行政院國家科學委員會專題研究計畫成果報告
不確定非線性系統具
不確定非線性系統具
不確定非線性系統具
不確定非線性系統具 H
∞∞∞∞追蹤規格之適應性模糊觀測器與控制器之設計
追蹤規格之適應性模糊觀測器與控制器之設計
追蹤規格之適應性模糊觀測器與控制器之設計
追蹤規格之適應性模糊觀測器與控制器之設計
Adaptive Fuzzy Observer and Controller for H
∞Tracking Performance in a Class
of Uncertain Nonlinear Systems
計畫編號：NSC 90-2213-E-030-010 執行期限：90 年 8 月 1 日至 91 年 7 月 31 日 主持人：王偉彥 輔仁大學電子工程系 共同主持人：呂藝光 華夏工商專校 電子工程系 參與人員：李宜勳、鄭智元、黃宏智 輔仁大學電子工程系 摘要 摘要摘要 摘要 本報告是以 H∞ 追蹤規格和 SPR-Lyapunov 設 計方法為基礎，提出一種針對非線性不確定系統以 H∞-追蹤為基礎的適應性模糊觀測器 (HAFO) 和以 H∞-觀測器為基礎的適應性模糊控制器(HAFC)的設 計方法。吾人假設在此系統中只有輸出量測得到並且 不必受被觀測系統的非線性項必須是輸出的函數的 限制。此種 HAFO 和 HAFC 均以模糊系統之權重 因子為參數獲得線上調整，並且 HAFO 和 HAFC 也 以 H∞追蹤的方法來壓制誤差和干擾。由於此觀測器 具良好的觀測特性，此輸出回授控制器將保證系統內 所有信號均為有界。最後，藉由例子的模擬，倒單擺 穩定的問題，來驗證本計畫所提的方法確實可行。 關鍵詞 關鍵詞關鍵詞 關鍵詞：模糊系統、非線性系統、 H∞追蹤規格、適 應性模糊控制 AbstractIn this report, based on the H∞ tracking technique
and the strictly positive real Lyapunov (SPR-Lyapunov) design approach, an H∞-tracking- based adaptive fuzzy
observer (HAFO) and an H∞-observer-based adaptive fuzzy controller (HAFC) for a class of uncertain nonlinear systems are developed. It is assumed that the total system states are not available for measurement and the uncertain system nonlinearities are not necessarily restricted to the system output. Both the HAFO and HAFC are tuned on-line by the weighting factors of the fuzzy system and they provide errors and disturbances attenuation with H∞ tracking performance. Based the good feature of the observer, the output feedback controller guarantees that all signals involved are bounded. Finally, an example is illustrated to show the effectiveness of the proposed methods.
Keywords: fuzzy system, nonlinear system, H∞tracking
performance, adaptive fuzzy control 一、
一、一、
一、Introduction
In real control systems, the plants are always uncertain and nonlinear. Since neural networks and fuzzy systems are universal approximators [1,2], the uncertain nonlinear functions of real plants can be approximated by neural networks or fuzzy systems. In [3]-[6], the adaptive control schemes of nonlinear systems incorporating the techniques of fuzzy systems or neural networks have been proposed. In [3], L. X. Wang proposed an approach that combines the approximate mathematical model, linguistic model description, and
linguistic control rules into an adaptive fuzzy controller. In [5,6,17], the fuzzy control algorithms for adaptive control were proposed by Leu et al.. They demonstrated how to design the adaptive controller based on the fuzzy system.
In the past decade, the nonlinear H∞ optimal
control theory has been introduced to tackle the robust performance design problem of nonlinear systems. In [7,8], H∞ control approach provides both robust
stability and disturbance attenuation with H∞-norm bound for closed-loop uncertain linear and nonlinear systems. Designers have to solve a Hamilton-Jacobi equation that is a nonlinear partial differential equation. However, only special nonlinear systems have closed-form solutions. As far as practical control system design is concerned, the methods are not suitable. To satisfy the purpose of practical applications, adaptive fuzzy control with H∞ tracking performance for
uncertain nonlinear systems has been proposed by Chen et al. [9,10].
The problems of constructing an observer for a nonlinear system had been studied by many researchers [11]-[16]. In [11,12], a coordinate transformation approach is presented. The fundamental idea of their approach is to transform a nonlinear system into an observer canonical form by appropriate coordinate changes. The extension results of this approach are proposed in [13,14]. Based on the filter transformations, an adaptive observer for a nonlinear system has been presented in [15,16]. They assume that the nonlinear terms in the transformed system are restricted to the system output only. The limitation has been relaxed in [17] and the observer-based adaptive fuzzy control approach is introduced in [17]. The observer presented in [17] is used to estimate the tracking error vector e, not directly to estimate the real state vector x. To directly estimate real states, the observer is redesigned in this report. Therefore, not only controller but also observer are constructed by the fuzzy system. We adopt the H∞
tracking design technology in [9] in which the lumped uncertainty only is bounded rather than known. Therefore, the H∞ tracking performance for the
uncertain nonlinear system is combined into the design algorithms of the observer and controller. The influences of modeling errors and disturbances are attenuated to a desired value for the adaptive fuzzy control system satisfied the H∞ tracking performance.
二、 二、二、
二、Problem Formulation
This section describes the preliminaries required to derive the design approaches of the HAFO and HAFC. 2-1. System Model
Consider the nth order nonlinear dynamical system as follows: , , ) g( ) f( ) ( x y d u xn = + + = x x (1) or D (f ( ) g( ) ), , x Ax B x x C x = + + + = u d y T (2) where = 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 l l l l l l l l l A , = 1 0 0 0 o B , = 0 0 0 1 o C , x=[ ,D, , (−)] x x xn T l 1 = ∈ℜ [x x, , ,xn] T n 1 2 l is a vector of
states, d is the external bounded disturbance, and u∈ℜ and y∈ℜ are the control input and system output, respectively. The functions,f ( )x and g( )x , are uncertain nonlinear functions. Only the system output y
is assumed to be measurable. 2-2. Fuzzy Approximator [5,6]
The thl fuzzy IF-THEN rule is written as
) (l R ：if x1 is l A1 and ... and xn is l n A , then y is B , l where l i A and l
B are fuzzy sets with the membership functions ( )
i Ail x
µ and µBl( y), respectively. By using product inference, center-average and singleton fuzzifier, the output, o(x), of the fuzzy approximator can be expressed as
∑
∏
∑
∏
= = = = = h i n j j A h i n j j A i x x y o i j i j 1 1 1 1 ) ( ) ( ) ( µ µ x =θTψ(x), (3) where ( ) j Ai x jµ is the membership function value of the fuzzy variable xj, h is the number of the total IF-THEN rules, and yi is the point at which
1 ) ( i = Bi y
µ . θ =[y1y2myh]T is an adjustable parameter
vector, and ψ=[ψ1ψ2mψh]T is a fuzzy basis vector,
where ψi is defined as
∑
∏
∏
= = = = h i n j j A n j j A i x x i j i j 1 1 1 ) ( ) ( ) ( µ µ ψ x . (4)The fuzzy approximator is applied to approximate the uncertain nonlinear functions f ( )x and g( )x in (2). Let f( )x and g( )x be the estimation functions of the uncertain nonlinear functions, f ( )x and g( )x , respectively, and xˆ denotes the estimate of x . We
replace the estimation functions, f( )x and g( )x , by the outputs of the fuzzy approximator as follows:
), ˆ ( ) | ˆ ( fˆx θf θfψx T = (5) and ), ˆ ( ) | ˆ ( gˆx θg =θTgψx (6) where f
θ and θ are the adjustable parameter vectors. g 2-3. Mathematical Background
In order to derive the proposed results, many assumptions and lemma are given first.
