Tài liệu Hệ thống điều khiển mờ - Thiết kế và phân tích P4 ppt

Likewise, a systematic design method of fuzzy regulators and fuzzy observers plays an important role for fuzzy control systems.. This chapter presents the concept of fuzzy observers and

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

ISBNs: 0-471-32324-1 Hardback ; 0-471-22459-6 Electronic

CHAPTER 4

FUZZY OBSERVER DESIGN

In practical applications, the state of a system is often not readily available Under such circumstances, the question arises whether it is possible to determine the state from the system response to some input over some

w x period of time For linear systems, a linear observer 1 provides an affirma-tive answer if the system is observable Likewise, a systematic design method

of fuzzy regulators and fuzzy observers plays an important role for fuzzy control systems This chapter presents the concept of fuzzy observers and two

design procedures for fuzzy observer-based control 2, 3 In linear system theory, one of the most important results on observer design is the so-called separation principle, that is, the controller and observer design can be carried out separately without compromising the stability of the overall closed-loop system In this chapter, it is shown that a similar separation principle also holds for a large class of fuzzy control systems

4.1 FUZZY OBSERVER

Up to this point we have mainly dealt with LMI-based fuzzy control designs involving state feedback In real-world control problems, however, it is often the case that the complete information of the states of a system is not always available In such cases, one need to resort to output feedback design methods such as observer-based designs This chapter presents fuzzy ob-server design methodologies involving state estimation for T-S fuzzy models Alternatively, output feedback design can be treated in the framework of dynamic feedback, which is the subject of Chapter 12

83

w x

As in all observer designs, fuzzy observers 4 are required to satisfy

xŽ t y x t ™ 0ˆ Ž as t ™⬁,

Ž

converges to 0 As in the case of controller design, the PDC concept is employed to arrive at the following fuzzy observer structures:

CFS

Obser©er Rule i

IF z t is M and1 i1 ⭈⭈⭈ and z t is M p i p

THEN

ˆxŽ t s A x t q B u t q KŽ Ž ŽyŽ t y y tŽ ,

yŽ t s C x t ,Ž i s 1, 2, , r Ž4.1.

DFS

Obser©er Rule i

IF z t is M and1 i1 ⭈⭈⭈ and z t is M p i p

THEN

xŽt q 1 s A x t q B u t q K Ž Ž ŽyŽ t y y tŽ ,

yŽ t s C x t ,Ž i s 1, 2, , r Ž4.2.

The fuzzy observer has the linear state observer’s laws in its consequent

next section

4.2 DESIGN OF AUGMENTED SYSTEMS

This section presents LMI-based designs for an augmented system containing both the fuzzy controller and observer

The dependence of the premise variables on the state variables makes it necessary to consider two cases for fuzzy observer design:

Case A z t , , z t do not depend on the state variables estimated by a1 p

fuzzy observer

Ž Ž

Case B z t , , z t depend on the state variables estimated by a fuzzy1 p

observer

Obviously the stability analysis and design of the augmented system for Case A are more straightforward, whereas the stability analysis and design for Case B are complicated since the premise variables depend on the state variables, which have to estimated by a fuzzy observer This fact leads to

of fuzzy observer and controller

4.2.1 Case A

The fuzzy observer for Case A is represented as follows:

CFS

r

w z tŽ Ž A xˆ Ž t q B u t q KŽ ŽyŽ t y y tˆ Ž 4

is1

ˆ

xŽ t s

w z tŽ Ž

Ý i

is1 r

s Ýh z t iŽ Ž A x iˆ Ž t q B u t q K i Ž iŽyŽ t y y tˆ Ž 4, Ž4.3.

is1 r

yŽ t s h z tŽ Ž C xŽ t Ž4.4.

is1

DFS

r

w z tŽ Ž A xˆ Ž t q B u t q KŽ ŽyŽ t y y tˆ Ž 4

is1

xŽt q 1 s.

w z tŽ Ž

Ý i

is1 r

s Ýh z t iŽ Ž A x iˆ Ž t q B u t q K i Ž iŽyŽ t y y tˆ Ž 4, Ž4.5.

is1 r

yŽ t s h z tŽ Ž C xŽ t Ž4.6.

is1

Ž Ž

In the presence of the fuzzy observer for Case A, the PDC fuzzy controller

Ž takes on the following form, instead of 2.23 :

r

w z tŽ Ž F xˆ Ž t

is1

uŽ t s y r s yÝh z t iŽ Ž F x iˆ Ž t Ž4.7.

