線性不定輸出回授系統之混合H2/LMI控制及其應用

(1)

行政院國家科學委員會專題研究計畫成果報告

線性不定輸出回授系統之混合 H2/LMI 控制及其應用

計畫類別：個別型計畫計畫編號： NSC93-2218-E-151-003- 執行期間： 93 年 08 月 01 日至 94 年 07 月 31 日執行單位：國立高雄應用科技大學機械工程系計畫主持人：陳信宏報告類型：精簡報告處理方式：本計畫涉及專利或其他智慧財產權，2 年後可公開查詢

中華民國 94 年 8 月 12 日

(2)

行政院國家科學委員會專題研究計畫成果報告

線性不定輸出回授系統之 H2/LMI 控制及其應用

計畫編號：NSC 93-2218-E-151-003

執行期限：93 年 8 月 1 日至 94 年 7 月 31 日

主持人：陳信宏教授國立高雄應用科技大學機械系

一、中英文摘要本研究成果報告提出混合田口基因法來求解同時具有元素型態和範數界限型態參數不確定量之輸出回授系統的混合控制問題，使得閉迴路系統的強健穩定性能獲得保證且輸出信號的追蹤誤差為最小。本研究計畫案將首先針對同時具有元素型態和範數型態參數不確定量之輸出回授系統的強健穩定性進行分析，提出以線性矩陣不等式表示的充分條件，並以數值範例來和已知文獻的結果作保守性之比較。其次，結合田口與基因法來求解混合控制之最佳輸出回授控制器設計問題，並將之應用於直昇機的飛行高度控制。 LMI H /₂ LMI H /₂ 關鍵詞：強健穩定性，性能，線性矩陣不等式，參數不確定量。 2 H Abstract

This report proposes a hybrid Taguchi-genetic algorithm (HTGA) approach to solve the mixed optimal output feedback controller design problem of linear systems under both time-varying elemental (structured) and norm-bounded (unstructured) parameter uncertainties. A sufficient condition is proposed in terms of linear matrix inequalities (LMIs) for ensuring that the linear output feedback systems with both time-varying elemental and norm-bounded parameter uncertainties by directly considering the mixed quadratically-coupled uncertainties in the problem formulation are asymptotically stable. The proposed HTGA approach is effectively applied to solve the mixed optimal output feedback controller design problem of linear systems with time-varying elemental and norm-bounded parameter uncertainties. A

design example is given to illustrate the application of integrating the presented sufficient condition and the HTGA approach.

LMI / 2 H LMI / 2 H

Keywords: Stability robustness,

performance, linear matrix inequality, parameter uncertainties.

2

H

二、計畫緣由與目的

In general, a mathematical description is only an approximation of the actual physical system and deals with fixed nominal parameters. Usually, these parameters are not known exactly due to imperfect identification or measurement, aging of components and/or changes in environmental conditions. Thus, it is almost impossible to get an exact model for the system due to the existence of various parameter uncertainties. Here, we consider linear state-space systems with time-varying uncertain parameters in the system matrix, input matrix, and output matrix. Because the output feedback controller design is usually based on the nominal values of these system matrices, it is interesting to know whether the closed-loop system remains asymptotically stable in the presence of time-varying uncertain parameters. Applying those previous robust stability analysis results (Zhou and Khargonekar, 1987; Kolla et al., 1989; Weinmann, 1991; Gao and Antsaklis, 1993; Chou, 1994; Chen and Yang, 1997; Schreier et al., 1998; Chen, 1999; Chou and Chen, 2000; Costa and Oliveira, 2002; and references therein) to solve this problem is not easy, in that after output feedback, there will be coupled terms of parameters in the closed-loop system matrix because of the uncertain parameters in both input and output matrices (Su and Fong, 1993). Though we may regard these coupled terms as new independent

