The Final Value Theorem and Ramp Response Obtain the steady-state difference f (∞) − v(∞) between the input and output of the following model: [latex]\tau \dot{v} + v = bf (t)[/latex], where b is a constant and f (t) = mt. Assume that v(0) = 0 and that the model is stable ([latex]\tau > 0[/latex]).

The transform of the response is [latex]V(s) = \frac{b}{\tau s + 1} F(s) = \frac{b}{\tau s + 1} \frac{m}{s^{2}} [/latex] Use this with the final value theorem to find the steady-state difference: [latex]f(\infty) − v(\infty) = \lim_{s→0} [s F(s)] − \lim_{s→0} [sV(s)] = \lim_{s→0} s[F(s) − V(s)] [/latex] [latex]= \lim_{s→0} s \left(\frac{m}{s^{2}} − \frac{b}{\tau s + 1} \frac{m}{s^{2}} \right) [/latex] [latex]= \lim_{s→0} \frac{m}{s} \left(\frac{\tau s + 1 − b}{\tau s + 1} \right) [/latex] [latex] = \left \{ \begin{matrix} \infty b \neq 1 \\ m \tau b = 1 \end{matrix} \right.[/latex] Thus, the steady-state difference is infinite unless b = 1. Both the input and output approach straight lines at steady state. The preceding result shows that the lines diverge unless b = 1. If b = 1, the lines are a vertical distance mτ apart. This is the case shown in Figure 9.1.8.

Question 9.1.2: The Final Value Theorem and Ramp Response Obtain the steady-...

System Dynamics

The Final Value Theorem and Ramp Response

Obtain the steady-state difference f (∞) − v(∞) between the input and output of the following model: $\tau \dot{v} + v = bf (t)$ , where b is a constant and f (t) = mt. Assume that v(0) = 0 and that the model is stable ( $\tau > 0$ ).

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The transform of the response is
$V(s) = \frac{b}{\tau s + 1} F(s) = \frac{b}{\tau s + 1} \frac{m}{s^{2}}$
Use this with the final value theorem to find the steady-state difference:
$f(\infty) − v(\infty) = \lim_{s→0} [s F(s)] − \lim_{s→0} [sV(s)] = \lim_{s→0} s[F(s) − V(s)]$

= \lim_{s→0} s \left(\frac{m}{s^{2}}  −  \frac{b}{\tau s  +  1} \frac{m}{s^{2}} \right)

= \lim_{s→0} \frac{m}{s} \left(\frac{\tau s  +  1  −  b}{\tau s  +  1} \right)

$= \left \{ \begin{matrix} \infty b \neq 1 \\ m \tau b = 1 \end{matrix} \right.$
Thus, the steady-state difference is infinite unless b = 1. Both the input and output approach straight lines at steady state. The preceding result shows that the lines diverge unless b = 1. If b = 1, the lines are a vertical distance mτ apart. This is the case shown in Figure 9.1.8.