MEM 355 Performance Enhancement of Dynamical Systems Root Locus Analysis

Size: px
Start display at page:

Download "MEM 355 Performance Enhancement of Dynamical Systems Root Locus Analysis"

Transcription

1 MEM 355 Performance Enhancement of Dynamical Sytem Root Locu Analyi Harry G. Kwatny Department of Mechanical Engineering & Mechanic Drexel Univerity

2 Outline The root locu method wa introduced by Evan in the It remain a popular tool for imple SISO control deign. What i a root locu? Pole & Tranient Repone (Why do we care about pole?) The Root Locu Method Problem Definition The Two Key Formula Root Locu Rule Example: Flexible Spacecraft Robotic Arm General Aviation Aircraft Helicopter Pitch Control

3 Deigning a Feedback Control Sytem Firt we chooe a compenator There are many ueful compenator type.we have already een Proportional and Proportional plu Integral. Thi give u a tructure, i.e., a compenator tranfer function. The tranfer function will have one or more free parameter. The root locu method i one approach to elect the free deign parameter o a to achieve deired tranient behavior.

4 What i a Root Locu? On the right i a negative feedback loop We wih to examine the cloed loop pole a the gain K varie A K increae from zero the 4 pole move from the open loop value & trace 4 loci At any particular value of K there are 4 cloed loop pole In thi example there i a critical value of K at which the ytem become untable. y ye K K 0 G () H G H K G K K Im y Re at K critical tability i lot

5 Uing Matlab >> =tf(''); >> G=(+3)/(*(+1)*(+)*(+4)); >> rlocu(g) >> grid >> [K,Pole]=rlocfind(G) Select a point in the graphic window elected_point = i K = Pole = i i i i

6 Example 1 1, 1 CL 1 11 KK G G K H G G 1G 1K K K K K K Im K K K 4 K 0 Re

7 Tranient Repone Conider a ytem decribed by the open loop tranfer function n G K c pfd 1 0 d 1 There are three way to ae ytem tranient behavior: 1. time domain (output time trajectorie). pole (or eigenvalue) location 3. frequency repone (Bode or Nyquit plot) Here we conider pole location. The pole are the root of c The root locu method i concerned with adjuting the cloed loop pole poition pp p 1 q d 0 Complex root occur in complex conjugate pair. In thi cae there are p+q pole

8 Ideal Pole Location Im degree of tability, decay rate 1/ ideal region for cloed loop pole damping ratio in 1 Re Our goal i to deign a compenator o that the cloed loop pole lie in the haded region. We get to chooe the form of the compenator and elect it parameter.

9 Problem Definition y G () y H The cloed loop input repone tranfer function i G Gy 1 GH The error repone tranfer function i (recall e y y) 1GH G Ge1Gy 1 GH The pole of the cloed loop ytem are the root (zero) of 1GH 0

10 Problem Definition, Cont d Suppoe m n z z b b GH K K K d p p a a m m1 1 m1 0 n n1 1 n n1 0 n, d are completely known, but K i a parameter that we can adjut. Root Locu Problem: Generate a ketch in the complex plane of with varying gain K. the cloed loop pole

11 Solution Strategie We will do thi two way: The eay way: Have MATLAB olve for the root for each of a pecified lit of value for K and plot them. The hard (old) way: Generate a ketch by hand. Why do it the hard way at all? We need to know how to interpret the plot. We obtain inight concerning the choice of compenator. We learn how to et the compenator parameter other than the gain K.

12 Root Locu Method ~ 1 Magnitude equation d () Kn () 1GH0 0 d( ) Kn( ) 0 d () n () j(k1) 1GH 0 K 1 e, k 0, 1,, d () Thi mean n () n () K 1 and K (k1) d () d () for K 0 : K n () n () 1 and (k1) d () d () Angle equation

13 Root Locu Method ~ Our goal i to find value of that atify both of thee equation. Note that for any given, the magnitude equation i atified for ome value of K, i.e., K d n Note that the angle equation doe not depend on K at all. Strategy: Firt, find value of that atify the angle equation. Second, calibrate the plot uing the magnitude equatio n.

14 Root Locu ~ 3 Uing the Angle Formula tet point j3 Im n () (k 1) d () n d (k1) G H k Re for any integer j3 i not a point on the root locu The point i / i 180 k

15 Baic Rule ~ 1 1. Number of branche: The number of branche of the root locu equal the number of open loop pole. i the order of theorder of the polynomial d Kn d. Symmetry: The root locu i ymmetric about the real axi. Pole occur in complex conjugate pair. 3. Starting & ending point: The root locu begin at the open loop pole and end at the finite and infinite open loop zero. d Kn 0 d =0 a K 0 1 K d n 0n 0 a K if i bounded

16 Example: G H K Im Re

17 Baic Rule ~ 4. Real-axi egment: For K 0, real axi egment to the left of an odd number of finite real axi pole and/or zero are part of the root locu. Im Re Tet point 1

18 Baic Rule ~ 3 5. Behavior at infinity: The root locu approache infinity along aymptote with angle: (k 1), k 0, 1,, 3, # finite pole # finite zero Furthermore, thee aymptote interect the real axi at a common point given by finite pole finite zero # finite pole # finite zero

19 Baic Rule ~ 4 Angle part i eay: n m i1 i i1 d n z p j Take e. For, Then m n i1 i i1 n So k1 mn k1 d z p m n i i i

20 Baic Rule ~ 4 6. Real axi breakaway and break-in point: The root locu break away from the real axi where the gain i a (local) maximum on the real axi, and break into the real axi where it i a local minimum. To locate candidate break point d imply plot K on the axi egment n d d or olve 0 d n 7. j-axi croing: Ue Routh tet to determine value of K for which loci cro imaginary axi.

