A dynamic model of the thermal mixing system in Fig. $6.12$ was derived in Chapter 6 . Use this model to do the following:

(a) Derive an expression for the relative gain array.

(b) Design an ideal decoupling control system, assuming that the transmitters and control valves have negligible dynamics.

(c) Are these decouplers physically realizable? If not, suggest appropriate modifications.

Question

A dynamic model of the thermal mixing system in Fig. [latex]6.12[/latex] was derived in Chapter 6 . Use this model to do the following:

(a) Derive an expression for the relative gain array.

(b) Design an ideal decoupling control system, assuming that the transmitters and control valves have negligible dynamics.

(c) Are these decouplers physically realizable? If not, suggest appropriate modifications.

Accepted Answer

a) From Eq. 6-105

[latex]\begin{aligned}\left[\begin{array}{c}T^{\prime}(s) \H^{\prime}(s)\end{array}\right]=&\left[\begin{array}{cc}\frac{\left(\overline{T}_{h}-\overline{T}\right) / \overline{w}}{\tau s+1} & \frac{\left(\overline{T}_{c}-\overline{T}\right) / \overline{w}}{\tau s+1} \\frac{1 / A \rho}{s} & \frac{1 / A \rho}{s}\end{array}\right]\left[\begin{array}{l}W_{h}^{\prime}(s) \W_{c}^{\prime}(s)\end{array}\right] \+&\left[\begin{array}{cc}\frac{\overline{w}_{h} / \overline{w}}{\tau s+1} & \frac{w}{\tau s+1} / \overline{w} \0 & 0\end{array}\right]\left[\begin{array}{c}T_{h}^{\prime}(s) \T_{c}^{\prime}(s)\end{array}\right]\end{aligned}[/latex] (6-105)

[latex]\begin{aligned}&G_{p_{11}}(s)=\frac{\left.\overline{\left(T_{h}\right.}-\overline{T}\right) / \overline{w}}{\tau_{S}+1}, \quad G_{p_{12}}(s)=\frac{\left(\overline{T}_{c}-\overline{T}\right) / \overline{w}}{\tau S+1} \&G_{p_{21}}(s)=\frac{1 / A P}{S} \quad, \quad G_{p_{22}}(s)=\frac{1 / A P}{S} \&\text { Thus } K_{11}=\frac{\overline{T_{h}}-\overline{T}}{\overline{w}}, \quad K_{12}=\frac{\overline{T_{c}}-\overline{T}}{\overline{w}}\end{aligned}[/latex]

and since [latex]G_{p 21}, G_{p 22}[/latex] contain integrating elements,

[latex]\begin{aligned}&K_{21}=\lim _{s \rightarrow 0} s G_{P_{21}}(s)=\frac{1}{A P} \&K_{22}=\lim _{s \rightarrow 0} s G_{P_{22}}(s)=\frac{1}{A P}\end{aligned}[/latex]

Substituting into Eq. 18-34,

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

[latex]\lambda=\frac{1}{1-\frac{\overline{T_{c}}-\overline{T}}{\overline{T_{h}}-\overline{T}}}=\frac{\overline{T_{h}}-\overline{T}}{\overline{T_{h}}-\overline{T_{c}}}[/latex]

Hence [latex]0 \leq \lambda \leq 1[/latex], and the choice of pairing depends on whether [latex]\lambda>0.5[/latex] or not. The RGA is

[latex]^{T\[1cm] }_{\ h} \overset{w_h\hspace{30 pt} \quad w_c}{\left[\begin{array}{cc}{\overline{T} _{h}}-\overline{T} & \overline{T} -\overline{T}_{c} \\overline{\overline{T}_{h}-\overline{T}_{c}} & \overline{\overline{T}_{h}-\overline{T}_{c}} \{\overline{T} -\overline{T}_{c}} & \overline{T}_{h}-\overline{T} \\overline{\overline{T}_{h}-\overline{T}_{c}} & \overline{\overline{T}_{h}-\overline{T} _{c}}\end{array}\right]} [/latex]

b) Assume that [latex]\lambda \geq5[/latex] so that the pairing is [latex]T-w_{h}, h-w_{c}[/latex]. Assume valve gains to be unity. Then the ideal decoupling control system will be as in Fig.18.9 where [latex]Y_{1} \equiv T, Y_{2} \equiv h, U_{1} \equiv w_{h}, U_{2} \equiv w_{ c }[/latex], and using Eqs. 18-78 and 18-80,

[latex]T_{21}=-\frac{G_{p 21}}{G_{p 22}}[/latex] (18-78)

[latex]T_{21}=-\frac{K_{p 21}}{K_{p 22}} \quad \text { and } \quad T_{12}=-\frac{K_{p 12}}{K_{p 11}}[/latex] (18-80)

[latex]\begin{aligned}T_{21}(s)=&-\frac{(1 / A P) s}{(1 / A P) s}=-1 \T_{12}(s)=-& \frac{\left[\left(\overline{T_{c}}-\overline{T}\right) / \overline{w}\right] /(\tau s+1)}{\left[\left(\overline{T_{h}}-\overline{T}\right) / \overline{w}\right] /(\tau s+1)}=\frac{\overline{T}-\overline{T}_{c}}{\overline{T_{h}}-\overline{T}}\end{aligned}[/latex]

c) The above decouplers are physically realizable.

Question 18.8: A dynamic model of the thermal mixing system in Fig. 6.12 wa...

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