Dissertation Overview

The rest of this dissertation is organized as follows. Chapter 2 describes the design procedure of the RASFC scheme for nonaffine nonlinear systems. The structure learning phase performed by the SFS is introduced. The adaptive laws to tune the parameters, including the means and variances of membership functions and the single consequents of the fuzzy rules and, are derived. The stability analysis and example are also provided in this chapter. The DACHDNN is developed in Chapter 3. The adaptive laws to tune the synaptic weightings are derived. The stability analysis and examples are also provided in this chapter.

Finally, conclusions and future works are stated in Chapter 4.

Chapter 2

Robust Adaptive Self-structuring Fuzzy Control Design for Nonaffine Nonlinear Systems

Reviewing some literatures on nonaffine nonlinear system control, we find some problems left to be addressed. In [50], although the system stability is guaranteed in the Lyapunov sense, the un-measurable term in the adaptive law needs to be approximated. This will make the system stability questionable. Even the system stability can be guaranteed, the tracking error is only ultimately uniformly bounded. In [51], the tracking error is uniformly asymptotically stable, but the robust controller to compensate the external disturbance causes the chattering of control input. Although the authors in [50] suggested some remedies to reduce the chattering, the tracking error may not be UAS due to these remedies.

In this chapter, we aim at solving the control problem of SISO nonaffine nonlinear systems. An adaptive fuzzy control scheme is developed to achieve this goal, and the resulting structuring problem of fuzzy systems is also solved by a proposed self-structuring fuzzy system (SFS). The automatic rule pruning and growing functions of the SFS are discussed and separately illustrated in the Examples 2-1 and 2-2 to give more insights. Using the proposed SFS, we will show how a novel robust adaptive self-structuring fuzzy control (RASFC) scheme can remarkably reduce the computational burden without sacrificing the favorable control performance for SISO nonaffine nonlinear systems.

2.1 Problem Formulation

Consider a single-input and single-output (SISO) nonaffine nonlinear system d

u f

x⁽ⁿ⁾ = (x, )+ (2-1) where x=[xx&Kx⁽ⁿ⁻¹⁾]^Tis the measurable state vector of the system on a domain Ω_x ⊂Rⁿ,

( )

u R R

f x, :Ω_x× → is the smooth unknown nonlinear function, u is the control input, and d is the bounded external disturbance. Here the single output is x. It should be noted that f(x, u) is an implicit function with respect to u. Feedback linearization is performed by rewriting (2-1) as

d u zu

x⁽ⁿ⁾ = +∆(x, )+ (2-2) where z is a constant to be designed and ∆(x,u)= f(x,u)−zu. Here we assume that

u u f

∂

∂ (x, )is nonzero for all _{(x ),u ∈Ωx×R} with a known sign. Without losing generality, we

further assume that [51, 54-55]

) 0 , ( >

∂

∂ u

u

f x (2-3)

for all _{f(x ),u ∈Ωx ×R}. Note that for the nonaffine systems with property ^{( , )} _<0

∂

∂ u

u

f x , the control scheme can be easily defined with minor modifications discussed in section 4. The control objective is to develop a control scheme for the nonaffine nonlinear system (2-1) so that the output trajectory x can track a given trajectory xc closely. The tracking error is defined as

x x

e= _c − (2-4) If the system dynamics and the external disturbance are well known, the ideal feedback controller can be determined as

1[ ( , )]

u d

z u

u_id = _lc − −∆ x (2-5) where

e k^T

n c

lc x

u = ^{( )} + (2-6) with e=[ee&Ke⁽ⁿ⁻¹⁾]^T and k =[k_n k_n−₁Kk₁]^T . Applying (2-5) to (2-2) and using (2-4) yield the following error dynamics

e⁽ⁿ⁾ +k₁e⁽ⁿ⁻¹⁾+L+k_ne=0 (2-7) If ki, i=1, 2, …, n are chosen so that all roots of the polynomial _H(s)∆sⁿ _+k1sⁿ⁻¹_+L+kn lie strictly in the open left half of the complex plane, then lim ( )=0

∞

→ e t

t can be implied for any initial conditions. However, since ∆( ux, )and the external disturbance d may be unknown or perturbed, the ideal feedback controller uid in (2-5) cannot be implemented. Thus, to achieve

the control objective, an SFS is designed to estimate the system uncertainty ∆( ux, ) in (2-2).

