DESIGN USING TRANSFORMATION TECHNIQUE CLASSICAL METHOD


 Patrick Pearson
 3 years ago
 Views:
Transcription
1 206 Spring Semester ELEC733 Digital Control System LECTURE 7: DESIGN USING TRANSFORMATION TECHNIQUE CLASSICAL METHOD
2 For a unit ramp input Tz Ez ( ) 2 ( z ) D( z) G( z) Tz e( ) lim( z) z 2 ( z ) D( z) G( z) Tz Tz lim lim z( z )( D( z) G( z)) z( z ) D( z) G( z) Similarly, e k v Velocity error constant 2 ( ) lim z ( ) 2 z ( D ( z ) G ( z )) lim Tz Tz z 2 ( z ) D( z) G( z) 2 k a Acceleration error constant 24
3 Remark: ( Truxal's Rule ) Yz ( ) Hz ( ) : the overall transfer function Rz ( ) Assume ( z z)( z z2) ( z zn ) H( z) K ( z p )( z p ) ( z p ) 2 and H( z) results from a Type system of G( z). n For unit step, steadystate error must be zero E( z) R( z) H( z) R( z) 0 H()  () Y( z) E( z) Remark: H( z), Y( z) G( z) E( z), and R( z) R( z) G( z) 25
4 For a ramp input E( z) R( z)( H( z)) ( z ) 2 ( Hz ( )) Tz e( ) lim( z ) ( H( z)) z 2 ( z ) Hz ( ) lim  (2) Tk z z v Tz k v We can use L'Hopital's rule. 26
5 ( d / dz)( H( z)) dh( z) lim lim Tk z ( / )( ) z v d dz z dz d d d Note : ln H( z) H( z) H( z) since H() dz H( z) dz dz Tk v d lim ln Hz ( ) z dz d ( zz ) i lim lnk z dz ( z pi ) d lim ln( z zi ) ln( z pi ) lnk z dz d d lim ( z zi ) ( z pi ) z z zi dz z pi dz n n Tk p z v i i i i 27
6 n n Tk p z v i i i i Imz pole  0 zero Re z kv Errors are decreased pole 0 zero Large overshoot Poor dynamic response Small steadystate error against Good transient response 28
7 ex) Antenna system in p. 228 of Franklin s z G( z) where T sec ( z)( z0.9048) D( z) K ( proportional controller  static gain) KG( z) Hz ( ) KG( z) Characteristic polynomial z K 0 ( z )( z ) Design specifications : k, T 0sec ζ 0.5, T sec v s 4.6 σ r e T s
8 Imaginary Axis p /T 0.8 p /T Discrete root locus with and without compensation 0.7 p /T 0.6 p /T 0.5 p /T 0.4 p /T p /T p /T 0. p /T 0 p /T p /T p /T 0. p /T p /T 0.2 p /T p /T 0.3 p /T 0.6 p /T 0.4 p /T 0.5 p /T Real Axis without compensator D( Z) K ( p.0, p , z ) 2 z with D( z) 6.64, T sec (emulation) z ( p.0, p , z ) 2 30
9 Imz r ζ Re z The system goes unstable at K 9 where k v That means there is no value of gain that meets the steadystate specification. "Trial and error" based controller design 3
10 Dynamic compensation ( pp. 228 ~ 235 ) z D( z) 6.64 T sec (design by emulation) z Direct design using root locus (trial and error) z 0.8 i) D( z) 6 z 0.05 ii) D( z) 3 z 0.88 z 0.5 z 0.8 iii ) D( z) 9 (hidden ocillation) z 0.8 iv ) z 0.88 Dz ( ) 3 (delay) zz ( 0.5) 32
11 Imaginary Axis Root locus for antenna design p /T p /T p /T 0.9 p /T 0.8 p /T 0.8 p /T 0.7 p /T 0.7 p /T 0.6 p /T 0.6 p /T 0.5 p /T 0.5 p /T 0.4 p /T 0.4 p /T p /T p /T 0.2 p /T 0.2 p /T 0. p /T 0. p /T Real Axis z 0.8 Dz ( ) 6 z 0.05 ( p.0, p , p 0.05, z 0.8, z )
12 .4.2 OUTPUT, Y and CONTROL, U/ TIME (SEC) 34
13 Imaginary Axis Root locus for compensated Antenna Design p /T p /T p /T 0.8 p /T 0.7 p /T 0.6 p /T 0.5 p /T 0.4 p /T p /T p /T 0. p /T p /T 0. p /T p /T 0.2 p /T p /T 0.6 p /T 0.5 p /T 0.4 p /T 0.3 p /T Real Axis z 0.88 Dz ( ) 3 z 0.5 ( p.0, p , p 0.5, z 0.88, z )
