ANALYSIS OF TIME-VARYING SCALED SYSTEMS VIA GENERAL ORTHOGONAL POLYNOMIALS

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-32, NO. 7, JULY 1987 645 CoroI/ary 4.3: Assume (4.1), (4.8), and (4.13) are satisfied and

suppose K l and a, solve the nonzero set point problem with K l E S

;.

Then there exist n X n

Q,

P 2 0 such that

K , = - R F 1 6 , (4.14)

cul=R;’Bfa;rPy-R;’[Bfa;7(L7Ro-6TR;1RO:)-RO:]6 (4.15) and such that Q and P satisfy

O=(A-B1R;’@ O = A T P

Finally, setting

y=O,

we obtain the result of r21.

+ P A + R o - 6 r R , 6 . (4.17)

Coro1Iut-y 4.4: Assume (4.1), (4.8), (4.13), and (4.18) are satisfied and suppose K l and a, solve the nonzero set point problem with Kl E S Then there exists n x n P 2 0 such that

K , = -R;lB:P, (4.19)

( Y I = -R;’B;A,‘Ro6 (4.20) and such that P satisfies

O=ATP+PA+Rn-PCP. (4.21)

REFERENCES

K. Kwakernaak and R. Sivan, Linear Optimal Control Systems. New York: Wiley, 1972.

Z. Artstein and A. Leiarowitz, “Tracking periodic signals with the overtaking

criterion,” IEEE Trans. Aufomat. Contr., vol. AC-30, pp. 1123-1126, 1985. W. S. Levine and M. Athans, “On the determination of the optimal constant output feedback gains for linear multivariable systems,’’ IEEE Trans. Automat. Contr.. vol. AC-15, pp. 4 4 4 8 , 1970.

1. M a n i c , “On stabilization and optimization by output feedback,” in Proc. 12th Asilomar Conf. Circ., Sysf. Comp., Pacific Grove, CA, Nov. 1978, pp. 412416.

J . V. Medanic, “Asymptotic properties of dynamic controllers designed by projective controls.” in Proc. 24th IEEE Conf. Decision Contr., Fort Lauderdale. FL, Dec. 1985.

D. S. Bemstein, “The optimal projection equations for static and dynamic output S . Basuthakur and C. H. Knapp, “Optimal constant controllers for stochastic feedback: The singular case,” IEEE Trans. Automat. Contr., to be published. linear systems,” IEEE Trans. Automat. Confr., vol. AC-20, pp. 664-666,

1975.

W. M. Wonham, “Optimal stationaty control of a linear system with state- dependent noise,” SIAM J. Contr., vol. 5 , pp. 486-500, 1967.

D. Kleinman, “Optimal stationary control of linear systems with control- dependent noise,” IEEE Trans. Automat. Contr., vol. AC-14, pp. 673-677, 1969.

P. McLane, “Optimal stochastic control of linear systems with state and control- dependent disturbances,” IEEE Trans. Automat. Confr., vol. AC-16, pp. 793- 798, 1971.

J. Bismut, “Linearquadratic optimal stochastic control with random coeffi- cients,” SIAM J. Contr., vol. 14, pp. 419444, 1976.

D. S. Bernstein and D. C. Hyland, “The optimal projectiodmaximum entropy approach to designing low-order, robust controllers for flexible strucmres,” in Proc. 24th IEEE Conf. Decision Contr., Fort Lauderdale, FL, Dec. 1985, pp. 745-752.

D. S. Bernstein and S. W. Greeley, “Robust controller synthesis using the maximum entropy design equations,” IEEE Trans. Automat. Contr., vol. AC- 31, pp. 362-364. 1986.

spectives,” in Proc. Amer. Contr. Conf., Seattle, WA, June 1986, pp. 1818- --,“Robust output-feedback stabilization: Deterministic and stochastic per-

1821.

L. Arnold, Stochastic Differential Equations: Theory and Applications. New York: Wiley, 1974.

D. C. Hyland and D. S. Bernstein, “The optimal projection equations for fixed- order dynamic compensation,” IEEE Trans. Automat. Contr., vol. AC-29, pp. 1034-1037. 1984.

