ROBUST FUZZY CONTROLLER DESIGN FOR UNCERTAIN

DESCRIPTOR MARKOVIAN JUMP SYSTEMS

Wudhichai Assawinchaichote

Department of Electronic and Telecommunication Engineering

King Mongkut’s University of Technology Thonburi, 91 Prachautits Rd., Bangkok 10140, Thailand

Sing Kiong Nguang

Department of Electrical and Computer Engineering

The University of Auckland, Private Bag 92019 Auckland, New Zealand

Keywords:

TS Fuzzy model, H

∞

control, Markovian jumps, LMI.

Abstract:

This paper examines the problem of designing a robust H

∞

state-feedback controller for a class of uncertain

nonlinear descriptor Markovian jump systems described by a Takagi-Sugeno (TS) fuzzy model with Markov-

ian jumps. Based on a linear matrix inequality (LMI) approach, LMI-based sufﬁcient conditions for the un-

certain nonlinear descriptor Markovian jump systems to have an H

∞

performance are derived. The proposed

approach does not involve the separation of states into slow and fast ones and it can be applied not only to

standard, but also to nonstandard nonlinear descriptor systems. A numerical example is provided to illustrate

the design developed in this paper.

1 INTRODUCTION

Markovian jump systems, sometimes called hybrid

systems with a state vector, consists of two compo-

nents; i.e., the state (differential equation) and the

mode (Markov process). The Markovian jump sys-

tem changes abruptly from one mode to another mode

caused by some phenomenon such as environmental

disturbances, changing subsystem interconnections

and fast variations in the operating point of the sys-

tem plant, etc. The switching between modes is gov-

erned by a Markov process with the discrete and ﬁnite

state space. Over the past few decades, the Markovian

jump systems have been extensively studied by many

researchers; see (Kushner, 1967; Dynkin, 1965; Won-

ham, 1968; X. Feng and Chizeck, 1992; de Souza and

Fragoso, 1993; Boukas and Liu, 2001; Boukas and

Yang, 1999; Rami and Ghaoui, 1995; Shi and Boukas,

1997). This is due to the fact that jumping systems

have been a subject of the great practical importance.

For the past three decades, descriptor systems

or called singularly perturbed systems have been

intensively studied by many researchers; see (Shi

and Boukas, 1997; K. Benjelloun and Costa, 1997;

E. K. Boukas and Liu, 2001; V. Dragan and Boukas,

1999; Pan and Basar, 1993; Pan and Basar, 1994;

Fridman, 2001; Shi and Dragan, 1999; P. V. Koko-

tovic and O’Reilly, 1986). Singularly perturbed sys-

tems also known as multiple time-scale dynamic sys-

tems normally occur due to the presence of small

“parasitic” parameters, typically small time constants,

masses, etc. In state space, such systems are com-

monly modelled using the mathematical framework

of singular perturbations, with a small parameter, say

ε, determining the degree of separation between the

“slow” and “fast” modes of the system. However, it

is necessary to note that it is possible to solve the sin-

gularly perturbed systems without separating between

slow and fast mode subsystems. But the require-

ment is that the “parasitic” parameters must be large

enough. In the case of having very small “parasitic”

parameters which normally occur in the description

of various physical phenomena, a popular approach

adopted to handle these systems is based on the so-

called reduction technique. According to this tech-

nique the fast variables are replaced by their steady

states obtained with “frozen” slow variables and con-

trols, and the slow dynamics is approximated by the

corresponding reduced order system. This time-scale

is asymptotic, that is, exact in the limit, as the ratio of

the speeds of the slow versus the fast dynamics tends

to zero.

In the last few years, the research on singularly

perturbed systems in the H

∞

sense has been highly

recognized in control area due to the great practical

importance. H

∞

-optimal control of singularly per-

turbed linear systems under either perfect state mea-

surements or imperfect state measurements has been

91

Assawinchaichote W. and Kiong Nguang S. (2005).

ROBUST FUZZY CONTROLLER DESIGN FOR UNCERTAIN DESCRIPTOR MARKOVIAN JUMP SYSTEMS.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics, pages 91-97

DOI: 10.5220/0001156800910097

Copyright

c

SciTePress

investigated via differential game theoretics approach.

