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

Size: px
Start display at page:

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

Transcription

1 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/ / 33

2 Tentative schedule # Date Topic 1 Sept. 22 Introduction, Signals and Systems 2 Sept. 29 Modeling, Linearization 3 Oct. 6 Analysis 1: Time response, Stability 4 Oct. 13 Analysis 2: Diagonalization, Modal coordinates. 5 Oct. 20 Transfer functions 1: Definition and properties 6 Oct. 27 Transfer functions 2: Poles and Zeros 7 Nov. 3 Analysis of feedback systems: internal stability, root locus 8 Nov. 10 Frequency response 9 Nov. 17 Analysis of feedback systems 2: the Nyquist condition 10 Nov. 24 Specifications for feedback systems 11 Dec. 1 Loop Shaping 12 Dec. 8 PID control 13 Dec. 15 Implementation issues 14 Dec. 22 Robustness E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

3 Today s learning objectives Recognize different ways of writing transfer functions, and why. Graphical computation of g(s). Poles and their effects on the response. Zeros and their effects on the response. Zeros and derivative action Effects of non-minimum-phase zeros E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

4 Recap We know that the time response of a causal LTI system with state-space model (A, B, C, D) is t y(t) = Ce At x(0) + Ce A(t τ) Bu(τ) dτ + Du(t). 0 Assuming all the eigenvalues of A have negative real part (i.e., the system is stable), the steady-state response to an input of the form u(t) = e st is y ss (t) = G(s)e st where G(s) = C(sI A) 1 B + D. E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

5 Different ways to write transfer functions For causal, finite-dimensional SISO LTI systems, transfer functions take the form of proper rational functions, e.g., g(s) = N(s) D(s) = b ns n + b n 1 s n 1 + b n 2 s n b 0 s n + a n 1 s n a 0, There are other forms that are convenient. The first one, called the partial fraction expansion, is useful to compute transient responses, and to assess how much different modes contribute to the response. It has the form: g(s) = r 1 + r r n + r 0, s p 1 s p 2 s p n where p 1,..., p n are the poles, i.e., the roots of the characteristic polynomial det(si A), i.e., the eigenvalues of A. The numbers r 0,..., r n are called the residues. There are many ways to compute the residues, we will look at one later today. E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

6 Different ways to write transfer functions There are also slightly different ways to factorize the polynomials at the numerator and denominator: Root-locus form: This is useful to compute the value of g(s) by hand, and to use control design techniques like the root locus (next week) g(s) = k rl (s z 1)(s z 2)... (s z m) (s p 1)(s p 2)... (s p n) Bode form: This is useful to use control design techniques like the Bode plot (in 2-3 weeks) ( s z g(s) = k 1 + 1)( s z 2 + 1)... ( s Bode ( s p 1 + 1)( s p 2 z m + 1) + 1)... ( s p n + 1) In the above formulas, z 1,..., z n are called the zeros of g(s), and are the roots of the numerator. E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

7 Example Computing g(s) via factorization Consider an LTI system with transfer function s + 1 g(s) = 2 s 3 + 4s 2 + 6s + 4 What is the steady-state response to an input u(t) = sin(t) (i.e., s = j)? We know that y ss (t) = g(j) sin(t + g(j)) We could substitute s j in the transfer function and compute g(j), but that is not a nice/insightful calculation. Let s use graphical methods instead. Factorize the transfer function, and write in the root-locus form: with some creative eyeballing (or using matlab) we can find that the 3 poles of g(s) are at { 2, 1 ± j}. s + 1 g(s) = 2 (s + 2)(s j)(s + 1 j). E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

8 Computing the magnitude of g(s) Since a b = a b, we can write s + 1 g(s) = 2 s + 2 s j s + 1 j Graphically, s p is the length of the vector from p to s. E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

9 Computing the magnitude of g(s) Im j Re From the graph, we find: 2 g(j) = 2 = E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

10 Computing the phase (argument) of g(s) Since (a b) = ( a) + ( b), we can write g(s) = (2) + (s + 1) (s + 2) (s j) (s + 1 j) Graphically, (s p) is angle formed by the vector from p to s with the real axis. E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

11 Computing the phase of g(s) Im j Re From the graph, we find: g(j) = arctan(1/2) arctan(2) + 0 = 45. E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