21 Routh Stability Tet It i deired to determine the number of right hand plane root of a polynomial, ay: a a0 3 a3 a a a aa a a0 3 a3 a1 0 b1 b b3 1 c1 c 0 d1 b a3 a3 c 1 1 a 1 a0 a a a 0, b,... a b a 3 1 b 1 3,... The number of right half plane pole i equal to the number of ign change in the firt column. a

22 Uing MATLAB The baic MATLAB function are: rlocu rlocu(y) calculate and plot the root locu of the open-loop SISO model y. rlocfind [K,POLES] = RLOCFIND(SYS) i ued for interactive gain election from the root locu plot of the SISO ytem SYS generated by RLOCUS. RLOCFIND put up a crohair curor in the graphic window which i ued to elect a pole location on an exiting root locu. The root locu gain aociated with thi point i returned in K and all the ytem pole for thi gain are returned in POLES. iotool When invoked without input argument, iotool open a SISO Deign GUI for interactive compenator deign. Thi GUI allow you to deign a ingle-input/ingleoutput (SISO) compenator uing root locu and Bode diagram technique.

23 Flexible Spacecraft

24 Flexible Spacecraft K 1 1 G H K j K j With One Flex Mode Rigid 4 G H K K 4

25 4 3 dcl 1K 16K 10K8K Spacecraft K 8K 3 1 K 10K 0 Rigid Im Re K 6K 1 K K13K 6K 13K 6K 8K K K K 1 3K 6K 1 0, 3 6 0, 0 8K 0 0 Alway poitive (prove it) 1-3K+6 K^ 4 One Flex Mode Im K Re - untable egment -4

26 Example Robotic Arm Compenator Amplifier Motor/ Speed Controller K G 0K 0K

27 Robotic Arm Im Im Re Re? Im 45, 135, 5, Re

28 Robotic Arm 3 Plot K d, 4, 0.1 n Max, breakaway point at K Min, breakin point at

29 Robotic Arm 4 >> =tf(''); >> G=0*(+0.1)/(^*(+4)*(+5.5)^); >> rlocu(g) >> rlocfind(g) Select a point in the graphic window elected_point = i an =.111 elected_point = i an =

30 Robotic Arm 5 G cl 0K K K Routh Table K K K K K 71K 89.76K K K Governing Inequality K K K K , , 0 11 K K

31 Example: Piper Dakota Pitch Control e, error e Gc( ) Gp( ) elevator angle, Plant: G 160 p Lead compenator: G e pitch angle, c 3 K 0

32 Piper Dakota Im Im What do we expect? Re Re

33 Piper Dakota >> =tf(''); >> Gp=160*(+.5)*(+0.7)/((^+5*+40)*(^+0.03*+0.06)); >> Gc=(+3)/(+0); >> G=Gc*Gp; >> rlocu(g) >> grid >> [K,Pole]=rlocfind(G) Select a point in the graphic window elected_point = i K = Pole = i i i i >> Gcl=1.5*G/(1+1.5*G); >> tep(gcl)

34 Piper Dakota

35 Piper Dakota Open loop repone to elevator Cloed loop repone with: K 1.5

36 Example: Helicopter Pitch Control Notice untable dynamic K K 9 Pick the inner loop pitch control gain K o that the dominant inner loop pole have a damping ratio of Select an outer loop gain (tick enitivity) to place the pole. Determine ultimate error in repone to unit tep diturbance.

37 Inner (Stabilization) Loop 1GH 1 K The inner loop root locu i hown on the right. Chooe a gain K 1.5. The inner loop reolve to: G Gp 1 1.5GG c p j

38 Outer Loop Deign K j

39 Outer Loop On the bai of the root locu on the right, chooe a gain of 1. The cloed loop pole are: j j

40 Diturbance Repone Error G de G 1 G pilot G j j lim Gde

41 Summary How pole characterize tranient repone Oberving the influence of gain on cloed loop pole uing root locu plot Sketching the root locu: The magnitude and gain formula Baic rule of root locu ketching Uing MATLAB Example

The Root Locus Method

The Root Locus Method The Root Locu Method MEM 355 Performance Enhancement of Dynamical Sytem Harry G. Kwatny Department of Mechanical Engineering & Mechanic Drexel Univerity Outline The root locu method wa introduced by Evan

More information

ME2142/ME2142E Feedback Control Systems

ME2142/ME2142E Feedback Control Systems Root Locu Analyi Root Locu Analyi Conider the cloed-loop ytem R + E - G C B H The tranient repone, and tability, of the cloed-loop ytem i determined by the value of the root of the characteritic equation

More information

Root Locus Contents. Root locus, sketching algorithm. Root locus, examples. Root locus, proofs. Root locus, control examples

Root Locus Contents. Root locus, sketching algorithm. Root locus, examples. Root locus, proofs. Root locus, control examples Root Locu Content Root locu, ketching algorithm Root locu, example Root locu, proof Root locu, control example Root locu, influence of zero and pole Root locu, lead lag controller deign 9 Spring ME45 -

More information

CISE302: Linear Control Systems

CISE302: Linear Control Systems Term 8 CISE: Linear Control Sytem Dr. Samir Al-Amer Chapter 7: Root locu CISE_ch 7 Al-Amer8 ١ Learning Objective Undertand the concept of root locu and it role in control ytem deign Be able to ketch root

More information

Chapter 7. Root Locus Analysis

Chapter 7. Root Locus Analysis Chapter 7 Root Locu Analyi jw + KGH ( ) GH ( ) - K 0 z O 4 p 2 p 3 p Root Locu Analyi The root of the cloed-loop characteritic equation define the ytem characteritic repone. Their location in the complex

More information

EE Control Systems LECTURE 14

EE Control Systems LECTURE 14 Updated: Tueday, March 3, 999 EE 434 - Control Sytem LECTURE 4 Copyright FL Lewi 999 All right reerved ROOT LOCUS DESIGN TECHNIQUE Suppoe the cloed-loop tranfer function depend on a deign parameter k We

More information

Root Locus Diagram. Root loci: The portion of root locus when k assume positive values: that is 0

Root Locus Diagram. Root loci: The portion of root locus when k assume positive values: that is 0 Objective Root Locu Diagram Upon completion of thi chapter you will be able to: Plot the Root Locu for a given Tranfer Function by varying gain of the ytem, Analye the tability of the ytem from the root

More information

MODERN CONTROL SYSTEMS

MODERN CONTROL SYSTEMS MODERN CONTROL SYSTEMS Lecture 1 Root Locu Emam Fathy Department of Electrical and Control Engineering email: emfmz@aat.edu http://www.aat.edu/cv.php?dip_unit=346&er=68525 1 Introduction What i root locu?