Analysis of Time-Varying Scaled Systems Via General

Orthogonal Polynomials

TSU TIAN LEE AND

YIH

FONG CHANG

Abstruct-General orthogonal polynomials are introduced to analyze and approximate the solution of a class of scaled systems. Using the operational matrix of integration, together with the Operational matrix of linear transformation, the dynamical equation of a scaled system is reduced to a set of simultaneous linear algebraic equations. The coefficient vectors of the general orthogonal polynomials can be deter- mined recursively by the derived algorithm. An illustrative example is given to demonstrate the validity and applicability of the orthogonal polynomial approximations.

I. INTRODUCTION

An investigation of the dynamics of an overhead current collection mechanism for an electric locomotive by Ockendon and Taylor [I21 revealed that under certain conditions, the dynamics of the systems is characterized by a differential equation containing terms with a scaled argument of the form

d ( t ) = A X ( A t ) + B X ( t )

X ( 0 )

=xo

where X ( X t ) and X ( t ) are n-vectors and A and B are n X n matrices and the constant 0

<

X

<

1. This type of differential equation also plays an important role in several chemical processes [3], [13]. This equation was first studied by Fox et al. [ 1 11 with the intmduction of a finite difference method for 0

<

h

<

1. Recently, the solution of such a scaled system has been obtained by several different orthogonal functions, such as block- pulse functions [14], [2], [3], Walsh functions [l], delayed unit step

functions [4], Laguerre polynomials [5], Chebyshev polynomials [6], [7], and Legender polynomials [ 151. The common approach of these methods is the use of the operational matrix of integration together with the operational matrix of scaling to reduce the differential equation to a set of linear algebraic equations, which is more suitable for computer program- ming.

In this note we will employ the operational matrix of integration and product operational matrix of the general orthogonal polynomials, together with the operational matrix of linear transformation, which will be derived later, to obtain the solution of the scaled system. The operational matrix of linear transformation is derived based on the following properties, namely, the pure recurrence relation

~ l + ~ ( ~ ) = ( U i z + b i ) ~ i ( z ) - ~ ; ~ , - ~ ( ~ ) (1) with

&i(z)=1; ~ 1 ( z ) = w + b o and the differential recurrence relation

~ i ( z ) = A ; i ; , , ( z ) + B , ~ ; ( Z ) + C ; ~ ; - I ( Z ) (2) where recurrence coefficients a,, bi, ci and differential recurrence coefficients A;,

Bi,

and Ci, are specified by the particular orthogonal polynomials under consideration and some are listed in [9]. The aim of this paper is twofold: 1) to derive an operational matrix of linear transformation for general orthogonal polynomials so that the scaled Manuscript received December 6, 1985; revised April 14, 1986 and September 9, T. T. Lee is with the Department of Elecaical Engineering, University of Kentucky, Y. F. Chang is with the Institute of Electronics, National Chiao Tung University, IEEE Log Number 8714806.

1986.

Lexington, KY 405064M46. Taiwan, Republic of China.

646

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-32, NO. 7, JULY 1987 matrices derived by Hwang [ 4 ] , Hwang and Shih [ 5 ] , Chou and Horng

[7], and Shih and Kung [15] can be obtained by the derived operational matrix as its special case; and 2) to present a general solution of scaled

systems, whether it is time-varying or time-invariant, via orthogonal polynomials. This solution can certainly be reduced to different polynomi-

als approximation solutions of the specific problem, including Che- byshev, Jacobi, Legender, ultraspherical, and any other orthogonal polynomials that possess recurrence relations (1) and (2).

