Dynamic Systems and Control, Chapter 2: Linear System Theory Linear System Theory © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-1 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 © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-2 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 © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-3 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 © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-4 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 ≥ © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-5 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 © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-6 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 © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-7 Dynamic Systems and Control, Chapter 2: Linear System Theory Unit-Step Response of First-Order Systems We obtain Taking the inverse Laplace transform © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-8 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) → © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-9 Dynamic Systems and Control, Chapter 2: Linear System Theory Unit-Ramp Response of First-Order Systems We obtain Taking the inverse Laplace transform The error signal e(t) is then e(∞)=T © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-10 Dynamic Systems and Control, Chapter 2: Linear System Theory Second-Order Systems and Transient-Response Specifications Peak time calculation © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-27 Dynamic Systems and Control, Chapter 2: Linear System Theory Second-Order Systems and Transient-Response Specifications Maximum overshoot calculation © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-28 Dynamic Systems and Control, Chapter 2: Linear System Theory Second-Order Systems and Transient-Response Specifications Settling time calculation Underdamped system: Ovedamped system: Two definitions of ts © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-29 Dynamic Systems and Control, Chapter 2: Linear System Theory Settling time ts versus ζ curves Mp versus ζ curve © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-30 Dynamic Systems and Control, Chapter 2: Linear System Theory Impulse Response of Second-Order Systems Taking inverse Laplace transform yields © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-31 Dynamic Systems and Control, Chapter 2: Linear System Theory Impulse Response of Second-Order Systems © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-32 Dynamic Systems and Control, Chapter 2: Linear System Theory Transient Response of Higher-Order Systems Closed-loop poles are all real and distinct Transfer function with With unit-step input zeros poles © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-33 Dynamic Systems and Control, Chapter 2: Linear System Theory Transient Response of Higher-Order Systems Real poles and pairs of complexconjugate poles Transfer function with With unit-step input zeros poles © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-34 Dynamic Systems and Control, Chapter 2: Linear System Theory Stability in Complex Plane - If any of poles lie in the right-half s plane, a system become unstable If all closed-loop poles lie in the left-half s plane, a system is stable Whether a linear system is stable or unstable is a property of the system itself and does not depend on the input or driving function of the system The relative stability and transient-response performance of a closedloop control system are directly related to the closed-loop pole-zero configuration in the s plane © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-35 Dynamic Systems and Control, Chapter 2: Linear System Theory S-Plane and Transient Response © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-36 Dynamic Systems and Control, Chapter 2: Linear System Theory Routh’s Stability Criterion Consider the transfer function of the closed-loop system with Q: How to determine the stability of the system? Routh’s Stability Criterion: - Routh’s stability criterion tells us whether or not there are unstable roots in a polynomial equation without actually solving for them - The use of this criterion to collect information about absolute stability obtained directly from the coefficients of the characteristic equation © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-37 Dynamic Systems and Control, Chapter 2: Linear System Theory Routh’s Stability Criterion • This is for LTI systems with a polynomial denominator (without sin, cos, exponential etc.) • It determines if all the roots of a polynomial - lie in the open LHP (left half-plane), - or equivalently, have negative real parts • It also determines the number of roots of a polynomial in the open RHP (right half-plane) • It does NOT explicitly compute the roots © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-38 Dynamic Systems and Control, Chapter 2: Linear System Theory Routh’s Stability Criterion The number of roots of q(s) with positive real parts is equal to the number of sign changes in the first column © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-39 Dynamic Systems and Control, Chapter 2: Linear System Theory Examples © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-40 Dynamic Systems and Control, Chapter 2: Linear System Theory Examples If appears in the first column of a nonzero row in Routh array, replace it with a small positive number © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 0-41 ... nqchi@hcmut.edu.vn 0 -2 6 Dynamic Systems and Control, Chapter 2: Linear System Theory Second-Order Systems and Transient-Response Specifications Peak time calculation © 20 15 Quoc Chi Nguyen, Head of Control. .. nqchi@hcmut.edu.vn 0 -2 7 Dynamic Systems and Control, Chapter 2: Linear System Theory Second-Order Systems and Transient-Response Specifications Maximum overshoot calculation © 20 15 Quoc Chi Nguyen, Head of Control. .. nqchi@hcmut.edu.vn 0 -2 8 Dynamic Systems and Control, Chapter 2: Linear System Theory Second-Order Systems and Transient-Response Specifications Settling time calculation Underdamped system: Ovedamped system: