Week 2 Transfer Function PDF

Chapter 1: Introduction to Control System Ts Dr Mohamad Heerwan Bin Peeie Recap Chapter 1 2 Why learn Applied Control System? Building models...

Chapter 1: Introduction to Control System Ts Dr Mohamad Heerwan Bin Peeie Recap Chapter 1 2 Why learn Applied Control System? Building models Simulating predictions Dynamic interactions Filtering & rejecting noise Selecting and building Source: subaru-cyprus.com Source: quora.com by Zachary E. Fishbein hardware Testing ➔ It’s understanding your system Source: zerotohundred.com Source: spiraxsarco.com What are common difficulties in applied control system? Don't like working with things that you can't see Too abstract to be mentally pictured Gap between theory and practice Source: elprocus.com  Key to understand Applied Control System Ability to mental picture and imagine> Have a sense of physics. Understand principles of behavior: dynamics, fluid mechanics, heat transfer, electrical ➔ They are governed by similar principles Solid foundation in mathematics!  Basic Control System Concepts Definition : A control system consists of subsystems and processes (or plants) assembled for the purpose of obtaining a desired output with desired performance, Input / excitation Output / response Control (Desired System (Actual response) response)  Definition Control: Measuring the value of the controlled variable (output) of the system and applying accordingly the manipulated variable (input) to make the two as equal as necessary. A system: is a combination of components that act together and perform a certain objective. “A control system consists of subsystems and plant assembled for the purpose of controlling the outputs of the plants.” -> Norman S. Nise  Example : elevator Example Consider an elevator, when the third- floor button is pressed on the ground floor, the elevator rises to the third floor with a speed and floor-leveling accuracy designed for passenger comfort. Source: commons.wikimedia.org  Example : elevator Input / excitation Output / response Elevator 3rd floor System 3rd floor ±error 3 Basic Control Syste  Elements of control system A control system consists of these elements: 1. Input: the desired response of a control system 2. Output: the actual response of a control system 3. Subsystem: any system that helps controlling the output of the control system 4. Plant: a system where the output is the variable to be controlled. Elements of control system A control system consists of these elements: 1. Input: the desired response of a control system 2. Output: the actual response of a control system 3. Subsystem: any system that helps controlling the output of the control system 4. Plant/Process: a system where the output is the variable to be controlled. Example: air-conditioning Plant: The room Output: Actual luminosity Actuator: bulb Input: Desired luminosity Subsystems: Switch & bulb Types of control systems Types of control system : - OPEN LOOP SYSTEM -  Open loop vs. Closed loop Open loop Closed loop No continuous control over the output Continuous control over the output The input has predict / to take into Eventual disturbance will be auto- account eventual disturbance corrected No feedback loop Additional sensor and comparator need to be integrated into the feedback loop The input into the controller is constant The input into the controller is the error signal (which varies in function of disturbance)  Closed loop system: Example Taking the previous example of having a certain desired luminosity in a room, the open loop system can be transformed to closed loop as shown below  Control systems design process From a physical systems to control diagram, several steps are involved including the Steps below. Throughout the course, we will see those elements Determination of Drawing of functional Transforming physical physical systems and its block diagram system into schematic specifications Analyzing and Reduce the multiple Obtain signal flow by validation regarding blocks into single block connecting the diagram the specifications Modelling of dynamic System using Transfer Function 16  Lecture Outline Lesson 1. Lesson 2. Modelling of Dynamic System via Transfer Function Permanent Magnet DC Motor Modelling 2024 BTD2232: Applied Control System Block diagram (input, CS, output) BTD2232: Applied Control System  First Lesson We will cover these skills: Laplace Transform Transfer Function – Frequency Domain (FD) Inverse Laplace Transform 2024 BTD2232: Applied Control System Laplace Transform A technique to solve differential equation Transforming time domain function to frequency domain function No need to carry out differentiation or integration. 