14 OUTPUT, Y and CONTROL, U/0.5 Step Response of Compensated Antenna TIME (SEC) 36
15 Imaginary Axis 0.76 Root locus for Compensated Antenna Design Real Axis z 0.8 Dz ( ) 9 z 0.8 ( p.0, p , p 0.8, z 0.8, z )
16 OUTPUT, Y and CONTROL, U/0.5 Step response of compensated Antenna Design TIME (SEC) 38
17 Imaginary Axis Root locus for compensated.2 Antenna Design Real Axis z 0.88 Dz ( ) 3 z ( z 0.5) ( p.0, p , p 0.5, p 0, z 0.88, z )
18 OUTPUT, Y and CONTROL, U/0 2 Step response for compensated antenna Design TIME (SEC) 40
19
20
21
22
23 FREQUENCY RESPONSE METHODS. The gain/ phase curve can be easily plotted by hand. 2. The frequency response can be measured experimentally. 3. The dynamic response specification can be easily interpreted in terms of gain/ phase margin. k p 4. The system error constants and can be read directly from the low frequency asymptote of the gain plot. k v 5. The correction to the gain/phase curves can be quickly computed. 6. The effect of pole/ zero gain changes of a compensator can be easily determined. Note :, 5, 6 above are less true for discrete frequency response design using z  transform. 45
24 NYQUIST STABILITY CRITERION Continuous case ns ( ) KD( s) G( s) d( s) Hs ( ) KD( s) G( s) ns ( ) + ns ( ) ds ( ) ds ( ) ns ( ) ns ( ) ds ( ) n( s) d( s) ds ( ) characteristic equation openloop system known closedloop system zeros of the closedloop characteristic equation, n(s) + d(s) = poles of the closedloop system, n(s)+d(s) 46
25 Z (unknown) = # of unstable zeros (same direction) of + K D(s) G(s) ( or # of unstable poles of H(s) ) P (known) = # of unstable poles (opposite direction) of + K D(s) G(s) ( or # of unstable poles of KD(s)G(s)) N(known after mapping) = # of encirclement (same direction) of the origin of +KD(s)G(s) ( or  of KD(s)G(s) ) 47
26 Splane +KD(s)G(s)plane KD(s)G(s)plane unstable poles 0  D(s)G(s)plane Z P =N or Z = P + N /K 48
27 Nyquist stability criterion
28 9.3 Nyquist stability criterion
29 9.3 Nyquist stability criterion
30 9.3 Nyquist stability criterion
31 9.3 Nyquist stability criterion
32 Discrete case ( The ideas are identical ) Unstable region of the zplane is the outside of the unit circle Consider the encirclement of the stable region. N = { # of stable zeros }  { # of stable poles} = { n Z } { n P } = P Z Z = P N In summary,. Determine the number, P, of unstable poles of KDG. jωt 2. Plot KD(z)G(z) for the unit circle, z e and 0ωT 2π. 3. Set N equal to the net number of CCW encirclements of the point  on the plot 4. Compute Z = P N. This system is stable iff Z = 0. 54
33 55
34 ex) p. 24 (Franklin s) The unit feedback discrete system with the plant transfer function with sampling rate ½ Hz and zeroorder hold G( s), ZOH at T 2 s ( s).35 ( z 0.523) Gz ( ) ( z) ( z0.35) K P 0, N # of CCW encirclements of the point 0 Z 0 56
35 Imaginary Axis Nyquist plot from Example using contour Real Axis 57