12 Putting the results together The steady-state response of an LTI system with transfer function s + 1 g(s) = 2 s 3 + 4s 2 + 6s + 4 to an input of the form u(t) = sin(t) is given by y ss (t) sin(t 45 ). 1 Linear Simulation Results 0.8 y Amplitude Time (seconds) E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

13 What is the effect of poles and zeros? E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

14 Transient response to special inputs In order to understand the effect of poles and zeros, it is useful to apply some other standard test inputs to a system to evaluate its transient behavior (vs. the steady-state one). Typical test inputs include: Unit impulse u(t) = δ(t). This is not really a function, but a mathematical construct such that ε δ(t) dt = 1 for any ε > 0. 0 In particular, t f (τ)δ(τ) dτ = f (0), for any t > 0. 0 Unit step input u(t) = 1, for t 0. (Note this is the same as u(t) = e 0t.) Other less used test inputs include the unit ramp u(t) = t and higher order ramps. E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

15 Impulse response Assume D = 0, x(0) = 0, and u(t) = δ(t). Then:. y imp (t) = t 0 Ce A(t τ) Bδ(τ) dτ = Ce At B The impulse response is the same as the response to an initial condition x(0) = B. Remember: the initial-condition response is given by simple exponentials for real eigenvalues, and sinusoids with exponentially-changing magnitude for complex-conjugate eigenvalues. E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

16 Unit step response Assume D = 0, x(0) = 0, u(t) = 1 = e 0t, for t 0, and that A is invertible. Then: y step (t) = t 0 Ce A(t τ) B dτ = CA 1 B + CA 1 e At B The steady-state response is given by y ss (t) = G(0) = CA 1 B. For a scalar system, the step response then is simply computed as y step (t) = y ss (t)(1 e at ), i.e., the step response is the steady-state response minus the scaled impulse response. The impulse response totally defines the response of a system (it is in fact the inverse Laplace transform of the transfer function)! E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

17 First-order system A system with state-space model (A = a, B = b, C = c, D = 0) The transfer function is with r = bc. g(s) = r s + a The response to an unit impulse (or to an initial condition x(0) = b) has the form y(t) = re at. E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

18 Higher-order system If we write the partial fraction expansion of g(s), assuming no repeated poles, we get g(s) = r 1 + r r n. s p 1 s p 2 s p n The response to an impulse will then be y(t) = r 1 e p1t + r 2 e p2t +... r n e pnt. The effect of the poles is then clear: each pole p i generates a term of the form e p i t in the impulse response (and step response, etc.) As we know, these are simple exponentials if the pole p i is real, and are sinusoids with exponentially-changing amplitude for complex-conjugate pole pairs. E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

19 Response shapes as function of pole location Im Re Each pole p i = σ i + jω i with residue r i determines a term of the impulse response. Each term s magnitude is bounded by r i e σ i t and oscillates at frequency ω i. E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

20 Effects of zeros on the response How can we compute the residues r i? A convenient approach is the cover-up method. For a non-repeated pole p i this takes the form r i = lim s pi (s p i )g(s) which in practice means remove the factor (s p i ) from the denominator and compute g(p i ) only considering the other terms. While the exponents in the terms of the response only depend on the poles p i, the residues are affected by the zeros z i. E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

21 Example Consider g(s) = 1 (s + 1)(s j)(s + 1 j). Im Re Using the cover-up method we get g(s) = 1 s /2 s j + 1/2 s + 1 j. E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

22 Example - impulse response Impulse Response p1 p2,p3 combined 0.6 Amplitude Time (seconds) E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

23 Example adding a zero near a pole Consider g(s) = s ε (s + 1)(s j)(s + 1 j). Im Re Using the cover-up method we get g(s) ε s /2j s j + 1/2j s + 1 j. E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

24 Example - impulse response 0.35 Impulse Response Amplitude Time (seconds) A zero can reduce the residue (i.e., the effect) of a nearby pole. E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

25 Pole-zero cancellation What if a zero matches a pole exactly? g(s) = s + 1 (s + 1)(s j)(s + 1 j) = 1 (s j)(s + 1 j). One of the poles has been cancelled by the zero. Effectively its residue is zero, i.e., g(s) = 0 s /2j s j + 1/2j s + 1 j. Recall from the modal (diagonal) form that the residue is also given by r i = b i c i ; if the residue is zero, the i-th mode is either uncontrollable, unobservable, or both. This is ok if the i-th mode (i.e., p i ) is stable, but a big problem if it is unstable. Avoid unstable pole-zero cancellation! E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

