Adaptive Fuzzy Output-Feedback Sliding Mode Control for Switched Uncertain Nonlinear Systems with Input Saturation

Adaptive Fuzzy Output-Feedback Sliding Mode Control

for Switched Uncertain Nonlinear Systems with Input

Saturation

Chiang-Cheng Chiang*_{, Wei-Tse Kao}

Department of Electrical Engineering, Tatung University, Taipei, Taiwan, Republic of China

Abstract

The paper deals with the problem of adaptive fuzzy output-feedback sliding mode control for switched uncertain nonlinear systems with input saturation. Based on sliding mode control technique, an adaptive fuzzy sliding mode controller is designed for switched uncertain nonlinear systems by using fuzzy logic systems to approximate the uncertain functions. By means of state variable filters, a state observer is constructed to solve the problem of unmeasured states of the system. Moreover, input saturation which is one of the most important input constraints usually occurs in many industrial control systems. The controller is designed by choosing a suitable Lyapunov function. Actually, it is verified that all the signals in the closed-loop system can not only guarantee uniformly ultimately bounded, but also achieve well performance. Finally, some computer simulation results of a practical example are illustrated to show the performance of the proposed approach.

Keywords

Switched nonlinear systems, Lyapunov function, Sliding mode control, Input saturation, Fuzzy logic system

1. Introduction

Recently, the control problem of switched uncertain nonlinear systems has attracted much attention [1], [2]. For non-strict feedback uncertain switched nonlinear systems, an adaptive fuzzy output-feedback stabilization control method has been already studied in [1]. The adaptive fuzzy control problem of nonlinear switched stochastic pure feedback systems has been discussed in Yin et al. [2]. Also, a lot of decentralized controller design methods for switched nonlinear systems have been investigated and various successful control applications have been developed. Meanwhile, many research results of strict feedback form systems or non-strict feedback form systems have been proposed [3], [4]. In [3], an adaptive fuzzy controller has been presented for a class of switched uncertain nonlinear systems with strict-feedback form. Liu et al. [4] focus on backstepping-based adaptive neural control for switched nonlinear systems in nonstrict-feedback form.

Input saturation often occurs in practical engineering, which is a source of limiting the system performance severely and even leads to the system instability. Recently,

* Corresponding author:

[email protected] (Chiang-Cheng Chiang) Published online at http://journal.sapub.org/ajis

Copyright©2019The Author(s).PublishedbyScientific&AcademicPublishing This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

the problem of input saturation has attracted much attention. Zhou et al. [5] presents an adaptive fuzzy control approach for a class of uncertain non-strict feedback systems with input saturation and output constraint. In [6], a fuzzy backstepping output tracking control approach was developed for a class of single-input-single-output (SISO) strict feedback nonlinear systems with unmeasured states and input saturation.

Sliding mode controller (SMC) is a powerful technique that has been successfully applied to control nonlinear systems with external disturbances and parameter variations. The SMC as ordinary design tools for control systems have been well accepted, and the major benefits of the SMC are the following: 1) its robustness against for a class of system uncertainties or perturbations; 2) a fast response and a good transient performance. Based on the aforementioned advantages, sliding mode control has been applied widely [7-9]. In [8], adaptive sliding mode approach was presented to handle disturbances of unknown bounds in nonlinear systems. Zhang et al. [9] proposed the sliding mode controller to tackle the trajectory tracking problems for discrete non-affine nonlinear systems with disturbance.

sliding mode controller to treat the problem for a class of multi-input multi-output unknown nonlinear time-delay systems with input saturation By means of two effective technical measures, more relaxed designs of discrete-time Takagi–Sugeno fuzzy-model-based observers was presented in [12]. The observer-based robust adaptive fuzzy control for MIMO nonlinear uncertain systems with delayed output has been discussed in [16]. Hwang et al. [17] presented an adaptive fuzzy hierarchical sliding-mode control for the trajectory tracking of uncertain under-actuated nonlinear dynamic systems.

In this paper, a fuzzy sliding mode control approach based on an observer is presented for a class of switched nonlinear systems with input saturation. Comparing with the existing results, the main features and contributions of this paper can be listed below.

(i) Based on the adaptive fuzzy control method, input saturation is considered in the design procedure, which is more common for application in practical engineering.

(ii) By sliding mode control technique design, an adaptive fuzzy sliding mode controller is designed for the system via using fuzzy logic systems to approximate uncertain functions, and the fuzzy state observer is employed to estimate the unavailable state for measurement.

(iii) Choosing an appropriate Lyapunov function, it is theoretically guaranteed that all the signals in the closed-loop system are uniformly ultimately bounded and obtain good system performance under our designed controller.

