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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

1 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) Lecture 7: Control Systems I 3/11/ / 27

2 Tentative schedule # Date Topic 1 Sept. 22 Introduction, Signals and Systems 2 Sept. 29 Modeling, Linearization 3 Oct. 6 Analysis 1: Time response, Stability 4 Oct. 13 Analysis 2: Diagonalization, Modal coordinates. 5 Oct. 20 Transfer functions 1: Definition and properties 6 Oct. 27 Transfer functions 2: Poles and Zeros 7 Nov. 3 Analysis of feedback systems: internal stability, root locus 8 Nov. 10 Frequency response 9 Nov. 17 Analysis of feedback systems 2: the Nyquist condition 10 Nov. 24 Specifications for feedback systems 11 Dec. 1 Loop Shaping 12 Dec. 8 PID control 13 Dec. 15 Implementation issues 14 Dec. 22 Robustness E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

3 Today s learning objectives Standard feedback control cofiguration and transfer function nomenclature Well-posedness, Internal vs. I/O stability The root locus method. E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

4 Towards feedback control So far we have looked at how a given system, represented as a state-space model or as a transfer function, behaves given a certain input (and/or initial condition). Typically the system behavior may not be satisfactory (e.g., because it is unstable, or too slow, or too fast, or it oscillates too much, etc.), and one may want to change it. This can only be done by feedback control! The methods we will discuss next provide An analysis tool to understand how the closed-loop system (i.e., the system + feedback control) will behave for di erent choices of feedback control. A synthesis tool to design a good feedback control system. E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

5 Standard feedback configuration r e C(s) u P(s) y Transfer functions make it very easy to compose several blocks (controller, plant, etc.) Imagine doing the same with the state-space model! (Open-)Loop gain: L(s) =P(s)C(s) Complementary sensitivity: (Closed-loop) transfer function from r to y T (s) = L(s) 1+L(s) Sensitivity: (Closed-loop) transfer function from r to e S(s) = 1 1+L(s) E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

6 Concern #1: Well-posedness r e u y D 1 D 2 Assume both the plant and the controller are (static) gains. Then the denominator of the closed-loop transfer functions would be 1+D 2 D 1.IfD 2 D 1 = 1, the whole interconnections does not make sense it is not well posed. In general, never put in feedback blocks with a non-zero feed-through (i.e., non-zero D term). It is always a pain to deal with (Matlab complains of algebraic loops), and in certain conditions can make the system ill-posed. E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

7 Concern #2: (Internal) Stability It may be tempting to just check that the interconnection is I/O stable, i.e., check that the poles of T (s) have negative real part. However, consider what happens if C(s) = 1 s 1 and P(s) = s 1 s+1 : The interconnection is I/O stable: T (s) = 1. s+2 The closed-loop transfer function from r to u is unstable: S(s)C(s) = s+1 (s 1)(s+2). While the system seems to be stable, internally the controller is blowing up. The pole-zero cancellation made the unstable controller mode unobservable in the interconnection. Internal stability requires that all closed-loop transfer functions between any two signals must be stable. Never be tempted to cancel an unstable pole with a non-minimum-phase zero or viceversa. E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

8 How to determine closed-loop stability? Assuming the feedback interconnection is well-posed and internally stable, then all that remains to do is to design C(s) in such a way that all the poles of T (s) have negative real part. In principle one could just pick a trial design for C(s), go to matlab, and check what the closed-loop response (e.g., T (s)) looks like. However, it is desired to find a systematic way to choose C(s), while doing as little calculations as possible. The first automatic control engineers were working with paper, pencil, and possibly a slide-rule. All of classical control can be summarized in exploit the knowledge of the loop gain L(s) to figure out the properties of the closed-loop transfer functions T (s) and S(s) with the least e ort possible. E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

9 Classical methods for feedback control Remember: the main objective is to assess/design the properties of the closed-loop system by exploiting the knowledge of the open-loop system, and avoiding complex calculations. We have three main methods: 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) Di cult 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. Di cult 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. E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

10 Evan s Root Locus method Invented in the late 40s by Walter R. Evans. Useful to study how the roots of a polynomial (i.e., the poles of a system) change as a function of a scalar parameter, e.g., the gain. r y k L(s) Let us write the loop gain in the root locus form The sensitivity function is kl(s) =k N(s) D(s) = k (s z 1)(s z 2 )...(s z m ) (s p 1 )(s p 2 )...(s p n ) S(s) = 1 1+kL(s) = D(s) D(s)+kN(s) The closed-loop poles are the solutions of the characteristic equation D(s)+kN(s) =0. E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

