Process Control & Instrumentation (CH 3040)


 Ralf Murphy
 4 years ago
 Views:
Transcription
1 Firstorder systems Process Control & Instrumentation (CH 3040) Arun K. Tangirala Department of Chemical Engineering, IIT Madras January  April 010 Lectures: Mon, Tue, Wed, Fri Extra class: Thu A firstorder system a transfer function of the form: where Kp and τp are the gain and timeconstants of the system respectively The impulse, step and sinusoidal response are given by: where g(t) = K p τ p e t τp K p τ p s +1 = K p/τ p s +1/τ p y s (t) = K p (1 e t τp ) y f (t) = B sin(ω 0 t + φ) at steadystate B A = K p ; φ = tan τ 1 (τ p ω 0 ) p ω0 +1 Example s +1 u(t) = sin(π(0.)t) Sinusoidal (Frequency) Response Since sinusoids are also characterized by their frequencies, the steadystate response to such signals are also known as frequency response The LTI system, due to its linearity, will produce the same shape as the input, but at steadystate. The input will be, however, scaled and shifted The transient portion will be due to its inertia, which will die down if the system is stable. Consider a firstorder system G(s) excited with a sine input u(t) = Asin(ω0t) Y (s) = Kp Aω 0 τ ps+1 s +ω0 = y(t) =c 1 e t τp + c e jω0t + c 3 e jω0t The transient portion of the response dies off leaving behind the complex exponentials! Clearly c and c3 have to be complex conjugates of each other, i.e., c = c3* c = lim G(s) A A ; c = lim G(s) s jω0 jω 0 s jω0 jω 0 Thus, the steadystate response is, using polar representation: G(jω 0 )= G(jω 0 ) e j G(jω0) y s.s. (t) =(c e jω0t )=( A G(jω 0) e j G(jω0) e j π e jω 0t )=A G(jω 0 ) sin(ω 0 t + G(jω 0 ) Frequency response of LTI systems A very interesting result emerges: The steadystate response of an LTI system to a sinusoidal input is a scaled and shifted sine wave of the same frequency (as the input) u(t) = Asin(ω0t) LTI yss(t) = Bsin(ω0t + Φ) For any stable system, the steadystate output can be calculated by noting B A (ω 0) = G(jω 0 ) (Amplitude Ratio) ω0 refers to the input frequency φ(ω 0 ) = G(jω 0 ) (Phase shift) Observe that both amplification/attenuation and phase shift vary with the input frequency For causal systems, the phase shift Φ 0 (output can lag or be in phase with the input) Example (firstorder system): B A = K p ; φ = tan τ 1 (τ p ω 0 ) p ω
2 Firstorder system The amplitude ratio is usually plotted as decibels (db) (after Graham Bell) db = 0log10(AR). Thus a zero decibel corresponds to no attenuation/amplification ωc: Corner frequency Bode plots Plots of db vs. ω and phase shift vs. ω were first proposed by Bodé Hence these plots are popularly known as Bodé plots The ω on the xaxis refers to the input frequency and is on a logscale For a firstorder system, K p db = 0 log 10 τp ω0 +1 = 0 log 10 K p 10 log 10 (τ p ω 0 + 1) 0 log 10 K p when τ p ω log 10 K p 0 log 10 τ p ω 0 when τ p ω 0 1 φ = tan 1 (τ p ω 0 ) 0 when τ p ω 0 1 π when τ pω 0 1 Example s +1 The db value falls off at 0dB/octave at high freqs. The asymptotes of the db plot at low and high freqs. meet at a frequency  corner frequency The log scale for the xaxis helps in a linear curve for the db in the rolloff regime Bode plots of higherorder systems can be easily constructed from the knowledge of plots for lowerorder systems Any higherorder system can be expressed as first and/or secondorder systems in series The AR ratio for the overall system is a product of the ARs of the individual systems Consequently, dboverall = Sum of dbindividuals (due to the property of logarithms) All firstorder systems produce lagged outputs (lag systems) The corner frequency is, in fact, ωc = 1/τp A single pole will always shift high freq. inputs by 90º Phaseoverall = Sum of (phase shifts)individuals (angles add up in polar representations) 5 6 Frequency response function (FRF) The quantity G(jω) carries vital information on how the system treats inputs of different frequencies For this reason, it is known as the frequency response function In general, the response of an LTI system an input depends on its frequency! From this viewpoint, every LTI system is a (linear) filter (only certain inputs are passed ). In the firstorder case, the highfrequency inputs are relatively filtered out => firstorder systems are lowpass filters The FRF is of great value in communications, control, identification, etc. A good understanding of the frequencydomain behaviour of LTI systems is critical to design of filters, receivers, etc. In control and linear systems theory, FRFs provide powerful ways of handling delay systems, stability criteria, controller design, handling disturbances and modelling. In identification, the bandwidth of the system is useful in designing optimal input signals. FRF and Impulse Response The FRF is related to the impulse response through the Fourier Transform where y(t) = y(t)e jωt dt = g(τ)u(t τ) dτ = Y (ω) = G(ω)U(ω) G(ω) = Thus, knowing the FRF is equivalent to knowing the IR of the system Observe that Fourier Transform is a special case of Laplace transform => The quantity G(jω) is also known as the A.C. gain g(τ)u(t τ)e jωt dτ dt g(τ)e jωτ dτ = FRF = Fourier Transform of IR FRF = lim s jω G(s) The D.C. Gain is easily obtained from the A.C. gain by evaluating at ω = 0 (constant signal!) 7 8
