# Intro to Frequency Domain Design

Size: px
Start display at page:

Transcription

1 Intro to Frequency Domain Design MEM 355 Performance Enhancement of Dynamical Systems Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University

2 Outline Closed Loop Transfer Functions The gang of four Sensitivity functions Traditional Performance Measures Time domain Frequency domain Stability & Robustness Introduction role of sensitivity functions Nyquist Traditional gain/phase margins

3 0/4/208 Closed Loop Transfer Functions

4 Closed Loop Transfer Functons error control d disturbance r - G c u G p y output d 2 reference p c ( ) ( ) Y = G U + D U = G R Y D E = R Y 2 noise

5 Transfer Functions Output GG G GG Ys () = Rs () D() s + D() s + GG + GG + GG Error p c p p c 2 p c p c p c G GG Es () = Rs () D() s + D() s + GG + GG + GG Control p p c 2 p c p c p c G GG G U() s = R() s D () s D () s + GG + GG + GG c p c c 2 p c p c p c

6 Gang of Four Outputs Inputs Output Error Control Reference GG p c + GG p c Gc + GG p c + GG p c Disturbance Noise Gp G GG p p c + GG + GG + GG p c GG p c GG p c Gc + GG + GG + GG p c p p c c p p c c

7 Sensitivity Functions [ ] [ ] [ ] E() s = I + L R() s I + L G D () s + I + L LD () s where L: = GG p [ ] 2 sensitivity function: S: = I + L command error complementary sensitivity c function: For SISO systems Bode derived: dt T dl L = dt dl { 2 [ L] L [ L] } [ L T = = + + [ L] L L] = + = change in command S p L [ + L] [ ] T : = I + L L with respect to change in L L command output transfer function (normalized) output

8 0/4/208 Traditional Performance Criteria

9 Traditional Performance ~ Time Domain ultimate error, limit of e(t) as t approaches infinity rise time, T r, usually defined as the time to get from 0% to 90% of its ultimate (i.e., final) value. settling time, T s, the time at which the trajectory first enters an ε-tolerance of its ultimate value and remains there (ε is often taken as 2% of the ultimate value). peak time, T p, the time at which the trajectory attains its peak value. peak overshoot, OS, the peak or supreme value of the trajectory ordinarily expressed as a percentage of the ultimate value of the trajectory. An overshoot of more than 30% is often considered undesirable. A system without overshoot is overdamped and may be too slow (as measured by rise time and settling time).

10 TTrr rise timettpp peak timettss settling time Traditional Performance ~ Time Domain Cont d y p.4 Im.2 α degree of stability, decay rate /α 2ε ideal region for closed loop poles θ damping ratio θ = sin Re ρ T r T T p s rise time peak time settling time t T r T p T s Time response parameters Ideal pole locations

11 Bode Plot Given any transfer function describing a SISO system Y s = GsU s ( ) ( ) ( ) Suppose the input is a sinusoid u t = Asin ωt, ( ) ( ) then the output is a sinusoid y t = Bsin ωt+ θ ( ) ( ) B = G jω A, θ = G jω ( ) ( )

12 Traditional Performance ~ Frequency Domain Sensitivity Complementary sensitivity Desired Altitude - 2 s +.5s+ 0.5 K s disturbance ( 20s+ )( 0s+ )( 0.5s+ ) V-22 Osprey altitude control K=280

13 Sensitivity Functions, Cont d For unity feed back systems: S is the error response to command transfer function T is the output response to command transfer function ( ) Suppose L s is strictly proper m< n, then lim S( jω ) = lim = ω + L j ω lim S ( jω ) ω 0 ω 0 ( jω ) ω 0 ω 0 ( ω) = lim = + c L( jω ) ( ω) L( jω ) ( ω) L( jω ) L j lim T( jω ) = lim = 0 ω + ω limt 0 type L << type L= 0 L j type L = lim = + c type L= 0 We would like S=0 We would like T= A system is of type p if the transfer function has p free integrators in the denomintor, i.e. m m s + am s + + a0 L( s) = k s p s n p b s n p b ( + n p + + 0) L

14 Traditional Performance ~ Frequency Domain Bandwidth Definitions Sensitivity Function (first crosses / 2=0.707~-3db from below): { v S j 2 [0, v) } ω = max : ( ω) < ω BS v Complementary Sensitivity Function crosses ω BT 2 from above) = min{ v : T( jω) < 2 ω (, v ) } v Crossover frequency { v L j v } ω = max : ( ω) ω [0, ) c v (highest frequency where T