is1

w z tŽ Ž

Ý i

is1

tions:

CFS

xŽ t s h z tŽ Ž h z tŽ Ž  A y B F xŽ t q B F e tŽ 4,

is1 js1

eŽ t s h z tŽ Ž h z tŽ Ž A y K C e4 Ž t

is1 js1

DFS

xŽt q 1 s Ý Ýh z t iŽ Ž h z t jŽ Ž  ŽA y B F x i i j Ž t q B F e t i j Ž 4,

is1 js1

eŽt q 1 s Ý Ýh z t iŽ Ž h z t jŽ Ž A y K C e i i j4 Ž t

is1 js1

Therefore, the augmented systems are represented as follows:

CFS

x Ž t s h z tŽ Ž h z tŽ Ž G x Ž t

is1 js1 r

s Ýh z t iŽ Ž h z t iŽ Ž G x i i aŽ t

is1

q2Ý Ýh z t iŽ Ž h z t jŽ Ž x aŽ t , Ž4.8.

2

is1 i -j

DFS

x aŽt q 1 s Ý Ý h z t iŽ Ž h z t jŽ Ž G x i j aŽ t

is1 js1 r

s Ýh z t iŽ Ž h z t iŽ Ž G x i i aŽ t

is1

q2Ý Ýh z t iŽ Ž h z t jŽ Ž x aŽ t , Ž4.9.

2

is1 i -j

xŽ t

x aŽ t s ,

eŽ t

A y B F i i j B F i j

0 A y K C i i j

By applying Theorems 7 and 8 to the augmented system 4.8 and 4.9 , respectively, we arrive at the following theorems

THEOREM 16 CFS The equilibrium of the augmented system described by

Ž4.8 is globally asymptotically stable if there exists a common positi®e definite

matrix P such that

T

G q G i j ji G q G i j ji

ž 2 / ž 2 /

Proof. It follows directly from Theorem 7

THEOREM 17 DFS The equilibrium of the augmented system described by

Ž4.9 is globally asymptotically stable if there exists a common positi®e definite

matrix P such that

T

G q G i j ji G q G i j ji

ž 2 / ž 2 /

Proof. It follows directly from Theorem 8

Recall that Theorems 9 and 10 represent less conservative conditions than those of Theorems 7 and 8 Therefore, by applying Theorems 9 and 10 to

conditions:

w x

THEOREM 18 CFS The equilibrium of the augmented system described by

Ž4.8 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 i i P q PG q i i Žs y 1 Q. - 0, Ž4.15.

T

G q G i j ji G q G i j ji

ž 2 / ž 2 /

where s) 1

Proof. It follows directly from Theorem 9

THEOREM 19 DFS The equilibrium of the augmented system described by

Ž4.9 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 i i PG y P q i i Žs y 1 Q. - 0, Ž4.17.

T

G q G i j ji G q G i j ji

ž 2 / ž 2 /

where s) 1

Proof. It follows directly from Theorem 10

As a further refinement, we can incorporate the decay rate condition into the augmented systems as follows:

equivalent to

G T i i P q PG q i i Žs y 1 Q q 2. ␣ P - 0, Ž4.19.

T

G q G i j ji G q G i j ji

ž 2 / ž 2 /

equivalent to

G T i i PG y i i ␣2P qŽs y 1 Q. - 0, Ž4.21.

T

G q G i j ji G q G i j ji

2

ž 2 / ž 2 /

Next we consider the controller and observer design problem The ap-proach is to transform the conditions above for CFS and DFS into LMI ones

The transformation procedure can be similarly applied to all theorems in this section In the following, we present some representative results Other cases are left as exercises for the readers

Design Procedure for Case A: CFS Assume that the number of rules that

the decay rate that we can find using a quadratic Lyapunov function can be found by solving the GEVP

