RELATION BETWEEN IMPULSE RESPONSE AND AUTONOMOUS SOLUTION

It follows from (3.4,5a) that the matrix of impulse response functions H = [7i. w] is related to that of the transfer functions by

H(s) = G(s) (3.4,14)

Furthermore, from (3.2,23) we have that G(s) = B_1(s)C, so that

H(s) = B-!(s)C (3.4,15)

or B-!(s) = ЇВДС-1 (3.4,16)

Now in the autonomous case we have (3.3,4)

Подпись: or Подпись: y(s) =H(6')C-iy(0) У (t) = H(«)C-iy(0) Подпись: (3.4,17)

Substitution of (3.4,16) into (3.3,4) yields the result for the autonomous solution with initial condition y(0), i. e.

STEP-FUNCTION RESPONSE

This is like the impulse response treated above except that the input is the unit step function 1(1), with transform 1/s. The response in this case is called the indicial admittance, and is denoted It follows then that

= адї(«) = —

s

(a)

(3.4,18)

or

(b)

Since the initial values (at t = 0~) of hi}(t) and are both zero, the

Подпись: or RELATION BETWEEN IMPULSE RESPONSE AND AUTONOMOUS SOLUTION Подпись: (®) (b) Подпись: (3.4,19)

theorem (2.3,16) shows that

Thus can be found either by direct inversion of (3.4,186) (see examples

in Sec. 2.5) or by integration of By either method the results for first – and second-order systems are readily obtained, and are as follows (for a single input/output pair the indicial subscript is dropped) :

First-order system:

s/(t) = T( 1 – e~t/T) (3.4,20)

Second-order system:

Подпись: ^(t) = — COn1 — en<(eos cot — — sin a)t)l, £ < 1 (3.4,21)

со J

and for £ > 1, s/(t) is given by the r. h.s. of (2.5,5).

Graphs of the indicial responses are given in Figs. 3.96 and 3.11.

RELATION BETWEEN IMPULSE RESPONSE AND AUTONOMOUS SOLUTION

o>nt/2w

Fig. 3.11 Indicial admittance of second-order systems.

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