15 Example: Osprey Bandwidth Complementary sensitivity Open loop Sensitivity

16 Interpretation of Bode Plot G( s) Yresponse to input U: E( s) = GsU ( ) ( s) ( ω) = ( ω) ( ω) ( ω) = ( ω) ( ω), ( ω) = ( ω) + U ( jω) Any transfer function: Output Y j G j U j Y j G j U j Y j G j e sin ( ωt) 0.00 t 0. t e MAGNITUDE MAGNITUDE db 0 db rad sec input frequency distribution rad sec

17 A Fundamental Tradeoff [ ] [ ] Note that + L + + L L= S + T = Making S small improves tracking & disturbance rejection but makes system susceptible to noise S T : ( ) Es () = SsRs () () SsG () s D() s+ TsD () () s Typical design specifications S T ( jω ) < < ω [ 0, ω ] ( jω) << ω [ ω, ] And, there are other limitations ω > ω 2

18 Cauchy Theorem s-plane F(s) F-plane Any function of a complex number can be viewed as a map of points in one complex plane to another Theorem (Cauchy): Let C be a simple closed curve in the s-plane. F s is a rational function, having neither poles nor zeros on C. If C ( ) image of C under the map F s, then Z = P N where N the number of counterclockwise encirclements of the origin by C as s traverses C once in the clockwise direction. C ( ) ( ) C =Image(C) is the Z the number of zeros of F s enclosed by C, counting multiplicities. P the number of poles of F s enclosed by C, counting multiplicities. ( )

19 Nyquist return difference Sensitivity function Take F(s)=+L(s) (note: F=S - ) Choose a C that encloses the entire RHP Map into L-plane instead of F-plane (shift by -) X s-plane X s-plane L( s) L-plane X R X R X X Typical Nyquist contours Image of Nyquist contour

20 Nyquist Theorem Theorem (Nyquist): If the plot of L(s) (i.e., the image of the Nyquist contour in the L-plane) encircles the point -+j0 in the counterclockwise direction as many times as there are unstable open loop poles (poles of L(s) within the Nyquist contour) then the feedback system has no poles in the RHP. Z = P N closed loop poles in RHP = open loop poles in RHP - cc encirclements of -

21 Example X I s-plane R II L( s) L-plane X III ( ) L s = 2 2 ( s+ p)( s + 2ζω0s+ ω0 ) jθ ( ρ ) I: s = jω L jω =, 0 < ω < ( ) 2 2 ( jω + p)( ω + 2ζω0jω + ω0 ) jθ π π II: s = ρe, ρ, = 2 2 L e = 0 III: s = jω, III I jθ 2 j2θ jθ 2 ( ρ e + p )( ρ e + 2ζωρ e + ω ) * X ρ

22 Example 2 ( ) L s = 2 2 ( + 2ζω0s+ ω0 ) s s X X X I IV III 2 2 ( + j ) ( ) * ( ) s-plane R I: s = jω L( jω) =, 0 < ω < jω ω ζω ω ω jθ π π II: s = ρe, ρ, θ = 2 2 jθ L( ρ e ) = 0 jθ j θ jθ ρ ρ e ρ e 2ζω ρ e ω III: s = jω, III I jθ π π IV : s = εe, ε, θ = 2 2 jθ L( ε e ) = jθ j θ jθ ε e ε e 2ζω ε e ω εω II ε e L( s) jθ L-plane

23 Example 3 ( ) G s = ( ) ( s+ ) s s >> s=tf( s ); >> G=/(s*(s^2+2*0.*s+)); >> nyquist(g) >> s=tf( s ); >> G=/((s+0.0)*(s^2+2*0.*s+)); >> nyquist(g)

24 Gain & Phase Margin Gain margin 0 LL jjjj γ m Gain, db 0-0 γ m frequency 0 Phase margin φ m Phase, deg -80 φ m LL jjjj Nyquist plot L( jω ) Assumption: the nominal system is stable. Bode plot

25 Robustness From Sensitivity Functions M M Sensitivity peaks are related to gain and phase margin. Sensitivity peaks are related to overshoot and damping ratio. S T = max S( jω) ω = max T( jω) ω Unit circle centered at - Im L plane S > S < = ρ = + L= + ρ ( ) jθ S e L e jθ S - = a a L( jω) Re constant S prescribes circle S = + L

