Evaluation of generalised quadratic cost functionals for linear systems with many commensurate time delays

C. Hwang and F.-P. Chang

Abstract: A general analytic method for the evaluation of generalised quadratic integrals of the form Ji,,?(u) = J'YB'exp (.r)(d'r(f)/dt''j2df, where p and 11 arc integers and e(f) is the error variable of a control system having multiple commensurate time delays is presented. More precisely, the authors derive an analytic solution to the problem of evaluating J,.,(a) from the equivalent frequency-domain complex integral. The method is based on using a mathematically rigorous procedure to perform partial fraction decompositions of the integrand of the complex integral. I t involves solving sets of symbolic linear equations and the factorisation of I-D polynomials. The results presented for the evaluation of complex integrals involving the delay operator e?' are of direct interest to the parametric optimisation and balanced realisation and reduction of linear time-delay systems.

1 Introduction

Due to the mathematical treat, a quadratic cost functional is commonly used as the performance measure for optimum controller designs [l-31. A typical quadratic functional is the integral of the squared error:

Here the error is defined as

where r(r) and ?( I ) arc, respectively, the reference input and actual output of a control system. To place additional constraints, such as the relativc stability and the magnitude of overshoot on the designed fcedhack control system, the class of generalised quadratic cost functionals of the form:

in which i t and I' are integers, is often used as alternative performance index.

I t follows from Parseval's theorem [4] that the cost functional ( I ) can he written in the form:

IEE, 2002 1EE Pmeedlngs online no. 20020728

DOI: 1n.1049iip-~ta:2002~728 Paper first received 12th December 2000 and in rcvised form 14th August 2002 The authors are with the Depanment of Chcmical Enginccring, National Chung Chcng University, Chia-Yi 621, Taiwan

iF.5 Pmc -Co,xtrol 7/wog Appl., b l . I I Y , No. 6, li70rt~eiber- 2002

where

is the Laplace transform of the error variable e(t), and A(s) and B(s) in (5) are polynomials in s with deg B(s) < deg A ( s ) and A(s) being Hunvitz stable. For delay-free systems, the quadratic cost functional I can he easily evaluated from the coefficients of E(s) with a recursive algorithm [5-71. In particular, Newton rf al. [6] derived the algorithm by splitting the integrand into two parts:

B(s)B(-s) F(-s) F(s ) +- (6) A(s)A(-s) - A(s) A(-s) -

where F(s) is a polynomial in s of degree, at most, one less than that of A(s). For systems having a single time delay T, the function E(s) has the general form:

B(s) + D(s)e-"

A(s) + C(s)e-" E(s) = (7)

where A($), B(s), C(s), and D(s) are finite-degree polyno- mials with real coefficients. With E(s) as above, several authors r8-121 derived a closed-form expression for I by considering the zeros of A(s) + C(s)e-", at which

and

(9 )

547

By subtracting the above t e m fomi E(-s), they obtained the expression for I as follows:

1 A(s)B(-s) - C(s)D(-s)

A(s)A(-s) - C(s)C(-s)

A(s)B(-s) - C(s)D(-s) +E(s) A(s)A(-s) - C(s)C(-s)

(10) A(s)B(-s) - C(s)D(-r)

X A(s )A( - s ) - C(s)C(-s)'

where res(&); z) denotes the residue of F(s) at its pole z, and the index k runs for all distinct right-half-plane (RHP) zeros .SA and the finite-degree polynomial A ( s ) A ( - s ) - C(s)C(-s).

Based on the partial fraction decomposition of the form:

where the notation 2? stands for X(-s) , and XP(s) and Y,(-s) are rational functions of s, several authors [13-18] have derived analytic or closed-form solutions to the problem of evaluating the complex integral (4) for systems having multiple commensurate delays. The key step of these method lies in the determination of the (2n + 2 ) unknown rational functions Xk(s), Yk(-s) , k = 0, I, . . . , I ) . According to the decomposition (1 l ) , the unknown rational functions X&) and Yk(-.?) satisfy the relation:

Expanding the above equation and equating the coefficients of like powers zk, k = -11, -11 + I,. . . , -1,O, I , . . . , n - I , n, wc can obtain a set of 2n + 1 linear symbolic equations in the unknowns Xk and Pk. I t is noted that such a set of 2n + I linear equations does not allow one to obtain a unique solution for the 2 n + 2 unknowns X,, FA, k = O , I , . . . ,n , although the solution to the complex integral (4) is unique. Scvcral approaches have been proposed to solve this non- uniqueness problem. Gu et ul. [I31 solely considered the case where B,,(.s)=O and set X,(s)=O, Y,,(-s)=O and Y P = X k , k=O, I , . . .,n - I . Walton and Marshall [I41 solved the problem by letting p~=&, k = O , 1,. . . , n, and supplementing one equation by letting X,(s) = Y!,(-s). In [I51 and [17], it is assumed that FP=xP and they proposed an ad hoc procedure to obtain Xk(-s), k=O, I , . . . , n . The procedure exploits the symmetry property of the equations derived from (12) and it is difficult to extend it to a general case. Very recently, Hwang ef al. [I81 performed the partial fraction decomposition (1 I) using the symbolic Euclid algorithm without specifying a special relation between Fk and &. However, their algorithm is difficult to implcment and may cause a numerical instability problem in the arithmetic operations of polynomials.

As for the evaluation of the generalised quadratic cost functional J,<, ,,(u) defined in (3), methods based on using the Euclid algorithm have been proposed by Talbot [19, 201 for systems without a time delay. By extending the approach of deriving (IO), Marshall el ul. 1161 and Walton ef a/. [21]

. .

548

gave an analytic expression for the evaluation of generalised quadratic functionals in the frequency domain for systems having a single delay. Based on the infinite-tcrm partial fraction expansion of E(s), Gorecki and Popek 1221 and Sularz 1231 derivcd closed-form expression for cvaluating the generalised quadratic functionals J,,,,(O) associated with single-delay systems. For systems having multiplc non- commensurate or commensurate timc delays a numerical approach has been proposed by Hwang et ul. [24] and Guo et a/. [25] to evaluate a general complex integral of the form:

The approach is bascd on applying the variable change s = j tan(O/2) to convert the above improper integral into the definite integral:

I ' %(F(; tan(0/2))G(-;tan(8/2))) dO (14)

and computing this definite integral by solving a first-order differcntial equation with zero initial condition using a numerical integration scheme capable of accuracy control by automatically adjusting the step length. It has been dcmonstrated in [24, 251 that this numerical approach is quite effective for giving accurate solutions, particularly for systems that have multiple non-commensurate time delays.

In this paper we arc concerned with the analytic solution to the problem of evaluating the generalised quadratic cost functionals that are related to linear time-invariant systems with multiple commcnsurate time delays. In particular, we focus on the derivation of a closed-form exprcssion for the complex integral (13) with

J = A 1 +cosu

and

The approach adopted in this papcr is also based on the product-to-sum decomposition of the integrand. However, it avoids the non-uniqueness problem by following a mathe- matically rigorous procedure given [18], which is used in the solution of a Diophantine equation with the Euclid algo- rithm. Instead of using the symbolic Euclid algorithm, which is difficult to implement and may cause numerical instability problems .in the arithmetic operations of poly- nomials, we perform the decomposition by forming a set of n+ni symbolic linear equations for exactly the same number of unknowns, which are in gcneral rational func- tions of s. In actual computation, accurate solution to such a set of symbolic linear equations can be effectively obtained either using a mathematical software capable of symbolic processing, such as Mathematica [26] and Mapplc [27]_ or using a DFT interpolation scheme 1281. To evaluate the generalised quadratic cost functional J,Jo), it is shown that a sequential decomposition scheme can be used so as to avoid the problem of solving a large system of symbolic equations. In addition to the application in parametric optimisation of controllers for time-delay systems, the results presented in this paper are also of direct interest to the balanced realisation and reduction of linear time-delay systems. To demonstrate the effectiveness of the presented

IEE Proc.~Contrd Theory A p p l , Cbi 14Y. iVo. 6, Vowmbw 2002

results for evaluating generalised quadratic cost functionals for linear time-delay systems, a nunierical example with the system having two commensurate time delays is provided.