More information

Control Systems Analysis and Design by the Root-Locus Method

Control Systems Analysis and Design by the Root-Locus Method 6 Control Sytem Analyi and Deign by the Root-Locu Method 6 1 INTRODUCTION The baic characteritic of the tranient repone of a cloed-loop ytem i cloely related to the location of the cloed-loop pole. If

More information

Automatic Control Systems. Part III: Root Locus Technique

Automatic Control Systems. Part III: Root Locus Technique www.pdhcenter.com PDH Coure E40 www.pdhonline.org Automatic Control Sytem Part III: Root Locu Technique By Shih-Min Hu, Ph.D., P.E. Page of 30 www.pdhcenter.com PDH Coure E40 www.pdhonline.org VI. Root

More information

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

Delhi Noida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubaneswar Kolkata Patna Web:     Ph: Serial :. PT_EE_A+C_Control Sytem_798 Delhi Noida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubanewar olkata Patna Web: E-mail: info@madeeay.in Ph: -4546 CLASS TEST 8-9 ELECTRICAL ENGINEERING Subject

More information

Chapter 13. Root Locus Introduction

Chapter 13. Root Locus Introduction Chapter 13 Root Locu 13.1 Introduction In the previou chapter we had a glimpe of controller deign iue through ome imple example. Obviouly when we have higher order ytem, uch imple deign technique will

More information

Control Systems Engineering ( Chapter 7. Steady-State Errors ) Prof. Kwang-Chun Ho Tel: Fax:

Control Systems Engineering ( Chapter 7. Steady-State Errors ) Prof. Kwang-Chun Ho Tel: Fax: Control Sytem Engineering ( Chapter 7. Steady-State Error Prof. Kwang-Chun Ho kwangho@hanung.ac.kr Tel: 0-760-453 Fax:0-760-4435 Introduction In thi leon, you will learn the following : How to find the

More information

CONTROL SYSTEMS. Chapter 5 : Root Locus Diagram. GATE Objective & Numerical Type Solutions. The transfer function of a closed loop system is

CONTROL SYSTEMS. Chapter 5 : Root Locus Diagram. GATE Objective & Numerical Type Solutions. The transfer function of a closed loop system is CONTROL SYSTEMS Chapter 5 : Root Locu Diagram GATE Objective & Numerical Type Solution Quetion 1 [Work Book] [GATE EC 199 IISc-Bangalore : Mark] The tranfer function of a cloed loop ytem i T () where i

More information

Linear System Fundamentals

Linear System Fundamentals Linear Sytem Fundamental MEM 355 Performance Enhancement of Dynamical Sytem Harry G. Kwatny Department of Mechanical Engineering & Mechanic Drexel Univerity Content Sytem Repreentation Stability Concept

More information

Control Systems. Root locus.

Control Systems. Root locus. Control Sytem Root locu chibum@eoultech.ac.kr Outline Concet of Root Locu Contructing root locu Control Sytem Root Locu Stability and tranient reone i cloely related with the location of ole in -lane How

More information

Control Systems. Root locus.

Control Systems. Root locus. Control Sytem Root locu chibum@eoultech.ac.kr Outline Concet of Root Locu Contructing root locu Control Sytem Root Locu Stability and tranient reone i cloely related with the location of ole in -lane How

More information

Lecture 12: Examples of Root Locus Plots. Dr. Kalyana Veluvolu. Lecture 12: Examples of Root Locus Plots Dr. Kalyana Veluvolu

Lecture 12: Examples of Root Locus Plots. Dr. Kalyana Veluvolu. Lecture 12: Examples of Root Locus Plots Dr. Kalyana Veluvolu ROOT-LOCUS ANALYSIS Example: Given that G( ) ( + )( + ) Dr. alyana Veluvolu Sketch the root locu of 1 + G() and compute the value of that will yield a dominant econd order behavior with a damping ratio,

More information

Stability. ME 344/144L Prof. R.G. Longoria Dynamic Systems and Controls/Lab. Department of Mechanical Engineering The University of Texas at Austin

Stability. ME 344/144L Prof. R.G. Longoria Dynamic Systems and Controls/Lab. Department of Mechanical Engineering The University of Texas at Austin Stability The tability of a ytem refer to it ability or tendency to eek a condition of tatic equilibrium after it ha been diturbed. If given a mall perturbation from the equilibrium, it i table if it return.

More information

Figure 1: Unity Feedback System

Figure 1: Unity Feedback System MEM 355 Sample Midterm Problem Stability 1 a) I the following ytem table? Solution: G() = Pole: -1, -2, -2, -1.5000 + 1.3229i, -1.5000-1.3229i 1 ( + 1)( 2 + 3 + 4)( + 2) 2 A you can ee, all pole are on

More information

Feedback Control Systems (FCS)

Feedback Control Systems (FCS) Feedback Control Sytem (FCS) Lecture19-20 Routh-Herwitz Stability Criterion Dr. Imtiaz Huain email: imtiaz.huain@faculty.muet.edu.pk URL :http://imtiazhuainkalwar.weebly.com/ Stability of Higher Order

More information

ROOT LOCUS. Poles and Zeros

ROOT LOCUS. Poles and Zeros Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity ROOT LOCUS The Root Locu i the ath of the root of the characteritic equation traced out in the - lane a a ytem

More information

Stability Criterion Routh Hurwitz

Stability Criterion Routh Hurwitz EES404 Fundamental of Control Sytem Stability Criterion Routh Hurwitz DR. Ir. Wahidin Wahab M.Sc. Ir. Arie Subiantoro M.Sc. Stability A ytem i table if for a finite input the output i imilarly finite A

