Course Background. Loosely speaking, control is the process of getting something to do what you want it to do (or not do, as the case may be).

Size: px
Start display at page:

Download "Course Background. Loosely speaking, control is the process of getting something to do what you want it to do (or not do, as the case may be)."

Transcription

1 ECE4520/5520: Multivariable Control Systems I. 1 1 Course Background 1.1: From time to frequency domain Loosely speaking, control is the process of getting something to do what you want it to do (or not do, as the case may be). The something can be almost anything. Some examples: aircraft, spacecraft, cars, machines, robots, radars, etc. Some less obvious examples: energy systems, the economy, biological systems, the human body... The something is called the system that we would like to control. DEFINITION: Control is the process of causing a system variable to conform to some desired value, called a reference value (e.g., variable D temperature for a climate-control system). DEFINITION: Feedback is the process of measuring the controlled variable and using that information to influence its value. Disturbance Ref. Value Compensator Plant Output Feedback is not necessary for control. But, it is necessary to caterfor system uncertainty, which is the principal role of feedback. Open-loop control is also possible.

2 ECE4520/5520, Course Background 1 2 Goals of feedback control: Change dynamic response of a system to have desired properties t p M p Output of system tracks reference input. Reject disturbances. 0.1 t r t s Control design requires mathematical sets of equations (called a t model) that describes the system being controlled. Classical feedback techniques (cf., ECE4510) use frequency-domain (Laplace) models and tools to analyze and design control systems. Involves moving the poles of the closed-loop transfer function. Multivariable, state-space control instead: Primarily uses time-domain matrix representations of systems. Very powerful. Can often place poles of closed-loop system anywhere we want! Can make fast, smooth, etc. Same methods work for single-input, single-output (SISO) or multi-input, multi-output (MIMO) systems. Advanced techniques (cf., ECE5530) allow design of optimal linear controllers with a single MATLAB command! This course is a bridge between classical control and topics in advanced linear systems and control. We now review some of the concepts of classical linear systems and control which we will use...

3 ECE4520/5520, Course Background 1 3 Dynamic response Our primary objective is to be able to understand and learn how to control linear time-invariant (LTI) systems. We will also spend some time investigating nonlinear and linear time-varying (LTV) systems. LTI dynamics may be specified via models expressed as linear, constant-coefficient ordinary differential equations (LCCODE). Examples include: Mechanical systems: Use Newton s laws. Electrical systems: Use Kirchoff s laws. Electro-mechanical systems (generator/motor). Thermodynamic systems. Fluid-dynamic systems. EXAMPLE: Second-order system in standard form : Ry.t/ C 2! n Py.t/ C! 2 n y.t/ D!2 n u.t/: 4 dy.t/ u.t/ is the input, y.t/ is the output, Py.t/ D,and Ry.t/ D 4 d2 y.t/. dt dt 2 Laplace Transform The Laplace transform is a tool to help analyze dynamic systems. Y.s/ D H.s/U.s/,where Y.s/ is Laplace transform of output, y.t/; U.s/ is Laplace transform of input, u.t/; H.s/ is transfer function the Laplace tx of impulse response, h.t/.

4 ECE4520/5520, Course Background 1 4 Lf Py.t/g D sy.s/ y.0/ in general, and Lf Py.t/g D sy.s/ for a system initially at rest. EXAMPLE: Transfer function for second-order system: s 2 Y.s/C 2! n sy.s/ C! n 2 Y.s/ D!2 n U.s/ Y.s/ D! 2 n U.s/: s 2 C 2! n s C! n 2 Transforms for systems with LCCODE representations can be written as Y.s/ D H.s/U.s/,where H.s/ D b 0s m C b 1 s m 1 CCb m 1 s C b m a 0 s n C a 1 s n 1 CCa n 1 s C a n ; where n m for physical systems. These can be represented in MATLAB using vectors of numerator and denominator polynomials: num=[b0 b1 ::: bm]; den=[a0 a1 ::: an]; sys=tf(num,den); Can also represent these systems by factoring the polynomials into zero-pole-gain form: Q m id1.s i/ H.s/ D K Q n id1.s p i/ : sys=zpk(z,p,k); % in MATLAB

5 ECE4520/5520, Course Background 1 5 Input signals of interest include the following: u.t/ = kı.t/ ::: U.s/ = k impulse u.t/ = k1.t/ ::: U.s/ = k=s step u.t/ = kt 1.t/ ::: U.s/ = k=s 2 ramp k u.t/ = k exp. t/ 1.t/ ::: U.s/ = exponential s C k! u.t/ = k sin.!t/ 1.t/ : : : U.s/ = sinusoid s 2 C! 2 ks u.t/ = k cos.!t/ 1.t/ : : : U.s/ = cosinusoid s 2 C! 2 u.t/ = ke at k! sin.!t/ 1.t/ : : : U.s/ = decaying sinusoid.s C a/ 2 C! 2 u.t/ = ke at k.s C a/ cos.!t/ 1.t/ : : : U.s/ = decaying cosinusoid.s C a/ 2 C! 2 MATLAB s impulse, step, and lsim commandscanbeusedto find output time histories. The final value theorem states that if a system is stable and has a final, constant value, then lim x.t/ D lim sx.s/: t!1 s!0 Useful when investigating steady-state errors in a control system. The initial value theorem states that the initial value of a signal may be found using lim sx.s/ D s!1 ( x.0 / D x.0 C /; x.t/ continuous at t D 0I x.0 C /; otherwise: We will see a use for this later in the semester when studying system controllability.

