AN ADAPTIVE SLIDING-MODE FUZZY CONTROL (ASMFC)

APPROACH FOR A CLASS OF NONLINEAR SYSTEMS

Jian-Hua Zhang and Johann F. Böhme

Signal Theory Group

Department of Electrical Engineering and Information Sciences, Ruhr-Universität Bochum

Bochum 44780, Germany

Keywords: Sliding mode control, Fuzzy controller, TSK fuzzy model, Feedback linearizable systems, Adaptive fuzzy

identification

Abstract: This paper uses the concept of sliding-mode control (SMC), as a special approach in nonlinear control theory,

in aiding the design of a fuzzy controller. The mathematical specifics of the presented approach are given

along with its performance analysis. It was concluded that the new approach with distinctive characteristics

holds potential for coping with difficult control problems for a class of complex (generally nonlinear)

systems.

1 INTRODUCTION

In the previous literature, fuzzy control, especially

adaptive and self-learning fuzzy control, has been

successfully applied for complex nonlinear control

problems (Driankov, Hellendoorn and Reinfrank,

1993; Passino and Yurkovich, 1998; Wang, 1993;

Hwang and Lin, 1992; Takagi and Sugeno, 1983,

1985; Jang, 1992a, 1992b). One of the most

attractive features of adaptive fuzzy control is that

linguistic knowledge elicited from domain expert or

available input-output data set can be conveniently

incorporated into the design process of fuzzy

controller.

In a sliding-mode controller (SMC), the sliding

region is generally a hyper-plane. In the simple case

of 2-D, the sliding region is simply a line. Separated

by this sliding line, control force is switched to its

maximum at one side and minimum at another. In

the theory of SMC, it is usually presumed that the

SMC controller can switch from one extreme to

another extreme arbitrarily fast. Based on this

assumption, the trajectory can remain along this line

once it reaches it. In practice, nevertheless, it is well

known in SMC theory that the trajectory of the

system always chatter around this sliding line, rather

than sliding strictly along it (Hung, Gao and Hung,

1993; Slotine and Li, 1991). Thus the output of the

SMC controller alternates its sign along the switch

line.

The synergism of fuzzy control and SMC has

also been a hot research topic (Palm, 1992; Palm,

Driankov and Hellendoorn, 1996; Palm and Stutz,

2003). One reason, from the perspective of the basic

property of a control system—stability property,

may be that the mathematically strict stability

analysis for a fuzzy controller is hard to establish

and guarantee in general cases, contrarily that for a

sliding-mode controller can be well resolved.

Another advantage offered by the SMC method

includes its capability for decoupling high-

dimensional systems into a body of lower-

dimensional sub-systems to achieve the

dimensionality reduction for a complex multi-input

multi-output (MIMO) control system (Hung, Gao

and Hung, 1993). This advantage may be beneficial

for avoiding the curse of dimensionality inherent in

a fuzzy inference system (FIS) even with moderately

number of input variables (Jang, 1993; Chen and

Tsao, 1989).

In this paper, to improve the transient

performance of fuzzy controller, the state-space of

control system is partitioned into a number of local

cells, across individual cell state-space the sliding

hyper-plane of SMC controller within its cell is

designed separately in an adaptive fashion. The

paper is organized in the following way. Firstly

some basics of SMC are briefly introduced. In

section III, the detailed approach of adaptive sliding-

mode fuzzy control (ASMFC) is developed. Finally

its performance and unique features are discussed.

193

Zhang J. and Böhme J. (2004).

AN ADAPTIVE SLIDING-MODE FUZZY CONTROL (ASMFC) APPROACH FOR A CLASS OF NONLINEAR SYSTEMS.

