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## Q. 2.6.7

A First-Order Model with Numerator Dynamics

Obtain the transfer function and investigate the response of the following model in terms of the parameter a. The input g(t) is a unit-step function.

$5\overset{.}{x}+10x=a\overset{.}{g}(t)+10g(t) \quad x(0-)=0$

## Verified Solution

Transforming the equation with x(0−) = 0 and solving for the ratio X(s)/G(s) gives the transfer function:

$\frac{X(s)}{G(s)}=\frac{as+10}{5s+10}$

Note that the model has numerator dynamics if a ≠ 0.

For a unit-step input, G(s) = 1/s and

$X(s)=\frac{as+10}{s(5s+10)}=\frac{1}{s}+\frac{a-5}{5}\frac{1}{s+2}$

Thus, the response is

$x(t)=1+\frac{a-5}{5}e^{-2t}$

From this solution or the initial-value theorem, we find that x(0+) = a/5, which is not equal to x(0−) unless a = 0 (which corresponds to the absence of numerator dynamics). The plot of the response is given in Figure 2.6.4 for several values of a. The initial condition is different for each case, but for all cases the response is essentially constant for t >2 because the term $e^{−2t}$ becomes small.