26 More effects of zeros... E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

27 Integrator u(t) y(t) = t u(τ) dτ If the input is u(t) = e st, then the output will be y(t) = 1 s est. Hence, the transfer function of an integrator is G(s) = 1 s. You can verify from the state-space model: ẋ(t) = u(t), y(t) = x(t) A = 0, B = 1, C = 1, D = 0, and G(s) = C(sI A) 1 B + D = s 1. E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

28 Differentiator u(t) d dt y(t) = du(t) dt If the input is u(t) = e st, then the output will be y(t) = se st. Hence, the transfer function of an integrator is G(s) = s. Note that you cannot obtain this from a state-space model, because a differentiator is not a causal operator! E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

29 Zeros as derivative action If we have a transfer function g(s) = (s + z) g(s), we can decompose it into. g(s) = z g(s) + s g(s) If the impulse response of g(s) is given by ỹ(t), and the impulse response of g(s) is y(t), then remembering that s is the transfer function of a differentiator, we can write y(t) = zỹ(t) + ỹ(t). In other words, the zero is effectively adding a derivative term to the output. This typically has an anticipatory effect. E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

30 With and without a zero / derivative 1 Impulse Response 0.35 Impulse Response 0.8 p1 p2,p3 combined Amplitude 0.4 Amplitude Time (seconds) Time (seconds) E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

31 Non-minimum-phase zeros We know that poles with positive real part result in an unstable system. (The output diverges over time.) What happens when zeros have positive real part? The stability of the system is preserved (since the growth/decay of the terms in the response is not affected by the zeros only the respective residues) However, a zero in the right half plane effectively means a negative derivative action. This is the opposite of anticipatory indeed the output will tend to move in the wrong direction initially. These are called non-minimum phase zeros and are typically very bad news for control engineers, they make our work much harder. (Typically the presence of non-minimum-phase zeros depends on the choice of the output to make your life easier, choose another output and/or move the sensors!) E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

32 Minimum-phase vs. non-minimum-phase zeros Impulse Response no zero z = -1 z= Amplitude Time (seconds) E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

33 Today s learning objectives Recognize different ways of writing transfer functions, and why. Graphical computation of g(s). Poles and their effects on the response. Zeros and their effects on the response. Zeros and derivative action Effects of non-minimum-phase zeros E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/ / 33

### 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:

### Control Systems I. Lecture 5: Transfer Functions. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich

Control Systems I Lecture 5: Transfer Functions Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 20, 2017 E. Frazzoli (ETH) Lecture 5: Control Systems I 20/10/2017

### Control Systems I. Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback. Readings: Emilio Frazzoli

Control Systems I Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 13, 2017 E. Frazzoli (ETH)

### Control Systems I Lecture 10: System Specifications

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

### Control Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Guzzella 9.1-3, Emilio Frazzoli

Control Systems I Lecture 7: Feedback and the Root Locus method Readings: Guzzella 9.1-3, 13.3 Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 3, 2017 E. Frazzoli (ETH)

### Control Systems I. Lecture 9: The Nyquist condition

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

### Control Systems I. Lecture 9: The Nyquist condition

Control Systems I Lecture 9: The Nyquist condition Readings: Åstrom and Murray, Chapter 9.1 4 www.cds.caltech.edu/~murray/amwiki/index.php/first_edition Jacopo Tani Institute for Dynamic Systems and Control

### Control Systems I. Lecture 2: Modeling. Suggested Readings: Åström & Murray Ch. 2-3, Guzzella Ch Emilio Frazzoli

Control Systems I Lecture 2: Modeling Suggested Readings: Åström & Murray Ch. 2-3, Guzzella Ch. 2-3 Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich September 29, 2017 E. Frazzoli

### Control Systems I. Lecture 1: Introduction. Suggested Readings: Åström & Murray Ch. 1, Guzzella Ch. 1. Emilio Frazzoli

Control Systems I Lecture 1: Introduction Suggested Readings: Åström & Murray Ch. 1, Guzzella Ch. 1 Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich September 22, 2017 E. Frazzoli

### Control Systems I. Lecture 2: Modeling and Linearization. Suggested Readings: Åström & Murray Ch Jacopo Tani

Control Systems I Lecture 2: Modeling and Linearization Suggested Readings: Åström & Murray Ch. 2-3 Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich September 28, 2018 J. Tani, E.