In Proceedings of the First International Conference on Informatics in Control, Automation and Robotics, pages 193-197

DOI: 10.5220/0001124801930197

Copyright

c

SciTePress

2 BASICS OF SMC METHOD

Let us consider a class of continuous-time nonlinear

dynamical system which is feedback linearizable

and of the canonical form:

()

()

()

()

(

)

() () ()

[]

()

() ()

⎪

⎩

⎪

⎨

⎧

=

+=

−−

txty

tbutxtxtxtxftx

nnn

,,,,

21

&

L

(1)

where

[]

⋅f is an unknown continuous function

(generally nonlinear), b>0 is the controller gain,

()

ℜ∈tx is the system’s state variable, and

() ()

ℜ∈ℜ∈ tytu , are the input variable and output

variable of the system, respectively. Our goal is to

force the state vector of the system (1) (where the

superscript

τ

denotes the vector transpose)

() () ()

()

()

[]

τ

txtxtx

n 1

,,,

−

= L

&

tx

to follow a predefined reference trajectory

() () ()

()

()

[]

τ

txtxtx

n

rrrr

1

,,,

−

= L

&

tx .

Define the tracking error vector as the difference

between the actual states and desired states, i.e.,

() () ()

ttt

r

xxe −= (2)

then the control problem can be formalized as: find a

control law

()

tu such that

()

0lim =

∞→

t

t

e

.

A candidate of such a control law is

() ()

[]

(

)

() () ()

twttxFtu

n

r

+++Θ−= emtx

τ

, (3)

where

()

tw is an auxiliary control input to be

determined,

[]

⋅F is a proper function with

sufficiently rich parameter set

Θ used to well

approximate unknown function

[]

⋅f in eqn. (1), i.e.,

fF

ˆ

= may be implemented by an adaptive fuzzy

model (Jang, 1992a; Jang, 1992b; Jang, 1993), and

[]

1,,,

1

L

−

=

nn

mm

τ

m is an properly chosen vector

that controls the performance of the closed-loop

system with the control law (3). With this control

law, the resulting closed-loop system is a linear one

as

()

()

(

)

() ()() ()()

[]

()

tw

Ffte

n

m

n

emt

n

e

+

Θ−=++

−

+ ,

1

1

txtxL

(4)

Our suggested control approach is formulated as

the following procedures:

1. Use a parameterized adaptive fuzzy model to

approximate

[]

⋅f , i.e., adaptively update the

parameter vector of fuzzy model such that for

()

n

ℜ∈∀ tx and an upper bound of error 0>

ε

,

(

)

[

]

(

)

[

]

.,

ε

≤−Θ txtx fF (5)

2. Apply the SMC approach to design

(

)

tw to

guarantee the global stability property of the

close-loop system.

Using the standard SMC design approach, define

an error measure below:

(

)

(

)

(

)

()

()

etentets

nn

++−+=

−−

L

21

)1(

λλ

(6)

where constant

0>

λ

. Then the equation

()

0

=

ts is

called a switching surface in space

n

ℜ on which

(

)

te approaches to zero exponentially, i.e.,

asymptotical tracking performance is achieved.

For simplicity, introduce a kind of differential

operator to express the above differential polynomial

as

() ()

[]

()

.1,,)1(,

21

1

tente

dt

d

ts

nn

n

L

−−

−

−=

⎟

⎠

⎞

⎜

⎝

⎛

+=

λλλ

(7)

The control law in eqn. (3) guarantees the system

state trajectory, whatever the initial condition may

be, will approach and subsequently maintain on the

sliding surface

(

)

0

=

ts , if the condition

(

)

(

)

(

)

tststs

η

−≤⋅

&

(8)

holds. Here

η

is a positive constant, which restricts

that the state trajectory hits the sliding surface in a

finite time (Hung, Gao and Hung, 1993; Slotine and

Li, 1991). Thus

(

)

0→te exponentially with a time

constant

(

)

λ

1

−

n .

