Impulse Response

The system is specified to be initially quiescent and at time zero is subjected to a sin­gle impulsive input

eft) = 8(t) (7.3,1)

The Laplace transform of the ith component of the output is then

x,(s) = Gfs)8(s)

which, from Table (A. l), item 1, becomes

= Gifs)

This response to the unit impulse is called the impulse response or impulsive admit­tance and is denoted hft). It follows that

hfs) = G^s) (a)

that is, G(s) is the Laplace transform of h(t)

Gfs) = f hft)e”dt

•in

From the inversion theorem, (A.2,11) lift) is then given by

Kf t) = ^7 j[ G,/s)eH ds (7.3,3)

Now if the system is stable, all the eigenvalues, which are the poles of G„(.v) lie in the left half of the і plane, and this is the usual case of interest. The line integral of

(7.3,3) can then be taken on the imaginary axis, s = і to, so that (7.3,3) leads to

h,/t) = -1- f eiM%(ico) doj (7.3,4)

that is, it is the inverse Fourier transform of G, y(/a>). The significance of Си(іш) will be seen later.

For a first-order component of Fig. 7.4 with eigenvalue A the differential equa­tion is

x — Ax = c (7.3,5)

for which we easily get

. 1

G(s) = h(s) = ——— (7.3,6)

s — A

The inverse is found directly from item 8 of Table A. 1 as

h(t) = ex’

For convenience in interpretation, A is frequently written as A = — 1/Г, where T is termed the time constant of the system. Then

h(t) = e-^ (7.3,7)

A graph of h{t) is presented in Fig. 1.5a, and shows clearly the significance of the time constant T.

For a second-order component of Fig. 7.4 the differential equation is

У + Ч^пУ + = c

where x = [y yY is the state vector. It easily follows that

Let the eigenvalues be A = n± iu>, where

n =

to (Г2)"2

then h(s) becomes

1

(s — n — ia))(s — n + ico) 1

(s — n)2 + ш2

and the inverse is found from item 13, Table A. l to be

1

h(t) = — ent sin cot (7.3,11)

CO

For a stable system n is negative and (7.3,11) describes a damped sinusoid of fre­quency co. This is plotted for various £ in Fig. 7.6. Note that the coordinates are so chosen as to lead to a one-parameter family of curves. Actually the above result only applies for I < 1. The corresponding expression for £ > 1 is easily found by the same method and is

where

a,’ = con(£2 – 1 )1/2

Graphs of (7.3,12) are also included in Fig. 7.6, although in this case the second-or­der representation could be replaced by two first-order elements in series.

7.4 Step-Function Response

This is like the impulse response treated above except that the input is the unit step function I(t), with transform ls (Table A. l). The response in this case is called the step response or indicial admittance, and is denoted It follows then that

(a)

(7.4Д)

(b)

Since the initial values (at t = 0 ) of h^t) and Ml ^t) are both zero, the theorem (A.2,4) shows that

Thus siyit) can be found either by direct inversion of (7.4,lb) or by integration of hjft). By either method the results for first – and second-order systems are readily ob­tained, and are as follows (for a single input/response pair the subscript is dropped):

Second-order system:

For £ > 1, see Appendix A.2.

Graphs of the indicial responses are given in Figs. 7.5b and 7.7.

The asymptotic value of M(t) as t —> °° is called the static gain K. Applying the final value theorem (A.2,12) (7.4,1) yields

lim sd(t) = lim s. A(s) = lim G(s)

К = lim G(s)

s—*0

f—>oo s—>0 s—*0

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