2 Partial fraction decomposition

Owing to its key role in the evaluation of the complex integrals arising in linear time-delay systems and two- dimensional digital systems, the partial fraction decomposi- tion of the product of two two-variable polynomials is the main objective of this Section. More precisely, we present an etf'ective and niatheinatically rigorous procedure to perform the following partial fraction decomposition:

where the two-variable A(s, z), 5(s, 2). C(s, z ) and D(.Y, z ) are dctincd in (15) and (16) with z=c'-? It is noted that due to the lack of a general theory for the existence of a two- variable polynomial factorisation, the factor coefficients X,(s) and Y,(s) are, in general, rational functions of s rather than polynomials in s. As indicated in the proceeding Section, the direct manipulation of polynomials in z allows one to obtain only n +nr + I equations for the n + nr + 2 unknowns: Xj(.s), i = 0, I,. . . , n ; Yk(-s), k = 0, 1,. . . , m. As a result, the non-uniqueness problem arises in the determi- nation of the decomposition (17).

To avoid this non-uniqueness problem, we perform first the following decompositions:

Bk(s) = Bk(s)A, , ( s ) - Ak(s)B,,(s) (18) D(-s, z - ' ) Du(-.s) ~ ~ = , b k ( - s ) z - ' -~ - C(-s.z-l) Co(-s) + C,(-s)C(-s,z-l)

Dk(-s) = Da(-s)Cu(-.s) - Ck(-s)D,,(-.s) (19)

and then consider' the decomposition:

r;:;Bk(s)$ X E;=, b k ( 5 ) Z - k

AG, z) C(-s. 2-1)

In the above expansion the unknowns &s) and kk(-.s) are rational functions and satisfy the following relation:

This is a Diophantine equation whose solution is unique [29] . To obtain its solution, we expand the products of the polynomials in z and equate the coefficients of like powers zk for k = 0, I , . . . , n +ni - I . This procedure allows us to

IEE Proc.-Co,~lroI Theuq Appl., l 'oi i4Y. No. 6. Nowmhw 2002

where the symbols A! and C, denote A,($) and C,,,(.s), respectively, and

(23) I = 0. I . . . . , n + i n - 1

The set of linear equations with literal coefficients in (22) can he solved with symbolic manipulation software such as Mathematica or Maple, or numerically solved by an efficient I-D DFT interpolation algorithm [28].

After solving the set of symbo!ic linear equations (22) for Xk(s), k = O , I ,..., n - 1 and Y/(-s), / = 1,2 ,..., m, we obtain the partial fraction decomposition (19). Now, substituting (18) and (19) into the left-hand side of (17), and using (20), we obtain:

5(x, 2 ) D(-s, 2-1)

A(s, 2) C(-s. 2-1) -x

Comparing this decomposition with (17), we have:

549

It is noted that in the gouping of terms in (24), we have partitioned the term (5,,Co)/(A,,&J into two equal parts and distributed them into two groups. Since this partition is arbitrary, it become clear that the non-uniqueness of the decomposition ( I 7) $rises from the non-unique partition of the term (B,,Co)/(A,Do).

3 Evaluation of generalised quadratic cost functionals

In this Section we consider the evaluation of the generalised quadratic cost functional Ji,,Ju). For the existence of the integral JI,,Ju) defined in (3) , we assume that the following conditions are satisfied: (i) all poles of E(s) and E(s - u) lie in the open left-half of the s-plane; (ii) the Laplace transform of eiV1(f) is strictly propcr; and (iii) lim,- ,e"e'"(r)=O for

Before presenting the main results, it is helpful to recall the following three Laplace transform properties, which will be used in the subsequent derivations:

k = O , 1 ,..., 1'.

d"E(s) L(rI'e(r)) = (-1)l'-

dsi' L(e"'e(t)) = E(s - 0) (29)

For easy presentation, we consider in the following evalua- tion of JP,$,(u) for [ I =0, 1,2. As will be seen, the extension to the case where 1' > 2 is quite strai htfonvard.