6 ECE4520/5520, Course Background : From frequency to time domain The inverse Laplace transform (ILT) converts X.s/! x.t/. Here we assume that X.s/ is a ratio of polynomials in s. Thatis, X.s/ D C.s/ A.s/ : If X.s/ is not a proper rational function, wemustfirstperformlong division. (In a proper rational function, the degree of C.s/ is less than the degree of A.s/.) In general, partial-fraction inversion begins by writing where K.s/ is of the form X.s/ D K.s/ C B.s/ A.s/ We inverse transform K.s/ using D K.s/ C F.s/; K.s/ D k 0 C k 1 s CCk L s L : d n ı.t/ dt n s n : The remaining problem is to find the inverse transform of F.s/. F.s/ D b 0s m C b 1 s m 1 CCb m s n C a a s n 1 CCa n Q m id1.s i/ D kq n id1.s p i/ D r 1 s p 1 C r 2 s p 2 CC r n s p n.zeros/.poles/ so;.s p 1 /F.s/ D r 1 C r 2.s p 1 / s p 2 CC r n.s p 1 / s p n : if fp i g distinct:

7 ECE4520/5520, Course Background 1 7 Let s D p1.then, r 1 D.s p 1 /F.s/j sdp1 : Similarly, and f.t/ D nx id1 r i D.s p i /F.s/j sdpi r i e p it 1.t/ since L e kt 1.t/ D 1 s k : 5 EXAMPLE: F.s/ D s 2 C 3s C 2 D 5.s C 1/.s C 2/. ˇ r 1 D.s C 1/F.s/ D 5 ˇ D 5 ˇsD 1 s C 2 ˇsD 1 ˇ r 2 D.s C 2/F.s/ D 5 ˇ ˇsD 2 s C 1 Repeated poles f.t/d.5e t 5e 2t /1.t/: ˇsD 2 D 5 If F.s/ has repeated roots, we must modify the procedure. e.g., fora pole repeated 3 times: F.s/ D k.s p 1 / 3.s p 2 / D r 1;1 s p 1 C r 1;2.s p 1 / 2 C r 1;3.s p 1 / 3 C r 2 s p 2 C r 1;3 D.s p 1 / 3 F.s/ˇˇsDp1 r 1;2 D d ds.s p1 / 3 F.s/ ˇˇˇˇsDp1

8 ECE4520/5520, Course Background 1 8 r 1;1 D 1 d 2.s p1 / 3 F.s/ ˇ 2 ds 2 r x;k i D 1 d i.s pi / k F.s/ ˇ iš ds i ˇsDp1 s C 3 EXAMPLE: Find ILT of.s C 1/.s C 2/ 2. ans: f.t/d.2e t 2e 2t ƒ te 2t /1.t/: from repeated root. TEDIOUS. Use MATLAB. e.g., F.s/ D 5 s 2 C 3s C 2. Example 1. Example 2. ˇsDpi : >> Fnum=[0 0 5]; >> Fnum=[ ]; >> Fden=[1 3 2]; >> Fden=conv([1 1],conv([1 2],[1 2])); >> [r,p,k]=residue(fnum,fden); >> [r,p,k]=residue(fnum,fden); r = -5 r = p = p = -2 k = [] -2-1 k = [] When you use residue and get repeated roots, BE SURE to type help residue to correctly interpret the result. Complex-conjugate poles The theory developed thus far works for either real or complex poles.

9 ECE4520/5520, Course Background 1 9 It may be easier to handle complex-conjugate poles separately. Consider F.s/ D B.s/.s p 1 /.s p 1 /Q.s/ D K 1 s p 1 C K 2 s p 1 C R.s/ Q.s/ : Expand R.s/=Q.s/ using previous methods. Expand the first part as K 1 s p 1 C K 2 s p 1 D as C b.s 1 / 2 C! 1 2 : It can be shown that a D 2R.K1 /, b D 2ŒR.K 1 / 1 C I.K 1 /! 1, 1 D R.p 1 /,and! 1 D I.p 1 /. EXAMPLE: As a specific problem consider X.s/ D 2s 2 C 6s C 6.s C 2/.s 2 C 2s C 2/ D r 1 s C 2 C Using the simple-pole formula we find r 1 D 2s2 C 6s C 6 ˇ s 2 C 2s C 2 ˇsD 2 D as C b.s C 1/ 2 C 1 : We will substitute values for s to obtain a and b C C 2 D 1: Let s D D 1 2 C b 2 ) b D 2: Let s D 1. 2 C 6 C C 2 C 2/ D 1 3 C a C 2 5 ) a D 1:

10 ECE4520/5520, Course Background 1 10 Finally, X.s/ D 1 s C 2 C s C 2.s C 1/ 2 C 1 Taking the inverse-laplace transform D 1 s C 2 C s C 1.s C 1/ 2 C 1 C 1.s C 1/ 2 C 1 : x.t/ D e 2t C e t cos.t/ C e t sin.t/ 1.t/: Symbolic Laplace transforms using MATLAB MATLAB incorporates part of the Maple symbolic toolbox. The commands of interest to us here are: laplace, ilaplace, ezplot and pretty. F=laplace(f) is the Laplace transform of symbolic fn f. f=ilaplace(f) is the inverse-laplace transform of F. ezplot(f,[xmin xmax]) plots symbolic function f. pretty(s) displays symbolic S in a pretty way.

