混合強健/最佳PID控制器設計及其應用

行政院國家科學委員會專題研究計畫 成果報告

混合強健/最佳 PID 控制器設計及其應用

研究成果報告(精簡版)

計 畫 類 別 ： 個別型 計 畫 編 號 ： NSC 95-2221-E-151-023- 執 行 期 間 ： 95 年 08 月 01 日至 96 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 計畫參與人員： 碩士班研究生-兼任助理：汪東家 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 96 年 09 月 05 日

行政院國家科學委員會補助專題研究計畫

■ 成 果 報 告

□期中進度報告

混合強健/最佳 PID 控制器設計及其應用

計畫類別：▓ 個別型計畫 □ 整合型計畫

計畫編號：NSC95

-2221-E-151-023

執行期間：

95 年 8 月 1 日至 96 年 7 月 31 日

計畫主持人：

陳信宏 教授

共同主持人：

計畫參與人員：

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國立高雄應用科技大學機械系

中 華 民 國 96 年 9 月 5 日

行政院國家科學委員會專題研究計畫成果報告

混合強健/最佳 PID 控制器設計及其應用

計畫編號：NSC95-2221-E-151-023

執行期限：95 年 8 月 1 日至 96 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 本研究成果報告應用正交函數法、田口 法與基因演算法來求解具有時變結構型態 不確定量之 PID 控制系統的混合強健/最佳 控制問題，使得閉迴路系統的強健穩定能獲 得保證且 性能為最小。首先將具時變結 構型之 PID 控制系統，轉換成具時變結構型 與二次式結構型不確定量之靜態輸出回授 控制之廣義系統。利用矩陣測度的基本性 質，推導出廣義系統同時具有結構型態與二 次式結構型態之參數不確定量的情況時，其 強健區域特徵値叢集之判定條件。其次，結 合正交函數法、田口法與基因演算法來求解 混合強健/最佳控制之最佳 PID 控制器設計 問題並作效能評估。 2 H 關鍵詞：PID 控制系統，強健穩定性， 性 能，結構型不確定量，廣義系統。 2 H Abstract

This reportuses the orthogonal function approach, Taguchi method and genetic algorithm to solve the mixed robust/optimal control problem of linear continuous-time systems with PID controllers as well as both time-varying structured and unstructured parameter uncertainties by directly considering the quadratically-coupled uncertainties in the problem formulation. First, the regional eigenvalur-clustering robustness problem of PID controlled systems is converted to the regional eigenvalues-clustering robustness problem of singular systems controlled by static output feedback controller.

B

ased on the OFA, an algorithm only involving algebraic computation is derived for the quadratic-optimal controller of nominal closed-loop singular systems. Using some

essential properties of matrix measures, a new sufficient condition is proposed for ensuring that the resulting singular systems with structured and quadratically-coupled structured parameter uncertainties is regular, impulse-free and all the finite eigenvalues retained within the specified region. Next, the hybrid Taguchi-genetic algorithm (HTGA) approach is applied to design the robust/optimal PID controller for the linear systems with time-varying structured parameter uncertainties. A design example is given to illustrate the application of integrating the presented sufficient condition, OFA and HTGA approach.

Keywords: Regional eigenvalues-clustering

robustness, PID controller, performance, singular systems, structured parameter uncertainties.

2 H

二、計畫緣由與目的

The PID (proportional-integral-derivative) controller is the most common form of feedback in use today, and is successfully used for a wide range of application: process control, motor drives, magnetic and optic memories, automotive, flight control, instrumentation and so on (Tan et al., 1999; Isaksson and Hagglund, 2002). But the problem for the performance analysis and design of linear multivariable PID control systems is still a real challenge to control system engineers (Saeki, 2006). Therefore, recently, some researchers have proposed some approaches to study the performance analysis and design problems of linear multivariable PID control systems (see, for example, Tan et al., 2002; Zheng et al., 2002; Lin et al., 2004; Kim and Cho, 2005; Xue et al., 2005; He et al., 2005; Xue et al., 2006;

Saeki, 2006; Ahn and Nguyen, 2007; and references therein).

