Contribution to the problem of initial conditions of linear dynamical systems at a discontinuous driving function

182

A~znales de l'Associa/io, J,ler,zaliomde pore" le Calcu/ m~alogique

N" 4 -.- Oc/obre 1970

CONTRIBUTION TO THE PROBLEM OF INITIAL CONDITIONS OF LINEAR DYNAMICAL SYSTEMS AT A. DISCONTINUOUS DRIVING FUNCTION * by ,laromir K R E M E N and Petr D E M E L

SUMMARY. --- The article is a c.ntribution to the methods of solution of the differential equation "

IL

£ll

*

v"' (l) : : ~ /,, ~'~" (0

i ~o

• •

I -o

with discontinuous driving function x O) and with generally non-zero initial conditions y(i) (0), i,--- 0, 1. . . . . n - 1. At first the concept of <is introduced. Furthermore a relation between the generalized solution of the differential equation and its solution on analogue computer is shown. As a result we can conclude that the existence of the analogue solution closely depends on the existence of the generalized solution. The rigorous answer concerning the existence conditions of the generalized solution is given by Caratheodory's theorem. It is proved in the article that the construction of Laplace transform of the above mentioned equation requests the knowledge of the left-hand initial conditions only as oppose to the demands in the existing literature. The proof is based on the continuity of the generalised solution. In the present article are also derived two methods of easy and rapid analogue simulation of the above mentioned differential equation with non-zero left-hand initial conditions. Theoretical conclusions are illustrated on a simple example.

1. Introduction. It is well known that the Laplace transformation is a convenient tool for investigation of linear dynamical systems which are exposed to the action of a driving function with discontinuity at time t = 0. Another method is analogue simulation. The system, to be investigated, is usually defined by a differential equation with constant coefficients of the order ,~, including the derivatives of both the solution and the driving function, and by initial conditions, describing the system energy immediately before the instant t = 0. W e shall call them .the left-hand initial conditions. If both the driving function and its derivatives up to the order n - - 1 are continuous functions at time t = 0, then the initial values of the solution and of its derivatives up to t:he order n - - 1 (i.e. the values for t = 0) are equal to ,these left-hand initial conditions. If, however, the driving function at time t = 0 is discontinuous, the initial values of the so,lution and of its derivatives (we shall call these values the right-hand initial values) generally differ from the left-hand initial conditions and this change at time I = 0. has the character o,t: a step. This stepwise change is obviously a part of the solution.

* Manuscript received 3rd September 1969 ** Institute of Computation Technique . Czechoslovakia

If we construct the Laplace transform of the differential equation of a system with the knowledge gained from the theory of Laplace transformation, we come to rather inconvenient conclusions. According to this theory, the finding-out of the Laplace transform requires the knowledge (which is unnecessary, as we shall see later) o.f unconditionally .the right-hand initial values of both the driving function and its derivatives and also the knowledge o.f right-hand initial values of the solution and o.f its derivatives. These requirements render the determination of the transform of a given equation rather difficult. This article yields, however, a proof .that the transform of a differential equation can be constructed from the mere knowledge of its original and o.f the left-hand initial conditions of both the so.lution and its , 1 derivatives, this being analogous to classical analysis at fl~e work with continuous functions. An attempt .to prove this theorem can be found in [1]. W e also quote two methods enabling simple simulation, on an analogue computer, o.f linear differential equations with constant coefficients with non-zero initial conditions of the solution, given by the initial system energy, and with right side in form of a linear combination of driving function derivatives. The methods, mentioned above, eliminate the drawbacks of existing methods which require either two models [21] or ,'datively complicated auxiliary computations and an unnecessarily great number of operational units [3].

]. Kremen and P. Deme/

."

ln#ia/ cond#ions oi linear d),namJca/ s),s/ems

Prior to go to the solution of the problems, mentioned above, let us discuss some partial questions. We shall investigate the behaviour of systems which can be described by differential equations of the type

The equation (2) is only meaningful in such a time interval in which the derivative x' (I) exists. We shall call the solution of equation (2) the <>. More generally valid is the equation

a. y " ' (t) + a._, y"-'> (t) + ... + a,, y (t) =

b. x " " (t) +

y (o-) = y' (o-) = y<,,-" ( o ) =

b,,_, .,.'" " (t)