### 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...........

### 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 =

### 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

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

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

### 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

### Plan of the Lecture. Goal: wrap up lead and lag control; start looking at frequency response as an alternative methodology for control systems design.

Plan of the Lecture Review: design using Root Locus; dynamic compensation; PD and lead control Today s topic: PI and lag control; introduction to frequency-response design method Goal: wrap up lead and

### 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

### 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

### ECEN 420 LINEAR CONTROL SYSTEMS. Lecture 2 Laplace Transform I 1/52

1/52 ECEN 420 LINEAR CONTROL SYSTEMS Lecture 2 Laplace Transform I Linear Time Invariant Systems A general LTI system may be described by the linear constant coefficient differential equation: a n d n

### 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

### Systems Analysis and Control

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

### Course Summary. The course cannot be summarized in one lecture.

Course Summary Unit 1: Introduction Unit 2: Modeling in the Frequency Domain Unit 3: Time Response Unit 4: Block Diagram Reduction Unit 5: Stability Unit 6: Steady-State Error Unit 7: Root Locus Techniques

### Control Systems Design

ELEC4410 Control Systems Design Lecture 13: Stability Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 13: Stability p.1/20 Outline Input-Output

### Transfer function and linearization

Transfer function and linearization Daniele Carnevale Dipartimento di Ing. Civile ed Ing. Informatica (DICII), University of Rome Tor Vergata Corso di Controlli Automatici, A.A. 24-25 Testo del corso:

### 6.241 Dynamic Systems and Control

6.241 Dynamic Systems and Control Lecture 12: I/O Stability Readings: DDV, Chapters 15, 16 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology March 14, 2011 E. Frazzoli

### Lecture 1: Feedback Control Loop

Lecture : Feedback Control Loop Loop Transfer function The standard feedback control system structure is depicted in Figure. This represend(t) n(t) r(t) e(t) u(t) v(t) η(t) y(t) F (s) C(s) P (s) Figure

### ECE504: Lecture 9. D. Richard Brown III. Worcester Polytechnic Institute. 04-Nov-2008

ECE504: Lecture 9 D. Richard Brown III Worcester Polytechnic Institute 04-Nov-2008 Worcester Polytechnic Institute D. Richard Brown III 04-Nov-2008 1 / 38 Lecture 9 Major Topics ECE504: Lecture 9 We are

### Systems Analysis and Control

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

### 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

### 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

### 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

### Explanations and Discussion of Some Laplace Methods: Transfer Functions and Frequency Response. Y(s) = b 1

Engs 22 p. 1 Explanations Discussion of Some Laplace Methods: Transfer Functions Frequency Response I. Anatomy of Differential Equations in the S-Domain Parts of the s-domain solution. We will consider

### CDS 101: Lecture 4.1 Linear Systems

CDS : Lecture 4. Linear Systems Richard M. Murray 8 October 4 Goals: Describe linear system models: properties, eamples, and tools Characterize stability and performance of linear systems in terms of eigenvalues

### CDS 101/110: Lecture 3.1 Linear Systems

CDS /: Lecture 3. Linear Systems Goals for Today: Describe and motivate linear system models: Summarize properties, examples, and tools Joel Burdick (substituting for Richard Murray) jwb@robotics.caltech.edu,

### 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

### Recitation 11: Time delays

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

### 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

### 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

### 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...................

### Control Systems. Frequency domain analysis. L. Lanari

Control Systems m i l e r p r a in r e v y n is o Frequency domain analysis L. Lanari outline introduce the Laplace unilateral transform define its properties show its advantages in turning ODEs to algebraic

### 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

### 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

### 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

### 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

### Design Methods for Control Systems

Design Methods for Control Systems Maarten Steinbuch TU/e Gjerrit Meinsma UT Dutch Institute of Systems and Control Winter term 2002-2003 Schedule November 25 MSt December 2 MSt Homework # 1 December 9

### GATE EE Topic wise Questions SIGNALS & SYSTEMS

www.gatehelp.com GATE EE Topic wise Questions YEAR 010 ONE MARK Question. 1 For the system /( s + 1), the approximate time taken for a step response to reach 98% of the final value is (A) 1 s (B) s (C)

### 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 +

### Learn2Control Laboratory

Learn2Control Laboratory Version 3.2 Summer Term 2014 1 This Script is for use in the scope of the Process Control lab. It is in no way claimed to be in any scientific way complete or unique. Errors should

