International Journal of Uncertainty,

Fuzziness and Knowledge-Based Systems

Vol. 20, No. 3 (2012) 339−353

© World Scientific Publishing Company

DOI: 10.1142/S0218488512500171

339

ADAPTIVE FUZZY-BASED TRACKING CONTROL FOR A CLASS OF

STRICT-FEEDBACK SISO NONLINEAR TIME-DELAY SYSTEMS

WITHOUT BACKSTEPPING

HASAN A. YOUSEF*, ‡, MOHAMED HAMDY†, § and MUHAMMAD SHAFIQ*

*Department of Electrical and Computer Engineering, College of Engineering,

Sultan Qaboos University, Sultanate of Oman, Muscat 123, PO Box 33 †Department of Industrial Electronics and Control Engineering,

Faculty of Electronic Engineering, Menofia University, Menof-32952, Egypt ‡hyousef@squ.edu.om

§mhamdy72@hotmail.com

Received 22 August 2010

Revised 12 April 2012

In this paper, an adaptive fuzzy tracking control is presented for a class of SISO nonlinear strict

feedback systems with unknown time delays. The proposed algorithm does not use the backstepping scheme rather it converts the strict-feedback time-delayed system to the normal form. The Mamdani-

type fuzzy system is employed to approximate on-line the lumped uncertain system nonlinearity.

The developed controller guarantees uniform ultimate boundedness of all signals in the closed-loop system. The designed control law is independent of the time delays and has a simple form with only

one adaptive parameter vector needed to be updated on-line. As a result, the proposed control

algorithm is considerably simpler than the previous ones based on backstepping. The Lyapunov stability of the fuzzy system parameters and the filtered tracking error is employed to guarantee

semiglobal uniform boundedness for the closed loop system. Simulation results are presented to

verify the effectiveness of the proposed approach.

Keywords: Adaptive fuzzy control; strict-feedback nonlinear systems; time-delay; stability.

1. Introduction

Systems with delays frequently appear in engineering applications. Typical examples of

time-delay systems are electrical networks, chemical processes, rolling mill systems,

teleportation systems, underwater vehicles and so on. The existence of time delays may

degrade the control performance and make the stabilization problem more difficult. So

far, the stability analysis and robust control for these dynamic time-delay systems have

attracted a number of researchers over the past years; see Ref. 1, and 2 for example and

the references therein. Stability analysis and synthesis of time-delay systems are

important issues addressed by many authors and for which surveys can be found in many

articles.3-7

Recently, by combining Lyapunov–Krasovskii functional and backstepping tech-

nique, an adaptive neural tracking control scheme was proposed in Ref. 8 for a class of

strict-feedback nonlinear time-delay systems with unknown virtual control coefficients.

H. A. Yousef, M. Hamdy & M. Shafiq 340

The suggested controller guarantees the uniform ultimate boundedness of the adaptive

closed-loop system, while the output tracking is achieved. Further improvements were

given in Refs. 9 and 10. Robust stabilization methods were presented via the

approximation capability of neural network.11-13

In Refs. 10, 14 and 15, observer-based fuzzy adaptive control for strict-feedback

nonlinear systems and fuzzy adaptive observer backstepping control for MIMO nonlinear

systems are developed. In Ref. 16, the adaptive H∞ control was addressed via

backstepping and neural networks technique. The observer-based adaptive neural

controller was designed for a special class of nonlinear time-delay systems.17 However,

these adaptive neural control methods8-17

require a large number of neural weights to be

adapted online simultaneously. This makes the learning time unacceptably large. In view

of this disadvantage, several adaptive fuzzy control schemes have been developed in

Refs. 18−25 for nonlinear delay-free systems. The advantage of these control schemes is

that the proposed controllers require much less parameters to be updated online. The T-S

fuzzy model combines the flexibility of fuzzy logic theory and rigorous mathematical

analysis tools into a unified framework. Linear models are employed for the consequent

parts which makes it simple to use conventional linear system theory for the analysis. The

advantage of T-S fuzzy modeling is its greater accuracy, dimensionality, and also the

simplification of the structure of nonlinear systems. It provides an effective

representation of complex nonlinear systems. Stability analysis and control synthesis of

T-S fuzzy delayed systems were proposed in Refs. 26−33. An approximation-based

adaptive control has also been addressed for nonlinear systems with time-delay. An

attempt for using Mamdani-type fuzzy logic systems to design an adaptive fuzzy tracking

controller by using the backstepping technique and Lyapunov–Krasovskii functional

appeared in Ref. 29. Most of the previous adaptive fuzzy control algorithms for strict-

feedback nonlinear systems were based on the backstepping scheme, which makes the

control law and stability analysis very complicated. The main advantage of the results

proposed in Ref. 29 is that the developed fuzzy controllers contain less adaptation laws.

The adaptive fuzzy controllers based on backstepping design method have some

drawbacks. First, the determination of virtual control terms and their time derivatives

requires tedious and complex analysis. As pointed out in Ref. 33, for the relatively simple

application to dc motor control, the regression matrix in the adaptive backstepping

control almost covers an entire page.34 The complexity exhibits an exponential increase

as the order of the controlled system, i.e., n, grows. However, this complexity must be

avoided for practical implementation. Recently, in Ref. 35, an attempt for using an

adaptive neural control for SISO nonlinear systems without backstepping is proposed.

Based on the above observation, a novel systematic design procedure is developed in

this paper to synthesize a stable adaptive fuzzy controller for a class of strict feedback

nonlinear time-delay systems without backstepping. Fuzzy logic systems are employed to

approximate the lumped uncertain system nonlinearity, and then, the adaptive law of

adjustable parameters is obtained. The universal approximation property of fuzzy systems

developed in Ref. 36 is utilized to generate estimation for the lumped uncertain

parameters appeared in the control law. The main advantage of the proposed method is

that for an nth order strict-feedback nonlinear system, only one parameter vector is

needed to be estimated on-line and therefore, the computation burden is significantly

reduced and the algorithm is easily realized in practice.

Adaptive Fuzzy-Based Tracking Control 341

The paper is organized as follows. The system under investigation and its

transformation to normal form is outlined in Sec. 2. The adaptive fuzzy control design is

presented in Sec. 3. The main result is presented in Sec. 4. Simulation results are provided

in Sec. 5, with conclusions given in Sec. 6.

2. Problem Formulation

Consider the SISO nonlinear time-delay dynamic systems in the following strict feedback

form:

, , , ...., ,

) ) ,

i i i i i i i i i

n n n n n n n n

x g x x f x h x t i n

x g x u t f x h x t

y x

τ

τ+= + + − = −

= + + −

=

ɺ

ɺ

(1)

where [ , ,....., ]Ti ix x x x= , [ , ,....., ]T nnx x x x R= ∈ ∈∈∈∈R

n is the system state vector, u ∈R

u ∈∈∈∈R and y ∈R denote system control input and output, respectively. The functions

,i i ig f h are unknown smooth functions and τi is an unknown time delay of the

state variables, i = 1,… , n.

It is assumed that (1) has measurable state vector x. With

regard to controllability, the following assumption must be made.

Assumption 1. For 1 ≤ i ≤ n, the signs of ( )i ig x are known. Without loss of generality, it

is assumed that ( )i ig x > .

The control objectives are to design an adaptive fuzzy tracking controller for the

system (1) such that the system output y tracks a desired reference signal yd and to assure

the boundedness of all signals in the closed-loop system.

The first step in the design is to convert the original system (1) into the normal form

with respect to the newly defined state variables [ , ,....., ]Tnz z z= .

Let yz∆=1 and z z g x f d

∆= = + +ɺ where ( ( ))i i i id h x t τ= − . The time derivative of z2

is derived as

( )

( )( ) ( )

( ) ( ) ( ( ))j

f g dz x x x g x x t

x x x

f g dx f g x d g f g g x g d x t

x x x

a x b x x l x t∆

τ

τ

τ

∂ ∂ ∂= + + + −

∂ ∂ ∂

∂ ∂ ∂= + + + + + + + −

∂ ∂ ∂

= + + −

ɺ ɺ ɺ ɺɺ

ɺ

(2)

where ( ) (( ))( )f g

a x x f g x g fx x

∂ ∂= + + +

∂ ∂, ( )b x g g=

and

( ( )) (( )) ( )j

f g dl x t x d g d x t

x x xτ τ

∂ ∂ ∂− = + + + −

∂ ∂ ∂ɺ

H. A. Yousef, M. Hamdy & M. Shafiq 342

Again, let ( ) ( ) ( )z a x b x x l x= + + and time derivative is induced by

( )

( )( ) ( ) ( )

( ) ( ) ( ( ))

j j j j

j j jj j j

j j j j j jj j jj j

j

a b lz x x x b x x t

x x x

a b lx f g x d b f g x d x t

x x x

a x b x x l x t∆

τ

τ

τ

= = =

+= =

∂ ∂ ∂= + + + −

∂ ∂ ∂

∂ ∂ ∂= + + + + + + + −

∂ ∂ ∂

= + + −

∑ ∑ ∑

∑ ∑

ɺ ɺ ɺ ɺɺ

ɺ

(3)

where ( ) ( )( )j j jj jj

a ba x x f g x b f

x x+

=

∂ ∂= + + +

∂ ∂∑

, ( )b x b g g g g= = and

( ) ( ) j jj j jj j

a b ll x x d x b d

x x x= =

∂ ∂ ∂= + + +

∂ ∂ ∂∑ ∑ ɺ

In general, by induction, if we define a f= , b g= and l d= , the following is

satisfied for i = 2,..,n:

( ) ( ) ( ( ))

( ) ( ) ( ( ))

i i i i i i i i i

i i i i i i i i i

z a x b x x l x t

z a x b x x l x t

∆ τ

τ− − − − − − −

+ −

= + + −

= + + −ɺ

(4)

where

( ) ( )( )

( )

( ( )) (( ) )

ii i

i i i j j j i ij jj

i

i i i i j

j

ii i i

i i i i j j i ij j jj

a ba x x f g x b f

x x

b x b g g

a b ll x t x d x b d

x x xτ

−− −

+ −=

−=

−− − −

−=

∂ ∂= + + +

∂ ∂

= =

∂ ∂ ∂− = + + +

∂ ∂ ∂

∑

∏

∑ ɺ

(5)

As a result, the strict-feedback time-delayed system (1) can be represented as the

following normal form with respect to the new state variables iz :

( ) ( ) ( )

i i

n d

z z

z a x b x u l x

y z

+=

= + +

=

ɺ

ɺ

(6)

where ( ) ( ), ( ) ( ), ( ) ( ) and [ ( ) ( )]n n d n d d n na x a x b x b x l x l x x x t x tτ τ= = = = − −⋯

with

( )di i ix x t τ= − . It should be noted that the functions ( ), ( ), and ( )da x b x l x are unknown

functions of x and time delays.

Assumption 2. The unknown function ( )dl x satisfies the following

( ) ( ) ( ( ))n n

d i i di i i i i

i i

l x x x tρ φ ρ φ τ= =

= = −∑ ∑

where ( )i dixφ are known positive function with ( ( ( ))n

i i

i

x t Cφφ=

≤∑ and iρ are unknown

constants.

Adaptive Fuzzy-Based Tracking Control 343

From Assumption 1, it is noted that a constant b > exists such that ( ) , nb x b x R≥ ∀ ∈ .

This assumption poses a controllable condition on system (6).

3. Brief Description of Fuzzy Approximator

The fuzzy system considered in this paper has a center-average defuzzifier, product

inference and singleton fuzzifier. This type of fuzzy logic system is given by:

( ( ))

( )

( ( ))

i

i

nM

iFi

nM

iFi

q x

q x

x

µ

µ

= =

= =

=∑ ∏

∑ ∏

ℓ

ℓ

ℓ

ℓ

ℓ

(7)

where M is the number of IF-THEN rules in the fuzzy rule base. The If-Then rules take

the following form for , ,...,M=ℓ :

: If is and is and and Then isn nR x F x F x is F q Gℓ ℓ ℓ ℓ ℓ⋯

where ℓ

iF and ℓG are the fuzzy sets with membership functions iF

µ ℓand

Gµ ℓ

respectively, and q is the linguistic variables which can be considered as output of the

fuzzy logic system. The parameter q ℓ is the point at which ( )G

qµ ℓ

ℓ

achieves its

maximum value and we assume that ( )G

qµ =ℓ

ℓ . Equation (7) can be rewritten as

( ) ( )Tq x xψ ζ= (8)

where [ , ,...., ]TMψ ψ ψ ψ= is a parameter vector, and ( ) [ ( ), ( ),...., ( )]M Tx x x xζ ζ ζ ζ= is a

regressive vector with the regressor ( )xζ ℓ defined as fuzzy basis function (FBF) of the

form

( )

( )

( ( ))

i

i

n

iFi

nM

iFi

x

x

x

µ

ζ

µ

=

= =

=∏

∑ ∏

ℓ

ℓ

ℓ

ℓ

Property. For any given real continuous function ( )inxβ on a compact set inx

Ω and an

arbitrary small numberreε > , there exists a fuzzy system ˆ( )inxβ in the form of (8) and an

optimal parameter vector *ψ such that *ˆsup ( ) ( )in xin

in in rex

x xΩ

β β ψ ε∈

− < .

4. Adaptive Fuzzy Controller Design

In this section an adaptive fuzzy tracking controller for system (1), or equivalently (6), is

designed. First the ideal controller is determined assuming the functions ( ),a x ( )b x and

( )dl x are known. Then the adaptive control law is designed using the approximation

capability of fuzzy logic system.

H. A. Yousef, M. Hamdy & M. Shafiq 344

4.1. Ideal controller

It is assumed that the desired output trajectory and its derivatives ( )[ , ,..., ]n Td d d dY y y y −= ɺ

are measurable and bounded and ( )ndy

− denotes the (n-1)th derivative of dy , with respect

to time. Define the error vector e as

[ , ,...., ]Td ne z Y e e e= − =

(9)

and the filtered tracking error se as

( ) [ ] n T

s

de e e

dtλ Λ−= + =

(10)

where [ , ( ) ,..., ( ) ]n n Tn nΛ λ λ λ− −= − − , and λ > 0 is a positive constant to be specified by

the designer.

Assumption 3. The desired trajectory vector dy is continuous, available, and

satisfies ( )[ , ]nT T nd d dy Y y R += ∈ .

From (6) and (10), one obtains the following differential equation for the tracking

error

( ) ( ) ( )n

s i i di

i

e a x b x u x vρ φ=

= + + +∑ɺ (11)

where ( )[ ] nTdv e yΛ= − .

Using triangular inequality, we have

( ( ) ( ) ) ( )n

Ts s s s i di

i

e e e a x b x u v e xρ ρ φ=

≤ + + + +∑ɺ (12)

with Tnρ ρ ρ= …

Though the functions ( )i dixφ are known, it could not be used for the construction of

control law as dix is not available due to the unknown time delays. To tackle this

problem, let us introduce the following Lyapunov-Krasovskii functional

( ( ))

i

tn

U i i

i t

V x d

τ

φ λ λ= −

= ∑ ∫

The time derivative of UV is

( ( ( )) ( ))n

U i i i di

i

V x t xφ φ=

= −∑ɺ (13)

in which the non-positive term can be used to compensate for the terms with unknown

time delay.

Lemma 1. Considering that (6) satisfies Assumption 1, the filtered tracking error se is

uniformly ultimately bounded if the ideal control law is designed as

* *

*

( )

( )( )

( )

s ad

ad

u ke u w

a w vu w

b w

= − −

+=

(14)

where k > 0 is a design parameter and [ ]Tw x v= .

Adaptive Fuzzy-Based Tracking Control 345

Proof. Consider the Lyaunov function

se s UV e V= +

(15)

Using (13), the time derivative of (15) along (11) is

( ( ( )) ( ))s

n

e s s i i i di

i

V e e x t xφ φ=

= + −∑ɺ ɺ .

(16)

Substitution of (12) and (14) in (16) yields

( )( ( ) ( )( )) ( ) ( ( ( )) ( ))

( )s

n nT

e s s s i di i i i di

i i

a x vV e a x b x ke v e x x t x

b xρ ρ φ φ φ

= =

+≤ + − − + + + + −∑ ∑ɺ

which reduces to

s

Te s sV kbe e Cφρ ρ≤ − + +ɺ

( )

se s s

CV e e

φδ≤ − − − +ɺ (17)

where ( )Tkbδ ρ ρ= − > . Therefore, for all s

Ce

φ≥ , Eq. (15) becomes ( )se sV eδ≤ − −ɺ

which concludes that se is uniformly ultimately bounded.

4.2. Adaptive control law

Using the fuzzy system of (8), the fuzzy-logic based control input is designed as follows:

ˆ ( )

ˆˆ ( ) ( )

s ad

Tad

u ke u w

u w wψ ζ

= − −

= (18)

where ψ is the estimation of the parameter vector and can be updated as follows:

ˆ ˆ ˆ( ( ) ( ) )s sw e eψ γ ζ σ ψ ψ= −ɺ

(19)

In (19) γ is positive learning rate and the switching function ˆ( )σ ψ chosen as (see

Ref. 38)

ˆ, if ˆ( )

, otherwise

Cζψ

ψψ ε

εσ ψ

>

=

(20)

with ψε being a design constant and Cζζ ≤ .Then ˆ ψψ ε≤ .

It should be noted that the switching function (20) is employed to retain the learned

information of FBF and to prevent the loss of information if ψε is chosen sufficiently

large such that *ψψ ε< while guaranteeing the boundedness of ψ .

Remark. One parameter vector ψ that contains M adjustable parameters is required to

synthesis the control law (18) where M is the number of fuzzy sets. It is worth-noting to

mention that some adaptive fuzzy controllers for non-delayed systems which require less

number of adjustable parameters have appeared recently.38-40

H. A. Yousef, M. Hamdy & M. Shafiq 346

5. Main Result

In this section, the boundedness of the closed-loop signals is proved using the Lyapunov

function approach.

Theorem 1. For the adaptive system comprising (1) under Assumption 1, the controller

(18) and update law (19), the filtered error se is uniformly ultimately bounded.

Proof. Let us consider the following Lyapunov function candidate:

ˆ ˆs

TeV V ψ ψ

γ= + (21)

The time derivative of (21) is given by

ˆ ˆ ˆ( ( ) ( ) )

s

Te in s sV V x e eψ ζ σ ψ ψ= + −ɺ ɺ

(22)

where

( ( ) ( ) ))

s

Te s sV e a x b x u v e Cφρ ρ≤ + + + +ɺ (23)

Using (18) in (23) to get

ˆ( ( ) ( )( ) ))

s

T Te s s sV e a x b x ke v e Cφψ ζ ρ ρ≤ + − − + + +ɺ

Now equation (22) becomes

* * ˆ ˆ ˆ( ) ( ) ( ) ( ( ) )T T T T

s s s ad s s sV b x ke be b x e u e e e Cφψ ζ ψ ζ ψ ζ σ ψ ψ ρ ρ≤ − + + − + − + +ɺ ɶ