The remainder of this paper is organized as follows. Section II describes the problem formulation and preliminaries together with the definition of fuzzy logic systems (FLSs) are given. In Section III, adaptive fuzzy sliding mode controller and observer are designed for a class of switched uncertain nonlinear systems with input saturation and the stability analysis of the whole system is also verified by Lyapunov functional. Some computer simulation results of a practical example are indicated to show the performance of the proposed approach in Section IV. Finally, the conclusion is drawn in Section V.

2. Problem Statement and Preliminaries

2.1. System Description

Consider a class of switched uncertain nonlinear systems with input saturation in strict-feedback from [15]:

1 2 1, (t) 1 1, (t)

1 1, (t) 1 1, (t)

(t) n, (t) , (t)

1

( ) ( , )

( ) ( , )

( (t)) ( ) ( , )

n n n n n

n n

x x f x d t

x x f x d t

x sat u f d t

y x

σ σ

σ σ

σ σ σ

− − − −

= + +

  

 _{= + +}



 _{= + +}



 ₌



x

x

x x

  



(1)

where xi =[ ,x x_{1 2,}, ] ,xi T ∈ R ii =1,2,..., ( = ) n x xn is

the system state vector which is assumed to be unavailable for measurement, uσ(t) ∈R and y( )t ∈Rn _{are input and} output of the system output, respectively. The function

(t) :[0, ) M {1,2,...,m}

σ ∞ → = , is a switching signal which is assumed to be a piecewise continuous (from the right) function of time. If σ

( )

t = k, then we say the k-th switched subsystem is active and the remaining switched subsystems are inactive. fi k i, ( ),(i 1,2,...,n,k M)x = ∈ are unknown smooth nonlinear functions,

, ( , ),(i 1,2,...,n,

i k

d xt = k M)∈ are unknown external bound disturbances. sat(u (t))_k denotes the nonlinear saturation characteristic.

Eq. (1) can be rewritten as

(t) (t) (t) (t)

1

[ ( ) ( , )] ( (t)) n

i i, i i, n

i f x d t sat u

y

σ σ σ σ

= 

= + + +

   = 

∑

x A x B x B

Cx



(2) where

(t)

0 1 0 0

0 0 1 0

,

0 0 0 1

0 0 0 0

n n

R × for all Aσ A

 

 

 

 

= ∈ =

 

 

 

 

A

      

 

 1

{

}

[0 0 1 0 0]T n , 1,2,3,...,

i

i R i n

×

= ∈ =

B 

[

_{1 0 0}

]

_R1×n

= ∈

C 

Assumption 1: 0< d_{i k,} ( , )xt ≤h_{i k,} ( )x < ∞ , where , ( )

i k

h x are unknown functions.

In order to deal with the control constraints for convenience, the saturation function 𝑠𝑠𝑠𝑠𝑠𝑠(𝑢𝑢(𝑠𝑠)) can be rewritten as

𝑠𝑠𝑠𝑠𝑠𝑠(𝑢𝑢(𝑠𝑠)) = 𝜒𝜒(𝑢𝑢(𝑠𝑠)) ∙ 𝑢𝑢(𝑠𝑠) (3) where 𝜒𝜒(𝑢𝑢(𝑠𝑠)) be expressed as

𝜒𝜒(𝑢𝑢(𝑠𝑠)) = � 𝑢𝑢1 , if −𝑢𝑢𝑚𝑚/𝑢𝑢 , if 𝑢𝑢 > 𝑢𝑢𝑚𝑚 ≤ 𝑢𝑢 ≤ 𝑢𝑢𝑚𝑚 𝑚𝑚 −𝑢𝑢𝑚𝑚/𝑢𝑢, if −𝑢𝑢𝑚𝑚 > 𝑢𝑢

(4)

Control Objective: Design a controller for (1) such that the states of the system (t)x will converge asymptotically to a neighborhood of zero.

2.2. Description of Fuzzy Logic Systems

The fuzzy logic system performs a mapping from

n

U⊂R to V ⊂R. Let U U= ₁× ×_ U_n where U_i ⊂R, 1,2, ,

( )

1 1 2 2

: IF is , and is , and and, is THEN is , for 1,2, , .

l l l l

n n

l

R x F x F x F

y G l= M



 (5)

in which x=

[

x x_{1 2}, , , x_n

]

T∈U and y V∈ ⊂R are the input and output of the fuzzy logic system, F_il and Gl are fuzzy sets in Ui and V , respectively. The fuzzifier maps a

crisp point x=

[

x x_{1 2}, , , x_n

]

T into a fuzzy set in U . The fuzzy inference engine performs a mapping from fuzzy sets in U to fuzzy sets inV , based upon the fuzzy IF-THEN rules in the fuzzy rule base and the compositional rule of inference. The defuzzifier maps a fuzzy set in V to a crisp point in V .