More information

Homework 12 Solution - AME30315, Spring 2013

Homework 12 Solution - AME30315, Spring 2013 Homework 2 Solution - AME335, Spring 23 Problem :[2 pt] The Aerotech AGS 5 i a linear motor driven XY poitioning ytem (ee attached product heet). A friend of mine, through careful experimentation, identified

More information

ME 375 FINAL EXAM SOLUTIONS Friday December 17, 2004

ME 375 FINAL EXAM SOLUTIONS Friday December 17, 2004 ME 375 FINAL EXAM SOLUTIONS Friday December 7, 004 Diviion Adam 0:30 / Yao :30 (circle one) Name Intruction () Thi i a cloed book eamination, but you are allowed three 8.5 crib heet. () You have two hour

More information

Module 4: Time Response of discrete time systems Lecture Note 1

Module 4: Time Response of discrete time systems Lecture Note 1 Digital Control Module 4 Lecture Module 4: ime Repone of dicrete time ytem Lecture Note ime Repone of dicrete time ytem Abolute tability i a baic requirement of all control ytem. Apart from that, good

More information

EE Control Systems LECTURE 6

EE Control Systems LECTURE 6 Copyright FL Lewi 999 All right reerved EE - Control Sytem LECTURE 6 Updated: Sunday, February, 999 BLOCK DIAGRAM AND MASON'S FORMULA A linear time-invariant (LTI) ytem can be repreented in many way, including:

More information

SKEE 3143 CONTROL SYSTEM DESIGN. CHAPTER 3 Compensator Design Using the Bode Plot

SKEE 3143 CONTROL SYSTEM DESIGN. CHAPTER 3 Compensator Design Using the Bode Plot SKEE 3143 CONTROL SYSTEM DESIGN CHAPTER 3 Compenator Deign Uing the Bode Plot 1 Chapter Outline 3.1 Introduc4on Re- viit to Frequency Repone, ploang frequency repone, bode plot tability analyi. 3.2 Gain

More information

Wolfgang Hofle. CERN CAS Darmstadt, October W. Hofle feedback systems

Wolfgang Hofle. CERN CAS Darmstadt, October W. Hofle feedback systems Wolfgang Hofle Wolfgang.Hofle@cern.ch CERN CAS Darmtadt, October 9 Feedback i a mechanim that influence a ytem by looping back an output to the input a concept which i found in abundance in nature and

More information

376 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD. D(s) = we get the compensated system with :

376 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD. D(s) = we get the compensated system with : 376 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Therefore by applying the lead compenator with ome gain adjutment : D() =.12 4.5 +1 9 +1 we get the compenated ytem with : PM =65, ω c = 22 rad/ec, o

More information

March 18, 2014 Academic Year 2013/14

March 18, 2014 Academic Year 2013/14 POLITONG - SHANGHAI BASIC AUTOMATIC CONTROL Exam grade March 8, 4 Academic Year 3/4 NAME (Pinyin/Italian)... STUDENT ID Ue only thee page (including the back) for anwer. Do not ue additional heet. Ue of

More information

Lecture 8. PID control. Industrial process control ( today) PID control. Insights about PID actions

Lecture 8. PID control. Industrial process control ( today) PID control. Insights about PID actions Lecture 8. PID control. The role of P, I, and D action 2. PID tuning Indutrial proce control (92... today) Feedback control i ued to improve the proce performance: tatic performance: for contant reference,

More information

Linear-Quadratic Control System Design

Linear-Quadratic Control System Design Linear-Quadratic Control Sytem Deign Robert Stengel Optimal Control and Etimation MAE 546 Princeton Univerity, 218! Control ytem configuration! Proportional-integral! Proportional-integral-filtering! Model

More information

Chapter #4 EEE Automatic Control

Chapter #4 EEE Automatic Control Spring 008 EEE 00 Chapter #4 EEE 00 Automatic Control Root Locu Chapter 4 /4 Spring 008 EEE 00 Introduction Repone depend on ytem and controller parameter > Cloed loop pole location depend on ytem and

More information

Lecture 4. Chapter 11 Nise. Controller Design via Frequency Response. G. Hovland 2004

Lecture 4. Chapter 11 Nise. Controller Design via Frequency Response. G. Hovland 2004 METR4200 Advanced Control Lecture 4 Chapter Nie Controller Deign via Frequency Repone G. Hovland 2004 Deign Goal Tranient repone via imple gain adjutment Cacade compenator to improve teady-tate error Cacade

More information

The state variable description of an LTI system is given by 3 1O. Statement for Linked Answer Questions 3 and 4 :

The state variable description of an LTI system is given by 3 1O. Statement for Linked Answer Questions 3 and 4 : CHAPTER 6 CONTROL SYSTEMS YEAR TO MARKS MCQ 6. The tate variable decription of an LTI ytem i given by Jxo N J a NJx N JN K O K OK O K O xo a x + u Kxo O K 3 a3 OKx O K 3 O L P L J PL P L P x N K O y _

More information

1 Routh Array: 15 points

1 Routh Array: 15 points EE C28 / ME34 Problem Set 3 Solution Fall 2 Routh Array: 5 point Conider the ytem below, with D() k(+), w(t), G() +2, and H y() 2 ++2 2(+). Find the cloed loop tranfer function Y () R(), and range of k

More information

into a discrete time function. Recall that the table of Laplace/z-transforms is constructed by (i) selecting to get

into a discrete time function. Recall that the table of Laplace/z-transforms is constructed by (i) selecting to get Lecture 25 Introduction to Some Matlab c2d Code in Relation to Sampled Sytem here are many way to convert a continuou time function, { h( t) ; t [0, )} into a dicrete time function { h ( k) ; k {0,,, }}

More information

Digital Control System

Digital Control System Digital Control Sytem - A D D A Micro ADC DAC Proceor Correction Element Proce Clock Meaurement A: Analog D: Digital Continuou Controller and Digital Control Rt - c Plant yt Continuou Controller Digital

