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Chapter 20

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)

Step-by-Step

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