Taking

[

]

λλλ

τ

,,)1(,,0

21

L

−−

−=

nn

nm (9)

differentiating eqn. (6), and inserting eqn. (5) into it

yield

(

)

(

)

(

)

(

)

(

)

(

)()

., twFfts +

Θ

−

=

txtx

&

(10)

and

(

)

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

[]

()

.sgn, stwFf

dt

tsd

+Θ−= txtx (11)

Then condition (8) always maintains if we choose

(

)

(

)

(

)

.sgn stw

η

ε

+

−

=

(12)

By substituting eqns. (9) and (11) into eqn. (3),

eventually we have the control law

(

)

(

)

(

)

[

]

(

)

()()

.sgn

,,)1(,,0,

21

s

tenxFtu

nnn

r

ηε

λλλ

+−

−++Θ−=

−−

Lx

(13)

then the closed-loop system (4) can asymptotically

track the reference state trajectory specified

beforehand with guarantee of global stability

property (Jang, 1992b).

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194

3 CASE STUDY: A 2-D FEEDBACK

LINEARIZYBLE NONLINEAR

SYSTEM

Consider a 2nd-order system described by state-

space equation

() () ()

[]

)(, tbutxtxftx +=

&&&

(14)

One feasible control law of SMC for 2

nd

-order

system eqn. (14) may be chosen as:

vsgku +⋅−= )(

φ

(15)

where v is an equivalent control used when the

system state lies in the sliding mode, constant

0>k

represents the maximum output of SMC controller.

According to eqn. (6), the switching hyper-plane is

.0=+= ees

λ

&

(16)

There are many ways to define

)(sg

φ

in eqn. (15)

for different purposes. Three candidate functions for

define

)(sg

φ

are given here:

1. Sign function, i.e.,

()

⎩

⎨

⎧

>

<−

==

0for ,1

0for ,1

)sgn(

s

s

ssg

φ

(17)

Introduction of the sign function

)sgn(s often causes

chattering problem for a SMC controller. One way

to alleviate the problem is to use another nonlinear

function below.

2. Saturation function, i.e.,

()

⎪

⎪

⎩

⎪

⎪

⎨

⎧

>

≤

==

1for ),sgn(

1for ,

)(

φφ

φφ

φ

φ

ss

ss

s

satsg

(18)

where

φ

is a constant that determines the width of

the boundary layer around the switching surface.

In actuality, the control law resulting from this

selection of

)(sg

φ

is a continuous approximation of

the ideal relay control (Hung, Gao and Hung, 1993;

Slotine and Li, 1991). Another possible variant is as

follows.

3. Hyperbolic tangent function, i.e.,

()

)tanh(

φ

φ

s

sg =

(19)

In all the above three cases, provided sufficiently

large k, SMC controller of form (15) has been shown

to be asymptotically stable (Hung, Gao and Hung,

1993; Slotine and Li, 1991).

For a 2-D system, the controller structure and the

corresponding control surface are illustrated in

Figure 1.

s

u

k

k−

s

u

k

s

u

φ

−

k−

φ

k

k−

Figure 1: Three examples of SMC controllers in 2-D case.

4 ADAPTIVE SLIDING-MODE

FUZZY CONTROL (ASMFC)

From the perspective of optimal control theory,

SMC falls into the category of time sub-optimal

control. As is well known in optimal control theory,

the result of a time optimal control problem for a

regulator with set-point input is a type of Bang-Bang

control with respect to a nonlinear switching curve

shown in Figure 2. Figure 2 also illustrates the

control surface resulting from the nonlinear

switching function.

Since a fuzzy inference system (FIS) can

integrate and coordinate different control algorithms

in a seamless way by using fuzzy decision-making

logic according to the available fuzzy knowledge

and data base, we can directly incorporate the design

conception of SMC into the development of a fuzzy

controller without causing any undesirable effects.

In Takagi-Sugeno-Kang (TSK) fuzzy model, the

output of each fuzzy if-then rule is explicitly and

generally expressed as a linear combination of

controller inputs plus a constant term (Takagi and

Sugeno, 1983; Takagi and Sugeno, 1985; Hoffmann

and Nelles, 2001).