2.2 Self-structuring Fuzzy System 2.2.1 Description of Fuzzy System

FSs are attractive candidates for the systems that are structurally difficult to model due to inherent non-linearity and model complexities. Typically, a FS includes four well-known stages: a fuzzifier, a rule base, an inference engine, and a defuzzifier. The rule base is the collection of fuzzy rules which characterize the simple input-output relation of the system.

Note that the self-structuring algorithm introduced in this section is applicable to multi-input and multi-output (MIMO) FS. However, without losing generality and to simplify the notation, a multi-input and single-output (MISO) FS is adopted to describe the algorithm. A MISO FS can be are expressed as [19]:

im singleton consequent; F is the fuzzy sets characterized by the fuzzy membership function _j^ij

) ( _j

i

j X

F ^j , with ij∈

{

1 ,2,K ,Nj

}

being the ordinal number of membership functions of Xj. Define a set Ω which collects all possible fuzzy rules

{

i i i i N i N im Nm

}

linear combination of fuzzy basis functions defined as

∑ Π

That is, (2-10) can be rewritten as chapter, a Gaussian membership function is defined as

⎪⎭

2.2.2 Structure Learning Algorithm

The developed self-structuring algorithm consists of two parts: growing and pruning of fuzzy rules. Effective membership functions in the input spaces can be generated and ineffective fuzzy rules can be pruned automatically by the self-structuring algorithm, and thus a concise rule base can be obtained. In order to construct the fuzzy rule base, every input space S(Xj) is partitioned into several overlapping clusters to construct the fuzzy sets of Xj. It can happen that for some incoming Xj, the degree of belongings to all its fuzzy sets are quite small, i.e., F_j^ij(X_j), i_j =1 L,2, ,N_j are quite small, as depicted in Fig. 2-1(a). This means that the input space S(Xj) is not properly clustered. Hence, the fundamental concept of the growing of fuzz rules is developed to adjust the inappropriate clustering. Initially, create one initial fuzzy rule with the given initial state as

,1

The SFS will start operating from this single rule. Define the growing criterion as

g j < Θ

µmax , j=1, 2, …, m (2-16)

jth state variable

jth state variable at time t

Fig. 2-1 (a) Improper fuzzy clustering of input variable Xj; (b) Newly created membership function initial mean and standard deviation are

)

q

Nj

j ⁺¹ =

σ (2-18) where q>0 can be arbitrarily chosen, and it will be tuned by the adaptive law introduced in later section. The created membership function is shown in Fig. 2-1(b). For the case that one new membership function is created at some time, N1×K×N^j₋1×N^j₊1×K×N^m new fuzzy rules will be generated according to the new membership function as:

1

Assume that the growing criterion for X1 is satisfied at time t. Then, a new membership function

is created, and two rules are grown according to the new membership function as

1

Rule : IF X2, 1 is F and X₁² 2 is F THEN y is ₂¹ α_2,1

Rule : IF X1 is _2,2 F₁² and X2 is F THEN y is ₂² α_2,2 (2-21) A self-structuring FS with only rule generation algorithm may suffer from the computational load or learning failure caused by an overly large rule base which includes both effective and redundant fuzzy rules. In the following, the strategy to prune redundant rules is developed to solve this problem. Recall that there are n existing fuzzy rules, and then express (2-12) as collections of the singleton consequents and the fuzzy basis functions of the rest of fuzzy rules, respectively. Thus, the contribution made by kth rule on the output y can be defined as

follows:

∑

=

= _n

k k

k k

y C y

1

, k=1, 2, …, n (2-23)

where y_k =α_kξ_k. Now, we are ready to introduce the significance index which can help us to decide whether or not to prune a fuzzy rule. Significance index is a measurement of the importance of every fuzzy rule. Sk, which represents the significance index of the kth fuzzy rule, is updated as follows:

⎩⎨

⎧

≥

= <

if ,

if ,

β β τ

k rc

k

k rc

k

k S C

C

S S , k=1, 2, …, n (2-24)

whereS is the most recent S_k^rc k, 1)τ∈(0 , is a decay constant, and β∈(0 ,1) is a given constant. All Sk, k=1, 2, …, n, are initialized from ones. According to (2-18), if the contribution Ck is equal or larger than β, Sk keeps invariant; if Ck is smaller than β, Sk will be attenuated. An invariant significance implies that the associated rule is still important and should be remained; a decaying significance index implies that the associated rule is becoming less and less important and thus should be pruned. The selection of τ will affect the rate of pruning the fuzzy rules. The smaller the τ is (or the larger the β is), the faster the significance index Sk decays, and thus the faster the ineffective fuzzy rules will be pruned.