The fuzzy systems with center-average defuzzifier, product inference and singleton fuzzifier are of the following form:

( ) T

y=θ ξ x (6) where T _{= }θ1, ,θB



θ  with each variable θl as the point at which the fuzzy membership function of Gl

achieves the maximum value and ξ x( ) [ ( ),...,= ξ1 x

T ( )] M

ξ x with each variable ξl( )x as the fuzzy basis function defined as

1

1 1

( ) ( )

( )

n

l i

i F

l i

M _n

l i i _Fi l

x

x µ ξ

µ

=

= =

=

 

 

 

∏

∑ ∏

x (7)

where _l( )_i Fi x

µ is the membership function of the fuzzy set.

3. Controller Design and Stability

Analysis

3.1. Observer Design

First, let the unknown nonlinear functions f x_{i,k i}( ) can be approximated over a compact set Γ, by the fuzzy logic systems as follows:

ˆ _{( |} ˆ ₎ ˆT _{( )} i,k i fi f_i i

f x θ =θ ξ x (8)

i,k

ˆ _{( |ˆ} ˆT₎ ˆT _{( )ˆ} i h_i h_i i

h x θ =θ ξ x (9) where ( )ξ x is the fuzzy basis vector, ˆT_f ˆ_hT

i and i

θ θ are the corresponding adjustable parameter vectors of each fuzzy logic systems.

Owing to the unavailable states of the system and the unavailable elements of the output error vector in many practical systems, the fuzzy logic systems (8) and (9) are not used to control nonlinear systems whose states are not obtained for measurement. Therefore, we must employ an

observer to estimate. Let ˆx be defined as the estimates of x. Then, we can obtain the following fuzzy logic systems as

i,k

ˆ _{( |ˆ} ˆ ₎ ˆ _{( )ˆ} ˆ _{( |ˆ} ˆ ₎ ˆ _{( )ˆ}

T T

i,k i f_i f_i i

T T

i h_i h_i i

f x x

h x x

θ θ ξ

θ θ ξ =

= (10) In order to estimate the state, the observer for system (2) can be chosen as follows:

, 1

ˆ ˆ

ˆ ˆ [ ( |ˆ ) ] ( (t)) (y y)ˆ ˆ ˆ

n

i i,k i _fi i k n k k

i

f x θ v B sat u

= 

= + + + + −

   = 

∑

x Ax B L

y Cx



(11) where,

1 _{1 1}

1 2 1 ,

[ ]T n ,

k l,k l ,k ln ,k ln,k R vi k

× _×

−

= ∈ ∈

L  　 　R .

k

L is the observer gain matrix to guarantee the characteristic polynomial of A - L C_k to be Hurwitz. Let us define the estimation error vector as x x x= −ˆ andy = y - y 　ˆ, then by (2) and (11), we obtain

, 1

ˆ _ˆ ˆ

( k ) n i i,k i[ ( ) i,k i( | _fi)+ i,k( , ) i k]

i f x f x d t v

θ =



= − + −

   = 

∑

x A L C x B x

y Cx

 

 

－

(12) It is assumed that x,x θˆ, ˆ_f and θˆ_h belong to compact sets

ˆ ˆ

ˆ

, , and

x x _{θf θh}

Ω Ω Ω Ω respectively, which is defined as

{

}

:Rn N

Ω =_x x∈ x ≤ _x< ∞ (13)

{

}

ˆ :ˆ Rn ˆ Nˆ

Ω =_x x∈ x ≤ _x < ∞ (14)

{

}

ˆ_f ˆf RM : ˆf Nf

Ω_θ = θ ∈ θ ≤ < ∞ (15)

{

}

ˆ_h ˆh RM : ˆh Nh

Ω_θ = θ ∈ θ ≤ < ∞ (16) where Nx,Nxˆ,Nf andNh are the designed parameters,

and M is the number of fuzzy inference rules. Let us define the optimal parameter vector θ*_f and θ_h* as follows:

*

ˆ _ˆ ˆ ˆ ˆ

arg min sup ( ) ( | )

f_i ,k i i,k i f_i

f _f f x f x

θ θ

∈Ω _∈Γ

 

= _{ }

 

θ _θ x ｉ － (17)

*

,

ˆ _ˆ ˆ ˆ ˆ

arg min sup ( ) ( | )

h_i i k i i,k i h_i

h _h h x h x

θ θ

∈Ω ∈Γ

 

= _{ }

 

θ _{θ x} － (18)

where θ*_fi and θ_hi* are bounded in the suitable closed set ˆ

θf

Ω and Ω_θhˆ , respectively.

