A Simplified Method of Finding Equivalence Conditions in a Class of Distributed Parameter Systems

A Simplified Method of Finding Equivalence Conditions in a Class of Distributed Parameter Systems by

G. B. MAHAF’ATRA

Department of Electrical Engineering, Roorkee University, Roorkee, India 247672

ABSTRACT:A simplified method along with a proof is proposed in this paper to find the equivalence conditions in a class of linear distributed parameter systems. The proposed method is illustrated for (i) diffusion equations, (ii) wave equations and (iii) lateral vibration of bars; all with three spatial coordinates.

I. Introduction Homogeneous differential equations with non-homogeneous boundary conditions are essentially equivalent to non-homogeneous differential equations with homogeneous boundary conditions, provided certain continuity and daerentiability conditions, not necessarily made for the original problem, are maintained in the transformed problem (2). This is known as equivalence conditions in distributed parameter systems (DPS). Brogan (4) solved optimal control problems of a class of linear DPS by transferring boundary control directly to partial dif?erential equations (PDE) in terms of symbolic functions. This equivalence condition was obtained by using an extended definition operator (1). A disadvantage of this approach when applied to higher order DPS is that the transformed problem becomes ill-posed, although the original system is well-posed (1). Therefore, formulation of state equations in control studies becomes difficult by considering physical quantities such as displacement, velocity and acceleration as state variables. To overcome this, Brogan (4) has defined state variables as changes of displacement both in the time and space coordinates which is not true physically. Moreover, it is not yet clear why equivalence conditions cannot be directly expressed in terms of eigenfunctions, an important parameter of DPS. A simplified method is proposed in this paper to determine equivalence conditions by transferring boundary control directly to PDE in terms of system-eigenfunctions. Formulation of state equations satisfying wellposedness conditions for higher order PDE is also discussed when actual quantities, such as displacement, velocity and acceleration are defined as state variables in control studies. The proposed method is illustrated for (i) diffusion equations, (ii) wave equations, and (iii) lateral vibration of bars. All have three spatial coordinates and can be easily applied in optimal control studies of DPS with boundary control.

0

The Franklin

Institute

0016-0032~9/0401-0209$02.00/0

209

G. B. Mahapatra II. Proposed Method The proposed method to determine for three typical examples of DPS.

Example

an equivalence

condition

is illustrated

1. (Diffusion equation’)

Consider the three-dimensional

linear diffusion equation in normalized form: f3= 0(x1,

;=$I~$+a,e,

x2,

x3,0;

w@,

0,

(1)

I

with zero initial condition and boundary conditions: $--e+U(t)=O,

for

x1=0;

$=O,

for

x1=1

1

1

and 8~0,

for

and for

x,=x3=0

x,=x3=1.0.

When Eqs. (l)-(3) are solved by the Laplace transform peated application of the separation of variable method, 8 = ($)z;F($)(&)

cos~~s(lr-x’)sin

m

m

(3) technique

with re-

krx2sin krx3

I

f

exp (smkl(t - t’)) Vt’) dt’,

(4)

0

where r,,, is obtained from the solution of r, tan r, = 1.0

(5)

and s,~~=(Y~-(Y~T~-~‘(~(~~~+Q~~~);

k, 1= 1,3,5,

m=l,2,...; Assuming uniform convergence

....

(6)

conditions hold good for differentiation,

from

Eq. (4),

exp (~,,,~((t- t’))U(t’) dt’

. sin 17rx2- sin km, +($)~~~(~)(&)

m

cos~~,“,-xl) m

* sin 17rx2- sin krx3 - U(t)

(7)

and

. cosrm(1-d

cos r, 210

.

sin

lTx2

* sin krx3

f

exp (s,,,lcl(t- t’))U(t’) dt’.

Joumd

(8)

of The Franklin lusitute Pergamm Ress Ltd.

Equivalence Conditions in Distributed Parameter Systems On inspection of Eqs. (l), (7) and (8), it appears that 8 could be thought of as a solution of a non-homogeneous partial differential equation:

but with zero initial condition and homogeneous boundary conditions: -$--e=o,

for

x,=0;

g=O,

1

for

x,=1.0

1

(10)

and 8=0,

for

andfor

+=x3=0

x2=x3=1.0.

(11)

The distributed control appearing on the R.H.S. of Eq. (9) is given by: F(xI,X2,X3, m=($);&)(&), m

. cos r,(l-x,) cos r,

. sin IWX,- sin k7rx,U(t).

(12)

Proof: Taking the Laplace transform of Eq. (9) with respect to t and imposing the conditions in Eqs. (10) and (ll), we obtain

eh,

x2, x3,

4=

($)ET(j$)($) m

cos r,,,(1 -x1) . sin lrx, * sin krx3 WJ. (13) cos r, (s -siThA When Eq. (13) is solved, the expression for 8 in Eq. (4) is obtained. Therefore Eqs. (9)-(ll), with non-homogeneous PDE having homogeneous boundary conditions, are equivalent to the original equations (l)-(3) with homogeneous PDE and non-homogeneous boundary conditions. This completes the proof for the equivalence conditions. Example

2. (Wave equation)

Consider a DPS in three dimensions governed by wave equations in the normalized form: (14) with initial conditions: 8=0

and E=O

at

t=O

(1%

and with boundary conditions:

e =o,

for

X1=X2=X3=1.0,

e=o,

for

x2=x3=0

Vol. 307, No. 4. April 1979 Printed in Northern Ireland

and

(16)

8 = U(t),

for

x1 =O.

(17) 211

G. B. Mahapatra

ae

Introducing displacement 8 = 19~and velocity z = e2 as state variables in Eq. (14), the state equations are written as:

w t- - e2

(18)

and

with initial conditions: el(i, 0) = f&(i,0) = 0 and boundary conditions on & are given by Eqs. (16) and (17). Formulation of such state equations (18-19) have been considered by Kim and Erzberger (5) and Vinter and Fallside (6) who justified well-posedness of such formulations by defining proper norms. It can be seen that formulation of state equations using Brogan’s method (4) creates ill-posedness and therefore it cannot be used in control studies. Following the same line of thought as in the case of difhrsion equations, the equivalent system for Eqs. (18-19) with non-homogeneous boundary conditions as in Eqs. (16-17) can be obtained as:

ah -= at

e2;

~=i~~~

+~4el+~5e2+F(X1,

x2, x3,

W,

(20)

I

with zero initial conditions in Eq. (19) and homogeneous boundary conditions: &=O,

for

x1=x2=x3=0

and

1.0.

(21)

Equivalent distributed control F(xl, x2, x3, U) appearing on the R.H.S. of Eq. (20) is given by: nx1,xz,x3,

u=

sin17rx2

~+)~~~(~)sinnmxl

. sinkvx3U(t),

m=1,2,....;

k,l=1,3,5

,....

(22)

Proof for the equivalent systems can be obtained by following the previous method. Example 3. (Lateral vibration of bars) Consider the lateral vibration of prismatical bars (3) in three dimensions governed by PDE, i.e.

a48

(23

g=-

with initial conditions:

e=~=?.$=$=Oat 212

(24)

t=O

Journal

of The Franklin Institute PerQamon Reas Ltd.

Equivalence Conditions in Distributed Parameter Systems and with boundary conditions: for

O=U(t),

x1=0;

8=0,

for

xz=xJ=O and

$=O,

for

x1=x2=x3=0

and

1.0;

x1 =x2 = xg = 1.0; i = 1,2,3.

I

Introducing displacement 8 = 8,; velocity f = t&; acceleration $

(25) (26)

= e3 and rate

3

of change of acceleration &$ = 0, as state variables in Eq. (23), the state equations are written as: i = 1,2,3

and ae,_ yg-

-[

(27) i,+,$$-i4+4$$+a,Bp],

i=l

1

I

i=l

with initial conditions:

ei( ,o)=o;

i = 1,2,3,4.