II. GENERAL ORTHOGONAL POLYNOMIALS ON FINITE INTERVALS

The orthogonal polynomials &(z) with respect to the weight function w ( z ) over the interval a 5 z 5 b are defined as of degree precisely i in z and satisfy the condition [9]

and the recurrence relation (1). The general shifted orthogonal polynomi- als may be obtained by letting z = p t

+

q which transform domain [ a , b ] into domain [ a ' , b ' ] , where b'

>

a ' , a' and b' are both finite, p = (a

- b )/ ( a ' - b ' ) and q = (a'b - ab' )/ (a' - b ' ) . Thus, the shifted general orthogonal polynomial becomes

d;=,(t)=[al*r+b,*]+,*(r)-c:~:-,*(t) (4)

wheri

b,?+

= sip*, b: = b;

+

aiq, c: = ci, for i = 0 , 1 , * * a , with

4g(t)

= 1; 4 : ( t ) = agt

+

b ; . The new polynomials + : ( t ) with the recurrence relation (4) are orthogonal with respect to weight function w*(t) = w ( p t

+

q ) over the interval [ a ' , b ' ] . It has been shown that if C ( T ) is an n X r matrix time function, C ( t ) can be expanded by general orthogonal polynomials as [8]

where

Ci

is an n X r coefficient matrix

is called the general orthogonal polynomial matrix, C', an n X mr matrix, is called the general orthogonal coefficients matrix, Ir, a r X r identity matrix, IXI denotes a Kronecker product, and ~ ( t ) is an r X 1 time function,

u(t) =

v#*(r)

then

C(t)u(t)=[CoUC,U

...

C,-,U]L+*(t) (5) whereL = [i,,o.&,l - * - : Z , , , . . 1 & , , - 1 , 1

. - *

lm-l,m-l]Tiscalled

the operational matrix of product, li,j are the expansion coefficients

vectorsoftheproductof4,?(I)andb;(f),fori,j = 0 , 1 ,

* . * , m

- 1. Furthermore, it has also been shown that [SI

w h e r e D i f o r i = 1 , 2 ; . . , m - lcanbecalculatedfrom

I&= - A , ~ ~ + l ( 0 ) - B i + ~ ( O ) - C , 9 ~ l ( O ) . (8)

m.

OPERATTONAL MATRIX OF LINEAR TRANSFORMATION Define

that still satisfies the recurrence relation

for any ts even outside the interval [a I , b

'1

Substituting (9) into (lo), we have

"+I " - 1

d , - l , i + : C t ) = c d n . l I a n ( a r - P ) + b ~ 1 9 ) ( t ) - ~ cndn-l,,6:(r). (11)

,=o i = O , = O

Since the first term of the right-hand side of (1 l), after simple manipulations, can be expanded as

Substituting (12) into (1 1) and equating the like coefficients of the general

orthogonalplynomials{4F(t)},i= 0, 1;*.,n

+

1,wecanobtainthe

recurrence relation of the form

f o r i = 0 , 1 , '-0; n = 0, 1 ,

. - - ,

with do,o = 1 ; dl,o = bo(l - a ) - a&; d l , I = a ; d,,,; = 0 (i

>

n or i

<

0).

Thus, we derive a matrix termed as the operational matrix of linear transformation T to relate general orthogonal polynomials to their transformed forms as

and T is of the form

For

m

terms approximation, P is of the form

When a = h

>

0 and

0

= 0, then Tbecomes the scaled matrix

S,

i.e.,

p = - 1 P

-

B,-q A0 0 0 * * * C,+D, BI A1 0

...

4

C2 A2 B2

...

Dm-2 0 0 0 0.. Dm-l 0 0 0

-

The scaled matrices proposed by Hwang [6], Hwang and Shih [ 5 ] , Chou and Horng 171, and Shih and Kung [ 151, are the special cases of the present operational matrix. This operational matrix, together with the operational matrix of integration and product operation matrix, plays an

IEEE TRANSACTIONS OK AUTOMATIC CONTROL, VOL. AC-32, NO. 7, N L Y 1987

647

important role in reducing the scaled systems to a set of algebraic

equations.

IV. ANALYSIS OF SCALED SYSTEMS Consider the time-varying scaled system

i ( t ) = A ( t ) x ( h t ) + B ( t ) x ( t )

+

c ( t ) u ( t ) (17)

with x(0) is given where x ( t ) is an n X I state vector, u ( t ) is a r X

I

input vector, and A ( t ) , B ( t ) , and C ( t ) are time-varying matrices of appropriate dimensions.