11 ECE4520/5520, Course Background : Dynamic properties of LTI systems Block diagrams Useful when analyzing systems comprised of a number of sub-units. U.s/ H.s/ Y.s/ Y.s/ D H.s/U.s/ U.s/ H 1.s/ H 2.s/ Y.s/ Y.s/ D ŒH 1.s/H 2.s/ U.s/ U.s/ H 1.s/ H 2.s/ Y.s/ Y.s/ D ŒH 1.s/ C H 2.s/ U.s/ R.s/ U 1.s/ Y 2.s/ H 1.s/ H 2.s/ U 2.s/ Y.s/ Y.s/ D H 1.s/ 1 C H 2.s/H 1.s/ R.s/ Dynamic response versus pole locations The poles of H.s/ determine (qualitatively) the dynamic response of the system. The zeros of H.s/ quantify the relationship. If the system has only real poles, each one is of the form: H.s/ D 1 s C :

12 ECE4520/5520, Course Background 1 12 If >0,thesystemisstable,andh.t/ D e t 1.t/. Thetimeconstant is D 1=; and the response of the system to an impulse or step decays to steady-state in about 4 or 5 time constants. 1 impulse([0 1],[1 1]); 1 step([0 1],[1 1]); h.t/ e t 1 e y.t/ K K.1 e t= / System response. K D dc gain Response to initial condition t D Time (sec ) t D Time (sec ) If a system has complex-conjugate poles, each may be written as: H.s/ D! 2 n : s 2 C 2! n s C! n 2 We can extract two more parameters from this equation: D! n and! d D! n p 1 2 : I.s/ plays the same role as above it specifies D sin 1./ decay rate of the response.!d is the oscillation frequency of the output. Note:! d! n unless D 0.! n R.s/ is the damping ratio and it also plays a! d role in decay rate and overshoot.

13 ECE4520/5520, Course Background 1 13 Impulse response h.t/ D!n e t sin.! d t/ 1.t/. Step response y.t/ D 1 e t cos.! d t/c sin.! d t/.! d 1 Impulse Responses of 2nd-Order Systems D 0 2 Step Responses of 2nd-Order Systems D 0 y.t/ D 1 0:8 0:2 0:4 0:6 y.t/ :2 0:4 0: :8 1: ! n t ! n t Asummarychartofimpulseresponsesandstepresponsesversus pole locations is: I.s/ I.s/ R.s/ R.s/ Impulse responses vs. pole locations Step responses vs. pole locations

14 ECE4520/5520, Course Background 1 14 Time-domain specs. determine where poles should be placed in the s-plane t p M p Step-response specs often given. 0.1 t r t s t Rise time tr D time to go from 10 % to 90 % of final value. Settling time ts D time until permanently within 1 % of final value. Overshoot Mp D maximum percent overshoot t r 1:8=! n :::! n 1:8=t r Mp, % t s 4:6= ::: 4:6=t s M p e = p1 2 ::: fn.m p / I.s/ I.s/ I.s/ I.s/! n sin 1 R.s/ R.s/ R.s/ R.s/

15 ECE4520/5520, Course Background 1 15 Basic feedback properties r.t/ D.s/ G.s/ y.t/ Y.s/ R.s/ D D.s/G.s/ 1 C D.s/G.s/ D T.s/: Stability depends on roots of denominator of T.s/: 1 C D.s/G.s/ D 0. Routh test used to determine stability. Steady-state error found from E.s/ D.1 T.s//R.s/. ess D lim t!1 e.t/ D lim s!0 se.s/ if the limit exists. System type D 0 iff e ss is finite for unit-step reference-input 1.t/. System type D 1 iff e ss is finite for unit-ramp reference-input r.t/. System type D 2 iff e ss is finite for unit-parabola ref.-input p.t/... Some types of controllers Proportional ctrlr: u.t/ D Ke.t/: D.s/ D K: Integral ctrlr u.t/ D K Z t e.t/ dt: D.s/ D K T I 1 T I s Derivative ctrlr. u.t/ D KT D Pe.t/ D.s/ D KT D s Combinations: PI: D.s/ D K 1 C 1 I T I s PD: D.s/ D K.1C T D s/i PID:D.s/ D K 1 C 1 T I s C T Ds : Lead: D.s/ D K TsC 1 T s C 1 ; Lag: D.s/ D K TsC 1 T s C 1 ; < 1 (approx PD) > 1 (approx PI; often, K D )

16 ECE4520/5520, Course Background 1 16 Lead/Lag: D.s/ D K.T 1s C 1/.T 2 s C 1/. 1T 1 s C 1/. 2T 2 s C 1/ ; 1 <1; 2 >1: Integral can reduce/eliminate steady-state error. Derivatives can reduce/eliminate oscillation. Proportional term can speed/slow response. Lead control approximates derivative control, but reduces amplification of noise. Lag control approximates integral control, but is easier to stabilize. Where to from here? We have reviewed some important concepts from classical control theory, which uses a transfer-function approach. We now begin to examine state-space models.

