Sampled Data System and the z-Transform Thanh Vo-Duy Department of Industrial Automation thanh.voduy@hust.edu.vn Content • The Sampling Process • The z-Transform • Pulse Transfer Function and Manipulation of Block Diagram • Exercises Prior to Lecture • Conventional Control System Sampled data control system The Sampling Process • A Sampler is a switch that closes every T seconds The Sampling Process • The closure time is much smaller than T The Sampling Process • In ideal sampling, closure time can be neglected The Sampling Process                                  The Sampling Process • Now   for            • Taking Laplace transform            The Sampling Process • From sampled   to continuous output  Use Zero-Order Hold circuit. The Sampling Process • Transfer function:      • Taking the Laplace Transform:                Impulse response of ZOH [...]... ratio of the z-transform of the sampled output and input at the sample instants • A ratio of the z-transform of the output and input when both input and output are trains of pulse • … • Pulse Transfer Function = Discrete Transfer Function Pulse Transfer Function and Manipulation of Block Diagram Definition • Let’s sample the system below we obtain: 𝑦∗ 𝑠 = 𝑒∗ 𝑠 𝐺 𝑠 ∗ → 𝑦 ∗ 𝑠 = 𝑒∗ 𝑠 𝐺 ∗ 𝑠 and 𝑦 𝑧 = 𝑒... Sampling Process • Example The z-Transform Definition • From 𝑅 ∗ 𝑠 = ∞ 𝑛=0 𝑟 𝑛𝑇 𝑒 −𝑠𝑛𝑇 , define: 𝑍 = 𝑒 𝑠𝑇 ∞ 𝑟 𝑛𝑇 𝑧 −𝑛 → 𝑅 𝑧 = 𝑛=0 • z-Transformation in sampled data system • Laplace Transformation in continuous-time systems The z-Transform Commonly used functions • Unit step function 0, 𝑟 𝑛𝑇 = 1, 𝑛 1 The z-Transform Commonly used functions... 𝑧 → 𝐺(𝑧) is call Pulse Transfer Function Pulse Transfer Function and Manipulation of Block Diagram Open-Loop Systems • Example 1: Figure below shows an open-loop sampled data system Derive the z-transform of the output 𝐺1 (𝑠) 𝑒(𝑠) 𝑒 ∗ (𝑠) 1 𝑠 𝐺2 (𝑠) 𝑎 𝑠+ 𝑎 𝑦(𝑠) 𝑦 ∗ (𝑠) Pulse Transfer Function and Manipulation of Block Diagram Open-Loop Systems • Solution for Example 1 𝑦 𝑠 = 𝑒 ∗ 𝑠 𝐺1 𝑠 𝐺2 𝑠 = 𝑒 ∗ 𝑠... 𝑒 𝑧 𝑧 − 1 𝑧 − 𝑒 −𝑎𝑇 𝑎 = 𝑍 𝑠 𝑠+ 𝑎 Pulse Transfer Function and Manipulation of Block Diagram Open-Loop Systems • Example 2: Figure below shows an open-loop sampled data system Derive the z-transform of the output 𝐺1 (𝑠) 𝑒(𝑠) 𝑒 ∗ (𝑠) 1 𝑠 𝑥(𝑠) 𝑥 ∗ (𝑠) 𝐺2 (𝑠) 𝑎 𝑠+ 𝑎 𝑦(𝑠) 𝑦 ∗ (𝑠) Pulse Transfer Function and Manipulation of Block Diagram Open-Loop Systems • Solution for Example 2: ∗ 𝑥 𝑠 = 𝑒 ∗ 𝑠 𝐺1 𝑠 → 𝑥... http://lpsa.swarthmore.edu/LaplaceZTable/LaplaceZFuncTable.html The z-Transform z-Transform of Laplace Transfer Function • Method 1: G(s) → g(t) → G(z) • Method 2: G(s) → G(z) by equivalent z-Transform • Method 3: 𝐺 𝑠 = 𝑁(𝑠)/𝐷(𝑠) 𝑝 𝑁 𝑥𝑛 1 → 𝐺 𝑧 = 𝐷 ′ 𝑥 𝑛 1 − 𝑒 𝑥 𝑛 𝑇 𝑧 −1 𝑛=1 where: 𝐷 ′ = 𝜕𝐷/𝜕𝑠 