In fact, the rule output can also be a more

generally nonlinear function of the rule input

variables. In this section, we express the output of

each fuzzy rule, i.e., the control output when the

states enter into a local cell space, a switching

AN ADAPTIVE SLIDING-MODE FUZZY CONTROL (ASMFC) APPROACH FOR A CLASS OF NONLINEAR

SYSTEMS

195

function of state vector. In this way, we carefully

design a new adaptive fuzzy controller by borrowing

the notion of SMC, which actually leads to an

adaptive sliding-mode fuzzy control approach

presented in this short paper. In our approach, the

parameters in the output of each fuzzy rule that

covers different cell of state space are determined by

different SMCs that operate over the corresponding

cell state-space, whose concept was proposed by

Chen and Tsao (1989).

e

e

&

sliding line

trajectory

0

trajectory

[state plane]

Figure 2: Nonlinear switching curve and control surface.

For the ASMFC controller, the error and the rate

of error are taken as the its inputs. Its l-th fuzzy if-

then rule in the rule base takes the format of

l

R : if e is

l

F

1

and e

&

is

l

F

2

,

then

)(

l

ll

l

cee

ksatu

φ

λ

++

=

&

(20)

where

l

F

1

and

l

F

2

represent the linguistic label, i.e.,

input fuzzy set, which can be characterized by

proper parameterized membership function defined

over the corresponding universe of discourse.

With only a small number of if-then rules,

ASMFC can generate a complex nonlinear switching

function, which is difficult to achieve by standard

SMC method. Also note that the rule output in

expression (20) need not to be a saturation function,

it could be either a sign function or hyperbolic

tangential function described before.

Figure 3: Fuzzification of error e and its rate e

&

.

In the case of a 2-D system, the switching line

can be either a function of

e , or a function of e

&

. In

this case only a very small number of fuzzy rule

patches are required to cover the switching function

of single variable. Therefore in an ASMFC

controller the number of if-then fuzzy rules is

reduced to a reasonable and manageable amount and

thus the curse of dimensionality arising from multi-

variable fuzzy controller can be avoided.

To approximate the switching curve shown in

Figure 2, we assign 3 linguistic labels (described by

their own properly-parameterized membership

functions) to input variable

e and e

&

, respectively.

The fuzzification of

e and e

&

is illustrated in Figure

3, where symbols ‘ZO’, ’NS’, ’PS’ represent the

corresponding linguistic terms ‘zero’, ‘negative

small’, and ‘positive small’, respectively. In this

case, we partition the universe of discourse of both

input variables into 3 overlapping fuzzy subsets, and

hence we have 9 fuzzy rules in the rule-base of

fuzzy controller and the state space is partitioned

into 9 localized cells. Extensive simulation

experiments have demonstrated that the 9-rule base

suffices to well approximate the desired switching

curve. The control surface and sliding surface are

shown in Figure 4.

e

&

e

PS

ZO

N

S

N

S

ZO

PS

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196

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

error

error rate

Figure 4: Control surface and sliding surface for an

ASMFC controller with only 9 fuzzy if-then rules.

5 CONCLUSIONS

In this short article an adaptive sliding mode fuzzy

control approach is proposed with some analysis of

its property for addressing nonlinear control

problems. This approach combines the concept of a

branch of nonlinear control theory, namely SMC,

and that of a fuzzy inference system that can

uniformly approximate any nonlinear function with

arbitrary degree of accuracy. In this sense, global

stability of the control system designed by this

approach can be mathematically established (Jang,

1992b; Hung, Gao and Hung, 1993; Slotine and Li,

1991). Nevertheless, be aware that the presented

approach seems only applicable to the class of

nonlinear systems over which the feedback

linearization technique can be performed.

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AN ADAPTIVE SLIDING-MODE FUZZY CONTROL (ASMFC) APPROACH FOR A CLASS OF NONLINEAR

SYSTEMS

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