11 The root locus rules What can we say about the closed-loop poles? Since the degree of D(s)+kN(s) is the same as the degree of D(s), the number of closed-loop poles is the same as the number of open-loop poles. For k! 0, D(s)+kN(s) D(s), and the closed-loop poles approach the open-loop poles. For k!1, and the degree of N(s) is the same as the degree of D(s), then 1 k D(s)+N(s) N(s), and the closed-loop poles approach the open-loop zeros. If the degree of N(s) is smaller, then the excess closed-loop poles go to infinity (we will look into this more). The closed-loop poles need to be symmetric wrt the real axis (i.e., either real, or complex-conjugate pairs). E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

12 The angle and magnitude rules Let us rewrite the closed-loop characteristic equation as N(s) D(s) = 1 k The angle rule Take the argument on both sides: \(s z 1 )+\(s z 2 )+...+ \(s z m ) \(s p 1 ) \(s p 2 )... \(s p n )= 180 (±q 360 )ifk > 0 0 (±q 360 )ifk < 0 The magnitude rule Take the argument on both sides: s z 1 s z 2... s z m s p 1 s p 2... s p n = 1 k E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

13 Graphical interpretation All points on the complex plane that could potentially be a closed-loop pole (i.e., the root locus) have to satisfy the angle condition which is essentially THE rule for sketching the root locus. Im s \s p 2 \s z 1 Re \s p 1 The sum of the angles (counted from the real axis) from each zero to s, minus the sum of the angles from each pole to s must be equal to 180 (for positive k) or0 (for negative k), ± an integer multiple of 360. E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

14 All points on the real axis are on the root locus. All points on the real axis to the left of an even number of poles/zeros (or none) are on the negative k root locus All points on the real axis to the right of an odd number of poles/zeros are on the positive k root locus. When two branches come together on the real axis, there will be breakaway or break-in points. E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

15 Asymptotes So what happens when k!1and there are more open-loop poles than zeros? We can see, e.g., from the magnitude condition, that the excess closed-loop poles will have to go to to infinity. Since this is the complex plane, we need to identify in which direction they go towards infinity. This is were we use the angle rule again. If we zoom out su ciently far, the contributions from all the finite open-loop poles and zeros will all be approximately equal to \s, andthe angle rule is approximated by (m n)\s = \ k ± q360. In other words, as k!1, the excess poles will go to infinity along asymptotes at angles of \s = \ k ± q360 = \ k ± q360 m n n m These asymptotes meet in a center of mass lying on the real axis at P n i=1 s com = p P m i j=1 z j n m (note that poles are positive unit masses, and zeros are negative unit masses in this analogy) E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

16 Examples r k 1 s 1 y Im Re E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

17 Examples r k 1 (s 1)(s 2) y Im Re E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

18 Examples r k 1 s 2 +s+1 y Im Re E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

19 Examples r k s+1 s 2 +s+1 y Im Re E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

20 Examples r k 1 (s+1)(s 2 +s+1) y Im Re E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

21 Examples r k s+1 s 2 +s+1 y Im Re E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

22 Examples r k 1 (s+1)(s 2 +s+1) y Im Re E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

23 Stabilizing an inverted pendulum The equations of motion for an inverted pendulum (see lecture 3) can be written as ml 2 = mgl + u; The transfer function is given by G(s) = 1 s 2 g/l. How does the pendulum s behavior change if I attach a spring to the pendulum? E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

24 Proportional control of an inverted pendulum r k 1 s 2 g/l y Im Re E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

25 Proportional-derivative control of an inverted pendulum r k p + k d s 1 s 2 g/l y Im Re E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

26 Root locus summary (for now) Great tool for back-of-the-envelope control design, quick check for closed-loop stability. Qualitative sketches are typically enough. There are many detailed rules for drawing the root locus in a very precise way: if you really need to do that, just use matlab or other methods. Closed-loop poles start from the open loop poles, and are repelled by them. Closed-loop poles are attracted by zeros (or go to infinity). Here you see an obvious explanation why non-minimum-phase zeros are in general to be avoided. Remember that the root locus must be symmetric wrt the real axis. E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

27 Today s learning objectives Standard feedback control cofiguration and transfer function nomenclature Well-posedness, Internal vs. I/O stability The root locus method. E. Frazzoli (ETH) Lecture 7: Control Systems I 3/11/ / 27