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

Dr Ian R. Manchester

Dr Ian R. Manchester Week Content Notes 1 Introduction 2 Frequency Domain Modelling 3 Transient Performance and the s-plane 4 Block Diagrams 5 Feedback System Characteristics Assign 1 Due 6 Root Locus 7 Root Locus 2 Assign

More information

Time Response Analysis (Part II)

Time Response Analysis (Part II) Time Response Analysis (Part II). A critically damped, continuous-time, second order system, when sampled, will have (in Z domain) (a) A simple pole (b) Double pole on real axis (c) Double pole on imaginary

More information

Dr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Review

Dr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Review Week Date Content Notes 1 6 Mar Introduction 2 13 Mar Frequency Domain Modelling 3 20 Mar Transient Performance and the s-plane 4 27 Mar Block Diagrams Assign 1 Due 5 3 Apr Feedback System Characteristics

More information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture : Different Types of Control Overview In this Lecture, you will learn: Limits of Proportional Feedback Performance

More information

Basic Procedures for Common Problems

Basic Procedures for Common Problems Basic Procedures for Common Problems ECHE 550, Fall 2002 Steady State Multivariable Modeling and Control 1 Determine what variables are available to manipulate (inputs, u) and what variables are available

More information

Lecture 12. Upcoming labs: Final Exam on 12/21/2015 (Monday)10:30-12:30

Lecture 12. Upcoming labs: Final Exam on 12/21/2015 (Monday)10:30-12:30 289 Upcoming labs: Lecture 12 Lab 20: Internal model control (finish up) Lab 22: Force or Torque control experiments [Integrative] (2-3 sessions) Final Exam on 12/21/2015 (Monday)10:30-12:30 Today: Recap

More information

STABILITY. Have looked at modeling dynamic systems using differential equations. and used the Laplace transform to help find step and impulse

STABILITY. Have looked at modeling dynamic systems using differential equations. and used the Laplace transform to help find step and impulse SIGNALS AND SYSTEMS: PAPER 3C1 HANDOUT 4. Dr David Corrigan 1. Electronic and Electrical Engineering Dept. corrigad@tcd.ie www.sigmedia.tv STABILITY Have looked at modeling dynamic systems using differential

More 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

Dynamic Response. Assoc. Prof. Enver Tatlicioglu. Department of Electrical & Electronics Engineering Izmir Institute of Technology.

Dynamic Response. Assoc. Prof. Enver Tatlicioglu. Department of Electrical & Electronics Engineering Izmir Institute of Technology. Dynamic Response Assoc. Prof. Enver Tatlicioglu Department of Electrical & Electronics Engineering Izmir Institute of Technology Chapter 3 Assoc. Prof. Enver Tatlicioglu (EEE@IYTE) EE362 Feedback Control

More information

APPLICATIONS FOR ROBOTICS

APPLICATIONS FOR ROBOTICS Version: 1 CONTROL APPLICATIONS FOR ROBOTICS TEX d: Feb. 17, 214 PREVIEW We show that the transfer function and conditions of stability for linear systems can be studied using Laplace transforms. Table

More information

Control System. Contents

Control System. Contents Contents Chapter Topic Page Chapter- Chapter- Chapter-3 Chapter-4 Introduction Transfer Function, Block Diagrams and Signal Flow Graphs Mathematical Modeling Control System 35 Time Response Analysis of

More information

Video 5.1 Vijay Kumar and Ani Hsieh

Video 5.1 Vijay Kumar and Ani Hsieh Video 5.1 Vijay Kumar and Ani Hsieh Robo3x-1.1 1 The Purpose of Control Input/Stimulus/ Disturbance System or Plant Output/ Response Understand the Black Box Evaluate the Performance Change the Behavior

More information

EE Experiment 11 The Laplace Transform and Control System Characteristics

EE Experiment 11 The Laplace Transform and Control System Characteristics EE216:11 1 EE 216 - Experiment 11 The Laplace Transform and Control System Characteristics Objectives: To illustrate computer usage in determining inverse Laplace transforms. Also to determine useful signal

More information

Control Systems I. Lecture 6: Poles and Zeros. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich

Control Systems I. Lecture 6: Poles and Zeros. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich Control Systems I Lecture 6: Poles and Zeros Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 27, 2017 E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/2017

More information

STABILITY ANALYSIS. Asystemmaybe stable, neutrallyormarginallystable, or unstable. This can be illustrated using cones: Stable Neutral Unstable

STABILITY ANALYSIS. Asystemmaybe stable, neutrallyormarginallystable, or unstable. This can be illustrated using cones: Stable Neutral Unstable ECE4510/5510: Feedback Control Systems. 5 1 STABILITY ANALYSIS 5.1: Bounded-input bounded-output (BIBO) stability Asystemmaybe stable, neutrallyormarginallystable, or unstable. This can be illustrated

More information

Dr. Ian R. Manchester

Dr. Ian R. Manchester Dr Ian R. Manchester Week Content Notes 1 Introduction 2 Frequency Domain Modelling 3 Transient Performance and the s-plane 4 Block Diagrams 5 Feedback System Characteristics Assign 1 Due 6 Root Locus

More information

Topic # Feedback Control Systems

Topic # Feedback Control Systems Topic #1 16.31 Feedback Control Systems Motivation Basic Linear System Response Fall 2007 16.31 1 1 16.31: Introduction r(t) e(t) d(t) y(t) G c (s) G(s) u(t) Goal: Design a controller G c (s) so that the

More information

Chapter 6: The Laplace Transform. Chih-Wei Liu

Chapter 6: The Laplace Transform. Chih-Wei Liu Chapter 6: The Laplace Transform Chih-Wei Liu Outline Introduction The Laplace Transform The Unilateral Laplace Transform Properties of the Unilateral Laplace Transform Inversion of the Unilateral Laplace

More information

ECE317 : Feedback and Control

ECE317 : Feedback and Control ECE317 : Feedback and Control Lecture : Routh-Hurwitz stability criterion Examples Dr. Richard Tymerski Dept. of Electrical and Computer Engineering Portland State University 1 Course roadmap Modeling

More information

Unit 2: Modeling in the Frequency Domain Part 2: The Laplace Transform. The Laplace Transform. The need for Laplace