36 DESIGN SPEC. IN THE FREQUENCY DOMAIN Gain Margin (GM) : The factor by which the gain can be increased before the system to go unstable Phase Margin (PM): A measure of how much additional phase ex) p. 243 lag or time delay can be tolerated in the loop before instability results. G( s) with ZOH at T 0.2 sec 2 s ( s) ( ) ( z3.38) ( z0.242) Gz ( ) ( 0.887) 2 z z K 58
37 Bode plot Nyquist plot GM=.8, PM=8 59
38 4. INTRODUCTION An openloop (direct) system operates without feedback and directly generates the output in response to an input signal. Open loop system with a disturbance input An closedloop system uses a measurement of the output signal and a comparison with the desired output to generate an error signal that is used by the controller to adjust the actuator. Closed loop system with a disturbance input
39 4.2 ERROR SIGNAL ANALYSIS When H(s) =, Loop gain Sensitivity function Complementary sensitivity function Error minimization S(s), C(s): function of G c (s) and G(s) How? Design compromise
40 SENSITIVITY FUNCTION Tracking error performance Stability robustness stability E( s) R( s) D( s) G( s) E( jω) R( jω) D( jω) G( jω) S( jω) R( jω) sensitivity function Remarks: i) T( jω) Djω ( ) Gjω ( ) Djω ( ) Gjω ( ) complementary sensitivity function ii) T( jω) S( jω) 62
41 VGM Im D( jω) G( jω) plane ( DG) S GM PM  Re VGM : vector gain margin S S min( DG) the distance of the closest point from  VGM S VGM max S S S 63
42 PERFORMANCE E S R eb where eb is an error bound R S e b Define W ( ω) R performance frequency function e b S W W DG S Remark: At every frequency, the point DG W  on the Nyquist plot lies outside the disk of the center , radius W ( jω). DG 64
43 If the loop gain is large DG S where S DG DG W DG ex) e 0.005, R below 00Hz b 0 ω 2πf 200π R e b = W W ( ω) DG 200 ω The higher the magnitude curve at low frequency, the lower the steadystate errors 65
44 ROBUST STABILITY G( jω) G ( jω)[ w ( ω) ( jω)] 0 2 G( jω) G ( jω) w ( ω) ( jω) 0 2 multiplicative uncertainty additive uncertainty where G ( s) : nominal transfer function w ( ω) : magnitude ( jω) : phase ( jω) w ( ω) W ( ω) upper bound W ( ω) : robust stability frequency function 66
45 ( ) s B ω K Gs s s s B ω ω s ω K s s s B ω ω 2 ( ) s ω w ω s s B ω ω ex) p. 250 small for low frequencies and large for high frequencies
46 0 2 Fig.7.26 Model uncertainty for disk read/write head assembly Fig.7.27 Plot of typical plant uncertainty
47 Define T( jω) S( jω) DG 0 complementary sensitivity function Assume that the nominal system is stable, DG0 0 S DG 0 (for stability robustness) DG [ w ] DG DG 0 ( DG0) w2 0 DG0 ( DG )( Tw ) 0 Suff & Nec. Cond. Tw not to be zero any w W T w T W
48 T DG since DG DG W 2 is small for high frequencies. DG 0 W robust performance 200 Bode plot robust stability W 2 ω 70
49 Remark: WT 2 W DG 2 0 DG 0 ω W DG DG Im  DG 0 W DG 2 0 Re Remark: At every frequency, the critical point, , lies outside the disk of the center DG, radius W ( jω) Djω ( ) G( jω). 2 7
50 Robust Performance W Im  DG 0 W DG 2 0 Re Remark: For each frequency ω, construct two closed disks; one with center , radius radius W ( jω) ; the other with center DG, radius W ( jω) D( jω) G( jω). 2 Then robust performance holds iff for each ω, these two disks are disjoint. 72
