi= 0 0 ... 0 1 ; w = ˆf& ˆ+!u1 ... ˆ+!um : (22)
In the following paragraph, a gradient weight adaptation scheme which shutdown the self-tuning mechanism
(i.e. aLRi= 0) and a robust weight adaptation scheme which applies the self-tuning mechanism (i.e., aLRi = 0)
are proposed and compared.
Gradient weight adaptation scheme: For aLRi= 0, the tracking error represented by (21) allows us to use
the parameter adaptation law in [10,13]
˙i=
0 if eTPbbTPe6d2
0;
(I − d&ii⊥Ti⊥)Ri−1bTiPieiw otherwise (23)
with d&i=
0 if Ti⊥(R−1
i bTiPieiw)60;
min[1; dist(i; M&i)=-&] otherwise;
(24)
where Ri is a diagonal matrix with positive diagonal elements, b = Block diag[b1; : : : ; bm], P =
Block diag[P1; : : : ; Pm], and Pi= PiT is the solution ofthe following Riccati-like equation:
PiAi+ ATiPi+%12PibibTiPi+ Qi= 0 (25)
with the design parameters Qi¿ 0 and 0 ¿ 0; M&i- denotes the union of M&i and its boundary layer with
thickness -&, the preHx @ denotes the boundary, and i⊥= i=|i| is the unit normal vector. The adaptation
is turned-o5 when the tracking error is smaller than some threshold. The deadzone is to stop updating the parameters when the excitation is insuKcient to distinguish between the regression signal and the noise. By referring to the result in [10], this adaptive law has the following property:
Theorem 1. Let * = [*1; : : : ; *m]T; Q = Block diag[Q1; : : : ; Qm] and assume that there exists Q* = Supx; t *i 2;
and ei ¡21; ˜i ¡22; 21 and 22 are small enough; the adaptive law (23) guarantees
(1) i and the control input u are bounded.
(2) e converges to the residual set {e: eTQe602Q* or eTPbbTPe6d2
0}. Moreover; in the special case that
* 6(1=202)d0; e converges to the dead-zone {e: eTPbbTPe6d2 0}.
Robust weight adaptation scheme: To counteract the equivalent uncertainty, the self-tuning mechanism
aLRifˆLR is employed. The parameter aLRi is chosen as aLRi(#i) = #i tanh(bTiPieifˆLR=-) where #i is an auxiliary
adjustable parameter and - is a small positive constant. Using the following assumption:
Assumption 1. There exists the smallest non-negative parameter values #∗
i¿0 such that for all x ∈ Rn and
t ∈ R+
And let M#i= {#i: |#i|¡#i; max} be the bound of #i, M#-i be the union of M#i and its boundary layer of
thickness -#. We propose the following smooth robust weight adaptation scheme:
˙i=
0 if eTPbbTPe6d2
0;
(I − d&ii⊥Ti⊥)R−1i [wbTiPiei− 41(i− i0)] otherwise
(27) with d&i = 0 if T i⊥[wbTiPiei− 41(i− i0)]60;
min[1; dist(i; M&i)=-&] otherwise
(28) and ˙#i= 0 if eTPbbTPe6d2 0; (1 − d#i)r#−1i [w ibTiPiei− 42(#i− #i0)] otherwise (29) with d#i = 0 if #i[wibTiPiei− 42(#i− #i0)]60;
min[1; dist(#i; M#i)=-#] otherwise:
(30) w i = ˆfLRtanh bT iPieifˆLR - ; (31)
where Ri is a diagonal matrix with positive diagonal elements, Pi is a symmetric positive-deHnite matrix
satisfying the Lyapunov equation AT
iPi+ PiAi= − Qi, with the design parameters Qi¿0, and 41 and 42 are
chosen small but positive constant to keep i and #i from growing unbounded.
Theorem 2. Consider the nonlinear composite system (1) with controller (18); the parameter adaptation schemes (27) and (29) operating in the bounded state x ∈ 6. Then
(1) i; #i and the control input u are uniformly ultimately bounded.