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 structured parameter uncertainties in control system analysis and design. To ensure both stability robustness and certain performance robustness, it is important to guarantee that the eigenvalues of a linear time-invariant multivariable system under parameter uncertainties remain in a specified region. Recently, the authors (Chen et al., 2006) have discussed the robust analysis problems of eigenvalue-clustering in specified regions for the linear multivariable PID control systems with parameter uncertainties. But, only the robust stability is often not enough in control design. The optimal performance is also considered in many practical control engineering applications. Hence, the robust quadratic-optimal control designs are needed for regional eigenvalues-clustering robustness and quadratic-optimal performance design for linear multivariable output feedback PID control systems with structured parameter uncertainties. The robust quadratic-optimal control design is to find a stabilizing output feedback PID controller that minimizes the quadratic performance index subjects to the regional eigenvalues-clustering robustness constraint.

Besides, the orthogonal function approach (OFA) has received considerable attention in recent years as a convenient and sharp tool to deal with the analysis and optimization problems of dynamic systems (see, for example, Chou and Horng, 1985, 1986, 1987; Chou, 1987; Lewis and Mertrios, 1987; Nagurka and Wang, 1993; Datta and Mohan, 1995; Patra and Rao, 1996; Jaddu, 2002; Pacheco, 2002; Elbarbary and El-Kady, 2003; Akyuz and Sezer, 2003; and references therein). The main characteristic of such a technique is that it converts these problems to those of solving a system of algebraic equations, and hence the solution and optimization procedures are either greatly

reduced or much simplified accordingly. Therefore, the purpose of this paper is to propose a new method for finding the optimal output feedback PID controllers of the linear multivariable control systems with structured and quadratically-coupled structured parameter uncertainties such that the control objective of minimizing a quadratic performance index subject to the regional eigenvalues-clustering robustness constraint is achieved. The proposed new method is an integrative approach which integrates the robust regional eigenvalues-clustering sufficient condition, the OFA and the hybrid Taguchi-genetic algorithm (HTGA). In this project, we transform the robust quadratic-optimal control design problem of linear multivariable output feedback PID uncertain control system to the robust quadratic-optimal control design problem of linear uncertain singular control system with static output feedback controller.

三、研究方法與成果

Consider the linear uncertain systems described by

x&(t)= Ax(t)+Bu(t), (1) y(t)=Cx(t), (2) with PID controller of the form

) ( ) ( ) ( ) ( 0y d K y t K t y K t u D t I P + + & =

∫

θ θ , (3)

where is the state vector, is the output vector, is the input vector,

n R x∈ y∈Rp r R u∈

A , B and are the system matrix, the input matrix and the output matrix,

respectively. The matrices , respectively, are the

proportional feedback gain matrix, the integral feedback gain matrix and the derivative feedback gain matrix of the output feedback PID controller. C D I, K ∈ p × r R , P K K

Let a new state variable be

T T t _T T t x d x t x t x( ) [ ( ), ( ) , ( )] 0 &

∫

= θ θ and the new

output be T t T T T t d y t y t y( ) [ ( ), ( ) ,y ( )] 0 &

∫

= θ θ ,

then the system in (1) and (2) with the PID controller in (3) can be expressed as the

following uncertain closed-loop singular system: Ex&(t)= Ax(t), (4) where ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = 0 0 0 0 0 0 0 n n I I E , ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − + = n D I P n BK C BK C BK I 0 0 0 n I C A I A 0 ,

and I_n denote the n×n identity matrix. Equation (4) is called a singular system, generalized state-space system, implicit system, semi-state system, or descriptor system (Dai, 1989). Here the matrix E may be a singular matrix with rank(E)≤3n.

The quadratic-optimal control problem considered in this paper is to find the optimal output feedback controllers for the linear singular system in Eq. (4) such that the linear quadratic performance index

∫

+ = qtf uu T xx T dt t u R t u t x Q t x J 0 [ ( ) ( ) ( ) ( )] ,

∑∫

− = + + = 1 0 ) 1 ( )] ( ) ( ) ( ) ( [ q k t k kt uu T xx T f f dt t u R t u t x Q t x ,(5)

is minimized, where denotes a small time interval which is chosen for the independent variable , is a positive integer specified

by designer, Q is a symmetric

positive-semidefinite matrix, and is a symmetric positive-definite matrix. By using the PID controller in Eq. (3) and variable transformation, the quadratic performance index in Eq. (5) can be rewritten as

f t xx t q uu R

∑∫

₌− + + = 1 0 ) 1 ( )]) ( ( )) ( ( ) ( ) ( [ q k t k kt uu T T f f dt t y K R t y K t x Q t x J