26 Example: Osprey Sensitivity function plots for K=280, 500, 000 MAGNITUDE db rad sec

27 Example: Osprey K=280 5 Increased gain yields higher bandwidth (and reduced settling time) but higher sensitivity peak (lower damping ratio and more overshoot) K=000

28 V-22 with PID disturbance Desired Altitude - K s 2 +.5s+ 0.5 s ( 20s+ )( 0s+ )( 0.5s+ )

29 V-22 with PID db MAGNITUDE rad sec db MAGNITUDE rad sec db MAGNITUDE rad sec db MAGNITUDE rad sec Complementary sensitivity sensitivity

30 V-22 with PID >> s=tf('s'); >> G=280*((s^2+.5*s+0.5)/s) /((20*s+)*(0*s+)*(0.5*s+)); >> margin(g) Notice that gain margin is negative because system loses stability when gain drops.

31 V-22 PID+Lead disturbance Desired Altitude - K 2 s s s s s+ ( 20s+ )( 0s+ )( 0.5s+ )

32 V-22 PID+Lead Notice that sensitivity peak is reduced from 20 db to 0 db Complementary sensitivity Sensitivity MAGNITUDE MAGNITUDE db rad sec db rad sec -60 MAGNITUDE MAGNITUDE db db rad sec rad sec Disturbance to output K=50 Command to control

33 V-22 PID+Lead Gain margin is infinite because the system is never unstable for any K>0.

34 Summary Need to consider 2-4 transfer functions to fully evaluate performance Bandwidth is inversely related to settling time Sensitivity function peak is related to overshoot and inversely to damping ratio Gain and phase margins can be determined from Nyquist or Bode plots Sensitivity peak is inversely related to stability margin Design tools: Root locus helps place poles Bode and/or Nyquist helps establish robustness (margins) & performance (sensitivity peaks)

### 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

### 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

### 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

### 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

### 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

### 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

### ECSE 4962 Control Systems Design. A Brief Tutorial on Control Design

ECSE 4962 Control Systems Design A Brief Tutorial on Control Design Instructor: Professor John T. Wen TA: Ben Potsaid http://www.cat.rpi.edu/~wen/ecse4962s04/ Don t Wait Until The Last Minute! You got

### 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

### 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

### 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

### MAS107 Control Theory Exam Solutions 2008

MAS07 CONTROL THEORY. HOVLAND: EXAM SOLUTION 2008 MAS07 Control Theory Exam Solutions 2008 Geir Hovland, Mechatronics Group, Grimstad, Norway June 30, 2008 C. Repeat question B, but plot the phase curve

### 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

### 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

### Discrete Systems. Step response and pole locations. Mark Cannon. Hilary Term Lecture

Discrete Systems Mark Cannon Hilary Term 22 - Lecture 4 Step response and pole locations 4 - Review Definition of -transform: U() = Z{u k } = u k k k= Discrete transfer function: Y () U() = G() = Z{g k},

### Robust and Optimal Control, Spring A: SISO Feedback Control A.1 Internal Stability and Youla Parameterization

Robust and Optimal Control, Spring 2015 Instructor: Prof. Masayuki Fujita (S5-303B) A: SISO Feedback Control A.1 Internal Stability and Youla Parameterization A.2 Sensitivity and Feedback Performance A.3

### 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

### 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

### Let the plant and controller be described as:-

Summary of Fundamental Limitations in Feedback Design (LTI SISO Systems) From Chapter 6 of A FIRST GRADUATE COURSE IN FEEDBACK CONTROL By J. S. Freudenberg (Winter 2008) Prepared by: Hammad Munawar (Institute

### 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

### 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

### Return Difference Function and Closed-Loop Roots Single-Input/Single-Output Control Systems

Spectral Properties of Linear- Quadratic Regulators Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 2018! Stability margins of single-input/singleoutput (SISO) systems! Characterizations

### (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

### 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,

### 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

### Automatic Control 2. Loop shaping. Prof. Alberto Bemporad. University of Trento. Academic year

Automatic Control 2 Loop shaping Prof. Alberto Bemporad University of Trento Academic year 21-211 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 21-211 1 / 39 Feedback

### 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

### 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

### Systems Analysis and Control

Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 21: Stability Margins and Closing the Loop Overview In this Lecture, you will learn: Closing the Loop Effect on Bode Plot Effect