Although many researchers have studied the H

∞

con-

trol design of linear singularly perturbed systems for

many years, the H

∞

control design of nonlinear sin-

gularly perturbed systems remains as an open re-

search area. This is due to, in general, nonlinear sin-

gularly perturbed systems can not be decomposed into

slow and fast subsystems.

Recently, a great amount of effort has been made on

the design of fuzzy H

∞

for a class of nonlinear sys-

tems which can be represented by a Takagi-Sugeno

(TS) fuzzy model; see (Nguang and Shi, 2001; Han

and Feng, 1998; B. S. Chen and He, 2001; K. Tanaka

and Wang, 1996). Recent studies (Nguang and Shi,

2001; Han and Feng, 1998; B. S. Chen and He, 2001;

K. Tanaka and Wang, 1996; H. O. Wang and Grifﬁn,

1996) show that a fuzzy model can be used to approx-

imate global behaviors of a highly complex nonlinear

system. In this fuzzy model, local dynamics in differ-

ent state space regions are represented by local linear

systems. The overall model of the system is obtained

by “blending” of these linear models through nonlin-

ear fuzzy membership functions. Unlike conventional

modelling which uses a single model to describe the

global behavior of a system, fuzzy modelling is essen-

tially a multi-model approach in which simple sub-

models (linear models) are combined to describe the

global behavior of the system. Employing the existing

fuzzy results (Nguang and Shi, 2001; Han and Feng,

1998; B. S. Chen and He, 2001; K. Tanaka and Wang,

1996; H. O. Wang and Grifﬁn, 1996) on the singularly

perturbed system, one ends up with a family of ill-

conditioned linear matrix inequalities resulting from

the interaction of slow and fast dynamic modes. In

general, ill-conditioned linear matrix inequalities are

very difﬁcult to solve.

What we intend to do in this paper is to design

a robust H

∞

fuzzy state-feedback controller for a

class of uncertain nonlinear singularly perturbed sys-

tems with Markovian jumps. First, we approximate

this class of uncertain nonlinear singularly perturbed

systems with Markovian jumps by a Takagi-Sugeno

fuzzy model with Markovian jumps. Then based on

an LMI approach, we develop a technique for design-

ing a robust H

∞

fuzzy state-feedback controller such

that the L

2

-gain of the mapping from the exogenous

input noise to the regulated output is less than a pre-

scribed value. To alleviate the ill-conditioned linear

matrix inequalities resulting from the interaction of

slow and fast dynamic modes, these ill-conditioned

LMIs are decomposed into ε-independent LMIs and

ε-dependent LMIs. The ε-independent LMIs are not

ill-conditioned and the ε-dependent LMIs tend to zero

when ε approaches to zero. If ε is sufﬁciently small,

the original ill-conditioned LMIs are solvable if and

only if the ε-independent LMIs are solvable. The

proposed approach does not involve the separation of

states into slow and fast ones, and it can be applied

not only to standard, but also to nonstandard singu-

larly perturbed systems.

This paper is organized as follows. In Section 2,

system descriptions and deﬁnition are presented. In

Section 3, based on an LMI approach, we develop

a technique for designing a robust H

∞

fuzzy state-

feedback controller such that the L

2

-gain of the map-

ping from the exogenous input noise to the regulated

output is less than a prescribed value for the system

described in Section 2. The validity of this approach

is demonstrated by an example from a literature in

Section 4. Finally, conclusions are given in Section 5.

2 SYSTEM DESCRIPTIONS AND

DEFINITIONS

The class of nonlinear uncertain singularly perturbed

system with Markovian jumps under consideration

is described by the following TS fuzzy model with

Markovian jumps:

E

ε

˙x(t) =

P

r

i=1

µ

i

(ν(t))×

h

[A

i

(η(t)) + ∆A

i

(η(t))]x(t)

+[B

1

i

(η(t)) + ∆B

1

i

(η(t))]w(t)

+[B

2

i

(η(t)) + ∆B

2

i

(η(t))]u(t)

i

,

z(t) =

P

r

i=1

µ

i

(ν(t))×

h

[C

1

i

(η(t)) + ∆C

1

i

(η(t))]x(t)

