The time response of a control system consists of two parts: the transient response and the steadystate response • By transient response, we mean that which goes from the initial state to the final state • By steadystate response, we mean the manner in which the system output behaves as time approaches infinityThe time response of a control system consists of two parts: the transient response and the steadystate response • By transient response, we mean that which goes from the initial state to the final state • By steadystate response, we mean the manner in which the system output behaves as time approaches infinityThe time response of a control system consists of two parts: the transient response and the steadystate response • By transient response, we mean that which goes from the initial state to the final state • By steadystate response, we mean the manner in which the system output behaves as time approaches infinity
Dynamic Systems and Control, Chapter 2: Linear System Theory Linear System Theory © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-1 Dynamic Systems and Control, Chapter 2: Linear System Theory Response Analysis of Systems • In analyzing and designing control systems, we must have a basis of comparison of performance of various control systems • This basis may be set up by specifying particular test input signals and by comparing the responses of various systems to these input signals • Many design criteria are based on the response to such test signals • The use of test signals enables one to compare the performance of many systems on the same basis © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-2 Dynamic Systems and Control, Chapter 2: Linear System Theory Typical Test Signals • The commonly used test input signals are step functions, ramp functions, impulse functions, sinusoidal functions, and white noise • Which of these typical input signals to use for analyzing system characteristics may be determined by the form of the input that the system will be subjected to most frequently under normal operation © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-3 Dynamic Systems and Control, Chapter 2: Linear System Theory Transient Response and Steady-State Response • The time response of a control system consists of two parts: the transient response and the steady-state response • By transient response, we mean that which goes from the initial state to the final state • By steady-state response, we mean the manner in which the system output behaves as time approaches infinity Step response Harmonic response © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-4 Dynamic Systems and Control, Chapter 2: Linear System Theory Step response Example Harmonic response © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-5 Dynamic Systems and Control, Chapter 2: Linear System Theory Bounded Input Bounded Output Stability • A system is BIBO (bounded-input bounded-output) stable if every bounded input produces a bounded output A SISO system is BIBO stable if and only if its impulse response g(t) is absolutely integrable in the interval [0,∞), i.e., ∞ � 𝑔𝑔(𝜏𝜏) 𝑑𝑑𝜏𝜏 ≤ 𝑀𝑀 for some finite constant M ≥ © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-6 Dynamic Systems and Control, Chapter 2: Linear System Theory First-Order System The input-output relationship is given by the differential equation 𝑑𝑑𝑑𝑑(𝑡𝑡) + 𝑐𝑐 𝑡𝑡 = 𝑟𝑟 𝑡𝑡 𝑑𝑑𝑑𝑑 𝑇𝑇 Ex: RC circuit and thermal system The transfer function is © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-7 Dynamic Systems and Control, Chapter 2: Linear System Theory Unit Impulse Response of First-Order Systems-Order System The output of the system can be obtained as Taking inverse Laplace transform gives © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-8 Dynamic Systems and Control, Chapter 2: Linear System Theory Unit-Step Response of First-Order Systems We obtain Taking the inverse Laplace transform © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-9 Dynamic Systems and Control, Chapter 2: Linear System Theory Unit-Step Response of First-Order Systems Block diagram Error signal 𝑒𝑒 𝑡𝑡 = 𝑟𝑟 𝑡𝑡 − 𝑐𝑐 𝑡𝑡 = 𝑒𝑒 −𝑡𝑡⁄𝑇𝑇 𝑡𝑡 → ∞, 𝑒𝑒(𝑡𝑡) → © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-10 Dynamic Systems and Control, Chapter 2: Linear System Theory The Simulation of Systems Using MATLAB © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-80 Dynamic Systems and Control, Chapter 2: Linear System Theory The Simulation of Systems Using MATLAB © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-81 Dynamic Systems and Control, Chapter 2: Linear System Theory The Simulation of Systems Using MATLAB © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-82 Dynamic Systems and Control, Chapter 2: Linear System Theory The Simulation of Systems Using MATLAB © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-83 Dynamic Systems and Control, Chapter 2: Linear System Theory The Simulation of Systems Using MATLAB © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-84 Dynamic Systems and Control, Chapter 2: Linear System Theory The Simulation of Systems Using MATLAB © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-85 Dynamic Systems and Control, Chapter 2: Linear System Theory The Simulation of Systems Using MATLAB © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-86 Dynamic Systems and Control, Chapter 2: Linear System Theory The Simulation of Systems Using MATLAB © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-87 Dynamic Systems and Control, Chapter 2: Linear System Theory The Simulation of Systems Using MATLAB © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-88 Dynamic Systems and Control, Chapter 2: Linear System Theory The Simulation of Systems Using MATLAB © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-89 Dynamic Systems and Control, Chapter 2: Linear System Theory The Simulation of Systems Using MATLAB © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-90 Dynamic Systems and Control, Chapter 2: Linear System Theory The Simulation of Systems Using MATLAB © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-91 Dynamic Systems and Control, Chapter 2: Linear System Theory The Simulation of Systems Using MATLAB error Num4=[0.1]; © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-92 Dynamic Systems and Control, Chapter 2: Linear System Theory The Simulation of Systems Using MATLAB © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-93 Dynamic Systems and Control, Chapter 2: Linear System Theory The Simulation of Systems Using MATLAB © 2016 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 2-94