### 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

### 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)

### 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

### 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.

### LABORATORY INSTRUCTION MANUAL CONTROL SYSTEM I LAB EE 593

LABORATORY INSTRUCTION MANUAL CONTROL SYSTEM I LAB EE 593 ELECTRICAL ENGINEERING DEPARTMENT JIS COLLEGE OF ENGINEERING (AN AUTONOMOUS INSTITUTE) KALYANI, NADIA CONTROL SYSTEM I LAB. MANUAL EE 593 EXPERIMENT

### Additional Closed-Loop Frequency Response Material (Second edition, Chapter 14)

Appendix J Additional Closed-Loop Frequency Response Material (Second edition, Chapter 4) APPENDIX CONTENTS J. Closed-Loop Behavior J.2 Bode Stability Criterion J.3 Nyquist Stability Criterion J.4 Gain

### 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

### CDS 101/110: Lecture 10.3 Final Exam Review

CDS 11/11: Lecture 1.3 Final Exam Review December 2, 216 Schedule: (1) Posted on the web Monday, Dec. 5 by noon. (2) Due Friday, Dec. 9, at 5: pm. (3) Determines 3% of your grade Instructions on Front

### CDS 101/110a: Lecture 8-1 Frequency Domain Design

CDS 11/11a: Lecture 8-1 Frequency Domain Design Richard M. Murray 17 November 28 Goals: Describe canonical control design problem and standard performance measures Show how to use loop shaping to achieve

### 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

### 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

### Homework 7 - Solutions

Homework 7 - Solutions Note: This homework is worth a total of 48 points. 1. Compensators (9 points) For a unity feedback system given below, with G(s) = K s(s + 5)(s + 11) do the following: (c) Find the

### 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

### 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

### 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

### 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

### 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

### Due Wednesday, February 6th EE/MFS 599 HW #5

Due Wednesday, February 6th EE/MFS 599 HW #5 You may use Matlab/Simulink wherever applicable. Consider the standard, unity-feedback closed loop control system shown below where G(s) = /[s q (s+)(s+9)]

### VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur

VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur 603 203. DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING SUBJECT QUESTION BANK : EC6405 CONTROL SYSTEM ENGINEERING SEM / YEAR: IV / II year

### x(t) = x(t h), x(t) 2 R ), where is the time delay, the transfer function for such a e s Figure 1: Simple Time Delay Block Diagram e i! =1 \e i!t =!

1 Time-Delay Systems 1.1 Introduction Recitation Notes: Time Delays and Nyquist Plots Review In control systems a challenging area is operating in the presence of delays. Delays can be attributed to acquiring

### 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

### CDS 101/110a: Lecture 10-1 Robust Performance

CDS 11/11a: Lecture 1-1 Robust Performance Richard M. Murray 1 December 28 Goals: Describe how to represent uncertainty in process dynamics Describe how to analyze a system in the presence of uncertainty

### AN INTRODUCTION TO THE CONTROL THEORY

Open-Loop controller An Open-Loop (OL) controller is characterized by no direct connection between the output of the system and its input; therefore external disturbance, non-linear dynamics and parameter

### Digital Control: Summary # 7

Digital Control: Summary # 7 Proportional, integral and derivative control where K i is controller parameter (gain). It defines the ratio of the control change to the control error. Note that e(k) 0 u(k)

### MAE143a: Signals & Systems (& Control) Final Exam (2011) solutions

MAE143a: Signals & Systems (& Control) Final Exam (2011) solutions Question 1. SIGNALS: Design of a noise-cancelling headphone system. 1a. Based on the low-pass filter given, design a high-pass filter,

### 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)

### Table of Laplacetransform

Appendix Table of Laplacetransform pairs 1(t) f(s) oct), unit impulse at t = 0 a, a constant or step of magnitude a at t = 0 a s t, a ramp function e- at, an exponential function s + a sin wt, a sine fun

### Tradeoffs and Limits of Performance

Chapter 9 Tradeoffs and Limits of Performance 9. Introduction Fundamental limits of feedback systems will be investigated in this chapter. We begin in Section 9.2 by discussing the basic feedback loop

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering Dynamics and Control II Fall 2007

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering.4 Dynamics and Control II Fall 7 Problem Set #9 Solution Posted: Sunday, Dec., 7. The.4 Tower system. The system parameters are

### Control of Manufacturing Processes

Control of Manufacturing Processes Subject 2.830 Spring 2004 Lecture #19 Position Control and Root Locus Analysis" April 22, 2004 The Position Servo Problem, reference position NC Control Robots Injection