∑∫

− = + + = 1 0 ) 1 ( ) ( ] )[ ( q k t k kt uu T T T f f dt t x C K R K C Q t x , (6 re ) whe ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = 0 0 0 0 0 0 0 0 xx Q Q , ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = C C C C 0 0 0 0 0 0 , and K =

[

Kp Ki Kd

]

.

Now, assume that all elements of x(t) are absolutely integrable w

f f t k t kt ( 1) ithin + ≤

≤ , and let us define

t=kt_f +η , (7) and _xk =_x(ktf) , hich (8) in w k =0,1,2,...,q−1, and 0≤η ≤t_f . Then, the state vector x(t) , within

f

f t k t

kt ≤ ≤( +1) , can be appro ed by the

as

ximat

truncated orthogonal function representation

( ) ( ) ~( ) ( ) 1 0 ) ( t T x t T x t x k m i i k i = =

∑

− = , (9) where m is the number of terms required for

the orthogonal functions,

[

]

T m t T t T t T t T()= ₀(), ₁(),..., ₋₁() denotes the m×1 ( 0 ,1 ,..., 1

orthogonal basis vector, T_i(t) −

= m

i ) denote the orthog

fu 0,1,..., −1 onal nctio (k) i x ( ns, i= m ) are the 1 ×

n coefficient vector, and

[

( ) ( )

]

,..., ,x1k x k1 is the n m ) ) ( ~ m k x = 0( ₋ k x × cally he la

value of _m , the more accurate are the approximate solutions.

Substituting Eq. (4)

coefficient matrix. Theoreti , t r

and the truncated ger the

orthogonal function representation of x(t) in Eq. (9) into the performance index in Eq. (6), the performance index J becomes

[

]

∑

−1 ~~() ~ q T T k = + = 0 ) ( ) )( ( ) ( k k uu T x C K R K C Q x W trace J , (10)

where denotes the

-integr -matri rthogo

m to

W~

ation

product x of two o nal

basis vectors. The constant matrix W

depends on the particular choice of t orthogonal function vector T(t).

Integrating Eq. (5) fro

he f kt t= t= t within kt_f ≤t≤(k+1)t_f , we obtain

∫

= − t kt f f dt t x A kt x E t x E ( ) ( ) ( ) . (11)

Using the following integral property of the orthogonal functions:

t T(t)dt HT(t)

kt_f =

∫

, (12) and applying Eqs. (8) and

cast into the form

(9), Eq. (11) can be

[

x

]

Ax H E x E k k k) ( ) ( ~ 0 ,..., 0 , 0 , ~ _{− = , (13)}

in which H is the operational matrix of integration for the orthogonal functions (Chou and Horng, 1986).

Eq. (13) can be rewritten as

) ( ) ( ) ( ~ ~ ~ k k Q H x A = − , (14) k x E where ~(k)

[

,0,0,...,0

]

x Q =E _k is a n×m matrix.

Making use of the Kronecker product, the explicit form for the coefficient matrix ~(k)

x

comes directly from Eq. (15) as

[

]

1 ( ) ) ( _ˆ ) ( ˆ T k m k Q A H E I x = ⊗ − ⊗ − , (1 where denotes 5) the identity m I T k) , m m× matrix,

[

k

]

m T k k x x x xˆ()= (_{0 1}() ,_L , (₋)₁T T,

[

T T T

]

T k x E Q ) ,0 ,0T ,_L ,0 , mplies that k) ( ( ˆ ₌

denotes the Kronecker product and

(Barne 79). This i

⊗

t, 19 ~(k)

x can be obtained from Eq. (15).

Now, if the output feedback gain matrix

K is given, then ~x(k) (k =0,1 ,2,...,q−1)

alg in ving algebrai

Give a small time interval

Step 2:

Step Com te by using can be calculated from the following

orithm only vol c

computation.