3 Use of FRFs in control A quote Stability criteria are most generally provided in frequencydomain Bode s stability criterion (easily handles delays unlike the traditional rootlocus techniques) Nyquist s stability criterion (the most general criterion for linear systems) Frequency response techniques have been increasingly proven to be powerful tools in control system design and analysis A quote due to A.C. Hall Robustness analysis Using Nyquist s or Bode s stability criteria, one can determine the margins of stability once a controller has been designed Uncertainty/disturbance/modelling error descriptions are easily characterized in frequency domain (lowfrequency, highfrequency disturbances, mismatch in the FRFs, etc.) Ability to decouple the effects of delays, timeconstants and gains is a very powerful feature of the frequency response functions. Gain does not affect phase of the system, but only affects the magnitude plot Delay does not affect the magnitude plot, while only shapes the phase plot Source: Frequency Response (R. Oldenburger, ed.), p. 4, MacMillan, New York, Timeconstants affect both the magnitude and phase plots 9 10 Secondorder systems Secondorder systems (two poles/states) are classified according to their pole locations: Responses of secondorder systems The step responses of secondorder systems are shown below Poles (location) Type Impulse response Purely imaginary (IA) Complex, in the LHP Clearly all can be seen as variants of a purely oscillatory secondorder system The general transfer function is: Oscillatory Underdamped Sinusoidal with frequency = natural frequency of the system ωn Damped sinusoid (damping not enough to suppress the oscillatory nature) Repeated real, in the LHP Critically damped Exponential (damping just about sufficient) Real, in the LHP Overdamped Sum of exponentials (damping more than sufficient) Overdamped 100s + 30s +1 Underdamped 100s + 10s +1 ζ =0 (no damping); ζ < 1 (underdamped); ζ =1 (critical damping); ζ > 1 (overdamping); 11 1
4 Frequency response of underdamped systems Frequency response of overdamped systems Underdamped systems exhibit a peak in the magnitude plot For zero damped systems, resonance occurs when the input frequency coincides with the natural frequency of the system The db rolloff is 40 db/octave at high frequencies, while the overall phase shift at high frequencies is 180º. The peak in magnitude plot is only observed when damping ratio is Zero damping AR max = K p ζ 1 ζ, ω r = ω n 1 ζ 0 < ζ < 1 The overdamped system behaves like two firstorder systems in series Hence the values of db and phase are merely the sum of db and phase values of the two respective firstorder systems The overall rolloff rate is 40 db/ octave at high frequencies τ p1 = 6.18, ω c1 =0.038, 100s + 30s +1 τ p =3.8 sec ω c =0.618 rad/sec One observes two corner frequencies, one at ωc = 1/τ1 and the other at ωc = 1/τ A pure delay system is described by y(t) = u(t  D), D! 0 The transfer function is e Ds How is delay defined? Delay systems Delay is the time taken for the system to respond for the first time after the input is changed. Delay is NOT settling time or timeconstant It is a characteristic of the system that is different from its inertia Why do we encounter delays in systems? Transportation lags: Change in input is introduced at a distance away from the system Measurement lags: Output is measured at a distance away from the point of change Higherorder dynamics: Higherorder systems exhibit such sluggishness that appears as delay The first two are true delays present in a system, while the third one is an apparent delay u(t) 0 y(t) 0 D t t Bode plots of delay systems Delays in systems only influence the phase plot of the systems Delays cause a linear rolloff in phase Delays present serious challenges and limitations in controller design Shower example: The shower head is usually about 1 m from the point of mixing => measurement delay. Assume the process is initially at steadystate. A hotter stream is desired so that more hot water is let in, but no immediate change is observed. Unaware of the measurement delay, the hot water valve is opened further, by when the effect of the earlier mixing arrives. After the delay, the hotter stream arrives, forcing the hot water valve to be closed partially, but due to delay, the valve is closed more than by the necessary amount!  this continues forever producing oscillatory response G(jω) = e jdω = G(jω) = 1 and G(jω) = Dω A delay system is an allpass filter 15 16