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

Control Systems I. Lecture 9: The Nyquist condition

Control Systems I. Lecture 9: The Nyquist condition Control Systems I Lecture 9: The Nyquist condition Readings: Åstrom and Murray, Chapter 9.1 4 www.cds.caltech.edu/~murray/amwiki/index.php/first_edition Jacopo Tani Institute for Dynamic Systems and Control

More information

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 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

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 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

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

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

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

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

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

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

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

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

Root Locus U R K. Root Locus: Find the roots of the closed-loop system for 0 < k < infinity

Root Locus U R K. Root Locus: Find the roots of the closed-loop system for 0 < k < infinity Background: Root Locus Routh Criteria tells you the range of gains that result in a stable system. It doesn't tell you how the system will behave, however. That's a problem. For example, for the following

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

Dr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Root Locus

Dr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Root Locus Week Content Notes 1 Introduction 2 Frequency Domain Modelling 3 Transient Performance and the s-plane 4 Block Diagrams 5 Feedback System Characteristics Assign 1 Due 6 Root Locus 7 Root Locus 2 Assign

More information

Välkomna till TSRT15 Reglerteknik Föreläsning 4. Summary of lecture 3 Root locus More specifications Zeros (if there is time)

Välkomna till TSRT15 Reglerteknik Föreläsning 4. Summary of lecture 3 Root locus More specifications Zeros (if there is time) Välkomna till TSRT15 Reglerteknik Föreläsning 4 Summary of lecture 3 Root locus More specifications Zeros (if there is time) Summary of last lecture 2 We introduced the PID-controller (Proportional Integrating

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

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

CHAPTER # 9 ROOT LOCUS ANALYSES

CHAPTER # 9 ROOT LOCUS ANALYSES F K א CHAPTER # 9 ROOT LOCUS ANALYSES 1. Introduction 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

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

Example on Root Locus Sketching and Control Design

Example on Root Locus Sketching and Control Design Example on Root Locus Sketching and Control Design MCE44 - Spring 5 Dr. Richter April 25, 25 The following figure represents the system used for controlling the robotic manipulator of a Mars Rover. We

More information

Automatic Control (TSRT15): Lecture 4

Automatic Control (TSRT15): Lecture 4 Automatic Control (TSRT15): Lecture 4 Tianshi Chen Division of Automatic Control Dept. of Electrical Engineering Email: tschen@isy.liu.se Phone: 13-282226 Office: B-house extrance 25-27 Review of the last

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

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

School of Mechanical Engineering Purdue University. DC Motor Position Control The block diagram for position control of the servo table is given by:

School of Mechanical Engineering Purdue University. DC Motor Position Control The block diagram for position control of the servo table is given by: Root Locus Motivation Sketching Root Locus Examples ME375 Root Locus - 1 Servo Table Example DC Motor Position Control The block diagram for position control of the servo table is given by: θ D 0.09 See

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

MAK 391 System Dynamics & Control. Presentation Topic. The Root Locus Method. Student Number: Group: I-B. Name & Surname: Göksel CANSEVEN

MAK 391 System Dynamics & Control. Presentation Topic. The Root Locus Method. Student Number: Group: I-B. Name & Surname: Göksel CANSEVEN MAK 391 System Dynamics & Control Presentation Topic The Root Locus Method Student Number: 9901.06047 Group: I-B Name & Surname: Göksel CANSEVEN Date: December 2001 The Root-Locus Method Göksel CANSEVEN

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

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

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

I What is root locus. I System analysis via root locus. I How to plot root locus. Root locus (RL) I Uses the poles and zeros of the OL TF

I What is root locus. I System analysis via root locus. I How to plot root locus. Root locus (RL) I Uses the poles and zeros of the OL TF EE C28 / ME C34 Feedback Control Systems Lecture Chapter 8 Root Locus Techniques Lecture abstract Alexandre Bayen Department of Electrical Engineering & Computer Science University of California Berkeley

More information

Alireza Mousavi Brunel University

Alireza Mousavi Brunel University Alireza Mousavi Brunel University 1 » Control Process» Control Systems Design & Analysis 2 Open-Loop Control: Is normally a simple switch on and switch off process, for example a light in a room is switched

