Control Systems I. Lecture 9: The Nyquist condition

Save this PDF as:

Size: px
Start display at page:

Download "Control Systems I. Lecture 9: The Nyquist condition"

Transcription

1 Control Systems I Lecture 9: The Nyquist condition Readings: Åstrom and Murray, Chapter Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 16, 2018 J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

2 Tentative schedule # Date Topic 1 Sept. 21 Introduction, Signals and Systems 2 Sept. 28 Modeling, Linearization 3 Oct. 5 Analysis 1: Time response, Stability 4 Oct. 12 Analysis 2: Diagonalization, Modal coordinates 5 Oct. 19 Transfer functions 1: Definition and properties 6 Oct. 26 Transfer functions 2: Poles and Zeros 7 Nov. 2 Analysis of feedback systems: internal stability, root locus 8 Nov. 9 Frequency response 9 Nov. 16 Analysis of feedback systems 2: the Nyquist condition 10 Nov. 23 Specifications for feedback systems 11 Nov. 30 Loop Shaping 12 Dec. 7 PID control 13 Dec. 14 State feedback and Luenberger observers 14 Dec. 21 On Robustness and Implementation challenges J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

3 Recap A system is a function mapping input signals into output signals. An LTI system can be described by a transfer function. u L(s) y L(s) = N(s) D(s) D(s) is the characteristic polynomial of the matrix A. Poles: the roots of D(s), zeros: the roots of N(s) u(t) = sin(ωt) y ss (t) = L(jω) sin(ωt + L(jω)). For any input U(s) Y (s) = L(s)U(s) Stable system: a system that does not blow up. The system L(s) is BIBO-stable if all of its poles are on the LHP. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

4 Recap Feedback control allows one to Stabilize an unstable system; Handle uncertainties in the system; Reject external disturbances. r e u y k L(s) The closed-loop transfer function is: kl(s) 1 + kl(s). It is also called the complimentary sensitivity function. The closed-loop poles: the zeros of 1 + kl(s). The poles of 1 + kl(s) are identical to the poles of L(s). The closed-loop system is stable if all of its poles are on the LHP. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

5 The phase rule and the magnitude rule G(s) = k (s z 1)(s z 2 )... (s z m ) (s p 1 )(s p 2 )... (s p n ) Im jω Re G(s) = k s z 1 s z 2... s z m s p 1 s p 2... s p n G(s) = k + (s z 1 ) + (s z 2 ) (s z m ) (s p 1 ) (s p 2 )... (s p n ) J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

6 Classical methods for feedback control The main objective here is to assess/design the properties of the closed-loop system by exploiting the knowledge of the open-loop system, and avoiding complex calculations. Stability analysis of closed-loop system: Routh-Hurwitz criterion: a mathematical evaluation of the characteristic equation of the closed-loop system. There are three geometric methods to find out the stability of a systems: They are useful both in analysis and synthesis. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

7 Classical methods for feedback control Root Locus Quick assessment of control design feasibility. The insights are correct and clear. Can only be used for finite-dimensional systems (e.g. systems with a finite number of poles/zeros) Difficult to do sophisticated design. Hard to represent uncertainty. Nyquist plot The most authoritative closed-loop stability test. It can always be used (finite or infinite-dimensional systems) Easy to represent uncertainty. Difficult to draw and to use for sophisticated design. Bode plots Potentially misleading results unless the system is open-loop stable and minimum-phase. Easy to represent uncertainty. Easy to draw, this is the tool of choice for sophisticated design. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

8 Towards Nyquist s theorem J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

9 The goal r e u y k L(s) Our goal is to count the number of RHP poles (if any) of the closed-loop transfer function T (s) = kl(s) 1 + kl(s) based on the frequency response of the open-loop transfer function L(s). J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

10 The principle of variation of the argument If we take a complex number in the s-plane and substitute it into a function G(s), it results in another complex number which could be plotted in the G(s)-plane. Let Γ be a simple closed curve in the s-plane, which does not pass through any pole of a function G(s). As s moves along the closed curve Γ, G(s) describes another closed curve. Im G(s) Im D Re Re Remarkable fact: The number of times G(s) encircles the origin, or, equivalently, the total variation in its argument G(s), as s moves along Γ, counts the number of zeros and poles of G(s) in D. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

11 Phase change and encirclements Consider a clockwise closed contour, Γ, not passing through the origin 0, traversed by s. The origin is either inside or outside Γ. What is the net change in s as s traverses Γ? The phase change as a s traverses a closed path Γ is equal to 2πN, where N is the number of clockwise encirclements of 0 by Γ. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

12 Phase change and encirclements Consider a simple closed contour, Γ, traversed clockwise by s. A fixed complex number r is either inside or outside Γ. What is the net phase change in (s r) as s traverses Γ? J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

13 Principle of argument Find the image of Γ under G(s) = k (s z 1)(s z 2 )... (s z m ) (s p 1 )(s p 2 )... (s p n ) At any value of s, the angle of G(s) is: G(s) = k + (s z 1 ) (s z m ) (s p 1 )... (s p n ) 2π(number of clockwise encirclements of 0 by G(Γ)) = = net change in G(s) as s traverses Γ = = net change in (s z 1 ) net change in (s z m ) net change in (s p 1 )... net change in (s p n ) 2πN = 2πZ ( 2πP) J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

14 The general case Theorem (Variation of the argument [Proof in A&M, pp ]) The number N of times that G(s) encircles the origin of the complex plane as s moves along the simple closed curve Γ satisfies N = Z P, where Z and P are the numbers of zeros and poles of G(s) in D, respectively. Note that the encirclements are counted positive if in the same direction as s moves along Γ, and negative otherwise. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