5 Apparent Delays Approximations of Delays Higherorder systems produce such sluggishness that can appear as delays Traffic flow example: At the lights, the signal changes from red to green (step change). Assuming N (large) vehicles in front of your vehicle, to an observer (usually behind you) observing only the signal change and your movements, he/she feels that you have responded only after a finite time. None among the N drivers exhibit a delay, but this is due to collective inertia. N tanks connected in series: Imagine N tanks connected in series and an observer stands at the last tank to observe the level change for a change in the inlet flow to the first tank. To the observer, the response of the N th tank is delayed. From a mathematical viewpoint, delays are infiniteorder systems e Ds = 1 e Ds = 1 Statespace viewpoint: = 1+Ds + D! s + 1 lim 1+ Ds N N In the standard statespace form, a delay system is described by infinite states! N A model containing delays may have to be approximated as a lowerorder model for purposes of controller design For this purposes, very often different approximation of delays are used Singlezero approximation: e Ds 1 Ds Singlepole approximation: e Ds 1 1+Ds 1 D Pade s firstorder approximation: e Ds s 1+ D s Pade s secondorder approximation: e Ds 1 D s + D 1 s 1+ D s + D 1 s (For small delays) (Error of O(s 3 )) (Error of O(s 5 )) Each of these approximations has its use in controller design The above approximations are used to approximate parts of higherorder systems as delays Revisiting phase plots It is useful to examine the Bode plots of certain basic elements so that for a given system, one can construct an approximate overall Bode plot Constructing a Bode plot: Example (s + 1) (10s + 1)(4s + 1)(s + 1) e 3s Consider, the Bode plot can be easily constructed from its parts  a lead, a constant (gain element), a delay and three lags. Constant system Firstorder (Lag element) Firstorder neg. gain FRF of unstable systems have little meaning  because they do not exist theoretically! Firstorder unstable Lead element Integrator 19 0
6 Putting them together Effects of Zeros The Bode diagrams of the individual elements can now be added up to arrive at the overall Bode plot for the thirdorder (plus delay) system Until now we have examined the effects of pole locations on LTI systems Poles are a characteristic of the systems, arising due to inertia (independent of the inputs) However, zeros can have a significant influence on the response depending on the locations What are zeros anyway? They tell us which inputs are blocked by the system (also known as transmission zeros) Why do they arise? Due to the way (i) the inputs interact with the system (the input derivatives affect the output), (ii) two or more subsystems affect each other Due to approximations of delays They can drastically limit the performance and stability margins of a controller A study of effect of zeros is useful in controller design In filter and controller design, zeros are used to nullify or weaken the effects of poles 1 Firstorder systems with a zero Consider a firstorder system with a zero The system can be viewed as a sum of two subsystems (1/5) + 4/5 Thus, both feedthrough (static) and inertial effects The net effect is a discontinuity in the step response Such systems are known as jump systems The zero affects both db and phase plots (s + 1) Example: Zero located closer to the origin (10s + 1) The zero is located closer to the origin than the pole is. The system exhibits highpass filter characteristics Since a LHP zero produces lead, while an LHP pole induces lag, firstorder systems are known as leadlag systems An LHP zero introduces a 90º lead at high frequencies An RHP zero introduces a 90º lag at high frequencies The location of the zero strongly determines the filtering characteristics u(t) = sin(0.t) u(t) = sin(10t) 3 4