More information

MM1: Basic Concept (I): System and its Variables

MM1: Basic Concept (I): System and its Variables MM1: Baic Concept (I): Sytem and it Variable A ytem i a collection of component which are coordinated together to perform a function Sytem interact with their environment. The interaction i defined in

More information

Given the following circuit with unknown initial capacitor voltage v(0): X(s) Immediately, we know that the transfer function H(s) is

Given the following circuit with unknown initial capacitor voltage v(0): X(s) Immediately, we know that the transfer function H(s) is EE 4G Note: Chapter 6 Intructor: Cheung More about ZSR and ZIR. Finding unknown initial condition: Given the following circuit with unknown initial capacitor voltage v0: F v0/ / Input xt 0Ω Output yt -

More information

NAME (pinyin/italian)... MATRICULATION NUMBER... SIGNATURE

NAME (pinyin/italian)... MATRICULATION NUMBER... SIGNATURE POLITONG SHANGHAI BASIC AUTOMATIC CONTROL June Academic Year / Exam grade NAME (pinyin/italian)... MATRICULATION NUMBER... SIGNATURE Ue only thee page (including the bac) for anwer. Do not ue additional

More information

ECE 3510 Root Locus Design Examples. PI To eliminate steady-state error (for constant inputs) & perfect rejection of constant disturbances

ECE 3510 Root Locus Design Examples. PI To eliminate steady-state error (for constant inputs) & perfect rejection of constant disturbances ECE 350 Root Locu Deign Example Recall the imple crude ervo from lab G( ) 0 6.64 53.78 σ = = 3 23.473 PI To eliminate teady-tate error (for contant input) & perfect reection of contant diturbance Note:

More information

State Space: Observer Design Lecture 11

State Space: Observer Design Lecture 11 State Space: Oberver Deign Lecture Advanced Control Sytem Dr Eyad Radwan Dr Eyad Radwan/ACS/ State Space-L Controller deign relie upon acce to the tate variable for feedback through adjutable gain. Thi

More information

Advanced D-Partitioning Analysis and its Comparison with the Kharitonov s Theorem Assessment

Advanced D-Partitioning Analysis and its Comparison with the Kharitonov s Theorem Assessment Journal of Multidiciplinary Engineering Science and Technology (JMEST) ISSN: 59- Vol. Iue, January - 5 Advanced D-Partitioning Analyi and it Comparion with the haritonov Theorem Aement amen M. Yanev Profeor,

More information

S_LOOP: SINGLE-LOOP FEEDBACK CONTROL SYSTEM ANALYSIS

S_LOOP: SINGLE-LOOP FEEDBACK CONTROL SYSTEM ANALYSIS S_LOOP: SINGLE-LOOP FEEDBACK CONTROL SYSTEM ANALYSIS by Michelle Gretzinger, Daniel Zyngier and Thoma Marlin INTRODUCTION One of the challenge to the engineer learning proce control i relating theoretical

More information

ME 375 FINAL EXAM Wednesday, May 6, 2009

ME 375 FINAL EXAM Wednesday, May 6, 2009 ME 375 FINAL EXAM Wedneday, May 6, 9 Diviion Meckl :3 / Adam :3 (circle one) Name_ Intruction () Thi i a cloed book examination, but you are allowed three ingle-ided 8.5 crib heet. A calculator i NOT allowed.

More information

Massachusetts Institute of Technology Dynamics and Control II

Massachusetts Institute of Technology Dynamics and Control II I E Maachuett Intitute of Technology Department of Mechanical Engineering 2.004 Dynamic and Control II Laboratory Seion 5: Elimination of Steady-State Error Uing Integral Control Action 1 Laboratory Objective:

More information

Function and Impulse Response

Function and Impulse Response Tranfer Function and Impule Repone Solution of Selected Unolved Example. Tranfer Function Q.8 Solution : The -domain network i hown in the Fig... Applying VL to the two loop, R R R I () I () L I () L V()

More information

Lecture 5 Introduction to control

Lecture 5 Introduction to control Lecture 5 Introduction to control Tranfer function reviited (Laplace tranform notation: ~jω) () i the Laplace tranform of v(t). Some rule: ) Proportionality: ()/ in () 0log log() v (t) *v in (t) () * in

More information

CHAPTER 4 DESIGN OF STATE FEEDBACK CONTROLLERS AND STATE OBSERVERS USING REDUCED ORDER MODEL

CHAPTER 4 DESIGN OF STATE FEEDBACK CONTROLLERS AND STATE OBSERVERS USING REDUCED ORDER MODEL 98 CHAPTER DESIGN OF STATE FEEDBACK CONTROLLERS AND STATE OBSERVERS USING REDUCED ORDER MODEL INTRODUCTION The deign of ytem uing tate pace model for the deign i called a modern control deign and it i

More information

Gain and Phase Margins Based Delay Dependent Stability Analysis of Two- Area LFC System with Communication Delays

Gain and Phase Margins Based Delay Dependent Stability Analysis of Two- Area LFC System with Communication Delays Gain and Phae Margin Baed Delay Dependent Stability Analyi of Two- Area LFC Sytem with Communication Delay Şahin Sönmez and Saffet Ayaun Department of Electrical Engineering, Niğde Ömer Halidemir Univerity,

More information

Analysis of Stability &

Analysis of Stability & INC 34 Feedback Control Sytem Analyi of Stability & Steady-State Error S Wonga arawan.won@kmutt.ac.th Summary from previou cla Firt-order & econd order ytem repone τ ωn ζω ω n n.8.6.4. ζ ζ. ζ.5 ζ ζ.5 ct.8.6.4...4.6.8..4.6.8

More information

Solutions. Digital Control Systems ( ) 120 minutes examination time + 15 minutes reading time at the beginning of the exam

Solutions. Digital Control Systems ( ) 120 minutes examination time + 15 minutes reading time at the beginning of the exam BSc - Sample Examination Digital Control Sytem (5-588-) Prof. L. Guzzella Solution Exam Duration: Number of Quetion: Rating: Permitted aid: minute examination time + 5 minute reading time at the beginning

More information

G(s) = 1 s by hand for! = 1, 2, 5, 10, 20, 50, and 100 rad/sec.

