Analysis and optimal control of time-varying system via wavelets

ANALYSIS AND OPTIMAL CONTROL OF TIME-VARYING SYSTE...

14th World Congress of IFAC

Copyright:G' 1999 IFAC 14th Triennia! World Congress, Beijing~ P.R. China

F-2d-07-6

ANALYSIS AND OPTIMAL CONTROL OF TIME-VARYING SYSTEM VIA WAVELETS

Liu Xiaogqian

Zhang Lin

Department of Automation ~ Tsinghua University, Beijing

~

100084, P. R. China

[email protected]. edu.cn, zhlin@mail. au. tsinghua. edu. en

Abstract: In this paper, Haar wavelet integral operational matrix is introduced and applied to analyze and optimal control linear time-varying system. The method converts the original problems to the solution of linear algebraic equations..Hence, computational difficulties are considerably reduced. Based on the property of time-frequency localization of Haar wavelet bases) the solution of a system includes both the frequency information and the time information. Other orthogonal functions do not have this property. Examples are given, and the results are shown to be very accurate and satisfactory. Copyright © 1999 IFAC

Keywords: Approximate analysis, Algebraic approaches, Boundary value problem, Time-varying systems, Optimal control,

1. INTRODUCTION Orthogonal functions are frequently used by many researchers as a convenient and powerful tool for obtaining the approximate s.olutions of physical systmes(Mohan and Datta,1995; Lee, L. T and Tsay, Y.F.,1986; Wang, M.L. et al., 1986;). The main feature of this technique is to convert a differential equation into an algebraic equation, and hence the solution of the control problem is either greatly reduced or much simplified accordingly. All previous orthogon~1 functions, however, are supported on the whole interval a ~ t ~ b . This kind of global support is evidently a drawback for certain analysis work, particularly systems involving abrupt variations or a local function vanishing outside a short interval of time or space. In recent years, Wavelets and waveletbased analysis have found their way into many different fields of science and engineering (Chen,C.F. and Hsiao,C.H~,1997~ Gu,J~S. and Jiang,W~S., 1996; Daubechies, 1992). It could be a possible tool for overcoming the difficulty in communication, physics and image processing. In this paper, Haar wavelet integral operational matrix is introduced and then applied to the analysis and optimal control of linear time-varying systems. The extension is achieved through representing the

product of two wavelet series in a new wavelet series. The method reduces the original problems to solve linear algebraic equations and avoids solving a system of coupled nvo-point boundary-value problems. Therefore computational difficulties arc greatly reduced. Moreover, in the process of solving practical problems, the different resolution bases can be selected on the local time domain in order to understand more clearly system behavior on the local time domain. This advantage results from the property of Haar wavelet localization ~ and other orthogonal functions do not have this property because they are supported on the whole interval. Examples, which demonstrate the results, are given.

2. HAAR WAVELET INTEGRAL OPERATIONAL MATRlX

In this section the integral operational matrix P wiH be obtained, it is the basement of the following analysis. For any function I(t) E L2 (R) ,the discrete wavelet expansion of f(t) is represented as f(t) = Lbrtd~VFrn.~(t)

= b7

9(l)

1

(1)

m,!:

2989

Copyright 1999 IFAC

ISBN: 0 08 043248 4

ANALYSIS AND OPTIMAL CONTROL OF TIME-VARYING SYSTE...

14th World Congress of IFAC

\vhere b is Haar wavelet coefficient vector, 8 (I) is a set of Haar wavelets

¥/mJ, (t) (m,k

E

Z) which is

A(l)x(t)

i

XHm~

chosen according to necessity. The two vectors are defined by b = [bp b2 ,·· ·~bm Y ~ (2)

=

= [lI'l~'I'~,···, W,'1'Y' , (3) where w. (i = t2... ,m) is some ~m,k (t) . Haar wavelet integral operational matrix P is defined as 9(t)dt = P8(t) ~ (4)

H = ,

where the element P(j~j) of the integral operational matrix P is obtained by the same algorithm as the discrete wavelet expansion, Le. 0

r

•

.

[~",J~d 1 .

.(14)

Similarly

I

\)

\

.(13)

where

J

= (f'lj/·dt'lI/l')

tAi XH ; 9{t) i=)

o(t)

P(i,j)

XH

= [Ai ... Am]l : '].a(t)

B(t)u(t)

= [B I

Bm][U~' Jl S(/)

•..

Ull

=

iBiUH i a(t).(I 5) 1=1

m

Integration of equation (6) gives

(5)

x(t)-x(O)= l~A(t)X(t)dl+ J:B(t)U(t)dt

Of courSE, jf different bases are chosen, different integral operational matrices \vill be got. It is the advantage of wavelets. In other words, different bases can be chosen according to different interested intervals. Other orthogonal functions do not have this property.