(24)

where TTT ψψψ ˆ~ * −= . Equivalently, (24) can be put in the following form

ˆ ˆ ˆ( ) ( ( ) )T

s s s re sV bke e C b e C C e Cφ ψ ζ ζρ ρ ε ψ σ ψ ψ≤ − + + + + − −ɶɺ (25)

where T C Cψ ζψ ζ <ɶ

ɶ . Now if ˆ ψψ ε> , then according to (20), the term

ˆ ˆ( ( ) )Cζσ ψ ψ − will be a positive constant Cσ and (25) becomes

ˆ( ) ( )s s s re s

CV e e b e C C e C C

φψ ζ σψδ ε≤ − − − + + + −ɶ

ɺ (26)

Therefore, for all ˆ( ) )s re

s

Ce b C C C C

e

φψ ζ σψε≥ + + −ɶ

, Eq. (26) will be ( ) sV eδ≤ − −ɺ

and hence the filtered error se is uniformly ultimately bounded.

6. Simulation Results

In this section, we demonstrate the effectiveness of the proposed adaptive fuzzy control

algorithm using the following illustrative examples.

Example 1. In this example, we consider a second order time–delay system of the form:9

.( ) ( )

( cos( )) . ( ) sin ( )

xx x x x e x t

x x x u x x x t x t

y x

τ

τ τ

−= + + + −

= + + + − −

=

ɺ

ɺ (27)

Adaptive Fuzzy-Based Tracking Control 347

In this example, secτ τ= = and the desired reference signal is chosen as

. (sin sin( . )dy t t= + .6

The above system is simulated under the proposed adaptive fuzzy control given by

(18) and (19) where [ , , ]Tinx x x v= and dv e yλ= − ɺɺ . In this case, seven Gaussian

membership functions with centers evenly spaced between [−1.5, 1.5] for each variable , , ,i inW x i∈ = are chosen as follows:

4

)5.1(5.0

4

)1(5.0

4

)5.0(5.0

4

)(5.0

4

)5.0(5.0

4

)1(5.0

4

)5.1(5.0

2

7

2

6

2

5

2

4

2

3

2

2

2

1

,,

,,,,

−−−−−−

−+−+−+−

===

====

i

i

i

i

i

i

i

i

i

i

i

i

i

i

W

F

W

F

W

F

W

F

W

F

W

F

W

F

eee

eeee

µµµ

µµµµ

(28)

The FBFs used to generate the approximation of )(ˆinad xu take the following form:

( ) ( )

( ) , ,i i

T

i iF Fi i

i

W W

WD D

µ µζ = =

=

∏ ∏1 7…

where ( )i

iFi

D Wµ= =

=∑∏ ℓ

ℓ

.

The controller parameters are selected as ; ; and k λ γ= = = and zero initial

conditions for the model dynamics as well as for the vector ˆ.ψ

Simulation results are

depicted in Figs. 1 and 2.

0 2 4 6 8 10 12 14 16 18 20-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

ou

tpu

t y a

nd

re

fere

nce

sig

nal yd

sec

y

yd

Fig. 1. Tracking performance.

H. A. Yousef, M. Hamdy & M. Shafiq 348

0 2 4 6 8 10 12 14 16 18 20-2

0

2

4

6

8

10

12

co

ntr

ol in

pu

t

sec

Fig. 2. Control signal.

Example 2. In this example, a two-stage chemical reactor with delayed recycle streams is

simulated with the proposed controller. The reactor model is given by:41

. . sin( ) ( ),

. ( ) . sin( ) ( ),

,

x x x t x t

x u x x t t x t

y x

τ

τ τ

= − + + −

= − + − + −

=

ɺ

ɺ (29)

where the delay times are secτ τ= = . The desired reference signal is assumed

asdy = . For the vector [ , , ]Tinx x x v= seven Gaussian membership functions of the form

(28) are used. Simulation results are shown in Figs. 3 and 4 for ; . ; and k λ γ= = =

and initial conditions of [ ] ˆ( ) , and (0)=0x ψ= − .