Assumption 1 [18]: Let x and >x belong to compact
sets U and x Uxˆ,respectively, where
} : { ∈ ≤ <∞ = x X x x U Rn m , {ˆ : ˆ } ˆ ˆ = ∈ ≤ x<∞ X x x U Rn m ,
and m and x m are designed parameters. It is known xˆ
a prior that the optimal parameter vectors, ∗
f
θ and ∗
g
θ , lie in some convex regions
} : { M f f f f θ θ θ R θ m h ≤ ∈ = , (7) and } : { Mθg θg R θg mθg h ≤ ∈ = , (8) where the radiuses
g θ
m
and f θm
are constant, ] ) | ˆ ( fˆ ) f( sup [ min arg f ˆ , M f ˆ f f θ x x θ x x θ x U x U θ − = ∈ ∈ ∈ ∗ , (9) and ] ) | ˆ ( gˆ ) g( sup [ min arg g ˆ , M g ˆ g g θ x x θ x x θ xU x U θ − = ∈ ∈ ∈ ∗ . (10)Assumption 2 [3]: The parameter vector
g
θ is such that gˆ(xˆ|θg) is bounded away from zero.□
Lemma 1 [19] : If H s T s
( )=C ( I A− )−1B with A a
Hurwitz matrix is an SPR(strictly proper rational) function, then for any given L=LT>0, there exists a scalar ξ >0, a vector q and a matrix P=PT>0 such that A P PA qq L PB C T + = − T− = ξ , . (11)
Lemma 2 [19]: Consider the linear time-invariant
system
D( ) ( ) ( ), ( )
x t =Axt +Bu t x 0 =x0 (12)
where x( )t ∈ℜn , u( )t ∈ℜm , A∈ℜn n× , B∈ℜn m× . Suppose that A is a Hurwitz matrix and u( )t ∈L2e. Let
α0 and λ0 be the positive constants that satisfy
eA(t−τ) ≤λe−α(t−τ) 0
0 . For any constant α∈[ ,0α)
1 , where 0<α1<2α0, then x( )t e t x B u t ≤ + − − λ λ α α α α 0 0 0 0 2 0 2 , (13) where =
∫
t − t− T t 0e d ) ( 2 2 1 ) ) ( ) ( ( α τ τ τ τ α u u u and 2 1 ) ( ) ( 2 =∫
ot T t u τ uτ dτ u . We say that u(t)∈L2α if α 2 tu exists, and that u(t)∈L2e if ut 2 exists for
any finite t. 三、 三、三、
Consider the following observer that estimates the states in (2) as , ˆ ˆ ), ˆ ( ] ) ˆ ( gˆ ) ˆ ( fˆ [ ˆ ˆ x C x x C K x x B x A x T T o r y v u = − + − + + = (14)
where yˆ is the estimate of the output y, xˆ denotes the estimate of the state vector x, <f( < )x and <g( <)x are the estimation functions of the uncertain nonlinear functions, f ( )x and g( )x , respectively, and
T o n o o o=[k1k2mk ]
K is the observer gain vector, chosen such that the characteristic polynomial of A K C− o
T is
strictly Hurwitz because ( , )C A is observable. The robust term vr is employed to compensate the external
disturbance d and the estimation errors. To evaluate the error of observing, we define the observation errors
x x x= − and x~1=x1−xˆ1=y−yˆ. Subtracting (14) from (2), we have . ~ ~ ], )) ˆ ( gˆ ) (g( ) ˆ ( fˆ ) [f( ~ ) ( ~ 1 C x x x x x B x C K A x T r T o x v d u = + + − + − + − = (15)
Furthermore, the dynamics of (15) can be given as ], )) ˆ ( gˆ ) (g( ) ˆ ( fˆ ) )[f( ( ~ 1 H s u v d x = x − x + x − x + r+ (16) where H s T s o T ( )=C ( I−(A K C− ))−1B is the transfer function of (15) and chosen as a stable transfer function for K . o
Lemma 3: If the robust term r
v is defined as vr =−η~x1
where η is chosen as 1+ηH(s)≠0 and η>0, then
1
~
x is bounded.