JILv(u) can be written as Let ,f(f) = d"""'l(r) and g(r) = /'e c I 4 ( r ) . Then the integral

cc J , , , X ~ ) = Jo f(r)r(r)dt

The use of Parseval's theorem leads to the following equivalcnt expression:

.J,,,v(u) = - F(s)G(-s)ds (30) 2rrj Ja -ice

where F(s) and G(s) are the Laplace transforms offit) and g(r), respectively. In order to avoid the use of excessive notation. we represent the Laplace transform of e("'(r), denoted by E,,(s), as

Do(s) + DI(s)e-" + . . . + D,,(s)e-"" ~ a D(s, z)

Co(s) + C , ( S ) ~ - ~ > + . . . + C,,(s)e-nZs C(s, z ) E,(s) = -

(31)

Then, we have:

(33)

In particular, we have:

(34)

For p = 0, the product F(s)G(-s) can he decomposed as

where X(s , z) and Y(s, z) are polynomials in s and z. With this decomposition, the integral Jll,v(~) can be expressed as

It is noted that forthe existence ofthe integral Jl<, , , (~) , we have assumed Iimt-=J(f) = lim,- me"e""(r) = 0 and lim,+ , g(r)= hi+ ,f"dv1(t) = 0. This implies that the integrands in (36) vanish as s on the circle of infinity radius. Since the quasi-polynomial A(s, e-'") has no zeros on the right-half of the s-plane and C(-s, e") has no zeros on the left-half of the s-plane, we close the contour of integration for the first integral in the last equation of (36) with a semi-circle of infinite radius so that the closed contour of integration encloses no zcros ofA(s, e-'"). With the same argument, we close the contour of integration or the second integral in (36) with a semi-circle of infinitc radius so that thc closed contour of integration does not enclose any zeros of the quasi- polynomial C(-s, e'"). Using closed contours of integration rR and r,. shown in Fig. 1, and Cauchy's residue theorem, we can obtain the following expression for the integral Jo,,(u):

Y(-s, e") ,

A ( ~ ) c ( - s . e")' + 23 res (37)

where s,:* and sLk denote the roots of the polynomial A(s) that lie in the regions enclosed by r, and rL, respectively.

Now, let us consider the evaluation of JI,.(u). Using the middle equation in (34), we can expand the product F(s)G(-s) as -

B(s, i) D(')(-s. z-') B(s. z) F(s)G(-s) = -- +- A ( s , z ) C(-s.z- I ) A(s.2)

550

For the evaluation of J2,,(c), we perform the following partial fraction decompositions sequentially:

Using the procedure described in the previous Section, we can obtain the following decompositions:

B(s, z ) D(' ) ( - s . z - I ) __ A(s . 2 ) C(-s. z-1)

(39) - X,(s,z) Y,(-s,z-l) -

A(sjA(s, z ) + A(s)C(-s. z-I) X(s. 3) d l ' ( - s . z-1) A(s, 2) '~ C(-s. z-1)

(40) - X2(s.z) Y2(-s.z-Ij -

A(.s)A(s. ?) + A(s)C(-s, 2-1)

Substituting the decompositions (35), (39) and (40) into (38), we obtain:

Y2(-s,z-I) + X,(% 4 + A'(s)A(s. z ) A2(.s)C(-s, z-I)

With the above decomposition and the contours of integra- tion shown in Fig. l , the integral J,.,,(G) can be evaluated as

where sp,k and sLk denote the zeros of A(s) inside the regions enclosed by rR and rL, respectively.

/EE P!r,c.-Grii,ol Theor?, Appl.. 161. 149, No. 6. r\'oi"&er 2002

where A stands for A(s), and for other symbols, X and P stand for X(s, e-'") and Y(-.s, e'"), respectively.

It follows from the last equation in (34) and the above decompositions that the product F(s)G(-s) can be decom- posed as

x, i; Y,C( ' j XI, PI, F(.s)G(-s) = -+- - 2- - 2- - 2-

M A C AC2 A ~ A A2?

Using thc above decomposition and closing the contours as shown in Fig. 1, we can apply the Cauchy's residue theorem to obtain the integral J2.,((i).

4 An illustrative example

To demonstrate thc effectivcness of the proposed proccdure to evaluate the generalised quadratic performance index, we consider the evaluation of the integrals J0,0(-0.2) and Jl,,(-0.2) associated with the error variable e(t) whose Laplace transform is givcn by

k(k - l)e-2rs - (k2 - 4)(l + rsje-"

wherez=e-",k=I.I,r=I,and T=0.2.Itcanheverified that E(s) is analytic for 9tjsf > 0.

