www.elsevier.com/locate/fss
Self-organizing fuzzy control of multi-variable systems
using learning vector quantization network
Wei-Song Lin
∗, Chih-Hsin Tsai
Department of Electrical Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd, Taipei 106, Taiwan, ROC Received 22 July 1998; received in revised form 7 March 2000; accepted 6 April 2000
Abstract
Using learning vector quantization (LVQ) network to construct a self-organizing fuzzy controller (SOFC) for multi-variable nonlinear composite systems is developed in this paper. The LVQ network is used to provide information about the better locations of the IF-part membership functions through un-supervised learning. The generated fuzzy rule base is applied to the SOFC and updated by a self-learning procedure. Using Lyapunov stability methods, the proposed adaptive scheme is proven to provide the SOFC some degree of robust properties and guarantee uniform ultimate boundedness in the presence of disturbances, measurement noise and perturbed initialization error. The e9ectiveness of the proposed controller has been demonstrated numerically by applying to control a two-link manipulator. c 2001 Elsevier Science
B.V. All rights reserved.
Keywords: Fuzzy control; Self-organizing; Multi-variable; Nonlinear
1. Introduction
Fuzzy logic has emerged as a practical and successful alternative in the control of complex or ill-de>ned systems [17]. Conventionally, the fuzzy inference is based on rules constructed according to the experience of experts [2,14]. But there are situations, due to safety, complexity or beyond reach that the experts are not allowed to experience the detailed changes of the dynamic system to extract the fuzzy rules. Or sometimes the constructed fuzzy rule base can only support the experienced conditions other than new environment of the system. Procyk and Mamdani [12] addressed this problem by introducing a linguistic self-organizing controller, which is capable of generating and modifying the control protocol by a learning process based on performance measurement. Recently, many other researchers focused on combining the learning ability of neural network with fuzzy logic system to create=adapt the proper fuzzy rule base [3,4,7,9]. Lin [3] and Lee [4]
∗Corresponding author. Tel.: +886-2-3635251 ext. 413; fax: +886-2-3638247.
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)00117-2
proposed a reinforcement neural-network-based fuzzy logic control system, which not only performs the fuzzy inference operations but also extracts the fuzzy rules and membership functions from learning examples. Jang [7] presented an enhanced fuzzy controller with temporal backpropagation methodology, which can either re>ne or automatically derive the fuzzy if-then rules. Nie and Linkens [10,11] proposed a fuzzi>ed CMAC network and a fuzzi>ed RBF network to act as adaptive controllers with the feature of self-organizing association cells and the further ability of self-learning the required teacher signals in real-time. Despite these considerable achievements, the development of systematic approaches, which perform self-organization of control knowledge, is still far from being completely and satisfactorily resolved. Speci>cally, in the cases of multi-variable nonlinear control systems, few researchers discuss the stability of the closed-loop system and the convergence of the tracking error in the procedure of self-organization and self-learning.
In this paper, a self-organizing fuzzy controller (SOFC), capable of self-organizing its own structure and self-learning the required knowledge to control multi-variable nonlinear composite systems, is proposed. Design of the SOFC is a systematic approach that uses the learning vector quantization network (LVQ) [6,8] in the self-organizing level and incorporates a robust adaptation scheme in the task execution level. A LVQ network is trained to obtain the information about the better locations of the IF-part membership functions. With the generated fuzzy rule base, the proposed robust adaptive scheme is proven to provide the SOFC some degree of robust properties and guarantee uniform ultimate boundedness in the presence of disturbances, measurement noise and perturbed initialization error.
This paper is organized as follows. In Section 2, the SOFC system and its architecture is presented. The self-organizing level using LVQ network and the coordination level are proposed in Sections 3.1 and 3.2, respectively. In Section 3.3, the task execution level is described. The robust parameter adaptation scheme is proposed in Section 3.4. Also, in this section, the robust property and the convergence of output tracking error are analyzed. In Section 4, the performance of controlling a two-link robot arm carrying a heavy load is evaluated and demonstrated. Section 5 is the conclusion.