Now, we define the approximation error as
u wo ≡fˆ(xˆ|θf)−f(x)+(gˆ(xˆ|θg)−g(x))
∗
∗ , (17)
where the approximation error w is that the difference o of the optimal estimate functions, fˆ(ˆ| f)
∗ θ x and ) | ˆ (
gˆx θg∗ , and the uncertain nonlinear functions, f ( )x and g( )x , with the optimal parameter vectors ∗
f
θ and
∗
g
θ in (9)~(10), respectively. Thus, the observation error dynamics (15) can be rewritten as
. ~ ~ ], )) | ˆ ( gˆ ) | ˆ ( gˆ ( ) | ˆ ( fˆ ) | ˆ ( fˆ [ ~ ) ( ~ 1 g g f f x C θ x θ x θ x θ x B x C K A x T r o T o x v d w u = + + + − + − + − = ∗ ∗ D (18)
According to assumption 1, (5) and (6), i.e., ) ˆ ( ) | ˆ ( fˆx θ∗f =θf∗Tψx , ) ˆ ( ) | ˆ ( gˆx θg θgψx T ∗ ∗ = , fˆ(ˆ| ) (ˆ) f f θ ψx θ x = T , and gˆ(xˆ|θg) θgψ(xˆ) T
= , the observation error dynamics (18) becomes as , ~ ~ ], ) ˆ ( ~ ) ˆ ( ~ [ ~ ) ( ~ 1 g f x C x ψ θ x ψ θ B x C K A x T T T m r T o x u w v = + + + + − = D (19) where θ~f =θf−θ∗f , = − ∗ g g g ~ θ θ θ , and wm=wo+d represents the effects of the modeling error and the external disturbance. Since only the output Cx1 in (18) is
assumed to be measurable, we use the SPR-Lyapunov design approach to analyze the stability of the observation error dynamics (19) and to generate the adaptive laws for θf and
g
θ .
First, the output error dynamics (19) can be rewritten as ], ) ˆ ( ~ ) ˆ ( ~ )[ ( ~ g f 1 H s v w u x = r + m+θTψx +θTψx (20) where H s s s k s k T o T n o n n o ( ) ( ( )) . = − − = + + + − − C I A K C 1B 1 1 1 m (21)
The transfer function H s( ) is chosen as a stable transfer function for K . In order to apply the o SPR-Lyapunov design approach, (20) must be written as
)], ) ˆ ( ~ ) ˆ ( ~ )[ ( ) ( ~ g f 1 H s L s v w u x T T f r + +θ Φx +θ Φx = (22) where ), ) ˆ ( ~ ) ˆ ( ~ )( ( ) ˆ ( ~ ) ˆ ( ~ , ) ( ), ˆ ( ) ( ) ˆ ( g f g f 1 1 r T T r T T m f v u s L v u w w w s L w s L + + − + + + = = = − − x Φ θ x Φ θ x ψ θ x ψ θ x ψ x Φ
and L s( ) is chosen so that L−1 s
( ) is a proper stable transfer function and H s L s( ) ( ) is a proper SPR transfer function. Suppose that L s sm b sm b sm b m ( )= + 1 −1+ − + + 2 2 , where m<n, such that H s L s( ) ( ) is a proper SPR transfer function. Therefore, the state-space realization of (22) can be written as , ~ ~ ], ) ˆ ( ~ ) ˆ ( ~ [ ~ ~ 1 g f x C x Φ θ x Φ θ B x A x T c T T f r c c x u w v = + + + + = (23) where ( T) nn, o c × ℜ ∈ − = A K C A [00 1 2 ] n, m T c = bb b ∈ℜ B , ] 0 0 1 [ n T c = ∈ℜ C and H s L s c s T c c ( ) ( )=C ( I A− )−1B .