P1, P , Y , Q2 22, M , N1i 2i

P A1 T iyM1T i B T i qA P y B M q i 1 i 1i Žs y 1 Y q 2. ␣ P - 0,1

A T i P y C2 T i N2T iqP A y N C q2 i 2i i Žs y 1 Q. 22q2␣ P - 0,2

P A1 T iyM1T j B i TqA P y B M y i 1 i 1j 2 Y q 4 ␣ P1

qP A1 T jyM1T i B T j qA P y B M j 1 j 1i- 0,

A T i P y C2 T j N2T iqP A y N C y2 i 2i j 2 Q22q4␣ P2

qA T j P y C2 i T N2T jqP A y N C2 j 2j i- 0,

optimization techniques involving LMIs if they exist The feedback gains and

The design conditions above address decay rate and relaxed stability

Y s 0 , and Q22s0

The design problem for discrete systems can be handled similarly

Design Procedure for Case A: DFS

P1, P2) 0,

P1 P A y M B1 i 1i i

T T T

P2 A P y C N i 2 i 2i

T

P A y N C2 i 2i i P2

P A y M B1 i 1j i

4 P1 žqP A y M B1 T j 1T i T j /

A P y B M i 1 i 1j

P1

qA P y B M

A P y C N i 2 j 2i

4 P2 žqA P y C N T j 2 i T 2T i/

T

P A y N C2 i 2i j

P2

T

žqP A y N C2 i 2i j/

Remark 15 Note that in the designs above the controller gains and the observer gains can be determined separately This powerful result is similar

to the well-known separation principle for linear systems Unfortunately, such a separation principle only holds for Case A and does not hold for Case

w x

B 3

Finally, we would like to point out, as in Chapter 3, that a variety of control performance specifications can be incorporated into the LMI-based observer and controller design

4.2.2 Case B

Ž

unknown since they depend on the state variables to be estimated by fuzzy

The fuzzy observers for Case B are of the following forms, instead of 4.3

or 4.5 :

CFS

r

ˆ

xŽ t s h z tŽ Ž A xŽ t q B u t q KŽ ŽyŽ t y y tŽ 4, Ž4.27.

is1 r

yŽ t s h z tŽ Ž C xŽ t

is1

DFS

r

xŽt q 1 s. h z tŽ Ž A xŽ t q B u t q KŽ ŽyŽ t y y tŽ 4, Ž4.28.

is1 r

yŽ t s h z tŽ Ž C xŽ t

Ž Accordingly, instead of 4.7 , the PDC fuzzy controller becomes

r

w z tŽ ˆ Ž F xˆ Ž t

is1

uŽ t s y r s yÝh z t iŽ ˆ Ž F x iˆ Ž t Ž4.29.

is1

w z tŽ ˆ Ž

Ý i

is1

Then the augmented systems are obtained as follows:

CFS

x Ž t s h z tŽ Ž h z tŽ Ž h ŽzŽ t .G x Ž t

is1 js1 ks1

s Ý Ý h z t iŽ Ž h z t jŽ ˆ Ž h z t jŽ ˆ Ž G i j j x aŽ t

is1 js1

q2Ý Ý h z t iŽ Ž h z t jŽ ˆ Ž h kŽzˆ Ž t . x aŽ t Ž4.30.

2

is1 j -k

DFS

x aŽt q 1 s Ý Ý Ýh z t iŽ Ž h z t jŽ ˆ Ž h kŽzˆ Ž t .G i jk x aŽ t

is1 js1 ks1

s Ý Ýh z t iŽ Ž h z t jŽ ˆ Ž h z t jŽ ˆ Ž G i j j x aŽ t

is1 js1

q2Ý Ýh z t iŽ Ž h z t jŽ ˆ Ž h kŽ ˆzŽ t . x aŽ t ,

2

is1 j -k

4.31

where

xŽ t

x aŽ t s ,

eŽ t

eŽ t s x t y x t ,Ž ˆ Ž

A y B F i i k B F i k

S i jk S i jk

S i jk1 sŽA y A i j.yŽB y B F q K i j. k jŽC y C k i.,

Ž The following stability theorem for the augmented system 4.30 can be derived from Theorem 7

THEOREM 20 CFS The equilibrium of the augmented system described by

Ž4.30 is globally asymptotically stable if there exists a common positi®e definite

matrix P such that

T

G i jkqG i k j G i jkqG i k j

ž 2 / ž 2 /

Proof. It follows directly from Theorem 7

The following stability theorem for the augmented system 4.31 can be derived from Theorem 8

THEOREM 21 DFS The equilibrium of the augmented system described by

Ž4.31 is globally asymptotically stable if there exists a common positi®e definite

matrix P such that

T

G i jkqG i k j G i jkqG i k j

ž 2 / ž 2 /

Proof. It follows directly from Theorem 8

Remark 16 Consider the common C matrix case, that is, C s C s1 2 ⭈⭈⭈ s

C s C r In this case,

S i jk1 sŽA y A i j.yŽB y B F i j. k,

S i jk2 sA y K C q B y B F j j Ž i j. k

The conditions of Theorems 20 and 21 imply those of Theorems 18 and 19, respectively

