Adaptive Mamdani Fuzzy Backstepping Control for a Class of Strict-feedback Nonlinear Time-varying Delay Systems

Adaptive Mamdani Fuzzy Backstepping Control for a Class of Strict-feedback Nonlinear Time-varying Delay Systems

Adaptive Mamdani Fuzzy Backstepping Control for a Class of Strict-feedback Nonlinear Time-varying Delay Systems M. Hamdy, G. El-Ghazaly, M. Ibrahim De...

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Adaptive Mamdani Fuzzy Backstepping Control for a Class of Strict-feedback Nonlinear Time-varying Delay Systems M. Hamdy, G. El-Ghazaly, M. Ibrahim Department of Industrial Electronics and Control Engineering, Faculty of Electronic Engineering, Menofia University, 32952 Menof, Egypt (E-mail: [email protected], [email protected],) Abstract: This paper presents an adaptive Mamdani fuzzy control scheme for a class of uncertain strictfeedback nonlinear systems with unknown time-varying delays. Within this scheme, fuzzy logic systems are used to approximate nonlinear functions and the developed adaptive fuzzy controller is recursively designed via backstepping technique and Lyapunov-Krasovskii functionals, which not only significantly compensate for the unknown time-varying delays but also, avoids controller singularity problem. The proposed adaptive fuzzy controller guarantees that all signals in the closed-loop system are uniformly ultimately bounded, while tracking error converges to a small neighborhood of the origin. The main advantages of the proposed controller are that the designed control law is time-delay independent and only one adaptive parameter is required to be updated online. Simulation results are presented to verify the effectiveness of the proposed scheme. Keywords: Adaptive fuzzy control, backstepping, nonlinear time-varying delay systems, stability

1. INTRODUCTION Recently, adaptive fuzzy-based control techniques have represented an alternative approach to solve many complex control problems. The most useful property of adaptive fuzzy systems is their ability to approximate any nonlinear function provided that enough rules are used. Based on the universal approximation theorem by combining fuzzy logic systems and adaptive control schemes, stable direct and indirect adaptive fuzzy control schemes were first developed to control uncertain nonlinear systems by Lyapunov synthesis approach (Wang, 1994). Based on the initial results in (Wang, 1994), an adaptive fuzzy control schemes have been developed for SISO nonlinear systems (Poursamad et. al., 2009). The corresponding research has been extended to MIMO nonlinear systems (Tong et. al., 2005) and large scale interconnected nonlinear systems (Yousef et. al., 2009). A key restriction on these nonlinear systems is to be feedback linearizable. So, in order to guarantee global stability, systems nonlinearities needs to satisfy matching conditions, extended matching conditions, or growth conditions. If the nonlinear systems do not satisfy these conditions, the adaptive fuzzy control approaches mentioned above cannot be implemented. These restrictions were subsequently relaxed by the development backstepping approach (Kristic et. al., 1995). Backstepping is one of the most popular design methods for a large class of nonlinear systems. Following the development of adaptive and robust backstepping design methodologies (Polycarpou et. al., 1996), the backstepping approach has been used to design adaptive fuzzy (Yang et. al., 2004, Tong et. al., 2009) and neural network-based (Li et. al., 2004) control schemes for unknown nonlinear systems. The main advantages of these schemes are that the parameters and the nonlinear functions can be unknown and the uncertainties in these systems need not to satisfy matching conditions. However, the nonlinear systems considered in these approaches are delay-free systems.

Dynamic systems with delays frequently appear in engineering applications. Electrical networks, chemical processes, rolling mill systems, are among many others. The existences of time-delays in dynamic systems render the control problem much more complexity and difficulty and may degrade the control performance and destabilize such systems. So far, the stability analysis and robust control for these dynamic time-delay systems have attracted a number of researchers to extensively address these problems over the past three decades; see, for example, (Richard , 2003) and the references therein. Stability analysis and controller design of these methods are based on Lyapunov-Krasovskii functionals or Razumikhin lemma. Recently, backstepping design is combined with either Lyapunov-Krasovskii functionals or Razumikhin lemma to design robust control schemes to stabilize nonlinear timedelay systems. A robust controller is designed via backstepping for a class of time-delay nonlinear system with block triangular structure in (Nguang , 2000). The closedloop system is shown to be uniformly ultimately bounded using Lyapunov stability theorem. In (Hau et. al., 2008) , a robust controller for a class of SISO nonlinear time-delay is designed using backstepping and Razumikhin Lemma. However, the results cannot be constructively obtained when system nonlinearities are unknown. The need for knowledge of system nonlinearities is relaxed with the use of neural networks or fuzzy logic systems. Many adaptive neural control schemes have been designed via backstepping to stabilize unknown time-delay nonlinear systems have been reported in (Ge et. al., 2003, 2004, 2007), where neural networks are utilized to approximate unknown nonlinear functions in the systems. However, these proposed methods require a large number of adaptation parameters. An attempt to fuzzy logic system to design an adaptive fuzzy controller using the backstepping design and construction LyapunovKrasovskii functionals appeared in (Wang et. al., 2007). The main advantage of (Wang et. al., 2007). is that the proposed adaptive fuzzy control scheme lies in that the number of online adaptive parameters is not more than the order of the