2. The self-organizing fuzzy control system
Consider a nonlinear composite plant, such as a robot, governed by
y(r)= f (x) + G(x)u + z(x; t);
Jx = x + nx;
Jy = y + ny;
(1)
where y = [y1; : : : ; ym]T, Jy; x = [x1; : : : ; xn]T= [y1; : : : ; y1(r1−1); : : : ; ym; : : : ; y(rmm−1)]T, and Jx(·) denote the output,
measured output, state and measured state vectors, respectively, y(r) ≡ [y(r1)
1 ; y2(r2); : : : ; ym(rm)]T, r = [r1; : : : ; rm]
and mi=1ri= n denote the plant relative degree, u = [u1; : : : ; um]T is the plant input, f = [f1; : : : ; fm]T, G =
Block diag[g1; : : : ; gm] and fi and gi are smooth functions, z(x; t) = [z1(x; t); : : : ; zm(x; t)]T denotes the
ag-gregation of unknown interaction and time-varying disturbance. The exogenous signals nx= [nx1; : : : ; nxn]T,
ny= [ny1; : : : ; nyn]T and z(x; t) are assumed to have the properties of standard smoothness and boundedness.
Additionally, the exogenous signals are assumed to satisfy nyi∈ Cri. Let yMi, and vi denote the reference
output and input, respectively. The aim of control is to make each subsystem of (1) asymptotically tracking a linear reference model of the following form:
y(ri)
Fig. 1. A robust multi-variable tracking control system using the SOFC.
Fig. 2. Diagrammatic representation of the hierarchical SOFC.
where the constants i1; : : : ; iri are selected so that (2) is asymptotically stable. Since the plant operates
repetitively in the presence of disturbance, measurement noise and perturbed initialization error, the ability of self-organization of the proposed controller is critical.
Fig. 1 shows the architecture of the proposed SOFC for robust tracking control of repetitive nonlinear plants described by (1). The learning process is comprised of a “training samples storage” that allows the controller to store measured state variable samples in each trial. The SOFC is a hierarchically intelligent control system [13]. Fig. 2 shows the conceptual structure of the hierarchical SOFC that is composed of three main levels of intelligence. The three levels are
(1) the organization level represents the brain of the system with functions dominated by a LVQ network to plan and make decisions about the fuzzy rule base;
(2) the coordination level is the interface between high and low levels of intelligence with functions coordi-nating the original rule base and the robust parameter adaptation law for the SOFC; and
(3) the execution level is the lowest level that requires high precision and has functions dominated by multi-layer fuzzy control system.
The details of each level will be described in the following sections.
3. Design of the self-organizing fuzzy controller
Since fuzzy controllers are usually realized by using digital input and output, a particular class of fuzzy systems with the singleton fuzzi>ed, algebraic product T-norm, the sup star compositional operator [15] and the local mean-of-maximum [1] method is considered.
3.1. Organization level 3.1.1. Fuzzy rule base
A multivariable system can be controlled by the following M + N + 1 linguistic rules
Rj: IF x
1 is Aj1 AND : : : AND xn is Ajn
THEN u1 is B1j AND : : : AND um is Bjm for j = 1; : : : ; M + N + 1;
where M denotes the number of regular rule partitions and N denotes the number of rule partitions positioned
near the trial trajectory. The fuzzy sets Ajk and Bij are linguistic terms characterized by the fuzzy membership
functions Aj k(xk) = exp(− (xk− m j k)2=ajk) (3) and Bj i(ui) = (1 + ((cij− ui)=aLi)2)−1; if ui6cij; (1 + ((ui− cij)=aRi)2)−1; if ui¿ cij; (4)
where {akj; mkj} and {aLi; aRi; cij} are referred to the premise and consequence parameters, respectively.
