fuzzy control design for switched nonlinear systems

8/9/2019 Fuzzy Control Design for Switched Nonlinear Systems

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Fuzzy Control Design for Switched Nonlinear Systems

Song-Shyong Chen1, Yuan-Chang Chang

2, Jenq-Lang Wu

3, Wen-Chang Cheng

4and Shun-Feng Su

5

1,4Department of Information Networking Technology, Hsiu-Ping Institute of Technology(1E-mail: [email protected], 4E-mail: [email protected])

2Department of Electrical Engineering, Lee-Ming Institute of Technology

(E-mail:[email protected])3Department of Electrical Engineering, National Taiwan Ocean University

(E-mail: [email protected])5Department of Electrical Engineering, National Taiwan University of Science and Technology

(E-mail:[email protected])

Abstract: Recently, many researchers have devoted themselves to the study of methods of designing controllers forswitched nonlinear systems with the use of the switched Takagi-Sugeno (T-S) fuzzy model. The main feature of

switched T-S fuzzy models is that they characterize the local dynamics of each fuzzy rule by a linear model. The appealof the switched T-S fuzzy model in control design is that the stability and performance characteristics of a system can beverified by using a Lyapunov function approach. Nevertheless, it should be noted that in a switched T-S fuzzy model,

the consequence could be any functions. In this study, we attempt to study the control design problem of switched T-Sfuzzy models, which have nonlinear consequence functions. The paper presents a novel switching fuzzy control designapproach based on control Lyapunov function. The proposed approach can design stable controllers for a switched T-S

fuzzy model of which the consequents are affine nonlinear state dynamic equations. The proposed switching fuzzycontroller guarantees the stability of the closed loop switched systems. The Sontag’s formula developed for affinenonlinear control systems is employed to construct a switching T-S fuzzy controller. Based on a control Lyapunovfunction approach, we derive a sufficient condition to ensure the stability of the closed loop switched fuzzy systems.

Two examples are given to show the advantage of the presented method.

Keywords: switched nonlinear systems, control Lyapunov function, Sontag’s formula, Takagi-Sugeno (T-S) fuzzymodel.

1. Introdution

A switched system is a hybrid dynamical systemthat composes of a family of continuous timesubsystems and a rule orchestrating the switching between the subsystems [1-3]. In the last two decades,there has been increasing interest in stability analysis

and control design for switched systems.The investigated problems focus on the stability

analysis and design of switched systems. So, our purpose is to identify a stable switched system or find a

stabilizable switching signal [4-5]. In this article, weconsider the “switched T-S fuzzy system”, whichconsists of several T-S fuzzy models. A sufficient

condition is proposed to stabilize the switched T-Sfuzzy system. A design method is also proposed tostabilize the switched T-S fuzzy system.

The motivation for study of the switched T-S fuzzysystem [15] is from the fact that many practical systemsare inherently multi-model in the sense that severaldynamical subsystems are required to describe their

behavior which may depend on various environmentalfactors, and that the methods of intelligent controldesign are based on the idea of switching between

different controllers. Moreover, there are situationswhere continuous stabilizing controllers do not exist,which make switching control techniques especiallysuitable.

The basic idea of existing control designmethodology is to design a feedback gain for each local

model and then to construct a global controller fromthese local gains so that the global stability of theoverall fuzzy system is guaranteed. Such a control

design approach easily leads to linear system problems[16], which can be solved through various linear systemtechniques, such as linear matrix inequalities (LMI)toolbox built in Matlab. However, it is easy to see that

when the number of rules become large, the problemmay become difficult to solve. For example, when there

are many fuzzy rules in the considered system, thenumber of linear matrix inequalities needed to be

satisfied simultaneously is also large, and then LMItoolbox may not be able to find the desired solution. In

fact, the more complicated nonlinear systems is, themore rules is required in the T-S model with linearconsequent parts to describe the system under a

sufficient accuracy. Thus, this paper attempts to studythe control design problem, in which T-S models withaffine nonlinear dynamic systems are used to representthe considered systems. Hence, for a complicatednonlinear system, a switched T-S model with the affinenonlinear consequent parts may need fewer rules tomodel.

