Consider the transfer function model of Example 18.2 but with a gain of $-2$ in $G_{p11}(s)$ and a time delay of unity in each transfer function:

$G_{p}(s)= \begin{bmatrix}\frac{ -2e^{-s} }{10s+1} & \frac{1.5e^{-s} }{s+1} \\ \frac{1.5e^{-s} }{s+1} & \frac{ 2e^{-s} }{10s+1} \end{bmatrix}$ (18-51)

Use the RGA approach to determine the recommended controller pairing based on steady-state considerations. Do dynamic considerations suggest the same pairing?

Question

Consider the transfer function model of Example 18.2 but with a gain of [latex]−2[/latex] in [latex]G_{p11}(s)[/latex] and a time delay of unity in each transfer function:

[latex]G_{p}(s)= \begin{bmatrix}\frac{ -2e^{-s} }{10s+1} & \frac{1.5e^{-s} }{s+1} \ \frac{1.5e^{-s} }{s+1} & \frac{ 2e^{-s} }{10s+1} \end{bmatrix}[/latex] (18-51)

Use the RGA approach to determine the recommended controller pairing based on steady-state considerations. Do dynamic considerations suggest the same pairing?

Accepted Answer

The corresponding steady-state gain matrix is

[latex]K= \begin{bmatrix} -2 & 1.5 \ 1.5 & 2 \end{bmatrix}[/latex] (18-52)

Using the formula in Eq. 18-34, we obtain [latex]\lambda _{11} =0.64[/latex]. Thus, the RGA analysis indicates that the 1-1/2-2 pairing should be used. However, the off-diagonal time constants in Eq. 18-51 are only one-tenth of the diagonal time constants. Thus, [latex]y_{1}[/latex] responds 10 times faster to [latex]u_{2}[/latex] than to [latex]u_{1}[/latex];similarly, [latex]y_{2}[/latex] responds 10 times faster to [latex]u_{1}[/latex] than to [latex]u_{2}[/latex]. Consequently, the 1-2/2-1 pairing is favored based on dynamic considerations, and a conflict exists between steadystate and dynamic considerations that should be resolved by dynamic simulation studies. Extensions of the RGA to include process dynamics are discussed next.

[latex]\lambda _{11} = \frac{1}{1-\frac{K_{12}K_{21} }{K_{11}K_{22} } } [/latex] (18-34)

Question 18.5: Consider the transfer function model of Example 18.2 but wit...

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