51 Bode plot 73
52 Remark: T st z e z 2 T 2 2 z ω T z s s bilinear rule There exists big difference in the Bode diagram at the high frequency. Types of Compensator: phaselead ( high pass ~ PD) transient response phaselag ( low pass ~ PI ) steadystate response PID : a special case of a phase leadlag compensator 74
53 Remark: Transient response PM ς 00 Steadystate response 75 Type 0 system 20 log K at the lowfrequency magnitude in Bode logmagnitude plot p Type system K v is the intersection (extension of the initial 20 db/decde) with frequency axis in Bode logmagnitude plot Type 2 system K a is the intersection (extension of the initial 40 db/decde) with frequency axis in Bode logmagnitude plot
54 DESIGN PROCEDURE IN FREQUENCY. 2. Bode plot 3. Gs ( ) hold ν jω G( ν ) Gjω ( ) T ν z 2 bilinear transformation T ν Gz ( ) 2 G( ν) Assuming that the low frequency gain of D( ω) is unity, design D( ω). 2 z ω T z D( ω) Dz ( ) 4. 76
55 ex) p (Ogata's) Gs ( ) Ts e 0 s s0 δ T st e 0 s s 0 Ts ( ) ( z ) G z e 0 0 z z s s 0 s( s 0) z
56 z ( T / 2) v 0.05v ( T / 2) v 0.05v ( v) Gv ( ) 0.05v v 0.05v 0.05v 9.24 v
57 ex) p.236 (Ogata's) e Gs ( ) s Ts K s( s ) Design Specifications: phase margin: 50 velocity error constant sampling period T gain margin: at least 0db K v : 2sec Gz ( ) Ts e K z s s( s ) Kz ( ) ( z )( z 0.887) K(0.0873z ) 2 z.887z
58 z ( T / 2) v 0.v ( T / 2) v 0.v 0.v K ( ) v Gv 0. ( ) 0.v 2 0.v ( ).887( ) v 0.v v v 2 K( )( ) K( v v ) v v vv ( ) How to determine the gain K? 80
59 K lim vg ( v) G( v) K 2 v v 0 Gv ( ) D 2 2( v v ) v v v v 2( )( ) vv ( ) o From Bode plot, PM 30 and GM 4.5dB PM should be properly adjusted. What type of compensator is needed? τv GD ( v), 0 α (phaselead compensator) ατv 8
60 82
61 Review ( p.628 ιn Νιse's): τv GD ( v) ατv φ tan τν tan ατν dφ τ ατ 2 dν ( τν) ( ατν) v φ G max ma D G D 2 τv ( τ and break frequencies from GD ( v) ) () τ α ατv α α x tan sin ( α from the required phase) (2) 2 α α j jτωmax α ( jωmax ) jατω j α ( jω max ) where G ( jω ) G( jω ) at ω. D α max max (magnitude at the peak of phase curve) (3) max Solving (2), we can obtain α. Using max (3) and (), we can determine τ. 83
62 How to design G ( v)? D φ G max D tan sin ( jω ) max α α α 2 α α α 20 log db α 0.36 o sin 28 α 0.36 At v.7, G( jω ) db v max max max.7 τ τ α τ α G D τv v ( v) GD ( v). ατv v 84
63 85
64 G G D D v ( v) v z (0 ) ( z) z z (0 ) z z.9387 z make it a difference equation! G D ( z) G( z) z ( z0.9356) z ( z )( z 0.887) z 0.008z z 2.377z.8352z C( z) 0.089z 0.008z Rz ( ) z z.8460z ( z0.9357)( z0.845) ( z 0.826)( z j0.396)( z j0.396) 86
65 87
66 HOMEWORK 4 (DUE DATE: MAY 7TH) B4 B43 B45 B46 B48 B49 B43 B44
Digital Control Systems
Digital Control Systems Lecture Summary #4 This summary discussed some graphical methods their use to determine the stability the stability margins of closed loop systems. A. Nyquist criterion Nyquist
More information6.1 Sketch the zdomain root locus and find the critical gain for the following systems K., the closedloop characteristic equation is K + z 0.