YA y'A , y a (n-l) ,

F ... +

b,, u ( O ,

(t)

initial values for t approaching zero from. the left side, i.e. t --> O-,

where x, x', ..., x (") is the driving function and its derivatives with respect to independent variable t, y, y', .... y"'> is the solution and its deriwttives, bo, h~, ..., b, 3 are constant coefficients, a, =/= O, t is the independent variable (dine). In case of systems of this sort a somewhat idealized driving function is frequentty considered, viz. a discontinuous quantity x (inp.ut of the system under investigation), for instance Heaviside's unit function, Dirac's delta function, function having the properties x(t) = 0 for t < 0, x(t) = cos o,1 for t > 0, etc. The equation (1) in the form, mentioned above, is only then meaningful within a certain time interval, if there exist the derivatives on both its sides. It is known that these derivatives exist almost in all instances where the equation (1) describes a real natural process. Serious problem may arise only with the idealization, mentioned above (discontinuity of driving function for t = 0). But, on the other side, this idealization frequently results in a considerable simplification of the problem set up, which is free of negligible and sometimes also of parasitic influences, is quite common and frequently used. Complications, resulting from the problem idealization, appear, however, only in case of an analytical solution of equation (1) in the form, given above. In case of solution on an analogue computer they usually do not exist and it might seem on the first glance that the non-existence o,f this problem is rooted in parasitic effects o.f the computer which render all the quantities continuous. We shall now demonstrate that this is not the case and that the essential fact consists in that the computer yields a so-called generalized solution. This concept will be more thoroughly explained in the section to follow. 2. Generalized solution of the differential equation, Let us demonstrate the concept of generalized solution on an example. We shall consider the equation

0, (t) + y (t) = y (o-) =

x' (t), y,;.

183

(2)

J['

y (,) ,], + 0,

-

0, (o) -- .+ (]) -- ,-(o),

O)

where both the initial values x (0), y (0) .'ire considered either from the left or from the right side. The equation (3) was gained by integrating the equation (2) within the limits < 0, t > and the problem of existence of the derivative x' (t) does not exist in equation (3) any more. The equation (3) has a solution even at points at which the derivative x' (t) does not exist and at which, consequently, the equation (2) has no meaning. We shall call the solution of equation (3) the <>* [4]. The introduction of the concept of generalized solution enables the requirement of existence of the derivative x' (/) within an enclosed interwd < 0, ! > to be l'eplaced by the requirement of existence of a (Lebesgue) integral of the derivative x' (t) within the interval < 0, t >. Let us now follow the relation between the above defined generalized solution of differential equation and its computation on an analogue computer. Let us solve the equation (2) on an analogue con> puter. We shall exclude the possibility of using the differentiators and shall suppose that the computer setup only consists of summing amplifiers and integrators. In order to be able to draw the diagram of the computer setup, we shall transform the equation (2) into the form. (3) (i.e. to carry out its formal integration). The block diagram is shown in figure 1.

-x

ff)

e-


The derivative of the driving function x (i) does not appear in the block diagram at all. Similarly, if we develop the computer set-up of the type (1) differential equation of the n+th order with the use of any arbitrary integral method, we can easily demon*) The conditions of existence of generalized solution and of its relation to classical sohltion are deternained by Carath6odory's theorenl [5].

184

Annales de I'Association internationa/e pont le Calcu/ analogique

stt'ate that the explicite derivatives of drMng functioll will appear nowhere in tile correspondin,g block diagrain. This, of course, refers to ,:in :malogt, e computer, in which the differential relations are solved by means of integrators and .this means in other words that tile computer solves the given differential equation in an integral form and yields a generalized solution of tile given equation. An analogue computer, in which the differential relations are solved by means of differentiators, yields a <> of the given equation, which is accompanied with equal difficulties as in case of analytical solution. In the text, to follow, we shall, therefore, always suppose that the computer is equipped with integrators for solution Of differential relations. Let us now concentrate upon the first problem, mentioned above, the initial values of Laplace transfoma of the solution of differential equation (1). 3. Initial values of the solution of a differential equation in Laplace transformation at a driving function with discontinuity at time t --- 0. After having applied the Laplace transformation to equation (1), we obtain the expression (4), This necessitates the existence of transforms of the derivatives x(i) (t) and yI*)(t), i = 0, 1. . . . , n . * * 11--1

N (p). Y (p) = M (p). X (p) + 2 -,,_,-, p',

(4)

i=0

where X (p) =

Z" {x(t)},

Y(p) =

~" {y(t)}, 11

N ( p ) = Z al p t , i=O

11

M(p) = JE= 0 ,~, P~, n-t

<'<,,-.~-i =

E

a,++, y"<-+> (o+)

--/,,++1","<-'>

(o+),

R= i

i =

0, 1 .... , n--1..

(4a)

Let us now introduce the concept of a relaxed system. We shall call a dynamical system, described by equation (1), a relaxed system, if a'll the given initial conditions of the equation (1) will equal zero, i.e. y,~ = O, y',~ -- O, ..., y~("-a) = 0.. W e shall denote the variables of a relaxed system with index R. **) If we denote the above mentioned derivatives generally ]'m (t), i == 0, 1. . . . . n, then ]{o (t) must fulfil the well known theorem on the transform of a derivative: Let

I (t), /' (/) ..... /o-~) (l) be, for t 7> 0, continuous functions of the order eat as t tends to infinity and also let ](i) be, for t ~> 0, a sectionally continuous fu,actlon. Then the transform of ]o) (t) exists when Re p Z> ce and it can be written as follows : £ [1<'>(t)] = l" t: (p) - - t,'+' f(0 +) - - ... - - 1'-' (0 +)

N" 4 --

Octobre 1970

lu case the equation (4) describes a relaxed system, the mcinbcr It I

E -.

i°'

i =13

is usually omitted, which means that I1-1.

c,,,_l_i pi , ~

0

i =0

is assumed. If we now insert concrete values into the equation (4a), we can prove that this assumption is justified. A correct proof for general values is not known to the authors as yet. We shall try to find this proof in the text to follow. As already mentioned in the introduction we take the stepwise changes of the solution and of its derivatives at time t = 0 as a part of tile results of solution. In the theory of one-sided Laplace transformation, these stepwise changes are not, however, included in the concept <>. Here it is understood that solution means the events occurring in time t > O, while the values at time t = 0 are given by righthand limits, i.e. by the results of action of stcpwise changes. Consequently, the values (limits for I -~ 0+), which we have to insert into the relations for the transform of a differential equation, are usually unknown, since they represent a partial result of tile process, to be investigated. We could rather expect that the Laplace transformation will enable these values to be computed. Thus we have to answer the question whether we can find tile relations which would enable the determination of both the initial values of the solution and its n - - 1 right-hand derivatives from the knowledge of the equation (1) and of the respective left-hand initial conditions. We must, therefore, study the behaviour of the solution and of its n - - 1 derivatives in the region adjacent to point t = O. In our case the satisfactory region is I C < r , co >, r < O. It is known from the theory of differential equations that an equivalent system of n differential equations of the first order can be found to the differential equation (1) of the n-th order. Since we try to find the generalized solution of equation (1), we shall operate with an integral equation or with a system of n integral equations of the first order, respectively, which correspond to equation (it). 'We obtain this system of equations by n successive integrations of the equation (1). Below, we shall demonstrate the procedure of finding out the system of integral equations. We sh.'dl try to decompose the equation (1) into a system of integral equations of the first order, where yo (t), yt (t) ..... y,,_, (t) are the newly introduced variables : t

Y,, (t) = y,, (r) -I- J [

f,, (y<,, y, ..... y , , , x) dr,

s, (0 = y, (~) +

A (yo, > ..... y,,_,, .) dr,

Y. i (I) ~-, yn i (r) -[- c t f,,-, (Y,,, .Y, ..... Y,,-1, x) dl. +%

(5)

]. KreJ,len and P. Denlel .' in/t/a~ condilions of linear d),naJll;ca/ systems

is integrated withir~ the limits (r, t), 1 > O, the integrals being meant in the sense of Lebesgue. Simultaneously, both sides of the modified equation will be supposed to be equal to a new variable ),,_~ (1):

Here, f,,, fl ..... f,,-t are the meanwhile unknown linear functions of variables )',,, y l , ..., y...~, x which will be found late*" on. The relations for the new variables yo, )'t . . . . , y,_~ will be successively defined so that the equation (1)

),..,(t)

= a,, ),,.-,, (t)

+

=

-t- ... +

a,, y,,,-1, (,)

...

t- ~,, a,

185

-/,1

y(t) --/~,, .,., ..... ,(t) --...

y6")

x,,,-,, (.,.)

--/,,

...--

--

.,(,')

/,, ..,-(T) +

L

-F ; ~

[b,) x(t) - - a,) y(t)] d t .

(6)

The formula (6) represents two relations for y . , , (t). The first of them, i.e. a,, y(n-,) (t) -t- ... -1- at y(l)

y. , (I) =

-

/,,, .x"'"-') (I).--

-

... - - h ,

x(/),

determines of how the variable y,, , (t) depends on original wtriables x(l), y (1) and on their derivatives. second relation is the sought integral equation of the system (5) : ystem i 5 ) :

(7) The

1

Y.-,

(t) = y,,_, 09 + /

lZ,,, ., (t)

"Jr

--

,,,

(8)

y (t)l ,et.

Let us now proceed to the next step of decomposition. The procedure applied to equation (1) will now be used for equation (7), i.e. the equation will be integrated within the limits (r, l) and transformed into the form (9) and both its sides will be put equal to the variable y,,_~ (t): y,,_., (t)

y (t)

=

a,, y(,, ~)(t)

-t-

... +

a,.

=

,,, y , , , - ~ , ( , )

+

... +

,,., y ( , )

--

b,, ~'"

'-'> U) --

... --

--

/,,, ,:,,,--'-',(,) _+....--

/-'.: .'. (t)

/,, .,.(,)

+

L

+ £

[I.,, x(t)

--a,

y (t)

+

)',,_1U)I

(9)

dr.

For y,_., (1) we obtain the formulae :

>,-.-, U) = a,, y(,,--' (t) y,,_~ (t)

=

y. ,, (,)

+

~

+

... +

~., y (t) --

a,, ,,,,--', (t) --...

