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Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach
Kazuo Tanaka, Hua O. Wang
Copyright ᮊ 2001 John Wiley & Sons, Inc.
Ž. Ž .
ISBNs: 0-471-32324-1 Hardback ; 0-471-22459-6 Electronic
CHAPTER 3
LMI CONTROL PERFORMANCE
CONDITIONS AND DESIGNS
The preceding chapter introduced the concept and basic procedure of
parallel distributed compensation and LMI-based designs. The goal of this
chapter is to present the details of analysis and design via LMIs. This chapter
forms a basic and important component of this book. To this end, it will be
shown that various kinds of control performance specifications can be repre-
sented in terms of LMIs. The control performance specifications include
stability conditions, relaxed stability conditions, decay rate conditions, con-
strains on control input and output, and disturbance rejection for both
wx
continuous and discrete fuzzy control systems 1᎐3. Other more advanced
control performance considerations utilizing LMI conditions will be pre-
sented in later chapters.
3.1 STABILITY CONDITIONS
In the 1990’s, the issue of stability of fuzzy control systems has been
wx
investigated extensively in the framework of nonlinear system stability 1᎐18 .
Today, there exist a large number of papers on stability analysis of fuzzy
control in the literature. This section discusses some basic results on the
stability of fuzzy control systems.
In the following, Theorems 5 and 6 deal with stability conditions for the
open-loop systems. Theorem 5 can be readily obtained via Lyapunov stability
wx
theory. The proof of Theorem 6 is given in 4, 7 .

49
LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS
50
wx Ž.
THEOREM 5 CFS The equilibrium of the continuous fuzzy system 2.3 with
Ž.
u t s 0 is globally asymptotically stable if there exists a common positi®e definite
matrix P such that
A
T
P q PA - 0, i s 1,2,...,r ,3.1
Ž.
ii
that is, a common P has to exist for all subsystems.
wx Ž.
THEOREM 6 DFS The equilibrium of the discrete fuzzy system 2.5 with
Ž.
u t s 0 is globally asymptotically stable if there exists a common positi®e definite
matrix P such that
A
T
PA y P - 0, i s 1,2,...,r ,3.2
Ž.
ii
that is, a common P has to exist for all subsystems.
Next, let us consider the stability of the closed-loop system. By substituting
Ž . Ž. Ž. Ž. Ž.
2.23 into 2.3 and 2.5 , we obtain 3.3 and 3.4 , respectively.
CFS
rr

x t s h z thz t A y BF x t .3.3
Ä4
Ž. Ž. Ž. Ž. Ž .
Ž.Ž.
˙
ÝÝ
ij iij
i
s1 js1
DFS
rr
x t q 1 s h z thz t A y BF x t .3.4
Ä4
Ž . Ž. Ž. Ž. Ž .
Ž.Ž.
ÝÝ
ij iij
i
s1 js1
Denote
G s A y BF.
ij i i j
Ž. Ž. Ž. Ž.
Equations 3.3 and 3.4 can be rewritten as 3.5 and 3.6 , respectively.
CFS
r
x t s h z thz t Gxt
Ž. Ž. Ž. Ž.
Ž.Ž.
˙

Ý
ii ii
i
s1
r
G q G
ij ji
q 2 h z thz t x t .3.5
Ž. Ž. Ž. Ž .
Ž.Ž.
ÝÝ
ij
½5
2
i
s1 i-j
DFS
r
x t q 1 s h z thz t Gxt
Ž . Ž. Ž. Ž.
Ž.Ž.
Ý
ii ii
i
s1
r
G q G
ij ji
q 2 h z thz t x t .3.6
Ž. Ž. Ž. Ž .

