In Figs. 18.6 and 18.7, look at the different pairings of controllers. Which one has the larger stability region? How does this compare with the preferred pairing indicated by the RGA(is it the same or is it different)? Can you suggest a reason for the results (dynamic vs. steady-state effects)?

Question

In Figs. [latex]18.6[/latex] and [latex]18.7[/latex], look at the different pairings of controllers. Which one has the larger stability region? How does this compare with the preferred pairing indicated by the RGA (is it the same or is it different)? Can you suggest a reason for the results (dynamic vs. steady-state effects)?

Accepted Answer

[latex]1-2 / 2-1[/latex] controller pairing has a larger stability region compared with [latex]1-1 / 2-2[/latex].

RGA:

[latex]\begin{aligned}&\lambda_{11}=\frac{1}{1-\frac{K_{12} K_{21}}{K_{11} K_{22}}}=\frac{1}{1-\frac{1.5 imes 1.5}{2 imes 2}}=2.28 \&{\left[\begin{array}{cc}2.28 & -1.28 \-1.28 & 2.28\end{array} ight]}\end{aligned}[/latex]

Based on RGA, controller pairing should be [latex]1-1 / 2-2[/latex] to avoid negative values.

Stability analysis is based on dynamic effects and employs the numerical region of controller gain to get a stable closed-loop response. RGA is based on static process gain [latex]\left(K_{i j} ight)[/latex] analysis, which only show the open loop steady state behavior.

For this problem, 1-2/2-1 pairing has a larger stability region, which means choice of [latex]K_{c 1}[/latex] and [latex]K_{c 2}[/latex] has a larger margin with guaranteed stability. However, around the steady state, the negative RGA indicates control loop "fighting", which may be vulnerable to process noise. Thus, 1-2/2-1 pairing should be avoided in this case.

Question 18.24: In Figs. 18.6 and 18.7, look at the different pairings of co...

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