7 Minimumphase and Nonminimum phase systems Two systems can have the same amplitude ratio but a different phase For a given AR, the system with the least phase is said to be minimum phase Delays and RHP zeros cause the systems to be nonminimum phase Example: (s + 1) What problems arise due to nonminimum phase? vs. (s 1) Secondorder systems with zero The responses of secondorder systems with zero depends on the location of the zero (i.e., in LHP or RHP) (5s + 1) (10s + 1)(4s + 1) = 5/3 10s /3 4s +1 = y s(t) = (5/3)(1 e 0.1t )+(1/3)(1 e 0.5t ) ( 5s + 1) (10s + 1)(4s + 1) = 5 10s s +1 = y s (t) = 5(1 e 0.1t ) 3(1 e 0.5t ) Nonminimum phase is due to RHP zero(s) and/or delays The ideal controller is the inverse of process Nonminimum phase systems are only partially invertible (RHP zeros / delays will cause unstable poles / noncausal parts in process inverse) Thus, noninvertible portions cannot be controlled and result in performance limitations 5 6 Inverse Response FRFs of systems with and w/o inverse response u(t) K 1 τ 1 s +1 K τ s +1 Secondorder systems with a zero in RHP exhibit a phenomenon known as the inverse response when a step input is applied. +  y(t) K 1 K > τ 1 τ > 1 The net phase shift is larger for systems with inverse response (70º) than that of those without the inverse response Once again this is due to the nonminimum phase behaviour Observe how the phase plot begins from π and ends up at π/ radians A system is said to exhibit inverse response when the first direction of step response is in a direction opposite to that of the final response (slope of the step response at t = 0) The slope of the response at t = 0 can be calculated from the initial value theorem. The inverse response is due to the fact that a secondorder system with RHP zero is equivalent to two competing subsystems operating at different time scales Inverse response leads to nonminimum phase behaviour Any LTI system will exhibit inverse response if it has odd number of RHP zeros 7 8
8 Higherorder systems Higherorder systems produce more sluggish step responses (larger no. of inertial systems) The initial sluggishness appears as delay to an observer of such systems The larger lags at high frequencies for higherorder systems explains the initial sluggishness Comparison of responses with increase in order Comparison of Bode plots with increase in order 9
Dynamic circuits: Frequency domain analysis
Electronic Circuits 1 Dynamic circuits: Contents Free oscillation and natural frequency Transfer functions Frequency response Bode plots 1 System behaviour: overview 2 System behaviour : review solution
More informationRadar Dish. Armature controlled dc motor. Inside. θ r input. Outside. θ D output. θ m. Gearbox. Control Transmitter. Control. θ D.
Radar Dish ME 304 CONTROL SYSTEMS Mechanical Engineering Department, Middle East Technical University Armature controlled dc motor Outside θ D output Inside θ r input r θ m Gearbox Control Transmitter
More informationRaktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Frequency ResponseDesign 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 informationFrequency domain analysis
Automatic Control 2 Frequency domain analysis Prof. Alberto Bemporad University of Trento Academic year 20102011 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 20102011
More informationDynamic Response. Assoc. Prof. Enver Tatlicioglu. Department of Electrical & Electronics Engineering Izmir Institute of Technology.
Dynamic Response Assoc. Prof. Enver Tatlicioglu Department of Electrical & Electronics Engineering Izmir Institute of Technology Chapter 3 Assoc. Prof. Enver Tatlicioglu (EEE@IYTE) EE362 Feedback Control
More informationPlan 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 frequencyresponse design method Goal: wrap up lead and
More informationHomework 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
More informationFrequency 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 informationCourse roadmap. Step response for 2ndorder system. Step response for 2ndorder system
ME45: Control Systems Lecture Time response of ndorder systems Prof. Clar Radcliffe and Prof. Jongeun Choi Department of Mechanical Engineering Michigan State University Modeling Laplace transform Transfer
More informationSinusoidal Forcing of a FirstOrder Process. / τ
Frequency Response Analysis Chapter 3 Sinusoidal Forcing of a FirstOrder Process For a firstorder transfer function with gain K and time constant τ, the response to a general sinusoidal input, xt = A
More information( ) Frequency Response Analysis. Sinusoidal Forcing of a FirstOrder Process. Chapter 13. ( ) sin ω () (
1 Frequency Response Analysis Sinusoidal Forcing of a FirstOrder Process For a firstorder transfer function with gain K and time constant τ, the response to a general sinusoidal input, xt = A tis: sin
More information8.1.6 Quadratic pole response: resonance
8.1.6 Quadratic pole response: resonance Example G(s)= v (s) v 1 (s) = 1 1+s L R + s LC L + Secondorder denominator, of the form 1+a 1 s + a s v 1 (s) + C R Twopole lowpass filter example v (s) with
More informationOutline. 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 informationLecture 7:Time Response PoleZero Maps Influence of Poles and Zeros Higher Order Systems and Pole Dominance Criterion
Cleveland State University MCE441: Intr. Linear Control Lecture 7:Time Influence of Poles and Zeros Higher Order and Pole Criterion Prof. Richter 1 / 26 FirstOrder Specs: Step : Pole Real inputs contain
More informationTime Response Analysis (Part II)
Time Response Analysis (Part II). A critically damped, continuoustime, second order system, when sampled, will have (in Z domain) (a) A simple pole (b) Double pole on real axis (c) Double pole on imaginary
More informationSystem Identification (CH 5230)
Classification of Models System Identification (CH 5230) Arun K. Tangirala Department of Chemical Engineering, IIT Madras January  April 2010 Lecture Set 1 Models can assume various forms depending on
More informationINTRODUCTION TO DIGITAL CONTROL
ECE4540/5540: Digital Control Systems INTRODUCTION TO DIGITAL CONTROL.: Introduction In ECE450/ECE550 Feedback Control Systems, welearnedhow to make an analog controller D(s) to control a lineartimeinvariant
More informationSystems 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
More informationSystems 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 informationThe FrequencyResponse
6 The FrequencyResponse Design Method A Perspective on the FrequencyResponse Design Method The design of feedback control systems in industry is probably accomplished using frequencyresponse methods
More informationDynamic System Response. Dynamic System Response K. Craig 1
Dynamic System Response Dynamic System Response K. Craig 1 Dynamic System Response LTI Behavior vs. NonLTI Behavior Solution of Linear, ConstantCoefficient, Ordinary Differential Equations Classical
More informationSchool of Mechanical Engineering Purdue University
Case Study ME375 Frequency Response  1 Case Study SUPPORT POWER WIRE DROPPERS Electric train derives power through a pantograph, which contacts the power wire, which is suspended from a catenary. During
More informationControl Systems I. Lecture 5: Transfer Functions. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control DMAVT ETH Zürich
Control Systems I Lecture 5: Transfer Functions Readings: Emilio Frazzoli Institute for Dynamic Systems and Control DMAVT ETH Zürich October 20, 2017 E. Frazzoli (ETH) Lecture 5: Control Systems I 20/10/2017
More informationDr 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 splane 4 27 Mar Block Diagrams Assign 1 Due 5 3 Apr Feedback System Characteristics
More informationR. W. Erickson. Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder
R. W. Erickson Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder 8.1. Review of Bode plots Decibels Table 8.1. Expressing magnitudes in decibels G db = 0 log 10
More informationTransient Response of a SecondOrder System
Transient Response of a SecondOrder System ECEN 830 Spring 01 1. Introduction In connection with this experiment, you are selecting the gains in your feedback loop to obtain a wellbehaved closedloop
More informationMAS107 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
More informationResponse to a pure sinusoid
Harvard University Division of Engineering and Applied Sciences ES 145/215  INTRODUCTION TO SYSTEMS ANALYSIS WITH PHYSIOLOGICAL APPLICATIONS Fall Lecture 14: The Bode Plot Response to a pure sinusoid
More informationAPPLICATIONS FOR ROBOTICS
Version: 1 CONTROL APPLICATIONS FOR ROBOTICS TEX d: Feb. 17, 214 PREVIEW We show that the transfer function and conditions of stability for linear systems can be studied using Laplace transforms. Table
More informationROOT LOCUS. Consider the system. Root locus presents the poles of the closedloop 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 closedloop system when the gain K changes from 0 to 1+ K G ( s)
More informationFrequency 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 informationSTABILITY. Have looked at modeling dynamic systems using differential equations. and used the Laplace transform to help find step and impulse
SIGNALS AND SYSTEMS: PAPER 3C1 HANDOUT 4. Dr David Corrigan 1. Electronic and Electrical Engineering Dept. corrigad@tcd.ie www.sigmedia.tv STABILITY Have looked at modeling dynamic systems using differential
More informationCourse 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: SteadyState Error Unit 7: Root Locus Techniques
More informationControl Systems I. Lecture 6: Poles and Zeros. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control DMAVT ETH Zürich
Control Systems I Lecture 6: Poles and Zeros Readings: Emilio Frazzoli Institute for Dynamic Systems and Control DMAVT ETH Zürich October 27, 2017 E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/2017
More informationA system that is both linear and timeinvariant is called linear timeinvariant (LTI).