Integrating (17) from t' = 0 to t' = t , we obtain

- 1

x ( t ) - x ( O ) =

1

A ( t ' ) x ( h r ' ) d t ' +

Ir

B ( f ' ) x ( f ' ) d t ' +

If

C ( r ' ) u ( t ' ) dr'.

- 0 0 .

(18)

Expanding x ( t ) , u ( t ) , A ( t ) , B ( t ) , and C ( t ) by general orthogonal polynomials, we have x ( f ) = [ x , x,

...

x,- I]$*(t) =X$*@) (19) u ( t ) = [ u o UI

...

u,-,]$*(t).=

u$*(t)

(20) A ( t ) = [ A o A I

...

A,-1lA(t) (21) B ( ~ ) = [ B o BI

...

Bm-~l$n(t) (22)

c(t)=IG

CI

...

C m - ~ l $ A t ) * (23)

Applying (16), x ( X t ) can be expanded as

x(Xt)=XS$*(t). (24)

Notice that initial condition x(0) can be expanded into

x(0) = [x(O) 0

. .

.O]+*(t) =X0$*(t). (25)

Substituting (19)-(25) into (18) and using (5) and ( 6 ) , we obtain X$*(t)-X&*(t)=[AoXS AIKS Am-&S]LP $*(t )+[B& BI X

...

Bm-lX]LP$*(t)+[CoU CIU

...

C,-JJ]LP$*(t). (26) Equating the coefficient matrices of general orthogonal polynomials

vector yields

X-Xo=[AoXS AlXS

...

Am_lXS]LP+[Br,X BIX

...

B m_l X]LP +[CoU ClLJ

...

Cm_lU]LP. (27)

LettingL = [LiLT

...

L:_,]', whereL; = [li,&

...

li,m-l]T, f o r i = 0, 1,

...,

rn - 1, then (27) becomes

m - I m - I m - I

X-Xo= A,XSL,P+ B,XLiP+ C,ULiP. (28)

, = O , = O i = O

Letting Wi = [ w , , w ; ~

...

wi,,-,] = C;UL;P, X = [xgI e*- x,,-l], X,,

= [x(O)0

. . .

01 and defining2 = [x,'.;

. . .

xi-,]',

4

= [xT(0)0'

...

07'

and G, = [w;wi7;

.. .

w ~ - ~ ] ' , then (28) can be solved by P={I,,- [ A , E (SLiP)'+B; E4 (L ,P )']} -'[&+ tt i]. (29)

" - 1 m - I

,=O , = O

Note that if A ( t ) , B(r) are constant matrices, and C ( t ) = 0, which is the case considered by Hwang [5], the result of Hwang can be obtained by substituting C ( t ) = 0 into (29). If B ( t ) = 0, and A ( t ) and C ( t ) are constant matrices, which is the case considered by Chen [2] and Chou and Horng [7], the approximate solutions can be obtained by substituting B ( t ) = 0 into (29) which then will yield the results of Chou and Horng [7] and Chen [2]. If A ( t ) , B(t), and C ( t ) are all constant matrices, which is the

TABLE I

THE APPROXIMATION SOLUTION OF ~ ( t ) FOR DIFFERENT ORTHOGONAL POLYNOMIAL EXPANSIONS

6'dt) Q ' N d.41) d*&) O',IC) e'&) .91157 .08W5 42407 -.KO75 ,03077 -.W ,81817 ,08018 -.01299 -Ow88 . W 1 -."oo ,78681 ,31026 4 7 S O Q -.WE99 .WmI -.@X04 .8034i ,15863 -.03805 -.wS17 .00154 - . w w Z ,79108 31447 -.OQ974 -.01115 ,00557 -.woo8 ,58074 33883 ,18821 ,092M .08M)5 w 8 7 9 .78l80 - 10228 -.E565 .X0435 - W 2 1 7 .w230

All Jarobi 1-c polynomids areshilled to domaio [ I , 1:.