In most fuzzy logic control systems, the fuzzy rule space is partitioned into a number of domains of equivalent sizes, called regular rule partition as shown in Fig. 3(a), and a rule is stored in each domain. If higher tracking accuracy is desired, a partition consisting of small intervals is required. However, the problem of computational complexity will worsen as the dimensionality of the state space, n, increases. In this paper, only a few, M, fuzzy rules are constructed according to the regular rule partition to ensure that the ability to perform basic function approximation is presented in the complete space. In addition, the N rule partitions are positioned near the trial trajectory, as shown in Fig. 3(b), to achieve the desired tracking accuracy. This results in a signi>cant reduction in the number of fuzzy rules. In this section, a new scheme using a LVQ network is proposed to accomplish this task.
3.1.2. Locating of membership functions by LVQ network
Vector quantization is a classical method that produces an approximation to a continuous probability density
function p(x) of the vector input variable x using a >nite number of neurons with weights mj= [mj
1; m2j; : : : ;
mnj]T, j = 1; : : : ; N. One kind of optimal placement of the mj is to minimize E, the expected rth power of the
reconstruction error: E =
x − mj
Fig. 3. (a) Regular rule partition, (b) rule partitions depend on the trial trajectory.
where dx is the volume di9erential in the x space, and the index w = w(x) of the best-matching neuron vector (“winner”) is a function of the input vector x:
x − mj
w = minj {x − mj}: (6)
Once the LVQ network is trained, the neurons will be appropriately associated with their weights mj, and
each of which corresponds to the location of IF-part membership function as in (3). To implement this idea in
the sense of discrete-time notation, consider a controlled system with measured state variable samples { Jx(Ti)},
which are generated at every sampling time Ti. In the >rst operation, one can obtain a sequence of them by
the given reference model. For consecutive trials the state samples can be obtained from the “training samples
storage” which is generated by the response of the controlled plant. De>ning a neighborhood set Nw that
shrinks monotonically with time, the updating process may read
mj(k + 1) = mj(k) + (k)( Jx(k) − mj(k)); if j ∈ N w; mj(k); if j =∈ N w; (7)
with (k) being a suitable, monotonically decreasing sequence of scalar-valued “adaptation gain”, 0¡(k)¡1
and Jx(k) is the randomly selected sample from { Jx(T
i)}.
3.1.3. Providing samples with conditional weights
In order to achieve higher tracking accuracy, the rule space around the region of large tracking error should be partitioned with smaller intervals to obtain better function approximation. This can be done by LVQ
network if each sample, Jx(Ti), is provided with conditional weight that depend on the following performance
index
Ji= yM(Ti) − Jy(Ti)
2
iyM(Ti) − Jy(Ti)2; (8)
where yM(Ti)− Jy(Ti) denotes the tracking error at sampling time Ti. Then, through a roulette wheel with slots
weighted in proportion to the performance index Ji, the training samples { Jx(Ti)} are presented to the LVQ
network. The idea of roulette wheel is inspired by the reproduction process in the genetic algorithms [5]. 3.2. Coordination level
The intermediate structure, i.e. the coordination level, serves as an interface between the organization and execution levels. It dispatches an organized rule base, gives rule credit assignment, and presents robust parameter adaptation laws to the execution level. The parameter adaptation law is analyzed in Section 3.4.
3.2.1. Rule credit assignment
The basic idea of the rule credit assignment is to reward good rules by increasing the con>dence of the
consequent fuzzy sets and the recommended fuzzy output of this rule. Denote !ij¿1 (or !ij¡1) as a reward
(or a punishment) o9ered to the jth rule in the ith knowledge rule base, then the consequent membership function (4) can be reshaped into
˜Bj i(ui) = (1 + (!ij(cij− ui)=aLi)2)−1; if ui6cij; (1 + (!ij(ui− cij)=aRi)2)−1; if ui¿ cij; (9) and the recommended fuzzy output of each rule is determined in singleton form as follows,
!ij· I(j( Jx); ˜Bj i(ui)) = !ji· j( Jx); for u i= ˜cji; 0 otherwise; (10)
where “·” in the multiplication operation, I is the implication function [15], j( Jx) =
Aj
1( Jx1) · · · Ajn( Jxn) denotes
the matching degree, respectively, and ˜cji denotes the location of the singleton implication fuzzy set
de>ned as
˜cij= the centroid of the set {ui: ˜Bj
i(ui)¿
j( Jx)}: (11)
Using (9) and (11) can be resolved into ˜cij= cij− aLRij
(j)−1− 1=!j
i; (12)
where aLRi= (aLi− aRi)=2.