Ž

Remark 17 We can no longer apply the relaxed conditions Theorems 9 and

4.3 DESIGN EXAMPLE

Consider the following nonlinear system:

x t s xŽ Ž t q sin x Ž t q x2Ž t q 1 u t ,Ž

x Ž t s x t q 2 xŽ Ž t ,

x Ž t s x2Ž t x Ž t q x t ,Ž

x Ž t s sin x Ž t ,

y t s x1Ž Ž 1 2Ž t q 1 x. 4Ž t q x2Ž t ,

y2Ž t s x2Ž t q x3Ž t

Ž

x t are estimated using a fuzzy observer It is also assumed that4

x t g ya, a ,1Ž x3Ž t g yb, b ,

2Ž

Ž

sin x t The nonlinear terms can be represented as3

x1 2Ž t s M1 1Žx t1Ž ⭈ a2qM1 2Žx t1Ž ⭈ 0,

sin x3Ž t s M2Žx3Ž t .⭈ 1 ⭈ x t q M x t ⭈3Ž 2Ž 3Ž ⭈ x t ,3Ž

b

where

M1Žx t1Ž , M1Žx t1Ž , M2Žx3Ž t ., M2Žx3Ž t .g 0, 1 ,

M1 1Žx t1Ž qM1 2Žx t1Ž s1, M2 1Žx3Ž t .qM2 2Žx3Ž t .s1

By solving the equations, they are obtained as follows:

x2 1

M1Žx t1Ž s a2,

x2Ž t

M1Žx t1Ž s1 yM1Žx t1Ž s1 y 2 ,

a

°b ⭈ sin x t y sin b ⭈ x t3Ž 3Ž

M2Žx3Ž t .s 3

Ž

M2 2Žx3Ž t .s1 yM2 1Žx3Ž t .

°b ⭈ x t y sin x tŽ 3Ž 3Ž .

Ž

3

where

x t g ya, a ,1Ž x3Ž t g yb, b

func-tions of fuzzy sets By using these fuzzy sets, the nonlinear system can be represented by the following T-S fuzzy model:

Model Rule 1

IF x t is M and x t is M ,1 1 3 2

xŽ t s A x t q B u t ,Ž Ž

THEN½yŽ t s C x t 1 Ž Ž4.37.

Model Rule 2

IF x t is M and x t is M ,1 1 3 2

xŽ t s A x t q B u t ,Ž Ž

THEN½yŽ t s C x t 2 Ž Ž4.38.

Model Rule 3

IF x t is M and x t is M ,1 1 3 2

xŽ t s A x t q B u t ,Ž Ž

THEN½yŽ t s C x t 3 Ž Ž4.39.

Model Rule 4

IF x t is M and x t is M ,1 1 3 2

xŽ t s A x t q B u t ,Ž Ž

THEN½yŽ t s C x t Ž Ž4.40.

T

xŽ t s x t1Ž x2Ž t x3Ž t x4Ž t ,

2

2

2

2

Note that it exactly represents the nonlinear system under the condition

x t g ya, a ,1Ž x3Ž t g yb, b

Case A is adapted since the premise variables are independent of the

Figure 4.1 shows a simulation result, where the dotted lines denote the

optimization technique involving LMIs

The designed fuzzy controller stabilizes the overall control system The fuzzy observer estimates the states of the nonlinear system without

steady-Fig 4.1 Simulation result.

state errors for the range

x t g y0.8, 0.8 ,1Ž x3Ž t g y0.6, 0.6

REFERENCES

1 R E Kalman, ‘‘On the General Theory of Control Systems,’’ in Proc IFAC,

Vol 1, Butterworths, London, 1961, pp 481᎐492.

2 K Tanaka and H O Wang, ‘‘Fuzzy Regulators and Fuzzy Observers: A Linear Matrix Inequality Approach,’’ 36th IEEE Conference on Decision and Control, Vol 2, San Diego, 1997, pp 1315 ᎐1320.

3 K Tanaka, T Ikeda, and H O Wang, ‘‘Fuzzy Regulators and Fuzzy Observers,’’

Ž

IEEE Trans Fuzzy Syst., Vol 6, No 2, pp 250᎐265 1998

4 K Tanaka and M Sano, ‘‘On the Concept of Fuzzy Regulators and Fuzzy Observers,’’ Proceedings of Third IEEE International Conference on Fuzzy Systems,

Vol 2, June 1994, pp 767 ᎐772.