original system. However, the systems considered in this work are with constant time-delays and the disturbance inputs are functions of the system states. In this paper, an approximation-based control scheme is developed for a class of nonlinear systems with unknown time-varying delays. By combining adaptive fuzzy backstepping and, the proposed control scheme guarantees that all closed-loop signals are uniformly ultimately bounded, while tracking error converges to a small neighborhood of the origin. The key features of the proposed controller are that the designed control law is time-delay independent and only one adaptive parameter is required to be estimated online which enhances its practical importance. In addition, the system considered in this paper contains unknown timevarying delays and the disturbance terms are less restrictive than those considered in (Wang et. al., 2007).

function f (x) defined on some compact set. Using singleton fuzzifier, product inference, and center-average defuzzifier to deduce the following fuzzy rules: Ri : IF x1 is F1i and x 2 is F2i and ...... and x n is Fni THEN y is B i for i = 1, 2,..., N , where x = [ x1 , x2 , L , xn ]T ∈ R n and y ∈ R are the input and the

output of fuzzy system, respectively, Fi j and Bi are fuzzy sets in R . The output of this fuzzy logic system (Wang, 1994) can be formulated as:

∑ y (∏ y( x) = ∑ (∏ N

i

i =1 N

j =1

µ F i ( x j )) j

(2)

n

µ i ( x j )) j =1 F

i =1

where

n

j

y i is the center of the fuzzy set

Bi at which

µ B i achieves its maximum value, and we assume that

2. PROBLEM FORMULATION AND FUZZY SYSTEM Consider the following unknown nonlinear time-varying delay systems: x&i (t ) = f i ( xi (t )) + g i ( xi (t )) xi +1 (t ) + hi ( xi (t − τ i (t ))) + d i (t ), x& n (t ) = f n ( xn (t )) + g n ( xn (t ))u (t ) + hn ( xn (t − τ n (t ))) + d n (t ), (1) y (t ) = x1 (t ) , 1 ≤ i ≤ n − 1

µ Bi ( y i ) = 1 .

Φ = [ y 1 , L , y N ]T and

Define

P ( x ) = [ p1 ( x ) , L, p N ( x) ]T with pi (x) given by:

fuzzy

basis

function

∏ j=1 µ F ( x j ) pi ( x ) = n N ∑i=1 ( ∏ j=1 µ F ( x j )) n

i j

i j

where x = [ x1 , x2 , L , xn ]T ∈ R n , u ∈ R and y ∈ R denote system state vector, system control input and output, xi = [ x1 , x2 , L , xi ]T , respectively. i = 1 , 2 , L , n −1 . Functions f i (⋅) , g i (⋅) and hi (⋅) are unknown nonlinear smooth functions. τ i (t ) are unknown time-varying delays,

d i (t ) denotes the external disturbance inputs which is assumed to satisfy di (t ) ≤ d i with d i being a constant, i =1, 2 ,L, n . Remark 1. The system (1) represents many physical processes such as recycled reactors, recycled storage tanks and cold rolling mills (Nguang , 2000). Since system (1) contains the external disturbance inputs and unknown timevarying delays. Obviously, the system considered in this paper is more general than the ones considered in (Ge et. al., 2004, Wang, 2007). The main objective of this paper is to design an adaptive fuzzy tracking controller for system (1) such that the system output y follows a desired trajectory yd , while all signals in the closed loop system remain bounded. (1) (i ) T i =1, 2 ,L, n . Define ydi = [ yd , yd , L , yd ] , Now, introducing the following assumptions for system (1). Assumption 1. The desired trajectory yd and their time derivatives up to nth order are continuous and bounded. Assumption 2. The signs of g i (⋅) are known and there exist an unknown constant bm and bM such that 0 ≤ bm ≤ gi (⋅) ≤ bM , i =1, 2 ,L, n . Assumption 3. There exist positive functions Q ij ( xi (t − τ i (t )))

The fuzzy logic system (2) can be written as: y ( x ) = ΦT P( x)

(3) If all membership functions are chosen as Gaussian functions, it has been shown in (Wang, 1994) that the above fuzzy logic system is capable of approximating any continuous nonlinear function over a compact set Ω x with any degree of accuracy. This property is shown by the following lemma. Lemma 1. (Wang, 1994) For Let f (x) be a continuous function defined on a compact set Ω . Then for any given constant ε > 0 , there exist a fuzzy logic system (3) such that: sup f ( x) − y ( x) ≤ ε x∈Ω

3. ADAPTIVE FUZZY CONTROL DESIGN This section is devoted to developing an adaptive fuzzy control design procedure based on the recursive backstepping technique. The design procedure is composed of n design steps. In each step, the fuzzy logic system ΦiT P( Zi ) will be used to approximate the unknown nonlinear function f i ( Z i ) . Let us define an unknown constant as: 1 Φi  bm

θ = max 

where bm is

2

a

 :1 ≤ i ≤ n 

constant

, defined

in

Assumption

2,

function f i and vector Zi will be specified in each step. Furthermore, for each step i , i = 1 , 2 , L , n − 1 , the virtual control law is constructed as: 

1 2

1 ˆ T θ zi Pi ( Z i ) Pi ( Z i ) 2ai2

for j = 1 , L , i such that hi ( xi (t − τ i (t ))) ≤ ∑ j =1 Qij ( xi (t − τ i (t ))) .

α i = − ki +  zi −

Remark 2. Assumption 2, implies that the unknown functions g i (⋅) are either strictly positive or negative. Without loss of generality, it is further assumed that g i (⋅) > bm > 0 . In addition, because the constants bm and bM are not used for controller design, they can be unknown. Fuzzy logic system will be used to approximate an unknown continuous

where ki > 0 and ai > 0 are design parameters, θˆ is the estimation of the unknown constant θ , zi = xi − α i −1 , and α o is equal to yd . Let us choose the adaptive law as:

i



&

n

θˆ = ∑ i =1

r 2 T zi Pi ( Z i ) Pi ( Z i ) − bθˆ , 2ai2

i =1, 2 ,L, n

(4)

(5)

where r > 0 and b > 0 are design parameters. In the following, for the purpose of simplicity, the time variable t will be omitted in state variables except the delayed state variables xi (t − τ i (t )) . Then, system (1) can be expressed as: x&i = f i ( xi ) + g i ( xi ) xi+1 + hi ( xi (t − τ i (t ))) + d i , 1 ≤ i ≤ n − 1 x& n = f n ( xn ) + g n ( xn )u + hn ( xn (t − τ n (t ))) + d n , (6) y = x1

Now, we propose the following backstepping design procedure through the following steps: Step 1: Define tracking error as z1 = x1 − yd . Then, its time derivative along the first subsystem (6) is given by (7) z&1 = f1 ( x1 ) + g1 ( x1 )α1 − y& d + h1 ( x1 (t − τ 1 (t ))) + d1 Consider the following Lyapunov function 1 2 bm ~ 2 z1 + θ (8) 2 2r ~ where θ = θ − θˆ . Then, the time derivative of Vz1 is given by Vz1 =

V&z1 = z1 ( f1 ( x1 ) + g1 ( x1 )α1 − y& d + h1 ( x1 (t − τ 1 (t ))) + d1 ) + z1 g1 ( x1 ) z 2 −

(9)

bm ~ ˆ& θθ r

1 2 1 2 2 z1 + ρ d1 , 2 2ρ 2

(10)

By using assumption 3, and completion of squares, one can obtain z1h1 ( x1 (t − τ 1 (t ))) ≤ z1 Q11 ( x1 (t − τ 1 (t ))) (11) 2 1 1 ≤ z12 + Q11 ( x1 (t − τ 1 (t ))) 2

]

Substituting inequalities (10) and (11) into (9) give that. 1 1 V&z1 ≤ z1 ( f1 ( x1 ) + g1 ( x1 )α1 − y& d + (1 + 2 ) z1 ) 2 ρ 2 b ~& 1 1 + z1g1 ( x1 ) z2 + Q11 ( x1 (t − τ 1 (t ))) + ρ 2 d12 − m θ θˆ 2 2 r

[

]

V1 = Vz1 + VU 1 with VU1

[

z1 g1 ( x1 )α1 ≤ −

& b ~ r ε2 1 V&1 ≤ − k1bm z12 + (a12 + 1 + ρ 2 d12 + m θ ( 2 z12 P1T ( Z1 ) P1 ( Z1 ) − θˆ)) 2 bm r 2a1 z (17) + z1 g1 ( x1 ) z 2 + [1 − 2 tanh 2 ( 1 )]U 1 η1 The coupling term z1 g1 ( x1 ) z2 will be handled in the next step,

and the last term in (17) will be considered later. Step i: Similarly, for each step i (2 ≤ i ≤ n − 1) , the dynamics of zi - subsystem is given by (18) z&i = fi ( xi ) + g i ( xi ) xi +1 + hi ( xi (t − τ i (t ))) + d i − α& i −1

Differentiating Vz i yields V&zi = zi ( fi ( xi ) + gi ( xi ) xi +1 + hi ( xi (t − τ i (t ))) + d i − α&i −1 ) i −1

where α& i −1 = ∑ j =1



i −1 j =1

(13)

∂α i −1 h j ( x j (t − τ j (t ))) + ∂x j



i −1 j =0

∂α i −1 ( j +1) ∂α i −1 &ˆ yd + θj ∂yd( j ) ∂yθˆ j

(20)

i

∂α i −1 h j ( x j (t − τ j (t ))) ≤ j =1 ∂x j

i −1

and zi ∑ +

1

i −1

j

j =1

k =1 2

∑ ∑

zi

[Q ( x (t − τ j k

k



∂α i −1 1 d j ) ≤ 2 zi2 j =1 ∂x ρ 2 j

i −1

And zi d i ≤ and η is a

1 z tanh 2 ( 1 ) is well defined at z1 = 0 . Therefore, it z1 η

can be approximated by fuzzy logic system. Thus, the fuzzy logic system (Wang, 1994) can be used to approximate f1 such that for given ε1 > 0 , δ1 ( Z1 ) ≤ ε1

∂α i −1 ( f j ( x j ) + g j ( x j ) x j +1 + d j ) + ∂x j

i −1

∑ j=1 ∑

k (t )))

]

1 2  ∂α i −1  z   k =1 2 i  ∂x j 

(21) 2

j

2

(22)

Also, from completion of squares, the following inequalities hold

positive constant. As pointed out by Remark 5 in [35], the

f1 ( Z1 ) = Φ1T P1 ( Z1 ) + δ1 ( Z1 ) ,

(19)

∑ j =1 zi Qij ( xi (t − τ i (t ))) i i 1 2 1 i ≤∑ z +∑ [Q j ( xi (t − τ i (t )))]2 j =1 2 i j =1 2

Differentiating V1 and then using (12), the inequality below can be easily obtained.

function

(16)

Now, substituting (15) (16) into (13) give that

(12)

]

1 V&1 ≤ z1 ( f1 ( Z1 ) + g1 ( x1 )α1 ) + z1 g1 ( x1 ) z 2 + ρ 2 d12 2 bm ~ ˆ& 2 z1 − θ θ + [1 − 2 tanh ( )]U1 r η1 1 where Z1 = [ x1 , y d , y& d ]T , U1 = [Q11 ( x1 )]2 , 2 1 1 2 z f1 ( Z1 ) = f1 ( x1 ) − y& d + (1 + 2 ) z1 + tanh 2 ( 1 )U1 2 z1 η1 ρ

bm ˆ 2 T 1 θz1 P1 ( Z1 ) P1 ( Z1 ) − (k1 + )bm z12 2 2 2a1

zi hi ( xi (t − τ i (t ))) ≤

2 1 1 = Q1 ( x1 ( s ) ds t −τ 1 (t ) 2



From (5), it can be verified that for any initial condition θˆ(t0 ) ≥ 0 , θˆ(t ) ≥ 0 for all t ≥ 0 . Consequently, it can be easily to verify that

By using Assumption 3 and completion of squares for the term zi hi ( xi (t − τ i (t ))) yields

To compensate the unknown time-varying delay term in (12), consider a Lyapunov-Krasovskii functional as follows t

(15)

Consider the following Lyapunov function Vz i = zi2

ρ >0

[ 2

bm 2 T 1 2 a2 b ε1 z θP ( Z1 ) P1 ( Z1 ) + 1 + m z12 + 2 1 1 2 2 2bm 2a1

z1 f1 ( Z1 ) ≤

1 2

By using completion of squares for the term z1d1 z1d1 ≤

where fuzzy approximation error. δ1 ( Z1 ) denotes Consequently, from (14), the completion of squares, and the definition of θ , it can be easily to verify that

(14)

2



 ∂α i −1  1 2   + ρ j =1 ∂x 2  j  

i −1

1 2 1 2 2 zi + ρ d i , 2 2ρ 2

i −1

∑ j =1 d j2

ρ >0

(23) (24)

Substituting (20) - (24) into (19) produces V&z i ≤ zi ( f i ( xi ) + g i ( xi ) xi +1 +



1 z + j =1 2 i

i

i −1

∑ j =1 ∑

2

+

1 2ρ 2

zi

i −1



∂α i−1 ( j +1) yd ) + ∂yd( j )





+

∑ ∑

j =0

 ∂α i −1  1 z −   + 2 i j =1 ∂x ρ 2 j   

i −1

1

i −1

j

j =1

k =1 2



i

[Q ( x (t −τ j k

k

1

j =1 2

i −1

∑ j=1

1 z k =1 2 i j

 ∂α i −1     ∂x j 

2

∂α i −1 ( f j ( x j ) + g j ( x j ) x j +1 ) − ∂x j

[Q ij ( xi (t − τ i (t )))]2

] + z ∂∂αyθˆ

k (t )))

2

i −1

i

j

&

1 2

θˆ j + ρ 2 ∑ j =1 d j2 (25) i

To compensate for the unknown time-varying delay terms in (25), consider the following Lyapunov-Krasovskii functional. Vi = Vz i + VU i with VU i =

Vn = Vz n + VU n With Vz n = VU n =

2

[Qkj ( xk (s))] ds ∑ j =1 ∫t −τ (t ) 2 [Qij ( x j (s))] ds + ∑∑ ∫ t −τ (t ) 2 j =1 k =1 1

t

i

i −1

2

j

1

t

j

k

Differentiating VU i results in. V&U i = U i −

= zi

∑ j =1 2 [Qij ( x j (t − τ j (t ))] −∑ j =1 ∑k =1 2 [Qkj ( xk (t − τ k (t ))] 1

i

 z z  2 tanh 2 ( i )U i + 1 − 2 tanh 2 ( i )U i − ηi η zi i  

[

i −1

∑ ∑

j



j

1

2

∑ j =1 2 [Qij ( x j (t − τ j (t ))] i

1

[

i −1

j =1

[

1 2 ρ 2

∑ j =1 d j2 i

(27)

where



1 z + j =1 2 i

i

i −1

1 z k =1 2 i

∑ ∑ j =1

j

 ∂α i−1     ∂x j 

2

2

 ∂α i −1  i −1 ∂α   − j =1 i −1 ( f j ( x j ) + g j ( x j ) x j +1 ) 2 j =1 ∂x ∂x j 2ρ j    i −1 ∂α i −1 ( j +1) z 1 2 + zi − y d + tanh 2 ( i )U i − ϕ i j =0 ηi zi 2ρ 2 ∂y d( j )

+

1

zi

i −1







(28)

where

+

1 2ρ 2 n −1

zn

n −1  ∂α n−1 

by introducing a function ϕi to compensate for this term. The function ϕi will be specified later in the sense that all signals in the closed-loop system remains bounded. Fuzzy logic system (3) can be used to approximate the unknown function fi ( Z i ) such that for given ε i > 0 , there exist f i ( Z i ) = ΦiT Pi ( Z i ) + δ i ( Z i ) , δ i ( Z i ) ≤ ε i

(29) where δ1 ( Z1 ) denotes fuzzy approximation error. Thus, using following the same procedure from (15) to (16). Consequently, (27) can be written as i b ~ r 2 T ε2 1 V&i = −k i bm zi2 + (ai2 + i + ρ 2 ∑ d j2 ) + m θ z P ( Z i ) Pi ( Z i ) 2 i i j =1 r

2 ai

 ∂α i−1 ˆ& 2 zi  θ ) + zi g i ( xi ) z i+1 (30) 1 − 2 tanh ( )U i + zi (ϕ i − η ∂θˆ i   Step n: In this step, the true control u will be constructed. The dynamics of the zn subsystem is given by (31) z&n = f n ( xn ) + g n ( xn ) xn +1 + hn ( xn (t − τ n (t ))) + d n − α& n −1

Consider the following Lyapunov-Krasovskii functional.

[

2

]

1 j Qk ( xk ( s )) ds t −τ k (t ) 2

∑∑ ∫

t

1 2 ρ 2

∑ j =1 d j2 n



1 z + j =1 2 n

n

∂α

d

1  ∂α n−1  z   k =1 2 n  ∂x j 

∑ j =1 ∑

2

j

n−1 ∂α n −1 1 z − ( f j ( x j ) + g j ( x j ) x j +1 ) −  + 2 n j =1 ∂x ∂ x ρ 2 j  j   2 z y d( j +1) + tanh 2 ( n )U n − ϕ n (33) ηn zn

∑ j=1 

∑ j=0 ∂y (nj−)1

n −1

2



where Z n = [ xn , yd ,L, yd( n ) ]T . Fuzzy logic system (3) can be used to approximate the unknown function f n ( Z n ) such that for given ε n > 0 , there exist f n ( Z n ) = ΦnT Pn ( Z n ) + δ n ( Z n ) ,

δ n (Z n ) ≤ ε n where fuzzy approximation δ n ( Z n ) denotes Consequently, if we choose the control law u as

(34) error. (35)