+[D

12

i

(η(t)) + ∆D

12

i

(η(t))]u(t)

i

(1)

where E

ε

=

I 0

0 εI

, ε > 0 is the singular per-

turbation parameter, ν(t) = [ν

1

(t) ··· ν

ϑ

(t)] is

the premise variable that may depend on states in

many cases, µ

i

(ν(t)) denote the normalized time-

varying fuzzy weighting functions for each rule, ϑ

is the number of fuzzy sets, x(t) ∈ ℜ

n

is the

state vector, u(t) ∈ ℜ

m

is the input, w(t) ∈

ℜ

p

is the disturbance which belongs to L

2

[0, ∞),

z(t) ∈ ℜ

s

is the controlled output, the matrix

functions A

i

(η(t)), B

1

i

(η(t)), B

2

i

(η(t)), C

1

i

(η(t)),

D

12

i

(η(t)), ∆A

i

(η(t)), ∆B

1

i

(η(t)), ∆B

2

i

(η(t)),

∆C

1

i

(η(t)) and ∆D

12

i

(η(t)) are of appropriate di-

mensions. {η(t))} is a continuous-time discrete-

state Markov process taking values in a ﬁnite set

S = {1, 2, ··· , s} with transition probability matrix

P r

∆

= {P

ık

(t)} given by

P

ık

(t) = P r(η(t + ∆) = k|η(t) = ı)

=

λ

ık

∆ + O(∆) if ı 6= k

1 + λ

ıı

∆ + O(∆) if ı = k

(2)

ICINCO 2005 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

92

where ∆ > 0, and lim

∆−→0

O(∆)

∆

= 0. Here λ

ık

≥ 0

is the transition rate from mode ı (system operating

mode) to mode k (ı 6= k), and

λ

ıı

= −

s

X

k=1,k6=ı

λ

ık

. (3)

For the convenience of notations, we let µ

i

∆

=

µ

i

(ν(t)), η = η(t), and any matrix M (µ, ı)

∆

=

M(µ, η = ı). The matrix functions ∆A

i

(η),

∆B

1

i

(η), ∆B

2

i

(η), ∆C

1

i

(η) and ∆D

12

i

(η) repre-

sent the time-varying uncertainties in the system and

satisfy the following assumption.

Assumption 1

∆A

i

(η) = F (x(t), η, t)H

1

i

(η),

∆B

1

i

(η) = F (x(t), η, t)H

2

i

(η),

∆B

2

i

(η) = F (x(t), η, t)H

3

i

(η),

∆C

1

i

(η) = F (x(t), η, t)H

4

i

(η),

and ∆D

12

i

(η) = F (x(t), η, t)H

5

i

(η),

where H

j

i

(η), j = 1, 2, ··· , 5 are known matri-

ces which characterize the structure of the uncertain-

ties. Furthermore, there exists a positive function

ρ(η) such that the following inequality holds:

kF (x(t), η, t)k ≤ ρ(η). (4)

We recall the following deﬁnition.

Deﬁnition 1 Suppose γ is a given positive number. A

system of the form (1) is said to have the L

2

-gain less

than or equal to γ if

E

"

Z

T

f

0

{z

T

(t)z(t) −γ

2

w

T

(t)w (t)} dt

#

≤ 0, (5)

where x(0) = 0 and E [·] stands for the mathematical

expectation, for all T

f

and all w(t) ∈ L

2

[0, T

f

].

Note that for the symmetric block matrices, we use

(∗) as an ellipsis for terms that are induced by sym-

metry.

3 ROBUST H

∞

FUZZY

STATE-FEEDBACK CONTROL

DESIGN

This section provides the LMI-based solutions to

the problem of designing a robust H

∞

fuzzy state-

feedback controller that guarantees the L

2

-gain of the

mapping from the exogenous input noise to the regu-

lated output to be less than some prescribed value.

First, we consider the following H

∞

fuzzy state-

feedback which is inferred as the weighted average of

the local models of the form:

u(t) =

r

X

j=1

µ

j

K

j

(ı)x(t). (6)

Then, we describe the problem under our study as

follows.

Problem Formulation: Given a prescribed H

∞

per-

formance γ > 0, design a robust H

∞

fuzzy state-

feedback controller of the form (6) such that the in-

equality (5) holds.

Before presenting our ﬁrst main result, we recall

the following lemma.