i.e., z = -2t + 1

T A B L E I1

COMPARISON OF THE DIFFERENT POLYNOMIALS APPROXIMATION FOR

xu)

t Jacob, Lltra Chebyshev Chebyshev Lcgendrc L a g u m Hermice Runge Im=6I h = 8 I 1st [m-6) Znd[m=6) ( m d ) [ m d ) ( m 4 ) Kutta 0 0 0.2

I

:::

1 m 9 ,97752 .go225 ,77032

]Ow39 I.owO6 l . W 2 I.wO14 1.22448 ,86885 I.WM0

,97752 m 7 5 1 ,37752 97752 1 . ~ 3 5 4 .ea338 37751 .7iO32 ,77026 ,77028 77028 .R4398 .78523 .77@3l 90228 ,80224 .Bo226 .M?5 ,81085 ,81433 .90226

0.8 ,59203 59209 ,59214 ,59210 59211 50061 ,64591 .59209

1.0 39484 383E .39380 39378 39388 37884 ,17338 ,38356

case considered by Shih and Kung [ 151, the approximate solutions of Shih and Kung can be obtained by substituting A ( t ) = A , B ( t ) = B , and

C

( t )

=

C

into (29).

V. ILLUSTRATIVE EXAMPLE

Example I : Consider the scaled system

i ( t ) = - tx(0.8t) - t * x ( t ) x(0) = 1

The expansion coefficients of x ( t ) for rn = 6 and tf = 1 for different orthogonal polynomials are given in Table I. Some classical orthogonal polynomials approximation of x ( t ) for rn = 6 and tf = 1, together with the solution obtained by the Runge-Kutta method is shown in Table II. It is clear that, in general, the agreement is very satisfactory. In particular, the Jacobi type polynomials approximation converges faster than the others. Noted the poor quality of results obtained via either the Laguerre or Hermite polynomial. This is due to the fact that the zeros of the Laguerre polynomial and the Hermite polynomial are widely spread over the interval of [O, a] and [ - a,

001,

respectively. Hence, in general, these two polynomials require more t e r n than the Jacobi type polynomi- als in order to yield similar results as that of Jacobi type polynomials within a small interval. In this example, rn = 6 is not large enough for these two polynomials.

VI. CONCLUSIONS

The operational matrix of linear transformation for general orthogonal polynomials is first introduced, and a systematic method is presented to analyze a class of time-varying scaled systems. The operational matrix of linear transformation, together with the operational matrix of integration, are applied to reduce the differential equation to a set of linear algebraic equations which is very convenient for digital computation. Illustrative example shows that only a small number of terms are required to obtain accurate approximations. Moreover, in general, the Jacobi type ortho- gonal polynomials solution converges faster than the Hermite polynomials solution and the Laguerre polynomials solution.

REFERENCES

[ l ] G . P. Rao and K. R. Palanisamy, "Walsh strech matrices and functional differential equation," IEEE Trans. Automar. Contr., vol. AC-27, no. 1, pp.

648 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-32, NO. 7, JULY 1987

1141

W. L. Chen, “Block pulse series analysis of scaled systems,” Int. J. Syst. Sei.,

vol. 12, no. 7. pp. 885-891, 1981.

C. Hwang and Y. P. Shih. ‘‘Solution of population balance equations via block pulse functions,” Chem. Eng. J., vol. 25, pp. 3 9 4 5 , 1982.

C. Hwang, “Solution of a functional differential equation via delayed unit step functions,” Int. J. Syst. Sei., vol. 14, 1983.

C. Hwang and Y. P. Shih, “Laguent series solution of a functional differential

C. Hwang, “Solution of a scaled system via Chebyshev polynomials.” J. equation.” Int. J. Syst. Sci., vol. 14, no. 7, pp. 783-788, 1982.

J . H. Chou and I. R. Homg, “Chevyshev series analysis and identification of

Franklin Inst., vol. 318, no. 4, pp. 233-241, 1984.

Y. F. Chang and T. T. Lee, “General orthogonal polynomials approximation of scaled systems,” Int. J . Syst. Sci., 1985.

the linearquadratic-Gauian control design,” Int. J. Conlr., vol. 43, pp. 1879-

1895, 1986.

G. SzegB, Orthogonal PoIynomiak, 4th ed. New York: American Mathemati-

cal Society, 1975.