And follow the same procedure from (15) - (16) we can easily obtain.  z  d j2 ) + 1 − 2 tanh 2 ( n )U n η n   (36) bm ~ r 2 T ∂α n−1 &ˆ + θ z P ( Z ) P ( Z ) + z ( − θ ) ϕ n n n n n n n r 2an2 ∂θˆ

ε 1 V&n = −k nbm z n2 + (an2 + n + ρ 2 bm 2 2

& Remark 3. The adaptive law (5) shows that θˆ is a function of ∂α i −1 &ˆ θ in (27) cannot the whole state variables, thus the term ∂θˆ be combined as a part of fi ( Z i ) . This problem can be solved

bm

j

1 1  u = − kn +  zn − 2 θˆ z n PnT ( Z n ) Pn ( Z n ) 2 2 an  

where Z i = [ xi , yd ,L, yd(i ) ]T

2

+ z n ( f n ( Z n ) + g n ( xn )u ) +

f n ( Z n ) = f n ( xn ) + z n−1 g n ( xn ) +

]

2 1 j Qk ( xk ) . k =1 2 j

substituting (25) and (26) into the derivative of LyapunovKrasovskii functional, the delay terms are cancelled and yields ∂α &   z   V&i ≤ zi  ϕ i − i−1 θˆ  + 1 − 2 tanh 2 ( i )U i + zi ( f i ( Z i ) ˆ η ∂ θ    i 

f i ( Z i ) = f i ( xi ) + zi−1 g i ( xi ) +

]

(26)

] ∑ ∑

+ g i ( xi )α i ) + zi g i ( xi ) zi +1 +

t

Following the same procedure from (19) to (27) with i = n , Thus, we have. &  ∂α z   V&n ≤ z n  ϕ i − n−1 θˆ  + 1 − 2 tanh 2 ( n )U n ˆ η ∂ θ    n  (32)

2

]

2 1 j Qk ( xk (t − τ k (t )) j =1 k =1 2 2 i 1 where U j = j =1 Q ij ( x j ) + 2



i −1

2

[

2 n −1 1 n Q j ( x j ( s )) ds + t −τ j (t ) 2 j =1 k =1

∑ j =1 ∫ n

1 2 zn and 2



n

j =1

4. STABILITY ANALYSIS In this section, the stability analysis of the closed-loop system will be carried out. The boundness of all signals in the closed loop system will be proved. The following Theorem summarize our main results. Theorem 1. Consider the closed-loop system (1) under the assumptions 1-3, the control law (35) and parameter adaptation law (5). Suppose that the unknown functions f i (Z i ) , i = 1,K, n , can be approximated by fuzzy logic systems in the sense that approximation errors are bounded. Then, the closed-loop systems with the bounded initial condition θˆ(t0 ) ≥ 0 exhibits boundness to all signals in the closed-loop. Proof. For the stability analysis of the closed-loop system, let n us choose a Lyapunov function candidate V = ∑i =1Vi . Thus, from (17), (30) and (36), it can be clearly seen that the time derivative of V satisfies

V& = −bm +

bm ~  θ r 

+

1

ε2

∑i=1 ki zi2 + 2 ∑i=1 (ai2 + bmi + ρ 2 ∑ j =1 d j2 ) n

n

i

&

r

n

n

z 



∂α & z (ϕ i − i −1 θˆ) i =2 i ∂θˆ





∑i=1 2ai2 zi2 PiT (Z i ) Pi (Z i ) − θˆ + ∑i=1 1 − 2 tanh 2 (ηii )U i

n

(37)

The first step in the proof is devoted for obtaining ϕi such that the following inequality holds n ∂α & (38) ∑i = 2 zi (ϕi − ∂θiˆ−1 θˆ) ≤ 0 & with 0 < PiT Pi ≤ 1 and the definition of θˆ in (5), we have n n ∂α i −1 ˆ& ∂α − ∑ zi θ = ∑i= 2 bziθˆ i −1 i =2 ˆ ∂θ ∂θˆ n n ∂α i −1 r 2 T (∑ z P ( Z j ) Pj ( Z j )) − ∑ zi i =2 j =1 2a 2 j j ∂θˆ j n n i −1 r ∂α ∂α i −1 = ∑ bziθˆ i −1 − ∑ zi (∑ z 2 PT ( Z j ) Pj ( Z j ) i =2 i =2 j =1 2a 2 j j ˆ ˆ ∂θ ∂θ j +

r

∑ j =i 2a 2j z 2j PjT (Z j ) Pj (Z j )) n

=







n i=2

∂α bziθˆ i −1 − ∂θˆ

n z i=2 i

∂α i −1 ( ∂θˆ





n j =i

n z i =2 i

∂α i −1 ( ∂θˆ

∑i=2 n

zi

∂α i −1 ( ∂θˆ





i −1

r

j =1 2 a 2 j

z 2j PjT ( Z j ) Pj ( Z j ))

r 2 T z j Pj ( Z j ) Pj ( Z j )) 2a 2j

(39)

r 2 zj) j =i 2a 2j

∂α1 r 2 ∂α r 2 ∂α r 2 z + z2 1 z + L + z2 1 z 2 2 2 3 2 n ˆ ˆ ∂θ 2a2 ∂θ 2a3 ∂θˆ 2an