More information

6.302 Feedback Systems Recitation 7: Root Locus Prof. Joel L. Dawson

6.302 Feedback Systems Recitation 7: Root Locus Prof. Joel L. Dawson To start with, let s mae sure we re clear on exactly what we mean by the words root locus plot. Webster can help us with this: ROOT: A number that reduces and equation to an identity when it is substituted

More information

Root Locus. Signals and Systems: 3C1 Control Systems Handout 3 Dr. David Corrigan Electronic and Electrical Engineering

Root Locus. Signals and Systems: 3C1 Control Systems Handout 3 Dr. David Corrigan Electronic and Electrical Engineering Root Locus Signals and Systems: 3C1 Control Systems Handout 3 Dr. David Corrigan Electronic and Electrical Engineering corrigad@tcd.ie Recall, the example of the PI controller car cruise control system.

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

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

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

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

Root locus Analysis. P.S. Gandhi Mechanical Engineering IIT Bombay. Acknowledgements: Mr Chaitanya, SYSCON 07

Root locus Analysis. P.S. Gandhi Mechanical Engineering IIT Bombay. Acknowledgements: Mr Chaitanya, SYSCON 07 Root locus Analysis P.S. Gandhi Mechanical Engineering IIT Bombay Acknowledgements: Mr Chaitanya, SYSCON 07 Recap R(t) + _ k p + k s d 1 s( s+ a) C(t) For the above system the closed loop transfer function

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

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 6: Generalized and Controller Design Overview In this Lecture, you will learn: Generalized? What about changing OTHER parameters

More information

Lecture Sketching the root locus

Lecture Sketching the root locus Lecture 05.02 Sketching the root locus It is easy to get lost in the detailed rules of manual root locus construction. In the old days accurate root locus construction was critical, but now it is useful

More information

Dr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Review

Dr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Review Week Date Content Notes 1 6 Mar Introduction 2 13 Mar Frequency Domain Modelling 3 20 Mar Transient Performance and the s-plane 4 27 Mar Block Diagrams Assign 1 Due 5 3 Apr Feedback System Characteristics

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

The loop shaping paradigm. Lecture 7. Loop analysis of feedback systems (2) Essential specifications (2)

The loop shaping paradigm. Lecture 7. Loop analysis of feedback systems (2) Essential specifications (2) Lecture 7. Loop analysis of feedback systems (2). Loop shaping 2. Performance limitations The loop shaping paradigm. Estimate performance and robustness of the feedback system from the loop transfer L(jω)

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

Exam. 135 minutes + 15 minutes reading time

Exam. 135 minutes + 15 minutes reading time Exam January 23, 27 Control Systems I (5-59-L) Prof. Emilio Frazzoli Exam Exam Duration: 35 minutes + 5 minutes reading time Number of Problems: 45 Number of Points: 53 Permitted aids: Important: 4 pages

More information

EE302 - Feedback Systems Spring Lecture KG(s)H(s) = KG(s)

EE302 - Feedback Systems Spring Lecture KG(s)H(s) = KG(s) EE3 - Feedback Systems Spring 19 Lecturer: Asst. Prof. M. Mert Ankarali Lecture 1.. 1.1 Root Locus In control theory, root locus analysis is a graphical analysis method for investigating the change of

More information

Step input, ramp input, parabolic input and impulse input signals. 2. What is the initial slope of a step response of a first order system?

Step input, ramp input, parabolic input and impulse input signals. 2. What is the initial slope of a step response of a first order system? IC6501 CONTROL SYSTEM UNIT-II TIME RESPONSE PART-A 1. What are the standard test signals employed for time domain studies?(or) List the standard test signals used in analysis of control systems? (April

More information

27. The pole diagram and the Laplace transform

27. The pole diagram and the Laplace transform 124 27. The pole diagram and the Laplace transform When working with the Laplace transform, it is best to think of the variable s in F (s) as ranging over the complex numbers. In the first section below

More information

20. The pole diagram and the Laplace transform

20. The pole diagram and the Laplace transform 95 0. The pole diagram and the Laplace transform When working with the Laplace transform, it is best to think of the variable s in F (s) as ranging over the complex numbers. In the first section below

More information

ESE319 Introduction to Microelectronics. Feedback Basics

ESE319 Introduction to Microelectronics. Feedback Basics Feedback Basics Stability Feedback concept Feedback in emitter follower One-pole feedback and root locus Frequency dependent feedback and root locus Gain and phase margins Conditions for closed loop stability

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

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

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

Bangladesh University of Engineering and Technology. EEE 402: Control System I Laboratory