6. Sketch the zdomain root locus and find the critical gain for the following systems K (i) Gz () z 4. (ii) Gz K () ( z+ 9. )( z 9. ) (iii) Gz () Kz ( z. )( z ) (iv) Gz () Kz ( + 9. ) ( z. )( z 8. ) (i)
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 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 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 informationH(s) = s. a 2. H eq (z) = z z. G(s) a 2. G(s) A B. s 2 s(s + a) 2 s(s a) G(s) 1 a 1 a. } = (z s 1)( z. e ) ) (z. (z 1)(z e at )(z e at )
.7 Quiz Solutions Problem : a H(s) = s a a) Calculate the zero order hold equivalent H eq (z). H eq (z) = z z G(s) Z{ } s G(s) a Z{ } = Z{ s s(s a ) } G(s) A B Z{ } = Z{ + } s s(s + a) s(s a) G(s) a a
More informationSTABILITY ANALYSIS TECHNIQUES
ECE4540/5540: Digital Control Systems 4 1 STABILITY ANALYSIS TECHNIQUES 41: Bilinear transformation Three main aspects to controlsystem design: 1 Stability, 2 Steadystate response, 3 Transient response
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 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 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 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 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 informationSystems 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 informationOutline. Classical Control. Lecture 5
Outline Outline Outline 1 What is 2 Outline What is Why use? Sketching a 1 What is Why use? Sketching a 2 Gain Controller Lead Compensation Lag Compensation What is Properties of a General System Why use?
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 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 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 informationChapter 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 Steadystate Steadystate errors errors Type Type k k systems systems Integral Integral
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 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 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 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 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 information1 (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 informationDIGITAL CONTROLLER DESIGN
ECE4540/5540: Digital Control Systems 5 DIGITAL CONTROLLER DESIGN 5.: Direct digital design: Steadystate accuracy We have spent quite a bit of time discussing digital hybrid system analysis, and some
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 informationRoot Locus Design Example #4
Root Locus Design Example #4 A. Introduction The plant model represents a linearization of the heading dynamics of a 25, ton tanker ship under empty load conditions. The reference input signal R(s) is
More informationKINGS COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING
KINGS COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING QUESTION BANK SUB.NAME : CONTROL SYSTEMS BRANCH : ECE YEAR : II SEMESTER: IV 1. What is control system? 2. Define open
More information1 (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 informationEECS C128/ ME C134 Final Thu. May 14, pm. Closed book. One page, 2 sides of formula sheets. No calculators.
Name: SID: EECS C28/ ME C34 Final Thu. May 4, 25 58 pm Closed book. One page, 2 sides of formula sheets. No calculators. There are 8 problems worth points total. Problem Points Score 4 2 4 3 6 4 8 5 3
More informationCourse 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 informationSystems 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 informationECSE 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
More informationSystems 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 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 informationRobust Performance Example #1
Robust Performance Example # The transfer function for a nominal system (plant) is given, along with the transfer function for one extreme system. These two transfer functions define a family of plants
More 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 informationMAE 143B  Homework 9
MAE 43B  Homework 9 7.2 2 2 3.8.6.4.2.2 9 8 2 2 3 a) G(s) = (s+)(s+).4.6.8.2.2.4.6.8. Polar plot; red for negative ; no encirclements of, a.s. under unit feedback... 2 2 3. 4 9 2 2 3 h) G(s) = s+ s(s+)..2.4.6.8.2.4
More informationControl 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 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 informationELECTRONICS & COMMUNICATIONS DEP. 3rd YEAR, 2010/2011 CONTROL ENGINEERING SHEET 5 LeadLag Compensation Techniques
CAIRO UNIVERSITY FACULTY OF ENGINEERING ELECTRONICS & COMMUNICATIONS DEP. 3rd YEAR, 00/0 CONTROL ENGINEERING SHEET 5 LeadLag Compensation Techniques [] For the following system, Design a compensator such
More informationToday (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:
More informationControls Problems for Qualifying Exam  Spring 2014
Controls Problems for Qualifying Exam  Spring 2014 Problem 1 Consider the system block diagram given in Figure 1. Find the overall transfer function T(s) = C(s)/R(s). Note that this transfer function