Unit 2: Modeling in the Frequency Domain Part 2: The Laplace Transform. The Laplace Transform. The need for Laplace Unit : Modeling in the Frequency Domain Part : Engineering 81: Control Systems I Faculty of Engineering & Applied Science Memorial University of Newfoundland January 1, 010 1 Pair Table Unit, Part : Unit,

More information

Time Response of Systems

Time Response of Systems Chapter 0 Time Response of Systems 0. Some Standard Time Responses Let us try to get some impulse time responses just by inspection: Poles F (s) f(t) s-plane Time response p =0 s p =0,p 2 =0 s 2 t p =

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

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

Outline. Classical Control. Lecture 2

Outline. Classical Control. Lecture 2 Outline Outline Outline Review of Material from Lecture 2 New Stuff - Outline Review of Lecture System Performance Effect of Poles Review of Material from Lecture System Performance Effect of Poles 2 New

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

(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

Lecture 7:Time Response Pole-Zero Maps Influence of Poles and Zeros Higher Order Systems and Pole Dominance Criterion

Lecture 7:Time Response Pole-Zero Maps Influence of Poles and Zeros Higher Order Systems and Pole Dominance Criterion Cleveland State University MCE441: Intr. Linear Control Lecture 7:Time Influence of Poles and Zeros Higher Order and Pole Criterion Prof. Richter 1 / 26 First-Order Specs: Step : Pole Real inputs contain

More information

Raktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Dynamic Response

Raktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Dynamic Response .. AERO 422: Active Controls for Aerospace Vehicles Dynamic Response Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. . Previous Class...........

More information

7.1 Introduction. Apago PDF Enhancer. Definition and Test Inputs. 340 Chapter 7 Steady-State Errors

7.1 Introduction. Apago PDF Enhancer. Definition and Test Inputs. 340 Chapter 7 Steady-State Errors 340 Chapter 7 Steady-State Errors 7. Introduction In Chapter, we saw that control systems analysis and design focus on three specifications: () transient response, (2) stability, and (3) steady-state errors,

More information

Notes for ECE-320. Winter by R. Throne

Notes for ECE-320. Winter by R. Throne Notes for ECE-3 Winter 4-5 by R. Throne Contents Table of Laplace Transforms 5 Laplace Transform Review 6. Poles and Zeros.................................... 6. Proper and Strictly Proper Transfer Functions...................

More information

Raktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Basic Feedback Analysis & Design

Raktim 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

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

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 6: Generalized and Controller Design Overview In this Lecture, you will learn: Generalized? What about changing OTHER parameters

More information

Laplace Transforms Chapter 3

Laplace Transforms Chapter 3 Laplace Transforms Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first. Laplace transforms play a key role in important

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

ECE 3793 Matlab Project 3

ECE 3793 Matlab Project 3 ECE 3793 Matlab Project 3 Spring 2017 Dr. Havlicek DUE: 04/25/2017, 11:59 PM What to Turn In: Make one file that contains your solution for this assignment. It can be an MS WORD file or a PDF file. Make

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

Lecture 25: Tue Nov 27, 2018

Lecture 25: Tue Nov 27, 2018 Lecture 25: Tue Nov 27, 2018 Reminder: Lab 3 moved to Tuesday Dec 4 Lecture: review time-domain characteristics of 2nd-order systems intro to control: feedback open-loop vs closed-loop control intro to

More information

Part IB Paper 6: Information Engineering LINEAR SYSTEMS AND CONTROL. Glenn Vinnicombe HANDOUT 5. An Introduction to Feedback Control Systems

Part IB Paper 6: Information Engineering LINEAR SYSTEMS AND CONTROL. Glenn Vinnicombe HANDOUT 5. An Introduction to Feedback Control Systems Part IB Paper 6: Information Engineering LINEAR SYSTEMS AND CONTROL Glenn Vinnicombe HANDOUT 5 An Introduction to Feedback Control Systems ē(s) ȳ(s) Σ K(s) G(s) z(s) H(s) z(s) = H(s)G(s)K(s) L(s) ē(s)=

More information

Linear State Feedback Controller Design

Linear State Feedback Controller Design Assignment For EE5101 - Linear Systems Sem I AY2010/2011 Linear State Feedback Controller Design Phang Swee King A0033585A Email: king@nus.edu.sg NGS/ECE Dept. Faculty of Engineering National University

More information

Analysis and Design of Control Systems in the Time Domain

Analysis and Design of Control Systems in the Time Domain Chapter 6 Analysis and Design of Control Systems in the Time Domain 6. Concepts of feedback control Given a system, we can classify it as an open loop or a closed loop depends on the usage of the feedback.

More information

Module 02 Control Systems Preliminaries, Intro to State Space

Module 02 Control Systems Preliminaries, Intro to State Space Module 02 Control Systems Preliminaries, Intro to State Space Ahmad F. Taha EE 5143: Linear Systems and Control Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha August 28, 2017 Ahmad

More information

Radar Dish. Armature controlled dc motor. Inside. θ r input. Outside. θ D output. θ m. Gearbox. Control Transmitter. Control. θ D.