The paper presents a novel switching fuzzy control

design approach based on control Lyapunov function.The proposed approach can design stable controllers for

a switched T-S fuzzy model of which the consequents


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are affine nonlinear state dynamic equations. The proposed switching fuzzy controller guarantees the

stability of the closed loop switched systems. TheSontag formula developed for affine nonlinear controlsystems is employed to construct a switched T-S

switching fuzzy controller. Based on a control Lyapunov

function approach, we derive a sufficient condition toensure the stability of the closed loop switched fuzzysystems. Moreover, the proposed condition leads to

control Lyapunov function nonlinear standpoints [15]that can find desired switching fuzzy controllesr directlyand avoid solving simultaneous matrix inequality, which

usually must be solved through numerical methods.The objective of our study is to design switching

fuzzy controllers that can be a nonlinear controller for

each switched fuzzy rule and can guarantee the stabilityof the closed loop switched nonlinear systems based onthe SWITCHED T-S fuzzy model, which may also be

characterized by nonlinear dynamic systems locally. In

our control design, we adapted the idea of controlLyapunov function to design switching controllers forswitched nonlinear systems. The used switching fuzzycontroller is parallel distributed control (PDC), in whichthe number of the rules is the same as that of theswitched T-S fuzzy model of the system and the

premises of the controller are the same as those of themodel. The Sontag formula technique is employed tosolve the control design problem directly.

2. The switched Takagi-Sugeno fuzzy models

Takagi and Sugeno [6] proposed an elegant modeling

method, which is often referred to as the T-S fuzzymodel, to represent or approximate a nonlineardynamical system.

Consider the “switched T-S fuzzy system” whichconsists of a family of T-S fuzzy models and a rule thatorchestrates the switching among them. The switchedT-S fuzzy system is described as follows

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

( ( )) ( ) ( ) N

x t j x t j x t j

j

x t h x t f x g u t σ

σ σ σ

=

= + (1)

Where ( ) { }, :R 1, 2, ,n x t R N σ +× → is the switching

rule.

The above switched T-S fuzzy system (1) consists of N

T-S fuzzy models

S1: ( ) ( )1

1 1 1

1

( ( )) ( ) ( ) N

j j j

j

x t h x t f x g u t =

= +

S2: ( ) ( )2

2 2 2

1

( ( )) ( ) ( ) N

j j j

j


= +

S N-1: ( ) ( )1

( 1) ( 1) ( 1)

1

( ( )) ( ) ( ) N N

N j N j N j

j

x t h x t f x g u t −

− − −

=

= +

S N: ( ) ( ) N

N N N

1

( ( )) ( ) ( ) N

j j j

j


= +

The switching rule ( , ) x t iσ = implies that the T-S

fuzzy model Si is activated.

In each fuzzy rule, a linear or nonlinear model can be used to describe the system locally. The overall

system is described by fuzzily “blending” those localmodels according to the defuzzification process used.The Ni rules synthesizing the ith T-S fuzzy model S i is

expressed as follows.

In general, the ith rule of the T-S fuzzy model S i isrepresented as

Rule j: IF1 1( ) is and ... ( ) isi i

n jn x t M x t M

THEN ( ) ( ( )) ( ( )) ( )ij ij x t f x t g x t u t = + ,

j=1,2,…,Ni. (2) where 1( ) ( )n x t x t are the premise variables, and

i

jk M is the corresponding fuzzy set for k =1, …, n,

j=1,2,…,Ni, [ ]1 2( ) ( ), ( ), , ( ) T

n x t x t x t x t = is the state

vector, and ( ) mu t ∈ ℜ is the input vector. Note that the

consequence of (2) is an affine nonlinear dynamic

function. In our study, we require thatij

f andij

g

be smooth vector fields and (0) 0ij

f = for all i,j. Thus,

by using the center of gravity method for

defuzzification, the final output of the ith T-S fuzzymodel Si is inferred as:

N

1

1

1

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

( )

( ( ))

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

i

i

i

ij ij ij

j

N

ij

i

N

ij ij ij

i

w x t f x t g x t u t

x t

w x t

h x t f x t g x t u t

=

=

=

+

=

= +

(3)

where1

( ( )) ( ( ))n

i

ij jk k

k

w x t M x t =

= ∏ , ( ( ))i jk k M x t is the

fuzzy membership grade of ( )k

x t belonging to i jk M ,

1

( ( ))( ( ))

( ( ))i

ij

ij N

ij

j

w x t h x t

w x t =

=

. It is assumed that ( ( )) 0ij

w x t ≥ ,

i1, 2, , N j = and

1

( ( )) 0i N

ij

j

w x t =

> for all t . Therefore,

( ( )) 0ijh x t ≥

, i1, 2, , N j =

and 1 ( ( )) 1

i N

ij j h x t =

=

, forall t .

In current research, switched T-S fuzzy models

usually possess linear consequence functions. Since ourapproach can also work for linear consequence functions,we also introduce its representation as follows. A typical

IF-THEN rule of the switched T-S fuzzy model withlinear consequences is represented as

Rule j: IF1 1( ) is and ... ( ) isi i

n jn x t M x t M

THEN ( ) ( ) ( )ij ij

x t A x t B u t = + , (4)

Then, the overall system is


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1

1

1

( ( ))( ( ) ( ))

( )

( ( ))

( ( ))( ( ) ( ))

N

ij ij ij

j

N

ij

j

N

ij ij ij j

w x t A x t B u t

x t

w x t

h x t A x t B u t

σ

σ

σ

=

=

=

+

=

= +

(5)

It is obvious that (5) is a special form of (3) if

( )ij ij

f x A x= and ( )ij ij

g x B= .

3. Switched fuzzy design via control Lyapunov

function

Switched fuzzy controllers for stabilizing the fuzzy

system (1) can be designed via parallel-distributed

control (PDC) [6-9]. In PDC, fuzzy controllers share the

same premise parts with (1). Since the consequent parts

of switched T-S fuzzy models in (1) are described byaffine nonlinear dynamic equations, the nonlinear

control theory is used to design the consequent parts of

a fuzzy controller. In our study, the controller for the ith

rule of the T-S fuzzy model Si is

IF1 1( ) is and ... ( ) isi i

n jn x t M x t M

THEN ( ) ( ( ))ij

u t u x t = , 1, , N, 1,i

i j N = = .

(6)

Then, the overall output of the fuzzy controller is

1

( ) ( ( )) ( ( )) N

j j

j

u t h x t u x t σ

σ σ

=

= , (7)

where ( ( )) jh x t σ is the same as that of the jth rule of

the switched fuzzy system (1). By substituting (7) into

(1), we get

1 1

( ) ( ( )) ( ( )) ( ( )) ( ( )) ( ( )) N N

j j j i i

j i

x t h x t f x t g x t h x t u x t σ σ

σ σ σ σ

= =

= +

{ } N

1 1 2 2

1

j j j j N j N

j

h f h g u h g u h g uσ

σ σ σ σ σ σ σ σ σ σ σ σ σ

=

= + + + +

. (8)

Now, consider a control Lyapunov function V ( x). In

the control design analysis [13], it is required that V ( x)

exist and satisfy the following inequality:

( ) ( ) ( )( ) 0 j j j j

T T f f g g

L V L V L V L V σ σ σ σ

+ + < . (9)

Let ( ) 0 j I x j hσ σ = ≠ . (10)

Denote

( ) j j f

j I x

a h L V σ

σ

σ

∈

= (11)

( ) j j g

j I x

b h L V σ

σ

σ

∈

= (12)

Then, we can construct a nonlinear controllers uσ

, for

( ) { }, 1, 2, , x t N σ → j=1, 2, …, N σ

as

2 4( )

0 and ( )

0 otherwise

T

T

T j j

a a bbb if b j I x

u h bb σ

σ σ

+ +− ≠ ∈

=

. (13)

The state-based switching rule can be defined as

( ) ( )

1( , ) arg min L V L V

i

ij ij

N

f x g x iji

j x t uσ

=

= +

where ( )( ) ( ) iL V L V


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Let ( ) T

V x x P xσ σ

= . Then we obtain

0T T

i i i i A P P A Q P B R B P σ σ σ σ σ σ σ σ σ σ + + + < (18)

where , Q, and R are symmetry positive define

matrices. Denote

( )

T

i ii I x

a h x P A xσ

σ σ σ

∈

= (19)

( )

T

i ii I x

b h x P Bσ

σ σ σ

∈

= (20)

2 4( )

0

0 0

T

T

T i i

a a bbb if b

u h bb

if b

σ σ

+ +− ≠

=

=

(21)

Then, we have the following theorem.

Theorem 2: If there is a smooth control Lyapunov

function V ( x), Eq. (18) is satisfied and let nonlinear

controllers iuσ , for i=1, 2,…, Nσ , be Eq. (21),

then the equilibrium of the closed-loop system (17)is asymptotically stale.

Proof: Consider a Lyapunov function as ( ) T

V x x P xσ

= .

By taking the time derivation of1

( )2

V x and using Eq.

(17), we have

{ }1 1 2 2 N N1

1 1( ) ( ( ) )

2 2

r

x i i i i i

i

V x V h A x t h B u h B u h B uσ σ

σ σ σ σ σ ω σ σ σ σ σ

=

= + + + +

{ }1 1 2 2 N N1

r T T T T

i i i i i

i

h x P A x h x P B u h x P B u h x P B uσ σ

σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ

=

= + + + +

1 1 1 1 1 1 1 2 1 2 1 N 1 N

T T T T h x P A x h h x P B u h h x PB u h h x PB uσ σ

σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ = + + + +

2 2 2 1 2 1 2 2 2 2 2 N 2 N

T T T T h x P A x h h x P B u h h x PB u h h x P B u

σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ

+ + + + +

N N N 1 N 1 N 2 N 2 N N N N

T T T T h x PA x h h x PB u h h x PB u h h x PB uσ σ σ σ σ σ σ σ σ σ

σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ + + + + +

Assume that V ( x) is a control Lyapunov function for the

closed-loop fuzzy systems (17).

We have

0T

i x P A xσ σ < if there exists 0 and 0T

i j i x h h x P B

σ σ σ σ ∀ ≠ = (22)

It is obvious that Eq. (18) is satisfied. And let nonlinear

controllersi

uσ

, for i=1, 2,…, Nσ

, be Eq. (21). We

obtain ( ) 0V x


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the initial condition (0) [2 1]T

x = − for [0,20]t ∈ .

From these simulation results, it is evident that the

designed switched T-S fuzzy model based the proposed

fuzzy controller can stabilize the nonlinear system.

5. Conclusions

In this paper, we have proposed a novel controller

design methodology of designing controllers for a

switched T-S fuzzy model. The method is very simple.

Simulation results have verified and confirmed the

effectiveness of the new approach for designing

controllers for nonlinear systems.

Acknowledgements

The authors would like to thank the National ScienceCouncil of the Republic of China for financiallysupporting this research under Contract No. NSC

96-2221-E-164 -011.

References

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[2]

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[7]

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Fig. 1 Membership function of the two examples.

-5 5

1

0

1

1 M

2

1 M

x1


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0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x(t)

t

Fig. 2 State response of the switched closed-loop systemfor x(0) =2.

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

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1x(t)

t

Fig. 3 State response of the switched closed-loop systemfor x(0) =-1.

0 5 10 15 20-2

-1

0

1

2

3

4

5

6

7

x1

x2

x(t)

t

Fig. 4 State response of the switched closed-loop systemfor x(0) =[-2 3]T.

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x1

x2

x(t)

t

Fig. 5 State response of the switched closed-loop system

for x(0) =[2 1].

fuzzy control design for switched nonlinear systems

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