Algorithm

Step 1: t , _f q ,

the specified positive integer the initial state vector x(0), and set k=0. Calculate ˆ(k) x from Eq. (15). 3: pu xk+1 ) ) 1 (( ~ ) ((k x(k)T k t x x_k+1= +1)t_f = + _f .

Step 4: Set k = k+1. If k > q−1, then stop; otherwise go to Step 2.

From th ious

he output feedback gain matrix

e above algorithm, it is obv

that if t K is

specified, then ~(k)

x (k =0 ,1 ,2,...,q−1) can be determined, and thus the value of the performance index in Eq. (6) corresponding to this output feedback gain matrix K can be calculated. Given another output feedback

gain matrix K , there obtains another value of the performance index in Eq. (6). That is, the value of the performance index in Eq. (6) is actually dependent on the output feedback gain matrix K which means

J = F(K₁₁,K₁₂,_L,K_rs), (16) where K (_ij i =1,2,_L ,r and j=1 ,2,_L3p), respectively, denote the elements of the output feedback g in matrix K . Hena n problem of the optimal output feedback controller for the linear singular system is to search for the optimal K such that the _ij

performance index in Eq. (6) is minimized.

Robust Regional Eigenvalue-Clustering

ce, the desig

Analysis

s

described by

Consider the linear uncertain system

∑

∑

+ + + = m m i t u B k t Bu t x A k t Ax t x&() () () () () , = i i i i i 1 (17) =1

∑

= + = m i i iC x t k t y t Cx 1 ) ( ) ( ) ( , (18) ( where k_i i=1,2,_L,m) are and ( the elemental uncertainties; A , _i B_i C_i i=1,2,_L,m

are, respectively, the given n×n,

)

r n× and

n

p× constant ma es wh a d

prior to denote the line dent

ation on elemental uncertainties _i

_'

_s

tric ich a re rly ibe prescr depen inform ε ;

m is the number of independent uncertain

parameters.

Hence, the system in (17) and (18) with the PID controller in (3) can be expressed as the following uncertain closed-loop singular system:

∑∑

∑

= = = + + = Ax t m k Ax t m m kk A x t t x E&( ) ( ) ( ) () i j ij j i i i i 1 1 1 ) ( ) (t + Ax t x A Δ = , (19 where )

∑∑

∑

= = = + m m Δ = i i j ij j i m i iA k k A k A 1 1 1 , ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ + + + + = C K B C BK C K B C BK C K B C BK A A D i i D I i i I P i i P i i 0 0 0 0 0 0 ,

and ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ + + + = i D j j D i j I j j I i i P j j P i ij C K B C K B C K B C K B C K B C K B A 0 0 0 0 0 0 .

It is well known that for the nominal lar system described by

singu Ex&(t)= Ax(t) (or the pair (E,A)), the regularity condition

0 )

det(sE− A ≠ guaranteed the existence an uniqueness of solutions on , the

d )

, 0

[ ∞

condition rank(E)=degreeofdet(sE−A)

ensures that the system is imp ,

there is no im ehavior in the system, and the eigenvalue-clustering in a specified region the finite eigenvalues of the nominal system

ulse-free, i.e. pulse b

means that all

) ,

(E A lie within a specified

region (D .

Hence, the regional eigenvalue-clustering robustness analysis problem of linear multivariable output feedback PID uncertain control system in Eqs. (17), (18) and (3) can be converted t robustness analysis problem of the regularity, impulse-immunity and regional eigenvalues-clustering of the linear uncertain singular system in Eq.(6). Our problem can be formulated as: under the assumption that a set of PID feedback gain matrix K =[K_P,K_I,K_D] has been designed to make the linear nominal singular system

ai, 1989; Chou and 1998)

o the

Liao,

) ,

(E A be regular and impulse-free, and to make all its finite eigenvalues be located within a specified region D , determine the condition such that the linear uncertain singular system (20) denoted by (E,A+ΔA) is still regular and impulse-free, and has all its

eigenvalues retained in the same specified region as the nominal system finite

) ,

(E A does. That is, determ ne the condition under which the linear multivariable output feedback PID uncertain control sy l its eigenvalues kept within the same specified region as the linear multivariable output feedback PID nominal control system does.

