Digital Control Systems

Save this PDF as:

Size: px
Start display at page:

Download "Digital Control Systems"

Transcription

1 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 criterion is a graphical method that allows to determine the number of poles of the closed loop system that are outside the unit circle. It can be used to test stability in the frequency domain. The method is applied to the open loop transfer function allows obtaining information about the closed loop system. The Nyquist method uses the Cauchy principle of arguments. In addition to stability, the Nyquist criterion allows to deduce stability margins. Stability margins provide good information for frequency domain controller design. We assume we know the number of open loop poles outside the unit circle; we denote this number by P. The number of closed loop poles outside the unit circle is denoted by Z is unknown. Let N be the number of counterclockwise encirclements of point (, ) in the complex plane, ( N) the number of clockwise encirclements of the same point. The following theorem is based on the Cauchy principle of arguments: Theorem: Nyquist criterion: Assuming the open loop transfer function C(z)G za (z) has P poles outside the unit circle the number of counterclockwise encirclements is N (thus ( N) is the number of clockwise encirclements). The closed loop system has Z poles outside the unit circle with B. Examples Z = P N () The Nyquist diagrams for these examples are shown in figure. ) System : The open loop system: z +.5 () The open loop pole is.5, thus, P =. From the Nyquist diagram: N =. Thus Z = the system is unstable. This can be verified from the closed loop transfer function, which is given by (3) z +.5 The closed loop pole is.5, which is clearly outside the unit circle. ) System : The open loop system: z.5 (4) The open loop pole is.5, thus, P =. From the Nyquist diagram: N =. Thus Z = the system is stable. This can be verified from the closed loop transfer function, which is given by (5) z +.5 The closed loop pole is.5, which is clearly inside the unit circle. 3) System 3: The open loop system: z.5z +. (6) The open loop poles are.5 ± j.9, thus, P =. From the Nyquist diagram: N =. Thus Z = the system is unstable. This can be verified from the closed loop transfer function, which is given by z (7).5z +. The closed loop poles are.5 ±.86i, which are outside the unit circle because their magnitude is.48. 4) System 4: The open loop system: z.5z. (8) The open loop poles are.653,.53, thus, P =. From the Nyquist diagram we have: N =. Thus Z = the system is stable. This can be verified from the closed loop transfer function, which is given by z (9).5z +.9 The closed loop poles have magnitude.9487, the poles are inside the unit circle. The Nyquist diagrams are shown in figure. In system 4, you noticed that the system is barely stable (closed loop poles magnitude at.95). This leads us to the notion of stability margins. C. Stability margins Stability margin is a measure of the relative stability of the closed loop system. It has two components: a gain margin a phase margin. A system may be specified to have minimum stability margins. For example, we want the system to have a phase margin of at least 5 o,

2 Digital Control, spring 8 Summary 4 System System Real Axis Real Axis System 3 System 4 Real Axis Real Axis 3 Fig.. Nyquist Examples Fig.. Top: illustration of the disturbance, bottom: A simple proportional control or the gain margin of at least 5db. The gain margin phase margin are defined as follows: Gain margin G m : The gain margin can be defined as the gain disturbance that makes the system marginally stable. Phase margin P m : The phase margin can be defined as the phase disturbance that makes the system marginally stable. With reference to figure, the disturbance is given by L(z), which could be a pure gain, or a complex function. It is possible to obtain the phase margin the gain margin from the Nyquist plot (right click then go to characteristics). The gain margin corresponds to 8 o crossing, which corresponds to the intersection with the negative real axis. The system is less stable when the crossing is closer to (, ). Thus the gain margin is measured based on the distance along the negative real axis to point (, ). Phase margin corresponds to the angle measured along the unit circle to reach point (, ). Gain phase margins can be deduced from the Bode plot as well. The gain margin is the gain at angle 8 of the system to reach the db line. The phase margin is the phase at db to reach 8 o. Examples of gain phase margins are shown in figure 3, where the gain margin corresponds to point P the phase margin corresponds to point P. Left click on these points shows the numerical values for G m P m. Note that G m is in db. D. Root locus With reference to figure bottom, the root locus can be define as a plot of the closed loop poles as the gain varies (from zero to ). The starting points of the root locus are the open loop poles.