15 How do we use these results for feedback control? J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

16 The Nyquist or D contour For closed-loop stability, the closed-loop poles, which corresponds to the roots (i.e., zeros!) of 1 + kl(s), must have negative real part. The poles of 1 + kl(s) are also the poles of L(s). Construct the region D as a D-shaped region containing an arbitrarily large (but finite) part of the complex right-half plane. As s moves along the boundary of this region, 1 + kl(s) encircles the origin N = Z P times, where Z is the number of unstable closed-loop poles (zeros of 1 + kl(s) in the rhp); P is the number of unstable open-loop poles (poles of 1 + kl(s) in the rhp); J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

17 The Nyquist plot The previous statement can be rephrased: As s moves along the boundary of this region, L(s) encircles the 1/k point N = Z P times, where Z is the number of unstable closed-loop poles (zeros of 1 + kl(s) in the Nyquist contour); P is the number of unstable open-loop poles (poles of 1 + kl(s) in the Nyquist contour); Symmetry of poles/zeros about the real axis implies that L( jω) = L(jω), hence the plot of L(s) when s moves on the boundary of the Nyquist contour is just the polar plot + its symmetric plot about the real axis. This is what is called the Nyquist plot. The key feature of the Nyquist plot is the number of encirclements of the 1/k point. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

18 The Nyquist contour Segment 1 corresponds to s = jω, where ω : 0. On this segment, L(Γ) is just L(jω): frequency response. Segment 2 corresponds to s = Re jθ, where R and θ : π 2 π 2. On this segment, L(Γ) collapses on a single point, since R is very large. Segment 3 corresponds to s = jω, where ω : 0. On this segment, L(Γ) is just L( jω), where ω : 0. L( jω) is complex-conjugate of L(jω), so L( jω) is the reflection of L(jω) about the real axis. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

19 The Nyquist condition Theorem Consider a closed-loop system with loop transfer function kl(s), which has P poles in the region enclosed by the Nyquist contour. Let N be the net number of clockwise encirclements of 1/k by L(s) when s moves along the Nyquist contour in the clockwise direction. The closed loop system has Z = N + P poles in the Nyquist contour. In particular: If the open-loop system is stable, the closed-loop system is stable as long as the Nyquist plot of L(s) does NOT encircle the 1/k point. If the open-loop system has P poles, the closed-loop system is stable as long as the Nyquist plot of L(s) encircles the 1/k point P times in the negative (counter-clockwise) direction. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

20 Counting encirclements Draw a line out from the 1/k point in any arbitrary direction. Count the number of times that the Nyquist path crosses the line in the clockwise direction, and subtract the number of times it crosses in the counterclockwise direction. You get the number of clockwise encirclements of the 1/k point. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

21 Nyquist plot when L(s) has no pole on the jω axis 1 Image of Segment 1: Plot L(jω) for ω : 0. This is also called the polar plot of L(s). There is no special rules for drawing it. 2 Image of Segment 3: Reflect it about the real axis to draw L(jω) for ω : 0. 3 Image of Segment 2: This segment maps onto a point, in the case of physically realizable systems. For a strictly proper systems: if s, then L(s) maps onto origin. For a proper systems: if s, then L(s) would be a constant. 4 The points where the Nyquist plot crosses the real axis and the unit circle are of importance. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

22 Nyquist condition single real, stable pole Im L(s) = 2 s + 1 Im Re Re J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

23 Nyquist condition open-loop unstable system Im L(s) = s + 2 s 2 1 Im Re Re J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

24 Dealing with open-loop poles on the imaginary axis Im If there are open-loop poles on the imaginary axis, make small indentations in the Nyquist contour, e.g., leaving the imaginary poles on the left. Be careful on how you close the Nyquist plot at infinity: If moving CCW around the poles, then close the plot CW. Re J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

25 Nyquist poles on the imaginary axis L(s) = 2 (s 2 + 1)(s + 1) Im Im Re Re J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

26 Summary In this lecture, we learned: How to sketch a Nyquist plot. The Nyquist condition to determine closed-loop stability using a Nyquist plot. How to check the Nyquist condition.. J. Tani, E. Frazzoli (ETH) Lecture 9: Control Systems I 16/11/ / 26

Control Systems I. Lecture 9: The Nyquist condition

Control Systems I. Lecture 9: The Nyquist condition Control Systems I Lecture 9: The Nyquist condition adings: Guzzella, Chapter 9.4 6 Åstrom and Murray, Chapter 9.1 4 www.cds.caltech.edu/~murray/amwiki/index.php/first_edition Emilio Frazzoli Institute

More information

Control Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Guzzella 9.1-3, Emilio Frazzoli

Control Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Guzzella 9.1-3, Emilio Frazzoli Control Systems I Lecture 7: Feedback and the Root Locus method Readings: Guzzella 9.1-3, 13.3 Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 3, 2017 E. Frazzoli (ETH)

More information

Control Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Jacopo Tani. Institute for Dynamic Systems and Control D-MAVT ETH Zürich

Control Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Jacopo Tani. Institute for Dynamic Systems and Control D-MAVT ETH Zürich Control Systems I Lecture 7: Feedback and the Root Locus method Readings: Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 2, 2018 J. Tani, E. Frazzoli (ETH) Lecture 7:

More information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 23: Drawing The Nyquist Plot Overview In this Lecture, you will learn: Review of Nyquist Drawing the Nyquist Plot Using