More informationControl System Design
ELEC ENG 4CL4: Control System Design Notes for Lecture #11 Wednesday, January 28, 2004 Dr. Ian C. Bruce Room: CRL229 Phone ext.: 26984 Email: ibruce@mail.ece.mcmaster.ca Relative Stability: Stability
More informationLinear Control Systems Lecture #3  Frequency Domain Analysis. Guillaume Drion Academic year
Linear Control Systems Lecture #3  Frequency Domain Analysis Guillaume Drion Academic year 20182019 1 Goal and Outline Goal: To be able to analyze the stability and robustness of a closedloop system
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 informationEssence 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 setup, namely for the case when the closedloop
More informationDistributed RealTime Control Systems
Distributed RealTime Control Systems Chapter 9 Discrete PID Control 1 Computer Control 2 Approximation of Continuous Time Controllers Design Strategy: Design a continuous time controller C c (s) and then
More informationIC6501 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
More informationMEM 355 Performance Enhancement of Dynamical Systems
MEM 355 Performance Enhancement of Dynamical Systems Frequency Domain Design Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University 5/8/25 Outline Closed Loop Transfer Functions
More informationIntroduction. Performance and Robustness (Chapter 1) Advanced Control Systems Spring / 31
Introduction Classical Control Robust Control u(t) y(t) G u(t) G + y(t) G : nominal model G = G + : plant uncertainty Uncertainty sources : Structured : parametric uncertainty, multimodel uncertainty Unstructured
More informationEE402  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 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 informationLecture 11. Frequency Response in Discrete Time Control Systems
EE42  Discrete Time Systems Spring 28 Lecturer: Asst. Prof. M. Mert Ankarali Lecture.. Frequency Response in Discrete Time Control Systems Let s assume u[k], y[k], and G(z) represents the input, output,
More informationClassify a transfer function to see which order or ramp it can follow and with which expected error.
Dr. J. Tani, Prof. Dr. E. Frazzoli 505900 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 informationDr Ian R. Manchester
Week Content Notes 1 Introduction 2 Frequency Domain Modelling 3 Transient Performance and the splane 4 Block Diagrams 5 Feedback System Characteristics Assign 1 Due 6 Root Locus 7 Root Locus 2 Assign
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 informationExercises for lectures 13 Design using frequency methods
Exercises for lectures 13 Design using frequency methods Michael Šebek Automatic control 2016 31317 Setting of the closed loop bandwidth At the transition frequency in the open loop is (from definition)
More informationNyquist Criterion For Stability of Closed Loop System
Nyquist Criterion For Stability of Closed Loop System Prof. N. Puri ECE Department, Rutgers University Nyquist Theorem Given a closed loop system: r(t) + KG(s) = K N(s) c(t) H(s) = KG(s) +KG(s) = KN(s)
More informationThe Nyquist criterion relates the stability of a closed system to the openloop frequency response and open loop pole location.
Introduction to the Nyquist criterion The Nyquist criterion relates the stability of a closed system to the openloop frequency response and open loop pole location. Mapping. If we take a complex number
More informationRobust Control 3 The Closed Loop
Robust Control 3 The Closed Loop Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University /2/2002 Outline Closed Loop Transfer Functions Traditional Performance Measures Time
More informationVALLIAMMAI 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
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 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 informationStability 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
More informationCHAPTER 7 STEADYSTATE RESPONSE ANALYSES
CHAPTER 7 STEADYSTATE RESPONSE ANALYSES 1. Introduction The steady state error is a measure of system accuracy. These errors arise from the nature of the inputs, system type and from nonlinearities of
More informationAnalysis 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 informationCDS 101/110a: Lecture 81 Frequency Domain Design
CDS 11/11a: Lecture 81 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
More informationECE317 : Feedback and Control
ECE317 : Feedback and Control Lecture : Steadystate error Dr. Richard Tymerski Dept. of Electrical and Computer Engineering Portland State University 1 Course roadmap Modeling Analysis Design Laplace
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 informationDue 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, unityfeedback closed loop control system shown below where G(s) = /[s q (s+)(s+9)]
More informationIf you need more room, use the backs of the pages and indicate that you have done so.