F. B. Hildebrand, Introduction to Numerieaf Ana&%$ 2nd ed. New York: McGraw-Hill, 1974.

L. Fox, D. F. Mayers, J . R. Ockendon, and A. B. Taylor, “On a functional

J. differential equation,” R. Ockendon and A. B. Taylor, “The dynamics of a current collection systems J . Inst. Math. Appl., vol. 8, pp. 271-307, 1971.

for an electric locomotive,” Proc. Roy. SOC. London A., vol. 322, pp. 447468,

1971.

A. D. Randolph, “Effect of crystal breakage on crystal size distribution in mixed

suspension crystallizer,” Ind. Eng. Chem. Fundmentals, vol. 8, pp. 58-63, 1969.

G. P. Rao and T. Srinivasan, “An optimal methcd for solving differential equations characterizing the dynamics of a current collection systems for an D. H. Shih and F. C. Kung. “Analysis and parameter estimation of scaled systems elecuic locomotive,” J. Inst. Math. Appl., vol. 2 5 , pp. 329-342, 1980. via shifted Legendre polynomials,” Int. J. Syst. Sci., vol. 17, 1986.

The Operational Matrices of Integration and

Differentiation for the Fourier Sine-Cosine

and Exponential Series

P. N.

PARASKEVOPOULOS

Abstract-For the Fourier sine-cosine series basis vector p(t) and the Fourier exponential series basis vector $(t), a linear nonsingular transfor- mation Tis determined such that $(t) = T&).

This

result is then used to show that the operational matrices of integration P and Q for p(t) and

I,&), respectively, are related by the expression TP = QT. Analogous

results are derived for the corresponding operational matrices of

differentiation D and R . General expressions are derived for T, P , Q , D , and R .

I. INTRODUCTION

Recently, orthogonal series have been used for studying various problems in system analysis and synthesis. The key idea involved is based on the integral expression

16

d u ) d u = P v ( t ) , where a ( t ) = I ~ d t ) ~ ~ ~ ( t ) , . . . I ~ , - ~ ( t ) l (1)

is the orthogonal basis vector and P i s an r X r constant matrix called the operational matrix of integration. The matrix P has already been

determined for many types of orthogonal series, such as Walsh [ 11, block- pulse [2], Laguerre [3], [4], Chebyshev [SI, Legendre [6],

[A,

Hermite [8], Jacobi [9], Bessel [ 101, Fourier sine-cosine series [ 1 11, and the Haar functions [12]. Most of the problems that have been studied may be

Manuscript received June 25, 1986; revised February 23. 1987.

The author is with the Division of Computer Science, Department of Electrical IEEE Log Number 8714807.

Engineering, National Technical University of Athens, Zographou. Greece.

summarized as follows. State-space analysis [13]-[18], optimal control 1191-[22], identification 1231-[27], sensitivity analysis [28]-[30], ob- server design [31]-[33], model simplification [34]-[36], solution of integral and variational problems [37]-[42], etc.

In this note the Fourier sine-cosine series are revisited and they are studied in conjunction with the Fourier exponential series. Let p(t) and

$(t) denote the orthogonal basis vectors for the sine-cosine series and the exponential series, respectively. Also let P and Q denote the respective operational matrices of integration. Then, it will be shown that a nonsingular transformation matrix T exists such that

T P = Q T . (3)

Since det T # 0, it follows that (2) and (3) are very useful since one may go from one set of orthogonal functions to another. In particular, (3) is used to derive

0

knowing P . Similar results are derived for the operational matrices of differentiation.

II. THE OPERATIONAL MATRICES OF INTEGRATION

Consider the Fourier sine-cosine series basis vector ~ ( t ) having the form

where

Also consider the Fourier exponential series basis vector $ ( t ) having the

form

where

where j =

f i

that p(f) and $(t) are related as follows:

Making use of the relation efi = cos 8

+

j sin 8 one may readily show

where Tis an 2r

+

1 square nonsingular transformation rnamx having the form

where Z, is the r X r unit matrix. Furthermore,

v(r)=

T-’Jl(t) where