Ž.Ž.
ÝÝ
ij
½5
2
i
s1 i-j
STABILITY CONDITIONS
51
Ž
By applying the stability conditions for the open-loop system Theorems 5
.Ž. Ž.
and 6 to 3.5 and 3.6 , we can derive stability conditions for the CFS and
the DFS, respectively.
wx
THEOREM 7 CFS The equilibrium of the continuous fuzzy control system
Ž.
described by 3.5 is globally asymptotically stable if there exists a common
positi®e definite matrix P such that
G
T
P q PG - 0,3.7
Ž.
ii ii
T
G q GGq G
ij ji ij ji
P q P F 0,
ž/ž/
22

i - j s.t. h l h /
␾
.3.8
Ž.
ij
Proof. It follows directly from Theorem 5.
For the explanation of the notation i - j s.t. h l h /
␾
, refer to
ij
Chapter 1.
wx
THEOREM 8 DFS The equilibrium of the discrete fuzzy control system
Ž.
described by 3.6 is globally asymptotically stable if there exists a common
positi®e definite matrix P such that
G
T
PG y P - 0,3.9
Ž.
ii ii
T
G q GGq G
ij ji ij ji
P y P F 0,
ž/ž/
22
i - j s.t. h l h /
␾
. 3.10

Ž.
ij
Proof. It follows directly from Theorem 6.
Ž.
The fuzzy control design problem is to determine F ’s j s 1,2,...,r
j
which satisfy the conditions of Theorem 7 or 8 with a common positive
definite matrix P.
Consider the common B matrix case, that is, B s B s иии s B . In this
12 r
case, the stability conditions of Theorems 7 and 8 can be simplified as
follows.
COROLLARY 1 Assume that B s B s иии s B . The equilibrium of the
12 r
Ž.
fuzzy control system 3.5 is globally asymptotically stable if there exists a
Ž.
common positi®e definite matrix P satisfying 3.7 .
COROLLARY 2 Assume that B s B s иии s B . The equilibrium of the
12 r
Ž.
fuzzy control system 3.6 is globally asymptotically stable if there exists a
Ž.
common positi®e definite matrix P satisfying 3.9 .
LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS
52
In other words, the corollaries state that in the common B case, G
T
P q
ii

PG - 0 implies
ii
T
G q GGq G
ij ji ij ji
P q P F 0
ž/ž/
22
and G
T
PG y P - 0 implies
ii ii
T
G q GGq G
ij ji ij ji
P y P F 0
ž/ž/
22
To check stability of the fuzzy control system, it has long been considered
difficult to find a common positive definite matrix P satisfying the conditions
wx
of Theorems 5᎐8. A trial-and-error type of procedure was first used 4, 7, 9 .
wx
In 19 , a procedure to construct a common P is given for second-order fuzzy
systems, that is, the dimension of the state is 2. It was first stated in
wx
11, 12, 17 that the common P problem for fuzzy controller design can be
solved numerically, that is, the stability conditions of Theorems 5᎐8 can be
expressed in LMIs. For example, to check the stability conditions of Theorem
7, we need to find P satisfying the LMIs

P ) 0, G
T
P q PG - 0,
ii ii
T
G q GGq G
i j ji i j ji
P q P F 0, i - j s.t. h l h /
␾
,
ij
ž/ž/
22
or determine that no such P exists. This is a convex feasibility problem. As
shown in Chapter 2, this feasibility problem can be numerically solved very
efficiently by means of the most powerful tools available to date in the
mathematical programming literature.
3.2 RELAXED STABILITY CONDITIONS
We have shown that the stability analysis of the fuzzy control system is
reduced to a problem of finding a common P.If r, that is the number of
IF-THEN rules, is large, it might be difficult to find a common P satisfying
Ž.
the conditions of Theorem 7 or Theorem 8 . This section presents new
stability conditions by relaxing the conditions of Theorems 7 and 8. Theorems
wx
9 and 10 provide relaxed stability conditions 1᎐3 . First, we need the
following corollaries to prove Theorems 9 and 10.
COROLLARY 3
rr
1