EE 343 Exam II Ahmad F. Taha Spring 206 Your Name: Your Signature: Exam duration: hour and 30 minutes. This exam is closed book, closed notes, closed laptops, closed phones, closed tablets, closed pretty
More informationFEEDBACK CONTROL SYSTEMS
FEEDBAC CONTROL SYSTEMS. Control System Design. Open and ClosedLoop Control Systems 3. Why ClosedLoop Control? 4. Case Study  Speed Control of a DC Motor 5. SteadyState Errors in Unity Feedback Control
More information1 x(k +1)=(Φ LH) x(k) = T 1 x 2 (k) x1 (0) 1 T x 2(0) T x 1 (0) x 2 (0) x(1) = x(2) = x(3) =
567 This is often referred to as Þnite settling time or deadbeat design because the dynamics will settle in a Þnite number of sample periods. This estimator always drives the error to zero in time 2T or
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 informationTopic # Feedback Control
Topic #4 16.31 Feedback Control Stability in the Frequency Domain Nyquist Stability Theorem Examples Appendix (details) This is the basis of future robustness tests. Fall 2007 16.31 4 2 Frequency Stability
More 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 information1 An Overview and Brief History of Feedback Control 1. 2 Dynamic Models 23. Contents. Preface. xiii
Contents 1 An Overview and Brief History of Feedback Control 1 A Perspective on Feedback Control 1 Chapter Overview 2 1.1 A Simple Feedback System 3 1.2 A First Analysis of Feedback 6 1.3 Feedback System
More informationProportional plus Integral (PI) Controller
Proportional plus Integral (PI) Controller 1. A pole is placed at the origin 2. This causes the system type to increase by 1 and as a result the error is reduced to zero. 3. Originally a point A is on
More information(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 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 informationRoot Locus Design Example #3
Root Locus Design Example #3 A. Introduction The system represents a linear model for vertical motion of an underwater vehicle at zero forward speed. The vehicle is assumed to have zero pitch and roll
More informationDigital 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)
More informationData 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
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 informationFEL3210 Multivariable Feedback Control
FEL3210 Multivariable Feedback Control Lecture 5: Uncertainty and Robustness in SISO Systems [Ch.7(8)] Elling W. Jacobsen, Automatic Control Lab, KTH Lecture 5:Uncertainty and Robustness () FEL3210 MIMO
More information100 (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
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 informationR a) Compare open loop and closed loop control systems. b) Clearly bring out, from basics, Forcecurrent and ForceVoltage analogies.
SET  1 II B. Tech II Semester Supplementary Examinations Dec 01 1. a) Compare open loop and closed loop control systems. b) Clearly bring out, from basics, Forcecurrent and ForceVoltage analogies..
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 informationCDS 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
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 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 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 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 informationSolutions to SkillAssessment Exercises
Solutions to SkillAssessment Exercises To Accompany Control Systems Engineering 4 th Edition By Norman S. Nise John Wiley & Sons Copyright 2004 by John Wiley & Sons, Inc. All rights reserved. No part
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 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 informationModule 5: Design of Sampled Data Control Systems Lecture Note 8
Module 5: Design of Sampled Data Control Systems Lecture Note 8 Laglead Compensator When a single lead or lag compensator cannot guarantee the specified design criteria, a laglead compensator is used.
More informationRaktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Basic Feedback Analysis & Design
AERO 422: Active Controls for Aerospace Vehicles Basic Feedback Analysis & Design Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University Routh s Stability
More information