More information

Control Systems I Lecture 10: System Specifications

Control Systems I Lecture 10: System Specifications Control Systems I Lecture 10: System Specifications Readings: Guzzella, Chapter 10 Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 24, 2017 E. Frazzoli (ETH) Lecture

More information

Control Systems I. Lecture 6: Poles and Zeros. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich

Control Systems I. Lecture 6: Poles and Zeros. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich Control Systems I Lecture 6: Poles and Zeros Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 27, 2017 E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/2017

More information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 22: The Nyquist Criterion Overview In this Lecture, you will learn: Complex Analysis The Argument Principle The Contour

More information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 23: Drawing The Nyquist Plot Overview In this Lecture, you will learn: Review of Nyquist Drawing the Nyquist Plot Using the

More information

Control Systems I. Lecture 5: Transfer Functions. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich

Control Systems I. Lecture 5: Transfer Functions. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich Control Systems I Lecture 5: Transfer Functions Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 20, 2017 E. Frazzoli (ETH) Lecture 5: Control Systems I 20/10/2017

More information

Control Systems I. Lecture 2: Modeling. Suggested Readings: Åström & Murray Ch. 2-3, Guzzella Ch Emilio Frazzoli

Control Systems I. Lecture 2: Modeling. Suggested Readings: Åström & Murray Ch. 2-3, Guzzella Ch Emilio Frazzoli Control Systems I Lecture 2: Modeling Suggested Readings: Åström & Murray Ch. 2-3, Guzzella Ch. 2-3 Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich September 29, 2017 E. Frazzoli

More information

Topic # Feedback Control

Topic # Feedback Control Topic #4 16.31 Feedback Control Stability in the Frequency Domain Nyquist Stability Theorem Examples Appendix (details) This is the basis of future robustness tests. Fall 2007 16.31 4 2 Frequency Stability

More information

Control Systems I. Lecture 2: Modeling and Linearization. Suggested Readings: Åström & Murray Ch Jacopo Tani

Control Systems I. Lecture 2: Modeling and Linearization. Suggested Readings: Åström & Murray Ch Jacopo Tani Control Systems I Lecture 2: Modeling and Linearization Suggested Readings: Åström & Murray Ch. 2-3 Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich September 28, 2018 J. Tani, E.

More information

Lecture 15 Nyquist Criterion and Diagram

Lecture 15 Nyquist Criterion and Diagram Lecture Notes of Control Systems I - ME 41/Analysis and Synthesis of Linear Control System - ME86 Lecture 15 Nyquist Criterion and Diagram Department of Mechanical Engineering, University Of Saskatchewan,

More information

Control Systems. Frequency Method Nyquist Analysis.

Control Systems. Frequency Method Nyquist Analysis. Frequency Method Nyquist Analysis chibum@seoultech.ac.kr Outline Polar plots Nyquist plots Factors of polar plots PolarNyquist Plots Polar plot: he locus of the magnitude of ω vs. the phase of ω on polar

More information

Course Outline. Closed Loop Stability. Stability. Amme 3500 : System Dynamics & Control. Nyquist Stability. Dr. Dunant Halim

Course Outline. Closed Loop Stability. Stability. Amme 3500 : System Dynamics & Control. Nyquist Stability. Dr. Dunant Halim Amme 3 : System Dynamics & Control Nyquist Stability Dr. Dunant Halim Course Outline Week Date Content Assignment Notes 1 5 Mar Introduction 2 12 Mar Frequency Domain Modelling 3 19 Mar System Response

More information

Control Systems I. Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback. Readings: Emilio Frazzoli

Control Systems I. Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback. Readings: Emilio Frazzoli Control Systems I Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 13, 2017 E. Frazzoli (ETH)

More information

ECE 345 / ME 380 Introduction to Control Systems Lecture Notes 8

ECE 345 / ME 380 Introduction to Control Systems Lecture Notes 8 Learning Objectives ECE 345 / ME 380 Introduction to Control Systems Lecture Notes 8 Dr. Oishi oishi@unm.edu November 2, 203 State the phase and gain properties of a root locus Sketch a root locus, by

More information

r + - FINAL June 12, 2012 MAE 143B Linear Control Prof. M. Krstic

r + - FINAL June 12, 2012 MAE 143B Linear Control Prof. M. Krstic MAE 43B Linear Control Prof. M. Krstic FINAL June, One sheet of hand-written notes (two pages). Present your reasoning and calculations clearly. Inconsistent etchings will not be graded. Write answers

More information

Linear Control Systems Lecture #3 - Frequency Domain Analysis. Guillaume Drion Academic year

Linear Control Systems Lecture #3 - Frequency Domain Analysis. Guillaume Drion Academic year Linear Control Systems Lecture #3 - Frequency Domain Analysis Guillaume Drion Academic year 2018-2019 1 Goal and Outline Goal: To be able to analyze the stability and robustness of a closed-loop system

More information

STABILITY ANALYSIS. Asystemmaybe stable, neutrallyormarginallystable, or unstable. This can be illustrated using cones: Stable Neutral Unstable

STABILITY ANALYSIS. Asystemmaybe stable, neutrallyormarginallystable, or unstable. This can be illustrated using cones: Stable Neutral Unstable ECE4510/5510: Feedback Control Systems. 5 1 STABILITY ANALYSIS 5.1: Bounded-input bounded-output (BIBO) stability Asystemmaybe stable, neutrallyormarginallystable, or unstable. This can be illustrated