Radar Dish. Armature controlled dc motor. Inside. θ r input. Outside. θ D output. θ m. Gearbox. Control Transmitter. Control. θ D. Radar Dish ME 304 CONTROL SYSTEMS Mechanical Engineering Department, Middle East Technical University Armature controlled dc motor Outside θ D output Inside θ r input r θ m Gearbox Control Transmitter

More information

e st f (t) dt = e st tf(t) dt = L {t f(t)} s

e st f (t) dt = e st tf(t) dt = L {t f(t)} s Additional operational properties How to find the Laplace transform of a function f (t) that is multiplied by a monomial t n, the transform of a special type of integral, and the transform of a periodic

More information

Performance of Feedback Control Systems

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

More information

9.5 The Transfer Function

9.5 The Transfer Function Lecture Notes on Control Systems/D. Ghose/2012 0 9.5 The Transfer Function Consider the n-th order linear, time-invariant dynamical system. dy a 0 y + a 1 dt + a d 2 y 2 dt + + a d n y 2 n dt b du 0u +

More information

ECE317 : Feedback and Control

ECE317 : Feedback and Control ECE317 : Feedback and Control Lecture : Stability Routh-Hurwitz stability criterion Dr. Richard Tymerski Dept. of Electrical and Computer Engineering Portland State University 1 Course roadmap Modeling

More information

Control Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Jacopo Tani. Institute for Dynamic Systems and Control D-MAVT ETH Zürich

Control Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Jacopo Tani. Institute for Dynamic Systems and Control D-MAVT ETH Zürich Control Systems I Lecture 7: Feedback and the Root Locus method Readings: Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 2, 2018 J. Tani, E. Frazzoli (ETH) Lecture 7:

More information

AN INTRODUCTION TO THE CONTROL THEORY

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

More information

Identification Methods for Structural Systems. Prof. Dr. Eleni Chatzi System Stability - 26 March, 2014

Identification Methods for Structural Systems. Prof. Dr. Eleni Chatzi System Stability - 26 March, 2014 Prof. Dr. Eleni Chatzi System Stability - 26 March, 24 Fundamentals Overview System Stability Assume given a dynamic system with input u(t) and output x(t). The stability property of a dynamic system can

More information

Introduction to Feedback Control

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

More information

MAT 275 Laboratory 7 Laplace Transform and the Symbolic Math Toolbox

MAT 275 Laboratory 7 Laplace Transform and the Symbolic Math Toolbox Laplace Transform and the Symbolic Math Toolbox 1 MAT 275 Laboratory 7 Laplace Transform and the Symbolic Math Toolbox In this laboratory session we will learn how to 1. Use the Symbolic Math Toolbox 2.

More information

Raktim Bhattacharya. . AERO 632: Design of Advance Flight Control System. Preliminaries

Raktim Bhattacharya. . AERO 632: Design of Advance Flight Control System. Preliminaries . AERO 632: of Advance Flight Control System. Preliminaries Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. Preliminaries Signals & Systems Laplace

More information

EE/ME/AE324: Dynamical Systems. Chapter 7: Transform Solutions of Linear Models

EE/ME/AE324: Dynamical Systems. Chapter 7: Transform Solutions of Linear Models EE/ME/AE324: Dynamical Systems Chapter 7: Transform Solutions of Linear Models The Laplace Transform Converts systems or signals from the real time domain, e.g., functions of the real variable t, to the

More information

Professor Fearing EE C128 / ME C134 Problem Set 7 Solution Fall 2010 Jansen Sheng and Wenjie Chen, UC Berkeley

Professor Fearing EE C128 / ME C134 Problem Set 7 Solution Fall 2010 Jansen Sheng and Wenjie Chen, UC Berkeley Professor Fearing EE C8 / ME C34 Problem Set 7 Solution Fall Jansen Sheng and Wenjie Chen, UC Berkeley. 35 pts Lag compensation. For open loop plant Gs ss+5s+8 a Find compensator gain Ds k such that the

More information

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

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

More information

School of Mechanical Engineering Purdue University. ME375 Dynamic Response - 1

School of Mechanical Engineering Purdue University. ME375 Dynamic Response - 1 Dynamic Response of Linear Systems Linear System Response Superposition Principle Responses to Specific Inputs Dynamic Response of f1 1st to Order Systems Characteristic Equation - Free Response Stable

More information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 8: Response Characteristics Overview In this Lecture, you will learn: Characteristics of the Response Stability Real

More information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 8: Response Characteristics Overview In this Lecture, you will learn: Characteristics of the Response Stability Real Poles

More information

Outline. Classical Control. Lecture 5

Outline. 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 information

C(s) R(s) 1 C(s) C(s) C(s) = s - T. Ts + 1 = 1 s - 1. s + (1 T) Taking the inverse Laplace transform of Equation (5 2), we obtain

C(s) R(s) 1 C(s) C(s) C(s) = s - T. Ts + 1 = 1 s - 1. s + (1 T) Taking the inverse Laplace transform of Equation (5 2), we obtain analyses of the step response, ramp response, and impulse response of the second-order systems are presented. Section 5 4 discusses the transient-response analysis of higherorder systems. Section 5 5 gives

More information

The Laplace Transform

The Laplace Transform The Laplace Transform Generalizing the Fourier Transform The CTFT expresses a time-domain signal as a linear combination of complex sinusoids of the form e jωt. In the generalization of the CTFT to the

More information

MATHEMATICAL MODELING OF CONTROL SYSTEMS

MATHEMATICAL MODELING OF CONTROL SYSTEMS 1 MATHEMATICAL MODELING OF CONTROL SYSTEMS Sep-14 Dr. Mohammed Morsy Outline Introduction Transfer function and impulse response function Laplace Transform Review Automatic control systems Signal Flow

More information

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

Course roadmap. Step response for 2nd-order system. Step response for 2nd-order system ME45: Control Systems Lecture Time response of nd-order systems Prof. Clar Radcliffe and Prof. Jongeun Choi Department of Mechanical Engineering Michigan State University Modeling Laplace transform Transfer

More information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 12: Overview In this Lecture, you will learn: Review of Feedback Closing the Loop Pole Locations Changing the Gain

More information

Laplace Transform Part 1: Introduction (I&N Chap 13)

Laplace Transform Part 1: Introduction (I&N Chap 13) Laplace Transform Part 1: Introduction (I&N Chap 13) Definition of the L.T. L.T. of Singularity Functions L.T. Pairs Properties of the L.T. Inverse L.T. Convolution IVT(initial value theorem) & FVT (final

More information

MODELING OF CONTROL SYSTEMS

MODELING OF CONTROL SYSTEMS 1 MODELING OF CONTROL SYSTEMS Feb-15 Dr. Mohammed Morsy Outline Introduction Differential equations and Linearization of nonlinear mathematical models Transfer function and impulse response function Laplace

More information

Step input, ramp input, parabolic input and impulse input signals. 2. What is the initial slope of a step response of a first order system?