The parameter estimation errors can be defined as * _{ˆ , for i 1,2,...,n}

fi fi fi

* _{ˆ , for i 1,2,...,n} hi hi hi

θ =θ −θ = (20) *

1 ,i k fi,k i( ) ( |x fˆi,k ixˆ _fi), for i 1,2,...,n

ω = － θ = ₍₂₁₎

*

2 ,i k h xi,k i( ) h xˆi,k i( |ˆ _hi), for i 1,2,...,n

ω = － θ = ₍₂₂₎

1,k =[ω11,k,ω12,k,...,ω1 ,n k]T

ω

2,k =[ω21,k,ω22,k,...,ω2 ,n k]T

ω

are the minimum approximation errors, which correspond to approximation errors obtained when optimal parameters are used.

Applying (19) and (21), we obtain

* *

1 _,

1 , ,

1

( )

ˆ ˆ

ˆ ˆ

[ ( ) ( | )

ˆ _{( |ˆ )} ˆ _{( |ˆ )+ ( , ) ]}

ˆ

( ) [ (x ) ( , ) ]

k

n _{i i,k i i,k i fi}

i _{i,k i fi i,k i fi i,k i k} n

T

k i _fi i i k i,k i k

i

f x f x

f x f x d t v

d t v

θ

θ θ

θ ξ ω

=

=

= −

 

 

+  

+ − − 

 

= − + + + −

=

∑

∑

x A L C x B

x

A L C x B x

y Cx

 

 

 

－

(23)

The output error dynamics of (23) can be expressed as follows:

1 , ,

ˆ

(s)[ T (x ) ( , ) ] , for i 1,2,3,...,n i _fi i i k i,k i k

H θ ξ ω d t v

= + + − =

y  x

(24) where

1

(s) (s ( )) , for i 1,2,3,...,n

i k i

H =C I A L C− − − B = (25) and s denotes the complex Laplace transform variable. As has been discussed, we could not obtain all elements of x, because not all states of the system are available for measurement. Hence, we could not obtain all elements of x. We will employ the state variable filters [14] to cope with this problem. First, we choose a stable filter ( )G s as the following form:

1 2

1 1

1

(s) _{n n}

n G

S − g S − g −

=

+ + +_ (26) where gi >0 1,2,...,n 1i= − are coefficients of the Hurwitz polynomial.

(s) n ₁ n-1 ₂ n-2 _n-1

p =s + g s + g s + + g (27) Introducing (26) into (24), we can obtain the steady-state equation

1

1 ˆ 1 , ,

(s) H (s) {x } (s)[ (x ) ( , ) ] for i 1,2,3,...,n

T

i _fi i i k i,k i k

G − ₌G θ ξ ₊ω ₊d t v₋

=

x

 

(28) Define a set of state variable filters

(s) G(s)s ,i i

T = i=1,2,...,n-1

The corresponding filtered signals are defined as follows: 1

(s){x } , 1,2,...,n 1

f_i i

x =T  i= − (29) 1,kf =T0(s){ 1,k}

ω ω (30) 0(s){ }

f T

ξ = ξ (31) , 0(s){v },

i kf i k

v =T (32) and

, ( , t) T (s){0 , ( , t)}

i kf i k

d x = d x (33) Then, Eq. (23) can be rewritten as follows:

1 , ,

1

( )

ˆ

[ (x ) ( , ) ]

f k f

n T

i _fi f i i kf i,kf i kf i

f

d t v

θ ξ ω

=

 = − 

_{+ + + −}

   = 

∑

x A L C x

B x

y Cx

 



 

(34)

where

1 2 3

[ , , , , ]T n

f = x f x f x f xnf ∈R

x      1, 2,

n 1, n,

1 0 0

0 1 0

0 0 1

0 0 0

k k

n n k

k k l l

R l

l

×

− −

 

 ₋ 

 

 

− = ∈

₋ 

 

 ₋ 

 

A L C

 

    

 



{

}

[

]

1 1

,

1 0 0

[0 0 1 0 0]T 1,2,3,...,

i

n i

n _i R

R

n

×

×

=

= ∈ =

∈ B

C 



Then, we define

ˆ

f f f

ω =ω −ω ₍₃₅₎ and

ˆ

h h h

ω =ω −ω (36) where ˆω_f and ˆω_h are the estimated of ω_f and ω_h, respectively. The following assumptions are satisfied

1,kf ≤ωf

ω (37) 2,k ≤ωh

ω (38) Based on Lyapunov stable theorem, we can obtain the robust compensation term v_{i kf,} and the parameter update laws as follows:

, 1 1 1

1 1 1

ˆ ˆ

ˆ ˆ ˆ

= + ( )

n n n

T T T

i k f i k f i k f

i kf _n f _n h _{n hi}

T T T

f k i f k i f k i

i i i

i i i

v = ω = ω = h θ

= = =

+

∑

∑

∑

∑

∑

∑

B P x B P x B P x x x P B x P B x P B

  

   (39)

ˆ _{(x )ˆ} T f_i f f_{i i} i f k i

ˆ _{(x )ˆ} T h_i h h_{i i} i f k i

θ =γ ξ x P B (41)

1 ˆ_{f f} T_{f k} n _i

i

ω

ω γ

=

= x P

∑

B

 _{ (42)}

1 ˆ_{h h} T_{f k} n _i

i

ω

ω γ

=

= x P

∑

B

 _{ (43)}

whereγ_fi,γ_hi,γω_f and γω_hare positive constants.

Remark 1: Without loss of generality, the adaptive laws used in this paper are assumed that the parameter vectors are within the constraint sets or on the boundaries of the constraint sets but moving toward the inside of the constraint sets. If the parameter vectors are on the boundaries of the constraint sets but moving toward the outside of the constraint sets, we have to use the projection algorithm to modify the adaptive laws such that the parameter vectors will remain inside of the constraint sets. Readers can refer to reference [13]. The proposed adaptive law (40)-(43) can be modified as the following form:

{

}

ˆ

ˆ

(x ) , if ( N ) or ( = N

ˆ and (x ) 0)

ˆ

(x ) , if ( = N

fi

T

f f_{i i} i f k i f f

f f

T T

f k i f f_i i T

f f_{i i} i f k i f f P θ γ ξ ξ γ ξ < = ⋅ ≥

x P B θ θ x P B θ x P B θ





 

ˆ and T_{f k i} T_{f f}ξ_i(x ) < 0) _i           _⋅

 x P B θ

(44)

where P

{

γ ξ_{f fi i}(x )ˆ_i x PT_{f k}

}

is defined as

{

}

2 ˆ (x ) ˆ ˆ ˆ ˆ

= (x ) + (x )

ˆ

T f f_{i i} i f k

T f f

T T

f f_{i i} i f k i f_i f k i f_i i f

P γ ξ

γ ξ γ ξ

x P

θ θ x P B x P B

θ



  (45)

{

}

ˆ

ˆ

(x ) , if ( N ) or ( = N

ˆ and (x ) 0)

ˆ

(x ) , if ( = N

hi

T

h h_{i i} i f k i h h

h h

T T

f k i h h_i i T

h h_{i i} i f k i h h

P θ γ ξ ξ γ ξ < = ⋅ ≥

x P B θ

θ

x P B θ

x P B θ





 

ˆ

and T_{f k i} T_{h h}ξ _i(x ) < 0) _i           _⋅

 x P B θ

(46) where P

{

γ ξ_{h hi i}(x )ˆ_i x PT_{f k}

}

is defined as

{

}

2 ˆ

(x ) =

ˆ ˆ

ˆ ˆ

(x ) + (x )

ˆ T

h h_{i i} i f k

T

T T h h

h h_{i i} i f k i h f k i_i h_i i h

P γ ξ

γ ξ γ ξ

x P

θ θ x P B x P B

θ



  (47)

The main result of robust adaptive fuzzy observer scheme is summarized on the following theorem.

Theorem 1: Consider a class of switched uncertain nonlinear systems with input saturation in strict-feedback from (1). The robust adaptive fuzzy observer is defined by (11) with adaptation laws given by (40)-(43). For the given positive definite matrix Qk , if there exist symmetric

positive definite matrix P _k such that the following Lyapunov equation

(A L C P− _k )T _k +P A L C_k( − _k )= −Q_k (48) is satisfied, then all the closed-loop signals are bound, and the estimation errors converge to a neighborhood of zero.

Proof. Consider the Lyapunov function candidate

1,

1

2 2

1

1_[ 1

2

1 1 1 _]

n

T T

k f k f f f_{i i}

f

i i

n T

f h

h h_{i i} h

i i f h

V ω ω θ θ γ θ θ ω ω γ γ γ = = = + + + +

∑

∑

x P x   

  _{ } (49)

2 2,k 1₂ k

V = S

(50) By the time derivate of V_1,k and the facts

ˆ _; ˆ _{; ˆ ; ˆ}

f_i f_i h_i h_i f f h h

θ = −θ θ = −θ ω  = −ω ω = −ω we obtain

1,

1 1

1 , ,

1

1 [ ]