3 Digital Control, spring 8 Summary 4.5 Bode Diagram 8 6 Phase (deg) Magnitude (db) P System: L Gain Margin (db): 3.5 At frequency (rad/sec): 3.4 System: L Phase Margin (deg): 75.5 Delay Margin (samples): At frequency (rad/sec):.3.5 P.5 System: L Gain Margin (db): 3.5 At frequency (rad/sec): 3.4 System: L Phase Margin (deg): 75.5 Delay Margin (samples): At frequency (rad/sec):.3 35 P P 8 Frequency (rad/sec) Real Axis Fig. 3. Gain phase margins using the Bode plot the Nyquist diagram ) Example: By h, sketch the root locus for L (z) = z +.5 () L (z) = () z.5 The first step to solve is to obtain the closed loop transfer function, solve for the poles as a function of the gain, then plot the poles as varies. The intersection of the root locus with the unit circle is of particular importance because it gives the value of for which the system becomes unstable. For example, for system (), the closed loop system is given by z the pole location is given by () z =.5 (3) To sketch the root locus, we simply vary plot the pole locations in the complex plane. From the root locus of systems () () shown in figures 4 5, these systems become marginally stable for =.5 =.5, respectively. STEADY STATE ERROR One of the goals of control systems is to minimize the steady state error. The steady state error is the difference as time goes to infinity between the desired input the output. An example is shown in figure 6 where the desired input is a unit step. In this particular case, the steady state error is e( ) =.4. Consider figure 7 showing a closed loop system with unity feedback. L(z) is the open loop transfer function. It includes the controller, the ZOH the analog system, that is C(z)G za (z) (4) The steady state error can be obtained using the final value theorem ( z ) E(z) (5) z or (z ) R(z) (6) z z ( + L(z)) Without loss of generality, we can write L(z) as follows where N(z) (z ) n D(z) n is a positive integer, n. N() D() (7) Here n defines the type of the system. For example when n =, the system is of type. When n =, the system is of type. Now we examine the effect of stard reference inputs on the steady state error. Combining (6) (7), we get or z z( + (z )R(z) N(z) (n ) n D(z) ) (8) z (z ) n+ D(z)R(z) (z ) n D(z) + N(z) (9) It is clear that the steady state error depends on the type of the system of course on the input. E. Sampled step input For a unit step input, we have R(z) = z z ) For n =, we have: e( ) = () D() D() + N() = + L() = () + p p is called the position error constant. 3

4 Digital Control, spring 8 Summary Root Locus System: L Gain:.5 Pole: Damping:.377 Overshoot (%): Frequency (rad/sec): 3.4 System: L Gain:.37 Pole:.87 Damping:.95 Overshoot (%): 87 Frequency (rad/sec): 3. System: L Gain: Pole:.5 Damping:.5 Overshoot (%): 5 Frequency (rad/sec): Real Axis Fig. 4. Root locus for system L in equation () Root Locus System: L Gain:.5 Pole:. Damping:.8 Overshoot (%): Frequency (rad/sec): 3.4 System: L Gain:.77 Pole:.49 Damping: Overshoot (%): Frequency (rad/sec): Real Axis Fig. 5. Root locus for system L in equation () 4

5 Digital Control, spring 8 Summary 4 Response.5.5 r(k) y(k) e( ) e(k) = r(k) y(k) 5 5 Time (seconds) Fig. 6. An illustration of the steady state error. The input r(k) is a unit step, the output is y(k). ) For n =, we have: z ) For n =, we have z thus (z ) D(z) T z (z ) D(z) + N(z) (z ) = (6) (z ) D(z) T z (z ) D(z) + N(z) (z ) (7) e( ) = T D() N() (8) The velocity error constant can be defined in this case as v = lim (z )L(z) (9) z T When the error for a ramp input is constant (different from zero), we have e( ) = / v. 3) For n z (z ) n+ D(z) T z (z ) n D(z) + N(z) (z ) (3) (z ) n D(z) z (z ) n D(z) + N(z) = (3) G. Example Consider the following system G za (z) = (z.5) (3) C(z) = (33) ) Find the steady state error for a unit step. Fig. 7. Block diagram of a unity feedback system ) For n, we have z z (z ) n+ D(z) z (z ) n D(z) + N(z) (z ) n D(z) z (z ) n D(z) + N(z) because n, we have () (3) e( ) = (4) Solution The system is of type zero. For a unit step we have [ ] z (z ) z z + (z.5) z z.5 z.5 + (34) (35) e( ) = =.5 (36).5 + The steady state error can be reduced by increasing the gain. For =, the closed loop response is shown in figure 8. For this value of the gain, e( ) =.33. This is clearly confirmed on figure 8. Recall that there is a constraint on the values of, we can use only values of for which the system maintains stability. F. Ramp input For a unit ramp input, we have R(z) = T z (z ) (5) 5 H. Example Consider the following system G za (z) = (z +.8)(z ) (37)