The study of assigning rule credits may be complicated, where the modi>cation of control rules is achieved by giving a credit or reward value to individual rules engaged in the problem solving process. Generally, for a fuzzy=neural system, these parameters are updated depending on its output value and the associated teacher signals. Since no direct error measurement of the neural=fuzzy system is possible in the considered control system, the teacher signals are not available and only the error information between the plant and the desired trajectory can be used. In this paper, our approach is to treat the entire problem in the context of Lyapunov-based adaptive systems theory. As shown in Section 3:4, following the context of adaptive control
technique, we provide on-line tuning rules for !ij.
3.3. Execution level
Using the center average defuzzi>cation, the output response of the fuzzy controller is
ui(t) = Fi( Jx; !ji; aLRi) = M+N+1 j=1 !ji · j· ˜cji M+N+1 j=1 !ji · j : (13)
In the rule base, the (M + N + 1)th rule is chosen to be of Takagi–Sugeno type and its consequent fuzzy set BM+N+1
i is singleton with support represented as the form of the synthesis input
c
i= i1Jyi+ i2˙Jyi+ · · · + iriJy(rii−1)+ vi: (14)
The curvature control parameter, aM+N+1
k , of its antecedent membership function is assumed to approach
in>nity so that this rule will be >red whatever Jx is. The credit assignment takes place in rules Rj; j = 1; : : : ;
M + N but assigned to be 1 for RM+N+1. Accordingly, using (12) and (14), the analytical formulation of the
multi-layer fuzzy system in Eq. (13) resolve into
u = ˆD−1(− (T+ c− a
where ˆD = Block diag(!T
1; : : : ; !Tm); !i and are (M + N + 1) × 1 column vectors composed of !ij and
j; (= [(
1; : : : ; (m] ∈ RN×m; (i and are (M + N) × 1 column vectors composed of !ijcij and j; c=
[c
1; : : : ; cm]T; aLR= [aLR1; : : : ; aLRm]T, and ) = M+N+1j=1 j(j)−1− 1.
3.4. Parameters adaptation and performance analysis
Let (i= [(Ti ; !Ti]T; M(i= {Vi(t): |Vi(t)|6Vi; Max} be the bounds of (i, and
(∗ i ≡ arg min( i∈M(i[sup |fi(x) − ( T i (x)|]: !∗
i ≡ arg min!i∈M
(i[sup |gi(x) − !
T
i(x)|];
be the best function approximation parameters. The adjustable parameter aLRi in (15) represents the
di9er-ence between the left and right spreads of the consequent membership functions. In the conventional fuzzy
logic control systems, aLi is set to be equivalent to aRi or the consequent membership is just in singleton
form [15]. In this paper, the SOFC is developed to facilitate robust property by tuning the parameter aLRi,
i.e. tuning the consequent membership functions of the fuzzy system. By this on-line tuning mechanism, the fuzzy system can e9ectively deal with the lump disturbances, measurement noise, and interconnection
compensation among subsystems that is discussed in greater detail in [16]. The parameter aLRi is chosen as
aLRi(#i) = #itanh(bTiPiei)( Jx)=.) where #i is an auxiliary parameter and . is a small positive constant.
Assumption 1. There exists the smallest non-negative parameter values #∗
i¿0 such that for all Jx ∈ Rn and
t ∈ R+
|/i|6#∗i)( Jx) (16)
where /i (see Appendix A) denotes the equivalent uncertainties which lump disturbance; measurement noise
and interconnection e<ect among subsystems together.