+ z3

∂α 2 r 2 ∂α 2 r 2 ∂α 2 r 2 z + z3 z + L + z3 z 2 3 2 4 2 n ∂θˆ 2a3 ∂θˆ 2a4 ∂θˆ 2an

+ zk



∑i=2 zi bθˆ

i



n



∂α j −1 ) ∂θˆ

∂α i−1 ∂α i −1 − ( ∂θˆ ∂θˆ

r + 2 zi ( 2 ai



∂α j −1 z j =2 j ∂θˆ

i

If we choose ϕi as ϕ i = −bθˆ −

∂α i −1 ∂α i −1 + ( ∂θˆ ∂θˆ

r zi ( 2ai2

∑ j =2 i

zj

r

i −1

∑ j=1 2a 2j z 2j PjT (Z j ) Pj (Z j ))  ) 

i −1

(41)

r

∑ j =1 2a 2j z 2j PjT (Z j ) Pj (Z j ))

∂α j −1 ) ∂θˆ

(42)

Then, (41) can be simplified to n n ∂α i −1 &ˆ − ∑ zi θ ≤ −∑i = 2 ziϕi (43) i =2 ∂θˆ This implies that (38) holds. Furthermore, for the third term in (37) and the definition of θˆ , the following can be easily verified & b b ~ zi2 PiT ( Z i ) Pi ( Z i ) − θˆ = m θ θˆ (44) r  b b~ For the term m θ θˆ , the following holds r bmb ~ ˆ bmb ~ 2 θθ ≤ ( −θ + θ 2 ) (45) r 2r i ε2 b b 1 n Let C = (ai2 + i + ρ 2 d 2 ) + m θ 2 , (38) can be = 1 i j =1 j 2 bm 2r



n

r

i =1 2 a 2 i





written as V& ≤ −bm



n k z2 i =1 i i

+C −

bmb ~ 2 θ + 2r



z 

∑i =1 1 − 2 tanh 2 (ηii )U i n

(46)

From (46), it can be seen that the first and third terms are negative, and that the second one is positive constant. However the last term may be positive or negative, depending on the value of zi . Following the same procedure used in proof of Theorem 1. in (Ge et. al. 2007)This implies that all signals in the closed loop system are all bounded.

∂α k −1 r 2 ∂α k −1 r z + zk zk2+1 + L 2 k 2 ∂θˆ 2ak ∂θˆ 2ak +1

5. SIMULATION RESULTS

∂α k −1 r 2 ∂α r z + L + z n−1 n−2 z n2−1 2 n 2 ∂θˆ 2a n ∂θˆ 2an−1

In this section, we demonstrate the effectiveness of the proposed adaptive fuzzy scheme through the following illustrative example.

∂α n−1 r 2 ∂α n−1 r 2 z + zn z 2 n 2 n ∂θˆ 2a n ∂θˆ 2a n ∂α j −1 n i r 2 z ( z ) i = 2 2a 2 i j =2 j ∂θˆ i

Example 1. In this example, we consider a two-stage chemical reactor system with delayed cycle streams. The reactor model is given by (Nguang , 2000).

+ z n−1 ≤



n

n

≤ z2

+ L + zk

∑i=2 2ai2 zi2 (∑ j=2 z j

bm ~  θ r 

Expanding the last term in (39) and rearranging the resulting sequence, yields n n ∂α i −1 r 2 T − ∑ zi (∑ z j Pj ( Z j ) Pj ( Z j )) i=2 j =i ˆ 2a 2j ∂θ ≤

r

+



(40)

Merging (39) and (40) results in n ∂α i −1 &ˆ − ∑ zi θ i=2 ∂θˆ n n i −1 r ∂α ∂α i −1 ≤ ∑ bziθˆ i −1 − ∑ zi (∑ z 2 PT ( Z j ) Pj ( Z j )) i=2 i=2 j =1 2a 2 j j ˆ ˆ ∂θ ∂θ j

x&1 = 0.1x12 + 2 x2 + 0.6 sin( x1 (t − τ 1 (t ))) + d1 (t ), x& 2 = u + x1 x22 + 0.2 x2 (t − τ 2 (t )) + 0.5( x12 (t − τ 2 (t )) +

x22 (t

(47)

− τ 2 (t ))) + sin( x2 (t − τ 2 (t ))) + d 2 (t ), y = x1

In the simulation, we choose the disturbance term as d1 (t ) = 0.2 cos(1.5t ) + 0.3 sin(0.5t ), τ 1 (t ) = 1 + 0.5 sin(1.5t ) , d 2 (t ) = 0.2 cos(0.5t ) + 0.2 sin(1.5t ) and τ 2 (t ) = 3 + 0.5 cos(t ) . Seven Gaussian fuzzy membership functions are chosen for simulation and are chosen as follows.