For / I = 0, 1 = 0 and (i T -0.2, we have:

where

A(s. z ) = -4[1 + 7(s - n)I2 + 0 x z + k2eZrUz2

B(s. z ) = (k - 4)[1 + 7(s - 0)12 - (kz - 4)[1 + 7(s - (i)] x eToz + k(k - I)eZr'"2

C(s. z ) = -4( 1 + rs)' + 0(z + k2)2 o(S, 2) = ( k - 4)( 1 + T.S)* - (k' - 4)( 1 + 7S)Z + k ( k - I)Z2

55 I

Following the procedure described in Section 2, we obtain thc product-to-sum decomposition:

I B(.s,z)D(-s.z-') F(s)G(-.s) = -____

s(s - u)A(s, 2 ) C(-s. z - ' ) = -[ X,(s) +X,(sjz+,Y*(.s)z~

A(s)A(r. 3)

A(s)C(-s, z-I) 1 + Y,(-s) + Y1(-s)2-' + Yz(-s)i-2

where

X&) = -7.2732 - 10.221s+7.5064s2 + 15.834s'

+ 1 . 7 9 0 9 ~ ~ - 5.5 179s' - 1.9707.~~

Xl(s) =4.0449+2.1328s- 7 . 1 2 8 3 ~ ~ -4.1477s'

+ 3. 1095s' + 1.9434s'

X,(s) = 0.04990 - 0.0767s - 0 . 0 8 6 8 7 ~ ~ + 0.014724.~'

+ 0 . 0 3 6 8 1 ~ ~

Y,(-s) = -0.17868 + 0.42064s + 0 . 0 0 5 8 6 ~ ~ - 0.61 164s'

+0.28473s4 +0.21091s' - 0 . 1 3 1 8 2 ~ ~

Yl(-s) = 3.5083 - 4.5821s - 4 .2141 .~~ + 6.0969.~~ + 1 . 2 1 3 7 ~ ~ - 2 . 0 2 2 8 ~ ~

Y,(-s) = -1.4198 +0.36098s+ 1.7843s' - 0.2066s'

- 0.5 I 638.s4

A(s) = s(s + 0.2)( 1.3555 - 0.48s - 2 . 3 6 ~ ~ + 0.4s' + s4)

= s(s - 0.85883)(s - I . 125)(s + 0.2)(.s + 1.0588) x (s + 1.3250)

q s - SI)(S - s&s - S3)(' - s4)(s - s5)(s - S 6 )

With the contour of integration shown in Fig. 1, the integral J0,0(-0.2) is evaluated as

J0,,(-0.2) = res

= 0.3827478712

The value of the integral J0,,(-0.2) computed from the definite integral (15) by solving a first-order differential equation is given by 0.3827478914. This verifies the corrcctness of the computed result.

Now we consider the evaluation of the integral ./l,0(-0.2). For this case, F(.Y) is the same as above and

dE(s) G(s) = - ~

d.S I D"'(s. z ) I D(s, z) C'')(.s. z)

-~ +-x-- - - x- s C(s, 2) s ? C(s. 2) C(s. 2)

where

D("(S, 7.) = 2 T ( k - 4)(1 + Tis) + [7(k2 - 4)(l + TS) - ~ ( k ' - 4)]z - 2Tk(k - 1)z2

The partial-fraction decomposition of F(.s)G(-s) is obtained as follows:

1 I D(-.s. z - ' ) C")(-s. 2 - ' ) +- .s2 C(-S, 2-1) C(-s. z-I)

X2(s. z) Y*(-s.z-I) + Az(s)A(.s. z) + A2(s)C(-s, z-1)

+ Y(-s, 2-')C(lI(-.s, 2-1)

A;(sj[C(-s. z-')I2

where AI(s) = - ( s - Oj(.s - I)(s - 0.85883)(s - l.125)

x (s + 0.2)(s + I .0588)(s + 1.325)

A2(s) = -(s - O)2(s - I)(s - 0.85883)'

x (s - I .125)'(s + 0.2)(s + I .0588)2

x (s + I .325)2

A,(s) = (r - O)'(s - 0.85883)(s - 1.125) x (s + 0.2)(s + I .0588)(s + 1.325)

X , ( s , z ) =X,,(s) +x,.,(s)z+Xl.2(s)2 Xl,"(s)= -10.737- 15.375s+ 12.4001s2+26.112s3

+ 2 . 3 1 0 4 ~ ~ - 1 0 . 6 6 2 ~ ~ - 3.9414s'

X , , , ( s ) = 7.2955 + 3.8783s - 1 3 . 3 4 ~ ~ - 7.8409s'

