Tài liệu 218023_dynamic system and control lecture 1

Dynamic Systems and Control, Chapter 1: Mathematical Models of Systems Mathematical Models of Systems © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 0-1 Dynamic Systems and Control, Chapter 1: Mathematical Models of Systems Dynamics of systems • Dynamics describes how the states evolves, as a function on the current state and any external inputs • Inputs describe the external excitation of the dynamics • Outputs describe the directly measured variables  Outputs are a function of the state and inputs ⇒ not independent variables  Not all states are outputs; some states can’t be directly measured © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 0-2 Dynamic Systems and Control, Chapter 1: Mathematical Models of Systems Dynamics of Mechanical Systems © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 0-3 Dynamic Systems and Control, Chapter 1: Mathematical Models of Systems Dynamics of Mechanical Systems © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 0-4 Dynamic Systems and Control, Chapter 1: Mathematical Models of Systems Dynamics of Electrical Systems © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 0-5 Dynamic Systems and Control, Chapter 1: Mathematical Models of Systems Dynamics of Thermal Systems Derive dynamic equation of the tank C: Thermal capacitance hi: Heat rate of input ho: : Change of temperature © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 0-6 Dynamic Systems and Control, Chapter 1: Mathematical Models of Systems What is your observation? • The dynamics of many systems, whether they are mechanical, electrical, thermal, and so on, may be described in terms of differential equations. • The differential equations may be obtained by using physical laws governing a particular system (e.g., Newton’s laws for mechanical systems and Kirchhoff’s laws for electrical systems). • Deriving reasonable mathematical models is the most important part of the entire analysis of control systems © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 0-7 Dynamic Systems and Control, Chapter 1: Mathematical Models of Systems System modeling Models are a mathematical representations of system dynamics: • Models allow the dynamics to be simulated and analyzed, without having to build the system • Models are never exact, but they can be predictive The model you use depends on the questions you want to answer • A single system may have many models • Time and spatial scale must be chosen to suit the questions you want to answer • Always formulate questions before building a model © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 0-8 Dynamic Systems and Control, Chapter 1: Mathematical Models of Systems The principle of causality The current output of the system (the output at time t = 0) depends on the past input (the input for t<0) but does not depend on the future input (the input for t>0). Examples of causal systems - Memoryless system: ∝ - Autoregressive filter: Examples of noncausal systems - Central moving average: - Time reversal: © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 0-9 Dynamic Systems and Control, Chapter 1: Mathematical Models of Systems Linear systems A system is called linear if the principle of superposition applies Given : → then Linear time invariant system (LTI) The coefficients are constants or functions only of the independent variable Linear timevarying system (LTV) The coefficients are functions of time: Ex: Spacecraft control system. (The mass of a spacecraft changes due to fuel consumption) © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 0-10 Dynamic Systems and Control, Chapter 1: Mathematical Models of Systems Transfer Function and Impulse Response Function Definition. The transfer function of a linear, time-invariant, differential equation system is defined as the ratio of the Laplace transform of the output (response function) to the Laplace transform of the input (driving function) under the assumption that all initial conditions are zero. Differential equation © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 0-11 Dynamic Systems and Control, Chapter 1: Mathematical Models of Systems Transfer Function and Impulse Response Function - The transfer function is a property of a system itself, independent of the magnitude and nature of the input or driving function. - The transfer function does not provide any information concerning the physical structure of the system. Differential equation © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 0-12 Dynamic Systems and Control, Chapter 1: Mathematical Models of Systems Laplace Transform - Laplace transform: A transformation from t (time) to s (Laplace variable) Pierre Simon Laplace - Definition: Laplace transformation . is used to map time domain function into domain function . - This mapping . : is defined as → © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 0-13 Dynamic Systems and Control, Chapter 1: Mathematical Models of Systems List of Common Transforms © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 0-14 Dynamic Systems and Control, Chapter 1: Mathematical Models of Systems Taking a Laplace transform: Transfer function © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 0-15 Dynamic Systems and Control, Chapter 1: Mathematical Models of Systems © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 0-16 Dynamic Systems and Control, Chapter 1: Mathematical Models of Systems Dynamics of Electrical Systems Taking a Laplace transform 11 11 Transfer function 1 1 © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 0-17 Dynamic Systems and Control, Chapter 1: Mathematical Models of Systems Derive dynamic equation of the tank Rewrite dynamic equation ↔ ;C ; C: Thermal capacitance hi: Heat rate of input ho: Heat rate of output : Change of temperature R: Thermal resistance c: Specific heat of liquid M: Mass of liquid Transfer function:  © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 1 0-18 Dynamic Systems and Control, Chapter 1: Mathematical Models of Systems How to find the output in time domain? The output Y(s) can be written as the product of G(s) and X(s) Taking an inverse Laplace transform gives the following convolution integral: If the input is an impulse Complete information of the system (the dynamic characteristics of the system) can be obtained by exciting it with an impulse input and measuring the response. © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 0-19 Dynamic Systems and Control, Chapter 1: Mathematical Models of Systems Matlab for Dynamic Systems and Control © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, [email protected] 0-20