Theorem:

Assume that a set of PID feedback gain matrix K [KP,KI,KD]

i

stem has al

= has been

previously designed to make the nominal system (E,A)

n singul

be regular and impulse-free,

ar system

and to have all its finite eigenvalues located inside a specified region D . The linear uncertai (E,A+ΔA) is still regular and impulse-free, and has all its finite eigenvalues retained within the same specified region as the nominal system (E,A) does, if both the following in ualities are simultaneously satisfied: eq 1 1 1 1 < +

∑∑

∑

= = = m i m j ij j i m i i i k k kϕ ϕ , (20a) and 1 1 1 1 < +

∑∑

∑

= = = m i m j ij j i m i i i k k kφ φ , (20b) where 0; < 0; ≥ ), ( ), ( for for i i i i i k k A J A J ⎩ ⎨ ⎧ − − = μ μ ϕ (21a) 0; < 0; ), ( ), ( for for j i j i ij ij k k k k A J A J ⎪⎩ ⎪ ⎨ ⎧ − −μ μ (21b) ≥ ij = ϕ 0; < 0; i k ≥ ), ) ( ( sup ), ) (( sup for for 1 1 i i q i q i k A A qE A A qE ⎪⎩ ⎪ ⎨ ⎧ − − − − = ₋ − μ μ φ (21c) 0; < 0; ), ) ( ( sup ), ) (( sup for for 1 1 j i j i ij q ij q ij k k k k A A qE A A qE _≥ ⎪⎩ ⎪ ⎨ ⎧ − − − − = ₋ − μ μ φ (21d )

in which J is a constant matrix part. of

1

)

(sE− A − ;q∈ and denotes the boundary Q

For the linear uncertain singular system

(19

eigenvalue cluste c

t HTGA to find the table and optimal output feedback controller

Q

of the specified region D .

in Eq. ) under the robust regional s- ring onstraint (20), we will apply both the OFA and he

s

to make the quadratic performance index in Eq. (9) be minimized. Therefore, the robust quadratic-optimal output feedback controllers design problem for linear uncertain singular system is, under the constraint (20), how to find the output feedback gain matrix K to achieve the optimal performance by

minimizing the performance index in (9). This is equivalent to the problem

min J =F(K₁₁,K₁₂,_L,K_ij) (22a) subject to ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ + ≤

∑∑

∑

m i m m ij ij k k k D K ϕ < +

∑∑

∑

= = = = = = 1 1 1 1 1 1 1 m i m j ij j i m i i i i j i i i k k kφ φ (22b) for and < 1 ij jϕ r i=1,2,_L, j =1,2,_L,3p , where real n

practical consideration, respectively. Hence, the design problem of robust

raric-optima t feedback co

design procedure can be described as

performance index in (10). continuous v

problems of the nonlinear functions with the

have shown th tter

and

ij

D are positive umbers given from the

quad l outpu ntrollers

is a constrained optimization problem, and the following:

Step 1: Check the constraint in (20) of regional eigenvalues-clustering robustness.

Step 2: Minimize the quadratic Eq. (22a) is a nonlinear function with

ariables. For the optimization continuous variables, Chou and his associates

at the HTGA may obtain be more robust results than those existing improved genetic algorithm reported in the literature (Chou and Liao et al., 1998; Tsai et al., 2004). Therefore, in this project, the HTGA is employed here to search for the quadratic-optimal output feedback gain matrix

K .

Detailed Steps: HTGA.

Step 1: Parameters setting.

Input: population size G ,

crossover rate , mutation rate nerations. (10), ac c p , and m p number of ge

Output: the value of J in and the output feedb k gain matrix K .

J in which is the fitne

Step 2: Initialization. By using the

(10), ss

incorporated with the

aforem lgebraic

function defined for the HTGA,

entioned a

algorithm in the Section 2, the fitness values of the initial population feasible for the constraint are then calculated, where the initial population with the chromosomes of the form T ij j j i K K K K K K , ,..., ,..., , ,... ] [ _{11 12 1 1 2} = θ is randomly generated, in which K (ij i=1,2,L,r and j =1,2,_L,3p ) are the elements of the gain matrix K .

ove operation. probability of at Step 3: ro Step 4: d

Selection operation using the ulette wheel approach.