The pruning criterion of the kth fuzzy rule is defined as follows based on this knowledge S_k <Θ_p, k=1, 2, …, n (2-25) where )Θ_p ∈(0 ,1 is a selected threshold. If the pruning criterion is satisfied for Sk, the associated kth rule is pruned.

Remark 2-1: It is a difficult task to determine the initial values of the singleton consequents of the newly generated fuzzy rules. Because an SFS is in general equipped with a parameter learning algorithm to automatically tune the parameters of the fuzzy rules, the initial values of the singleton consequents can simply set as zeros. However, from (2-10), we can see that this will cause abrupt variation of the fuzzy output y and may deteriorate the performance of the SFS for a short period. This phenomenon can be observed in Fig. 2-5(b). To fix this drawback, we maintain the approximation property of the SFS at the instant that new rules are generated.

Assume that at some time tg, an SFS has n fuzzy rules and the last h rules are just newly generated. Define yp as the “pseudo fuzzy output” of the original n-h rules if h new rules were not generated at tg. The initial consequents of those new rules are chosen so that y(t_g)= y_p.

Thus, we have

p h

n

k k k

n h n

k k

new

g y

t

y =

∑

+

∑

⁻ =

= +

−

= 1 1

)

( α ξ α ξ (2-26) where α_n−h+1 =α_n−h+2 =L=α_n =α_new. From (2-26), we can easily obtain

∑

∑

+

−

=

−

=

−

= _n

h n

k k

h n

k k k

p new

y

1 1

ξ ξ α

α (2-27)

In this way, not only the bad effect caused by the abrupt variation can be mitigated, but also the future performance of the SFS can be improved by the h new rules.

Remark 2-2: While controlling, a membership function is possible to be pruned if all fuzzy rules associated with this membership function are pruned sequentially.

Remark 2-3: In the implementations of practical systems, if computational burden is the issue having highest priority, the threshold Θ can be chosen large enough so that more fuzzy _p rules are pruned. Hence, the computational burden will be substantially reduced at the expense of less favorable system performance.

Fig. 2-2 shows the flowchart to summarize the self-structuring algorithm for the SFS.

The growing and pruning effects during the control period will be illustrated in later sections with excellent result.

p

Sk <Θ

mzx

µj

ξ α ˆˆ^T ufc =

σ c αˆ ,ˆ ,ˆ

n k S_k , =1 ,2,

m j

j , 1 ,2,

max =

µ

Fig. 2-2 The flowchart of the self-structuring algorithm for the SFS

2.3 Design of RASFC

Now, we are ready for developing a robust adaptive self-structuring fuzzy controller (RASFC) for the unknown nonaffine nonlinear systems. In the RASFC, an SFS is used to estimate the system uncertainty ∆(x,u) in (2-2). The control law u in the RASFC system is designed as

(

urac ufc

)

u= 1z −

(2-28) where urac is the robust adaptive controller to achieve a L2 tracking performance with a desired attenuation level and ufc is the self-structuring fuzzy controller to approximate unknown system dynamics ∆( ux, ). Substituting (2-28) into (2-2) and using (2-4) yield

[

^{u u u d}

]

x

e⁽ⁿ⁾ = _c⁽ⁿ⁾ − _rac − _fc +∆(x, )+

[ ]

{

^{u u u u d}

}

u

x_cⁿ − _lc− ∆ − _fc + _rac− _lc +

= ^{( )} (x, ) ( )

[ ]

{

^{u u u u d}

}

e _{fc rac lc}

T − ∆ − + − +

−

= k (x, ) ( ) (2-29) or e&=Ae−b

[

∆(x,^u)−^ufc+(^urac −^ulc)+^d

]

(2-30) where

⎥⎥

⎥⎥

⎦

⎤

⎢⎢

⎢⎢

⎣

⎡

−

−

−

=

−1 1

1 0 0

0 0 0

1 0

k k

k_{n n} L L

L L

O O O M

L

A and b=[00K1]^T

2.3.1 Fuzzy Approximation

The unknown nonlinear function ∆( ux, ) is approximated by an SFS with inputs x and u. In this way, the output of the SFS ufc should be directly fed back to produce u, which is one of the input of the SFS. This kind of fuzzy system is called a recurrent fuzzy system, as depicted in Fig. 2-3(a). However, a recurrent fuzzy system will lead to a fixed-point problem which must be solved at every time instant and thus imposes computational burden [51, 54-55]. Thus, the following Lemma 2-1 is stated to avoid this problem [51, 54-55].