-/~.:

>: (,),

t [at xi/) - - ¢/, y (1) @" Y,,-I (1)] dl.

(lo)

(ll)

At the n-th step of decomposition we obtain :

yo (t)

a,, y (t) -- b,, x (t),

=

(12)

~t

y,, (t) = yo (~) + J

V,,_, x (t) -

~,,,_1 >' (0 + y, (01 ,¢t

(;3)

1"

and generally : n-1

y,, ,-~ (0

= E a,,,, y'"-~' (t)

-- hi,,, x",-', (t),

ii4)

k= I

i =

O, 1, ..., ~ - - 1 , I

y,,_~_, (t) = y,,_,_, 09 + ~ =

V,, .,(0 -

a, y it) +

y,,_, (01 ~/t,

05)

0, 1, ..., ~ - - 1 ,

0'. U) '~-

o.

Hereby the decomposition of equation (1) into the system (5) is completed. The relation (15) shows that all the functions y,,_l_~(t), i = 0, 1.... , n - - 1 , are continuous functions within tile interval < r, t >, which means that the following equation can be written : lim y,,_~_~(t) = t--N)" i

-----

lira Y,..1-i (t), t--~0+ O, I,

..., n .... I .

(16)

186

A , , a / e s de /'Associalio, he/ernatio,a/e po//r le CalcM analogiq//e

N" 4 - - Octobre 1970

Thus we come to the conclusion that the linear combination (i4) of discontinuous functions .x"l~ (1) and 3,'~ (1) is a function which is continuous. N o w we can return to the Laplace transform (4) of differential equation (*). The formulae (4a) and (14) clearly show that n-I

,~,,_,_, =

lira

y,,_,_, (t) =

t ---~,0 ~

E ~,,,+, y'"-" (09 - - / < ,

.,'"-" ( o 9 ,

07)

k= 1

i -- 0, 1, ..., n - - 1 . But, with regard to relation (t6), also the following equation can be written : n-i

,x._,_l

=

lim t-+O"

i =

.Yn-.l-i (t)

=

7~ ati~, k=l

y¢l, i/ ( 0 - )

--

/211+l .x"(l'-i) ( 0 - ) ,

(17a)

0, 1.... , ; ' z - - t .

Since we assume that the driving function is applied at time t = 0, it is obviously x ¢.i~ (0-) = 0, j = 0 1 ..... n, and the relations 07) and (17a) can be simplified to 11-1

I1-1

~,,-.~ = E ,,,,. y ' " - " (o-) =

E ~,,+, y'"-~' (09 ........ /,,,..,:,k ~, ( o 9 ,

It= 1

i :

k= 1

(~8)

0, 1, ..., n - - 1 .

The equation (18) gives an important relation between the left-hand conditions of equation (t) and the right-hand initial values of the solution of the same equation. Taking into account the equation (tS), we can write the relation (4) as follows : ll--k

I1--1

N(p).Y(p)

= M(p).X(p)

q- E P' Z a,¢,, y ' " - " ( 0 - ) i =0

(19)

k= i

and the solution of equation (19) is : II- l

Y (p) -

M (p) --. N (p)

x (p) - -

II- ]

E P~ E a,,+~ y~"-" (09 , .... ,~:, (20)

N (p)

For a relaxed system y(~ (0-) = 0, i = 0, 1 ..... n - - 1 ,

and, consequently, with regard to (1.8) :

E a,,,~ yR'k-'~ (09 - - b,~+, ,:,~-,, (09 =

k=l

0,

(21)

i = 0, 1, ..., n - - l , ;rod the equations (19), (20) can be simplified to known formulae N(p).YR(p)

=

m(p).X(p),

YR(p) = Q ( p ) . X ( p ) where

Q(p)

M (p)

-

N (p)

.

(22) (23)

is the transfer function.

The relation (21) represents a system of n algebraic equations from which the initial values of the solution and of its n - - l derivatives for a relaxed system can be determined. Since we discuss a linear system, the law of superposition is valid, i.e. : y(U(0 +) = yrt(l)(0 +) -b y('~(0-), i =

(24)

0, 1. . . . , n - - 1 .

Thus we can determine the initial values of both the solution and its n - - 1 derivatives of an unrelaxed system from the values of relaxed system (21) to which the left-hand initial conditions according to equation (24) are added. N o w we shall demonstrate that the relation (23), which is a Laplace transform of the solution of equation (1) at zero initial conditions (relaxed system), includes information on initial values of the solution in dependence on initial values of tile driving function and, consequently, equal information as the system (21). This

]. Kremen and P. Demel ." Initial conditions of linear dynamical syslems

187

is why, for a relaxed system, the solution can be found by the mere consideration of the transfer function in conformity with equation (23). y.(o9

=

lira y . ( t )

=

t-+O +

--+

a,, y . ( o 9

y ' a ( o +) =

--

bll

-

lira p Q ( p ) . x ( p )

--

b,, x ( o 9

=

(25)

- - y.(O+)] =

+

b||

Sa (o+) -

all

.," (09 +

lira p

M (p). x(0 ~) - - N ( p ) . y (0')

p--,~

N (p)

x' (o+) +

lim p p___?~ b.,

lira p IQ (P) P , X ( p ) - - .x. (o+) -Fp--> :¢.