The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Lecture Notes: Time, Frequency & Transform Domains February 28, 2012 Signals & Systems Signals are mapped
More informationMASSACHUSETTS 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
More informationSTABILITY 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: Boundedinput boundedoutput (BIBO) stability Asystemmaybe stable, neutrallyormarginallystable, or unstable. This can be illustrated
More informationAndrea 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 informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 24: Compensation in the Frequency Domain Overview In this Lecture, you will learn: Lead Compensators Performance Specs Altering
More informationExercise 1 (A Nonminimum Phase System)
Prof. Dr. E. Frazzoli 559 Control Systems I (HS 25) Solution Exercise Set Loop Shaping Noele Norris, 9th December 26 Exercise (A Nonminimum Phase System) To increase the rise time of the system, we
More informationControl Systems Design
ELEC4410 Control Systems Design Lecture 13: Stability Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 13: Stability p.1/20 Outline InputOutput
More informationCHAPTER 7 : BODE PLOTS AND GAIN ADJUSTMENTS COMPENSATION
CHAPTER 7 : BODE PLOTS AND GAIN ADJUSTMENTS COMPENSATION Objectives Students should be able to: Draw the bode plots for first order and second order system. Determine the stability through the bode plots.
More informationLecture 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
More informationSIGNAL PROCESSING. B14 Option 4 lectures. Stephen Roberts
SIGNAL PROCESSING B14 Option 4 lectures Stephen Roberts Recommended texts Lynn. An introduction to the analysis and processing of signals. Macmillan. Oppenhein & Shafer. Digital signal processing. Prentice
More informationSystems 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 informationExercise 1 (A Nonminimum Phase System)
Prof. Dr. E. Frazzoli 559 Control Systems I (Autumn 27) Solution Exercise Set 2 Loop Shaping clruch@ethz.ch, 8th December 27 Exercise (A Nonminimum Phase System) To decrease the rise time of the system,
More informationAMME3500: System Dynamics & Control
Stefan B. Williams May, 211 AMME35: System Dynamics & Control Assignment 4 Note: This assignment contributes 15% towards your final mark. This assignment is due at 4pm on Monday, May 3 th during Week 13
More informationIndex. Index. More information. in this web service Cambridge University Press
Atype elements, 4 7, 18, 31, 168, 198, 202, 219, 220, 222, 225 Atype variables. See Across variable ac current, 172, 251 ac induction motor, 251 Acceleration rotational, 30 translational, 16 Accumulator,
More informationAutomatic 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 21211 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 21211 1 / 39 Feedback
More informationIntroduction to Feedback Control
Introduction to Feedback Control Control System Design Why Control? OpenLoop vs ClosedLoop (Feedback) Why Use Feedback Control? ClosedLoop Control System Structure Elements of a Feedback Control System
More informationLecture 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 informationControl 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 DMAVT ETH Zürich November 24, 2017 E. Frazzoli (ETH) Lecture
More informationDiscrete 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},
More informationEE 3054: Signals, Systems, and Transforms Summer It is observed of some continuoustime LTI system that the input signal.