Bangladesh University of Engineering and Technology. EEE 402: Control System I Laboratory Bangladesh University of Engineering and Technology Electrical and Electronic Engineering Department EEE 402: Control System I Laboratory Experiment No. 4 a) Effect of input waveform, loop gain, and system

More information

EECE 460. Decentralized Control of MIMO Systems. Guy A. Dumont. Department of Electrical and Computer Engineering University of British Columbia

EECE 460. Decentralized Control of MIMO Systems. Guy A. Dumont. Department of Electrical and Computer Engineering University of British Columbia EECE 460 Decentralized Control of MIMO Systems Guy A. Dumont Department of Electrical and Computer Engineering University of British Columbia January 2011 Guy A. Dumont (UBC EECE) EECE 460 - Decentralized

More information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 8: Response Characteristics Overview In this Lecture, you will learn: Characteristics of the Response Stability Real Poles

More information

5 Root Locus Analysis

5 Root Locus Analysis 5 Root Locus Analysis 5.1 Introduction A control system is designed in tenns of the perfonnance measures discussed in chapter 3. Therefore, transient response of a system plays an important role in 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

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 8: Response Characteristics Overview In this Lecture, you will learn: Characteristics of the Response Stability Real

More information

Inverted Pendulum. Objectives

Inverted Pendulum. Objectives Inverted Pendulum Objectives The objective of this lab is to experiment with the stabilization of an unstable system. The inverted pendulum problem is taken as an example and the animation program gives

More information

Root Locus Methods. The root locus procedure

Root Locus Methods. The root locus procedure Root Locus Methods Design of a position control system using the root locus method Design of a phase lag compensator using the root locus method The root locus procedure To determine the value of the gain

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

Procedure for sketching bode plots (mentioned on Oct 5 th notes, Pg. 20)

Procedure for sketching bode plots (mentioned on Oct 5 th notes, Pg. 20) Procedure for sketching bode plots (mentioned on Oct 5 th notes, Pg. 20) 1. Rewrite the transfer function in proper p form. 2. Separate the transfer function into its constituent parts. 3. Draw the Bode

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

ECE 380: Control Systems

ECE 380: Control Systems ECE 380: Control Systems Winter 2011 Course Notes, Part II Section 002 Prof. Shreyas Sundaram Department of Electrical and Computer Engineering University of Waterloo ii Acknowledgments Parts of these

More information

Essence of the Root Locus Technique

Essence of the Root Locus Technique Essence of the Root Locus Technique In this chapter we study a method for finding locations of system poles. The method is presented for a very general set-up, namely for the case when the closed-loop

More information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture : Different Types of Control Overview In this Lecture, you will learn: Limits of Proportional Feedback Performance

More information

CHAPTER 5 ROBUSTNESS ANALYSIS OF THE CONTROLLER

CHAPTER 5 ROBUSTNESS ANALYSIS OF THE CONTROLLER 114 CHAPTER 5 ROBUSTNESS ANALYSIS OF THE CONTROLLER 5.1 INTRODUCTION Robust control is a branch of control theory that explicitly deals with uncertainty in its approach to controller design. It also refers

More information

EE402 - Discrete Time Systems Spring Lecture 10

EE402 - Discrete Time Systems Spring Lecture 10 EE402 - Discrete Time Systems Spring 208 Lecturer: Asst. Prof. M. Mert Ankarali Lecture 0.. Root Locus For continuous time systems the root locus diagram illustrates the location of roots/poles of a closed

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

Unit 11 - Week 7: Quantitative feedback theory (Part 1/2)

Unit 11 - Week 7: Quantitative feedback theory (Part 1/2) X reviewer3@nptel.iitm.ac.in Courses» Control System Design Announcements Course Ask a Question Progress Mentor FAQ Unit 11 - Week 7: Quantitative feedback theory (Part 1/2) Course outline How to access

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

EQ: What are limits, and how do we find them? Finite limits as x ± Horizontal Asymptote. Example Horizontal Asymptote

EQ: What are limits, and how do we find them? Finite limits as x ± Horizontal Asymptote. Example Horizontal Asymptote Finite limits as x ± The symbol for infinity ( ) does not represent a real number. We use to describe the behavior of a function when the values in its domain or range outgrow all finite bounds. For example,

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

Problems -X-O («) s-plane. s-plane *~8 -X -5. id) X s-plane. s-plane. -* Xtg) FIGURE P8.1. j-plane. JO) k JO)