2

1 _ˆ 1 _ˆ 1 _ˆ 1 _ˆ

1 = [ ( - ) ( )) ] 2 ˆ + { [ ( ) ( , ) ]} T T

k f k f f k f

n n

T T

f f h h

f f_{i i} h h_{i i}

f h

i i i i f h

T T

f k k k k f

n

T T

f k i _fi f i i kf i,kf i kf i

V

x d t v

ω ω θ θ θ θ ω ω ω ω γ γ γ γ θ ξ ω = = = = + − − − − + − + + −

∑

∑

∑

x P x x P x

x A L C P P A L C x

x P B x

   _{   }   _{ }   _{ }     1 1 1 1 , 1 1 , 1

1 _ˆ 1 _ˆ 1 _ˆ 1 _ˆ

1 _ˆ = [ ( ) ]+ [ ( )] 2 [ ( , )] [ ] [ ]

n _T n _T

f f h h

f f_{i i} h h_{i i}

f h

i i i i f h

n

T T T

f k f f k i _fi f i i

n n

T T

f k i i,kf f k i i kf

i i

n T

f k i i kf i x d t v ω ω θ θ θ θ ω ω ω ω γ γ γ γ θ ξ ω = = = = = = − − − − − + + − −

∑

∑

∑

∑

∑

∑

x Q x x P B

x P B x x P B

x P B

  _{ }   _{ }        1 1 1 _ˆ

1 _ˆ 1 _ˆ 1 _ˆ

n _T

f f_{i i} f i i n

T

f f h h

h h_{i i} h

i i ωf ωh

FromAssumption 1, it yields

1,

1

, 1 ,

1 1 , 1 1 1 1 _ˆ = [ ( ) ]+ [ ( )] 2 [ ( )] [ ] 1 _ˆ [ ]

1 _ˆ 1 _ˆ 1 _ˆ

=

n

T T T

k f k f f k i _fi f i

i

n n

T T

f k i i k f k i i k f

i i

n n

T T

f k i i kf _{f fi i} f

i i i

n T

f f h h

h h_{i i} h

i i f h

V x h v ω ω θ ξ ω θ θ γ θ θ ω ω ω ω γ γ γ = = = = = = ≤ − + + − − − − −

∑

∑

∑

∑

∑

∑

x Q x x P B

x P B x x P B

x P B

  _{  }       _{ }  _{ } 1 * , 1 *

1 , ,

1 1

1_{[ ( ) ]+ [ ( )]ˆ} 2 ˆ _ˆ ˆ ˆ _ˆ ˆ [ ( )- ( | )+ ( | ) ˆ _ˆ ˆ ˆ _ˆ ˆ - ( | )+ ( | )] [ ] [ ] 1 n

T T T

f k f f k i _fi f i i

n T

f k i i k i,k h_i i,k h_i i

i,k i h_i i,k i h_i

n n

T T

f k i i kf f k i i kf

i i

f i i

x

h h x h x

h x h x

v θ ξ θ θ θ θ ω γ = = = = = − + + − −

∑

∑

∑

∑

x Q x x P B

x P B x

x P B x P B

       ｉ ｉ 1 1

1 1 1

ˆ ˆ _{ˆ ˆ}

n n

T T

f f h h

f f_{i i} h h_{i i} h

i i ωf ωh

θ θ θ θ ω ω ω ω

γ γ γ

=

− − −

∑

 

∑

  _  _ 

By employing (20) and (22), we can get

1, 1 2 , 1 1, , 1 1

1_{[ ( ) ]+ [ ( )]ˆ} 2 ˆ ˆ ˆ ˆ [ ( )+ω + ( | )] [ ] [ ] n

T T T

k f k f f k i _fi f i

i n

T T

f k_i i hi i i k i,k i hi

n n

T T

k i kf f k i i kf f

i i

V x

x h x

v θ ξ θ ξ θ = = = = ≤ − + + −

∑

∑

∑

∑

x Q x x P B

x P B

x P B ω x P B

  _{  }     1 1

1 _ˆ 1 _ˆ 1 _ˆ 1 _ˆ

n n

T T

f f h h

f f_{i i} h h_{i i}

f h

i i i i ωf ωh

θ θ θ θ ω ω ω ω γ γ γ γ = = −

∑

  −

∑

  −   −   1 1 1 1 1 1

1 _ˆ 1 _ˆ

[ ( ) ]+ ( ) 2 1 _ˆ ˆ ( ) 1 _ˆ + ˆ [ n n

T T T T

f k f f k i f_i f i f f_{i i} f

i i i

n n

T T T

f k i h_i i h h_{i i} h

i i i

n T

k i h h h

f i h n T k i f i x x h ω θ ξ θ θ γ θ ξ θ θ γ ω ω ω γ = = = = = =     ≤ −  −          +_ − _        ₋        +