Let M#i= {#i: |#i|¡#i; max} be the bound of #i; M#.i (or M(i.) be the union of M#i (or M(i) and its
boundary layer of thickness .# (or .(); (i⊥= (i=|(i| be the unit normal vector, and the pre>x @ denotes the
boundary. Denote ei= [yi−yid; ˙y− ˙yid; : : : ; yi(ri−1)−yid(ri−1)]T and e = [e1T; : : : ; eTm]T. A robust adaptive algorithm
for (i and #i motivated by an attempt to provide treatment to the equivalent uncertainties are proposed as
follows
˙(i(t) =
0; if eTPbbTPe6d2
0;
(I − d(i(i⊥(Ti⊥)R−1i [bTiPieiw − 21((i− (i0)]; otherwise;
(17) with d(i = 0; if (T i⊥[bTiPieiw − 21((i− (i0)]60;
min[1; dist((i; M(i)=.(]; otherwise;
(18) and ˙#i(t) = 0; if eTPbbTPe6d2 0; (1 − d#i)r#i−1[wibTiPiei− 22(#i− #i0)]; otherwise; (19)
with
d#i =
0; if #i[bTiPieiwi− 22(# − #0)]60;
min[1; dist(#i; M#i)=.#]; otherwise;
(20) w i = ) tanh bT iPiei)( Jx) . ; (21)
where Pi is a symmetric positive-de>nite matrix satisfying the Lyapunov equation ATiPi+ PiAi= − Qi, with
the design parameters Qi¿0, and 21 and 22 are chosen small but positive constant to keep (i and #i from
growing unbounded.
Theorem 1. Consider the nonlinear composite system (1) with the SOFC (15); the parameter adaptation
schemes (17) and (19) operating in the bounded state x ∈ 4. Then
(1) (i; #i and the control input u are uniformly ultimately bounded.
(2) Given any 5 satisfying 5∗¡5; where
5∗=
m
i=1[21((∗i − (i0)T((∗i − (i0) + 22(#∗i − #i0)2+ 26#Mi .]
minimin{7min(Qi)=7max(Pi); 21=7max(Ri); 22=8#} (22)
with #M
i ≡ max{#∗i; #i0} and 6 being a constant that satis>es 6 = e−(6+1); i.e. 6 = 0:2785; there exists
T such that for T6t6∞ the tracking error e converges to the residual set {e: eTPe65 or eTPbbTPe6d2
0}: (23)
Proof. Refer to Appendix A for details.
Remark1. From (22), the tracking error residual is determined by the design parameter 5∗. If the design
constants .; 21; 22; 8#; Qi; Pi and Ri are appropriately chosen, it is possible to make 5∗ as small as desired
and therefore better tracking performance can be achieved.
4. Simulation
A two-link robot manipulator is simulated to show the e9ectiveness of the proposed SOFC. The equations of motion of the robot can be expressed in 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 ; (24)
where m1, m2, J1, J2, r1= 0:5l1, and r2= 0:5l2 are the mass, the moment of inertia, the half-length of link
1 and 2, and c1≡ cos(q1), s12≡ sin(q1+ q2), etc. The combined e9ects of friction and the external torque
disturbance are given by
d1= 2:0 sin( ˙q1) + 2:5 sin( ˙q2) + 0:5 sin(t);
Fig. 4. Reference outputs of joints 1 and 2.