(2) Given any 0 satisfying 0∗¡0 where
0∗=
m
i=1[41(∗i − i0)T(∗i − i0) + 42(#∗i − #i0)2+ 27#Mi -]
minimin{8min(Qi)=8max(Pi); 41=8max(Ri); 42=9#} (32)
with #M
i ≡ max{#i∗; #i0} and 7 being a constant that satis<es 7 = e−(7+1); i.e.; 7 = 0:2785; there exists
T such that for T6t6∞ the tracking error e converges to the residual set
{e: eTPe60 or eTPbbTPe6d2
0}: (33)
Proof. Let V& and V# be positive-deHnite functions of the forms V&=12mi=1(iTi); V#=12mi=1#2. Their time
derivative are ˙V&=mi=1iT˙&i and ˙V#=mi=1#iT˙#i, respectively. Ifthe Hrst line of(28) is true then d&i= 0,
and the conclusion ˙V&60 is trivial. Ifthe second line of(28) is true then d&i¡1 and i∈ M&-i (but i =∈ @M&-i).
Therefore, either ˙V&60 or i∈ M&-i is obtained. Similarly we have either ˙V#¡0 or #i∈ M#-i. Therefore, the
by (1), (18), (27), and (29), we choose the following positive-deHnite functions: V = V1+ · · · + Vm; (34) where Vi= 1 2d20+12˜ T iRi˜i+12r#i˜# 2 i if eTPbbTPe6d20; 1 2eTiPiei+12˜TiRi˜i+12r#i˜# 2 i otherwise; (35)
˜#i= #i− #iM are the auxiliary adjustable parameter error and #iM≡ max{#∗i; #i0}. Taking the derivative of Vi
along the trajectories ofthe closed-loop system and taking (21), (27), and (29) into account we obtain: ˙Vi= 0
for eTPbbTPe6d2
0, and
˙Vi= eiTPi(Aiei− bi˜Tiw + bi(*i− aLRifˆLR)) + ˜
T
i(I − d&i⊥Ti⊥)[wbTiPiei
− 41(i− i0)] + ˜#i(1 − d#)[wibTiPiei− 42(#i− #i0)]
=1
2eTi(ATiPi+ PiAi)ei− eiTPibi˜Tiw + eiTPibi(*i− #iwi)
+ ˜TiwbT
iPiei− 41˜Ti(i− i0) + ˜#iwibTiPiei− 42˜#i(#i− #i0)
− d&˜Tii⊥Ti⊥[wbTiPiei− 41(i− i0)] − d#˜#i[wibTiPiei− 42(#i− #i0)] (36)
for eTPbbTPe¿d2
0. By (28), if i⊥T [wibTiPiei− 41(i− i0)]60, we have d&i= 0 and the last term ofthe above
equation is equal to zero. When T
i⊥[wibTiPiei−41(i−i0)]¿0, if i∈ M&i we also have d&i= 0 and the above
conclusion holds. If i =∈ M&i and suppose that M&i and M#i are appropriately selected such that i∗ and #∗i are
in the interior of M&i and M#i; respectively, we obtain
˜T ii⊥ = (i− ∗i)Ti=|i| = 1 2[(i− ∗i)T(i− ∗i) + Tii− ∗Ti ∗2i ]=|i| ¿ 0 (37) or ˜T ii⊥Ti⊥[wbTiPiei− 41(i− i0)]¿0: (38)
In a similar way, it can be shown that
˜#i[wibTiPiei− 42(#i− #i0)]¿0: (39)
Therefore,
˙Vi612eTi(AiTPi+ PiAi)ei+ eiTPibi(*i− #Mwi) − 41˜Ti(i− i0)
−42˜#i(#i− #i0): (40)
Using Assumption 1, the second term on the right-hand side satisHes the inequality
eT
iPibi(*i− #Mwi) 6 |eTiPibi|#∗ifˆLR− eTiPibi#Mwi
6 #M
= #M i |eT iPibifˆLR| − eiTPibi fˆLRtanh eT iPibifˆLR - 6 #M i 7-: (41)
Since the following fact can be shown easily by straightforward algebraic manipulation. Claim 1.