2
h z t y 2 h z thz t G 0,
Ž. Ž. Ž.
Ž. Ž.Ž.
ÝÝÝ
iij
r y 1
i
s1 is1 i-j
RELAXED STABILITY CONDITIONS
53
where
r
h z t s 1, h z t G 0
Ž. Ž.
Ž. Ž.
Ý
ii
i
s1
for all i.
Proof. It holds since
rr
1
2
h z t y 2 h z thz t
Ž. Ž. Ž.
Ž. Ž.Ž.
ÝÝÝ
iij

r y 1
i
s1 is1 i-j
r
1
2
s h z t y h z t G 0. Q.E.D.
Ž. Ž.
Ž. Ž.
Ä4
ÝÝ
ij
r y 1
i
s1 i-j
COROLLARY 4 If the number of rules that fire for all t is less than or equal to
s, where 1 - s F r, then
rr
1
2
h z t y 2 h z thz t G 0,
Ž. Ž. Ž.
Ž. Ž.Ž.
ÝÝÝ
iij
s y 1
i
s1 is1 i-j
where
r

h z t s 1, h z t G 0
Ž. Ž.
Ž. Ž.
Ý
ii
i
s1
for all i.
Proof. It follows directly from Corollary 3.
wx
THEOREM 9 CFS Assume that the number of rules that fire for all t is less
than or equal to s, where 1 - s F r. The equilibrium of the continuous fuzzy
Ž.
control system described by 3.5 is globally asymptotically stable if there exist a
common positi®e definite matrix P and a common positi®e semidefinite matrix Q
such that
G
T
P q PG q s y 1 Q - 0 3.11
Ž. Ž.
ii ii
T
G q GGq G
ij ji ij ji
P q P y Q F 0,
ž/ž/
22
i - j s.t. h l h /
␾
3.12

Ž.
ij
where s ) 1.
LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS
54
ŽŽ..
T
Ž. Ž.
Proof. Consider a candidate of Lyapunov function V x t s x t Px t ,
where P ) 0. Then,
rr
T
˙
V x t s h z thz t x t
Ž. Ž. Ž. Ž.
Ž. Ž.Ž.
ÝÝ
ij
i
s1 js1
=
T
A y BF Pq PAy BF x t
Ž.
Ž.Ž.
iij iij
r
2 TT
s h z t x t GPq PG x t
Ž. Ž. Ž.

Ž.
Ý
iiiii
i
s1
r
T
q 2h z thz t x t
Ž. Ž. Ž.
Ž.Ž.
ÝÝ
ij
i
s1 i-j
=
T
G q GGq G
ij ji ij ji
P q Pxt ,
Ž.
ž/ž/
22
where
G s A y BF.
ij i i j
Ž.
From condition 3.12 and Corollary 4, we have
r
2 TT
˙

V x t F h z t x t GPq PG x t
Ž. Ž. Ž. Ž.
Ž. Ž.
Ý
iiiii
i
s1
r
T
q 2h z thz t x t Qx t
Ž. Ž. Ž. Ž.
Ž.Ž.
ÝÝ
ij
i
s1 i-j
r
2 TT
F h z t x t GPq PG x t
Ž. Ž. Ž.
Ž.
Ý
iiiii
i
s1
r
2 T
q s y 1 h z t x t Qx t
Ž . Ž. Ž. Ž.
Ž.

Ý
i
i
s1
r
2 TT
s h z t x t GPq PG q s y 1 Qxt .
Ž. Ž. Ž . Ž.
Ž.
Ý
iiiii
i
s1
˙
Ž. ŽŽ.. Ž.
If condition 3.11 holds, V x t - 0at x t / 0. Q.E.D.
wx
THEOREM 10 DFS Assume that the number of rules that fire for all t is less
than or equal to s, where 1 - s F r. The equilibrium of the discrete fuzzy control
Ž.
system described by 3.6 is globally asymptotically stable if there exist a common
positi®e definite matrix P and a common positi®e semidefinite matrix Q such that
G
T
PG y P q s y 1 Q - 0, 3.13
Ž. Ž.
ii ii
T
G q GGq G
ij ji ij ji