Problems -X-O («) s-plane. s-plane *~8 -X -5. id) X s-plane. s-plane. -* Xtg) FIGURE P8.1. j-plane. JO) k JO) Problems 1. For each of the root loci shown in Figure P8.1, tell whether or not the sketch can be a root locus. If the sketch cannot be a root locus, explain why. Give all reasons. [Section: 8.4] *~8 -X-O

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

EECE 460 : Control System Design

EECE 460 : Control System Design EECE 460 : Control System Design SISO Pole Placement Guy A. Dumont UBC EECE January 2011 Guy A. Dumont (UBC EECE) EECE 460: Pole Placement January 2011 1 / 29 Contents 1 Preview 2 Polynomial Pole Placement

More information

and a where is a Vc. K val paran This ( suitab value

and a where is a Vc. K val paran This ( suitab value 198 Chapter 5 Root-Locus Method One classical technique in determining pole variations with parameters is known as the root-locus method, invented by W. R. Evens, which will be introduced in this chapter.

More information

An Internal Stability Example

An Internal Stability Example An Internal Stability Example Roy Smith 26 April 2015 To illustrate the concept of internal stability we will look at an example where there are several pole-zero cancellations between the controller and

More information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 12: Overview In this Lecture, you will learn: Review of Feedback Closing the Loop Pole Locations Changing the Gain

More information

UNIT 3. Recall From Unit 2 Rational Functions

UNIT 3. Recall From Unit 2 Rational Functions UNIT 3 Recall From Unit Rational Functions f() is a rational function if where p() and q() are and. Rational functions often approach for values of. Rational Functions are not graphs There various types

More information

UNIT 3. Rational Functions Limits at Infinity (Horizontal and Slant Asymptotes) Infinite Limits (Vertical Asymptotes) Graphing Rational Functions

UNIT 3. Rational Functions Limits at Infinity (Horizontal and Slant Asymptotes) Infinite Limits (Vertical Asymptotes) Graphing Rational Functions UNIT 3 Rational Functions Limits at Infinity (Horizontal and Slant Asymptotes) Infinite Limits (Vertical Asymptotes) Graphing Rational Functions Recall From Unit Rational Functions f() is a rational function

More information

Introduction to Root Locus. What is root locus?

Introduction to Root Locus. What is root locus? Introduction to Root Locus What is root locus? A graphical representation of the closed loop poles as a system parameter (Gain K) is varied Method of analysis and design for stability and transient response

More information

Answers for Homework #6 for CST P

Answers for Homework #6 for CST P Answers for Homework #6 for CST 407 02P Assigned 5/10/07, Due 5/17/07 Constructing Evans root locus diagrams in Scilab Root Locus It is easy to construct a root locus of a transfer function in Scilab.

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

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

Model Uncertainty and Robust Stability for Multivariable Systems

Model Uncertainty and Robust Stability for Multivariable Systems Model Uncertainty and Robust Stability for Multivariable Systems ELEC 571L Robust Multivariable Control prepared by: Greg Stewart Devron Profile Control Solutions Outline Representing model uncertainty.

More information

Compensation 8. f4 that separate these regions of stability and instability. The characteristic S 0 L U T I 0 N S

Compensation 8. f4 that separate these regions of stability and instability. The characteristic S 0 L U T I 0 N S S 0 L U T I 0 N S Compensation 8 Note: All references to Figures and Equations whose numbers are not preceded by an "S"refer to the textbook. As suggested in Lecture 8, to perform a Nyquist analysis, we

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

1 Closed Loop Systems

1 Closed Loop Systems Harvard University Division of Engineering and Applied Sciences ES 45 - Physiological Systems Analysis Fall 2009 Closed Loop Systems and Stability Closed Loop Systems Many natural and man-made systems

More information

Lecture 5 Classical Control Overview III. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

Lecture 5 Classical Control Overview III. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Lecture 5 Classical Control Overview III Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore A Fundamental Problem in Control Systems Poles of open

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

Control Systems II. ETH, MAVT, IDSC, Lecture 4 17/03/2017. G. Ducard

Control Systems II. ETH, MAVT, IDSC, Lecture 4 17/03/2017. G. Ducard Control Systems II ETH, MAVT, IDSC, Lecture 4 17/03/2017 Lecture plan: Control Systems II, IDSC, 2017 SISO Control Design 24.02 Lecture 1 Recalls, Introductory case study 03.03 Lecture 2 Cascaded Control

More information