In the control experiments described below, the kinematics and inertial parameters of the 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
be-tween m1 and m2 is to emphasize the load e9ect. As shown in Fig. 4, the trajectory to be
followed by the ith joint is given by two decoupled linear systems as
RqMi= i1qMi+ i2˙qMi+ vi; i = 1; 2: (26)
The model parameters are chosen as follows: 11= − 1:0, 12= − 1:0, 21= − 4:0, 22= − 2:0 and the
driving inputs to the reference model are sinusoidal functions v1= < sin(0:8<t), v2= 1:5< cos(<t). In (17)
and (19), the design parameters are given by Q1= Q2= 10I2×2, R1= Block diag [0:01I256×256, 20 000I256×256],
R2= Block diag [0:025I256×256, 20 000I256×256], r#1= r#2= 0:025, 21= 22= 0:002, and . = 0:005. The elements
in (
i and !i are chosen randomly within the interval (−10; 10) and (−2; 2), respectively. The membership
functions of state q1, ˙q1, q2, and ˙q2 (represented by generic variable xk) for M = 34= 81 regular rule
parti-tions are de>ned as {NB; ZE; PB} where NB: Aj
k(xk) = exp(− 4(xk+ 1:8) 2), ZE: Aj k(xk) = exp(− 4x 2 k), PB: Aj k(xk) = exp(− 4(xk− 1:8)
2). Meanwhile, N = 12, i.e. 12 rule partitions positioned near the trial trajectory
are de>ned as Aj
k(xk) = exp(− 4(xk−m
j
k)2). The locations of these membership functions, mj= [m1j; : : : ; mnj]T,
j = 1; : : : ; N, are self-organized through the LVQ network. The corresponding adaptation gain (k) and the
radius of Nw decreased linearly with time from 0.9 and 0.1 to zero, respectively. Two-hundred state variable
samples, Jx(Ti), i = 1; : : : ; 200 along the state trajectories were used in the training process. In the simulation,
for the purpose of demonstrating the adaptability and robustness of the SOFC, preceding systematic devel-opment is applied to the robot manipulator considering environments without and with initial state error and measurement noise. For both of the two environments, three trial trajectories are run and denoted by trial
A, B, C, and trial A, B, C, respectively. The results of the organization level for locating the membership
functions are shown in Fig. 5. The squares denote the locations of the IF-part membership functions, mj,
j = 1; : : : ; N. In Fig. 5, the solid lines of (b) denote the reference state trajectories, while the solid lines of
(c) and (c) denote the plant state trajectories.
Fig. 5(a) shows the locations of the membership functions, mj, before applying the self-organization
pro-cess and they were applied in trials A and A for cases 1 and 2 simulations; (b) shows that the LVQ
network is >rst trained to adjust mj base on the state trajectory of the reference model for trial B and B.
Fig. 5(c) and (c) shows that the LVQ network is trained again for trial C and C base on the plant state
trajectory generated by trial B and B, respectively. In Fig. 5(c) and (c), the training samples are provided
with conditional weights according to the performance index Ji. The dense partition phenomenon in the larger
Fig. 5. Locating membership functions by LVQ network.
Case 1: In Fig. 6, the simulation result of trials A; B, and C which are in the situation characterized by the same initial conditions on the reference model and the plant without measurement noise are presented.
The initial conditions are set to be q1(0) = − 1:5 rad; q2(0) = − 1:2 rad; ˙q1(0) = 0 rad=s; ˙q2(0) = 0 rad=s.
Case 2: In Fig. 7, the re-initialization error setting is satis>ed within an admissible deviation level, q1(0) −
Fig. 6. Tracking error of (a) joint 1, and (b) joint 2 without initial state error and measurement noise.
Jq2= q2+ 0:01 cos(10t) are applied. From the variation of the tracking error of the SOFC for di9erent trials,
we >nd that (i) For a trajectory tracking or model-reference control problem, the SOFC can be trained base on
the desired trajectory or reference model (like trials B and B) to improve the tracking performance; (ii) The
SOFC is particularly suitable for the control of repetitive operating processes that the state trajectories in the preceding period can be used for the training and more signi>cant improvement of the tracking performance
will be obtained (like trials C and C). The simulation results of cases 1 and 2 also demonstrated the robustness
and adaptability of the SOFC to initial state error and state measurement noise.
5. Conclusion
A hierarchical self-organizing fuzzy logic controller using LVQ network has been proposed for the robust tracking control of unknown nonlinear composite systems. A complementary algorithm for self-learning has been developed and demonstrated. Firstly, a set of regular rule partition makes it possible to avoid certain improper rule modi>cations. Secondly, the information about the better location of the IF-part membership function is obtained through an LVQ network. It has been proven that the overall fuzzy control system is able to guarantee the output tracking error to converge to a residual set ultimately. In addition, the system is also robust to the disturbances, measurement noise, and perturbed initialization error. The simulation results of robot control show that better results are obtained if the LVQ network is used for self-organizing the rule base. Consequently, in the context of organization level of the SOFC, by providing conditional weights according to the performance index to the training samples, the output trajectories follow the required path more closely and smoothly.
