# Find the range of variables based on the steady-state response

I have a transfer function as below:

$$G(s) = \dfrac{s(s+2)}{s^2+(b+2)s+ab}$$

Now with input as

$$u(t) = \sin{t}$$

The maximum steady-state response of y(t) is less than 1, I would like to find the range of a and b.

I have tried to use the Final Value theorem or try to use inverse Laplace transform, but all seems weird, I guess that is mostly because I am not giving the initial value to be zero.

$$\lim_{t\to ∞}g(t) = \lim_{s\to 0}sG(s)U(s) = \lim_{s\to 0}s\dfrac{s(s+2)}{s^2+(b+2)s+ab}\dfrac{1}{s^2+1}$$

OR

$$y(t) = \mathscr{L^{-1}}[\dfrac{s(s+2)}{s^2+(b+2)s+ab} \cdot \dfrac{1}{s^2 + 1}]$$

As far as I know, the response will also be sine waves since the input is a sine wave, and therefore, the only difference lies in the magnitude and phase, however, my attempts seem not working. Where should I revise?

• The final value theorem requires that the limits exist. The function sin(t) does not have a limiting value as t goes to infinity. Jul 15, 2023 at 15:02

\begin{align*} G_D(s) = \dfrac{s(s+2)}{s^2+(b+2)s+ab} \end{align*}
\begin{align*} & G_D(jω) = \dfrac{jω(jω+2)}{(jω)^2 + j(b+2)ω + ab} \\\ |G_D(jω)| & = \dfrac{-ω^2 + 2jω}{-ω^2 + j(b+2)ω + ab}|_{ω=1} = \dfrac{|-1+2j|}{|-1+ab+j(b+2)|} \\\ & = \dfrac{\sqrt{5}}{\sqrt{(ab-1)^2+(2+b)^2}} < 1 \end{align*}
$$(ab-1)^2 + (2+b)^2 > 5$$