Step input, ramp input, parabolic input and impulse input signals. 2. What is the initial slope of a step response of a first order system? IC6501 CONTROL SYSTEM UNIT-II TIME RESPONSE PART-A 1. What are the standard test signals employed for time domain studies?(or) List the standard test signals used in analysis of control systems? (April

More information

12.7 Steady State Error

12.7 Steady State Error Lecture Notes on Control Systems/D. Ghose/01 106 1.7 Steady State Error For first order systems we have noticed an overall improvement in performance in terms of rise time and settling time. But there

More information

Control Systems. Laplace domain analysis

Control Systems. Laplace domain analysis Control Systems Laplace domain analysis L. Lanari outline introduce the Laplace unilateral transform define its properties show its advantages in turning ODEs to algebraic equations define an Input/Output

More information

Transform Solutions to LTI Systems Part 3

Transform Solutions to LTI Systems Part 3 Transform Solutions to LTI Systems Part 3 Example of second order system solution: Same example with increased damping: k=5 N/m, b=6 Ns/m, F=2 N, m=1 Kg Given x(0) = 0, x (0) = 0, find x(t). The revised

More information

BASIC PROPERTIES OF FEEDBACK

BASIC PROPERTIES OF FEEDBACK ECE450/550: Feedback Control Systems. 4 BASIC PROPERTIES OF FEEDBACK 4.: Setting up an example to benchmark controllers There are two basic types/categories of control systems: OPEN LOOP: Disturbance r(t)

More information

PID controllers. Laith Batarseh. PID controllers

PID controllers. Laith Batarseh. PID controllers Next Previous 24-Jan-15 Chapter six Laith Batarseh Home End The controller choice is an important step in the control process because this element is responsible of reducing the error (e ss ), rise time

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

20.6. Transfer Functions. Introduction. Prerequisites. Learning Outcomes

20.6. Transfer Functions. Introduction. Prerequisites. Learning Outcomes Transfer Functions 2.6 Introduction In this Section we introduce the concept of a transfer function and then use this to obtain a Laplace transform model of a linear engineering system. (A linear engineering

More information

2.161 Signal Processing: Continuous and Discrete Fall 2008

2.161 Signal Processing: Continuous and Discrete Fall 2008 MIT OpenCourseWare http://ocw.mit.edu 2.6 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS

More information

School of Mechanical Engineering Purdue University. ME375 Feedback Control - 1

School of Mechanical Engineering Purdue University. ME375 Feedback Control - 1 Introduction to Feedback Control Control System Design Why Control? Open-Loop vs Closed-Loop (Feedback) Why Use Feedback Control? Closed-Loop Control System Structure Elements of a Feedback Control System

More information

Lecture 5: Linear Systems. Transfer functions. Frequency Domain Analysis. Basic Control Design.

Lecture 5: Linear Systems. Transfer functions. Frequency Domain Analysis. Basic Control Design. ISS0031 Modeling and Identification Lecture 5: Linear Systems. Transfer functions. Frequency Domain Analysis. Basic Control Design. Aleksei Tepljakov, Ph.D. September 30, 2015 Linear Dynamic Systems Definition

More information

Alireza Mousavi Brunel University

Alireza Mousavi Brunel University Alireza Mousavi Brunel University 1 » Control Process» Control Systems Design & Analysis 2 Open-Loop Control: Is normally a simple switch on and switch off process, for example a light in a room is switched

More information

Bangladesh University of Engineering and Technology. EEE 402: Control System I Laboratory

Bangladesh University of Engineering and Technology. EEE 402: Control System I Laboratory Bangladesh University of Engineering and Technology Electrical and Electronic Engineering Department EEE 402: Control System I Laboratory Experiment No. 4 a) Effect of input waveform, loop gain, and system

More information

An Introduction to Control Systems

An Introduction to Control Systems An Introduction to Control Systems Signals and Systems: 3C1 Control Systems Handout 1 Dr. David Corrigan Electronic and Electrical Engineering corrigad@tcd.ie November 21, 2012 Recall the concept of a

More information

Definition of the Laplace transform. 0 x(t)e st dt

Definition of the Laplace transform. 0 x(t)e st dt Definition of the Laplace transform Bilateral Laplace Transform: X(s) = x(t)e st dt Unilateral (or one-sided) Laplace Transform: X(s) = 0 x(t)e st dt ECE352 1 Definition of the Laplace transform (cont.)