6 Digital Control, spring 8 Summary 4 Step Response.4 Step Response. Amplitude.8.6 e( ) =.33 Amplitude.8.6 e( ) = Time (sec).5.5 Time (sec) Fig. 8. Closed loop step response for example Fig. 9. Closed loop step response for example C(z) = (38) ) Find the steady state error for a unit step. ) Find the steady state error for a unit ramp. 5 4 Linear Simulation Results Solution ) The system is of type one, for a unit step we have [ ] z (z ) (39) z z + (z )(z+.8) z (z +.8)(z ) (z )(z +.8) + (4) e( ) = (4) The steady state error is zero does not depend on the gain. The closed loop response is shown in figure 9, the error is zero which confirms the previous results. ) When the input is a unit ramp, we have [ T z (z ) z (z ) ] + (z )(z+.8) [ ] T z(z )(z +.8) z (z ) (z )(z +.8) + (4) (43) e( ) = T.8 (44) The steady state error can be reduced by increasing the gain or reducing the sampling time. The closed loop response for a ramp is shown in figure for a gain = a sampling time T =.s. Matlab code to plot the step response is shown below. Amplitude 3 e( ) Time (sec) Fig.. Closed loop ramp response for example I. In conclusion ) For a unit step { +L() e( ) = = + p for n = for n ) For a ramp input e( ) = with lim z (45) p = L() (46) for n = T (z ) v for n = for n (47) (z )L(z) v = lim (48) z T 3) When we talk about steady state error, we assume that the system is stable. Steady state error does not make 6

7 Digital Control, spring 8 Summary 4 sense if the system is unstable. 4) The fact that the time response of the system increases in figure does not imply instability of the system. 5) The analysis done here assumes a unity feedback. Thus the equations are not valid for a non-unity feedback. L = tf([],[ -.5],.) CL = feedback(l, ) step(cl) %Step response 7

Systems Analysis and Control

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

More information

Systems Analysis and Control

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

More information

DESIGN USING TRANSFORMATION TECHNIQUE CLASSICAL METHOD

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

More information

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

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

More information

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

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

More information

6.1 Sketch the z-domain root locus and find the critical gain for the following systems K., the closed-loop characteristic equation is K + z 0.

6.1 Sketch the z-domain root locus and find the critical gain for the following systems K., the closed-loop characteristic equation is K + z 0. 6. Sketch the z-domain 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 information

Systems Analysis and Control

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

More information

Module 6: Deadbeat Response Design Lecture Note 1

Module 6: Deadbeat Response Design Lecture Note 1 Module 6: Deadbeat Response Design Lecture Note 1 1 Design of digital control systems with dead beat response So far we have discussed the design methods which are extensions of continuous time design

More information

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

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

More information

MAS107 Control Theory Exam Solutions 2008

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

More information

(b) A unity feedback system is characterized by the transfer function. Design a suitable compensator to meet the following specifications:

(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 information

Digital Control: Summary # 7

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)

More information

Outline. Classical Control. Lecture 1

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

More information

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

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

More information

Control Systems I Lecture 10: System Specifications

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

More information

Systems Analysis and Control

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

More information

Control Systems. Frequency Method Nyquist Analysis.