More information

1 (s + 3)(s + 2)(s + a) G(s) = C(s) = K P + K I

1 (s + 3)(s + 2)(s + a) G(s) = C(s) = K P + K I MAE 43B Linear Control Prof. M. Krstic FINAL June 9, Problem. ( points) Consider a plant in feedback with the PI controller G(s) = (s + 3)(s + )(s + a) C(s) = K P + K I s. (a) (4 points) For a given constant

More information

Frequency methods for the analysis of feedback systems. Lecture 6. Loop analysis of feedback systems. Nyquist approach to study stability

Frequency methods for the analysis of feedback systems. Lecture 6. Loop analysis of feedback systems. Nyquist approach to study stability Lecture 6. Loop analysis of feedback systems 1. Motivation 2. Graphical representation of frequency response: Bode and Nyquist curves 3. Nyquist stability theorem 4. Stability margins Frequency methods

More information

Nyquist Criterion For Stability of Closed Loop System

Nyquist Criterion For Stability of Closed Loop System Nyquist Criterion For Stability of Closed Loop System Prof. N. Puri ECE Department, Rutgers University Nyquist Theorem Given a closed loop system: r(t) + KG(s) = K N(s) c(t) H(s) = KG(s) +KG(s) = KN(s)

More information

Analysis of SISO Control Loops

Analysis of SISO Control Loops Chapter 5 Analysis of SISO Control Loops Topics to be covered For a given controller and plant connected in feedback we ask and answer the following questions: Is the loop stable? What are the sensitivities

More information

Intro to Frequency Domain Design

Intro to Frequency Domain Design Intro to Frequency Domain Design MEM 355 Performance Enhancement of Dynamical Systems Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Outline Closed Loop Transfer Functions

More information

The Nyquist criterion relates the stability of a closed system to the open-loop frequency response and open loop pole location.

The Nyquist criterion relates the stability of a closed system to the open-loop frequency response and open loop pole location. Introduction to the Nyquist criterion The Nyquist criterion relates the stability of a closed system to the open-loop frequency response and open loop pole location. Mapping. If we take a complex number

More information

MEM 355 Performance Enhancement of Dynamical Systems

MEM 355 Performance Enhancement of Dynamical Systems MEM 355 Performance Enhancement of Dynamical Systems Frequency Domain Design Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University 5/8/25 Outline Closed Loop Transfer Functions

More information

STABILITY ANALYSIS TECHNIQUES

STABILITY ANALYSIS TECHNIQUES ECE4540/5540: Digital Control Systems 4 1 STABILITY ANALYSIS TECHNIQUES 41: Bilinear transformation Three main aspects to control-system design: 1 Stability, 2 Steady-state response, 3 Transient response

More information

K(s +2) s +20 K (s + 10)(s +1) 2. (c) KG(s) = K(s + 10)(s +1) (s + 100)(s +5) 3. Solution : (a) KG(s) = s +20 = K s s

K(s +2) s +20 K (s + 10)(s +1) 2. (c) KG(s) = K(s + 10)(s +1) (s + 100)(s +5) 3. Solution : (a) KG(s) = s +20 = K s s 321 16. Determine the range of K for which each of the following systems is stable by making a Bode plot for K = 1 and imagining the magnitude plot sliding up or down until instability results. Verify

More information

MEM 355 Performance Enhancement of Dynamical Systems

MEM 355 Performance Enhancement of Dynamical Systems MEM 355 Performance Enhancement of Dynamical Systems Frequency Domain Design Intro Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University /5/27 Outline Closed Loop Transfer

More information

Lecture 1 Root Locus

Lecture 1 Root Locus Root Locus ELEC304-Alper Erdogan 1 1 Lecture 1 Root Locus What is Root-Locus? : A graphical representation of closed loop poles as a system parameter varied. Based on Root-Locus graph we can choose the

More information

Classify a transfer function to see which order or ramp it can follow and with which expected error.

Classify a transfer function to see which order or ramp it can follow and with which expected error. Dr. J. Tani, Prof. Dr. E. Frazzoli 5-059-00 Control Systems I (Autumn 208) Exercise Set 0 Topic: Specifications for Feedback Systems Discussion: 30.. 208 Learning objectives: The student can grizzi@ethz.ch,

More information

CONTROL SYSTEM STABILITY. CHARACTERISTIC EQUATION: The overall transfer function for a. where A B X Y are polynomials. Substitution into the TF gives:

CONTROL SYSTEM STABILITY. CHARACTERISTIC EQUATION: The overall transfer function for a. where A B X Y are polynomials. Substitution into the TF gives: CONTROL SYSTEM STABILITY CHARACTERISTIC EQUATION: The overall transfer function for a feedback control system is: TF = G / [1+GH]. The G and H functions can be put into the form: G(S) = A(S) / B(S) H(S)

More information

6.241 Dynamic Systems and Control

6.241 Dynamic Systems and Control 6.241 Dynamic Systems and Control Lecture 17: Robust Stability Readings: DDV, Chapters 19, 20 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology April 6, 2011 E. Frazzoli

More information

Module 3F2: Systems and Control EXAMPLES PAPER 2 ROOT-LOCUS. Solutions

Module 3F2: Systems and Control EXAMPLES PAPER 2 ROOT-LOCUS. Solutions Cambridge University Engineering Dept. Third Year Module 3F: Systems and Control EXAMPLES PAPER ROOT-LOCUS Solutions. (a) For the system L(s) = (s + a)(s + b) (a, b both real) show that the root-locus

More information

Control Systems I. Lecture 1: Introduction. Suggested Readings: Åström & Murray Ch. 1, Guzzella Ch. 1. Emilio Frazzoli

