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International Journal of Fuzzy Systems, Vol. 9, No. 1, March 2007

H ∞ Fuzzy Control Design for Nonlinear Stochastic Fuzzy Systems Chung-Shi Tseng, Yen-Fang Li and Yi-Fan Chiang

Abstract1 This paper describes the robust output feedback

H ∞ fuzzy control design for a class of nonlinear stochastic systems. The system dynamic is modelled by Itoˆ − type stochastic differential equations. For general nonlinear stochastic systems, the H ∞ ontrol

can be obtained by solving a second-order nonlinear Hamilton-Jacobi inequality. In general, it is difficult to solve the second-order nonlinear Hamilton-Jacobi inequality. In this paper, using fuzzy approach (T-S fuzzy model), the H ∞ fuzzy control design for the nonlinear stochastic systems can be given via solving linear matrix inequalities (LMIs) instead of a second-order Hamilton-Jacobi inequality. Simulation example is provided to illustrate the design procedure and expected performance. Keywords: Output feedback, H ∞ fuzzy control, T-S fuzzy model, nonlinear stochastic systems, Itoˆ − type stochastic differential equations, second-order nonlinear Hamilton-Jacobi inequality, and LMIs.

1. Introduction The output feedback H ∞ control problem is to design a controller using output feedback only, which guarantees the L 2 gain less than a prescribed level [1]. On the other hand, the stochastic H ∞ control problems with system models expressed by Itoˆ − type stochastic differential equations have become a popular research topic, and have gained extensive attention [2-5]. Most of the above works are limited to the linear stationary stochastic systems, while [3] and [4] discussed the linear and nonlinear stochastic H ∞ control problems. In [6], the stabilization problem of stochastic fuzzy system with norm bounded uncertainties has been discussed. The state feedback controller, which stabilizes either the nominal system or the uncertain one, is synthesized via solving the LMI problem. However, in [6], the control performance like H ∞ performance was not mentioned and only static state feedback case was studied. Unlike the deterministic case, the Hamilton-Jacobi inequality Corresponding author: C.-S. Tseng is with the Department of Electrical Engineering, Ming Hsin University of Science and Technology, Hsin-Feng 30401, Taiwan. ([email protected]) Manuscript accepted Dec. 2006.

(HJI) associated with nonlinear stochastic H ∞ control is a second-order (not first-order) nonlinear partial differential inequality due to the effect of the diffusion term, which makes the stochastic H ∞ control problem more complex [5]. In general, it is very difficult to solve the second-order nonlinear Hamilton-Jacobi inequality. Recently, there have been many applications of fuzzy systems theory in various fields. In most of these applications, the fuzzy systems were thought of as universal approximators for any nonlinear systems. The Takagi and Sugeno (T-S) fuzzy model [7] which has been proved to be a very good representation for a certain class of nonlinear dynamic systems was extensively studied in control systems [8-10]. In this study, using (T-S fuzzy model) fuzzy approach, the output feedback H ∞ fuzzy control design for a class of nonlinear stochastic systems can be given via solving linear matrix inequalities (LMIs) instead of a second-order Hamilton-Jacobi inequality. First, a T-S fuzzy model is proposed to approximate a class of nonlinear stochastic systems. Next, based on the T-S fuzzy model, the output feedback H ∞ fuzzy control design for the nonlinear stochastic systems is characterized in terms of minimizing the attenuation level subject to some linear matrix inequalities (LMIs), which is also called eigenvalue problem (EVP) [11] and can be efficiently solved by the LMI toolbox in Matlab [12]. Simulation example is provided to illustrate the design procedure and expected performance. For convenience, we adopt the following notations throughout this paper: A ′ : the trace (transpose) of matrix A. L2 R+ , R l : the space of nonanticipative

(

)

stochastic processes y(t) with respect to filtration Ft satisfying

∞

2

y(t) L : = E∫ y(t) dt < ∞. 2

2

0

C2(U): class of

functions V(x) twice continuously differential with respect to x ∈ U except possibly at the origin.