.(l6)

Substituting equations (1 1), (12),(13) and (15) into equation (16) yields X 9(t) - [k,O·· ·,OJ 9{t)

= f~f A,XH

j

9(t)dt +

L::tB,UH; 9(t)dt . .=L

1=1

Substituting (4) into the above equation gets 3- ANALYSIS OF L·INEAR TIME-VARYING

X 9(t)- [k,0,···,0] 9(t) = fAjXH1P 9(/) + 'tB i UH jP9(t)

SYS1~EM

,=1

,,,,I

or Consider a linear time-varying system characterized by: x(t} = L4.(t)X(t) + B(t)u(t) , (6) x(O) = x\] , (7) where the state variable xCI) and the input variable u(t) are respectively n-vector and r-vector (Note that u(J) is known when the system is analyzed). The time-varying coefficient matrices ACt) and B{t) are

x- iA/XHjP=[k,O,. .. ,O]+ :tB1UHiP J"'J

;",,1

Therefore, the solution of equation (6) can be found by

and n x r matrices, respectively ~ X o is initial vector. Let Haar wavelet base approximations of A(t) , B(t) , u(t) , X O ) x(t) be respectively: nxn

(8)

B(t) =

f

.'-:--1

matrix

B~V'i

(10) (11) (12)

x(t) = X 9(/) , [k,O~-·· ,0]

where I! = [lT 1 X ==

[Xl

9(t) ,

Um

U;1 Xl·'·

'7 which is defmed by V

(9)

,

u(t) = U 8(t) ,

x:, =

where I denotes the identity matrix, ® denotes the Kronecker product, and Vi is the i-th column of

Xm

lE R lE R

m

n .

=

/1I

,

L = [I -

[A

cpjj

J'

(19)

~«HiPr & A{)J'tCCPTH;)@ B,)

(20)

Consequently, the solution of system (6) is

l

By decomposing

approximations

.

K [I - t(H,PY & A,

.~

nx

Then A(t)x(t) =

=[k,O,·· ',0]

Let X donates the vector obtained by putting matrix X into one column~ then X= K\' +LiT (18) where

V/,'JI j

into Haar wavelet base

XCI) = X 9(1) =

9(/) yields

[XI"

'X01J[~I] .

(2])

V'"m

2990

Copyright 1999 IFAC

ISBN: 0 08 043248 4

ANALYSIS AND OPTIMAL CONTROL OF TIME-VARYING SYSTE...

14th World Congress of IFAC

4. OPTIMAL CONTROL OF LINEAR TIMEVAR YING SYSTEMS

1~~

I le:! Qe~el Ye.re, MeJe\:_l)n.,..
T

T

2"l~~ ~ ~ [(Set~e(/_I),,+.~KeU_l)n+;)

J == _x (/I)Sx(1f) + 2 T

T

After putting the matrix into one column, then

T

-!-2 Jot! [x

T

T

T

~

r

® (e l N) + (QeSe(l_I)~+.~Keu_lll1+/) 1

o (TM) e -"2] (e,.e TM e,e(l~l)n+sKe(j_l)ll+i )7 ® (e rQe .• e

(22)

(t)Qx(t) + u i'(t)Ru(t)]dt

T

or'

i

The control of the linear time-varying system of (6) with respect to a quadratic performance index ]

)"i'

- - L. L..J (-e: Se,l e ; ·Ve.~el Ne} + 2 .h.1 I=~ 2

l'

t

1

is considered. The problem is to find the control vector u(t) to minimize the above cost functional: \vhere I r is the tinal time, the matrices Sand Q are

"21 (e_
T t

j

7-

NereU-I!I'1+GKeU_lllH,)

.-

T

symmetric positive semidefinite and matrix R is symmetric positive definite. Let Y = X'f = [V, Y2 .". from equation (11)

T

t )-

T-.

l2: (e.
1 T TIT +ir
T

r-

r

T

@(e,iRejc i )]U

j

=0

Denote

Yml,

]

Eij =

rr

nr

2' ~~[(SelSe~-I)n+,~KeO_JJ"'+i)T

(8)

(e;N) +

yield x(tI

T K eU-I)n+i )T ® (TM) (Q e"~e{I_I)nH et

(23)

= yT 9(1 f)

)

The cost functional (22) can be rewTitten as J =

l- B"(t/)YSyTO(tj)+.l 2'

2

Jo

(e~Se~e:)]

Ef (t)URV 9(t)]dt= - tr(SYSy T ) +

J

F'l = 2[(Me j

2

J...[tr(MYQYl') + tr(MU.RU T )]

(24)

2

r )

f,J,.

SCt)

er (t)dt ,

N = 9(1.r)

eT (t f)

(25)

Substituting equations (23),(25) and (18) into (24),the cost functional J becomes a function of U'i . Thus far,

necessary condition of optimization, then gives

~- = eU-i).'lHKeU-i)n-Oof

~~

s

(26)

vUij

oui3

, 7 tr(MU RU )

= e,.T R U Me

j

-

1 TM eJe,TUeje 2"e j

TR

i

e (28) j

!!

then yields

-

dJ

ou~.

nnr

l

8

0

Yts

OU jj

.\=1

'=1

0

~~tr{RLTTMLJ)=~tf[e~(NYS+MYQ)e.. oX

st

2

obtain

2 ,~-1

oUij

OU ii

1=1

r .., ~' (2"1 e.,.Se.~.e: Ye.•.e; Net +"2 e Qe,re; Ye e T

T

)

T

s

8

r

ox

_

I

J'

T

r

-(e RUMe J-- -e 2 21.Re}.e Ve J.e i Me;) j

I

E,-I .