(28)

Boundary conditions on & are governed by Eqs. (25)-(26). Well-posedness of state equations formulated in Eq. (27) corresponding to original PDE (23) can be justified by imposing a proper norm on state variables of the type discussed in (6). Following the method employed in the case of diffusion equations, an equivalent system for Eqs. (27)-(28) with non-homogeneous boundary conditions as in Eqs. (25)-(26) can be obtained as:

(29) a64 x=-

i [ i=l

%+I$$-

i I

$+4$+alO3]+F(Xl,

i=l

X2,

X3,

I

U).

(30)

With zero initial conditions in Eq. (28) and homogeneous boundary conditions, we have 8r=O,

for

x~=x~=x~=O

and

1.0

(31)

and

a2e1

s=O, I

for

x,=x2=x3=0

and

1.0;

i = 1,2,3.

(32)

Equivalent distributed control F(x,, x2, x3, U) appearing on the R.H.S. of Eq. Vol. 307, No. 4, April 1979 Printed in Northern Ireland

213

G. B. Mahapatra (30) is given by: Fbl,

x2,

x3>

u)

=

-(64/p) ;;F

(~)(A,b13’2+A,b23’2). * sin m7rxr - sin 17rx2- sin

km,U(t),

(33)

where bl = 1/2[a+Ja2-4b],

(34)

b2 = ?/??[a -Ja2-4b],

(35)

a = a,+7r2(m2a,+

12a6+ k2a7),

(36)

b = q4(m4a,+ 14a3 + k4a4),

(37)

(38) Yi = m2r2- bi(aJa2),

(39)

Jbi, a&&IJbi,

Z = [Pi(as/a,) - {[(aS.d) - (4/aJ]bi + 2(a - aJ/aJ]

(40)

W = [pi (as/a3 + {[(a:/af) - (4/az)]bi + 2(a -

(41)

pi = [[(a:/a$) - (4/a2)]b? - 4(a - r2m2a5)bJa2 - 4(b - n4m4a2)/a2]“2;

i = 1,2.

(42)

The proof for the equivalent systems is the same as in the case of dithision systems. It is to be noted that the procedure adopted here to find equivalence conditions is simpler than that of the extended definition operator technique (1). Further, the procedure takes into account the eigenfunctions of DPS and the formulation of state equations by defining displacement, velocity and acceleration, as physical quantities of state variables. III. Conclusion An alternative method is proposed in this paper, supported by proof, to determine equivalence conditions of a class of DPS in terms of eigenfunctions of the system. The method is illustrated, by three typical examples of DPS in three dimensions and is simpler than that obtained by the extended definition operator (1). References (1)B. Friedman, “Principles and Techniques of Applied Mathematics”,

John Wiley, New York, 1956. (2) R. Courant and D. Hilbert, “Methods of Mathematical Physics”, Vol. 1, John Wiley, New York, 1953. (3) S. Timeshenko, “Vibration Problems in Engineering”, D. Van Nostrand Co., New York, 1954. (4) W. L. Brogan, “Optimal control theory applied to systems described by partial

214

Journal of The Franklin htftute

Pergamon Plea Ltd.

Equivalence Conditions in Distributed Parameter Systems differential equations”, Advan. Cont. Syst., Vol. 6, 1968. (5) M. Kim and H. Erzberger, “On the design of optimum distributed parameter system with boundary control function”, IEEE Trans. Auto. Cont., Vol. AC-12, p. 22, 1967. (6) R. V. Vinter, and F. Fallside, “Minimum time control of a class of linear fixed domain systems with distributed input”, Proc. IEE, Vol. 117, p. 2294, 1970.

Vol. 307, No. 4, April 1979 F’rinted in Northern Ireland

215