(l n

-

-*

o.

lira p [p Q ( p ) . X ( p ) p--+~

-I- Q ( p ) . x (o+) - -

., (o+)

dln

p----*

P" V,. ~,.(o,) - ,,,, , , ( o ' ) 1 + p " - ' [/,.,_, .,.to,) - (,,,+++- y , ( o ' ) l -t-... -t- b,, x(o+) -.~,, y , (o+)

N (p)

x (09

+

b,, ..' ( o 9

-

~,,,_~ > (o+)

--->

dn

--0.,

s ' . (o,) +

a,,, s . (o+) - -

b, x ( o 9 + Y,d"-') (09 ---,

... +

b,, x' (o+) - -

b. x (,,-*) (09 - -

=

b.,

.,. (09 =

a, y . ( 0 ' )

--

(26)

o,

... - - a,,_t J.(n-u)(0')

all

a, y . (09 +

... +

a,, y , ? . ,, (o+) - -

b, x (o9 - - ... - - / <

The relations (25), (26, (27) are obviously identical with relations (21), which we wished to show. The problem which still remains to be solved is the determination of conditions for which y~') (0 +) = ) , " (0-), /=0,1

(28)

..... n - - 1 .

These equations will be fulfilled if the contribution of driving function to initial values of the solution and to its derivatives yR (j) (t), i -- 0, 1, ..., n ~ 1, will equal zero. To find the necessary and sufficient conditions for the validity of equations (28), we would have to solve the system (21) with respect to y{~) (0+), i =, o, i ..... n - - 1 . This problem will not be discussed any further. Generally, we can conclude: The theorem on the transform of a derivative, applied to n-th derivative f(") (1) of some function f(t), yields the expression z" ( I " (t)) = p,, r (p) - - p , , , f (09

...

f - - - . (o+)

while the known conditions of the existence of Laplace transform of a derivative are fulfilled and the limits f(~) (0~), i = 0, 1, ..., n - - l , mean, in this case, the right-hand limits. If, however, this theorem is applied to two functions, say x(/), y (t), related to each other by a differential equation of the type (t), then these relations also define the link between the initi;fl wdues x (E) (0+) and ),(~)(0'), i = 0, 1 . . . . , n - - 1 , in the following w a y : According to (24), the initial values ),(i)(0 +) have the components ya (~ (0 +) and y(~ (0-). These

, . " - " (o+) =

o.

(27)

values y,(i)(0 +) are related to the values ,x'(i) (0') by equations (21). The values y(i)(0-) represent initial conditions of the solution of equation (1) which are usually prescribed in the formulation of the given equation. Thus, if the left-hand initial conditions y(U (0-) are given, the initial values x (~ (0+) of the input need not be known in advance and the given equation can then be expressed in the form (19). It is equally well possible that in the problem formulation the right-hand initial values y(*)(0 +) can be specified. In this case, however, the values x (i) (0') must be known (which cannot be fulfilled in case of, for instance, the random quantities). Then the respective values c< ~ t can be computed from the relation (18) and these values can be inserted into the equation (4). In a relaxed system all the values y(i)(0-) vanish. In the Laplace transformation, the relation between x ~l)(t) and y~(i)(l), t E < 0 , ~ >, is then g~ven by the equation (23), i.e. by the transfer operator, and this, of course, also determines the relation between the values x ~) (0 *) and yrt (*) (0+). As already demonstrated above, the operator of initial conditions does not Jnclude any relations which could not be expressed from the transfer operator and it has zero wdue with regard to equation (21). In an unrelaxed system the operator of initial conditions depends on initial conditions y(t)(0-), defining the initial energy of the system. Generally, this operator has a value which differs from zero and it includes informadons which are not included in the relation