### Laplace Transforms and use in Automatic Control

Laplace Transforms and use in Automatic Control P.S. Gandhi Mechanical Engineering IIT Bombay Acknowledgements: P.Santosh Krishna, SYSCON Recap Fourier series Fourier transform: aperiodic Convolution integral

### 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

### 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

### Exam. 135 minutes + 15 minutes reading time

Exam January 23, 27 Control Systems I (5-59-L) Prof. Emilio Frazzoli Exam Exam Duration: 35 minutes + 5 minutes reading time Number of Problems: 45 Number of Points: 53 Permitted aids: Important: 4 pages

### EE C128 / ME C134 Fall 2014 HW 6.2 Solutions. HW 6.2 Solutions

EE C28 / ME C34 Fall 24 HW 6.2 Solutions. PI Controller For the system G = K (s+)(s+3)(s+8) HW 6.2 Solutions in negative feedback operating at a damping ratio of., we are going to design a PI controller

### Control Systems I. Lecture 1: Introduction. Suggested Readings: Åström & Murray Ch. 1. Jacopo Tani

Control Systems I Lecture 1: Introduction Suggested Readings: Åström & Murray Ch. 1 Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich September 21, 2018 J. Tani, E. Frazzoli (ETH)

### 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

### 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

### Identification Methods for Structural Systems

Prof. Dr. Eleni Chatzi System Stability 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 be defined from

### 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,

### Intro. Computer Control Systems: F8

Intro. Computer Control Systems: F8 Properties of state-space descriptions and feedback Dave Zachariah Dept. Information Technology, Div. Systems and Control 1 / 22 dave.zachariah@it.uu.se F7: Quiz! 2

### Topic # /31 Feedback Control Systems

Topic #7 16.30/31 Feedback Control Systems State-Space Systems What are the basic properties of a state-space model, and how do we analyze these? Time Domain Interpretations System Modes Fall 2010 16.30/31

### EE3CL4: Introduction to Linear Control Systems

1 / 17 EE3CL4: Introduction to Linear Control Systems Section 7: McMaster University Winter 2018 2 / 17 Outline 1 4 / 17 Cascade compensation Throughout this lecture we consider the case of H(s) = 1. We

### 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

### 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

### Some solutions of the written exam of January 27th, 2014

TEORIA DEI SISTEMI Systems Theory) Prof. C. Manes, Prof. A. Germani Some solutions of the written exam of January 7th, 0 Problem. Consider a feedback control system with unit feedback gain, with the following

### CYBER EXPLORATION LABORATORY EXPERIMENTS

CYBER EXPLORATION LABORATORY EXPERIMENTS 1 2 Cyber Exploration oratory Experiments Chapter 2 Experiment 1 Objectives To learn to use MATLAB to: (1) generate polynomial, (2) manipulate polynomials, (3)

### ECE 388 Automatic Control

Controllability and State Feedback Control Associate Prof. Dr. of Mechatronics Engineeering Çankaya University Compulsory Course in Electronic and Communication Engineering Credits (2/2/3) Course Webpage:

### 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

### Module 09 From s-domain to time-domain From ODEs, TFs to State-Space Modern Control

Module 09 From s-domain to time-domain From ODEs, TFs to State-Space Modern Control Ahmad F. Taha EE 3413: Analysis and Desgin of Control Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/

### The Laplace Transform

The Laplace Transform Syllabus ECE 316, Spring 2015 Final Grades Homework (6 problems per week): 25% Exams (midterm and final): 50% (25:25) Random Quiz: 25% Textbook M. Roberts, Signals and Systems, 2nd

### 1 (20 pts) Nyquist Exercise

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

### Poles, Zeros and System Response

Time Response After the engineer obtains a mathematical representation of a subsystem, the subsystem is analyzed for its transient and steady state responses to see if these characteristics yield the desired

### EL2520 Control Theory and Practice

So far EL2520 Control Theory and Practice r Fr wu u G w z n Lecture 5: Multivariable systems -Fy Mikael Johansson School of Electrical Engineering KTH, Stockholm, Sweden SISO control revisited: Signal

### 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

### Table of Laplacetransform

Appendix Table of Laplacetransform pairs 1(t) f(s) oct), unit impulse at t = 0 a, a constant or step of magnitude a at t = 0 a s t, a ramp function e- at, an exponential function s + a sin wt, a sine fun