G(s) = 1 s by hand for! = 1, 2, 5, 10, 20, 50, and 100 rad/sec. 6003 where A = jg(j!)j ; = tan Im [G(j!)] Re [G(j!)] = \G(j!) 2. (a) Calculate the magnitude and phae of G() = + 0 by hand for! =, 2, 5, 0, 20, 50, and 00 rad/ec. (b) ketch the aymptote for G() according

More information

ECE 345 / ME 380 Introduction to Control Systems Lecture Notes 8

ECE 345 / ME 380 Introduction to Control Systems Lecture Notes 8 Learning Objectives ECE 345 / ME 380 Introduction to Control Systems Lecture Notes 8 Dr. Oishi oishi@unm.edu November 2, 203 State the phase and gain properties of a root locus Sketch a root locus, by

More information

A Simplified Methodology for the Synthesis of Adaptive Flight Control Systems

A Simplified Methodology for the Synthesis of Adaptive Flight Control Systems A Simplified Methodology for the Synthei of Adaptive Flight Control Sytem J.ROUSHANIAN, F.NADJAFI Department of Mechanical Engineering KNT Univerity of Technology 3Mirdamad St. Tehran IRAN Abtract- A implified

More information

Lecture 10 Filtering: Applied Concepts

Lecture 10 Filtering: Applied Concepts Lecture Filtering: Applied Concept In the previou two lecture, you have learned about finite-impule-repone (FIR) and infinite-impule-repone (IIR) filter. In thee lecture, we introduced the concept of filtering

More information

7.4 STEP BY STEP PROCEDURE TO DRAW THE ROOT LOCUS DIAGRAM

7.4 STEP BY STEP PROCEDURE TO DRAW THE ROOT LOCUS DIAGRAM ROOT LOCUS TECHNIQUE. Values of on the root loci The value of at any point s on the root loci is determined from the following equation G( s) H( s) Product of lengths of vectors from poles of G( s)h( s)

More information

ECE-320 Linear Control Systems. Spring 2014, Exam 1. No calculators or computers allowed, you may leave your answers as fractions.

ECE-320 Linear Control Systems. Spring 2014, Exam 1. No calculators or computers allowed, you may leave your answers as fractions. ECE-0 Linear Control Sytem Spring 04, Exam No calculator or computer allowed, you may leave your anwer a fraction. All problem are worth point unle noted otherwie. Total /00 Problem - refer to the unit

More information

CHAPTER 8 OBSERVER BASED REDUCED ORDER CONTROLLER DESIGN FOR LARGE SCALE LINEAR DISCRETE-TIME CONTROL SYSTEMS

CHAPTER 8 OBSERVER BASED REDUCED ORDER CONTROLLER DESIGN FOR LARGE SCALE LINEAR DISCRETE-TIME CONTROL SYSTEMS CHAPTER 8 OBSERVER BASED REDUCED ORDER CONTROLLER DESIGN FOR LARGE SCALE LINEAR DISCRETE-TIME CONTROL SYSTEMS 8.1 INTRODUCTION 8.2 REDUCED ORDER MODEL DESIGN FOR LINEAR DISCRETE-TIME CONTROL SYSTEMS 8.3

More information

THE PARAMETERIZATION OF ALL TWO-DEGREES-OF-FREEDOM SEMISTRONGLY STABILIZING CONTROLLERS. Tatsuya Hoshikawa, Kou Yamada and Yuko Tatsumi

THE PARAMETERIZATION OF ALL TWO-DEGREES-OF-FREEDOM SEMISTRONGLY STABILIZING CONTROLLERS. Tatsuya Hoshikawa, Kou Yamada and Yuko Tatsumi International Journal of Innovative Computing, Information Control ICIC International c 206 ISSN 349-498 Volume 2, Number 2, April 206 pp. 357 370 THE PARAMETERIZATION OF ALL TWO-DEGREES-OF-FREEDOM SEMISTRONGLY

More information

Chapter 9: Controller design. Controller design. Controller design

Chapter 9: Controller design. Controller design. Controller design Chapter 9. Controller Deign 9.. Introduction 9.2. Eect o negative eedback on the network traner unction 9.2.. Feedback reduce the traner unction rom diturbance to the output 9.2.2. Feedback caue the traner

More information

EE 4443/5329. LAB 3: Control of Industrial Systems. Simulation and Hardware Control (PID Design) The Inverted Pendulum. (ECP Systems-Model: 505)

EE 4443/5329. LAB 3: Control of Industrial Systems. Simulation and Hardware Control (PID Design) The Inverted Pendulum. (ECP Systems-Model: 505) EE 4443/5329 LAB 3: Control of Indutrial Sytem Simulation and Hardware Control (PID Deign) The Inverted Pendulum (ECP Sytem-Model: 505) Compiled by: Nitin Swamy Email: nwamy@lakehore.uta.edu Email: okuljaca@lakehore.uta.edu

More information

Chapter #4 EEE8013. Linear Controller Design and State Space Analysis. Design of control system in state space using Matlab

Chapter #4 EEE8013. Linear Controller Design and State Space Analysis. Design of control system in state space using Matlab EEE83 hapter #4 EEE83 Linear ontroller Deign and State Space nalyi Deign of control ytem in tate pace uing Matlab. ontrollabilty and Obervability.... State Feedback ontrol... 5 3. Linear Quadratic Regulator

More information

Lecture 8 - SISO Loop Design

Lecture 8 - SISO Loop Design Lecture 8 - SISO Loop Deign Deign approache, given pec Loophaping: in-band and out-of-band pec Fundamental deign limitation for the loop Gorinevky Control Engineering 8-1 Modern Control Theory Appy reult

More information

Then C pid (s) S h -stabilizes G(s) if and only if Ĉpid(ŝ) S 0 - stabilizes Ĝ(ŝ). For any ρ R +, an RCF of Ĉ pid (ŝ) is given by