188

Amaales de l'Associat/o~ haternatioJmle pour le Cahul maalogique

for the transfer operator. Consequently, the unrelaxed system is fully described by both the transfer operator and the operator of (left-hand) initial conditions, or by an equivalent expression in form of a differential equation with the respective initial conditions, which are meant as the left-hand ones. Lee us now focus the attention on the solution of the given equation (1) on an analogue computer. 4. Analogue model of a generally unrelaxed system. Let us return, for a while, to the system (5). The solution of this system is a generalized solution of equation (t). The transition from equation (i) to the system (5) is by no means a unique transformation and it is not even apparent on the first glance, which transformation should be selected. On the other side we know, however, that the finding out of the system (5) is the first step to the realization of an analogue model. Each equation of the system is the equation for one integrator. Generally, the functions f i , i = 0, 1, ..., n ~ 1, are algebraic or transcendent functions of variables y~, x, t, i =, 0, 1, ..., n - - 1 , and - - in case of equation (1) - - they are linear functions of these variables y~, x, i ----- 0, 1, ..., n - - - 1 . The difficulties of model realization will generally concern, to a great extent, the possibility of realization of just these functions f~. Thus it seems that in a general case the analogue model exists, if we know how to realize the model of functions f~. But, speaking quite rigorously, the generalized solution also depends, in addition to that-, on the form of the driving function. An exact answer to the question of existence of a generalized solution gives the CarathSodory's theorem. Theoretically, there exist such driving functions that the functions fi do not fulfill the assumptions of existence of a generalized solution, i.e. in this case the computing network would cease to have any meaning whatsoever. Such functions do not occur in practical applications. From this point of view, we can say that, after having found a correct analogue model, we also found the solution. Let us now generally mention the initial conditions of the model, i.e. the initial conditions of integrators

N" 4 - - Oclobre 1970

of the computing network. The system (5) holds quite generally and it is obvious that the functions Yi (t), = 0, 1, ..., p a - - 1 , are always continuous functions of the variable t. Since, in the model, the functions y~ (t) are the output quantities of integrators, the values y~ (0) are the initial values of these integrators. Thus, to determine the values Yi (0) = Yi (0-), we must know the relations between the given initial conditions y('~) (0-), k = 0, 1. . . . , n - - 1, of the original system (1) and the initial conditions y, (0-), i = O, 1. . . . . n - - 1 , of the system (5). Let us now discuss the analogue model of equation (1). a) Method of sltccessit,e h~tegratiom The system (1.5) represents one possible decomposition of the integral expression of equation (i). Each equation of the system, applies to one integrator in the model. Since we are used to denote the instant at which the model begins to operate (integrate) as zero time, we shall set r--> 0 in relation (15). The initial conditions of integrators will be determined from equation (14), since this equation holds for t C < r, co >. In equation (1) the left-hand initial conditions y(~)(0-), i =, 0, 1 . . . . , n - - 1 , are given and we, therefore, shall also express the initial conditions )',,-~-i (0) of integrators by means of them, i.e. r--> 0-. Then we obtain for the model of equation (t) the following relations from equations (14) and

(,5):

I1-1

ce,,_~_i =

)'....,.i

(0)

=

~. a~,+~

y(l~-L, (0-),

(t) =>_,_,(0)

+ o

+

y,,_, (t)] dr,

i ~

o, 1, ..., n ,

(30)

y, , ~ 0 and now we can easily draw the block-type circuit diagram (fig. 2) of the model computing network.

x (t)

6F7 '

( )-<

61(.9~-e

y/q y~ J- 4) <''~ ___

~ Of,u"2

(29)

1,~=1

y°

+ "O#v"l

Fig. 2. - - Block diagram of successive-integration-method : n

~.... = k£= r ak.yCk-r~ (0-), I'-= 1, 2.... , n

o

/. Kremen and P. Demel ." Initial conditions of linear dynamkal systen/s ..t

b) Melhod of dh'ect simu&tion of in#ial conditiom. The equation (14) and the equation (29), resulting from the former one, are relatively complicated. In case the initial conditions of equation (1) vary many times, we need a great number of auxiliary computations to determine the initial conditions of integrators. W e shall, therefore, describe a transformation which offers such a set of integral equations, where the initial conditions of integrators are given by equations y, (O) = y(*> (0-), i = O, 1 . . . . . n - - l , i,e. where the initial conditions of integrators are equal to given initial conditions of equation (1).

= y (09 + j

l/~ * (~) ÷ y,-, q)l ~/t,

o

(35) yo (0 =

y (t) -

p,, x (t) [~, ,: (t) -F y, (1)1 dr.

(36) Then, generally It-f[ y,,_,, (t)

-

y(,,-k, (t) - - E / L

x,. ',,> (0

j=0

We shall again start with equation (t) and shall divide this equation by the coefficient a, : y", (t) -t+ a,,,

Y" ,, (t) +

ao

... +

dn

dn

t

-- y.-,,, (o-) + f~

I:f~,,.,,, 1

(#

dn

for k = I, 2 ..... n, and for k = 0 the relation (33) is valid. The equation (37) represents the following two relations for the function .Y.-k(1):

(3t)

W e shall introduce new variables Yi (t), i = O, 1..... n--l, so that

... +

an

(/o

(01 ~zt (37)

an

y . , (t) + a,,_, y,,-, (t) +

X (I) @" Yll k,l

y (t)

an

b , l " - (t) + ... + b . _b..,-(,,, (0 + ~- ~,. i, "x

_

189

n-k

J,_,, (t) = y("-")(t) - - X 5J x(n-k-J)(I),

(37a)