(3)

parameters if we insist on using those previous robust stability analysis results, Su and Fong (1993) and Tseng et al. (1994) have showed that a conservative analysis conclusion may be reached. Therefore, Su and Fong (1993) and Tseng et al. (1994) investigated the robust stability problem of linear systems with constant output feedback in the presence of time-varying elemental (structured) parameter uncertainties by directly considering the coupled terms in the problem formulation. Su and Fong (1993) used the Lyapunov method to analyze the robust stability of linear systems with quadratically-coupled elemental uncertainties. Tseng et al. (1994) applied the structured singular value technique to solve the robust stability analysis problem of linear systems with quadratically-coupled elemental uncertainties. On the other hand, it is well known that an approximate system model is always used in practice and sometimes the approximation error should be covered by introducing both elemental (structured) and norm-bounded (unstructured) parameter uncertainties in control system analysis and design (Zhou and Gu, 1992). That is, it is not unusual that at times we have to deal with a system simultaneously consisting of two parts: one part has only the elemental parameter uncertainties, and the other part has the norm-bounded parameter uncertainties. Therefore, very recently, based on the Lyapunov approach and some essential properties of matrix measures, Chen and Chou (2003) proposed a sufficient condition to study the problem of stability robustness for linear output feedback systems with both time-varying elemental and norm-bounded parameter uncertainties by directly considering the mixed quadratically-coupled uncertainties in the problem formulation. Here it should be noted that only the articles of Su and Fong (1993), Tseng et al. (1994), and Chen and Chou (2003) studied the robust stability of linear continuous-time systems with output feedback controllers and time-varying uncertain parameters by directly considering the coupled terms in the problem formulation. That is, the research on the

stability robustness of linear continuous-time systems with output feedback controllers as well as both time-varying elemental and norm-bounded uncertainties by directly considering the coupled terms in the problem formulation is considerably rare and almost embryonic. Besides, here it should be also noted that, for the case of only considering time-varying elemental parameter uncertainties, Chen and Chou (2003) have shown that their sufficient condition is less conservative than those of Su and Fong (1993) and Tseng et al. (1994).

On the other hand, only robust stability is often not enough in control design. The optimal tracking performance is also considered in many practical control engineering applications. Hence, the mixed

optimal control designs are quite useful for robust stability and performance design for systems under both time-varying elemental and norm-bounded parameter uncertainties. The mixed optimal control design is to find a stabilizing controller that minimizes performance index (i.e., the integral of the squared error (ISE) or the integral of the time-weighted squared error (ITSA)) subject to some stability robustness inequality constraints (i.e., LMI-based sufficient conditions). But, to the author’s best knowledge, the mixed optimal output feedback control design problem for linear systems with time-varying elemental and norm-bounded parameter uncertainties has not been discussed in the literature. Thus, the mixed optimal output feedback control design problem of linear systems with time-varying elemental and norm-bounded parameter uncertainties is worth investigating.

LMI / 2 H LMI / 2 H 2 H LMI / 2 H LMI / 2 H

Therefore, the purpose of this report is to use a robust and statistically sound approach, which is named the hybrid Taguchi-genetic algorithm (HTGA) (Tsai et al., 2004), to solve the mixed H₂ /LMI output feedback controller design problems of linear systems. In this report, we determine the optimal output feedback controller gain by applying the HTGA to directly minimize a defined

(4)

performance index. 三、研究方法與成果

Consider the linear uncertain system with the state-space model

) ( ) ( ) (t Ax t Bu t x& = + , (1) ) ( ) (t Cx t y = , (2)

where is the state vector,

is the output vector, is the input vector, and

n R t x( )∈ p R t y( )∈ u(t)∈Rq ) ( ~ ) ( 1 0 t A A t A A m j j j + + =

∑

= ε , (3a) ) ( ~ ) ( 1 0 t B B t B B m j j j + + =

∑

= ε , (3b) ) ( ~ ) ( 1 0 t C C t C C m j j j + + =

∑

= ε (3c) are the system matrices, in which ε_j(t) (ε _j ≤ε_j₍_t₎≤ε_j, and _j=₁_,₂_,...,_m) are the time-varying elemental uncertainties; , and ( ) are, respectively, the

given , and constant

matrices which are prescribed prior to denote the linearly dependent information on time-varying elemental uncertainties

j A B_j j C j=1,2,...,m n n× n×q p×n ) (t j ε ’s; and is the number of independent time-varying uncertain parameters. The time-varying norm-bounded uncertain matrices m ) ( ~ t

