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## Q. 20.2

A process (including sensor and control valve) can be modeled by the transfer function.

$G(s)=\frac{4(1-3 s)}{(15 s+1)(5 s+1)}$

(a) Derive an analytical expression for the response to a unit step change in the input.

(b) Suppose that the maximum allowable value for the model horizon is $N=30$. What value of the sampling period $\Delta t$ should be specified to ensure that the step-response model covers a period of at least $99 \%$ of the open-loop settling time? (That is, we require that $N \Delta t \geq t_{99}$ where $t_{99}$ is the $99 \%$ settling time. $)$ Use the analytical solution and this value of $\Delta t$ to obtain a step-response model in the form of Eq. 20-1.

$y(k+1)=y(0)+\sum\limits_{i=1}^{N} h_{i} u(k-i+1)$

## Verified Solution

a) Note that $G(s)=G_{v}(s) G_{p}(s) G_{m}(s) .$ From Figure $12.2$,

$\frac{Y_{m}(s)}{P(s)}=G(s)=\frac{4(1-3 s)}{(15 s+1)(5 s+1)}$            (1)

For a unit step change, $P(s)=1 / s$, and (1) becomes:

$Y_{m}(s)=\frac{1}{s} \frac{4(1-3 s)}{(15 s+1)(5 s+1)}$

Partial Fraction Expansion:

$Y_{m}(s)=\frac{A}{s}+\frac{B}{(15 s+1)}+\frac{C}{(5 s+1)}=\frac{1}{s} \frac{4(1-3 s)}{(15 s+1)(5 s+1)}$           (2)

where:

\begin{aligned}&A=\left.\frac{4(1-3 s)}{(15 s+1)(5 s+1)}\right|_{s=0}=4 \\&B=\left.\frac{4(1-3 s)}{s(5 s+1)}\right|_{s=-\frac{1}{15}}=-108 \\&C=\left.\frac{4(1-3 s)}{s(15 s+1)}\right|_{s=-\frac{1}{5}}=16\end{aligned}

Substitute into (2) and take the inverse Laplace transform:

$y_{m}(t)=4-\frac{36}{5} e^{-t / 15}+\frac{16}{5} e^{-t / 5}$                  (3)

b) The new steady-state value is obtained from (3) to be $y_{m}(\infty)=4$.

For $t=t_{99}, \quad y_{m}(t)=0.99 y_{m}(\infty)=3.96$. Substitute into (3)

$3.96=4-\frac{36}{5} e^{-t_{99} / 15}+\frac{16}{5} e^{-t_{99} / 5}$

Solving (4) for $t_{99}$ gives $t_{99} \approx 77.9 min$

Thus, we specify that $\Delta t =77.9 / 30 \approx 3 min$

Table S20.2. Step response coefficients

 k t (min) $S_{i}$ k t (min) $S_{i}$ k t (min) $S_{i}$ 1 3 -0.139 11 33 3.207 21 63 3.892 2 6 0.138 12 36 3.349 22 66 3.912 3 9 0.578 13 39 3.467 23 69 3.928 4 12 1.055 14 42 3.563 24 72 3.941 5 15 1.511 15 45 3.642 25 75 3.951 6 18 1.919 16 48 3.707 26 78 3.960 7 21 2.272 17 51 3.760 27 81 3.967 8 24 2.573 18 54 3.803 28 84 3.973 9 27 2.824 19 57 3.839 29 87 3.978 10 30 3.034 20 60 3.868 30 90 3.982