+ 6 . 1 2 0 2 ~ ~ + 3.8869s'

Xl,,(s) = 0.09979 - 0.03534s - 0.17374.~~

+ 0.02945,s' + 0.07362,s'

Y1(-s.2-I) = Yl,"(-s) + Y','(-s)z-~ + Yl .2 ( - s ) i -2

Yl.,(-s) = -0.35737 + 0.84128,s + 0.01 173s'

- 1.2233s' + 0.56945.~~ + 0.42182s'

- 0.26364s'

YI,~(-.S) = 3.7594 - 4.2343s - 4 .7376 .~~ + 4.9345s' + 2 . 0 5 7 8 ~ ~ - 1.3755s' - 0 . 4 0 4 5 5 ~ ~

YI,>(-s) = -2.2987 + 0.58795s + 3.3488.~~

- 0 . 4 5 1 8 3 ~ ~ - l.2114s4+0.0319s'

X2(s . z ) =X2,,(s) +X2,,(s)z+X2,z(sJ2

A'',&) = -9.6846 + 20.279,s + 63.934,s'

-58.168s'- 15I.19$+48.549s5

+ 165.518 +2.2004s7 - S4.963.~'

- 1 8 . 5 6 2 ~ ~ + 16.381.~'" + 5.7143s"

X , , ( s ) = 5.4830 - 16.320s - 2 3 . 3 7 6 ~ ~ + 56.598s'

+ 4 5 . 3 7 5 ~ ~ - 74.339s' - 47.983s'

+43.751s7 +26.358s8 -9.7172s'

- 5 . 8 3 0 3 ~ ~ "

X2.2(s) = 0.03382 - 0.125402s - 0.04166.~~ + 0.40220s' - 0 .03963~~ - 0.48842s'

+ 0 . 0 7 3 3 2 ~ ~ + 0.26650s' - 0 . 0 2 5 7 7 ~ ~

Y,(-.T.z-l) = Y2,"(-.T) + Y2,i(-s)z-l + Y2,2(-s)z-z

YZ,"(-s) = -0.121 1 I + 0.69129s - 0 . 8 7 0 0 7 ~ ~

- 1 . 2 8 9 6 ~ ~ +3.1718s4 + 0.02492s'

-3.6188s6+ 1.3199s7+ 1 . 7 3 8 5 ~ ~

- 0.941 19s9 - 0 . 3 0 3 1 9 ~ ~ ~ f 0 . 1 9 7 7 3 ~ ' ~

Y2.1(-s) = -2.9366s t 4 . 8 1 4 3 ~ ~ +4.6145s3

- 9 . 7 7 6 7 ~ ~ - 1.6235s' + 6.6327s'

- 0.45484s' - 1.50450~~ + 0 . 2 3 5 1 6 ~ ~

Y2,z(-~) = 0.03663 + 5.1335s - 4 . 0 5 9 4 ~ ~ - 1 4 . 0 3 3 ~ ~ + 9 . 2 7 7 0 ~ ~ + 14.567s'

- 7 . 5 1 5 2 ~ ~ - 6.778s' + 2.4302s'

+ 1.1801" - 0.23845s"

With the above decomposition and contour of integration shown in Fig. I , the integral JI,,)(-0.2) is then evaluatcd to he

J1,0(-0.2) = 0.2603313273

The value of the integral Jl,"(-0.2) computed from the definite integral (15) by solving a first-order differential equation is given by 0.2603312272. This also verifies the correctness of the computed result.

5 Conclusions

In this paper, an effective procedure has been proposed for the evaluation of gencralised integral performance indices for linear systems containing many commensurate delays. It is based on evaluating the equivalent complex integrals using Cauchy's residue theorem. The key step to applying Cauchy's residue theorem to evaluate complex integrals lies in the decomposition ofthe integrand into stable and unstable parts. A sequential product-to-sum decomposition has also been presented which avoids the solution of a large system of linear equations with literal coefticients. The effectiveness of the proposed procedure has been verified by numerical computations. It is believed that the proposed procedure has its great applications in parametric optimisation of controllers and balanced model reduction for linear time- invariant systems having many cominensuratc time delays.

6 Acknowledgment

This work was supported by the National Science Council of the Republic of China under Grants NSC 89-2214-E-194- 014 and 89-2214-E-194-018.

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