A , B~(t) and C~(t) are assumed to be bounded, i.e., 1 ) ( ~ _≤ _β t A , B~(t) ≤β₂, and C~(t) ≤β₃, (4) where 1 β , 2

β , and β₃ are non-negative real constant number, and • denotes any matrix norm.

In this report, we only discuss the static output feedback gain controllers. Thus, the closed-loop system equation of the linear uncertain system can be expressed as

⎢ ⎣ ⎡ + + + + =

∑

= m j j j j j t A BKC BKC KC B A t x 1 0 0 0 0 0 ()( ) ) ( ε & 0 0 1 1 ) ( ~ ) ( ~ ) ( ~ ) ( ) (t t BKC At BKCt Bt KC m j m k k j k j + + + +

∑∑

= = ε ε ) ( ) ) ( ~ ) ( ~ )( ( ) ( ~ ) ( ~ 1 t x KC t B t C K B t t C K t B m j j j j ⎥ ⎦ ⎤ + + +

∑

= ε ) ( ) ( ) ( ) ( ) ( 1 1 1 0 t E t t E Ft xt A m j m k jk k j m j j j ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + + =

∑

∑∑

= = = ε ε ε ,(5)

where K denotes the output feedback gain matrix, 0 0 0 0 A B KC A = + , 0 0KC B KC B A Ej = j + j + j , ) ( 2 1 j k k j jk B KC B KC E = + , and ) ( ~ ) ( ~ ) ( ~ ) ( ~ ) ( ~ ) (t A t B₀KC t B t KC₀ B t KC t F = + + +

∑

. (6) = + + m j j j j t BKC t B t KC 1 ) ) ( ~ ) ( ~ )( ( ε

From Eqs. (4) and (6), we can get that

β ≤ ) (t F , (7) in which K K B KC₀ ₃ ₀ ₂ ₃ 2 1 β β β β β β = + + +

∑

= + + m j j j j B K KC 1 2 3 ) ( ~ _β _β ε ,

and ε~_j =max{ε_j, ε_j}. Therefore, the norm-bounded uncertain matrix can be expressed by ) (t F ) ( ) (t t F =β∆ , (8) where∆(t) is unknown matrix function which is bounded by ∆₍_t₎∈Ω_:=

{

∆₍_t₎ ∆₍_t₎ ≤₁_, the element of ∆_(t₎ are Lebesgue measurable}.

In what follows, under the assumption that an output feedback gain matrix K has been previously designed to make A₀ a stable matrix, by using the Lyapunov approach, we present two new sufficient conditions in terms of linear matrix inequalities (LMIs), which can be efficiently solved by means of standard optimization procedures (Gahinet et al., 1995), for ensuring that the linear closed-loop uncertain system in (5) remains asymptotically stable.

Theorem 1:

The linear closed-loop uncertain system in (5) remains asymptotically stable, if there exist a symmetric positive definite matrix and a constant

P

0 >

(5)

LMIs are simultaneously satisfied 0 ) ( ) ( < ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + + + + I P P I KW V U P P KW V U_i _j _k T _i _j _k τ β β τ (9) where

∑

= = m j j j i t E U 0 ) ( ε j j j t ε

ε

ε ( )= or ; (10)

∑

= = m j j j j t B V 1 ) ( ε j j j t ε

ε

ε ( )= or ; (11)

∑

= = m j j j k t C W 1 ) ( ε j j j t ε

ε

ε ( )= or ; (12) in which E₀ = A₀, and ε₀(t)=ε₀ =ε₀ =1, and i ,j,k =1,2,...,2m. Theorem 2:

The linear closed-loop uncertain system in (5) remains asymptotically stable, if, for a specified constant α with 0<α <1, there exist a symmetric positive definite matrix P and a constant τ >0 such that the following LMIs are simultaneously satisfied:

0 < ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + + I P P I PU P U_iT _i τ β β τ , (13) and the same matrix P makes the following inequality hold:

∑∑

∑

= = = < + m j m k jk k j l l j j j t t t s 1 1 1 ~ ) ( ) ( ~ ) ( 1 θ ε ε φ ε , (14) where _i=₁_,₂_,...,₂hr ; 0 1 ~ A A =α ; 0 2 (1 ) ~ A A = −α ;

∑

= = hr h j j j i t E U 0 ) ( ε j j j t ε

ε

ε ( )= or ; (15) 2 2 ~ ~ A P P A Q=− T − ; (16) P E PE Pj = j + jT ; (17) P E PE P_jk = _jk + _jkT ; (18) 0; < ) ( 0; ) ( ), ( ), ( ~ for for 2 1 2 1 2 1 2 1 t t Q P Q Q P Q j j j j j ε ε µ µ φ ≥ ⎪ ⎩ ⎪ ⎨ ⎧ − − = − − − − (19) and 0; < ) ( ) ( 0; ) ( ) ( ), ( ), ( ~ for for 2 1 2 1 2 1 2 1 t t t t Q P Q Q P Q k j k j jk jk jk ε ε ε ε µ µ θ ≥ ⎪ ⎩ ⎪ ⎨ ⎧ − − = − − − − (20) in which 1 ~ 0 A Eh = , εh₀(t)=εh₀ =εh₀ =1 ,

{

ε (),ε (),...,ε ()

} {

ε (),ε (),...,ε ()

}

=φ 2 1 2 1 t h t hr t l t l t ls t h I ,

{

( ), ( ),..., ( )

} {

( ), ( ),..., ( )

}

2 1 2 1 t h t hr t l t l t ls t h ε ε ε ε ε ε _U

{

ε₁(t),ε₂(t),...,ε_m(t)

}

, = r+s =m, and φ

denotes the empty set.

Remark 1: The problem of determining

the robust stability of the linear output feedback system with both time-varying elemental and norm-bounded parameter uncertainties in (5) can be considered as LMI feasibility problems which are convex and can be effectively solved by corresponding LMI software (e.g., Gahinet et al., 1995).

Remark 2: If the linear output feedback

uncertain system in (5) does not include norm-bounded parameter uncertain matrix (i.e.,

0 ) (t =

F ), then the sufficient conditions (9) and (13), respectively, become to solve the following LMI feasibility problems

0 ) ( ) (U_i +V_jKW_k TP+PU_i +V_jKW_k < , for m, (21) k j i , , =1,2,...,2 and 0 < + _i T i P PU U , for _i =₁_,₂_,...,₂hr, (22) and check whether the same matrix obtained in (22) makes the inequality in (14) hold. Similarly, if we do not consider the elemental parameter uncertainties (i.e.,

P

0 ) (t =

j

ε for j =1,2,...,m ), then the

sufficient conditions of robust stability become only to solve the following LMI feasibility problem 0 0 0 < ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + + I P P I A P P AT τ β β τ . (23)

Remark 3: The robust stability analysis

methods for the linear output feedback uncertain system in (5) can be grouped into two categories: LMI-based approaches and non-LMI-based approaches. Since the proposed LMI-based sufficient condition in (9), (13) and (14) and the non-LMI-based sufficient condition given by Chen and Chou (2003) are derived by different methods, it is difficult to compare the conservatism