,i=o

k = 0, 1, ..., n,

),,, (t)

au

~--- ~ ° X ('l, (I) -k 5 ' .%.(11I, (el) @ ... -k /~,1-%' (t)

(32) where fit, i = O, 1, ..., n, are the meanwhile unknown constants.

this being the relationship between the variable Y,,-k(1) and the original variables x (t), y (t) and their derivatives, and t

Y.-k (t) = yU,-k, (0-) -t- ; 0

[/3,_,,, X(I) -F y,,_k+, (t)l dt,

(37b) k -----1, 2 .... , n,

In equation (32), we shall transfer the members with new variables y, (t) to the right side and the remaining members to the left side. We shall denote both sides of the equation with a new variable y. (t) : y(,,, (t) --/~,,

y,, (t) =

x,,, (t) --...

--/j,,

.,. (t)

which is the integral system to be determined. For k = 0 the equation (37b) does not hold and the corresponding Yl (t) = y, (t) can be expressed from equation (33): i

an_l

-

y,,_, (t) --

all

.....

~o

an,_,

---

...

y,,.~ (t) -

I1-1

E < > (t).

> (t) -

(38)

'111 t=O

(~n

y,, (t).

03)

The relation (38) is an algebraic equation supplementing the system of integral equ,'~tions (37b).

an

Now we shall integrate the equation (33) within the limits < r --> 0-, t >, while both sides of the equation will be made equal to the variable y , , (1). In view of the assumption x( ~' ( r ) ~ O, .; = O, 1, .... n - - 1 , for r < 0 we shall obtain y,,_, (t) =

y("-')

(t) --/2..x'("

') (t) -- ... --/2._,

= y,, ~> (o-) + .-iT,: 15..,'(0

+

x (t)

r (~)1 dr. (34)

By successive integration we obtain after the (n - - 1)-th and the n-th step :

)'t(t)

= y'(t)

--

fi(, x'(t)

--

fit x(l)

The constants /3j can be determined by inserting the relation (37a) into equation (32). We shall modify the equation, gained in this way, so that the derivatives yii) (t) i = 0, 1 . . . . . n, have equal coefficients as those in equation (3t). By comparing the coefficients of derivatives x (~), i = 0, 1, ..., n, with coefficients of equal derivatives in equation (31), we obtain n + 1 equations I I1-1 3~ -

(/,,,, -art

5: < B,i_,,).

(39)

j:n-i

The diagram corresponding to relations (37b) and (38) is illustrated in figure 3.

190

Anna/es

de / ' A s s o c i a t i o n internationa/e

Dour /e Calc/d analogiql/e

N" 4 --

O c l o b r e 1970

x (t) o

)/7.-a

y

~.¢-~)'~~

>,~.#d" ~ ~ ; _ y, (-'U .~

(

()vo

()v,

~>

__ % ( >,<-/(

-a~.,

Y#)

( )o0<,a

Fig. 3. - - Block diagram t)f inltial-conditit)ns-direct-programming method : 1

5, = - -

n i

(b,,~ - - E aj./3j .... ),

Finally, we should note that the mmsition from equation (31) to equation (32) is required from the point of view of model realization. If we would apply the procedure, described above, to equation (31), the function y, (t) from equation (38) would be given by a linear combination of derivatives y(~) (t), i = O, 1 ..... n - - 1 , which do not occur explicitely in the model. If we apply the procedure to equation (32), the function y,, (t) from equation (38) is given by a linear combination of auxiliary variables Yi (t), i = O, 1, ..., n - - 1 , which are at the disposal in the model. The first method (fig. 2) can preferably be used in applications where the initial conditions will not vary frequently, since the change of initial conditions involves the recalculation of values of the constants ~, (initial conditions of integrators). On the other hand, the second method (fig. 3) is suitable in instances where the initial conditions vary rather often, but where the constants a~, b~ remain unchanged, since their change necessitates the recalculation of the constants /7~. The following example will illustrate the deductions explained above. 5. Example, A servomechanism of the first type, &Lm,ped by deviation derivative [6], is described by differential equation 1

d~9,

a),,-'

dt "~

2a +


d~o dt

1 -t-

9, -

o)."

i ..... n

o,

simulating the computation on an analogue computer for both methods, described above, under the following assumptions : a) At the beginning, the system was at rest, i.e.

(o-)

df-"

o,

=

9,,,,

4 (o-)

=

¢..

Solution : The equation (40) will be modified to : 9,'(t) -b 2a~o,,9,' (t) + (o,V'~ =

9,1" (l).

(41)

a) For a system with zero left-hand initial conditions (relaxed system), the Laplace transform of the solution of equation (41) is

p

cl~ (p) =: p" +

2

@~ (p).