06|r| − r tanhr
-
67- (42)
for any 9 ∈ R. Furthermore, it can be readily shown that ˜T i(i− i0) = 12˜iT˜i+12(i− i0)T(i− i0) −12(∗i − i0)T(∗i − i0); ˜#i(#i− #i0) = 12˜# 2 i +12(#i− #i0)2−12(#∗i − #i0)2: (43) Therefore, ˙Vi6 −12eTi(Qi)ei−421 ˜Ti ˜i−422 ˜#2i +421(∗i − i0)T(∗i − i0) +422(#∗i − #i0)2+ 7#Mi -6 − aiVi+ 8i; (44) where ai ≡ min 8min(Qi) 8max(Pi); 41 8max(Ri); 42 9#i and 8i =421(∗i − i0)T(∗i − i0) +422(#i− #i0)2+ #Mi 7-or ˙ V6 − aV + 8 (45)
where a = miniai and 8 =mi=18i. The di5erential inequality (45) satisHes
06V(t)68 a + V(0) −8 a e−at: (46)
Therefore ei; i; #i are uniformly ultimately bounded. Let 0∗= 28=a then from (46) we readily obtain (33).
Remark 1. The gradient weight adaptation scheme in [10] requires the assumption that the tracking error and weight errors are initially bounded and suKciently small. In our construction ofthe robust weight adaptation
scheme by using the self-tuning mechanism, aLRifˆLR, the assumption ofsmall initial weight errors is released.
Remark 2. From (32), ifthe design constants -; 41; 42; 9#; Qi; Pi and Ri are appropriately chosen, whether
i0 and #i0 are close to ∗i and #∗i or not, it is possible to make 0∗ as small as desired and therefore better
5. Performance comparisons 5.1. Simulation setup
A two-link robot is simulated to compare the robustness ofthe gradient weight adaptation scheme and the proposed robust weight adaptation scheme.
The equations ofmotion ofthe arm can be expressed in the matrix form as follows: (m1+ m2)r12+ m2r22+ 2m2r1r2c2+ J1 m2r22+ m2r1r2c2 m2r22+ m2r1r2c2 m2r22+ J2 Rq1 Rq2 + − m2r1r2s2˙q1( ˙q1+ ˙q2) m2r1r2s2˙q22 + ((m1+ m2)l1c2+ m2l2c12)g (m2l2c12)g = u1 u2 + d1 d2 ; (47)
where m1; m2; J1; J2; r1= 0:5l1, and r2= 0:5l2 are the mass, the moment ofinertia, the halflength oflink 1
and 2, and c1 ≡ cos(q1), s12 ≡ sin(q1+ q2), etc. The combined e5ects offriction and the external torque
disturbance are
d1= 2:0 sin( ˙q1) + 2:5 sin( ˙q2) + 0:5 sin(t);
d2= 5:0 sin( ˙q1) + 4:0 sin( ˙q2) + 0:4 sin(t): (48)
In the control experiments described below, the kinematics and inertial parameters ofthe arm are chosen as
l1= 2:04 m; l2= 1:66 m; J1= J2= 4:5 kg m; m1= 0:60 kg; m2= 7:02 kg, respectively. The excessive ratio
between m1 and m2 is to emphasize the load e5ect. The robot is given the following target joint rotations:
q1d = =12 sin(0:5t)
q2d = 2:5=12 cos(0:5t) + 2:5=24 cos(0:5t) (49)
with the initial states q1(0) = 1:5 rad; q2(0) = − 1:2 rad; ˙q1(0) = 0 rad=s; ˙q2(0) = 0 rad=s.
In (27) and (29), the design parameter are given by Q1= Q2= 10I2×2; R1= Block diag[0:01I81×81;
32 000I81×81; 20 000I81×81]; R2= Block diag[0:025I81×81; 20 000I81×81; 3200I81×81]; 0=0:01; r#1=r#2=
0:025; 41= 42= 0:002, and - = 0:005. The membership functions of state q1; ˙q1; q2, and ˙q2 (represented by
generic variable xk) f or M = 34= 81 regular rule partitions are deHned as {NB; ZE; PB} where NB: Ajk(xk) =
exp(− 4(xk+ 1:8)2), ZE: Ajk(xk) = exp(− 4xk2), and PB: Ajk(xk) = exp(− 4(xk− 1:8)2).
For the purpose ofcomparison, computer simulations are also carried out using the proposed controller with and without partial knowledge about the robot, i.e., rough mathematical model (but unknown disturbances)
and nominal parameters l0
1= 2:0 m; l02= 1:6 m; J10= 4:8 kg; J20= 5:1 kg m; m01= 0:48 kg; m02= 6:30 kg:
In the case that the nominal robot parameters are known a priori, through the training data {x(k)}, the initial
parameters
i and !ij are chosen based on the element-by-element minimization ofthe following objective
function:
k
|f0
i(x(k)| nominal robot parameters) − iTˆf&|2;
k
| g0
ij(x(k)|nominal robot parameters) − !Tijˆ+!|2:
We choose 32 testing points either the sampled points along the desired trajectories or points near them,
for the training data x(k). Ifthere are no nominal robot parameters, the elements in
i and !ij are chosen
5.2. Results and discussion
Figs. 3 and 4 show the tracking performance using (a) the gradient weight adaptive scheme and (b) the robust weight adaptive scheme with and without nominal robot parameters. The solid and dashed lines correspond to the desired and controlled robot-arm angle trajectories, respectively. In comparing the set of response (a) with (b) in Figs. 3 and 4, one can conclude that the tracking behavior is much better in the robust weight adaptive scheme case. While the gradient weight adaptive scheme reaches a large error residual set, the error in the robust weight adaptive scheme continues to decay around zero. This is exactly the di5erence
between these two schemes. The gradient weight adaptive scheme guarantees that the quadratic error eTQe
will be smaller than a certain bound, 02Q*, however, Q* varies with the control e5ort u and the disturbance
d. To reduce the error, one has to reduce the design parameter 0 which will cause larger input e5ort and then yield larger Q*. It seems that there is some limitation on reducing the error residual set. Since the error
residual set 0∗ does not directly depend on the magnitude ofcontrol e5ort (see Eq. (33)), the robust adaptive
scheme requires the design constants -; 41; 42; r#; Qi; Pi and Ri to be just appropriately chosen such that 0∗
is reduced. This is the reason why a smaller tracking error residual set can be achieved. On the other hand, in comparing Fig. 3(a) with Fig. 4(a), it is clear and expected that the tracking performance is much better
in the case that
i and !ij are selected in advance according to nominal robot parameters. Nevertheless, as
shown in Fig. 3(b) and Fig. 4(b) signiHcant improvements of system tracking performance are achieved after applying the robust weight adaptation scheme even without a priori knowledge ofthe nominal parameters of the robot.
6. Conclusion
A novel robust neuro-fuzzy controller has been developed successfully for the tracking control of compos-ite multivariable systems with uncertainties. The controller uses a decoupling neural network to counteract the interaction among the interconnected subsystems. The system robustness against the e5ect ofequivalent uncertainties for unknown plant dynamics and disturbance is mainly obtained by self-tuning the consequent membership functions of the fuzzy part. A robust weight adaptation scheme with self-tuning on the con-sequent membership functions has been derived and shown to be able to considerably reduce the tracking error residual set by automatically adapting the controller to react to the uncertainties. The overall system has been proved to be uniform ultimate bounded. And the analysis gives an account of the relationship between control performance and the design parameters of the neuro-fuzzy controller, which was obscure previously in the theory offuzzy=neural control. Comparison ofthe proposed method with an adaptive neuro-fuzzy control system without self-tuning the consequent membership function has been carried out via theoretical analysis and simulations. The results show that the proposed method is superior in both tracking performance and robustness. The e5ectiveness ofthis design has been investigated and demonstrated by an example ofrobot control.
Acknowledgements
The authors would like to thank the anonymous referees for their valuable suggestions and comments. The Hnancial support for this research from the National Science Council of Taiwan, R.O.C. under NSC 88-2212-E-002-070 is gratefully acknowledged.
Fig. 3. Time responses ofthe robot control provided with the rough mathematical model and nominal parameters use (a) the gradient weight adaptive scheme, (b) the robust weight adaptive scheme.
Fig. 4. Time responses ofthe robot control without knowing the rough mathematical model and nominal parameters use (a) the gradient weight adaptive scheme, (b) the robust weight adaptive scheme.
Appendix A
The derivation of (7) is provided in this appendix. Referring to Fig. 5 and by (4), for the matching degree
Aj at the left intersection point uP we have
Aj= (1 + (!j
Fig. 5. The deHnition of˜cji, the centroid ofthe line segment ofheight Aj intercepted by the fuzzy membership function. or uP = cji−a!Lij ii 1 Aj − 1: (A.2)
Similarly at the right intersection point uQ, we have the following result:
uQ= cji+a!Rij
ii
1
Aj − 1: (A.3)
Then by deHnition (6), we can obtain (7) as
˜cji = (uP+ uQ)=2: (A.4)
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