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

More information

Nyquist Stability Criteria

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

More information

KINGS COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING

KINGS 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 information

Homework 7 - Solutions

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

More information

EE451/551: Digital Control. Chapter 3: Modeling of Digital Control Systems

EE451/551: Digital Control. Chapter 3: Modeling of Digital Control Systems EE451/551: Digital Control Chapter 3: Modeling of Digital Control Systems Common Digital Control Configurations AsnotedinCh1 commondigitalcontrolconfigurations As noted in Ch 1, common digital control

More information

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

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

More information

MEM 355 Performance Enhancement of Dynamical Systems

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

More information

Controls Problems for Qualifying Exam - Spring 2014

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

More information

Control Systems I. Lecture 9: The Nyquist condition

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

More information

Frequency Response Techniques

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

More information

EE402 - Discrete Time Systems Spring Lecture 10

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

More information

DIGITAL CONTROLLER DESIGN

DIGITAL CONTROLLER DESIGN ECE4540/5540: Digital Control Systems 5 DIGITAL CONTROLLER DESIGN 5.: Direct digital design: Steady-state accuracy We have spent quite a bit of time discussing digital hybrid system analysis, and some

More information

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 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 information

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 =!

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

More information

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

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

More information

R a) Compare open loop and closed loop control systems. b) Clearly bring out, from basics, Force-current and Force-Voltage analogies.

R a) Compare open loop and closed loop control systems. b) Clearly bring out, from basics, Force-current and Force-Voltage 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, Force-current and Force-Voltage analogies..

More information

VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur

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

More information

SAMPLE SOLUTION TO EXAM in MAS501 Control Systems 2 Autumn 2015

SAMPLE SOLUTION TO EXAM in MAS501 Control Systems 2 Autumn 2015 FACULTY OF ENGINEERING AND SCIENCE SAMPLE SOLUTION TO EXAM in MAS501 Control Systems 2 Autumn 2015 Lecturer: Michael Ruderman Problem 1: Frequency-domain analysis and control design (15 pt) Given is a

More information

Intro to Frequency Domain Design

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

More information

a. Closed-loop system; b. equivalent transfer function Then the CLTF () T is s the poles of () T are s from a contribution of a

a. Closed-loop system; b. equivalent transfer function Then the CLTF () T is s the poles of () T are s from a contribution of a Root Locus Simple definition Locus of points on the s- plane that represents the poles of a system as one or more parameter vary. RL and its relation to poles of a closed loop system RL and its relation

More information

CONTROL * ~ SYSTEMS ENGINEERING

CONTROL * ~ SYSTEMS ENGINEERING CONTROL * ~ SYSTEMS ENGINEERING H Fourth Edition NormanS. Nise California State Polytechnic University, Pomona JOHN WILEY& SONS, INC. Contents 1. Introduction 1 1.1 Introduction, 2 1.2 A History of Control

More information

1 (20 pts) Nyquist Exercise

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

More information

CONTROL SYSTEMS ENGINEERING Sixth Edition International Student Version

CONTROL SYSTEMS ENGINEERING Sixth Edition International Student Version CONTROL SYSTEMS ENGINEERING Sixth Edition International Student Version Norman S. Nise California State Polytechnic University, Pomona John Wiley fir Sons, Inc. Contents PREFACE, vii 1. INTRODUCTION, 1

More information

Department of Electronics and Instrumentation Engineering M. E- CONTROL AND INSTRUMENTATION ENGINEERING CL7101 CONTROL SYSTEM DESIGN Unit I- BASICS AND ROOT-LOCUS DESIGN PART-A (2 marks) 1. What are the

More information

ECE 486 Control Systems

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

More information

MEM 355 Performance Enhancement of Dynamical Systems

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

More information

Frequency Response Analysis

Frequency 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 information

CHAPTER 7 : BODE PLOTS AND GAIN ADJUSTMENTS COMPENSATION

CHAPTER 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 information

Table of Laplacetransform

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

More information

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

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

More information

Control System Design

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

More information

Control of Single-Input Single-Output Systems

Control of Single-Input Single-Output Systems Control of Single-Input Single-Output Systems Dimitrios Hristu-Varsakelis 1 and William S. Levine 2 1 Department of Applied Informatics, University of Macedonia, Thessaloniki, 546, Greece dcv@uom.gr 2

More information

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

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:

More information

STABILITY OF CLOSED-LOOP CONTOL SYSTEMS

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

More information

INTRODUCTION TO DIGITAL CONTROL

INTRODUCTION 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 linear-time-invariant

More information

1 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) =

1 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 information

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

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

More information

LABORATORY INSTRUCTION MANUAL CONTROL SYSTEM I LAB EE 593

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

More information

Dynamic Compensation using root locus method

Dynamic Compensation using root locus method CAIRO UNIVERSITY FACULTY OF ENGINEERING ELECTRONICS & COMMUNICATIONS DEP. 3rd YEAR, 00/0 CONTROL ENGINEERING SHEET 9 Dynamic Compensation using root locus method [] (Final00)For the system shown in the

More information

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

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},

More information

EE 4343/ Control System Design Project LECTURE 10

EE 4343/ Control System Design Project LECTURE 10 Copyright S. Ikenaga 998 All rights reserved EE 4343/5329 - Control System Design Project LECTURE EE 4343/5329 Homepage EE 4343/5329 Course Outline Design of Phase-lead and Phase-lag compensators using

More information

Analysis of SISO Control Loops

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

More information

ECE317 : Feedback and Control

ECE317 : Feedback and Control ECE317 : Feedback and Control Lecture : Steady-state error Dr. Richard Tymerski Dept. of Electrical and Computer Engineering Portland State University 1 Course roadmap Modeling Analysis Design Laplace

More information

Engraving Machine Example

Engraving Machine Example Engraving Machine Example MCE44 - Fall 8 Dr. Richter November 24, 28 Basic Design The X-axis of the engraving machine has the transfer function G(s) = s(s + )(s + 2) In this basic example, we use a proportional

More information

Lecture 11. Frequency Response in Discrete Time Control Systems

Lecture 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 information

Chapter 9 Robust Stability in SISO Systems 9. Introduction There are many reasons to use feedback control. As we have seen earlier, with the help of a

Chapter 9 Robust Stability in SISO Systems 9. Introduction There are many reasons to use feedback control. As we have seen earlier, with the help of a Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology c Chapter 9 Robust

More information

ECE382/ME482 Spring 2005 Homework 7 Solution April 17, K(s + 0.2) s 2 (s + 2)(s + 5) G(s) =

ECE382/ME482 Spring 2005 Homework 7 Solution April 17, K(s + 0.2) s 2 (s + 2)(s + 5) G(s) = ECE382/ME482 Spring 25 Homework 7 Solution April 17, 25 1 Solution to HW7 AP9.5 We are given a system with open loop transfer function G(s) = K(s +.2) s 2 (s + 2)(s + 5) (1) and unity negative feedback.

More information

Stability of CL System

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

More information

SECTION 5: ROOT LOCUS ANALYSIS

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

More information

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

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

More information

CHAPTER 7 STEADY-STATE RESPONSE ANALYSES

CHAPTER 7 STEADY-STATE RESPONSE ANALYSES CHAPTER 7 STEADY-STATE 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 information

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

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

More information

DIGITAL CONTROL OF POWER CONVERTERS. 3 Digital controller design

DIGITAL CONTROL OF POWER CONVERTERS. 3 Digital controller design DIGITAL CONTROL OF POWER CONVERTERS 3 Digital controller design Frequency response of discrete systems H(z) Properties: z e j T s 1 DC Gain z=1 H(1)=DC 2 Periodic nature j Ts z e jt e s cos( jt ) j sin(

More information

R10 JNTUWORLD B 1 M 1 K 2 M 2. f(t) Figure 1

R10 JNTUWORLD B 1 M 1 K 2 M 2. f(t) Figure 1 Code No: R06 R0 SET - II B. Tech II Semester Regular Examinations April/May 03 CONTROL SYSTEMS (Com. to EEE, ECE, EIE, ECC, AE) Time: 3 hours Max. Marks: 75 Answer any FIVE Questions All Questions carry

More information

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

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

More information

Topic # Feedback Control Systems

Topic # Feedback Control Systems Topic #19 16.31 Feedback Control Systems Stengel Chapter 6 Question: how well do the large gain and phase margins discussed for LQR map over to DOFB using LQR and LQE (called LQG)? Fall 2010 16.30/31 19

More information

Recitation 11: Time delays

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

More information

Frequency (rad/s)