Control Systems I. Lecture 1: Introduction. Suggested Readings: Åström & Murray Ch. 1, Guzzella Ch. 1. Emilio Frazzoli Control Systems I Lecture 1: Introduction Suggested Readings: Åström & Murray Ch. 1, Guzzella Ch. 1 Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich September 22, 2017 E. Frazzoli

More information

ECEN 605 LINEAR SYSTEMS. Lecture 20 Characteristics of Feedback Control Systems II Feedback and Stability 1/27

ECEN 605 LINEAR SYSTEMS. Lecture 20 Characteristics of Feedback Control Systems II Feedback and Stability 1/27 1/27 ECEN 605 LINEAR SYSTEMS Lecture 20 Characteristics of Feedback Control Systems II Feedback and Stability Feedback System Consider the feedback system u + G ol (s) y Figure 1: A unity feedback system

More information

STABILITY OF CLOSED-LOOP CONTOL SYSTEMS

STABILITY OF CLOSED-LOOP CONTOL SYSTEMS CHBE320 LECTURE X STABILITY OF CLOSED-LOOP CONTOL SYSTEMS Professor Dae Ryook Yang Spring 2018 Dept. of Chemical and Biological Engineering 10-1 Road Map of the Lecture X Stability of closed-loop control

More information

Controls Problems for Qualifying Exam - Spring 2014

Controls Problems for Qualifying Exam - Spring 2014 Controls Problems for Qualifying Exam - Spring 2014 Problem 1 Consider the system block diagram given in Figure 1. Find the overall transfer function T(s) = C(s)/R(s). Note that this transfer function

More information

1 (20 pts) Nyquist Exercise

1 (20 pts) Nyquist Exercise EE C128 / ME134 Problem Set 6 Solution Fall 2011 1 (20 pts) Nyquist Exercise Consider a close loop system with unity feedback. For each G(s), hand sketch the Nyquist diagram, determine Z = P N, algebraically

More information

ECE 486 Control Systems

ECE 486 Control Systems ECE 486 Control Systems Spring 208 Midterm #2 Information Issued: April 5, 208 Updated: April 8, 208 ˆ This document is an info sheet about the second exam of ECE 486, Spring 208. ˆ Please read the following

More information

ECE317 : Feedback and Control

ECE317 : Feedback and Control ECE317 : Feedback and Control Lecture : Routh-Hurwitz stability criterion Examples Dr. Richard Tymerski Dept. of Electrical and Computer Engineering Portland State University 1 Course roadmap Modeling

More information

Outline. Classical Control. Lecture 1

Outline. Classical Control. Lecture 1 Outline Outline Outline 1 Introduction 2 Prerequisites Block diagram for system modeling Modeling Mechanical Electrical Outline Introduction Background Basic Systems Models/Transfers functions 1 Introduction

More information

MAE 143B - Homework 9

MAE 143B - Homework 9 MAE 43B - Homework 9 7.2 2 2 3.8.6.4.2.2 9 8 2 2 3 a) G(s) = (s+)(s+).4.6.8.2.2.4.6.8. Polar plot; red for negative ; no encirclements of, a.s. under unit feedback... 2 2 3. 4 9 2 2 3 h) G(s) = s+ s(s+)..2.4.6.8.2.4

More information

Introduction. Performance and Robustness (Chapter 1) Advanced Control Systems Spring / 31

Introduction. Performance and Robustness (Chapter 1) Advanced Control Systems Spring / 31 Introduction Classical Control Robust Control u(t) y(t) G u(t) G + y(t) G : nominal model G = G + : plant uncertainty Uncertainty sources : Structured : parametric uncertainty, multimodel uncertainty Unstructured

More information

Some special cases

Some special cases Lecture Notes on Control Systems/D. Ghose/2012 87 11.3.1 Some special cases Routh table is easy to form in most cases, but there could be some cases when we need to do some extra work. Case 1: The first

More information

Control Systems I. Lecture 1: Introduction. Suggested Readings: Åström & Murray Ch. 1. Jacopo Tani

Control Systems I. Lecture 1: Introduction. Suggested Readings: Åström & Murray Ch. 1. Jacopo Tani Control Systems I Lecture 1: Introduction Suggested Readings: Åström & Murray Ch. 1 Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich September 21, 2018 J. Tani, E. Frazzoli (ETH)

More information

FEL3210 Multivariable Feedback Control

FEL3210 Multivariable Feedback Control FEL3210 Multivariable Feedback Control Lecture 5: Uncertainty and Robustness in SISO Systems [Ch.7-(8)] Elling W. Jacobsen, Automatic Control Lab, KTH Lecture 5:Uncertainty and Robustness () FEL3210 MIMO

More information

Robust Control 3 The Closed Loop

Robust Control 3 The Closed Loop Robust Control 3 The Closed Loop Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University /2/2002 Outline Closed Loop Transfer Functions Traditional Performance Measures Time

More information

Course roadmap. ME451: Control Systems. What is Root Locus? (Review) Characteristic equation & root locus. Lecture 18 Root locus: Sketch of proofs

Course roadmap. ME451: Control Systems. What is Root Locus? (Review) Characteristic equation & root locus. Lecture 18 Root locus: Sketch of proofs ME451: Control Systems Modeling Course roadmap Analysis Design Lecture 18 Root locus: Sketch of proofs Dr. Jongeun Choi Department of Mechanical Engineering Michigan State University Laplace transform

More information

Outline. Control systems. Lecture-4 Stability. V. Sankaranarayanan. V. Sankaranarayanan Control system