Assumption 3: Suppose that for any given Q=QT>0 ,
there exist a scalar ξ >0, a vector q, and a matrix
P=PT>
0 such that: (a) the equality PBc=Cc is
satisfied, and (b) the following Riccati-like equation
A P PAc PB B P PB B P Q T c c c T c c T + − ′ ′ + ′′ + ′ = 2 0 2 ξ γ ξ δ (24)
is satisfied, where Q′ =qqT+ξQ , ξ′ =ξλmin( )Q ,
′ =
δ δ ξλmin( )Q , δ >0 is a given attenuation constant,
and 2
2δ
γ′= ′ .□
Remark 1: For any given L=LT>0 , there exist a
scalar ξ >0, a vector q and a matrix P=PT >0 such that A P PA qq L PB C c T c T c c + = − − = ξ , , (25) since H s L s c s T c c ( ) ( )=C ( I A− )−1B with Ac a Hurwitz
matrix is SPR (Lemma 1). We choose
Q P B PB L + ′ − ′ = T c c γ γ δ γ 0 2 0 2 , which satisfies L L= T >0 if the inequality 0 2 0 2 0 + > ′ − ′ PB B P Q T c c γ γ δ γ (26)
is satisfied where γ0 =λmin(Q).□
Assumption 4 [9]: wm and wf is bounded and
wf( )t ∈L2e.□
Lemma 4[17]: Suppose that the adaptive laws are
< = ≥ = < − = 0 ) ˆ ( ~ and if )), ˆ ( ~ (-Pr ) 0 ) ˆ ( ~ and ( or if ), ˆ ( ~ f 1 f 1 1 f f 1 f f 1 1 f f f f x Φ θ θ x Φ x Φ θ θ θ x Φ θ θ θ θ T T x m x x m m x γ γ & (27) < = ≥ = < − = 0 ) ˆ ( ~ and if ), ) ˆ ( ~ (-Pr ) 0 ) ˆ ( ~ and ( or if , ) ˆ ( ~ g 1 g 1 2 g g 1 g g 1 2 g g g g u x m u x u x m m u x T T x Φ θ θ x Φ x Φ θ θ θ x Φ θ θ θ θ γ γ & (28) where γ1 and 2
γ are design parameters and the projection operators [3] Prf(-γ1~x1Φ(xˆ)) and
) ) ˆ ( ~ (-Prg γ1x1Φxu are given as f 2 f f 1 1 1 1 1 1 f ) ˆ ( ~ ) ˆ ( ~ )) ˆ ( ~ (-Pr θ θ x Φ θ x Φ x Φ T x x x γ γ γ =− + , and g 2 g g 1 2 1 2 1 2 g ) ˆ ( ~ ) ˆ ( ~ ) ) ˆ ( ~ (-Pr θ θ x Φ θ x Φ x Φ u x u x u x T γ γ γ =− + , respectively. Then f f θ θ ≤m , g g θ θ ≤m , and f 2 ~ f θ θ ≤ m , g 2 ~ g θ θ ≤ m .□
On the basis of the above discussion, the following theorem can be obtained.
Theorem 1: Consider the observation error dynamics
(23) in which the control input
u
is bounded and suppose that assumptions 1-4 are satisfied. Let θ and fg
θ be chosen as the adaptive laws (27) and (28) and δ >0 be a given attenuation constant. Thus, for
) , 0 [ ∞ ∈
∀t , the following H∞ tracking performance ([9],
[10]) is achieved: 2 2 2 g g 0 2 f f 0 1 0 2 2 1 (0) ~ ) 0 ( ~ 1 ) 0 ( ~ ) 0 ( ~ 1 ) 0 ( ~ ) 0 ( ~ 1 ~ t f T T T t w x δ γ γ γ γ γ + + + ≤ x Px θ θ θ θ (29) with 1 1 min( )~ ~ 1 x x vr γ ξ λ γ ′ ′ − = − = Q , (30) where =
∫
t t x d x 0 2 1 2 2 1 ( ) ~ ~ τ τ ,∫
= t f ft w d w 0 2 2 2 (τ) τ , γ0=λmin( )Q and γ γ ξ= ′ is a positive constant. □Lemma 5: The observation error vector ~ tx() in (19) or
(23) is bounded under lemma 3.□
Remark 2: From (29), when x ( )0 =0, θ~f(0)=0 and
0
θ~g(0)= , the H∞ tracking performance (29) becomes
sup . w L t f t f e x w ∈2 ≤ 1 2 2 δ (31)
Theorem 2: Consider the nonlinear system (2) in which
the states x are estimated by the observer in (14). When substituting (5)-(6) into (14), the HAFO is as follows: , ˆ ˆ , ~ ] ) ˆ ( ) ˆ ( [ ˆ ˆ f g 1 x C K x ψ θ x ψ θ B x A x T o r T T y x v u = + − + + = (32)
with the robust term v in (30). Suppose that the r control input u is bounded and assumptions 1-4 and lemma 5 are satisfied. Let θ and f θ be adjusted by g the adaptive laws (27)-(28) and δ >0 be a given attenuation constant. Thus the output yˆ of the observer
in (14) or (32) with the H∞ tracking performance (29)
is achieved.□
Proof: Since theorem 1 and x~=x−xˆ, it is obvious that
the H∞ tracking performance (29) is achieved.
四、 四、四、
四、H∞∞∞∞----Observer-Based Adaptive Fuzzy Controller
First, a reference vector ym, and an estimation
error vector e are defined as . ˆ , ] [ ( 1) x x e y − = = n− T m m m m y yC ly (33)
Based on the certainty equivalence approach, the control law is defined as ] ) ˆ ( ) ˆ ( fˆ [ ) ˆ ( gˆ 1 ( ) c m T c n m u y u= − x + +K y −x − x , (34) where c c T n c n c=[k k −1mk1]
K is the feedback gain vector, chosen such that the characteristic polynomial of
A BK− c
T is Hurwitz because ( , )A B is controllable.
The compensatory term uc is employed to compensate
the external disturbance and the estimation errors. The functions f ( )x and g( )x represent the estimates of the nonlinear functions f ( )x and g( )x , respectively. Subtracting (14) from (2), we have
, ], )) ˆ ( ) (g( ) ˆ ( ) [f( ) ( 1 f e C x ψ θ x x ψ θ x B e C K A e T r T g T T o e v d u = + + − + − + − = (35) where e is an estimation error vector and e1=x1−ˆx1
denotes the output tracking error. Similarly, the approximation error wo in (17) is substituted into (35). Thus, (35) becomes . ], )) ˆ ( ) | ˆ ( gˆ (( ) ˆ ( ) | ˆ ( fˆ [ ) ( 1 g f e C x ψ θ θ x x ψ θ θ x B e C K A e T o r T g T f T o e d w v u = + + + − + − + − = ∗ ∗ D (36) Therefore, (36) can be rewritten as
, ], ) ˆ ( ~ ) ˆ ( ~ [ ) ( 1 g f e C x ψ θ x ψ θ B e C K A e T m r T T T o e w v u = + + + + − = D (37) where f f f ~ θ θ θ = ∗− , g g g ~ θ θ θ = ∗ − , and w w d o m = +
represents effects of the modeling error and the external disturbance. Since only output e1 in (37) is assumed to
be measurable, we use the SPR-Lyapunov design approach to analyze the stability of the estimation error dynamics (37) and generate the adaptive law for tuning
f
θ and θg.
By comparing (37) with (19), there is the same dynamics. By similar derivatives, Theorem 1 can also be applied to (37). Therefore, the H∞ tracking
performance [9,10] is achieved as follows:
2 2 2 g 0 2 f f 0 1 0 2 2 1 (0) ~ ) 0 ( ~ 1 ) 0 ( ~ ) 0 ( ~ 1 ) 0 ( ) 0 ( 1 t f T g T T t w e δ γ γ γ γ γ + + + ≤ e Pe θ θ θ θ (38) with 1 min( ) 1 e vr λ Q γ −
= , where γ =0 λmin(Q) and ξ
γ
γ= ′ is a positive scale value.