 M 11 M T  12

∗   M 11 ∆ T M 22  M 12

M 12  . M 22 

2. H ∞ Settings for Nonlinear Stochastic Systems Consider a class of nonlinear stochastic systems governed by Itoˆ − type stochastic differential

© 2007 TFSA

C.-S. Tseng:

H ∞ Fuzzy Control Design for Nonlinear Stochastic Fuzzy Systems

equations dx (t ) = [ f ( x (t )) + g ( x (t ))v (t )]dt + h( x (t ))dW (t ) (1)   η (t ) = q( x (t )) where x ( t ) ∈ R n is the system state, y( t ) ∈ R r is the measurement, η( t ) ∈ R m is the controlled output, and

v ∈ L2 (R + , R n v ) stands for the exogenous disturbance signal. f ( x ( t )), g ( x ( t )), h ( x ( t )) , and q ( x ( t )) are smooth functions with f (0) = h ( 0) = q(0) = 0 . W(⋅) is a standard one-dimensional Wiener process defined on the probability space (Ω, F, Ρ) relative to an increasing family (Ft ) t∈R + of σ -algebras

39

3. Output Feedback H ∞ Fuzzy Control Design For Nonlinear Stochastic System A fuzzy dynamic model has been proposed by Takagi and Sugeno [7] to represent locally linear input/output relations for nonlinear systems. This fuzzy dynamic model is described by fuzzy If-Then rules and will be employed here to deal with the filtering design problem for a class of nonlinear stochastic systems governed by Itoˆ − type stochastic differential equations. The ith rule of the fuzzy model for the nonlinear stochastic systems is proposed as the following form: Rule i: z 1 ( t ) is Fi1 and z g ( t ) is Fig If

Ft ⊂ F. In (1), the state equation, in engineering terminology, can be written as [13] Then x ( t ) = f ( x ( t )) + g ( x ( t )) v( t ) + h ( x ( t )) w ( t ) (2) dx ( t ) = (A i x ( t ) + Bi u ( t ) + D1i v( t ))dt + E1i x ( t )dW( t ) where w is a stationary white noise. (6) The following Lemma is a special case of Proposition Y( t ) = (C i x ( t ) + D 2i v( t )) + E 2i x ( t ) w ( t ) 1 in [5], which plays an important role in this paper.

for i= 1,…,L where x ( t ) = [ x 1 ( t ),… , x n ( t )]T ∈ R n×1

Lemma 1: For system (1), if there exists a positive function V( x ) ∈ C 2 (R n ) and V (0) = 0 solving the following HJI ∂V′ 1 ∂V′ ∂V f + γ −2 ( g)(g ′ ) ∂x 2 ∂x ∂x (3) 2 1 1 ∂ V 2 + q(x(t )) + h ′ 2 h < 0, 2 2 ∂x then (i) The equilibrium point x ≡ 0 of the system (1) is globally asymptotically stable in probability in the case of v=0 and (ii) the following inequality

denotes

η( t )

2 L2

≤ 2E[V( x (0))] + γ v( t ) 2

2 L2

(4)

2 L2

≤ γ 2 v( t )

2 L2

, ∀v ∈ L2F (R + , R n v ), v ≠ 0

state

vector;

A i , B i , D1i , E1i , C i , D 2i and E 2i are known constant matrices with appropriate dimension; L is the number of If-Then rules; z 1 ( t ), … , z g ( t ) are the premise variables; W (⋅) is a standard one-dimensional Wiener process defined on the probability space (Ω, F, P) relative

to

an

increasing

family

(Ft ) t∈R +

of

( t ) is a stationary σ -algebras Ft ⊂ F ; and w ( t ) = W white noise. To obtain a tractable mathematical interpretation of Y(t) in (6), we introduce

∀v ∈ L2F (R + , R n v ), v ≠ 0 holds for some γ > 0 if initial state x (0) ≠ 0 and η( t )

Y( t ) ∈ R m×1 is the measurement output; v ∈ L2 ( R+ , R nv ) stands for the exogenous disturbance signal; Fij is the fuzzy set; the

(5)

holds for some γ > 0 if initial state x (0) = 0 . Proof. The proof is immediately followed from Proposition 1 and Lemma 1 in [5].

t

y( t ) = ∫ Y(s)ds 0

and thereby we obtain the stochastic representation Rule i: z 1 ( t ) is Fi1 and z g ( t ) is Fig If

(7) integral

Then Remark 1: In general, it is difficult to solve the dx ( t ) = (A i x ( t ) + B i u ( t ) + D1i v( t ))dt + E 1i x ( t )dW( t ) second-order nonlinear Hamilton-Jacobi inequality in (3). (8) dy( t ) = (C i x ( t ) + D 2i v( t ))dt + E 2i x ( t )dW( t ) In the next section, using fuzzy approach, the H ∞ fuzzy control design for the nonlinear stochastic systems can be given via solving linear matrix inequalities (LMIs) Remark 2 : The fuzzy model in (8), in engineering terminology, can be expressed as follows: instead of a second-order Hamilton-Jacobi inequality. Rule i: z 1 ( t ) is Fi1 and z g ( t ) is Fig If