∑

∑

∑

∑

∑

∑

x Q x x P B

x P B

x P B

x P B

              1 , 1 ˆ ˆ ( | )] 1 _ˆ [ ] i,k i _hi n

T

f k i f f f

i f

n T

f k i i kf i x v ω θ ω ω ω γ = =     + −      −

∑

∑

x P B

x P B

  



(51)

Using (40)-(43), we have

1,

1

1

,

1 1

1_{[ ( ) ]+ ( )ˆ} 2 ˆ _ˆ ˆ [ ( | )] ˆ + ( ) [ ] n T T

k f k f f k i h

i n

T

k i i,k i h

f i

i

n n

T T

f k i f f k i i kf

i i V h x v ω θ ω = = = = ≤ − + −

∑

∑

∑

∑

x Q x x P B

x P B

x P B x P B

 _{  }



 

(52)

Then employing the robust compensation term ν_{i kf,} (39), the above equation can be rewritten as

1,k 1 [ ( ) ]₂ Tf k f

V ≤ − x Q x  (53) Therefore, it can be concluded that V_1,k ≤0 from (53), and the estimation errors of the closed-loop system converges asymptotically to a neighborhood of zero based on Lyapunov synthesis approach. This completes the proof. 3.2. Controller Design

Next, we design the observer-based incremental sliding mode controller. Define the sliding surfaces as follows:

1, 1ˆ 2, 2ˆ 3, 3ˆ 1, ˆ 1 ˆ

=c c c c

k k k k n k n n

S x + x + x + + − x − +x , (54)

where c_{i k,} for i=1, 2,…, n-1, are positive constants. Differentiating S_k with respect to time, we have

1, 1 2, 2 3, 3 1, 1

, 1 , ,

1

, ,

ˆ ˆ ˆ ˆ ˆ

=c c c c

ˆ ˆ

ˆ ˆ ˆ

= [ (x ( | ) (y y)]

ˆ _ˆ ˆ _ˆ

[sat(u (t)) ( | ) (y y)]

k k k k n k n n

n

i k i i,k i _fi i k i k i

k n,k n _fn n k i k

S x x x x x

c f x v l

f x v l

θ θ − − + = + + + + + + + + − + + + + −

∑

      _ (55)

The following controller can be chosen as:

, 1 , ,

1

, ,

ˆ ˆ

ˆ ˆ ˆ

[ (x ( | ) (y y)]

ˆ _ˆ ˆ _ˆ

[ ( | ) (y y)] n

k i k i i,k i _fi i k i k i

n,k n _fn n k i k k

u c f x v l

f x v l KS

θ θ + = = − + + + − − + + − −

∑

(56) wherev_{i k,} can be obtained by backward from vi kf_, and

Kis a positive constant.

Theorem 2: Consider the switched uncertain nonlinear system (1). The proposed adaptive fuzzy sliding mode controller defined by (56) with adaptive laws (40)-(43) guarantees that all signals of the closed-loop system are bounded, and converge to a neighborhood of zero.

Proof: Consider the Lyapunov function candidate 2

2,k 1₂ k

V = S . Differentiating the and V2,k with respect to

2,

, 1 , ,

1

, ,

, 1 , ,

1

ˆ ˆ

ˆ ˆ ˆ

{ [ (x ( | ) (y y)]

ˆ _ˆ ˆ _ˆ

[sat( (t)) ( | ) (y y)]}

ˆ ˆ

ˆ ˆ ˆ

{ [ (x ( | ) (y y)]

ˆ [ ( (t)) (t)

k k k n

k i k i i,k i _fi i k i k i

k n,k n _fn n k i k

n

k i k i i,k i _fi i k i k i

k k

V S S

S c f x v l

u f x v l

S c f x v l

u u

θ

θ

θ

χ

+ =

+ =

=

= + + + −

+ + + + −

= + + + −

+ +

∑

∑

 

, , ˆ

ˆ ˆ

( | ) (y y)]}

n,k n _fn n k i k f x θ +v +l −

(57)

Since 0<χ( (t)) 1u_k ≤ according to the density property of a real number [7].

2, , 1 , ,

1

, ,

ˆ ˆ

ˆ ˆ ˆ

{ [ (x ( | ) (y y)]

ˆ _ˆ ˆ _ˆ

[ (t) ( | ) (y y)]} n

k k i k i i,k i _fi i k i k i

k n,k n _fn n k i k

V S c f x v l

u f x v l

θ

θ + =

≤ + + + −

+ + + + −

∑



(58)

Then substituting equivalent control u_k into (56), we have

2 2,k k

V ≤ −KS (59) Therefore, it can be concluded that V_2,k ≤0 from (59), and the all signals of the closed-loop system converges asymptotically to a neighborhood of zero based on Lyapunov synpaper approach. This completes the proof.

