--- title: "Closed-loop pulse TF with two samplers and H(s)" subject: "Digital Control System" chapter: 3 key: "semester_6/digital_control_system:closed-loop-pulse-tf-two-samplers" source: https://www.bctengineer.com/q/semester_6/digital_control_system/closed-loop-pulse-tf-two-samplers site: BCT Engineering --- # Closed-loop pulse TF with two samplers and H(s) Digital Control System · Chapter 3: Analysis of Discrete Time Control System - Asked in: 2072 Magh, 4 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/digital_control_system/closed-loop-pulse-tf-two-samplers - Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-3 ## Question Obtain the closed loop pulse transfer function of the system shown below. [Figure: R(s) → summing junction → E(s) → sampler → G1(s) → sampler → G2(s) → C(s); C(s) fed back through H(s) to the negative input of the summing junction] ## Short answer With samplers on the error and between G₁ and G₂, the closed-loop pulse transfer function is C(z)/R(z) = G₁(z)G₂(z) / [1 + G₁(z) G₂H(z)], where G₂H(z)=Z[G₂(s)H(s)]. ## Answer There is a sampler on the error (before $G_1$) and another between $G_1$ and $G_2$. Because $G_2$ and $H$ are not separated by a sampler, they appear together as $G_2H(z)$. ```text E _/ E* +----+ U _/ U* +----+ C(s) R(s)->(S)-o/ o-->| G1 |---o/ o--->| G2 |----+---> ^ - T +----+ T +----+ | | +-----+ | +--------------| H |<---------------+ +-----+ ``` #### Step 1: Signal equations $$ \begin{aligned} C(s) &= G_2(s)U^*(s) \\ U(s) &= G_1(s)E^*(s) \;\Rightarrow\; U^*(s) = G_1^*(s)E^*(s) \\ E(s) &= R(s) - H(s)C(s) = R(s) - G_2(s)H(s)U^*(s) \end{aligned} $$ #### Step 2: Sample the error equation $$ \begin{aligned} E^*(s) &= R^*(s) - (G_2H)^*(s)\,U^*(s) = R^*(s) - (G_2H)^*(s)\,G_1^*(s)E^*(s) \\ E^*(s) &= \frac{R^*(s)}{1 + G_1^*(s)(G_2H)^*(s)} \end{aligned} $$ #### Step 3: Output $$ C^*(s) = G_2^*(s)U^*(s) = G_2^*(s)G_1^*(s)E^*(s) $$ #### Closed-loop pulse transfer function $$ \frac{C(z)}{R(z)} = \frac{G_1(z)\,G_2(z)}{1 + G_1(z)\,G_2H(z)} $$ where - $G_1(z) = \mathcal{Z}[G_1(s)]$ - $G_2(z) = \mathcal{Z}[G_2(s)]$ - $G_2H(z) = \mathcal{Z}[G_2(s)H(s)]$, which is in general not equal to $G_2(z)H(z)$ Special case: with unity feedback ($H = 1$), this becomes $\dfrac{G_1(z)G_2(z)}{1+G_1(z)G_2(z)}$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In the loop with samplers before G1 and between G1 and G2, why does G2 and H appear as the single term G2H(z)? A. Because G2 and H are not separated by a sampler, so they must be combined before sampling B. Because H is always equal to G2 C. Because the ZOH merges them D. Because samplers cancel out in feedback paths Answer: A. Without a sampler between G2 and H, the signal passing through both is continuous, so the product G2(s)H(s) is transformed as one block. (Key point: Why G2H stays together) 2. Which expression correctly gives the continuous error E(s) in this system? A. E(s) = R(s) - G2(s)H(s)U*(s) B. E(s) = R(s) - G1(s)H(s)E*(s) C. E(s) = R*(s) - H(z)C(z) D. E(s) = R(s) + G2(s)H(s)U*(s) Answer: A. C(s) = G2(s)U*(s) and the feedback subtracts H(s)C(s), giving R(s) - G2(s)H(s)U*(s). (Key point: Error equation) 3. What is the closed-loop pulse transfer function C(z)/R(z)? A. G1(z)G2(z) / (1 + G1(z)G2(z)H(z)) B. G1(z)G2(z) / (1 + G1(z)G2H(z)) C. G1G2(z) / (1 + G1G2H(z)) D. G1(z)G2(z) / (1 - G1(z)G2H(z)) Answer: B. Sampling the error equation gives E* = R*/(1 + G1*(G2H)*), and C* = G2*G1*E*. (Key point: Final result) 4. Which statement about G2H(z) is correct? A. It always equals G2(z)H(z) B. It is the z-transform of the product G2(s)H(s), generally not equal to G2(z)H(z) C. It is the z-transform of G2(s) divided by H(s) D. It is always equal to G2(z) + H(z) Answer: B. The z-transform of a product of s-functions is not the product of z-transforms; G2H(z) = Z[G2(s)H(s)]. (Key point: G2H(z) vs G2(z)H(z)) ## Close variants set in other exams - [Closed-loop pulse TF and continuous output C(s)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aclosed-loop-pulse-tf-continuous-output) — Digital Control System, chapter 3, asked 2× ## Related in Digital Control System - [Closed-loop TF with digital PID, plant 1/(s(s+1))](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Apid-closed-loop-plant-s1) — Digital Control System, chapter 3, asked 2× - [Pulse TF for two sampler configurations](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Apulse-tf-sampler-configurations) — Digital Control System, chapter 3, asked 2× - [Pulse TF with PID controller, plant 1/(s+1)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Apid-pulse-tf-first-order) — Digital Control System, chapter 3, asked 2× - [Pulse TF with digital PID, plant 1/(s(s+2))](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Apid-pulse-tf-plant-s2) — Digital Control System, chapter 3 - [Pulse transfer function with first order hold](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Afoh-system-pulse-tf) — Digital Control System, chapter 3 - [Pulse transfer function matrix of a general companion form](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Apulse-tf-matrix-general-companion) — Digital Control System, chapter 5, asked 7× ## Related in other subjects - [Block diagram reduction with G1, G2, H1, H2, feedforward](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abdr-g1g2-h1h2-ff-unity) — Control Systems, chapter 3, asked 2× - [Transfer function with G3 feedforward, H1 and unity loop](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abdr-g1g2g3-h1-unity-2070o) — Control Systems, chapter 3 - [Mason's gain formula: G1–G4 with H1, H2, unity loop](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Amason-g1-g4-h1h2-2075a) — Control Systems, chapter 3, asked 2× - [Block diagram reduction: G1–G4, H1, H2, unity loop](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abdr-g1-g4-h1h2-2075a) — Control Systems, chapter 3, asked 4×