Then C pid (s) S h -stabilizes G(s) if and only if Ĉpid(ŝ) S 0 - stabilizes Ĝ(ŝ). For any ρ R +, an RCF of Ĉ pid (ŝ) is given by 9 American Control Conference Hyatt Regency Riverfront, St. Loui, MO, USA June -, 9 WeC5.5 PID Controller Synthei with Shifted Axi Pole Aignment for a Cla of MIMO Sytem A. N. Gündeş and T. S. Chang Abtract

More information

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

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

More information

Chapter 4 Interconnection of LTI Systems

Chapter 4 Interconnection of LTI Systems Chapter 4 Interconnection of LTI Sytem 4. INTRODUCTION Block diagram and ignal flow graph are commonly ued to decribe a large feedback control ytem. Each block in the ytem i repreented by a tranfer function,

More information

Chapter 8. Root Locus Techniques

Chapter 8. Root Locus Techniques Chapter 8 Rt Lcu Technique Intrductin Sytem perfrmance and tability dt determined dby cled-lp l ple Typical cled-lp feedback cntrl ytem G Open-lp TF KG H Zer -, - Ple 0, -, -4 K 4 Lcatin f ple eaily fund

More information

Figure 1 Siemens PSSE Web Site

Figure 1 Siemens PSSE Web Site Stability Analyi of Dynamic Sytem. In the lat few lecture we have een how mall ignal Lalace domain model may be contructed of the dynamic erformance of ower ytem. The tability of uch ytem i a matter of

More information

A PLC BASED MIMO PID CONTROLLER FOR MULTIVARIABLE INDUSTRIAL PROCESSES

A PLC BASED MIMO PID CONTROLLER FOR MULTIVARIABLE INDUSTRIAL PROCESSES ABCM Sympoium Serie in Mechatronic - Vol. 3 - pp.87-96 Copyright c 8 by ABCM A PLC BASE MIMO PI CONOLLE FO MULIVAIABLE INUSIAL POCESSES Joé Maria Galvez, jmgalvez@ufmg.br epartment of Mechanical Engineering

More information

16.400/453J Human Factors Engineering. Manual Control II

16.400/453J Human Factors Engineering. Manual Control II 16.4/453J Human Factor Engineering Manual Control II Pilot Input 16.4/453 Pitch attitude Actual pitch error 8 e attitude 8 Deired pitch attitude 8 c Digital flight control computer Cockpit inceptor Fly-by-wire

More information

SIMON FRASER UNIVERSITY School of Engineering Science ENSC 320 Electric Circuits II. Solutions to Assignment 3 February 2005.

SIMON FRASER UNIVERSITY School of Engineering Science ENSC 320 Electric Circuits II. Solutions to Assignment 3 February 2005. SIMON FRASER UNIVERSITY School of Engineering Science ENSC 320 Electric Circuit II Solution to Aignment 3 February 2005. Initial Condition Source 0 V battery witch flip at t 0 find i 3 (t) Component value:

More information

Multivariable Control Systems

Multivariable Control Systems Lecture Multivariable Control Sytem Ali Karimpour Aociate Profeor Ferdowi Univerity of Mahhad Lecture Reference are appeared in the lat lide. Dr. Ali Karimpour May 6 Uncertainty in Multivariable Sytem

More information

Singular Value Analysis of Linear- Quadratic Systems!

Singular Value Analysis of Linear- Quadratic Systems! Singular Value Analyi of Linear- Quadratic Sytem! Robert Stengel! Optimal Control and Etimation MAE 546! Princeton Univerity, 2017!! Multivariable Nyquit Stability Criterion!! Matrix Norm and Singular

More information

Department of Mechanical Engineering Massachusetts Institute of Technology Modeling, Dynamics and Control III Spring 2002

Department of Mechanical Engineering Massachusetts Institute of Technology Modeling, Dynamics and Control III Spring 2002 Department of Mechanical Engineering Maachuett Intitute of Technology 2.010 Modeling, Dynamic and Control III Spring 2002 SOLUTIONS: Problem Set # 10 Problem 1 Etimating tranfer function from Bode Plot.

More information

What lies between Δx E, which represents the steam valve, and ΔP M, which is the mechanical power into the synchronous machine?

What lies between Δx E, which represents the steam valve, and ΔP M, which is the mechanical power into the synchronous machine? A 2.0 Introduction In the lat et of note, we developed a model of the peed governing mechanim, which i given below: xˆ K ( Pˆ ˆ) E () In thee note, we want to extend thi model o that it relate the actual

More information

Root Locus Design. MEM 355 Performance Enhancement of Dynamical Systems

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

More information

7.2 INVERSE TRANSFORMS AND TRANSFORMS OF DERIVATIVES 281

7.2 INVERSE TRANSFORMS AND TRANSFORMS OF DERIVATIVES 281 72 INVERSE TRANSFORMS AND TRANSFORMS OF DERIVATIVES 28 and i 2 Show how Euler formula (page 33) can then be ued to deduce the reult a ( a) 2 b 2 {e at co bt} {e at in bt} b ( a) 2 b 2 5 Under what condition

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

Mathematical modeling of control systems. Laith Batarseh. Mathematical modeling of control systems

Mathematical modeling of control systems. Laith Batarseh. Mathematical modeling of control systems Chapter two Laith Batareh Mathematical modeling The dynamic of many ytem, whether they are mechanical, electrical, thermal, economic, biological, and o on, may be decribed in term of differential equation

More information

Design of Digital Filters

Design of Digital Filters Deign of Digital Filter Paley-Wiener Theorem [ ] ( ) If h n i a caual energy ignal, then ln H e dω< B where B i a finite upper bound. One implication of the Paley-Wiener theorem i that a tranfer function

More information

Control of Delayed Integrating Processes Using Two Feedback Controllers R MS Approach

Control of Delayed Integrating Processes Using Two Feedback Controllers R MS Approach Proceeding of the 7th WSEAS International Conference on SYSTEM SCIENCE and SIMULATION in ENGINEERING (ICOSSSE '8) Control of Delayed Integrating Procee Uing Two Feedback Controller R MS Approach LIBOR

