Question Consider the following interacting two tank system (Figure 1), 91 92 A A2 hi hz Ri R2 43 Figure 1. Interacting two tank system. The objective is to control the level in each tank (hu, h2) using the volumetric flows (91, 92). a) Develop the transfer functions describing the change in level in the two tanks with respect to 91,92 (use the following numerical values, A2 = 1 m2 , Az = 0.5 m2, R2 = 0.5 h m2, R2 = 2.0 h m2 and assume that the flow through any valve can be modelled as a linear relationship with respect to liquid level and flow through the valves). [60 marks] b) Draw a block diagram representation of this multi-input- multi-output (MIMO) system clearly showing the transfer functions derived in part a). [10 marks]

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Transcribed Image Text: Consider the following interacting two tank system (Figure 1), 91 92 A A2 hi hz Ri R2 43 Figure 1. Interacting two tank system. The objective is to control the level in each tank (hu, h2) using the volumetric flows (91, 92). a) Develop the transfer functions describing the change in level in the two tanks with respect to 91,92 (use the following numerical values, A2 = 1 m2 , Az = 0.5 m2, R2 = 0.5 h m2, R2 = 2.0 h m2 and assume that the flow through any valve can be modelled as a linear relationship with respect to liquid level and flow through the valves). [60 marks] b) Draw a block diagram representation of this multi-input- multi-output (MIMO) system clearly showing the transfer functions derived in part a). [10 marks]

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The balance on tanks (1 & 2) induding linear rexutance wiol give:\begin{array}{l}\text { at skady stater } \left.\begin{array}{rl}\left(q-q_{12}\right) & =\left(A_{1}\right)(0)=0 \\\& & \left(q-q_{2 s}\right)=0\end{array}\right\} \\\end{array}scolving all equations givatTank-ti \left(Q-Q_{1}\right)=\left(A_{1}\right)\left(d H_{1} / d t\right)-1valve-t?value-2, \quad Q_{2}=\left(\mathrm{H}_{2} \mid \mathrm{R}_{2}\right)Taking Laplace tranforms of (1) (2), (3) & (4) give:We have doriveds\frac{H_{2}(l)}{Q(e)}=\left[\left(T_{1} T_{2}\right) x^{2}+\left(T_{1}+T_{2}+A_{1} R_{2}\right) s+1\right]where, \left(\tau_{1}=A_{1} R_{1}\right) ;\left(\tau_{2}=A_{2} R_{2}\right)given,\begin{array}{l}A_{1}=1 \mathrm{~m}^{2} ; \quad A_{2}=0.5 \mathrm{~m}^{2} ; \quad R_{1}=0.5 \mathrm{hm}^{-2} ; \quad R_{2}=2 \mathrm{hm}^{-2} \\T_{1}=A_{1} R_{1}=(1 \times 0.5)=0.5 \mathrm{~h} \\T_{2}=A_{2} R_{2}=(0.5 \times 2)=\underline{1.0 h} \\\Leftrightarrow\left[\frac{\overline{H_{2}(s)}}{\bar{Q}(s)}=\frac{(2)}{\left[(0.5) s^{2}+(3.5) s+1\right]}\right. \\\end{array}Transfer function\overline{Q(8)} ...