### Control of Manufacturing Processes

Control of Manufacturing Processes Subject 2.830 Spring 2004 Lecture #18 Basic Control Loop Analysis" April 15, 2004 Revisit Temperature Control Problem τ dy dt + y = u τ = time constant = gain y ss =

### Root Locus Design. MEM 355 Performance Enhancement of Dynamical Systems

Root Locus Design MEM 355 Performance Enhancement of Dynamical Systems Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Outline The root locus design method is an iterative,

### Chapter 15 - Solved Problems

Chapter 5 - Solved Problems Solved Problem 5.. Contributed by - Alvaro Liendo, Universidad Tecnica Federico Santa Maria, Consider a plant having a nominal model given by G o (s) = s + 2 The aim of the

### Stability of CL System

Stability of CL System Consider an open loop stable system that becomes unstable with large gain: At the point of instability, K( j) G( j) = 1 0dB K( j) G( j) K( j) G( j) K( j) G( j) =± 180 o 180 o Closed

### 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

### Dr Ian R. Manchester

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

### Today (10/23/01) Today. Reading Assignment: 6.3. Gain/phase margin lead/lag compensator Ref. 6.4, 6.7, 6.10

Today Today (10/23/01) Gain/phase margin lead/lag compensator Ref. 6.4, 6.7, 6.10 Reading Assignment: 6.3 Last Time In the last lecture, we discussed control design through shaping of the loop gain GK:

### FREQUENCY-RESPONSE DESIGN

ECE45/55: Feedback Control Systems. 9 FREQUENCY-RESPONSE DESIGN 9.: PD and lead compensation networks The frequency-response methods we have seen so far largely tell us about stability and stability margins

### FEEDBACK CONTROL SYSTEMS

FEEDBAC CONTROL SYSTEMS. Control System Design. Open and Closed-Loop Control Systems 3. Why Closed-Loop Control? 4. Case Study --- Speed Control of a DC Motor 5. Steady-State Errors in Unity Feedback Control

### (a) Find the transfer function of the amplifier. Ans.: G(s) =

126 INTRDUCTIN T CNTR ENGINEERING 10( s 1) (a) Find the transfer function of the amplifier. Ans.: (. 02s 1)(. 001s 1) (b) Find the expected percent overshoot for a step input for the closed-loop system

### 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

### Chapter 7 - Solved Problems

Chapter 7 - Solved Problems Solved Problem 7.1. A continuous time system has transfer function G o (s) given by G o (s) = B o(s) A o (s) = 2 (s 1)(s + 2) = 2 s 2 + s 2 (1) Find a controller of minimal

### Automatic Control (TSRT15): Lecture 7

Automatic Control (TSRT15): Lecture 7 Tianshi Chen Division of Automatic Control Dept. of Electrical Engineering Email: tschen@isy.liu.se Phone: 13-282226 Office: B-house extrance 25-27 Outline 2 Feedforward

### Lecture 5: Frequency domain analysis: Nyquist, Bode Diagrams, second order systems, system types

Lecture 5: Frequency domain analysis: Nyquist, Bode Diagrams, second order systems, system types Venkata Sonti Department of Mechanical Engineering Indian Institute of Science Bangalore, India, 562 This

### 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

### Control System Design

ELEC ENG 4CL4: Control System Design Notes for Lecture #11 Wednesday, January 28, 2004 Dr. Ian C. Bruce Room: CRL-229 Phone ext.: 26984 Email: ibruce@mail.ece.mcmaster.ca Relative Stability: Stability

### Feedback Control of Linear SISO systems. Process Dynamics and Control

Feedback Control of Linear SISO systems Process Dynamics and Control 1 Open-Loop Process The study of dynamics was limited to open-loop systems Observe process behavior as a result of specific input signals

### 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

### Delhi Noida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubaneswar Kolkata Patna Web: Ph:

Serial : 0. LS_D_ECIN_Control Systems_30078 Delhi Noida Bhopal Hyderabad Jaipur Lucnow Indore Pune Bhubaneswar Kolata Patna Web: E-mail: info@madeeasy.in Ph: 0-4546 CLASS TEST 08-9 ELECTRONICS ENGINEERING

### Data Based Design of 3Term Controllers. Data Based Design of 3 Term Controllers p. 1/10

Data Based Design of 3Term Controllers Data Based Design of 3 Term Controllers p. 1/10 Data Based Design of 3 Term Controllers p. 2/10 History Classical Control - single controller (PID, lead/lag) is designed