2024 BTD2232: Applied Control System  Laplace Transform – Frequency Domain In control systems, transforming a function from the time domain to the frequency domain, such as using a Laplace Transform is useful for several reasons: Simplified Analysis: In the frequency domain, differential equations are converted into algebraic equations, which are easier to manipulate and solve. For example, the Laplace transform converts time-domain differential equations into simple polynomial equations. Insight into System Behavior: Frequency domain methods give insight into the system's stability, transient response, and steady-state behavior. Bode plots, Nyquist plots, and root-locus diagrams, which are frequency domain tools, help in understanding how the system responds to different frequency inputs. Control Design: Many control design techniques (e.g., PID tuning, compensator design) are easier to apply in the frequency domain. It helps in designing controllers that ensure desired behavior across a range of frequencies. Signal Filtering: In the frequency domain, it's easier to design filters that eliminate unwanted noise or signals, since noise often appears as high-frequency components that can be removed more easily in the frequency domain. System Stability: Frequency domain methods provide tools to assess system stability, such as using poles and zeros in transfer functions. Techniques like the Nyquist criterion and Bode stability criteria rely on frequency domain representations. Laplace Transform Table 2024 BTD2232: Applied Control System Laplace Transform Table 2024 BTD2232: Applied Control System Laplace Transform Theorem 2024 BTD2232: Applied Control System  Laplace Transform Theorem 2024 BTD2232: Applied Control System What is Transfer Function? Frequency domain mathematical model that separates input from output dc(t ) + 2c(t ) = r (t ) dt Who is input and output? 2024 BTD2232: Applied Control System What is Transfer Function? dc(t ) + 2c(t ) = r (t ) dt sC ( s ) + 2C ( s ) = R( s ) C (s) 1 = R( s) s + 2 2024 BTD2232: Applied Control System  What is Transfer Function? 𝑑𝑐(𝑡) + 2𝑐(𝑡) = 𝑟(𝑡) 𝑑𝑡 By taking the Laplace transform of both sides of this equation, we obtain the transfer function representation: C ( s) 1 = R( s ) s + 2 Where: C(s) is the Laplace transform of the output c(t) R(s) is the Laplace transform of the input r(t) The transfer function 1 / (s + 2) relates the output C(s) to the input R(s) in the Laplace domain (s-domain). The transfer function 1 / (s + 2) describes the relationship between the input r(t) and the output c(t) in the frequency domain, enabling analysis and design of the system's behavior. 2024 BTD2232: Applied Control System Converting differential equation to transfer function d 3c(t ) d 2 c(t ) dc(t ) dr 2 (t ) dr(t ) 3 +3 +7 + 5c(t ) = +4 + 3r (t ) dt dt 2 dt dt 2 dt 2024 BTD2232: Applied Control System Converting transfer function to differential equation In control systems and signal processing, G(s) typically represents the transfer function of a system in the Laplace domain (s-domain). The transfer function G(s) relates the output of a system to its input in the complex frequency domain (s-domain). It is defined as the ratio of the Laplace transform of the output signal to the Laplace transform of the input signal, assuming zero initial conditions. Mathematically, the transfer function G(s) is expressed as: G(s) = Y(s) / X(s) Where: Y(s) is the Laplace transform of the output signal X(s) is the Laplace transform of the input signal 2024 BTD2232: Applied Control System Converting transfer function to differential equation 2s + 1 C ( s) G( s) = 2 = s + 6s + 2 R( s ) 2024 BTD2232: Applied Control System  Inverse Laplace Transform To convert transfer function from frequency domain to time domain Response/output of the system 2024 BTD2232: Applied Control System Partial Fraction Expansion A mathematical technique to help taking Inverse Laplace Transform 2 F (s) = ( s + 1)( s + 2) By Partial Fraction Expansion, K1 K2 F (s) = + s +1 s + 2 L−1[ F ( s)] = f (t ) = K1e −t + K 2 e − 2t 2024 BTD2232: Applied Control System Real and distinct roots ( s + 2) A B Y ( s) = = + s( s + 5) s s + 5 ( s + 2) 2 ( s + 2) 3 A= = B= = ( s + 5) s →0 5 ( s) s → −5 5 2 3 Y ( s) = 5 + 5 s s+5 2 0 t 3 − 5t 2 3 − 5 t y (t ) = e + e = + e 5 5 5 5 2024 BTD2232: Applied Control System  First Lesson Summary Use Laplace Transform to convert signal from time domain to frequency domain by using Laplace Table and Theorem Determine the transfer function (TF) of the control system by separating output and input Convert the signal from control system (ie TF) from frequency domain to time domain by applying Inverse Laplace Transform also using Laplace Table and Theorem. 2024 BTD2232: Applied Control System  2024 THANK YOU! BTD2232: Applied Control System Question & Answer 42 THANK YOU 43