Frequency (rad/s) . The frequency response of the plant in a unity feedback control systems is shown in Figure. a) What is the static velocity error coefficient K v for the system? b) A lead compensator with a transfer

More information

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

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

More information

Robust Performance Example #1

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

More information

Control Systems. University Questions

Control Systems. University Questions University Questions UNIT-1 1. Distinguish between open loop and closed loop control system. Describe two examples for each. (10 Marks), Jan 2009, June 12, Dec 11,July 08, July 2009, Dec 2010 2. Write

More information

Systems Analysis and Control

Systems 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 information

Example on Root Locus Sketching and Control Design

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

More information

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

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

More information

INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad

INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad - 500 043 Electrical and Electronics Engineering TUTORIAL QUESTION BAN Course Name : CONTROL SYSTEMS Course Code : A502 Class : III

More information

Chapter 7. Digital Control Systems

Chapter 7. Digital Control Systems Chapter 7 Digital Control Systems 1 1 Introduction In this chapter, we introduce analysis and design of stability, steady-state error, and transient response for computer-controlled systems. Transfer functions,

More information

Module 5: Design of Sampled Data Control Systems Lecture Note 8

Module 5: Design of Sampled Data Control Systems Lecture Note 8 Module 5: Design of Sampled Data Control Systems Lecture Note 8 Lag-lead Compensator When a single lead or lag compensator cannot guarantee the specified design criteria, a laglead compensator is used.

More information

EEE 184: Introduction to feedback systems

EEE 184: Introduction to feedback systems EEE 84: Introduction to feedback systems Summary 6 8 8 x 7 7 6 Level() 6 5 4 4 5 5 time(s) 4 6 8 Time (seconds) Fig.. Illustration of BIBO stability: stable system (the input is a unit step) Fig.. step)

More information

Compensator Design to Improve Transient Performance Using Root Locus

Compensator Design to Improve Transient Performance Using Root Locus 1 Compensator Design to Improve Transient Performance Using Root Locus Prof. Guy Beale Electrical and Computer Engineering Department George Mason University Fairfax, Virginia Correspondence concerning

More information

Analysis of Discrete-Time Systems

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

More information

IC6501 CONTROL SYSTEMS

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

More information

Robust Control 3 The Closed Loop

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

More information

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

(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

More information

Cascade Control of a Continuous Stirred Tank Reactor (CSTR)

Cascade Control of a Continuous Stirred Tank Reactor (CSTR) Journal of Applied and Industrial Sciences, 213, 1 (4): 16-23, ISSN: 2328-4595 (PRINT), ISSN: 2328-469 (ONLINE) Research Article Cascade Control of a Continuous Stirred Tank Reactor (CSTR) 16 A. O. Ahmed

More information

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

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

More information

MAE 143B - Homework 9

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

More information

PD, PI, PID Compensation. M. Sami Fadali Professor of Electrical Engineering University of Nevada

PD, PI, PID Compensation. M. Sami Fadali Professor of Electrical Engineering University of Nevada PD, PI, PID Compensation M. Sami Fadali Professor of Electrical Engineering University of Nevada 1 Outline PD compensation. PI compensation. PID compensation. 2 PD Control L= loop gain s cl = desired closed-loop

More information

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

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)]

More information

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

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

More information

1 An Overview and Brief History of Feedback Control 1. 2 Dynamic Models 23. Contents. Preface. xiii

1 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 information

Analysis of Discrete-Time Systems

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

More information

The Nyquist Feedback Stability Criterion

The Nyquist Feedback Stability Criterion ECE137B notes; copyright 2018 The Nyquist Feedback Stability Criterion Mark Rodwell, University of California, Santa Barbara Feedback loop stability A () s AOL ( s) AOL ( s) 1 A ( s) ( s) 1 T ( s) Ns ()

More information

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

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

More information

Recursive, Infinite Impulse Response (IIR) Digital Filters:

Recursive, Infinite Impulse Response (IIR) Digital Filters: Recursive, Infinite Impulse Response (IIR) Digital Filters: Filters defined by Laplace Domain transfer functions (analog devices) can be easily converted to Z domain transfer functions (digital, sampled

More information