EE 34: Signals, Systems, and Transforms Summer 7 Test No notes, closed book. Show your work. Simplify your answers. 3. It is observed of some continuoustime LTI system that the input signal = 3 u(t) produces
More informationTransient 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
More informationLecture 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 informationThe requirements of a plant may be expressed in terms of (a) settling time (b) damping ratio (c) peak overshoot  in time domain
Compensators To improve the performance of a given plant or system G f(s) it may be necessary to use a compensator or controller G c(s). Compensator Plant G c (s) G f (s) The requirements of a plant may
More information(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 closedloop system
More informationFATIMA MICHAEL COLLEGE OF ENGINEERING & TECHNOLOGY
FATIMA MICHAEL COLLEGE OF ENGINEERING & TECHNOLOGY Senkottai Village, Madurai Sivagangai Main Road, Madurai  625 020. An ISO 9001:2008 Certified Institution DEPARTMENT OF ELECTRONICS AND COMMUNICATION
More informationGATE EE Topic wise Questions SIGNALS & SYSTEMS
www.gatehelp.com GATE EE Topic wise Questions YEAR 010 ONE MARK Question. 1 For the system /( s + 1), the approximate time taken for a step response to reach 98% of the final value is (A) 1 s (B) s (C)
More informationIntroduction to Vibration. Mike Brennan UNESP, Ilha Solteira São Paulo Brazil
Introduction to Vibration Mike Brennan UNESP, Ilha Solteira São Paulo Brazil Vibration Most vibrations are undesirable, but there are many instances where vibrations are useful Ultrasonic (very high
More informationTime Response of Systems
Chapter 0 Time Response of Systems 0. Some Standard Time Responses Let us try to get some impulse time responses just by inspection: Poles F (s) f(t) splane Time response p =0 s p =0,p 2 =0 s 2 t p =
More informationSTABILITY OF CLOSEDLOOP CONTOL SYSTEMS
CHBE320 LECTURE X STABILITY OF CLOSEDLOOP CONTOL SYSTEMS Professor Dae Ryook Yang Spring 2018 Dept. of Chemical and Biological Engineering 101 Road Map of the Lecture X Stability of closedloop control
More informationECEN 420 LINEAR CONTROL SYSTEMS. Lecture 2 Laplace Transform I 1/52
1/52 ECEN 420 LINEAR CONTROL SYSTEMS Lecture 2 Laplace Transform I Linear Time Invariant Systems A general LTI system may be described by the linear constant coefficient differential equation: a n d n
More informationReview of Linear TimeInvariant Network Analysis
D1 APPENDIX D Review of Linear TimeInvariant Network Analysis Consider a network with input x(t) and output y(t) as shown in Figure D1. If an input x 1 (t) produces an output y 1 (t), and an input x
More informationChapter 3. 1 st Order Sine Function Input. General Solution. Ce t. Measurement System Behavior Part 2
Chapter 3 Measurement System Behavior Part 2 1 st Order Sine Function Input Examples of Periodic: vibrating structure, vehicle suspension, reciprocating pumps, environmental conditions The frequency of
More informationRichiami 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 informationFREQUENCYRESPONSE DESIGN
ECE45/55: Feedback Control Systems. 9 FREQUENCYRESPONSE DESIGN 9.: PD and lead compensation networks The frequencyresponse methods we have seen so far largely tell us about stability and stability margins
More informationr +  FINAL June 12, 2012 MAE 143B Linear Control Prof. M. Krstic
MAE 43B Linear Control Prof. M. Krstic FINAL June, One sheet of handwritten notes (two pages). Present your reasoning and calculations clearly. Inconsistent etchings will not be graded. Write answers
More informationDESIGN 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 informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 2: Drawing Bode Plots, Part 2 Overview In this Lecture, you will learn: Simple Plots Real Zeros Real Poles Complex
More informationControl 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
More informationTable 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
More informationEE C128 / ME C134 Fall 2014 HW 8  Solutions. HW 8  Solutions
EE C28 / ME C34 Fall 24 HW 8  Solutions HW 8  Solutions. Transient Response Design via Gain Adjustment For a transfer function G(s) = in negative feedback, find the gain to yield a 5% s(s+2)(s+85) overshoot
More informationFrequency Response Analysis
Frequency Response Analysis Consider let the input be in the form Assume that the system is stable and the steady state response of the system to a sinusoidal inputdoes not depend on the initial conditions
More informationModule 3F2: Systems and Control EXAMPLES PAPER 2 ROOTLOCUS. Solutions
Cambridge University Engineering Dept. Third Year Module 3F: Systems and Control EXAMPLES PAPER ROOTLOCUS Solutions. (a) For the system L(s) = (s + a)(s + b) (a, b both real) show that the rootlocus
More informationRecitation 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 informationIntroduction to Process Control
Introduction to Process Control For more visit : www.mpgirnari.in By: M. P. Girnari (SSEC, Bhavnagar) For more visit: www.mpgirnari.in 1 Contents: Introduction Process control Dynamics Stability The
More informationControl for. Maarten Steinbuch Dept. Mechanical Engineering Control Systems Technology Group TU/e
Control for Maarten Steinbuch Dept. Mechanical Engineering Control Systems Technology Group TU/e Motion Systems m F Introduction Timedomain tuning Frequency domain & stability Filters Feedforward Servooriented
More informationCHAPTER 1 Basic Concepts of Control System. CHAPTER 6 Hydraulic Control System
CHAPTER 1 Basic Concepts of Control System 1. What is open loop control systems and closed loop control systems? Compare open loop control system with closed loop control system. Write down major advantages
More informationRaktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Dynamic Response
.. AERO 422: Active Controls for Aerospace Vehicles Dynamic Response Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. . Previous Class...........