40


Remark 3: In practice, the fuzzy estimator is implemented as follows

Then

x ( t ) = (A i x ( t ) + B i u ( t ) + D1i v( t )) + E 1i x ( t ) w ( t ) Y( t ) = (C i x ( t ) + D 2i v( t )) + E 2i x ( t ) w ( t )

(9)

L

ˆ ˆ x(t)= ∑ h i (z(t)){Ai x(t)dt+B i u(t)

The final output of the fuzzy system is inferred as follows: L

dx (t ) = ∑ hi ( z (t ))[( Ai x (t ) + Bi u (t )

(17)

ˆ +Li [Y(t)-Y(t)]} Hence, the fuzzy observer-based fuzzy controller is proposed as

i =1

+ D1i v( t ))dt + E1i x ( t )dW ( t )]

i=1

L

u(t ) = ∑ h j ( z (t )) K j xˆ (t )

(10)

and

(18)

j =1

L

dy (t ) = ∑ hi ( z (t ))[(Ci x (t ) + D2i v (t ))dt

where K j is the control gain for the jth controller rule. Let us denote the estimation errors as

i =1

+ E 2i x ( t )dW ( t )]

de( t ) = dx ( t ) − dxˆ ( t )

(11)

(19)

By substituting (10) and (16) into (19), we get

where

µ i (z( t ))

h i (z( t )) =

L

(12)

L

∑ µ (z( t ))

i=1 j=1

i

i =1 g

L

de(t)= ∑∑ h i (z(t))h j (z(t)){[(Ai -Li C j )e(t) +(D1i -Li D 2j )v(t)]dt+(E1i -Li E 2j )x(t)dW(t)}

µi ( z (t )) = ∏ Fij ( z j (t )), z(t ) = [ z1 (t ),.., z g (t )] j =1

After manipulation, the augmented system can be expressed as the following form:

in Fij . It is assumed that µ i (z( t )) ≥ 0 and

i =1

i

Therefore, we get

h i (z( t )) ≥ 0 , for i = 1,2,… L

(13)

and L

∑ h (z( t)) = 1 i

(14)

Based on the fuzzy model (8), the following fuzzy estimator is proposed to deal with the state estimation for the nonlinear stochastic systems Observer Rule i: z 1 ( t ) is Fj1 and z g ( t ) is Fjg If Then

dxˆ ( t ) = A i xˆ ( t )dt + B i u ( t )dt + L i [dy( t ) − dyˆ( t )] (15) where L i is the fuzzy observer gain for the ith observer rule and dyˆ( t ) =

L

∑ h (z( t ))C xˆ ( t)dt . i =1

i

i

+ Li [dy (t ) - dyˆ (t )]}

j=1

(21)

where

D1i    x ( t ) ~ , D ij  x (t ) =     e( t )  D1i − L i D 2 j  − Bi K j  A i + B i K j A ij =  0 A i − L i C j   E1i  E ij =  E 1i − L i E 2 j

0 0

(22)

The nonlinear stochastic H ∞ fuzzy control problem can be stated as follows: Find the controller gain K j and the observer gain L i ( for i,j…=1,…L ) such that x ( t ) ≡ 0 of the augmented (i) The equilibrium point ~ system (21) is globally asymptotically stable in probability in the case of v( t ) = 0 . (ii) For a given disturbance attenuation level γ > 0 ,

~ x(t)

L

i =1

i=1

the following relation ( H ∞ performance) holds.

The overall fuzzy estimator is written as

dxˆ (t ) = ∑ hi ( z (t )){ Ai xˆ (t )dt +Bi u (t )dt

L

+Dij v(t))dt+Eij x(t)dW(t)]

L

∑ µ (z(t )) > 0 .

L

dx(t)= ∑ h i (z(t))∑ h j (z(t))[(Aijx(t)

and Fij (z j ( t )) is the grade of membership of z j ( t )

i =1

(20)

(16)

2 L2

≤ 2E[V(~ x (0))] + γ 2 v( t )

2 L2

∀v( t ) ∈ L2F (R + , R n v ) , v(t) ≠ 0 , where V(~ x) ∈ C 2 (R n ) is a positive function. Then, we obtain the following result.