(6)

mathematically. Therefore, by using some numerical examples, we can compare numerically the proposed LMI-based sufficient condition in (9), (13) and (14) with the non-LMI-based sufficient condition given by Chen and Chou (2003). From numerical examples, we can see that the proposed LMI-based sufficient conditions in (9), (13) and (14) may obtain less conservative results than the non-LMI-based sufficient condition given by Chen and Chou (2003). The reason why the proposed sufficient LMI-based condition is less conservative is that the problem of determining the stability robustness by applying the LMI-based condition can be solved via the optimization procedures. Therefore, any sufficient non-LMI conditions may obtain more conservative results because they do not utilize the optimization procedures to solve the robust stability problem. On the other hand, from numerical examples, we can also see that the proposed LMI-based sufficient conditions in (13) and (14) may obtain less conservative results than the proposed sufficient LMI condition in (9). The reasons why the proposed sufficient LMI-based condition in (13) and (14) is less conservative are that (i) the freedom in choosing a suitable constant

α with ₀<α<₁ can provide an opportunity to improve the robust stability condition, and (ii) the numbers of LMIs can be greatly reduced and the LMI problem can be easily solved via Matlab LMI toolbox (Gahinet et al., 1995). Therefore, in the follows, we use the proposed sufficient LMI-based condition in (13) and (14) for the robust stability constraints to design the optimal output feedback controller.

With the exception of robust stability, the optimal tracking performance is also considered in many practical control engineering applications. Therefore, the mixed

output feedback controller design problem for linear systems with time-varying elemental and norm-bounded parameter uncertainties in (1) and (2) is, under the robust stability constraints (20) and (21), how to find

the controller parameters of output feedback controller LMI / 2 H ] [kij

K = and a specified constant

α to achieve the optimal tracking performance by minimizing the following

performance index: 2 H

∫

∞ = 0 (e (t)e(t))dt J T , (24) where e(t)= y_r(t)− y(t) is the tracking error vector, and in which is the desired output vector.

) (t y_r

Remark 4: From the proposed sufficient

LMI-based conditions in (13) and (14), we can see that the constant α affects the conservatism of the proposed sufficient LMI-based conditions (13) and (14). In order to check whether the uncertain closed-loop system is asymptotically stable or not, and in order to reduce the conservatism of the proposed sufficient LMI-based conditions (13) and (14), we use the hybrid Taguchi-genetic algorithm (HTGA) (Tsai et al., 2004) incorporated with a considerable amount of simulations to search for both the optimal output feedback controller parameters

] [k_ij

K = and the optimal α . The HTGA combines traditional genetic algorithms (TGA) (Gen and Cheng, 1997) with Taguchi method (Taguchi et al., 2000; Wu, 2000). In HTGA, the Taguchi method is inserted between crossover and mutation operations of a TGA. Then, by using two major tools (signal-to-noise ratio and orthogonal arrays) of the Taguchi method, the systematic reasoning ability of Taguchi method is incorporated in the crossover operations to systematically select the better genes to achieve crossover, and consequently enhance the genetic algorithms. The detailed steps of the HTGA are described in the following. The detailed description of the Taguchi method can be found in the books presented by Taguchi et al. (2000) and Wu (2000). In this report, the fitness function is defined in (24). In this mixed H₂ /LMI optimal control problem, i.e., how to minimize (24) subject to (13) and (14), a procedure is described as following:

(7)

Step 1: Check the LMI-based constraints in

(13) and (14) of robust stability is satisfied or not.

Step 2: Minimize the performance index in (24).

2

H

Besides, the additional details regarding the HTGA can be found in the works proposed by Chou and his associates (Chou et al., 1998; Tsai et al., 2004).

}

Design Example: The dynamics of a

helicopter in a vertical plane for an airspeed range of 60-170 knots has been given by Narendra and Tripathi (1973). There are four state variables: _{is the horizontal velocity} (knot/sec), is the vertical velocity (knot/sec), _{is the pitch rate (deg/sec), and}

is the pitch angle (deg); and two control variables: _{is the collective pitch control} and _{is the longitudinal cyclic pitch} control. For this airspeed range of 60 knots to 170 knots, the dynamic system with time-varying uncertain parameters

1 x 2 x 3 x 4 x 1 u 2 u

{

A,B,C ) (t j ε ( ) and controlled by the output feedback controller 2 , 1 = j K are described by ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − = 0 1 0 0 3229 . 1 707 . 0 2855 . 0 1002 . 0 0208 . 4 0024 . 0 01 . 1 0482 . 0 4666 . 0 0188 . 0 0271 . 0 0366 . 0 A ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ + ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ + 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ) ( 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ) ( ₂ 1 t ε t ε , ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ + ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − = 0 0 0 0 0 1 0 0 ) ( 0 0 99 . 4 52 . 5 5922 , 7 0447 . 3 1761 . 0 4422 . 0 3 t B ε ,

[

0 1 0 0

]

₄(t)

[

0 1 0 0 C = +ε

]

,

[

0.1 0.13

]

) ( 1 t ∈ − ε , ε₂(t)∈

[

−0.12 0.1

]

,

[

0.27 0.29

]

) ( 3 t ∈− ε and

[

0.2 0.6

]

) ( 4 t ∈ − ε .

That is, the significant changes occur only in element , , and . In this

example, output feedback controller

32

a a₃₄ b₂₁ c₁₂

[

]

T

k k

K = 11 12 is chosen to treat the mixed

optimal control problem. So the controller parameter vector is defined as

LMI / 2 H

[

]

T k k α

θ₃_×₁ = ₁₁ . ₁₂ . For some practical requirements, all of parameters are assumed to be in the following region:

{

−10≤ ₁_,₂ ≤10 and ₃∈(0 ,1)

}

.

=

Θ θi θ θ The

following evolutionary environments are used in this example: the population size is , the crossover rate is , the mutation rate is , the maximum generation is 40 , and let

30 9 . 0 0.5 ) ( ) ( 1 1 t t h ε ε = , εh₂(t)=ε2(t) , εl₁(t)=ε3(t)

and εl₂(t)=ε4(t). The computational results obtained by using the HTGA approach are the optimal output feedback gain , the specified constant

[

]

T K = −4 10 1472 . 0 = α , the

tracking performance , and a symmetric positive definite matrix as

2 H 3 10 166 . 1 × − = J P ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − − = 7060 . 2 5229 . 0 4060 . 0 2914 . 0 5229 . 0 4102 . 1 1471 . 1 4095 . 0 4060 . 0 1471 . 1 9914 . 0 3322 . 0 2914 . 0 4095 . 0 3322 . 0 8390 . 1 P .

And then, adopting the above matrix P in the sufficient condition (14) with the 2-norm-based matrix measure, we can obtain (i) ()~ () ( )~ 0.9131 1, 4 1 4 1 2 1 1 1 1 +

∑∑

≤ <

∑

= = = j k jk k j l l l tφ ε t ε tθ ε for ε₁(t)∈

[

−0.1 0.13

]

, ε₂(t)∈

[

−0.12 0.1

]

,

[

0 0.29

]

) ( 3 t ∈ ε , and ε₄(t)∈

[

0 0.6

]

;(25a) (ii) ()~ () ()~ 0.8862 1, 4 1 4 1 2 1 1 1 1 +

∑∑

≤ <

∑

= = = j k jk k j l l l tφ ε t ε tθ ε for ε₁(t)∈

[

−0.1 0.13

]

, ε₂(t)∈

[

−0.12 0.1

]

,

[

0.27 0

]

) ( 3 t ∈− ε , and ε₄(t)∈

[

0 0.6

]

; (25b) (iii) ()~ () ()~ 0.8759 1, 4 1 4 1 2 1 1 1 1 +

∑∑

≤ <

∑

= = = j k jk k j l l l tφ ε tε tθ ε for ε₁(t)∈

[

−0.1 0.13

]

, ε₂(t)∈

[

−0.12 0.1

]

,

[

0.27 0

]