 − 0.5( xi + 1.5) 2   4  

,

 − 0.5( xi + 0.5) 2   4  

,

µ F 1 = exp  i

µ F 3 = exp  i

 − 0.5( x i + 1) 2   4  

µ F 2 = exp  i

1 x1 yd

,

0.8

0.6

0.4

 − 0.5 xi 2  ,  4 

µ F 4 = exp  i

0.2

0

-0.2

 − 0.5( xi − 0.5) 2   4  

µ F 5 = exp  i

 − 0.5( xi − 1) 2  µ F 6 = exp   i 4  

-0.4

-0.6

-0.8

 − 0.5( xi − 1.5)   4  

µ F 7 = exp  i

,

2

(48)

For the virtual control law (4), the control law (35) and the adaptive law (5), the following design parameters are chosen. k1 = k 2 = 50 , a1 = a2 = 1 , r = 30 and b = 0.75 . The initial condition is chosen as [ x1 (0) , x2 (0) ,θˆ(0)]T = [0.5 , 0.5 , 0]T . Fig. 1 shows the system output y and the reference signal yd . Fig. 2 shows the response of the state variable x2 . Fig. 3 displays the system control input. signal u . Fig. 4 shows the boundness of the adaptive parameter θˆ . From the simulation results, all signals in the closed-loop system are bounded, and good tracking performance is achieved.

-1

0

10

20

30 time (seconds)

40

50

60

Fig. 1. System output y and desired trajectory yd 0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

0

10

20

30

40

50

60

Time ( Seconds )

Fig. 6. x2 state trajectory 5

4

3

REFERENCES

2

1

0

Wang, L. X. (1994). Fuzzy systems and control: Design and stability analysis, Englewood cliffs, NJ: Prentice-Hall. Poursamad, A. and Markazi, A. (2009) Robust adaptive fuzzy control of unknown chaotic systems. Applied Soft Computing, volume (9), 970-976. Tong, S. C., Bin, C., and Yongfu, W. (2005) Fuzzy adaptive output feedback control for MIMO nonlinear systems. Fuzzy Sets and Systems, volume (156), 285-299. Yousef, H., Hamdy, M., El-Madbouly, E., and Eteim, D. (2009) Adaptive fuzzy decentralized control for interconnected MIMO nonlinear subsystems. Automatica, volume (45), 456-462. Krstic, M., Kanellakopoulos, I., and Kokotovic, P. (1995) Nonlinear and adaptive control design, Wiley, New York. Polycarpou, M. and Ioannou, P. (1996) A robust adaptive nonlinear control design. Automatica, volume (32), 423427. Yang, Y.S., Feng, G., and Ren, J.S. (2004) A combined backstepping and small-gain approach to robust adaptive fuzzy control for strict-feedback nonlinear systems. IEEE Transactions on Systems, Man, and Cybernetics Part-A: Systems and Humans, volume (34), 406-420. Tong, S. C., Li, Y., and Shi, P. (2009) Fuzzy adaptive backstepping robust control for SISO nonlinear system with dynamic uncertainties, Information Sciences, volume (179), 1319-1332. Li, H.Y., Qiang, S., Zhuang, X.Y., and Kaynak, O. (2004) Robust and adaptive backstepping control for nonlinear systems using RBF neural networks. IEEE Transactions on Neural Networks, volume (15), 693–703. Gu, K., Kharitonov, V. L., and Chen, J. (2003) Stability of Time-Delay Systems, Birkhäuser, Berlin, Germany. Richard, J. P. (2003) Time-delay Systems: an overview of some recent advances and open problems. Automatica, volume (39), 1667-1694. Nguang, S. (2000) Robust Stabilization of a Class of TimeDelay Nonlinear Systems. IEEE Transactions on Automatic Control, volume (45), 1667-1694.

-1

-2

-3

-4

-5

0

10

20

30

40

50

60

Time ( Seconds )

Fig. 3. The control input u -3

x 10

12

10

8

6

4

2

0

-2

-4 0

10

20

30 Time ( Seconds )

40

50

60

Fig. 4. The adaptive parameter θˆ

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