Cross r The crossover is etermined by crossover r e c p . 5: Select a suitable t Step wo-level orthogonal array L_γ(2γ−1) for matrix experiments, where γ denotes the number of experimental runs, and γ −1 is

the number of col the

orthogonal array. The orthogonal arrays )(215

16

L is used in th

illustrative design example given in the next section.

6: Choose randomly two

hromosomes at a time to execute atrix ex ts.

Calculate fitness values of umns in e perimen Step c m Step 7: γ

periments in the orth ex

Step

Step

8.

ogonal array

incorporated with the c chromosom ) 2 ( γ−1 γ L by using Eq. (10) aforementioned algebrai algorithm, where we give a penalty

on the fitness value for the e violating the constraint.

8: Calculate the effects of the various factors.

9: One optimal chromosome is generated based on the results form Step

number

the expected G×pc

has been met.

The population via the Taguchi ethod is gene Step 11: rated. n is mutation rate as parents of the Step 16: 17.

Step the optimal f

(Kwakernaak

pace equatio is

m

Step 12: Mutation operation. The probability of mutatio

determined by _m

Step 13: Offspring population is generated.

p .

Step 14: Sort the fitness values in increasing order among parents and offspring populations.

Step 15: Select the better G

populations next generation.

Has the stopping criterion benn met? If yes, go to Step Otherwise, return to Step 3 and continue through Step 16.

17: Display

chromosome and the optimal itness value.

Design Example

Consider the stirred tank of Figure 1 and Sivan, 1972). The state n of the system s ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎥ ⎦ ⎢ ⎢ ⎣ − − + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎣ − ) ( ) ( ) ( ) ( 0 2 2 1 0 0 2 0 0 1 2 1 0 0 0 t u t u V c c V c c t x t x V F V ⎤ _,(23a) ⎢ ⎡− = 0 ) ( 0 F t x& ⎥ ⎡ ⎤ 1 1 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎥ ⎦ ⎤ ⎡ F ⎢ ⎢ ⎣ = ) ( ) ( 1 0 0 2 ) ( 2 1 0 0 t x t x V t y , (23b)

where is the volume of the fluid in the tank, is the concentration in the tank, are the flow rates. In this ple, we use the following numerical

, ) ( 1 t x ) ( 2 t x and u 1 0 V = ( 25 km k ) ( 1 t u exam and ) ( 2 t 2 c ≤ i ( i

the PID controller (3), where values: ₍ 3₎ m , 0.019≤F0 ≤0.0212(m3/s), ) / . 1 3 0 olm c = , 0.0094≤c₁≤1.0006(kmol/m3) ) / ( 0009 . 2 9992 .

1 ≤ kmol m3 . This results

in the system equation (23) having uncertain

parameters = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − + ⎤ ⎢ ⎣ ⎡ − − = 02 . 0 0 0 01 . 0 0 0 01 . 0 1 k A , 3 , 2 , 1 ) and controlled by ⎥ ⎦ 02 . 0 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = 01 . 0 0 0 0 0 01 . 0 0 0 75 . 0 25 . 0 1 1 3 2 k k B , ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 0 0 0 01 . 0 1 0 0 01 . 0 1 k C

[

0.05 0.06

]

1∈ − k , and

[

0.06 0.06

]

2 ∈ − k ,

[

0.08 0.09

]

3∈ − k .

The linear quadratic performance index is

∫

= J 0 [ + ) f t q uu T xx T dt t u R t u t x Q t x ( ) ( ) ( ( )] ,

∑∫

− + + 1 ) 1 ( )] ( ) ( ) ( ) ( [ q t k _{f T T} dt t u R t u t x Q t x = = 0 k kt uu xx f

∑∫

₌− + + = 1 0 ) 1 ( ) ( ] )[ ( q k t k kt uu T T T f f dt t x C K R K C Q t x ,(24) in which q =3000, t_f =0.01, Q_xx =I₂ and 2 I

R_uu = . The initial state vector is set to

[

]

T 1 . 0 01 all the un x(0)= 0. ting system . In this exam f ite eige ple, we n the assume that in lu

resul certain closed-loop singular va es of

) are located inside a

specified rectangular region ,

(E A+ΔA

{

(~+~~)−3≤~≤0,−1.5≤~≤1.5

}

= x yj x y

D , where ~j = −1.