Outline. Control systems. Lecture-4 Stability. V. Sankaranarayanan. V. Sankaranarayanan Control system Outline Control systems Lecture-4 Stability V. Sankaranarayanan Outline Outline 1 Outline Outline 1 2 Concept of Stability Zero State Response: The zero-state response is due to the input only; all the

More information

Frequency Response Techniques

Frequency Response Techniques 4th Edition T E N Frequency Response Techniques SOLUTION TO CASE STUDY CHALLENGE Antenna Control: Stability Design and Transient Performance First find the forward transfer function, G(s). Pot: K 1 = 10

More information

Frequency domain analysis

Frequency domain analysis Automatic Control 2 Frequency domain analysis Prof. Alberto Bemporad University of Trento Academic year 2010-2011 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011

More information

Plan of the Lecture. Goal: wrap up lead and lag control; start looking at frequency response as an alternative methodology for control systems design.

Plan of the Lecture. Goal: wrap up lead and lag control; start looking at frequency response as an alternative methodology for control systems design. Plan of the Lecture Review: design using Root Locus; dynamic compensation; PD and lead control Today s topic: PI and lag control; introduction to frequency-response design method Goal: wrap up lead and

More information

ROOT LOCUS. Consider the system. Root locus presents the poles of the closed-loop system when the gain K changes from 0 to. H(s) H ( s) = ( s)

ROOT LOCUS. Consider the system. Root locus presents the poles of the closed-loop system when the gain K changes from 0 to. H(s) H ( s) = ( s) C1 ROOT LOCUS Consider the system R(s) E(s) C(s) + K G(s) - H(s) C(s) R(s) = K G(s) 1 + K G(s) H(s) Root locus presents the poles of the closed-loop system when the gain K changes from 0 to 1+ K G ( s)

More information

7.4 STEP BY STEP PROCEDURE TO DRAW THE ROOT LOCUS DIAGRAM

7.4 STEP BY STEP PROCEDURE TO DRAW THE ROOT LOCUS DIAGRAM ROOT LOCUS TECHNIQUE. Values of on the root loci The value of at any point s on the root loci is determined from the following equation G( s) H( s) Product of lengths of vectors from poles of G( s)h( s)

More information

EE 380. Linear Control Systems. Lecture 10

EE 380. Linear Control Systems. Lecture 10 EE 380 Linear Control Systems Lecture 10 Professor Jeffrey Schiano Department of Electrical Engineering Lecture 10. 1 Lecture 10 Topics Stability Definitions Methods for Determining Stability Lecture 10.

More information

Control Systems. Root Locus & Pole Assignment. L. Lanari

Control Systems. Root Locus & Pole Assignment. L. Lanari Control Systems Root Locus & Pole Assignment L. Lanari Outline root-locus definition main rules for hand plotting root locus as a design tool other use of the root locus pole assignment Lanari: CS - Root

More information

Nyquist Stability Criteria

Nyquist Stability Criteria Nyquist Stability Criteria Dr. Bishakh Bhattacharya h Professor, Department of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc - Funded by MHRD This Lecture Contains Introduction to

More information

DESIGN USING TRANSFORMATION TECHNIQUE CLASSICAL METHOD

DESIGN USING TRANSFORMATION TECHNIQUE CLASSICAL METHOD 206 Spring Semester ELEC733 Digital Control System LECTURE 7: DESIGN USING TRANSFORMATION TECHNIQUE CLASSICAL METHOD For a unit ramp input Tz Ez ( ) 2 ( z ) D( z) G( z) Tz e( ) lim( z) z 2 ( z ) D( z)

More information

AA/EE/ME 548: Problem Session Notes #5

AA/EE/ME 548: Problem Session Notes #5 AA/EE/ME 548: Problem Session Notes #5 Review of Nyquist and Bode Plots. Nyquist Stability Criterion. LQG/LTR Method Tuesday, March 2, 203 Outline:. A review of Bode plots. 2. A review of Nyquist plots

More information

Richiami di Controlli Automatici

Richiami di Controlli Automatici Richiami di Controlli Automatici Gianmaria De Tommasi 1 1 Università degli Studi di Napoli Federico II detommas@unina.it Ottobre 2012 Corsi AnsaldoBreda G. De Tommasi (UNINA) Richiami di Controlli Automatici

More information

Lecture 1: Feedback Control Loop

Lecture 1: Feedback Control Loop Lecture : Feedback Control Loop Loop Transfer function The standard feedback control system structure is depicted in Figure. This represend(t) n(t) r(t) e(t) u(t) v(t) η(t) y(t) F (s) C(s) P (s) Figure

More information

ECE317 : Feedback and Control

ECE317 : Feedback and Control ECE317 : Feedback and Control Lecture : Stability Routh-Hurwitz stability criterion Dr. Richard Tymerski Dept. of Electrical and Computer Engineering Portland State University 1 Course roadmap Modeling

More information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 13: Root Locus Continued Overview In this Lecture, you will learn: Review Definition of Root Locus Points on the Real Axis

More information

Digital Control Systems

Digital Control Systems Digital Control Systems Lecture Summary #4 This summary discussed some graphical methods their use to determine the stability the stability margins of closed loop systems. A. Nyquist criterion Nyquist

More information

Control Systems Engineering ( Chapter 8. Root Locus Techniques ) Prof. Kwang-Chun Ho Tel: Fax:

Control Systems Engineering ( Chapter 8. Root Locus Techniques ) Prof. Kwang-Chun Ho Tel: Fax: Control Systems Engineering ( Chapter 8. Root Locus Techniques ) Prof. Kwang-Chun Ho kwangho@hansung.ac.kr Tel: 02-760-4253 Fax:02-760-4435 Introduction In this lesson, you will learn the following : The