### 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

### Ultimate State. MEM 355 Performance Enhancement of Dynamical Systems

Ultimate State MEM 355 Performance Enhancement of Dnamical Sstems Harr G. Kwatn Department of Mechanical Engineering & Mechanics Drexel Universit Outline Design Criteria two step process Ultimate state

### EE C128 / ME C134 Fall 2014 HW 6.2 Solutions. HW 6.2 Solutions

EE C28 / ME C34 Fall 24 HW 6.2 Solutions. PI Controller For the system G = K (s+)(s+3)(s+8) HW 6.2 Solutions in negative feedback operating at a damping ratio of., we are going to design a PI controller

### IC6501 CONTROL SYSTEMS

DHANALAKSHMI COLLEGE OF ENGINEERING CHENNAI DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING YEAR/SEMESTER: II/IV IC6501 CONTROL SYSTEMS UNIT I SYSTEMS AND THEIR REPRESENTATION 1. What is the mathematical

### Introduction to Feedback Control

Introduction to Feedback Control Control System Design Why Control? Open-Loop vs Closed-Loop (Feedback) Why Use Feedback Control? Closed-Loop Control System Structure Elements of a Feedback Control System

### Active Control? Contact : Website : Teaching

Active Control? Contact : bmokrani@ulb.ac.be Website : http://scmero.ulb.ac.be Teaching Active Control? Disturbances System Measurement Control Controler. Regulator.,,, Aims of an Active Control Disturbances

### 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

### 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,

### Chapter 9: Controller design

Chapter 9. Controller Design 9.1. Introduction 9.2. Effect of negative feedback on the network transfer functions 9.2.1. Feedback reduces the transfer function from disturbances to the output 9.2.2. Feedback

### Performance of Feedback Control Systems

Performance of Feedback Control Systems Design of a PID Controller Transient Response of a Closed Loop System Damping Coefficient, Natural frequency, Settling time and Steady-state Error and Type 0, Type

### 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

### Transient response via gain adjustment. Consider a unity feedback system, where G(s) = 2. The closed loop transfer function is. s 2 + 2ζωs + ω 2 n

Design via frequency response Transient response via gain adjustment Consider a unity feedback system, where G(s) = ωn 2. The closed loop transfer function is s(s+2ζω n ) T(s) = ω 2 n s 2 + 2ζωs + ω 2

### MEM 355 Performance Enhancement of Dynamical Systems

MEM 355 Performance Enhancement of Dynamical Systems State Space Design Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University 11/8/2016 Outline State space techniques emerged

### Design and Tuning of Fractional-order PID Controllers for Time-delayed Processes

Design and Tuning of Fractional-order PID Controllers for Time-delayed Processes Emmanuel Edet Technology and Innovation Centre University of Strathclyde 99 George Street Glasgow, United Kingdom emmanuel.edet@strath.ac.uk

### 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

### 100 (s + 10) (s + 100) e 0.5s. s 100 (s + 10) (s + 100). G(s) =

1 AME 3315; Spring 215; Midterm 2 Review (not graded) Problems: 9.3 9.8 9.9 9.12 except parts 5 and 6. 9.13 except parts 4 and 5 9.28 9.34 You are given the transfer function: G(s) = 1) Plot the bode plot

### Course roadmap. Step response for 2nd-order system. Step response for 2nd-order system

ME45: Control Systems Lecture Time response of nd-order systems Prof. Clar Radcliffe and Prof. Jongeun Choi Department of Mechanical Engineering Michigan State University Modeling Laplace transform Transfer

### State Space Design. MEM 355 Performance Enhancement of Dynamical Systems

State Space Design MEM 355 Performance Enhancement of Dynamical Systems Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Outline State space techniques emerged around

### MEM 255 Introduction to Control Systems: Analyzing Dynamic Response

MEM 55 Itroductio to Cotrol Systems: Aalyzig Dyamic Respose Harry G. Kwaty Departmet of Mechaical Egieerig & Mechaics Drexel Uiversity Outlie Time domai ad frequecy domai A secod order system Via partial

### Step Response Analysis. Frequency Response, Relation Between Model Descriptions

Step Response Analysis. Frequency Response, Relation Between Model Descriptions Automatic Control, Basic Course, Lecture 3 November 9, 27 Lund University, Department of Automatic Control Content. Step