More information

Chapter 2. Classical Control System Design. Dutch Institute of Systems and Control

Chapter 2. Classical Control System Design. Dutch Institute of Systems and Control Chapter 2 Classical Control System Design Overview Ch. 2. 2. Classical control system design Introduction Introduction Steady-state Steady-state errors errors Type Type k k systems systems Integral Integral

More information

Lecture 4 Stabilization

Lecture 4 Stabilization Lecture 4 Stabilization This lecture follows Chapter 5 of Doyle-Francis-Tannenbaum, with proofs and Section 5.3 omitted 17013 IOC-UPC, Lecture 4, November 2nd 2005 p. 1/23 Stable plants (I) We assume that

More information

Lecture 12. AO Control Theory

Lecture 12. AO Control Theory Lecture 12 AO Control Theory Claire Max with many thanks to Don Gavel and Don Wiberg UC Santa Cruz February 18, 2016 Page 1 What are control systems? Control is the process of making a system variable

More information

Introduction & Laplace Transforms Lectures 1 & 2

Introduction & Laplace Transforms Lectures 1 & 2 Introduction & Lectures 1 & 2, Professor Department of Electrical and Computer Engineering Colorado State University Fall 2016 Control System Definition of a Control System Group of components that collectively

More information

TRACKING AND DISTURBANCE REJECTION

TRACKING AND DISTURBANCE REJECTION TRACKING AND DISTURBANCE REJECTION Sadegh Bolouki Lecture slides for ECE 515 University of Illinois, Urbana-Champaign Fall 2016 S. Bolouki (UIUC) 1 / 13 General objective: The output to track a reference

More information

12/20/2017. Lectures on Signals & systems Engineering. Designed and Presented by Dr. Ayman Elshenawy Elsefy

12/20/2017. Lectures on Signals & systems Engineering. Designed and Presented by Dr. Ayman Elshenawy Elsefy //7 ectures on Signals & systems Engineering Designed and Presented by Dr. Ayman Elshenawy Elsefy Dept. of Systems & Computer Eng. Al-Azhar University Email : eaymanelshenawy@yahoo.com aplace Transform

More information

D(s) G(s) A control system design definition

D(s) G(s) A control system design definition R E Compensation D(s) U Plant G(s) Y Figure 7. A control system design definition x x x 2 x 2 U 2 s s 7 2 Y Figure 7.2 A block diagram representing Eq. (7.) in control form z U 2 s z Y 4 z 2 s z 2 3 Figure

More information

Course roadmap. ME451: Control Systems. Example of Laplace transform. Lecture 2 Laplace transform. Laplace transform

Course roadmap. ME451: Control Systems. Example of Laplace transform. Lecture 2 Laplace transform. Laplace transform ME45: Control Systems Lecture 2 Prof. Jongeun Choi Department of Mechanical Engineering Michigan State University Modeling Transfer function Models for systems electrical mechanical electromechanical Block

More information

Dr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Root Locus

Dr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Root Locus Week Content Notes 1 Introduction 2 Frequency Domain Modelling 3 Transient Performance and the s-plane 4 Block Diagrams 5 Feedback System Characteristics Assign 1 Due 6 Root Locus 7 Root Locus 2 Assign

More information

Laplace Transforms. Chapter 3. Pierre Simon Laplace Born: 23 March 1749 in Beaumont-en-Auge, Normandy, France Died: 5 March 1827 in Paris, France

Laplace Transforms. Chapter 3. Pierre Simon Laplace Born: 23 March 1749 in Beaumont-en-Auge, Normandy, France Died: 5 March 1827 in Paris, France Pierre Simon Laplace Born: 23 March 1749 in Beaumont-en-Auge, Normandy, France Died: 5 March 1827 in Paris, France Laplace Transforms Dr. M. A. A. Shoukat Choudhury 1 Laplace Transforms Important analytical

More information

Control of Manufacturing Processes

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

More information

MAE143A Signals & Systems - Homework 5, Winter 2013 due by the end of class Tuesday February 12, 2013.

MAE143A Signals & Systems - Homework 5, Winter 2013 due by the end of class Tuesday February 12, 2013. MAE43A Signals & Systems - Homework 5, Winter 23 due by the end of class Tuesday February 2, 23. If left under my door, then straight to the recycling bin with it. This week s homework will be a refresher

More information

Overview of the Seminar Topic

Overview of the Seminar Topic Overview of the Seminar Topic Simo Särkkä Laboratory of Computational Engineering Helsinki University of Technology September 17, 2007 Contents 1 What is Control Theory? 2 History

More information

Professional Portfolio Selection Techniques: From Markowitz to Innovative Engineering

Professional Portfolio Selection Techniques: From Markowitz to Innovative Engineering Massachusetts Institute of Technology Sponsor: Electrical Engineering and Computer Science Cosponsor: Science Engineering and Business Club Professional Portfolio Selection Techniques: From Markowitz to

More information

FREQUENCY-RESPONSE DESIGN

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

More information

ẋ n = f n (x 1,...,x n,u 1,...,u m ) (5) y 1 = g 1 (x 1,...,x n,u 1,...,u m ) (6) y p = g p (x 1,...,x n,u 1,...,u m ) (7)

ẋ n = f n (x 1,...,x n,u 1,...,u m ) (5) y 1 = g 1 (x 1,...,x n,u 1,...,u m ) (6) y p = g p (x 1,...,x n,u 1,...,u m ) (7) EEE582 Topical Outline A.A. Rodriguez Fall 2007 GWC 352, 965-3712 The following represents a detailed topical outline of the course. It attempts to highlight most of the key concepts to be covered and

More information

ECE 3793 Matlab Project 3 Solution

ECE 3793 Matlab Project 3 Solution ECE 3793 Matlab Project 3 Solution Spring 27 Dr. Havlicek. (a) In text problem 9.22(d), we are given X(s) = s + 2 s 2 + 7s + 2 4 < Re {s} < 3. The following Matlab statements determine the partial fraction

More information