In g, we will apply the

propose approach, which integrates the OFA and the HTGA, to find the robust

the followin

quadratic-optimal output feedback controller such that the performance index in Eq. (24) is minimized. In the OFA, the type of orthogonal function considered in this example is the shifted Chebyshev function. The evolutionary environments of the HTGA used in this example are: the population size is 30, the crossover rate is 0.9, the mutation rate is 0.5, the maximum generation is 30.

Using the proposed integrative approach with m=4 and K_ij ≤30 in which i=1,2 and j=1,2,_L,6 are the elements

atrix

of the output feedback gain m K , we can

obta

, and the robust l

,and ;(25a)

,and

in that the quadratic performance index is

791 0

=

J

quad output feedback PID gain

matrices are ⎥ ⎦ ⎤ ⎢ ⎡ − − − = 93757 . 5 500 6056 . 2 74356 . 6 P K , ⎤ ⎡− − = 4.64159 5.14905 I K , 00067 . ratic-optima ⎣ 5 ⎢ ⎣− 4 ⎢ ⎣ ⎡ − − = D K and we have (i) 3 1 3 1 +

∑∑

∑

= i j i i i kϕ fork₁∈

[

0 0.06

]

,k₂∈ 3 1 1

∑∑

∑

= i i i fork₁∈

[

0 0.06

]

, 2∈ k (x) 0.9645 1 3 1 3 1 3 1 < ≤ +

∑∑

∑

= = = i j ij j i i i i k k kφ φ ,

[

0 0.06

]

1∈ k ,

[

0 ,an

]

k₃∈

[

0 0.09

]

for k₂∈−0.06 d ;(25j) (xi) 0.9061 1, 3 1 3 1 3 1 < ≤ +

∑∑

∑

= = = i j ij j i i i i k k kφ φ 5 .

_[

_]

06 . 0 0 1∈ k , ₂∈

[

−0.060

]

⎥ ⎦ − 946377. 97923 . ⎥ ⎦ ⎤ − − 806 . 0 6831 . 0 20348 . 2 45965 . 1 1 0.0235 3 1 < ≤ = = i j ij k k ϕ ,

[

0 0.06

]

k3∈

[

0 0.09

]

(ii) 3 + kϕ 0.0247 1 3 1 < ≤ = = i j ij j ik k ϕ ,

[

−0.06 0

]

k₃∈

[

0 0.09

]

;(25b) ,and ;(25c) (iii) 3 1 3 1 +

∑

= i i i kϕ fork₁∈

[

0 0.06

]

,k₂∈ (iv) 3 1 3 1 +

∑

= i i i kϕ fork₁∈

[

0 0.06

]

, 1 0.0239 3 1 < ≤

∑∑

= = i j ij j ik k ϕ ,

[

−0.060

]

k3∈

[

−0.080

]

1 0.0227 3 1 < ≤

∑∑

= = i j ij j ik k ϕ ,

[

0 0.06

]

2∈ k 3 1 3 1 +

∑∑

∑

= i i i kϕ fork₁∈

[

−0.050

]

, 2∈ k 3 1 3 1 +

∑

= i i i kϕ fork₁∈

[

−0.050

]

,k₂∈ 3 3 1 +

∑

= i i i kϕ fork₁∈

[

−0.050

]

, 2∈ k 3 1 +

∑

= i i i kϕ fork₁∈

[

−0.05 0

]

, ,and ;(25d) ,and ; (25e) ,and ;(25f) ,and ; (25g)

[

0.08 0

]

3∈− k (v) 0.02336 1, 3 1 < ≤ = = i j ij j ik k ϕ

[

0 0.06

]

k₃∈

[

0 0.09

]

(vi) 0.0246 1, 3 1 < ≤

∑∑

= = i j ij j ik k ϕ

[

−0.06 0

]

k3∈

[

0 0.09

]

(vii) ϕ 0.0135 1, 1 3 1 < ≤

∑∑

= = i j ij j ik k

[

−0.060

]

k₃∈

[

−0.080

]

(viii) 0.0225 1, 3 1 3 1 < ≤

∑∑

= = i j ij j ik k ϕ

[

0 0.06

]

2∈ k 3 1 3 1 + = i i i kφ

[

0 0.06

]