EY + FU ::::; 0

:

(30)

l::

x(t) "" (

0

... t

(31)

(1)U Ct) ,

°IX(f) + 0) 1

wbere input u(t) is unit step

~

T

F 21

Example 1 ~ Consider the foUowing time-varying linear system:

I)

I

F1m

5. EXAMPLES

Me,) OU·~I]+

l r 1 -r T -2 (e; .e,, Me,) - - RUMe, - -e 2 J', RC,J-e:.Ue " .

T

Put (3 I) and (18) together and use Y =- X r then ,,~,x,U can be obtained. Therefore the corresponding state" and performance index J under the control u(t) can be obtained.

= -;- LL. . -(tr(SyT?\lY) + (r(QyTl\lY))~ + ~

7

® (e) Reje; )] ,

(29)

E 21

lE~

-- 1: .. ", n, t -- 1~"', nl (27)

where e, is the vector that i-th element is 1 and other elements are zeros. Using fonnuIa

)'

0

F=;

~

oYu =;- ,

"I

r i

E 2m

From (1 8) obtains eX,'l

=

ELm

E=

i=t-··~r,j=l,-"",m

,..

Let i go from 1 to r, j go from 1 to m, rnr equations will be obtained. Denote Ell F11

the optimization problem is reduced to that of finding H,) to minimize the cost functional J. From the

du.~,

]

EijY + FijU

a

~=O

'/'

® (e l R)- 2(e j e; Me

hence yields

where tr(A) is the trace of tnatrix A, and M =

T Me,eU_1)nH T 2I (e.~eJ .

Ke, )T@(TQ T'_..!.. r, K e(J-l)'1+..) T ® (J-:)n+? e s e.~e!) 2 (e.,.e{TNete(t~IJn+.~

f'/[tY(t)YQyT8(t) +

1

T

-

J

and

x(O) == (0

"i)~',

According to the section 3, 16 bases are chosen then the solution of the system are obtained. It is shown by Fig. 1(the top figure is the trajectory of x 1 and the below· figure is the trajectory of x2, the smooth curve is the exact solution and the un continuous line is the

2991

Copyright 1999 IFAC

ISBN: 0 08 043248 4

ANALYSIS AND OPTIMAL CONTROL OF TIME-VARYING SYSTE...

approximate solution). More bases are chosen and more accurate solution will be got. This example demonstrates the fast, flexible and convenient capabilities of the new method. Example 2: Consider the system:

xU) = .x(t) + uU), x(O) = I, xCI) = 0 ,

1[x (t) + u (t )]dt 1

2

2

0.6 0.6

(32) 0.4

and the cost functional

= -1

14th World Congress of IFAC

(33)

0.2

A.ccording to the section 4, 16 bases are chosen then the optimal control law satisfied the given conditions are obtained. It is shown by Fig 2(thc smooth curve is the exact solution and the uncontinous line is the approxilnate solution, the top figure is the trajectory of s)/stenl and the below one is the control law). This examp le shows the sharpness and effect of the approach.

o

J

2

0

D

0.5

2

1 .5

6. CONCLUSIONS In this paper the algorithm to get Haar \vavelet integral operational matrix is introduced, and the analysis and optimal control of linear time-varying systems are studied. Because of the property of timefrequency localization of Haar wavelet bases, the solution of a system includes both the frequency information and the time infonnation. This point is significant in control system analysis and synthesis. Furthermore, in order to study the system behavior in the local time domain very well, the different resolution bases can be selected in the local time

0.5 """'-

o

--"

---1

0.5

Fig.! Solution of the system in exampJe

j

0.8 0,6

0.4

domain. 0.2

REFERENCES D.S

1V1ohan, B.M., Datta, V.B_ (1995), Analysis of linear time-invariant time-delay systems via orthogonal functions, International Journal of Systeln Science, 26, 91-111

-1

Chen.~

C. F. and Hsiao , C. H., (1997), Haar \,"'avelet Method for solving lumped and distributedparameter systems, lEE Proc-ControJ Theory Application, 144, 97-94 Daubechies Ingrid (1992), Ten Lectures on Wavelets, SIAM Phjladephia, Gu, J.S. and Jiang, W.S. (1996), The Haar wavelets operational matrix of integration, international Journal ofSystern Science, 27, 623-628 Lee ~ L_ T. and Tsay, )'. F. Tsay (1986). Analysis and optimal contro I of discrete linear timevarying systems via discrete general orthogonal polynomials~ International Journal of' Control , 44,1427-1436 Wang, M. L. Chang, R. Y. and Yal1g ,S. Y.(1986), Analysis and optimal control of time-varying systems via generalized orthogonal polynomials, International Journal afControl , 44,,895-910 1

-1

.5 -2

-2.5 -3

o

0.5

1

Fig.2 Optimal trajectory and control law of the exmnpfe 2

1

2992

Copyright 1999 IFAC

ISBN: 0 08 043248 4