More information

Raktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Frequency Response-Design Method

Raktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Frequency Response-Design Method .. AERO 422: Active Controls for Aerospace Vehicles Frequency Response- Method Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. ... Response to

More information

Control Systems Engineering ( Chapter 6. Stability ) Prof. Kwang-Chun Ho Tel: Fax:

Control Systems Engineering ( Chapter 6. Stability ) Prof. Kwang-Chun Ho Tel: Fax: Control Systems Engineering ( Chapter 6. Stability ) Prof. Kwang-Chun Ho kwangho@hansung.ac.kr Tel: 02-760-4253 Fax:02-760-4435 Introduction In this lesson, you will learn the following : How to determine

More information

Software Engineering 3DX3. Slides 8: Root Locus Techniques

Software Engineering 3DX3. Slides 8: Root Locus Techniques Software Engineering 3DX3 Slides 8: Root Locus Techniques Dr. Ryan Leduc Department of Computing and Software McMaster University Material based on Control Systems Engineering by N. Nise. c 2006, 2007

More information

Nyquist Plots / Nyquist Stability Criterion

Nyquist Plots / Nyquist Stability Criterion Nyquist Plots / Nyquist Stability Criterion Given Nyquist plot is a polar plot for vs using the Nyquist contour (K=1 is assumed) Applying the Nyquist criterion to the Nyquist plot we can determine the

More information

(Continued on next page)

(Continued on next page) (Continued on next page) 18.2 Roots of Stability Nyquist Criterion 87 e(s) 1 S(s) = =, r(s) 1 + P (s)c(s) where P (s) represents the plant transfer function, and C(s) the compensator. The closedloop characteristic

More information

Analysis of Discrete-Time Systems

Analysis of Discrete-Time Systems TU Berlin Discrete-Time Control Systems 1 Analysis of Discrete-Time Systems Overview Stability Sensitivity and Robustness Controllability, Reachability, Observability, and Detectabiliy TU Berlin Discrete-Time

More information

Methods for analysis and control of. Lecture 4: The root locus design method

Methods for analysis and control of. Lecture 4: The root locus design method Methods for analysis and control of Lecture 4: The root locus design method O. Sename 1 1 Gipsa-lab, CNRS-INPG, FRANCE Olivier.Sename@gipsa-lab.inpg.fr www.lag.ensieg.inpg.fr/sename Lead Lag 17th March

More information

Class 13 Frequency domain analysis

Class 13 Frequency domain analysis Class 13 Frequency domain analysis The frequency response is the output of the system in steady state when the input of the system is sinusoidal Methods of system analysis by the frequency response, as

More information

Software Engineering/Mechatronics 3DX4. Slides 6: Stability

Software Engineering/Mechatronics 3DX4. Slides 6: Stability Software Engineering/Mechatronics 3DX4 Slides 6: Stability Dr. Ryan Leduc Department of Computing and Software McMaster University Material based on lecture notes by P. Taylor and M. Lawford, and Control

More information

Recitation 11: Time delays

Recitation 11: Time delays Recitation : Time delays Emilio Frazzoli Laboratory for Information and Decision Systems Massachusetts Institute of Technology November, 00. Introduction and motivation. Delays are incurred when the controller

More information

The stability of linear time-invariant feedback systems

The stability of linear time-invariant feedback systems The stability of linear time-invariant feedbac systems A. Theory The system is atrictly stable if The degree of the numerator of H(s) (H(z)) the degree of the denominator of H(s) (H(z)) and/or The poles

More information

Robust Performance Example #1

Robust Performance Example #1 Robust Performance Example # The transfer function for a nominal system (plant) is given, along with the transfer function for one extreme system. These two transfer functions define a family of plants

More information

Stabilizing the dual inverted pendulum

Stabilizing the dual inverted pendulum Stabilizing the dual inverted pendulum Taylor W. Barton Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail: tbarton@mit.edu) Abstract: A classical control approach to stabilizing a

More information

Unit 7: Part 1: Sketching the Root Locus

Unit 7: Part 1: Sketching the Root Locus Root Locus Unit 7: Part 1: Sketching the Root Locus Engineering 5821: Control Systems I Faculty of Engineering & Applied Science Memorial University of Newfoundland March 14, 2010 ENGI 5821 Unit 7: Root

More information

Design Methods for Control Systems

Design Methods for Control Systems Design Methods for Control Systems Maarten Steinbuch TU/e Gjerrit Meinsma UT Dutch Institute of Systems and Control Winter term 2002-2003 Schedule November 25 MSt December 2 MSt Homework # 1 December 9

More information

Analysis and Synthesis of Single-Input Single-Output Control Systems

Analysis and Synthesis of Single-Input Single-Output Control Systems Lino Guzzella Analysis and Synthesis of Single-Input Single-Output Control Systems l+kja» \Uja>)W2(ja»\ um Contents 1 Definitions and Problem Formulations 1 1.1 Introduction 1 1.2 Definitions 1 1.2.1 Systems

More information

The Nyquist Stability Test

The Nyquist Stability Test Handout X: EE24 Fall 2002 The Nyquist Stability Test.0 Introduction With negative feedback, the closed-loop transfer function A(s) approaches the reciprocal of the feedback gain, f, as the magnitude of

More information

Analysis of Discrete-Time Systems

Analysis of Discrete-Time Systems TU Berlin Discrete-Time Control Systems TU Berlin Discrete-Time Control Systems 2 Stability Definitions We define stability first with respect to changes in the initial conditions Analysis of Discrete-Time

More information

Robust fixed-order H Controller Design for Spectral Models by Convex Optimization

Robust fixed-order H Controller Design for Spectral Models by Convex Optimization Robust fixed-order H Controller Design for Spectral Models by Convex Optimization Alireza Karimi, Gorka Galdos and Roland Longchamp Abstract A new approach for robust fixed-order H controller design by

More information

Lecture 6 Classical Control Overview IV. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

Lecture 6 Classical Control Overview IV. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Lecture 6 Classical Control Overview IV Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Lead Lag Compensator Design Dr. Radhakant Padhi Asst.