Lemma 1 Consider the system (1). Given a pre-

scribed H

∞

performance γ > 0, for ı = 1, 2, ··· , s,

if there exist matrices P

ε

(ı) = P

T

ε

(ı), positive con-

stants δ(ı) and matrices Y

j

(ı), j = 1, 2, ··· , r such

that the following ε-dependent linear matrix inequal-

ities hold:

P

ε

(ı) > 0 (7)

Ψ

ii

(ı, ε) < 0, i = 1, 2, ··· , r(8)

Ψ

ij

(ı, ε) + Ψ

ji

(ı, ε) < 0, i < j ≤ r (9)

where

Ψ

ij

(ı, ε) =

Φ

ij

(ı, ε) (∗)

T

(∗)

T

(∗)

T

R(ı)

˜

B

T

1

i

(ı) −γR(ı) (∗)

T

(∗)

T

Υ

ij

(ı, ε) 0 −γR(ı) (∗)

T

Z

T

(ı, ε) 0 0 −P(ı, ε)

Φ

ij

(ı, ε) = A

i

(ı)E

−1

ε

P

ε

(ı) + E

−1

ε

P

ε

(ı)A

T

i

(ı)

+B

2

i

(ı)Y

j

(ı) + Y

T

j

(ı)B

T

2

i

(ı)

+λ

ıı

E

−1

ε

P

ε

(ı),

Υ

ij

(ı, ε) =

˜

C

1

i

(ı)E

−1

ε

P

ε

(ı) +

˜

D

12

i

(ı)Y

j

(ı),

R(ı) = diag {δ(ı)I, I, δ(ı)I, I},

Z(ı, ε) =

p

λ

ı1

E

−1

ε

P

ε

(ı) ···

q

λ

ı(ı−1)

E

−1

ε

P

ε

(ı)

q

λ

ı(ı+1)

E

−1

ε

P

ε

(ı) ···

p

λ

ıs

E

−1

ε

P

ε

(ı)

,

P(ı, ε) = diag

n

E

−1

ε

P

ε

(1), ··· , E

−1

ε

P

ε

(ı − 1),

E

−1

ε

P

ε

(ı + 1), ··· , E

−1

ε

P

ε

(s)

o

,

ROBUST FUZZY CONTROLLER DESIGN FOR UNCERTAIN DESCRIPTOR MARKOVIAN JUMP SYSTEMS

93

with

˜

B

1

i

(ı) = [

I I I B

1

i

(ı)

]

˜

C

1

i

(ı) =

h

γρ(ı)H

T

1

i

(ı)

√

2ℵ(ı)ρ(ı)H

T

4

i

(ı)

0

√

2ℵ(ı)C

T

1

i

(ı)

i

T

˜

D

12

i

(ı) =

h

0

√

2ℵ(ı)ρ(ı)H

T

5

i

(ı)

γρ(ı)H

T

3

i

(ı)

√

2ℵ(ı)D

T

12

i

(ı)

i

T

ℵ(ı) =

I + ρ

2

(ı)

r

X

i=1

r

X

j=1

h

kH

T

2

i

(ı)H

2

j

(ı)k

i

1

2

then the inequality (5) holds. Furthermore, a suitable

choice of the fuzzy controller is

u(t) =

r

X

j=1

µ

j

K

ε

j

(ı)x(t) (10)

where

K

ε

j

(ı) = Y

j

(ı)(P

ε

(ı))

−1

E

ε

. (11)

Proof: The desired result can be carried out by a

similar technique used in (D. P. de Farias and Costa,

2000), (Nguang and Shi, 2003), and (Nguang and Shi,

2001). Due to limited pages, the detail of the proof is

omitted for brevity.

Remark 1 The linear matrix inequalities given in

Lemma 1 becomes ill-conditioned when ε is sufﬁ-

ciently small, which is always the case for the sin-

gularly perturbed system. In general, these ill-

conditioned linear matrix inequalities are very difﬁ-

cult to solve. Thus, to alleviate these ill-conditioned

linear matrix inequalities, we have the following the-

orem which does not depend on ε.

Now we are in the position to present our ﬁrst re-

sult.

Theorem 1 Consider the system (1). Given a pre-

scribed H

∞

performance γ > 0, for ı = 1, 2, ··· , s,

if there exist matrices P (ı), positive constants δ(ı)

and matrices Y

j

(ı), j = 1, 2, ··· , r such that the fol-

lowing ε-independent linear matrix inequalities hold:

EP (ı) + P (ı)D > 0 (12)

Ψ

ii

(ı) < 0, i = 1, 2, ··· , r (13)

Ψ

ij

(ı) + Ψ

ji

(ı) < 0, i < j ≤ r (14)

where EP (ı) = P

T

(ı)E, P (ı)D = DP

T

(ı), E =

I 0

0 0

, D =

0 0

0 I

,

Ψ

ij

(ı) =

Φ

ij

(ı) (∗)

T

(∗)

T

(∗)

T

R(ı)

˜

B

T

1

i

(ı) −γR(ı) (∗)

T

(∗)

T

Υ

ij

(ı) 0 −γR(ı) (∗)

T

Z

T

(ı) 0 0 −P(ı)

Φ

ij

(ı) = A

i

(ı)P (ı) + P

T

(ı)A

T

i

(ı) + B

2

i

(ı)Y

j

(ı)

+Y

T

j

(ı)B

T

2

i

(ı) + λ

ıı

˜

¯

P (ı),

Υ

ij

(ı) =

˜

C

1

i

(ı)P (ı) +

˜

D

12

i

(ı)Y

j

(ı),

R(ı) = diag{δ(ı)I, I, δ(ı)I, I},

Z(ı) =

p

λ

ı1

˜

¯

P (ı) ···

q

λ

ı(ı−1)

˜

¯

P (ı)

q

λ

ı(ı+1)

˜

¯

P (ı) ···

p

λ

ıs

˜

¯

P (ı)

,

P(ı) = diag

n

˜

¯

P (1), ··· ,

˜

¯

P (ı −1),

˜

¯

P (ı + 1), ··· ,

˜

¯

P (s)

o

,

˜

¯

P (ı) =

P (ı) + P

T

(ı)

2

with

˜

B

1

i

(ı) = [

I I I B

1

i

(ı)

]

˜

C

1

i

(ı) =

h

γρ(ı)H

T

1

i

(ı)

√

2ℵ(ı)ρ(ı)H

T

4

i

(ı)

0

√

2ℵ(ı)C

T

1

i

(ı)

i

T

˜

D

12

i

(ı) =

h

0

√

2ℵ(ı)ρ(ı)H

T

5

i

(ı)

γρ(ı)H

T

3

i

(ı)

√

2ℵ(ı)D

T

12

i

(ı)

i

T

ℵ(ı) =

I + ρ

2

(ı)

r

X

i=1

r

X

j=1

h

kH

T

2

i

(ı)H

2

j

(ı)k

i

1

2

then there exists a sufﬁciently small ˆε > 0 such that

the inequality (5) holds for ε ∈ (0, ˆε]. Furthermore, a

suitable choice of the fuzzy controller is

u(t) =

r

X

i=1

µ

j

K

j

(ı)x(t) (15)

where

K

j

(ı) = Y

j

(ı)(P (ı))

−1

. (16)

Proof: Due to limited pages, the detail of the proof

is omitted for brevity.

4 ILLUSTRATIVE EXAMPLE

Consider a modiﬁed series dc motor model based on

(Mehta and Chiasson, 1998) as shown in Fig. 1 which

is governed by the following difference equations:

J

d˜ω(t)

dt

= K

m

L

f

˜

i

2

(t) − (D + ∆D)˜ω(t)

L

d

˜

i(t)

dt

= −R

˜

i(t) − K

m

L

f

˜

i(t)˜ω(t) +

˜

V (t)

(17)

where ˜ω(t) = ω(t) − ω

ref

(t) is the deviation of

the actual angular velocity from the desired angular

ICINCO 2005 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

94

velocity,

˜

i(t) = i(t) − i

ref

(t) is the deviation of

the actual current from the desired current,

˜

V (t) =

V (t) − V

ref

(t) is the deviation of the actual input

voltage from the desired input voltage, J is the mo-

ment of inertia, K

m

is the torque/back emf constant,

D is the viscous friction coefﬁcient, and R

a

, R

f

, L

a

and L

f

are the armature resistance, the ﬁeld wind-

ing resistance, the armature inductance and the ﬁeld

winding inductance, respectively, with R

∆

= R

f

+ R

a

and L

∆

= L

f

+ L

a

. Note that in a typical series-

connected dc motor, the condition L

f

≫ L

a

holds.