### Exam. 135 minutes, 15 minutes reading time

Exam August 15, 2017 Control Systems I (151-0591-00L) Prof Emilio Frazzoli Exam Exam Duration: 135 minutes, 15 minutes reading time Number of Problems: 44 Number of Points: 52 Permitted aids: Important:

### 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

### Control Systems. System response. L. Lanari

Control Systems m i l e r p r a in r e v y n is o System response L. Lanari Outline What we are going to see: how to compute in the s-domain the forced response (zero-state response) using the transfer

### Systems and Control Theory Lecture Notes. Laura Giarré

Systems and Control Theory Lecture Notes Laura Giarré L. Giarré 2017-2018 Lesson 7: Response of LTI systems in the transform domain Laplace Transform Transform-domain response (CT) Transfer function Zeta

### 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

### 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

### 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

### 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

### Frequency domain analysis

Automatic Control 2 Frequency domain analysis Prof. Alberto Bemporad University of Trento Academic year 2010-2011 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011

### Raktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Frequency Response-Design Method

.. AERO 422: Active Controls for Aerospace Vehicles Frequency Response- Method Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. ... Response to

### Dynamic circuits: Frequency domain analysis

Electronic Circuits 1 Dynamic circuits: Contents Free oscillation and natural frequency Transfer functions Frequency response Bode plots 1 System behaviour: overview 2 System behaviour : review solution

### Model Uncertainty and Robust Stability for Multivariable Systems

Model Uncertainty and Robust Stability for Multivariable Systems ELEC 571L Robust Multivariable Control prepared by: Greg Stewart Devron Profile Control Solutions Outline Representing model uncertainty.

### A system that is both linear and time-invariant is called linear time-invariant (LTI).

The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Lecture Notes: Time, Frequency & Transform Domains February 28, 2012 Signals & Systems Signals are mapped

### Introduction. Performance and Robustness (Chapter 1) Advanced Control Systems Spring / 31

Introduction Classical Control Robust Control u(t) y(t) G u(t) G + y(t) G : nominal model G = G + : plant uncertainty Uncertainty sources : Structured : parametric uncertainty, multimodel uncertainty Unstructured

### LTI Systems (Continuous & Discrete) - Basics

LTI Systems (Continuous & Discrete) - Basics 1. A system with an input x(t) and output y(t) is described by the relation: y(t) = t. x(t). This system is (a) linear and time-invariant (b) linear and time-varying

### 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.)

### Systems Analysis and Control

Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 13: Root Locus Continued Overview In this Lecture, you will learn: Review Definition of Root Locus Points on the Real Axis

### SIGNALS AND SYSTEMS LABORATORY 4: Polynomials, Laplace Transforms and Analog Filters in MATLAB

INTRODUCTION SIGNALS AND SYSTEMS LABORATORY 4: Polynomials, Laplace Transforms and Analog Filters in MATLAB Laplace transform pairs are very useful tools for solving ordinary differential equations. Most

### 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

### Process Control & Instrumentation (CH 3040)

First-order systems Process Control & Instrumentation (CH 3040) Arun K. Tangirala Department of Chemical Engineering, IIT Madras January - April 010 Lectures: Mon, Tue, Wed, Fri Extra class: Thu A first-order

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

Dr. J. Tani, Prof. Dr. E. Frazzoli 5-059-00 Control Systems I (Autumn 208) Exercise Set 0 Topic: Specifications for Feedback Systems Discussion: 30.. 208 Learning objectives: The student can grizzi@ethz.ch,

### Andrea Zanchettin Automatic Control AUTOMATIC CONTROL. Andrea M. Zanchettin, PhD Spring Semester, Linear systems (frequency domain)

1 AUTOMATIC CONTROL Andrea M. Zanchettin, PhD Spring Semester, 2018 Linear systems (frequency domain) 2 Motivations Consider an LTI system Thanks to the Lagrange s formula we can compute the motion of

### Lecture 4: Analysis of MIMO Systems

Lecture 4: Analysis of MIMO Systems Norms The concept of norm will be extremely useful for evaluating signals and systems quantitatively during this course In the following, we will present vector norms

### EE Control Systems LECTURE 9

Updated: Sunday, February, 999 EE - Control Systems LECTURE 9 Copyright FL Lewis 998 All rights reserved STABILITY OF LINEAR SYSTEMS We discuss the stability of input/output systems and of state-space