(23)


C.-S. Tseng:

Theorem 2 : For the augmented system in (21), if there exists a symmetric positive definite matrix P solving the following inequalities A ′ij + A ′ji  A ij + A ji  )+( )P + I ∗ ∗  P( 2 2   E E + ij ji  P( ) −P ∗  < 0 (24)   2   D ij + D ji ′ ( ) P 0 − γ 2 I  2   ~ for all i ≤ j (i,j = 1,…,L), then (i) x ( t ) ≡ 0 of the augmented system (21) is globally asymptotically stable in probability in the case of v = 0 and (ii) the following H ∞ performance

~ x (t )

2 L2

≤ E[~ x ′(0)P~ x (0)] + γ 2 v( t )

2

(25)

L2

∀v( t ) ∈ L2F (R + , R n v ) , v(t) ≠ 0 holds for some γ > 0 . Proof. By Lemma 1, if there exists a positive function V(~ x ) ∈ C 2 (R n ) and V(0) = 0 such that L   ∂V ′ L h z t h j ( z (t ) Aij ~ x (t )) ( ( ( ))  ~ ∑ i ∑ x ∂ = = 1 1 i j   1 L ∂V ′ L + γ −2 ( ~ ( ∑ hi ( z (t ))∑ h j ( z (t ) Dij ) ∂x i =1  2 j =1  L L × (( ∑ hm ( z (t )) ∑ hn ( z (t ) Dmn )′ ∂V )  m=1 x ∂~ n =1   1 ~ ′~ + 2 ( x (t )) ( x (t ))  L L ∂ 2V + 1 ( h ( z (t )) h ( z (t ) E ~ ′ x ( t ) ) ∑ i j ij  2 ∑ x2 ∂~ i =1 j =1  L  L ( h ( z ( t )) hn ( z (t ) E mn ~ x (t ))} < 0 × ∑  ∑ m n =1  m =1

P(

)+(

2 D ij + D ji

A ′ij + A ′ji 2

E ij + E ji

) ′P (

D ij + D ji

2 E ij + E ji 2

0 P22 

(30)

then (24) can be rewritten as the following LMIs ( P 11 A i + P 11 B i K j ) + ( P 11 A j + P 11 B j K i )  {  2   ( A ′i P 11 + K ′j B ′i P 11 ) + ( A ′j P 11 + K ′i B ′j P 11 ) + I} + 2  − K ′j B ′i P 11 − K ′i B ′j P 11   2  P 11 E 1 i   ( P 22 E 1 i − Y i E 2 j ) + ( P 22 E 1 j − Y j E 2 i )  2   D 1′ i P 11 ∗ ( P 22 A

+

− Y i C j ) + ( P 22 A

i

( A ′i P 22

j

− Y jC i )

2 − C ′j Y i′ ) + ( A ′j P 22 − C ′i Y j′ ) 2

0 0

(26)

( D 1′ i P 22 − D ′2 j Y i′ ) + ( D 1′ j P 22 − D ′2 i Y j′ ) ∗ ∗

∗ ∗

− P 11 0 0

∗ − P 22 0

+

i

(31)

2 ∗  ∗  ∗  < 0  ∗  2  − γ I

where Yi = P22 L i . Note that (31) implies that (27)

)P + γ −2 [P(

)′P] + I + ( 2 2 for all i ≤ j (i , j = 1, … L) . × [(

P P =  11 0

( P 11 A

By (27), (26) holds, if

A ij + A ji

Aij′ + A′ji  Aij + A ji  ∗ ∗  P( 2 ) + ( 2 ) P + I   Eij + E ji  −P ∗  < 0 (29) P( ) 2   Dij + D ji  2  ( ) ′P 0 −γ I  2   Therefore, the proof is complete. For the convenience of design, let

{

then (i) and (ii) hold. Let

1 V(~ x) = ~ x ′P~ x 2

41

+ P 11 B i K j ) + ( P 11 A

j

+ P 11 B j K i )

2 ( A ′i P 11 + K ′j B ′i P 11 ) + ( A ′j P 11 + K ′i B ′j P 11 ) 2

(32) + I < 0

And (32) is equivalent to

)] (28)

) 0 and (31)

P22

where (34)

T 22

Step 4 : Construct the fuzzy estimator in (17). Step 5 : Construct the fuzzy controller in (18).

4. Simulation Example To illustrate the proposed fuzzy control approach, a control problem of balancing an inverted pendulum on a cart is considered in this study. For this example, the state equations of the inverted pendulum are given by x 1 = x 2 x 2 =

