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The control performance specifications includestability conditions, relaxed stability conditions, decay rate conditions, con-strains on control input and output, and disturbance rejectio

Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach

Kazuo Tanaka, Hua O Wang Copyright 䊚 2001 John Wiley & Sons, Inc.

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 ofparallel distributed compensation and LMI-based designs The goal of thischapter is to present the details of analysis and design via LMIs This chapterforms a basic and important component of this book To this end, it will beshown that various kinds of control performance specifications can be repre-sented in terms of LMIs The control performance specifications includestability conditions, relaxed stability conditions, decay rate conditions, con-strains on control input and output, and disturbance rejection for both

control performance considerations utilizing LMI conditions will be sented in later chapters

pre-3.1 STABILITY CONDITIONS

In the 1990’s, the issue of stability of fuzzy control systems has been

Today, there exist a large number of papers on stability analysis of fuzzycontrol in the literature This section discusses some basic results on thestability of fuzzy control systems

In the following, Theorems 5 and 6 deal with stability conditions for theopen-loop systems Theorem 5 can be readily obtained via Lyapunov stability

theory The proof of Theorem 6 is given in 4, 7

49

w x 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

u t s 0 is globally asymptotically stable if there exists a common positi®e definite

matrix P such that

A T i PA y P i - 0, i s 1, 2, , r , Ž3.2.

that is, a common P has to exist for all subsystems.

Next, let us consider the stability of the closed-loop system By substituting

Ž2.23 into 2.3 and 2.5 , we obtain 3.3 and 3.4 , respectively Ž Ž Ž Ž

described by 3.5 is globally asymptotically stable if there exists a common

positi®e definite matrix P such that

Proof. It follows directly from Theorem 5

Chapter 1

THEOREM 8 DFS The equilibrium of the discrete fuzzy control system

described by 3.6 is globally asymptotically stable if there exists a common

positi®e definite matrix P such that

which satisfy the conditions of Theorem 7 or 8 with a common positive

definite matrix P.

case, the stability conditions of Theorems 7 and 8 can be simplified asfollows

COROLLARY 1 Assume that B s B s1 2 ⭈⭈⭈ s B The equilibrium of the r

fuzzy control system 3.5 is globally asymptotically stable if there exists a

common positi®e definite matrix P satisfying 3.7

COROLLARY 2 Assume that B s B s1 2 ⭈⭈⭈ s B The equilibrium of the r

In other words, the corollaries state that in the common B case, G T i i P q

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

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

expressed in LMIs For example, to check the stability conditions of Theorem

7, we need to find P satisfying the LMIs

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 veryefficiently by means of the most powerful tools available to date in themathematical programming literature

3.2 RELAXED STABILITY CONDITIONS

We have shown that the stability analysis of the fuzzy control system is

IF-THEN rules, is large, it might be difficult to find a common P satisfying

the conditions of Theorem 7 or Theorem 8 This section presents newstability conditions by relaxing the conditions of Theorems 7 and 8 Theorems

following corollaries to prove Theorems 9 and 10

RELAXED STABILITY CONDITIONS 53

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

qÝ Ý2h z t iŽ Ž h z t jŽ Ž x Ž t Qx tŽ

is1 i -j r

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

RELAXED STABILITY CONDITIONS 55

Corollary 4 is used in the proofs of Theorems 9 and 10 The use of

Remark 11 It is assumed in the derivations of Theorems 7᎐10 that the

Ž 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,

G T i j P q PG ji- 0

in the CFS case and

G T i j PG y P ji - 0

conditions may be regarded as robust stability conditions for premise part

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

the eigenvalues of the subsystems in the PDC Figures 3.1 and 3.2 show thefeasible 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

of Theorem 7 Figure 3.1 and Theorem 9 Figure 3.2 exists if and only if the

conservative results

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

the above inequalities yields

can be obtained as

from the solutions X and M

STABLE CONTROLLER DESIGN 59

A stable fuzzy controller design problem for the DFS can be defined fromthe conditions of Theorem 8 as well:

the above inequalities yields

From the relaxed stability conditions of Theorem 9, the design problem to

Fuzzy Controller Design Using Relaxed Stability Conditions: CFS Find

P s Xy 1, F s M X i i y 1, Q s PYP Ž3.26.