0 5 10 15-2

-1.5

-1

-0.5

0

0.5

ou

tpu

t y

sec

Fig. 3. Tracking performance.

Adaptive Fuzzy-Based Tracking Control 349

0 5 10 15-15

-10

-5

0

5

10

15

co

ntr

ol in

pu

t

sec

Fig. 4. Control signal.

Example 3. Consider a controlled Brusselator model with both external disturbances and

existence of time delays of the form:33

.

.

( ) ( , ) ( ( )),

( cos( )) ( , ) ( ( )),

,

x C D x x x d t x h x t

x Dx x x x u d t x h x t

y x

τ

τ

= − + + + + −

= − + + + + −

=

(30)

where x1 and x2 denote the concentrations of the reaction intermediates, C, D > 0 are

parameters which describe the supply of “reservoir” chemicals, u is the control input. The

terms ( , )d t x and ( , )d t x represent external disturbances which come from the modeling

errors and other types of unknown nonlinearities in the practical chemical reactions. The

functions ( ( ))h x t τ− and ( ( ))h x t τ− are unknown time-delayed functions.

In simulation, we choose the same parameters, disturbances and delay functions as

given in Ref. 33, namely,

, , ( ) , ( ) . sin( ), . cos( . ), . ( ) sin ( )C D h x x h x x x d x t d x x t= = = = = = +

and secτ τ= = . The reference trajectory is selected as sin( . ) . sin( . ).dy t t= + +

Seven Gaussian membership functions of the form (28) are used for the

vector [ , , ] .Tinx x x v= Simulation results are shown in Figs. 5 and 6 for k = 150;

λ = 2.5; and γ = 4000 and initial conditions of x(0) = [2.5, 1] and ˆ(0) = 0y .

Simulation results of the above three examples demonstrate the effectiveness of the

proposed adaptive fuzzy controller in achieving good tracking performance and smooth

control signal.

H. A. Yousef, M. Hamdy & M. Shafiq 350

0 2 4 6 8 10 12 14 16 18 201.5

2

2.5

3

3.5

4

4.5

ou

tpu

t y a

nd

re

fere

nce

sig

na

l yd

sec

y

yd

Fig. 5. Tracking performance.

0 2 4 6 8 10 12 14 16 18 20-400

-300

-200

-100

0

100

200

con

tro

l in

pu

t

sec Fig. 6. Control signal for Example 3.

7. Conclusion

Adaptive fuzzy control has been developed for a class of SISO strict-feedback nonlinear

systems with unknown time-delays. The key aspect of the proposed method is that the

state-feedback control problem of the strict-feedback system can be viewed as the output-

feedback control problem of the affine system in the normal form; this results in avoiding

backstepping in the controller design. It is also demonstrated that control law and

stability analysis is considerably simpler than the previous backstepping based

algorithms. In the developed control algorithm, fuzzy logic systems of Mamdani-type are

utilized to approximate on-line the unknown nonlinear functions. The proposed control

Adaptive Fuzzy-Based Tracking Control 351

scheme guarantees the semi-global boundedness of all the signals in the closed-loop

system and good tracking performance. Moreover, the suggested adaptive fuzzy

controller is simple. This makes the design scheme easier to be implemented in practical

applications. Extensive simulation examples are provided to validate the effectiveness of

the proposed controller.

Acknowledgements

The authors would like to thank the Editor and the anonymous reviewers for their

inspiring encouragement and constructive comments, which have contributed much to the

improvement of the clarity and presentation of this paper. The first and third authors

acknowledge the support of Sultan Qaboos University.

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