P y P y Q F 0,
ž/ž/
22
i - j s.t. h l h /
␾
, 3.14
Ž.
ij
where s ) 1.
RELAXED STABILITY CONDITIONS
55
ŽŽ..
T
Ž. Ž.
Proof. Consider a candidate of Lyapunov function V x t s x t Px t ,
where P ) 0. Then,
⌬V x t s V x t q 1 y V x t
Ž. Ž . Ž.
Ž.Ž .Ž.
rr rr
s h z thz thz thz t
Ž. Ž. Ž. Ž.
Ž.Ž.Ž.Ž.
ÝÝÝÝ
ijkl
i
s1 js1 ks1 ls1
=
TT
x t GPG y Pxt

Ž. Ž.
ij kl
rr rr
1
s h z thz thz thz t
Ž. Ž. Ž. Ž.
Ž.Ž.Ž.Ž.
ÝÝÝÝ
ijkl
4
i
s1 js1 ks1 ls1
=
T
T
x t G q GPGq G y 4 Pxt
Ž. Ž . Ž.
Ž.
ij ji kl lk
rr
1
TT
F h z thz t x t HPHy 4 Pxt
Ž. Ž. Ž. Ž.
Ž.Ž.
ÝÝ
i j ij ij
4
i
s1 js1

T
rr
HH
ij ij
T
s h z thz t x t P y Pxt
Ž. Ž. Ž. Ž.
Ž.Ž.
ÝÝ
ij
22
i
s1 js1
r
2 TT
s h z t x t GPGy Pxt
Ž. Ž. Ž.
Ž.
Ý
iiiii
i
s1
T
r
HH
ij ij
T
q 2 h z thz t x t P y Pxt ,
Ž. Ž. Ž. Ž.
Ž.Ž.

ÝÝ
ij
22
i
s1 i-j
where H s G q G .
ij ij ji
Ž.
From condition 3.14 and Corollary 4, the right side of the above
inequality becomes
r
2 TT
F h z t x t GPGy Pxt
Ž. Ž. Ž.
Ž.
Ý
iiiii
i
s1
r
T
q 2 h z thz t x t Qx t
Ž. Ž. Ž. Ž.
Ž.Ž.
ÝÝ
ij
i
s1 i-j
r
2 TT

F h z t x t GPGy Pxt
Ž. Ž. Ž.
Ž.
Ý
iiiii
i
s1
r
2 T
q s y 1 h z t x t Qx t
Ž . Ž. Ž. Ž.
Ž.
Ý
i
i
s1
r
2 TT
s h z t x t GPGy P q s y 1 Qxt .
Ž. Ž. Ž . Ž.
Ž.
Ý
iiiii
i
s1
Ž. ŽŽ.. Ž.
If condition 3.13 holds, ⌬V x t - 0at x t / 0. Q.E.D.
LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS
56
Corollary 4 is used in the proofs of Theorems 9 and 10. The use of

Corollary 3 would lead to conservative results because s F r.
Remark 11 It is assumed in the derivations of Theorems 7᎐10 that the
ŽŽ..
weight h z t of each rule in the fuzzy controller is equal to that of each
i
rule in the fuzzy model for all t. Note that Theorems 7᎐10 cannot be used if
Ž.
the assumption does not hold. This fact will show up again in a case case B
of fuzzy observer design given in Chapter 4. If the assumption does not hold,
the following stability conditions should be used instead of Theorems 7᎐10:
G
T
P q PG - 0
ij ji
in the CFS case and
G
T
PG y P - 0
ij ji
in the DFS case. These conditions imply those of Theorems 7᎐10. These
conditions may be regarded as robust stability conditions for premise part
wx
uncertainty 18 .
Fig. 3.1 Feasible area for the stability conditions of Theorem 7.
RELAXED STABILITY CONDITIONS
57
Fig. 3.2 Feasible area for the stability conditions of Theorem 9.
The conditions of Theorems 9 and 10 reduce to those of Theorems 7
and 8, respectively, when Q s 0.
Example 9 This example demonstrates the utility of the relaxed conditions