and 𝑥 𝑛 , 𝑛 = 1,2, … , 𝑝 are roots of equation 𝐷 𝑠 = 0 The z-Transform z-Transform of Laplace Transfer Function • Example Find z-Transform of G(s) 1 𝐺... Open-Loop Time Response • Solution (cont.) The time response in this case: Pulse Transfer Function and Manipulation of Block Diagram Open-Loop Time Response • Solution (cont.) Pulse Transfer Function and Manipulation of Block Diagram Closed-Loop System • Example 1: Derive the z-transform of closed-loop system below • Result: 𝑦 𝑧 𝐺 𝑧 = 𝑟 𝑧 1 + 𝐺𝐻 𝑧 ... − −2𝑇 𝑧− 𝑒 𝑧 − 𝑒 −3𝑇 𝑧 𝑒 −2𝑇 − 𝑒 −3𝑇 → 𝐺 𝑧 = 𝑧 − 𝑒 −2𝑇 𝑧 − 𝑒 −3𝑇 The z-Transform z-Transform of Laplace Transfer Function • Method 3: 𝐷 ′ 𝑠 = 2𝑠 + 5 𝐷 𝑠 = 0 → 𝑥1 = −2; 𝑥2 = −3 2 𝑁 𝑥𝑛 1 𝐺 𝑧 = 𝐷 ′ 𝑥 𝑛 1 − 𝑒 𝑥 𝑛 𝑇 𝑧−1 𝑛=1 1 1 1 1 → 𝐺 𝑧 = + −2𝑇 𝑧 −1 11 − 𝑒 −1 1 − 𝑒 −3𝑇 𝑧 −1 𝑧 𝑧 → 𝐺 𝑧 = − −2𝑇 𝑧− 𝑒 𝑧 − 𝑒 −3𝑇 The z-Transform Properties of z-Transform • Linearity 𝑍 𝑎 𝑓 𝑛𝑇 ± 𝑏 𝑔 𝑛𝑇 • Time Shifting • Attenuation... 𝑧 −𝑚 𝐹(𝑧) 𝑍 𝑒 −𝑎𝑛𝑇 𝑓 𝑛𝑇 = 𝐹(𝑧𝑒 𝑎𝑇 ) The z-Transform Inverse z-Transform • Method to inverse Y(z) to y(t) • Power Series (long division) 𝑌 𝑧 = 𝑦0 + 𝑦1 𝑧 −1 + 𝑦2 𝑧 −2 + ⋯ → 𝑦 𝑡 = 𝑦0 𝛿 𝑡 − 𝑇 + 𝑦1 𝛿 𝑡 − 2𝑇 + 𝑦2 𝛿 𝑡 − 3𝑇 + ⋯ • Expanding Y(z) into partial fraction • Inversion formula method: using inversion integral 1 𝑦 𝑛𝑇 = 2𝜋𝑗 𝑌 𝑧 𝑧 −1 𝑑𝑧 𝑟 Pulse Transfer Function and Manipulation of Block Diagram Definition... Function and Manipulation of Block Diagram Open-Loop Time Response • Consider the system with ZOH as below What will the output response if a unit step input is applied? 𝑢(𝑠) 𝑢 ∗ (𝑠) Assume that T = 1s 𝐺1 (𝑠) ZOH 𝐺2 (𝑠) 1 𝑠+1 𝑦(𝑠) Pulse Transfer Function and Manipulation of Block Diagram Open-Loop Time Response • Solution Transfer function of ZOH: 1 − 𝑒 −𝑇𝑠 𝐺1 𝑠 = 𝑠 Equivalent transfer function of system. .. Find z-Transform of G(s) 1 𝐺 𝑠 = 2 𝑠 + 5𝑠 + 6 • Solution 1 1 1 𝐺 𝑠 = = − 𝑠+2 𝑠+3 𝑠+2 𝑠+3 The z-Transform z-Transform of Laplace Transfer Function • Method 1: 𝑔 𝑡 = 𝐿−1 𝐺 𝑠 ∞ = 𝑒 −2𝑡 − 𝑒 −3𝑡 𝑒 −2𝑛𝑇 − 𝑒 −3𝑛𝑇 𝑧 −𝑛 → 𝐺 𝑧 = 𝑛=0 𝑧 𝑧 → 𝐺 𝑧 = − −2𝑇 𝑧− 𝑒 𝑧 − 𝑒 −3𝑇 𝑧 𝑒 −2𝑇 − 𝑒 −3𝑇 → 𝐺 𝑧 = 𝑧 − 𝑒 −2𝑇 𝑧 − 𝑒 −3𝑇 The z-Transform z-Transform of Laplace Transfer Function • Method 2: 1 1 𝐺 𝑠 = − 𝑠+2 𝑠+3 By looking for equivalent . Sampled Data System and the z-Transform Thanh Vo-Duy Department of Industrial Automation thanh.voduy@hust.edu.vn Content • The Sampling Process • The z-Transform • Pulse Transfer Function and. Example The z-Transform Definition • From            , define:             • z-Transformation in sampled data system • Laplace Transformation in continuous-time systems The. Manipulation of Block Diagram • Exercises Prior to Lecture • Conventional Control System Sampled data control system The Sampling Process • A Sampler is a switch that closes every T seconds The