1 [( M + m )( J + ml 2 ) − ( ml cos x 1 ) 2 ] × [( M + m ) mgl sin x

1

− ( mlx 2 ) sin x 1 cos x 1 − ml cos x 1 u 2

+ v - 10 -3 ( M + m ) x 2 ( 7 + w )] Y =

x 1 + 0 . 02 v + 10 − 4 x 1 w

(35)

where x 1 denotes the angle (rad) of the pendulum from the vertical,

g = 9.8m / s

Y = C 2 x + D 22 v + E 22 xw Rule 3 : IF x 1 is about ± 2π / 9 THEN x = A 3 x + B 3 u + D13 v + E 13 xw and Y = C 3 x + D 23 v + E 23 xw Rule 4 : IF x 1 is about ± π / 3 THEN x = A 4 x + B 4 u + D14 v + E 14 xw and Y = C 4 x + D 24 v + E 24 xw

x 2 is the angular velocity (rad/s),

2

is the gravity constant, m is the mass (kg) of the pendulum, M is the mass (kg) of the cart, w is the uncertain friction factor (N/rad/s) of the pendulum, which is assumed to be a stationary white noise, l is the length (m) from the center of mass of the pendulum to the shaft axis, J is the moment of inertia ( kgm 2 ) of the pendulum, u is the force (N) applied to the cart, and v is external disturbance. The parameters in this example are assumed to be m=1.0, M=10, l=0.5, and J=0.005, and v = 2e − t sin 5t . To minimize the design effort and complexity, we try to use as few rules as possible. Hence, we approximate the system by the following four-rule fuzzy model. Rule 1 : IF x 1 is about 0 THEN x = A1 x + B1 u + D11 v + E 11 xw and

Y = C1 x + D 21 v + E 21 xw Rule 2 : IF x 1 is about ± π / 9 THEN x = A 2 x + B 2 u + D12 v + E 12 xw and

1  0  0   A1 =  , B1 =   ,  21.0959 − 0.0301 − 0.1957 1 0  0    , B2 =  A2 =   ,  20.4362 − 0.0298  − 0.1818 1  0   0  A3 =  , B3 =   , 18.6688 − 0.0290  − 0.1441 1  0  0   A4 =  , B4 =   , 16.2534 − 0.0281 − 0.0912 0   0  0 D11 =  , E11 =   , 0.3914 0 − 0.0043 0   0  0 D12 =  , E12 =   , 0.3870 0 − 0.0043 0   0  0 D13 =  , E13 =   , 0.3762 0 − 0.0041 0  0  0  D14 =  , E14 =   , 0.3646 0 − 0.0040 C i = [1 0] , D 2i = 0.02 , E 2i = 10 −4 0

[

]

for i = 1,…,4. Triangle type membership functions are used for Rule 1-Rule 4. The LMIP in (33) and the EVP in (34) are solved using the LMI optimization toolbox in Matlab. In this case, γ 2min = (0.5016) 2 and

 0.2662 − 0.3870 Q11 =   − 0.3870 0.7290   124.2369 − 132.3597 P22 =   − 132.3597 142.1074  The observe gains are found to be

894.6338 879.3905 L1 =  , L2 =    853.2804 838.9958 880.0687 1007.106 L3 =  , L4 =    957.7221 839.1571


C.-S. Tseng:

43

The control gains are found to be

design for the nonlinear stochastic systems can be given via solving linear matrix inequalities (LMIs) instead of a K 1 = [213.6 70.2], K 2 = [228.9 76.4] second-order Hamilton-Jacobi inequality. K 3 = [276.5 96.5], K 4 = [355.8 132.2] This study extends the output feedback H ∞ control The trajectories of x 1 , xˆ 1 , and external design from linear stochastic systems to a class of disturbance v are shown in Figure 1 and the trajectories nonlinear stochastic systems using fuzzy techniques. of x 2 and xˆ 2 are shown in Figure 2. From the LMI-based design procedure for the output feedback simulation results, the proposed fuzzy control clearly H ∞ fuzzy control problems for the nonlinear stochastic systems is developed systematically. The proposed results in desired H ∞ control performance. design procedure is very simple and can be performed 2 efficiently using the LMI optimization toolbox in Matlab. angle(rad) estimated angle(rad) Simulation example is provided to illustrate the design 1.5 external disturbance procedure and expected performance. Therefore, the 1 proposed method is very suitable for practical applications in the nonlinear stochastic systems. 0.5 0

References

-0.5 -1 -1.5 -2

0

1

2

3

4

5 time(sec)

6

7

8

9

10

Figure 1. The trajectories of x 1 and xˆ 1 , and external disturbance v. 2 angular velocity(rad/sec) estimated angular velocity(rad/sec) external disturbance

1.5 1 0.5 0 -0.5 -1 -1.5 -2

0

1

2

3

4

5 time(sec)

6

7

8

9

10

Figure 2. The trajectories of x 2 and xˆ 2 .

5. Conclusions In this paper, based on a T-S fuzzy model, the output feedback H ∞ fuzzy control problems for a class of

nonlinear stochastic systems governed by Itoˆ − type stochastic differential equations are studied. Using fuzzy approach, the output feedback H ∞ fuzzy control

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