For a model-reference=-following, trajectory tracking or repetitive control problem, the information about the better location of the IF-part membership function can be obtained in the context of the organization level of the SOFC. However, the limitation of the proposed technique is mainly in operating the set-point control
Fig. 7. Tracking error of (a) joint 1, and (b) joint 2 with admissible initial state error and measurement noise.
problem. To lift this diTculty, our further work is to design the organization level of the SOFC in which the locations of the membership functions are trained according to the moving direction of the process state instead of the position of the process state. A future work about the fuzzy systems is to determine initially the numbers of fuzzy rules to ensure a pre-speci>ed uniform approximation capability. To adaptively modify the widths of the IF-part membership functions may be an alternative of modifying the locations of them; it is worthy to further investigate this trade-o9.
Appendix A
Proof of Theorem 1. Eq. (1) can be rewritten in terms of the measured output Jy and the ith component as
Jy(ri) i = fi(x) + gi(x)ui+ zi(x; t) + n(ryii) = (∗T i ( Jx) + /fi + (!∗Ti ( Jx) + /ig)ui+ zi(x; t) + n(ryii); (A.1) where /fi = fi(x) − (∗Ti (x) − (∗Ti U(x; nx); /gi = gi(x) − !∗Ti (x) − !∗Ti U(x; nx); (A.2)
and
U(x; n
x) = ( Jx) − (x);
U(x; nx) = ( Jx) − (x)
(A.3)
are measures of the sensitivity of the nominal model z(x; t) ≡ nx≡ ny≡ 0 with respect to the measurement
noise nx. By (2) and (A.1), it is then possible to derive the error equation as
Jy(ri) i − y(rMii)= − i1yMi− i2˙yMi− · · · − iriyMi(ri−1)− vi +(∗T i ( Jx) + !∗Ti ( Jx)ui+ /i; (A.4) where /i= /fi + /giui+ zi(x; t) + n(ry; ii): By (15), subtracting !T
i( Jx)ui from and adding −(Ti ( Jx) + i1JyMi + i2˙JyMi + · · · + iriJy(rMii−1)−
aLRi)( Jx) to the right-hand side of (A.4) obtains
Jy(ri)
i − y(rMii)= i1( Jyi− yMi) + i2( ˙Jyi− ˙yMi) + · · · + iri( Jy(rii−1)− y(rMii−1))
+ ((∗T
i − (iT)( Jx) + (!∗Ti − !Ti)( Jx)ui+ /i− aLRi)( Jx) (A.5)
or
˙ei= Aiei− bi˜(Tiw + bi(/i− aLRi)); (A.6)
where ˜(i= (i− (∗i denotes the parameter estimation error and
Ai= 0 1 0 · · · 0 0 0 1 · · · 0 ... ... ... ... ... 0 0 0 · · · 1 − ai1 − ai2 − ai3 · · · − airi ; bi= 0 0 ... 0 1 ; w = : (A.7)
Let V( and V# be positive-de>nite functions of the forms V(=12mi=1((iT(i) and V#=12mi=1#2, respectively,
and their time derivatives are ˙V(=mi=1(Ti ˙(iand ˙V#=mi=1#Ti ˙#i. If the >rst line of (19) is true then d(i= 0,
and the conclusion ˙V(60 is trivial. If the second line of (19) is true then d(i¡1 and (i∈ M(.i. Therefore,