(42)

2 a o,. p -t- o,,,~

b) For a system with initial conditions ~o(0-) = ~oA, 9" (0-) = 9,'A, the relation (20) will be used, in which we insert the following values : ]1 =

2

)

a,2

~

1,

dI =

2 a o),,,

ao --

1,

bl -

O,

b(, --= O.

o)n ~ )

For the solution transform we obtain : ,r, (/,)

=

~o(t) is the solution (deviation of position).

P r o b l e m : Compute the Laplace transform of the solution ¢ (p) = .£ [9, (t)} and find the diagram for

=

9, (0-)

b... = (40)

¢(o9

=

b) (Left-hand) initial conditions of the deviation 9, have values differing from zero :

d'-'~

where 9,, (l) is the driving function (datum position), and

i =

+

p

'2

p'-' + 2a ..... p + ,,,£

it, (p)

9,A' P + 2 no< 9,a -t- 9,'A

p~ + 2 a ,,,. p + o,,,'-'

+

(43)

]. Kremen and P. Deme/ ." Initial canal#ions of linear dynandca/ .¢.llytems D i a g r a m of simulation according to the first method (fig. 2) - - the relation (29) will be u s e d : I

g a,.~ ¢ " ' (0-) = < ~ (0-) + a., ~0' (o-) = k=0

2 a ,,,,, ~:A -t- ~',X, 1

,<, ¢" "

(o)

=

~.,_ ~ ( o )

=

~,,

k= I

D i a g r a m of simulation according to the second method we use the relation (39) : (fig. 3 ) b., flo

--

i

1,

aa

1 /'d~ -- --- - ( & - - g a J ae

'

#j_,)

-

b,

--

--

L ITERATUP, E

2 a m.,

1

E

In the second part of the article two methods are derived which substantially facilitate and speed-up the del:ermination of an analogue model of a linear differential equation with constant coefficients with nonzero left-hand initial conditions and with the right side in form of a linear combination of driving function derivatives.

a , fi,,

2 a m.. 1

b. --

These requirements make the computation of the operator of initial conditions considerably difficult since the right-hand initial values of the solution must usually be found with the aid of various deductions b~Lsed on physics. In case of a random driving function the solution is even more difficult. The theory, explained in this article, fully elim.inates these difficulties.

(t2

j:i

0

fi'-' =

191

aj # j

=

b,, - -

a.#.

--

at#,

=

[1] M. Salamon : Pncatecni podminky pri vypoctu obrazu vystupni veliciny Iinearni dynamick6 soustavy (Initial conditions at computation of the transform of output quantity of a linear dynamic system), In Czech. Slaboproudy obzor 27, 1966, pp. 401.-404.

j :. 0 0

:

--

(,),l ~ . 1

4 a e o,,,e - -

--

2 a ~)n ( 2 a ("n)

,,hie :

[2] S. Fifer : Anah~gue Computation. Volume lI, McGraw-Hill Book Company Inc., New York, Tl~ronto, London 1961.

=

o),,2 (4 a ' - ' - - t ) .

The constants flo, fit, ft.-, will be put in figure 3 for n = 2.

[3] K,S. Miller, J.B. Walsh : Initial Cnndithms in Computer Simulation. IRE Tr:msactions EC, Vol. EC 10 (196t), No. 1, pp. 78-80.

6. Conclusion.

[~] V. Dolezal : Prispevek k vysetreni transientu lineamich

In the first part of the article we derived the proof that the Laplace transform of the solution of differential equation (i) is the expression

[5] G. Sansone : Obyknovennyje diferencialnyje uravenija

I1-I £

Y (p) - -

M (p)

N (p)

X (p) +

syst6mu (Contribution to the investigation of transient phenomena in linear systems), in Czech. Slaboproudy obzor 19, 1959, No. 9, pp. 617-624.

~,/-1-i

/~i

i:0

N (p)

where %_r_i is given by equation (t8). This means that a generally unrelaxed linear system is fully described by the well known transfer operator and by the operator of initial conditions, where, however, the operator of initial conditions is determined by the left-hand initial conditions of the solution and by the coefficients of the left side of equation. This relation was occasionnally used in the past without the p r o o f of its validity. In most cases, however, the relations quoted for the operator of initial conditions required the knowledge of right-hand initial values of both the solution and the driving function [61, [7].

(Ordinary differential equations), in Russian. Innostrannaja literatura, Moscow 1954. Z. Trnka : Servomechanismy (Serwmwchanisms). SNTL, Prague 1.960, in Czech.

[7] J. Benes : Theorie servosyst,4mu (Theory of servo-systerns), in Czech. SNTL, Prague 1955.

[s] G. Dnetsch : Anleitung zum praktischen Gebraoch der Laplace-Transformation ([nstructions for practical use of Laplace transformation), ill German. R. OIdenbourgh, Munich 1961.

[9] Solodownikov : Grundlagen der selbstt~itigen Regelung (Principles of automatic control), in German. R. Oldenbourgh, Munich 1958.