Dynamic Systems and Control, Chapter 3: Feedback Control Theory Feedback Control Theory © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-1 Dynamic Systems and Control, Chapter 3: Feedback Control Theory Lịch học bù: Ngày: 21/3/2015 Phòng: 211 B1 Thời gian: Tiết 1-2 © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-2 Dynamic Systems and Control, Chapter 3: Feedback Control Theory Root-Locus Method - - The basic characteristic of the transient response of a closed-loop system is closely related to the location of the closed-loop poles If the system has a variable loop gain, then the location of the closedloop poles depends on the value of the loop gain chosen The closed-loop poles are the roots of the characteristic equation Finding the roots of the characteristic equation of degree higher than is laborious and will need computer solution (Matlab can it) However, just finding the roots of the characteristic equation may be of limited value, because as the gain of the open-loop transfer function varies, the characteristic equation changes and the computations must be repeated Root-locus method, is one in which the roots of the characteristic equation are plotted for all values of a system parameter By using the root-locus method the designer can predict the effects on the location of the closed-loop poles of varying the gain value or adding open-loop poles and/or open-loop zeros © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-3 Dynamic Systems and Control, Chapter 3: Feedback Control Theory ROOT-LOCUS PLOTS Angle and Magnitude Conditions Consider the negative feedback system The characteristic equation for this closed-loop system Angle condition Magnitude condition © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-4 Dynamic Systems and Control, Chapter 3: Feedback Control Theory ROOT-LOCUS PLOTS When G(s)H(s) involves a gain parameter K, characteristic equation may be written as Then, the root loci for the system are the loci of the closedloop poles as the gain K is varied from zero to infinity © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-5 Dynamic Systems and Control, Chapter 3: Feedback Control Theory RELATIONSHIP BETWEEN ZEROS-POLES AND ANGLEMAGNIGTUDE The angle of G(s)H(s) is The magnitude of G(s)H(s) for this system is © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-6 Dynamic Systems and Control, Chapter 3: Feedback Control Theory General Rules for Constructing Root Loci Consider the control system Illustrative example The characteristic equation Rearrange this equation in the form © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-7 Dynamic Systems and Control, Chapter 3: Feedback Control Theory General Rules for Constructing Root Loci Rule Locate the poles and zeros of G(s)H(s) on the s plane The root-locus branches start from open-loop poles and terminate at zeros (finite zeros or zeros at infinity) - The root loci are symmetrical about the real axis of the s plane, because the complex poles and complex zeros occur only in conjugate pairs - If the number of closed-loop poles is the same as the number of open-loop poles, then the number of individual root-locus branches terminating at finite open-loop zeros is equal to the number m of the open-loop zeros The remaining n-m branches terminate at infinity (n-m implicit zeros at infinity) along asymptotes Illustrative example The first step in constructing a rootlocus plot is to locate the open-loop poles, s = 0, s =–1, and s =–2, in the complex plane © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-8 Dynamic Systems and Control, Chapter 3: Feedback Control Theory General Rules for Constructing Root Loci Rule Determine the root loci on the real axis Root loci on the real axis are determined by open-loop poles and zeros lying on it - Each portion of the root locus on the real axis extends over a range from a pole or zero to another pole or zero - In constructing the root loci on the real axis, choose a test point on it If the total number of real poles and real zeros to the right of this test point is odd, then this point lies on a root locus Illustrative example Q: - If the test point is on the positive real axis, then - If a test point on the negative real axis between and –1, then - If a test point is selected between –1 and –2, then © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-9 Dynamic Systems and Control, Chapter 3: Feedback Control Theory General Rules for Constructing Root Loci Rule Determine the asymptotes of root loci The root loci for very large values of s must be asymptotic to straight lines whose angles (slopes) are given by Illustrative example Since the angle repeats itself as k is varied, the distinct angles for the asymptotes are determined as 60°, –60°, and 180° © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-10 Dynamic Systems and Control, Chapter 3: Feedback Control Theory © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-67 Dynamic Systems and Control, Chapter 3: Feedback Control Theory © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-68 Dynamic Systems and Control, Chapter 3: Feedback Control Theory © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-69 Dynamic Systems and Control, Chapter 3: Feedback Control Theory © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-70 Dynamic Systems and Control, Chapter 3: Feedback Control Theory Notes on general Nyquist plots ˆFor physical realizable systems, the order of the denominator is larger than or equal to that of the numerator of the transfer function - Type systems: - Finite starting point on the positive real axis - The terminal point is the origin tangent to one of the axis - Type systems - Starting at infinity asymptotically parallel to imaginary axis - Also the curve converges to zero tangent to one of the axis - Type systems - The starting magnitude is infinity and asymptotic to -180 - Also the curve converges to zero tangent to one of the axis © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-71 Dynamic Systems and Control, Chapter 3: Feedback Control Theory © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-72 Dynamic Systems and Control, Chapter 3: Feedback Control Theory Relative Stability Analysis using Nyquist Plot © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-73 Dynamic Systems and Control, Chapter 3: Feedback Control Theory Nyquist Stability Criteria © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-74 Dynamic Systems and Control, Chapter 3: Feedback Control Theory Nyquist Stability Criteria © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-75 Dynamic Systems and Control, Chapter 3: Feedback Control Theory Application of Nyquist Stability Criteria © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-76 Dynamic Systems and Control, Chapter 3: Feedback Control Theory © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-77 Dynamic Systems and Control, Chapter 3: Feedback Control Theory © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-78 Dynamic Systems and Control, Chapter 3: Feedback Control Theory © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-79 Dynamic Systems and Control, Chapter 3: Feedback Control Theory Obtain the phase and gain margins of the system shown for the two cases where K=10 and K=100 © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-80 Dynamic Systems and Control, Chapter 3: Feedback Control Theory © 2015 Quoc Chi Nguyen, Head of Control & Automation Laboratory, nqchi@hcmut.edu.vn 3-81 ... nqchi@hcmut.edu.vn 3-2 Dynamic Systems and Control, Chapter 3: Feedback Control Theory Root-Locus Method - - The basic characteristic of the transient response of a closed-loop system is closely... nqchi@hcmut.edu.vn 3-2 8 Dynamic Systems and Control, Chapter 3: Feedback Control Theory - The log magnitude of the first-order factor 1/(1 + jT) is - For low frequencies, such that