x

Σ

u

)) , (

( u

u

fc

≈ ∆ x u

rac

(a)

x u ( ( , u ))

fc

≈ ∆ x u

rac

(b)

Fig. 2-3 (a) The recurrent fuzzy system; (b) The static fuzzy system

Lemma 2-1: Let the constant c satisfies the condition

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

∂

> ∂ u z f

2

1 (2-31)

Then, there exist a unique u^*_fc which is a function of x and u so that _rac u^*_fc(x,u_rac)satisfies 0

) , ( ) , , ( ) , ,

(x u_rac u^*_fc =^∆∆ x u_rac u^*_fc −u^*_fc x u_rac =

ψ (2-32)

for all (x,u_rac)∈Ω_x×R.

The Proof of Lemma 1 can be found in [51].

According to Lemma 2-1, the feedback path in Fig. 2-3(a) can be removed.

Consequently, a static FS in Fig. 2-3(b) can be used to approximate ∆( ux, ), and thus we do not need to solve the fixed-point problem at every time instant. For the nonaffine systems with the property ^{∂f x(}_∂u^{,u) <0}, Lemma 2-1 can be satisfied as well by simply modifying

(2-31) as _⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛

∂

< ∂ u z f

2

1 .

Define the vectors c and σ as

T m] [c₁ c₂ c

c= L (2-33)

T m] [σ₁σ₂ σ

σ= L (2-34) where c_j=_{[c L}¹_{j c}^N_j^j_]and σ_j =_[σ L¹_j σ^N_j^j_] collect the means and standard deviations of the

Gaussian membership functions of Xj, j=1, 2, …, m, respectively. Rewrite (2-12) in the vector adaptive online-tuning of the parameters of fuzzy rules, and thus favorable approximation performance can be achieved in the presence of unexpected disturbances. To achieve this goal, the Taylor linearization technique is employed to transform the nonlinear fuzzy basis function into partially linear form as follows [25, 56]:

+o Substituting (2-44) into (2-42) yields

ε the lumped uncertainty. The higher order term o satisfies

σ Thus, the lumped uncertainty ε satisfies

ω ε = α~^T(ξ^T_c~c+ξ^T_σσ~+o)+αˆ^To+

ω bounded, the lumped uncertainty term ε is thus bounded. We can guarantee the boundness of Γ by Lemma 2-2 given in the next subsection.

2.3.2 Parameter Learning Algorithm

Substituting (2-47) into (2-30) yields

)]

Lemma 2-2 [3]: Suppose that the adaptive laws are chosen as (2-56)-(2-58), where Pr(⋅) is the projection operator, and the symmetric positive P satisfies the following Riccati-like equation

where Q is a positive definite symmetric matrix and ρ is an attenuation level which satisfies 1 0 is bounded. The following theorem shows the properties of the developed control system.

Theorem 2-1: Suppose the assumption (2-3) holds. Consider a SISO nonaffine nonlinear system (2-1) with the control law (2-28), where the self-structuring fuzzy controller is given as

The adaptive laws are chosen as (2-56)-(2-58):

The robust adaptive controller is given as

Pe Note that since A is designed to be stable in (2-30) and Q in (2-54) is a positive definite symmetric matrix, therefore P must be a positive definite symmetric matrix. Then, the RASFC system can guarantee the global stability and robustness of the closed-loop system and achieve the following L2 criterion [57-58]:

ηα

Proof: Define the Lyapunov function candidate as

σ

Differentiating (2-61) with respect to time and using (2-53) yield σ

α

Substituting (2-59) into (2-62), we obtain

)

Similarly, we have (2-67) and (2-68) by using (2-57) and (2-58) respectively.

⎪⎪

Consequently, for any possible condition in (2-56)-(2-58),G_α ≥0, G_c ≥0, and G_σ ≥0 are satisfied. Thus, we can rewrite (2-64) as

2 Assume that there exists a finite constant γ so that [58]

γ Substituting (2-61) into (2-72), we have the L2 criterion shown in (2-60). This completes the proof. Q.E.D.

From (2-72), we can see that because V(0) is finite, the effect of lumped uncertainty and external disturbance on tracking error can be eliminated as small as possible by choosing an arbitrarily small attenuation level ρ.. In other words, a smaller ρ. results in smaller tracking error, which implies better tracking performance. The following Theorem 2-2 will present an explicit formulation of tracking error.

Theorem 2-2: The tracking error e can be expressed in terms of the sum of lumped uncertainty and external disturbance as

)

From (2-71), with the knowledge 0

0 ≥ symmetric matrix, we have

Pe

γ ρ

λ_min(P)e ² ≤e^TPe≤2V(T)≤2V(0)+ ² (2-76) from (2-74)-(2-75). Therefore (2-76) can be rearranged to yield the following important formula

) ( ) 0 ( 2

min 2

e P

λ

γ ρ

≤ V +

(2-77) which explicitly describe the tracking error e in terms of the sum of lumped uncertainty

and external disturbance. Q.E.D.

If initial state V(0)=0, tracking error e can be made arbitrarily small by choosing adequate ρ. Unlike the results in [50-51], (2-77) is very crucial to show that the proposed RASFC will provide the closed-loop stability rigorously in the Lyapunov sense.

Remark 2-4: Affine systems can be viewed as a special kind of nonaffine systems [59].

Consider an SISO nonlinear affine system

d u G F

x⁽ⁿ⁾ = (x)+ (x) + (2-78) where x=[xx&...x⁽ⁿ⁻¹⁾]^T is the state vector of the system,F(x) and G(x) are unknown nonlinear mapping, u is the control input of the system, and d is a bounded external disturbance. By letting f(x,u)=F(x)+G(x)u, we can easily find that the nonlinear affine system (2-78) can be viewed as a special case of nonaffine nonlinear system (2-1). Thus, the proposed RASFC scheme can be directly applied to such a nonlinear affine system when necessary assumptions hold. The overall RASFC can be shown in Fig. 2-4.

urac

ufc

x

e

Σ ^u

j ,Sk

µmax

n

σ c αˆ ,ˆ ,ˆ

xc

δ x e

) (n

xc

Fig. 2-4 The block diagram of RASFC for nonaffine nonlinear systems

Table 2-1 Three conditions in Example 2-1 desired trajectory of tracking control: xc=sin(1.5t)

number of rules consequents of newly generated fuzzy rules Condition

1a fixed (4 rules)

Condition 1b

t < 5: the same 4 rules in Condition 1a are used.

t ≧5: rule growing is operated initialized from zeros Condition

1c

t < 5: the same 4 rules in Condition 1a are used

t ≧5: rule growing is operated

initialized according to (2-27)

2.4 Simulation Results

In this section, the simulations are performed using MATLAB under Windows XP. Four examples are presented. Approximations of unknown nonlinear functions are shown in Examples 2-1 and 2-2 to reveal the growing and pruning capabilities of the proposed self-structuring algorithm, respectively. Examples 2-3 and 2-4 are used to examine the applicability and effectiveness of the proposed RASFC system for nonaffine nonlinear control problems. Two cases are performed in Examples 2-3 and 2-4 for comparison purpose. Case 3a and Case 4a show the effectiveness of the SFS with both rules growing and pruning capabilities. In Case 3b, an adaptive FS with fixed number of rules is adapted, and the parameters of the FS are also tuned by adaptive laws (2-56)-(2-58). In Case 4b, only the growing of fuzzy rules by SFS is considered. It can be easily shown that the following examples of nonaffine system control satisfy ( , ) 0

∂ >

∂ u

u f x

. It should be emphasized that the development of the RASFC does not need to know the exact system dynamics of the controlled systems.

Example 2-1: Consider the following nonaffine nonlinear system [60]:

2

1 x

x& =

x&₂ =x₁²+0.15u³+0.1(1+x₂²)u+sin(0.1u) (2-79) In tracking control, the SFS is used to approximate an unknown function

cu u u

x u

x

u = + + + + −

∆(x, ) ₁² 0.15 ³ 0.1(1 ₂²) sin(0.1 ) . To illustrate the rule growing capability of the self-structuring algorithm, the approximation is performed under three conditions as shown in Table 2-1. Figures 2-5(a)-2-5(c) show the approximation results of Condition 1a, 1b and 1c, respectively, Fig. 2-5(d) shows the absolute value of the modeling error, u~, and Fig.