When one obtains a series-connected dc motor, we

have i(t) = i

a

(t) = i

f

(t). Now let us assume that

|∆J| ≤ 0.1J.

b

V

L

I

f

I

a

V

+

−

back emf

f

a

L

a

R

f

R

+

−

Dω

τ

Figure 1: A modiﬁed series dc motor equivalent circuit.

Giving x

1

(t) = ˜ω(t), x

2

(t) =

˜

i(t) and u(t) =

˜

V (t), (17) becomes

˙x

1

(t)

ε ˙x

2

(t)

=

−

D

(J+∆J)

K

m

L

f

(J+∆J)

x

2

(t)

−K

m

L

f

x

2

(t) −R

x

1

(t)

x

2

(t)

+

0

1

u(t) (18)

where ε = L represents a small parasitic parameter.

Assume that, the system is aggregated into 3 modes

as shown in Table 1:

Table 1: System Terminology.

Mode ı Moment of Inertia J(ı) ± ∆J(ı)

(kg·m

2

)

1 Small 0.0005 ±10%

2 Normal 0.005 ±10%

3 Large 0.05 ±10%

The transition probability matrix that relates the

three operation modes is given as follows:

P

ık

=

"

0.67 0.17 0.16

0.30 0.47 0.23

0.26 0.10 0.64

#

.

The parameters for the system are given as R =

10 Ω, L

f

= 0.005 H, D = 0.05 N·m/rad/s and K

m

=

1 N·m/A. Substituting the parameters into (18), we

get

˙x

1

(t)

ε ˙x

2

(t)

=

−

0.05

J(ı)

0.005

J(ı)

x

2

(t)

−0.005x

2

(t) −10

x

1

(t)

x

2

(t)

+

0 0

0.1 0

w(t) +

0

1

u(t)

+

−

0.05

∆J(ı)

0.005

∆J(ı)

x

2

(t)

0 0

x

1

(t)

x

2

(t)

z(t) =

1 0

0 1

x

1

(t)

x

2

(t)

+

0

1

u(t)

where x(t) = [x

T

1

(t) x

T

2

(t)]

T

is the state variables,

w(t) = [w

T

1

(t) w

T

2

(t)]

T

is the disturbance input,

u(t) is the controlled input and z(t) is the controlled

output.

The control objective is to control the state variable

x

2

(t) for the range x

2

(t) ∈ [N

1

N

2

]. For the sake of

simplicity, we will use as few rules as possible. Note

that Fig. 2 shows the plot of the membership function

represented by

M

1

(x

2

(t)) =

−x

2

(t) + N

2

N

2

− N

1

and M

2

(x

2

(t)) =

x

2

(t) − N

1

N

2

− N

1

.

Knowing that x

2

(t) ∈ [N

1

N

2

], the nonlinear system

1 2

1

0

M (x ) M (x )

N

2

1

N

x (t)

2

2 2

−3

3

0

Figure 2: Membership functions for the two fuzzy set.

(19) can be approximated by the following TS fuzzy

model

E

ε

˙x(t) =

r

i=1

µ

i

[A

i

(ı) + ∆A

i

(ı)]x(t)

+B

1

i

(ı)w(t) + B

2

i

(ı)u(t)

, x(0) = 0,

z(t) =

r

i=1

µ

i

C

1

i

(ı)x(t) + D

12

i

(ı)u(t)

,

where µ

i

is the normalized time-varying fuzzy weighting

functions for each rule, i = 1, 2, x(t) =

x

1

(t)

x

2

(t)