Besides, in order to guarantee that the closed-loop system controlled by the designed controller (34) is stable in the sense that all signals involved are bounded, we have the following theorem.
Theorem 3: Consider the nonlinear system (1) that
satisfies assumptions 1-4. Suppose that the HAFC is ], ) ˆ ( ) ˆ ( [ ) ˆ ( 1 () f g c m T c n m T T y u u= −θ ψ x + +K y −x − x ψ θ (39) with the adaptive fuzzy observer in (32) or (14), the adaptive laws in (27)-(28), and
1 min( ) 1 e v uc r λ Q γ − = = .
Let δ >0 be given attenuation constant. Then the H∞
tracking performance (38) is achieved.□
Proof: From theorem 1 and (38), the H∞ tracking
performance (38) is achieved. 五、
五、五、
五、An Illustrative Example
Example 1: Consider the inverted pendulum system:
[ ]
1 0 , ), g (f 1 0 0 0 1 0 2 1 2 1 2 1 = + + + = x x y d u x x x x C C where ) ( 3 4 cos sin ) ( cos sin f 1 2 1 1 1 2 2 m M l x ml x g m M x x mlx + − + − = , g x ml x l M m = − − + cos cos ( ) 1 2 1 4 3 .M is the mass of the cart, m is the mass of the rod, g=9 8m
2
. sec is the acceleration due to gravity, l is the
half length of the rod, and u is the control input. It is assumed that M=1kg, m=0 1. kg, l=0.5m and the external disturbance d t( ) is a square wave with the amplitude ±0 1. and the period 2π . Our control objective is to control the state x1 of the system to track
the reference trajectory ym=(π30) sin( )t when only the system output y is measurable. The design parameters
are selected as 2 2 1 1, 2 10 − × = = γ
γ . The feedback and
observer gain vectors are given as T c =[144 24]
K and
T o =[30225]
K , respectively. The filter L−1( )s is given as L−1 s = s+
1 2
( ) ( ). Because the estimation states are two, the total fuzzy rules are forty-nine. The initial values of adjustable parameters are selected to be
1 ) 0 ( , 0 ) 0 ( g fi = θ i = θ , i=1 2, ,m,49.
The initial states are chosen as x( ) [ .0 = −0 4 −0 4. ]T
, and >( ) [ . . ] x 0 = 0 1 0 1T , respectively. Choose 0005 . 0 , 05 . 0 =
δ and get γ =0.1,0.001, respectively. Fig. 1 shows the trajectories of the states x1 and ym for
different values of γ. The transient and steady-state responses of the control input u is shown in Fig. 2 (a) and (b) respectively, for different values of γ.
六 六六
六.... Conclusions
In this report, the H∞-tracking-based adaptive fuzzy
observer (HAFO) and H∞ -observer-based adaptive
fuzzy controller (HAFC) tuned on-line for a class of uncertain nonlinear systems are developed. The control law and update laws are derived to tune on-line the
weighting factors of the HAFO and HAFC. To directly estimate real states, the HAFO is designed for more fitting the practical applications. Not only HAFC but also HAFO are constructed by the fuzzy system. Moreover, the same update laws tune both the HAFO and HAFC.
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0 1 2 3 4 5 6 7 8 9 10 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
rad
time(sec)
: y
m: x
1 (a) γ=0.1 0 1 2 3 4 5 6 7 8 9 10 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5rad
time(sec)
: y
m: x
1 (b) γ=0.001Fig. 1. Trajectories of the state x
1and y
min example 1.
1 2 3 4 5 6 7 8 9 1 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 γ=0.1 γ=0.001 time(sec) N