) ( 3 t ∈− ε , and ε₄(t)∈

[

−0.2 0

]

; (254c) (iv) ()~ () ()~ 0.9033 1, 4 1 4 1 2 1 1 1 1 +

∑∑

≤ <

∑

= = = j k jk k j l l l tφ ε tε tθ ε

(8)

for ε₁(t)∈

[

−0.1 0.13

]

, ε₂(t)∈

[

−0.12 0.1

]

,

[

0 0.29

]

) ( 3 t ∈ ε , and ε₄(t)∈

[

−0.2 0

]

.(25d) From the computational results, it can be seen that the HTGA approach can give good performance. Figure 1 shows the tracking response of the helicopter system with the designed output feedback controller subjects to desired output

LMI / 2 H ) * 6 exp( 1 ) (t t y_r = − − under parameter uncertainties. Hence, we could conclude that the HTGA is very feasible to find the output feedback controller parameters of the mixed optimal control system under parameter uncertainties.

LMI /

2

H

四、結論與討論

This report investigates the mixed output feedback controller design problems of linear systems under both time-varying elemental and norm-bounded parameter uncertainties. Two LMI-based sufficient conditions have been proposed for ensuring the linear output feedback systems with both time-varying elemental and norm-bounded parameter uncertainties to be asymptotically stable. The problem of determining the stability robustness has been turned into LMI feasibility problems which can be easily solved by means of numerically efficient convex programming algorithms. For robust stability analysis, two numerical examples have shown that the proposed sufficient LMI-based conditions are less conservative than the sufficient non-LMI condition of Chen and Chou (2003). The reason why the proposed sufficient LMI-based conditions are less conservative has been also given in Remark 3. A design example by using the HTGA shows that the HTGA can obtain good tracking performance and makes the closed-loop systems keep the stability robustness. LMI / 2 H 五、研究成果自評本成果報告已達成申請計畫書中預期完成的成果目標。本研究計畫案之部份成果的發表情況如下所列：

1. S. H. Chen and J. H. Chou, “Robust

Stability of Linear Continuous/Discrete-Time Output Feedback Systems with Both Time-Varying Structured and Unstructured Uncertainties”, JSME International Journal, Series C, Vol.46, No.2, pp.705-712, 2003. 2. S. H. Chen and J. H. Chou, “Stability

Robustness of Linear Output Feedback Systems With Both Time-Varying Structured and Unstructured Uncertainties as well as Actuator Saturation”, Proceedings of the Institution of Mechanical Engineers, Part I, Journal of Systems and Control Engineering, Vol.218, No.I4, pp.269-275, 2004.

3. S. H. Chen and J. H. Chou, “LMI-based Sufficient Conditions for Robust Stability of Linear Output Feedback Systems with Both Time-Varying Elemental and Norm-Bounded Uncertainties”, Proc. of the 21th National Conference on Mech.

Eng. CSME, Taiwan, R.O.C.,

pp.1221-1226, 2004.

4. S. H. Chen, J. H. Chou and L. A. Zheng, “Stability Robustness of Linear Output Feedback Systems With Both Time-Varying Structured and Unstructured Parameter Uncertainties as well as Delayed Perturbations”, Journal of The Franklin Institute, Vol.342, No.2, pp.213-234, 2005.

5. S. H. Chen and J. H. Chou, “Mixed H2/LMI Optimal Output Feedback Controller Design of Linear Systems with Both Time-Varying Elemental and Norm-Bounded Uncertainties via Genetic Algorithms”, 2005/08 (submitted to International Journal).

六、參考文獻

1. Chen, C. L. and S. K. Yang, 1997, “Robust stability and performance analysis for the system with linear dependent perturbations”, J. of Mathematical Analysis and Applications, Vol.213, pp.583-600.

2. Chen, J. D., 1999, “Robust stability criterion for structured perturbations

(9)

continuous systems with LQ regulators”, Proc. of the 1999 Automatic Control Conference, Taiwan, pp.748-750.

