Let's break down the solution into **3 steps** and explain each with a final block diagram at the end. --- ### **Step 1: Identify Components and Their Transfer Functions** - **Controller:** PI Controller ⇒ Transfer Function: \( G_c(s) = K_c\left(1 + \frac{1}{\tau_I s}\right) \) - **Valve:** First Order ⇒ Transfer Function: \( G_v(s) = \frac{1}{\tau_v s + 1} \) - **Sensor:** Second Order ⇒ Transfer Function: \( G_s(s) = \frac{1}{\tau_s^2 s^2 + 2 \zeta \tau_s s + 1} \) - **Tank (Process):** Usually a first-order system, \( G_p(s) = \frac{K_p}{\tau_p s + 1} \) (assuming standard tank dynamics). **Explanation:** We need to represent each part (controller, valve, process, sensor) as a transfer function block for the block diagram. --- ### **Step 2: Arrange the Blocks in Sequence** The signal flow is: - **Setpoint \(h_{sp}\)** - Compare with measured value (\(h_m\)) to get error (\(\epsilon\)) - Error goes to **Controller \(G_c(s)\)** - Output goes to **Valve \(G_v(s)\)** - Valve output (flow) goes to **Process (Tank) \(G_p(s)\)** - Tank output (level \(h\)) goes to **Sensor \(G_s(s)\)** - Sensor output is the **measured level \(h_m\)**, which feeds back to the summing junction. **Explanation:** This step arranges the components in the order the signal flows, setting up the feedback loop structure. --- ### **Step 3: Draw the Block Diagram** The block diagram structure is: ``` +-------------------------------+ | | | v +----------+ +----------+ +----------+ +----------+ +----------+ | h_sp (+) |-->| G_c(s) |-->| G_v(s) |-->| G_p(s) |-->| G_s(s) | +-----|----+ +----------+ +----------+ +----------+ +----------+ | | |<--------------------- Feedback (h_m) <----------------------+ ``` Where: - \( G_c(s) \) = Controller (PI) - \( G_v(s) \) = Valve (First Order) - \( G_p(s) \) = Process/Tank (First Order) - \( G_s(s) \) = Sensor (Second Order) **Explanation:** This diagram visually connects all the transfer functions in a closed-loop feedback system, matching the physical system's control flow. --- ## **Final Answer (Block Diagram):** \[ \boxed{ \begin{array}{ccccccccc} h_{sp} & \xrightarrow{+} & \text{[Summing]} & \xrightarrow{\epsilon} & G_c(s) & \rightarrow & G_v(s) & \rightarrow & G_p(s) \\ & & \uparrow & & & & & & \downarrow \\ & & h_m \leftarrow G_s(s) \leftarrow \text{(feedback)} & & & & & & \end{array} } \] **Block order:** **Setpoint** → **[Summing Junction]** → **PI Controller** → **Valve (1st Order)** → **Tank (Process)** → **Sensor (2nd Order)** → **Output/Feedback** This is the required block diagram for the described liquid-level control system.