2-5(e) shows the number of fuzzy rules. The approximation performances under Conditions 1a and 1b are better than that under Condition 1a after t ≥ 5. In Fig. 2-5(b), the abrupt variations are marked by circles. These abrupt variations are obviously caused by the rule generation so that the approximation performance is affected for a short period. In Fig. 2-5(c), this phenomenon is mitigated by using (2-27) discussed in Remark 2-1. From Fig. 2-5(d), we can see the approximation performance under Condition 1c is the best among three conditions.

time (sec) (a)

) , ( ux

∆

c

uf

ufc

time (sec) (a)

) , ( ux

∆

c

uf

) , ( ux

∆

c

uf

ufc

time (sec) (b) ufc

) , ( ux

∆

c

uf

time (sec) (b) ufc

) , ( ux

∆

c

uf

) , ( ux

∆

c

uf

time (sec) (c) ufc

) , ( ux

∆

c

uf

time (sec) (c) ufc

) , ( ux

∆

c

uf

) , ( ux

∆

c

uf

time (sec)

(d) u~

*

Condition 1a Condition 1b Condition 1c

time (sec) u~

time (sec)

(d) u~

*

Condition 1a Condition 1b Condition 1c

*

Condition 1a Condition 1b Condition 1c

time (sec) u~

rule number

time (sec) (e)

*

Condition 1a Condition 1b Condition 1c

rule number

time (sec) (e)

*

Condition 1a Condition 1b Condition 1c

*

Condition 1a Condition 1b Condition 1c

Fig. 2-5 Approximation results in Example 2-1

Table 2-2 Two conditions in Example 2-2 desired trajectory of tracking control:

xc=1.5sin(t)

rule number Condition 2a fixed (40 rules)

Condition 2b t ≧0, rule pruning is operated

Example 2-2: A third-order Chua’s chaotic circuit is a simple electronic system that consists of one linear resistor (_Rc), two capacitors (C , ₁ C ), one inductor (₂ L), and one nonlinear resistor (η). It has been shown to own very rich nonlinear dynamics such as chaos and bifurcations. The dynamic equations of Chua’s circuit are written as [9-10]

)) ( ) 1 (

1 (

2 1

2 1

1

C C

C

C v v v

R

v& = C − −η )

) 1(

1 (

2 1 2

2

L C C

C v v i

R

v& =C − + )

1(

1 0 L

C

L v Ri

i& = L − − (2-80)

where the voltages

C1

v ,

C2

v and current i are state variables, _L R is a constant, and ₀ η denotes the nonlinear resistor, which is a function of the voltage across the two terminals of

C1.Here, _φ is defined as a cubic function as

3 2 1vC1 λ vC1

λ

φ = + (λ_{1 <0,}λ_{2 >0}). (2-81) The state equations in (2-80) are not in the standard canonical form. Therefore, a linear transformation is needed to transform them into the form of (2-1). Then, the dynamic equations of transformed Chua’s circuit can be rewritten as

2

1 x

x& =

3

2 x

x& = u F x&₃ = +

x1

y= (2-82)

where x=[x₁ x₂ x₃]^T is the state vector of the system which is assumed to be available; the

system dynamic function

3 3 2 1

3 2

1 )

95 7 361 (28 45

2 38

1 9025

168 1805

14 x x x x x x

F = − + − + + (2-83)

and u is the control input. The reference signal is y_r(t)=1.5sin(t). In tracking control, the SFS is used to approximate an unknown function ∆(x,u)=F+u−cu. To illustrate the rule pruning of the self-structuring algorithm, the approximation is performed under two conditions as shown in Table 2-2. Figures 2-6(a)-2-6(b) show the approximation results.

Figure 2-6(c) shows the approximation error E. Figure 2-6(d) shows the number of fuzzy rules. Taking the last pruned rule for example, we record the contribution and significance index of the rule pruned at t=2.28 in Fig. 2-6(e). Figures. 2-6(a)-2-6(c) show that the approximation performances of Conditions 2a and 2b are both quit well. However, the convergence speed of u~ under Condition 2b is faster than that of Condition 2a. This shows that the parameter training of a large number of fuzzy rules slow down the convergence speed of approximation, and the pruned rules under Condition 2b are redundant and ineffective to the approximation performance. In Fig. 2-6(e), we show the contribution and significance

Figure 2-6(c) shows the approximation error E. Figure 2-6(d) shows the number of fuzzy rules. Taking the last pruned rule for example, we record the contribution and significance index of the rule pruned at t=2.28 in Fig. 2-6(e). Figures. 2-6(a)-2-6(c) show that the approximation performances of Conditions 2a and 2b are both quit well. However, the convergence speed of u~ under Condition 2b is faster than that of Condition 2a. This shows that the parameter training of a large number of fuzzy rules slow down the convergence speed of approximation, and the pruned rules under Condition 2b are redundant and ineffective to the approximation performance. In Fig. 2-6(e), we show the contribution and significance

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