, E

ε

=

ROBUST FUZZY CONTROLLER DESIGN FOR UNCERTAIN DESCRIPTOR MARKOVIAN JUMP SYSTEMS

95

1 0

0 ε

, ∆A

1

(ı) = F (x(t), ı, t)H

1

1

(ı), ∆A

2

(ı) =

F (x(t), ı, t)H

1

2

(ı),

A

1

(1) =

−100 10N

1

−0.005N

1

−10

,

A

2

(1) =

−100 10N

2

−0.005N

2

−10

,

A

1

(2) =

−10 N

1

−0.005N

1

−10

,

A

2

(2) =

−10 N

2

−0.005N

2

−10

,

A

1

(3) =

−1 0.1N

1

−0.005N

1

−10

,

A

2

(3) =

−1 0.1N

2

−0.005N

2

−10

,

B

1

1

(ı) = B

1

2

(ı) =

0 0

0.1 0

,

B

2

1

(ı) = B

2

2

(ı) =

0

1

,

C

1

1

(ı) = C

1

2

(ı) =

1 0

0 1

,

and D

12

1

(ı) = D

12

2

(ı) =

0

1

,

with kF (x(t), ı, t)k ≤ 1. Then we have

H

1

1

(ı) =

−

0.05

J (ı)

0.05

J (ı)

N

1

0 0

and H

1

2

(ı) =

−

0.05

J (ı)

0.05

J (ı)

N

2

0 0

.

In this simulation, we select N

1

= −3 and N

2

= 3.

Using the LMI optimization algorithm and Theorem 1

with ε = 0.005, γ = 1 and δ(1) = δ(2) = δ(3) = 1,

we obtain the results given in Fig. 3 and Fig. 4.

Remark 2 Employing results given in (Nguang and

Shi, 2001; Han and Feng, 1998; B. S. Chen and He,

2001; K. Tanaka and Wang, 1996; H. O. Wang and

Grifﬁn, 1996) and Matlab LMI solver (S. Boyd and

Balakrishnan, 1994), it is easy to realize that when

ε < 0.005 for the state-feedback control design, LMIs

become ill-conditioned and Matlab LMI solver yields

an error message, “Rank Deﬁcient”. However, the

state-feedback fuzzy controller proposed in this paper

guarantee that the inequality (5) holds for the sys-

tem (19). Fig. 3 shows the result of the changing

between modes during the simulation with the initial

mode at mode 1 and ε = 0.005. The disturbance in-

put signal, w(t), which was used during simulation

is with magnitude 0.1 and frequency 1 Hz. The ra-

tio of the regulated output energy to the disturbance

input noise energy obtained by using the H

∞

fuzzy

controller is depicted in Fig. 4. The ratio of the regu-

lated output energy to the disturbance input noise en-

ergy tends to a constant value which is about 0.0094.

So γ =

√

0.0094 = 0.0970 which is less than the

prescribed value 1. Finally, Table 2 shows the perfor-

mance index, γ, for different values of ε.

5 CONCLUSION

This paper has investigated the problem of design-

ing a robust H

∞

fuzzy state-feedback controller for

a class of uncertainty Markovian jump nonlinear sin-

gularly perturbed systems that guarantees the L

2

-gain

from an exogenous input to a regulated output to be

less or equal to a prescribed value. First, we approx-

imate this class of uncertain Markovian jump non-

linear singularly perturbed systems by a class of un-

certain Takagi-Sugeno fuzzy models with Markov-

ian jumps. Then, based on an LMI approach, LMI-

based sufﬁcient conditions for the uncertain Markov-

ian jump nonlinear singularly perturbed systems to

have an H

∞

performance are derived. The proposed

approach does not involve the separation of states into

slow and fast ones and it can be applied not only to

standard, but also to nonstandard nonlinear singularly

perturbed systems. An illustrative example is used

to illustrate the effectiveness of the proposed design

techniques.

Table 2: The performance index γ for different values of ε.

The performance index γ

ε State-feedback control design

0.005 0.0970

0.10 0.4796

0.30 0.8660

0.40 0.9945

0.41 > 1

0 0.5 1 1.5 2 2.5 3

1

1.5

2

2.5

3

Mode

Time (sec)

Figure 3: The result of the changing between modes during

the simulation with the initial mode at mode 1.

ICINCO 2005 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

96

0 0.5 1 1.5 2 2.5 3

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Ratio of the regulated output energy to the disturbance energy

Time (sec)

Figure 4: The ratio of the regulated output energy to the dis-

turbance noise energy,

T

f

0

z

T

(t)z(t)dt

T

f

0

w

T

(t)w(t)dt

with ε = 0.005.

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