From the relaxed conditions of Theorem 10, the design problem for DFScan be defined as well

Fuzzy Controller Design Using Relaxed Stability Conditions: DFS Find

STABLE CONTROLLER DESIGN 61

By substituting M s F X and Y s XQX into the above inequality, we obtain i i

␣ ) 0 Therefore, the largest lower bound on the decay rate that we can findusing a quadratic Lyapunov function can be found by solving the following

DECAY RATE 63

Decay Rate Fuzzy Controller Design: DFS The condition that⌬V x t F

Ž␣ y 1 V x t2 Ž Ž w20 for all trajectories is equivalent tox

Remark 12 The decay rate fuzzy controller designs reduce to the stable

that satisfies the LMI conditions of 3.31 and 3.32 or 3.36 and 3.37 is a

stable fuzzy controller In other words, the LMI conditions of 3.15 and

Ž3.36 and 3.37 , respectively Ž

Decay Rate Controller Design Using Relaxed Stability Conditions: CFS

can find using a quadratic Lyapunov function can be found by solving the

Remark 13 A fuzzy controller that satisfies the LMI conditions of 3.39 and

satisfies the LMI conditions of 3.23 and 3.24 or 3.27 and 3.28

Remark 14 As illustrated in Example 9, the conditions of Theorems 9 and

10 lead to less conservative results for the stability of a given fuzzy controlsystem For the design of stabilizing fuzzy controllers, it is recommended touse the conditions of these theorems together with other control perfor-mance considerations such as pole placement LMI conditions

3.5 CONSTRAINTS ON CONTROL INPUT AND OUTPUT

3.5.1 Constraint on the Control Input

CONSTRAINTS ON CONTROL INPUT AND OUTPUT 67

wor 3.27 and 3.28 and 3.46 and 3.47 Ž . Ž .x Ž . Ž .

3.5.2 Constraint on the Output

wor 3.27 and 3.28 and 3.53 and 3.54 Ž . Ž .x Ž . Ž .

3.6 INITIAL STATE INDEPENDENT CONDITION

The above LMI design conditions for input and output constraints depend on

Ž

be again determined using the above LMIs if the initial states x 0 change.

This is a disadvantage of using the LMIs on the control input and output We

Ž

modify the LMI constraints on the control input and output, where x 0 is

DISTURBANCE REJECTION 69

THEOREM 13 Assume that x0 F␾, where x 0 is unknown but the

upper bound ␾ is known Then,

minimize ␥ in 3.59 can be obtained by sol®ing the following minimization

problem based on LMIs.

Proof. Suppose there exists a quadratic function V x t sx t Px t ,

Ž The left-hand side of 3.65 can be decomposed as follows:

r T

Ž Ž TŽ Ž

Proof. Suppose there exists a quadratic functionV x t s x t Px t , P) 0,

By multiplying both sides of 3.77 by block-diag X I I I , 3.72 is

A design example for disturbance rejection will be discussed in Chapter 8

3.8 DESIGN EXAMPLE: A SIMPLE MECHANICAL SYSTEM

Let us consider an example of dc motor controlling an inverted pendulum via

a gear train 22 Fuzzy modeling for the nonlinear system was done in 3 ,

DESIGN EXAMPLE: A SIMPLE MECHANICAL SYSTEM 77

Ž

common B matrix, that is, B s B The fuzzy controller design of the1 2common B matrix cases is simple in general To show the effect of the

as follows:

0

B s2 0 20

3.8.1 Design Case 1: Decay Rate

We first design a stable fuzzy controller by considering the decay rate Thedesign problem of the CFS is defined as follows:

DESIGN EXAMPLE: A SIMPLE MECHANICAL SYSTEM 79

design, there is a limitation of control input It is important to consider notonly the decay rate but also the constraint on the control input The designproblem that considers the decay rate and the constraint on the control input

Ž Ž Ž Ž

5 Ž 5

3.8.3 Design Case 3: Stability H Constraint on the Control Input

It is also possible to design a stable fuzzy controller satisfying the constraint

u t It can be found that max u t t 2s38.1-␮

Fig 3.4 Design examples 3 and 4.

considered in the fuzzy controller design To improve the response, we candesign a fuzzy controller by adding the constraint on the output.

The solution is obtained as

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