4. An Example and Simulation Results

In this section, a series of simulation results of a switched RLC circuit [15] as depicted in Fig. 1, described by

1 2 1, (t) 1 1, (t)

2 (t) 2, (t) 2, (t) 1

( ) ( , )

( (t)) ( ) ( , )

x x f x d t

x sat u f d t

y x

σ σ

σ σ σ

= + +

 

= + +

  ₌ 

x

x x



 (60)

where the nonlinear functions f_{1, (t) 1}σ ( ) 0x = ,

2, (t) 1 2

(t) 1

( ) R

f x x

C L

σ

σ

= − −

x , and for each k, k = 1, . . . ,N,

(t)

C_σ denotes the k-th capacitor, x= [qc, ]ϕLT denotes the charge in the capacitor and the flux in the inductance, respectively, and sat(uσ(t)(t)) denotes the voltage. In the simulation, parameters of the RLC circuit are

1 2, 5, 1, 22

C = C = L = R= The disturbances are assumed to be d_{1, (t)σ} ( , ) 0.1xt = ×x₁(t) cos(t)× and

2, (t)( , )

d _σ xt = 0.1×x₂(t) cos(t)× for (t) = 1, 2σ .

In the implementation, five fuzzy sets are defined over interval [-2, 2] for x and x_{1 2} with labels F F F F₁, , , , _{2 3 4}

5

and F and their membership functions are

2 2

1 2

ˆ ˆ

( 2) ( 1)

( ) exp[ ] ( ) exp[ ]

4 4

i i

F xi x F xi x

µ = − + µ = − +

2 2

3 4

ˆ ˆ

( 0) ( -1)

( ) exp[ ] ( ) exp[ ]

4 4

i i

F xi x F xi x

µ = − + µ = −

2 5

ˆ ( -2)

( ) exp[ ] for 1,2 4

i

F xi x i

µ = − =

Figure 1. The mass-spring-damper system

In this section, we apply the adaptive fuzzy robust observer-based sliding mode control approach in Section 3 to deal with the control problem. First, we select the observer gain as l_1,1=2, 1, 4, 12l_2,1= l_1,2 = l_2,2 = . The sample time is 0.01s. The sliding surfaces are selected as

1, 1ˆ ˆ2 =

k k

S c x x+ when k = 1, S c x x c_{1 1,1 1}= ˆ +ˆ_{2 1,1}, 2 = and when k = 2, S c x x c2 1,2 1= ˆ +ˆ2 1,2, 1 = . The initial values are chosen as

1,1 2,1 1,2 2,2

1,1 2,1 1,2 2,2

,1 ,2

1 1

,1 ,1

1 2

(0) 1, (0) 2, (0) 2, (0) 2, ˆ (0) 1, (0) 3, (0) 2.5, (0) 3,ˆ ˆ ˆ

(0) (0) [ 0.2,0.1, 0.2,0.3, 0.1],

(0) (0) [ 0.55, 0.55, 0.55, 0.55, 0.55], ˆ (0) 0, (ˆ

h h

f h

x x x x

x x x x

ω ω

= = = =

= = = =

= = − − −

= = − − − − −

=

θ θ

θ θ

ｆ ｆ

ˆ 0) 0, (0) 0= h =

The other parameters are selected as K =K =10, _{1 2} γ_f,1= 1000, γω_f,1=0.5, 10, 10, 0.5, γω_h,1= γf_,2 = γω_f,2=

,2

h

ω

Figure 2. Trajectories of x₁ and xˆ₁

Figure 3. Trajectories of x₂ and xˆ₂

Figure 4. The estimation error of x₁ and xˆ₁

Figure 5. The estimation error of x₂ and xˆ₂

Figure 6. Phase plane plot of the system states x₁ and x₂

Figure 8. Switch signal of the system

Figure 9. The sliding surface S_k

Figure 10. Control signal with input saturation

Figure 11. Original control signal

5. Conclusions

An adaptive fuzzy observer-based sliding mode control approach has been proposed for a class of switched uncertain nonlinear systems with input saturation in strict-feedback form. A fuzzy state observer has been designed for estimating the unmeasured states with the help of FLS approximating the unknown functions. Based on the designed sliding mode controller, it can ensure the boundedness of the proposed sliding surfaces, which realizes the stability of system. By choosing an appropriate Lyapunov function, the proposed controller is designed to demonstrate that all the signals in the closed-loop system can not only guarantee uniformly ultimately bounded, but also achieve good performance. Finally, some computer simulation results of a practical example are illustrated to show the performance of the proposed control approach.

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