More information

MM7. PID Control Design

MM7. PID Control Design MM7. PD Control Deign Reading Material: FC: p.79-200, DC: p.66-68. Propertie of PD control 2. uning Method of PD Control 3. Antiwindup echnique 4. A real cae tudy BO9000 0/9/2004 Proce Control . PD Feedback

More information

Today s Lecture. Block Diagrams. Block Diagrams: Examples. Block Diagrams: Examples. Closed Loop System 06/03/2017

Today s Lecture. Block Diagrams. Block Diagrams: Examples. Block Diagrams: Examples. Closed Loop System 06/03/2017 06/0/07 UW Britol Indutrial ontrol UFMF6W-0- ontrol Sytem ngineering UFMUY-0- Lecture 5: Block Diagram and Steady State rror Today Lecture Block diagram to repreent control ytem Block diagram manipulation

More information

CALIFORNIA INSTITUTE OF TECHNOLOGY Control and Dynamical Systems

CALIFORNIA INSTITUTE OF TECHNOLOGY Control and Dynamical Systems Control and Dynamical Sytem CDS 0 Problem Set #5 Iued: 3 Nov 08 Due: 0 Nov 08 Note: In the upper left hand corner of the econd page of your homework et, pleae put the number of hour that you pent on thi

More information

ECE382/ME482 Spring 2004 Homework 4 Solution November 14,

ECE382/ME482 Spring 2004 Homework 4 Solution November 14, ECE382/ME482 Spring 2004 Homework 4 Solution November 14, 2005 1 Solution to HW4 AP4.3 Intead of a contant or tep reference input, we are given, in thi problem, a more complicated reference path, r(t)

More information

A Simple Approach to Synthesizing Naïve Quantized Control for Reference Tracking

A Simple Approach to Synthesizing Naïve Quantized Control for Reference Tracking A Simple Approach to Syntheizing Naïve Quantized Control for Reference Tracking SHIANG-HUA YU Department of Electrical Engineering National Sun Yat-Sen Univerity 70 Lien-Hai Road, Kaohiung 804 TAIAN Abtract:

More information

Evolutionary Algorithms Based Fixed Order Robust Controller Design and Robustness Performance Analysis

Evolutionary Algorithms Based Fixed Order Robust Controller Design and Robustness Performance Analysis Proceeding of 01 4th International Conference on Machine Learning and Computing IPCSIT vol. 5 (01) (01) IACSIT Pre, Singapore Evolutionary Algorithm Baed Fixed Order Robut Controller Deign and Robutne

More information

Lecture #9 Continuous time filter

Lecture #9 Continuous time filter Lecture #9 Continuou time filter Oliver Faut December 5, 2006 Content Review. Motivation......................................... 2 2 Filter pecification 2 2. Low pa..........................................

More information

Linear-Quadratic Control System Design!

Linear-Quadratic Control System Design! Linear-Quadratic Control Sytem Deign! Robert Stengel! Optimal Control and Etimation MAE 546! Princeton Univerity, 217!! Control ytem configuration!! Proportional-integral!! Proportional-integral-filtering!!

More information

NONLINEAR CONTROLLER DESIGN FOR A SHELL AND TUBE HEAT EXCHANGER AN EXPERIMENTATION APPROACH

NONLINEAR CONTROLLER DESIGN FOR A SHELL AND TUBE HEAT EXCHANGER AN EXPERIMENTATION APPROACH International Journal of Electrical, Electronic and Data Communication, ISSN: 232-284 Volume-3, Iue-8, Aug.-25 NONLINEAR CONTROLLER DESIGN FOR A SHELL AND TUBE HEAT EXCHANGER AN EXPERIMENTATION APPROACH

More information

Bogoliubov Transformation in Classical Mechanics

Bogoliubov Transformation in Classical Mechanics Bogoliubov Tranformation in Claical Mechanic Canonical Tranformation Suppoe we have a et of complex canonical variable, {a j }, and would like to conider another et of variable, {b }, b b ({a j }). How

More information

Digital Control System

Digital Control System Digital Control Sytem Summary # he -tranform play an important role in digital control and dicrete ignal proceing. he -tranform i defined a F () f(k) k () A. Example Conider the following equence: f(k)

More information

EE 4343/ Control System Design Project

EE 4343/ Control System Design Project Copyright F.L. Lewi 2004 All right reerved EE 4343/5320 - Control Sytem Deign Project Updated: Sunday, February 08, 2004 Background: Analyi of Linear ytem, MATLAB Review of Baic Concept LTI Sytem LT I

More information

Lecture 21. The Lovasz splitting-off lemma Topics in Combinatorial Optimization April 29th, 2004

Lecture 21. The Lovasz splitting-off lemma Topics in Combinatorial Optimization April 29th, 2004 18.997 Topic in Combinatorial Optimization April 29th, 2004 Lecture 21 Lecturer: Michel X. Goeman Scribe: Mohammad Mahdian 1 The Lovaz plitting-off lemma Lovaz plitting-off lemma tate the following. Theorem

More information

Copyright 1967, by the author(s). All rights reserved.

Copyright 1967, by the author(s). All rights reserved. Copyright 1967, by the author(). All right reerved. Permiion to make digital or hard copie of all or part of thi work for peronal or claroom ue i granted without fee provided that copie are not made or

More information

Systems Analysis. Prof. Cesar de Prada ISA-UVA

Systems Analysis. Prof. Cesar de Prada ISA-UVA Sytem Analyi Prof. Cear de Prada ISAUVA rada@autom.uva.e Aim Learn how to infer the dynamic behaviour of a cloed loo ytem from it model. Learn how to infer the change in the dynamic of a cloed loo ytem

More information

FRTN10 Exercise 3. Specifications and Disturbance Models

FRTN10 Exercise 3. Specifications and Disturbance Models FRTN0 Exercie 3. Specification and Diturbance Model 3. A feedback ytem i hown in Figure 3., in which a firt-order proce if controlled by an I controller. d v r u 2 z C() P() y n Figure 3. Sytem in Problem

More information