1∈ k , k ,and ;(25h)

]

[

0.080

]

3∈− k and (ix)

∑

0.9655 1 3 1 < ≤

∑∑

= = i j ij j ik k φ , for ₂∈

[

0 0.06,andk3∈

[

0 0.09

]

; (25i)

for k ,andk₃∈

[

−0.08 0

]

;(25k) (xii) 0.9071 1, 3 1 3 1 3 1 < ≤ +

∑∑

∑

= = = i j ij j i i i i kk kφ φ

[

0 0.06

]

1∈ k , ₂∈

[

0 0.06

]

for k ,andk₃∈

[

−0.08 0

]

; (25l) (xiii) 0.9865 1, 3 1 3 1 3 1 < ≤ +

∑∑

∑

= = = i j ij j i i i i kk kφ φ

[

0.050

]

1∈− k ,

[

]

for k₂∈0 0.06,andk₃∈

[

0 0.09

]

;(25m) (xiv) 0.9858 1, 3 1 3 1 3 1 < ≤ +

∑∑

∑

= = = i j ij j i i i i k k kφ φ

[

0.050

]

1∈− k k₂∈

[

−0.060

]

for , ,andk3∈

[

0 0.09

]

;(25n) 1 0.5117 3 1 3 1 3 1 < ≤ +

∑∑

∑

= = = i j ij j i i i i k k kφ φ , (xv)

[

0.050

]

1∈− k

[

0

]

for ,k₂∈−0.06 ,andk₃∈

[

−0.08 0

]

;(25o)

(xvi) 0.9285 1, 3 1 3 1 3 1 < ≤ +

∑∑

∑

= = = i j ij j i i i i k k kφ φ

[

0.05

]

1 fork∈− 0,k₂∈

[

0 0.06

]

,andk₃∈

[

−0.080

]

;(25p) inear uncertain closed-loop sin lar tem

From the results in (25), we can conclude that

the l gu

sys (E, A+ΔA) is regular and impulse-free, and has all its finite eigenvalues retained within the same specified region D . That is, the designed linear multivariable output feedback PID uncertain control system has all its eigenvalues retained inside the same

specified region D as the lin r

multivariable output feedback PID nominal control system does. The state responses and control signals for the uncertain stirred tank

system with the designed robust

quadratic-optimal output feedback PID controller are, respectively, shown in Figures 2 and 3. From Figures 2 and 3, it can be clearly seen that the robust stabilizing property is also achieved. Hence, we could conclude that the proposed integrative method, which integrates the OFA and the HTGA, is very feasible to solve the robust quadratic-optimal output feedback PID controller for the uncertain stirred tank control systems.

份成果 發表情況如下所列： 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 的

1. S. H. Chen, W. H. Ho and J. H. Chou, “Robust Eigenvalue-Clustering Analysis of Linear Multivariable Output Feedback PID Control Systems with Uncertain Parameters”, Proc. of the 2006 Automatic

Control Conference (Paper Number:

K0008), Taiwan, R.O.C., November 2006. 2. S. H. Chen and J. H. Chou, “Regional

Eigenvalue-Clustering Robustness of Linear Multivariable Output Feedback PID Uncertain Control Systems”, Linear

Algebra and Its Applications, 2007

(submitted).

3. T. C. Wang, S. H. Chen and J. H. Chou, “Design of Robust Quadratic-Optimal PID Controllers for Linear Multivariable Uncertain Control Systems”, Proc. of the

5th Conference on Precision Machinery and Manufacturing Technology, (Paper

Number: B40) Taiwan, R.O.C., May 2007. 4. S. H. Chen, J. H. Chou and T. C. Wang,

“Design of Robust Quadratic-Optimal Controllers for Linear Multivariable Output Feedback PID Uncertain Control Systems Using Orthogonal Function Approach and Genetic Algorithm”, 2007 (submit to the International Journal).

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Figure 1. A stirred tank system.

T

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ondon.

an, W., T. Chen and H. J. Marqu Figure 2. The state responses of the uncertain

stirred tank system with the designed robust quadratic-optimal output feedback PID controller (solid line: controlled result; dash line: uncontrolled result).

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Figure 3. The control signals of the designed Example.