3. Chen, S. H. and J. H. Chou, 2003, “Robust stability of linear continuous/discrete-time output feedback systems with both time-varying structured and unstructured uncertainties”, JSME International Journal, Series C, Vol.46, pp.705-712.

4. Chou, J. H., 1994, “Robust stability bounds on structured time-varying independent perturbations for linear state-space models”, JSME International Journal, Series C, Vol.37, pp.733-738.

5. Chou, J. H. and S. H. Chen, 2000, “Stability robustness analysis of linear time-variant systems with both time-varying structured and unstructured perturbations”, JSME International Journal, Series C, Vol.43, pp.98-103.

6. Chou, J. H., W. H. Liao and J. J. Li, 1998, “Application of Taguchi-genetic method to design optimal grey-fuzzy controller of a constant turning force systems”, Proc. of the 15th CSME Annual Conference, Taiwan, pp.31-38.

7. Costa, E. F. and V. A. Oliveira, 2002, “On the design of guaranteed cost controllers for a class of uncertain linear systems”, Systems and Control Letters, Vol.46, pp.17-29.

8. Gahinet, P., A. Nemirovski, A. J. Laub and M. Chilali, 1995, LMI Control Toolbox, Massachusetts: The Math Works Inc.

9. Gao, Z. and P. J. Antsaklis, 1993, “Explict asymmetric bounds for robust stability of continuous and discrete-time systems”, IEEE Trans. on Automatic Control, Vol.38, pp.332-335.

10. Gen, M. and R. Cheng, 1997, Genetic Algorithms and Engineering Design, John Wiely and Sons, New York.

11. Kolla, S. R., R. K. Yedavalli and J. B. Farison, 1989, “Robust stability bounds on time-varying perturbations for state-space models of linear discrete-time systems”,

Int. J. of Control, Vol.50, pp.151-159. 12. Narendra, K. S. and S. S. Tripathi, 1973,

“Identification and optimization of aircraft dynamics”, J. of Aircraft, Vol.10, pp.193-199.

13. Schreier, G., J. Ragot, P. M. Frank and G. L. Gissinger, 1998, “Bounds of parameter uncertainties for robust stability of linear uncertain systems”, Int. J. of Systems Science, Vol.29, pp.85-90.

14. Su, J. H. and I. K. Fong, 1993, “Robust stability analysis of linear continuous/discrete time systems with output feedback controllers”, IEEE Trans. on Automatic Control, Vol.38, pp.1154-1158.

15. Taguchi, G., S. Chowdhury and S. Taguchi, 2000, Robust Engineering, McGraw-Hill, New York.

16. Tsai, J. T., T. K. Liu and J. H. Chou, 2004, “Hybrid Taguchi-genetic algorithm for global numerical optimization”, IEEE Trans. on Evolutionary Computation, Vo.8, pp.365-377.

17. Tseng, C. L., I. K. Fong and J. H. Su, 1994, “Analysis and applications of robust nonsingularity problem using the structured singular value”, IEEE Trans. on Automatic Control, Vol.39, pp.2118-2122. 18. Weinmann, A., 1991, Uncertain Models

and Robust Control, New York: Springer-Verlag.

19. Wu, Y., 2000, Taguchi Methods for Robust Design, The American Society of Mechanical Engineers, New York.

20. Zhou, K. and G. Gu, 1992, “Robust stability of multivariable systems with both real parametric and norm bounded uncertainties”, IEEE Trans. on Automatic Control, Vol.37, pp.1533-1537.

21. Zhou, K. and P. P. Khargonekar, 1987, “Stability robustness bounds for linear state-space models with structured uncertainties”, IEEE Trans. on Automatic Control, Vol.32, pp.621-623.

(10)

Figure 1. The desired output response (dash line) and actual output responses (solid line) of Design Example.

線性不定輸出回授系統之混合H2/LMI控制及其應用

行政院國家科學委員會專題研究計畫 成果報告