More informationThe DiscreteTime Fourier
Chapter 3 The DiscreteTime Fourier Transform 清大電機系林嘉文 cwlin@ee.nthu.edu.tw 035731152 Original PowerPoint slides prepared by S. K. Mitra 311 ContinuousTime Fourier Transform Definition The CTFT of
More informationAnswers to multiple choice questions
Answers to multiple choice questions Chapter 2 M2.1 (b) M2.2 (a) M2.3 (d) M2.4 (b) M2.5 (a) M2.6 (b) M2.7 (b) M2.8 (c) M2.9 (a) M2.10 (b) Chapter 3 M3.1 (b) M3.2 (d) M3.3 (d) M3.4 (d) M3.5 (c) M3.6 (c)
More informationChapter 7: Time Domain Analysis
Chapter 7: Time Domain Analysis Samantha Ramirez Preview Questions How do the system parameters affect the response? How are the parameters linked to the system poles or eigenvalues? How can Laplace transforms
More informationEC CONTROL SYSTEM UNIT I CONTROL SYSTEM MODELING
EC 2255  CONTROL SYSTEM UNIT I CONTROL SYSTEM MODELING 1. What is meant by a system? It is an arrangement of physical components related in such a manner as to form an entire unit. 2. List the two types
More informationMAE143a: Signals & Systems (& Control) Final Exam (2011) solutions
MAE143a: Signals & Systems (& Control) Final Exam (2011) solutions Question 1. SIGNALS: Design of a noisecancelling headphone system. 1a. Based on the lowpass filter given, design a highpass filter,
More informationLab # 4 Time Response Analysis
Islamic University of Gaza Faculty of Engineering Computer Engineering Dep. Feedback Control Systems Lab Eng. Tareq Abu Aisha Lab # 4 Lab # 4 Time Response Analysis What is the Time Response? It is an
More informationPerformance 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 Steadystate Error and Type 0, Type
More information(b) A unity feedback system is characterized by the transfer function. Design a suitable compensator to meet the following specifications:
1. (a) The open loop transfer function of a unity feedback control system is given by G(S) = K/S(1+0.1S)(1+S) (i) Determine the value of K so that the resonance peak M r of the system is equal to 1.4.
More informationClass 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 informationChapter 8: Converter Transfer Functions
Chapter 8. Converter Transfer Functions 8.1. Review of Bode plots 8.1.1. Single pole response 8.1.2. Single zero response 8.1.3. Right halfplane zero 8.1.4. Frequency inversion 8.1.5. Combinations 8.1.6.
More informationLABORATORY 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
More informationAppendix A: Exercise Problems on Classical Feedback Control Theory (Chaps. 1 and 2)
Appendix A: Exercise Problems on Classical Feedback Control Theory (Chaps. 1 and 2) For all calculations in this book, you can use the MathCad software or any other mathematical software that you are familiar
More informationMAE 143B  Homework 9
MAE 143B  Homework 9 7.1 a) We have stable firstorder poles at p 1 = 1 and p 2 = 1. For small values of ω, we recover the DC gain K = lim ω G(jω) = 1 1 = 2dB. Having this finite limit, our straightline
More informationDelhi 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: Email: info@madeeasy.in Ph: 04546 CLASS TEST 089 ELECTRONICS ENGINEERING
More informationDynamic Behavior. Chapter 5
1 Dynamic Behavior In analyzing process dynamic and process control systems, it is important to know how the process responds to changes in the process inputs. A number of standard types of input changes
More informationECE 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 informationController Design using Root Locus
Chapter 4 Controller Design using Root Locus 4. PD Control Root locus is a useful tool to design different types of controllers. Below, we will illustrate the design of proportional derivative controllers
More informationChapter 7. Digital Control Systems
Chapter 7 Digital Control Systems 1 1 Introduction In this chapter, we introduce analysis and design of stability, steadystate error, and transient response for computercontrolled systems. Transfer functions,
More informationControl 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 =
More information