either ˙V(60 or (i∈ M(.i is obtained. Similarly, one has either ˙V#¡0 or #i∈ M#.i. Therefore, the boundedness
of (i, #i, and u is guaranteed. To show the performance of the closed-loop system formed by (1), (15), (18),
and (20), one can choose the following positive-de>nite function:
V = V1+ · · · + Vm; (A.8) where Vi(t) = 1 2d20+12 ˜( T iRi˜(i+128#i˜# 2 i; if eTPbbTPe6d20; 1 2eTiPiei+12˜(TiRi˜(i+128#i˜# 2 i; otherwise;
where ˜#i= #i− #Mi is the auxiliary adjustable parameter error and #Mi ≡ max{#∗i; #i0}. Taking the derivative
of Vi and considering (A.6), (18), and (20), one obtains: ˙Vi= 0 for eTPbbTPe6d20, and
˙Vi(t) = eTiPi(Aiei− bi˜(Tiw + bi(/i− LRi))) + ˜(Ti(I − d((i⊥(Ti⊥)
[wbT iPiei− 21((i− (i0)] + ˜#i(1 − d#)[wibTiPiei− 22(#i− #i0)] = 1 2eTi(ATiPi+ PiAi)ei− eiTPibi˜(Tiw + eTiPibi(/i− #iwi) + ˜(TiwbT iPiei− 21˜(Ti((i− (i0) + ˜#iwibTiPiei− 22˜#i(#i− #i0)
− d#˜#i[wibTiPiei− 22(#i− #i0)] − d(˜(Ti(i⊥(i⊥T [wbTiPiei− 21((i− (i0)] (A.9)
for eTPbbTPe¿d2
0. By (19), if (Ti⊥[wbTiPiei− 21((i− (i0)]60, one has d(i= 0 and the last term of the above
equation is equal to zero. When (T
i⊥[wbTiPiei− 21((i− (i0)]¿0, one also has d(i= 0 for Vi∈ M(i and the
above conclusion holds. If Vi ∈ M(i and suppose that M(i and M#i are appropriately selected such that V∗i
and #∗
i are in the interior of M(i and M#i, respectively, one obtains
˜(T i(i⊥ = ((i− (∗i)T(i=|(i| = 1 2[((i− (∗i)T((i− (∗i) + (Ti(i− (∗Ti (∗i=|(i| ¿ 0; (A.10) or ˜(T
i(i⊥(Ti⊥[wbTiPiei− 21((i− (i0)]¿0: (A.11)
In a similar way, it can be shown that
˜#i[wibTiPiei− 22(#i− #i0)]¿0: (A.12)
Therefore,
˙Vi612eTi(ATiPi+ PiAi)ei+ eTiPibi(/i− #Mwi)
− 21˜(Ti((i− (i0) − 22˜#i(#i− #i0): (A.13)
Using Assumption 1, the second term on the right-hand side satis>es the inequality
eT iPibi(/i− #Mwi) 6 |eTiPibi|#∗i) − eTiPibi#Mwi 6 #M i (|eTiPibi|) − eTiPibiwi) = #M i (|eTiPibi)| − eTiPibi) tanh(eTiPibi)=.)) 6 #M i 6.: (A.14)
Since the following fact can be shown easily by straightforward algebraic manipulation. Claim.
06|8| − 8 tanh8
.
for any 8 ∈ R, and 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˜#2i +12(#i− #i0)2−12(#∗i − #i0)2: (A.16) Therefore, ˙Vi6 −12eTi(Qi)ei−221 ˜(Ti ˜(i−222 ˜#2i +221((∗i − (i0)T((∗i − (i0) +222(#∗i − #i0)2+ #Mi 6. 6 − aiVi+ 7i; (A.17) where ai ≡ min 7min(Qi) 7max(Pi); 21 7max(Ri); 22 8#i and 7i =221((i∗− (i0)T((i∗− (i0) +222(#∗i − #i0)2+ #Mi 6. (A.18) or ˙ V6 − aV + 7; (A.19)
where a = miniai and 7 =mi=17i. The di9erential inequality (A.19) satis>es
06V(t)67 + V(0) −7 e−at: (A.20)
Therefore, ei, (i, and #i are uniformly ultimately bounded. Let 5∗= 27=a then from (A.20), (24) is readily
obtained.
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