in the CFS case. Consider the CFS, where r s s s 2,
2 y10 1
A s , B s ,
11
10 0
a y10 b
A s , B s .
22
10 0
wx
The local feedback gains F and F are determined by selecting y2 y2as
12
the eigenvalues of the subsystems in the PDC. Figures 3.1 and 3.2 show the
feasible areas satisfying the conditions of Theorems 7 and 9 for the variables
a and b, respectively. In these figures, the feasible areas are plotted for
Ž.
a ) 2 and b ) 20. A common P and a common Q satisfying the conditions
Ž. Ž.
of Theorem 7 Figure 3.1 and Theorem 9 Figure 3.2 exists if and only if the
system parameters a and b are located in the feasible areas under a ) 2 and
b ) 20. It is found in these figures that the conditions of Theorem 7 lead to
conservative results.
LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS
58
3.3 STABLE CONTROLLER DESIGN
This section presents stable fuzzy controller designs for CFS and DFS.
We first present a stable fuzzy controller design problem which is to
determine the feedback gains F for the CFS using the stability conditions of
i
Ž. Ž.

Theorem 7. The conditions 3.7 and 3.8 are not jointly convex in F and P.
i
Now multiplying the inequality on the left and right by P
y1
and defining a
new variable X s P
y1
, we rewrite the conditions as
yXA
T
yAXq XF
T
B
T
q BFX) 0,
ii ii ii
TT
yXA yAXy XA y AX
ii j j
TT TT
qXF B q BFXq XF B q BFXG 0.
ji ij ij ji
Define M s FX so that for X ) 0 we have F s MX
y1
. Substituting into
ii i i
the above inequalities yields
yXA
T
yAXq M

T
B
T
q BM) 0,
ii ii ii
TT
yXA yAXy XA y AX
ii j j
TT TT
qMBq BMq MBq BMG 0.
ji ij ij ji
Using these LMI conditions, we define a stable fuzzy controller design
problem.
Ž.
Stable Fuzzy Controller Design: CFS Find X ) 0 and M i s 1,...,r
i
satisfying
TTT
Ž.
yXA yAXq MBq BM) 0, 3.15
ii ii ii
TT
yXA yAXy XA y AX
ii j j
TT TT
qMBq BMq MBq BMG 0,
ji ij ij ji
i - j s.t h l h /
␾
3.16

Ž.
ij
where
X s P
y1
, M s FX. 3.17
Ž.
ii
The above conditions are LMIs with respect to variables X and M . We can
i
find a positive definite matrix X and M satisfying the LMIs or determina-
i
tion that no such X and M exist. The feedback gains F and a common P
ii
can be obtained as
P s X
y1
, F s MX
y1
3.18
Ž.
ii
from the solutions X and M .
i
STABLE CONTROLLER DESIGN
59
A stable fuzzy controller design problem for the DFS can be defined from
the conditions of Theorem 8 as well:
Ž.
T y1

Ž.
XAy BF X A y BF Xy X - 0,
iii iii
T
A y BFq A y BF
iijjji
y1
XX
½5
2
A y BFq A y BF
iijjji
= X y X F 0.
½5
2
Define M s FXso that for X ) 0 we have F s MX
y1
. Substituting into
ii i i
the above inequalities yields
Ž.
T y1
Ž.
X y AXy BM X AXy BM ) 0,
iii iii
T
AXy BMq AXy BM
iijjji
y1
X y XX

½5
2
AXy BMq AXy BM
iijjji
= X G 0.
½5
2
Ž.
These nonlinear convex inequalities can now be converted to LMIs using
the Schur complement. The resulting LMIs are
TTT
XXAy MB
iii
) 0,
AXy BM X
iii
T
AXq AXy BMy BM
ijijji
X
½5
2
G 0
AXq AXy BMy BM
ijijji
X
½5
2
in X and F .
i

Ž.
Stable Fuzzy Controller Design: DFS Find X ) 0 and M i s 1,...,r
i
satisfying
TTT
XXAy MB
iii
) 0.
3.19
Ž.
AXy BM X
iii
T
AXq AXy BMy BM
ijijji
X
½5
2
G 0,
AXq AXy BMy BM
ijijji
X
½5
2
i - j s.t. h l h /
␾
,
3.20
Ž.
ij

LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS
60
where
X s P
y1
, M s FX. 3.21
Ž.
ii
The feedback gain F and a common P can be obtained as
i
P s X
y1
, F s MX
y1
3.22
Ž.
ii
from the solutions X and M .
i
From the relaxed stability conditions of Theorem 9, the design problem to
determine the feedback gains F for CFS can be defined as well.
i
Fuzzy Controller Design Using Relaxed Stability Conditions: CFS Find
Ž.
X ) 0, Y G 0, and M i s 1,...,r satisfying
i
TTT
Ž. Ž.
yXA y AXq MBq BMy s y 1 Y ) 0, 3.23
ii ii ii

TT
2Y y XA y AXy XA y AX
ii jj
TT TT
qMBq BMq MBq BMG 0,
ji ij ij ji
Ž.
i - j s.t. h l h /
␾
, 3.24
ij
where
X s P
y1
, M s FX, Y s XQX . 3.25
Ž.
ii
The above conditions are LMIs with respect to variables X, Y, and M .We
i
can find a positive definite matrix X, a positive semidefinite matrix Y, and M
i
satisfying the LMIs or determine that no such X, Y, and M exist. The
i
feedback gains F , a common P, and a common Q can be obtained as
i
P s X
y1
, F s MX
y1
, Q s PYP 3.26

Ž.
ii
from the solutions X, Y, and M .
i
From the relaxed conditions of Theorem 10, the design problem for DFS
can be defined as well.
Fuzzy Controller Design Using Relaxed Stability Conditions: DFS Find
Ž.
X ) 0, Y G 0, and M i s 1,...,r satisfying
i
TTT
Ž.
X y s y 1 YXAy MB
iii
) 0,
3.27
Ž.
AXy BM X
iii
T
1
Ä4
X q YAXq AXy BMy BM
ijijji
2
G 0,
1
Ä4
AXq AXy BMy BM X
ijijji

2
i - j s.t. h l h /
␾
,
3.28
Ž.
ij
STABLE CONTROLLER DESIGN
61
where
X s P
y1
, M s FX, Y s XQX .
ii
The feedback gain F , a common P, and a common Q can be obtained as
i
P s X
y1
, F s MX
y1
, Q s PYP
ii
from the solutions X, Y, and M .
i
Ž. Ž.
The conditions 3.27 and 3.28 can be obtained as follows: Multiplying
Ž.
y1
both sides of 3.13 by P gives
P

y1
G
T
PG P
y1
y P
y1
q s y 1 P
y1
QP
y1
- 0.
Ž.
ii ii
Therefore,
P
y1
y s y 1 P
y1
QP
y1
Ž.
T
y1 y1 y1 y1
y AP y BFP P AP y BFP ) 0.
Ž.Ž.
iii iii
Since P
y1
s X, we have

T
y1
X y s y 1 XQX y AXy BFX X AXy BFX ) 0.
Ž. Ž .Ž .
iii iii
Define M s FXand Y s XQX. By substituting into the above inequality,
ii
we obtain
T
y1
X y s y 1 Y y AXy BM X A Xy BM ) 0.
Ž.Ž.Ž.
iii iii
Ž.
It easily follows that the above inequality can be transformed into 3.27 by
the Schur complement procedure.
Ž.
Similarly, from 3.14 , we have
T
G q GGq G
ij ji ij ji
y1 y1 y1 y1 y1
PPPy P y PQP F 0.
ž/ž/
22
Therefore,
P
y1
q P
y1

QP
y1
T
1
y1 y1 y1 y1
y GP q GP PGP q GP G 0.
Ž.Ž.
ij ji ij ji
4
Since P
y1
s X, we have
T
11
y1
X q XQX y GXq GX X GXq GX
Ä4
Ž.Ž.
½5
ij ji ij ji
22
T
1
y1
s X q XQX y AXq AXy BFXy BFX X
Ž.
½5
ijijji
2
1

= AXq AXy BFXy BFX G 0.
Ä4
Ž.
ijijji
2