More information

Chapter 2. Classical Control System Design. Dutch Institute of Systems and Control

Chapter 2. Classical Control System Design. Dutch Institute of Systems and Control Chapter 2 Classical Control System Design Overview Ch. 2. 2. Classical control system design Introduction Introduction Steady-state Steady-state errors errors Type Type k k systems systems Integral Integral

More information

Stability of Feedback Control Systems: Absolute and Relative

Stability of Feedback Control Systems: Absolute and Relative Stability of Feedback Control Systems: Absolute and Relative Dr. Kevin Craig Greenheck Chair in Engineering Design & Professor of Mechanical Engineering Marquette University Stability: Absolute and Relative

More information

MCE693/793: Analysis and Control of Nonlinear Systems

MCE693/793: Analysis and Control of Nonlinear Systems MCE693/793: Analysis and Control of Nonlinear Systems Introduction to Describing Functions Hanz Richter Mechanical Engineering Department Cleveland State University Introduction Frequency domain methods

More information

Andrea Zanchettin Automatic Control AUTOMATIC CONTROL. Andrea M. Zanchettin, PhD Spring Semester, Linear systems (frequency domain)

Andrea Zanchettin Automatic Control AUTOMATIC CONTROL. Andrea M. Zanchettin, PhD Spring Semester, Linear systems (frequency domain) 1 AUTOMATIC CONTROL Andrea M. Zanchettin, PhD Spring Semester, 2018 Linear systems (frequency domain) 2 Motivations Consider an LTI system Thanks to the Lagrange s formula we can compute the motion of

More information

2.004 Dynamics and Control II Spring 2008

2.004 Dynamics and Control II Spring 2008 MT OpenCourseWare http://ocw.mit.edu.004 Dynamics and Control Spring 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Massachusetts nstitute of Technology

More information

2.004 Dynamics and Control II Spring 2008

2.004 Dynamics and Control II Spring 2008 MT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control Spring 2008 or information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Reading: ise: Chapter 8 Massachusetts

More information

Uncertainty and Robustness for SISO Systems

Uncertainty and Robustness for SISO Systems Uncertainty and Robustness for SISO Systems ELEC 571L Robust Multivariable Control prepared by: Greg Stewart Outline Nature of uncertainty (models and signals). Physical sources of model uncertainty. Mathematical

More information

H(s) = s. a 2. H eq (z) = z z. G(s) a 2. G(s) A B. s 2 s(s + a) 2 s(s a) G(s) 1 a 1 a. } = (z s 1)( z. e ) ) (z. (z 1)(z e at )(z e at )

H(s) = s. a 2. H eq (z) = z z. G(s) a 2. G(s) A B. s 2 s(s + a) 2 s(s a) G(s) 1 a 1 a. } = (z s 1)( z. e ) ) (z. (z 1)(z e at )(z e at ) .7 Quiz Solutions Problem : a H(s) = s a a) Calculate the zero order hold equivalent H eq (z). H eq (z) = z z G(s) Z{ } s G(s) a Z{ } = Z{ s s(s a ) } G(s) A B Z{ } = Z{ + } s s(s + a) s(s a) G(s) a a

More information

Methods for analysis and control of dynamical systems Lecture 4: The root locus design method

Methods for analysis and control of dynamical systems Lecture 4: The root locus design method Methods for analysis and control of Lecture 4: The root locus design method O. Sename 1 1 Gipsa-lab, CNRS-INPG, FRANCE Olivier.Sename@gipsa-lab.inpg.fr www.gipsa-lab.fr/ o.sename 5th February 2015 Outline

More information

Course Summary. The course cannot be summarized in one lecture.

Course Summary. The course cannot be summarized in one lecture. Course Summary Unit 1: Introduction Unit 2: Modeling in the Frequency Domain Unit 3: Time Response Unit 4: Block Diagram Reduction Unit 5: Stability Unit 6: Steady-State Error Unit 7: Root Locus Techniques

More information

Qualitative Graphical Representation of Nyquist Plots

Qualitative Graphical Representation of Nyquist Plots Qualitative Graphical presentation of Nyquist Plots R. Zanasi a, F. Grossi a,, L. Biagiotti a a Department of Engineering Enzo Ferrari, University of Modena and ggio Emilia, via Pietro Vivarelli 0, 425

More information

Remember that : Definition :

Remember that : Definition : Stability This lecture we will concentrate on How to determine the stability of a system represented as a transfer function How to determine the stability of a system represented in state-space How to

More information

SECTION 5: ROOT LOCUS ANALYSIS

SECTION 5: ROOT LOCUS ANALYSIS SECTION 5: ROOT LOCUS ANALYSIS MAE 4421 Control of Aerospace & Mechanical Systems 2 Introduction Introduction 3 Consider a general feedback system: Closed loop transfer function is 1 is the forward path

More information