Collocated versus non-collocated control [H04Q7]

Size: px
Start display at page:

Download "Collocated versus non-collocated control [H04Q7]"

Transcription

1 Collocated versus non-collocated control [H04Q7] Jan Swevers September

2 Contents Some concepts of structural dynamics Collocated versus non-collocated control Summary This lecture is based on parts taken from Vibration control of active structures (2nd edition) by A. Preumont, Kluwer Academic Publishers

3 Some concepts of structural dynamics Equation of motion of a lumped parameter system (i.e. discrete system) Vibration modes Modal decomposition Transfer function of collocated systems

4 Equation of motion of a lumped parameter system Consider the following system: Equations of motion: M m m ẍ 1 ẍ 2 ẍ 3 + c c 0 c 2c c 0 c c ẋ 1 ẋ 2 ẋ 3 + k k 0 k 2k k 0 k k x 1 x 2 x 3 = f 0 0

5 The general form of the equation of motion for a (non-gyroscopic) discrete flexible structure with a finite number of DOF s: Mẍ + Cẋ + Kx = f M, C, and K: mass, damping and stiffness matrices (symmetric and semi positive definite)

6 Vibration modes Free response of the undamped system : Mẍ + Kx = 0 If one tries a solution of the form x = φ i e jω it, φ i and ω i must satisfy the following eigenvalue problem: Properties: (K ω 2 i M)φ i = 0 the eigenvalues ω 2 i are real and non negative ω i are the natural frequencies φ i are the corresponding modes

7 Define the matrix of all mode shapes as: [ ] Φ = φ 1 φ 2... φ n Due to orthogonality conditions: Φ T MΦ = diag(µ i ) Φ T KΦ = diag(µ i ω 2 i ) µ i are the modal masses (modes can be scaled such that the µ i s are equal to one).

8 Modal decomposition Let us perform a change of variables from the physical coordinates x to modal coordinates z: x = Φz yielding: MΦ z + CΦż + KΦz = f Left multiplying by Φ T, and using orthogonality relationships: diag(µ i ) z + Φ T CΦż + diag(µ i ω 2 i )z = Φ T f If matrix Φ T CΦ is diagonal, the damping is said to be normal: the modal damping is defined as: Φ T CΦ = diag(2ξ i µ i ω i )

9 In case of modal damping, the modal equations are decoupled: with z + 2ξΩż + (Ω 2 )z = µ 1 Φ T f ξ = diag(ξ i ), Ω = diag(ω i ), µ = diag(µ i ) Frequency domain interpretation: dynamic flexibility matrix or yielding: or X = [ ω 2 M + jωc + K ] 1 F = G(ω)F { } 1 Z = diag µ i (ωi 2 Φ T F ω2 + 2jξ i ω i ω) { } 1 X = ΦZ = Φdiag µ i (ωi 2 Φ T F ω2 + 2jξ i ω i ω) G(ω) = with φ i the i-th column of φ. n i=1 φ i φ T i µ i (ω 2 i ω2 + 2jξ i ω i ω)

10 If the structure has NO rigid body modes, we can evaluate the system for ω = 0: n G(0) = K 1 φ = i φ T i µ i ωi 2 i=1 If we consider only frequencies ω < ω b << ω m, we get: G(ω) m i=1 m i=1 φ i φ T n i µ i (ωi 2 ω2 + 2jξ i ω i ω) + m+1 φ i φ T i µ i ω 2 i φ i φ T i µ i (ω 2 i ω2 + 2jξ i ω i ω) + K 1 m 1 φ i φ T i µ i ω 2 i The static contribution of the high frequency modes to the flexibility matrix is called the residual mode, denoted by R.

11 Structure with r rigid bodies (ω i = 0, for i = 1,...,r): G(ω) r i=1 T φ i φ m i µ i ω 2 + i=r+1 φ i φ T i µ i (ω 2 i ω2 + 2jξ i ω i ω) + R with R = n m+1 φ i φ T i µ i ω 2 i

12 Transfer function of collocated systems What is a collocated system? Dynamics are described by the diagonal elements of the dynamic flexibility matrix (for undamped system): G kk (ω) = r i=1 φ 2 i (k) m µ i ω 2 + i=r+1 φ 2 i (k) µ i (ω 2 i ω2 ) + R kk G kk is real and : dg kk (ω 2 ) dω 2 0 φ i (k) is the k-th element of the i-th column of matrix φ i.

13 In contrast to the resonance frequencies, the anti-resonance frequencies do depend on the actuator location. There will be just one anti-resonance between two consecutive resonances. For non-collocated actuator - sensor systems, the numerators of the various terms in the modal expansion of G kl (ω) become φ i (k)φ i (l): they can be positive or negative, such that the above properties are lost.

14 Pole/zero pattern of a structure with collocated actuator and sensor (a) undamped, (b) lightly damped Nyquist and Bode diagram of a lightly damped collocated system

15 each flexible mode introduces a circle in the Nyquist diagram more or less centered on the imaginary axis which is intersected at ω = ω i and ω = ω oi.

16 Collocated versus non-collocated control Collocated actuator and sensor pairs for lightly damped flexible structures leads to alternating poles and zeros near the imaginary axis. SISO control systems based on this: very robust (root locus techniques) This property does not hold for non-collocated control: root locus may experience severe alterations for small parameter changes, e.g. pole-zero flipping

17 Pole-zero flipping Root locus: locus of the solutions s of the closed-loop characteristic equation: when g goes from 0 to 1 + gg(s)h(s) = 0 G(s)H(s) = k m i=1 (s z i) n i=1 (s p i) Any point P on the locus is such that: m φ i i=1 n ψ i = l 360 i=1 with φ i and ψ i the angles of the vectors joining the zeros z i and poles p i to P respectively.

18 Departure angles from poles and arrival angles at zeros before and after zero-pole flipping: Pole-zero flipping may occur in two different ways: There are compensator zeros near to system poles (notch filter): if the actual poles of the system are different from those assumed in the compensator design, a pole-zero flipping may occur. Some actuator/sensor configurations may produce flipping within the system alone, for small parameter changes. This is not possible if the actuator and sensor are collocated.

19 Collocated control G 1 (s) = Y (s) F(s) = s 2 + 2ξω 0 s + ω 2 0 Ms 2 (s 2 + (1 + µ)(2ξω 0 s + ω 2 0 )) G 2 (s) = D(s) F(s) = 2ξω 0 s + ω 2 0 Ms 2 (s 2 + (1 + µ)(2ξω 0 s + ω 2 0 )) ω 2 0 Ms 2 (s 2 + (1 + µ)(2ξω 0 s + ω 2 0 )) with ω 2 0 = k m, µ = m M, 2ξω 0 = b m

20 The approximation for G 2 (s) is valid for low damping ξ << 1: the far away zero will not influence the closed-loop response. When the mass ratio µ is small, numerator and denominator of G 1 (s) are almost equal (except for Ms 2 ): pole-zero cancellation

21 Consider a collocated lead compensator: H(s) = g Ts + 1, (α < 1) αts + 1

22 The lead compensator always increases the damping of the flexible mode For several flexible modes: always as many flexible pole-pairs as flexible zero-pairs pole-zero excess remains 2: angles of asymptotes remains 90 deg lead compensator increases the damping of all flexible modes, but especially those having their natural frequency between the pole and the zero of the compensator

23 Consider the same lead compensator, applied in non-collocated control configuration The pole-zero excess is 3: the flexible modes are heading towards the asymptotes at ±60 deg in the right half plane

24 Even with small bandwidth, the gain margin is extremely small.

25 Pole-zero flipping in the structure Consider the first system: M m m ẍ 1 ẍ 2 ẍ 3 + c c 0 c 2c c 0 c c ẋ 1 ẋ 2 ẋ 3 + k k 0 k 2k k 0 k k x 1 x 2 x 3 = f 0 0 In Laplace form (with c = 0): Ms 2 + k k 0 k ms 2 + 2k k 0 k ms 2 + k X 1 X 2 X 3 = F 0 0

26 This yields: G 1 (s) = X 1(s) F(s) = s 4 + 3ω 2 0s 2 + ω 4 0 Ms 2 (s 4 + (3 + µ)ω 2 0 s2 + (1 + 2µ)ω 4 0 ) G 2 (s) = X 2(s) F(s) = ω 2 0(s 2 + ω 2 0) Ms 2 (s 4 + (3 + µ)ω 2 0 s2 + (1 + 2µ)ω 4 0 ) G 3 (s) = X 3(s) F(s) = ω 4 0 Ms 2 (s 4 + (3 + µ)ω 2 0 s2 + (1 + 2µ)ω 4 0 ) with ω 2 0 = k/m and µ = m/m.

27 Properties Poles: p 2 ω0 2 = (3 + µ) ± 5 2µ + µ 2 2 G 1 has two pairs of zeros, independent of the mass ratio For µ = 0, the poles and zeros of G 1 cancel each other. For µ > 0, the poles and zeros alternate on the imaginary axis (collocated system). G 2 has one pair of zeros, G 3 has no zeros.

28 Evolution of the poles with the mass ratio µ: The zeros do not depend on µ. There is a pole-zero flipping in G 2 at µ = 1. G 2 (s) ω 2 0(s 2 + ω 2 0) Ms 2 (s 2 + ω 2 0 (1+µ 2 ))(s2 + ω 2 0 (5+µ 2 ))

29

30 Effect on Bode plot : A pole-zero flipping near the imaginary axis produces a phase change of 360. m i=1 G(jω)H(jω) = k (jω z i) n i=1 (jω p i) The phase of GH for a specific value of jω is given by: m i=1 φ i n i=1 ψ i, with

31 Relation to the mode shapes Evolution of the zeros of a simple supported beam with a point force actuator at 0.1l

32 As the sensor moves away from the actuator, the zeros migrate along the imaginary axis. When the sensor reaches 0.2l, which is the nodal point of mode 5, the fourth zero becomes identical to jω 5. The third zero crosses jω 4 at 0.25l, node of mode 4. Etc...

33 Relation to the mode shapes: non-minimum phase systems Evolution of the zeros of a beam when the sensor moves away from the actuator

34 While the imaginary zeros migrate along the imaginary axis, every pair of zeros that disappears at infinity, reappears symmetrically at infinity on the real axis and moves towards the origin. Right-half plane zeros are non-minimum phase zeros. Non-minimum phase zeros do not cause difficulties if they lie well outside the desired bandwidth of the closed-loop system. They put severe restrictions on the control system if they interfere with the bandwidth.

35 Summary Difference between collocated and non-collocated systems. Influence on stability and robustness properties, pole/zero location. These properties play a role in feedback and feedforward control design.

36 Assignment 1: Collocated versus Non-collocated control and effect of pole-zero flipping Control of three mass-system of slide 3 Data: M = 1kg, ω 0 = = 1rad/s, c = 0.005Ns/m. Input: Force on first mass k m Task: Design a lead-compensator with phase-margin larger than 30 o, and maximal bandwidth for following systems: System 1: µ = m M System 2: µ = m M System 3: µ = m M System 4: µ = m M = 1.2 and Output: position of mass 1 (collocated) = 1.2 and Output: position of mass 2 (non-collocated) = 0.5 and Output: position of mass 1 (collocated) = 0.5 and Output: position of mass 2 (non-collocated) Discuss the results based on bode-diagram and root-locus plot.

37 Assignment 2: Pole-zero flipping in simply supported beam Reproduce figure of slide 30 for the following beam: l = 1m, b = 0.02m, h = 0.005m, ρ = 7500kg/m 3, E = N/m 2 Position of actuator: x a = 0.1m Useful formulas: Moment of inertia: I = bh3 12 Eigenfrequencies: ωn 2 = (nπ) 4 EI ml 4 Mode shapes: Φ n (x) = sin ( nπx l Frequency response function: G(s) = i=1 and x s the position of the sensor Discuss the results ) Φ i (x a )Φ i (x s ) µ(s 2 +ω 2 i ) with µ = ml 2 Hint: Calculate zeros of G(s), based on approximation with finite number of modes

Chapter 2 SDOF Vibration Control 2.1 Transfer Function

Chapter 2 SDOF Vibration Control 2.1 Transfer Function Chapter SDOF Vibration Control.1 Transfer Function mx ɺɺ( t) + cxɺ ( t) + kx( t) = F( t) Defines the transfer function as output over input X ( s) 1 = G( s) = (1.39) F( s) ms + cs + k s is a complex number:

More information

Dynamics of Structures

Dynamics of Structures Dynamics of Structures Elements of structural dynamics Roberto Tomasi 11.05.2017 Roberto Tomasi Dynamics of Structures 11.05.2017 1 / 22 Overview 1 SDOF system SDOF system Equation of motion Response spectrum

More information

Active Control? Contact : Website : Teaching

Active Control? Contact : Website :   Teaching Active Control? Contact : bmokrani@ulb.ac.be Website : http://scmero.ulb.ac.be Teaching Active Control? Disturbances System Measurement Control Controler. Regulator.,,, Aims of an Active Control Disturbances

More information

Identification Methods for Structural Systems. Prof. Dr. Eleni Chatzi Lecture March, 2016

Identification Methods for Structural Systems. Prof. Dr. Eleni Chatzi Lecture March, 2016 Prof. Dr. Eleni Chatzi Lecture 4-09. March, 2016 Fundamentals Overview Multiple DOF Systems State-space Formulation Eigenvalue Analysis The Mode Superposition Method The effect of Damping on Structural

More information

Lecture 5: Frequency domain analysis: Nyquist, Bode Diagrams, second order systems, system types

Lecture 5: Frequency domain analysis: Nyquist, Bode Diagrams, second order systems, system types Lecture 5: Frequency domain analysis: Nyquist, Bode Diagrams, second order systems, system types Venkata Sonti Department of Mechanical Engineering Indian Institute of Science Bangalore, India, 562 This

More information

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

2.004 Dynamics and Control II Spring 2008

2.004 Dynamics and Control II Spring 2008 MT OpenCourseWare http://ocw.mit.edu.004 Dynamics and Control Spring 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Massachusetts nstitute of Technology

More information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 24: Compensation in the Frequency Domain Overview In this Lecture, you will learn: Lead Compensators Performance Specs Altering

More information

(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

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 486 Control Systems

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

More information

ME scope Application Note 28

ME scope Application Note 28 App Note 8 www.vibetech.com 3/7/17 ME scope Application Note 8 Mathematics of a Mass-Spring-Damper System INTRODUCTION In this note, the capabilities of ME scope will be used to build a model of the mass-spring-damper

More information

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

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

More information

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

MAE 142 Homework #5 Due Friday, March 13, 2009

MAE 142 Homework #5 Due Friday, March 13, 2009 MAE 142 Homework #5 Due Friday, March 13, 2009 Please read through the entire homework set before beginning. Also, please label clearly your answers and summarize your findings as concisely as possible.

More information

Dynamic circuits: Frequency domain analysis

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

More information

Table of Laplacetransform

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

More information

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 : 0. LS_D_ECIN_Control Systems_30078 Delhi Noida Bhopal Hyderabad Jaipur Lucnow Indore Pune Bhubaneswar Kolata Patna Web: E-mail: info@madeeasy.in Ph: 0-4546 CLASS TEST 08-9 ELECTRONICS ENGINEERING

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

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

Control Systems I. Lecture 9: The Nyquist condition

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

More information

1 (20 pts) Nyquist Exercise

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

More information

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

EECS C128/ ME C134 Final Wed. Dec. 14, am. Closed book. One page, 2 sides of formula sheets. No calculators.

EECS C128/ ME C134 Final Wed. Dec. 14, am. Closed book. One page, 2 sides of formula sheets. No calculators. Name: SID: EECS C128/ ME C134 Final Wed. Dec. 14, 211 81-11 am Closed book. One page, 2 sides of formula sheets. No calculators. There are 8 problems worth 1 points total. Problem Points Score 1 16 2 12

More information

OPTIMAL PPF CONTROLLER FOR MULTIMODAL VIBRATION SUPPRESSION

OPTIMAL PPF CONTROLLER FOR MULTIMODAL VIBRATION SUPPRESSION Engineering MECHANICS, Vol. 15, 2008, No. 3, p. 153 173 153 OPTIMAL PPF CONTROLLER FOR MULTIMODAL VIBRATION SUPPRESSION Štefan Fenik, Ladislav Starek* Positive Position Feedback (PPF) is one of the most

More information

Analysis of Discrete-Time Systems

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

More information

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

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

More information

Advanced Vibrations. Elements of Analytical Dynamics. By: H. Ahmadian Lecture One

Advanced Vibrations. Elements of Analytical Dynamics. By: H. Ahmadian Lecture One Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian ahmadian@iust.ac.ir Elements of Analytical Dynamics Newton's laws were formulated for a single particle Can be extended to

More information

Structural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations.

Structural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations. Outline of Multi-Degree-of-Freedom Systems Formulation of Equations of Motions. Newton s 2 nd Law Applied to Free Masses. D Alembert s Principle. Basic Equations of Motion for Forced Vibrations of Linear

More information

EECS C128/ ME C134 Final Wed. Dec. 15, am. Closed book. Two pages of formula sheets. No calculators.

EECS C128/ ME C134 Final Wed. Dec. 15, am. Closed book. Two pages of formula sheets. No calculators. Name: SID: EECS C28/ ME C34 Final Wed. Dec. 5, 2 8- am Closed book. Two pages of formula sheets. No calculators. There are 8 problems worth points total. Problem Points Score 2 2 6 3 4 4 5 6 6 7 8 2 Total

More information

Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams.

Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Outline of Continuous Systems. Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Vibrations of Flexible Strings. Torsional Vibration of Rods. Bernoulli-Euler Beams.

More information

Chapter 7. Digital Control Systems

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

More information

SAMPLE SOLUTION TO EXAM in MAS501 Control Systems 2 Autumn 2015

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

More information

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

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

More information

EC6405 - CONTROL SYSTEM ENGINEERING Questions and Answers Unit - I Control System Modeling Two marks 1. What is control system? A system consists of a number of components connected together to perform

More information

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

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

More information

Motion System Classes. Motion System Classes K. Craig 1

Motion System Classes. Motion System Classes K. Craig 1 Motion System Classes Motion System Classes K. Craig 1 Mechatronic System Design Integration and Assessment Early in the Design Process TIMING BELT MOTOR SPINDLE CARRIAGE ELECTRONICS FRAME PIPETTE Fast

More information

Software Engineering 3DX3. Slides 8: Root Locus Techniques

Software Engineering 3DX3. Slides 8: Root Locus Techniques Software Engineering 3DX3 Slides 8: Root Locus Techniques Dr. Ryan Leduc Department of Computing and Software McMaster University Material based on Control Systems Engineering by N. Nise. c 2006, 2007

More information

Analysis of Discrete-Time Systems

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

More information

Control for. Maarten Steinbuch Dept. Mechanical Engineering Control Systems Technology Group TU/e

Control for. Maarten Steinbuch Dept. Mechanical Engineering Control Systems Technology Group TU/e Control for Maarten Steinbuch Dept. Mechanical Engineering Control Systems Technology Group TU/e Motion Systems m F Introduction Timedomain tuning Frequency domain & stability Filters Feedforward Servo-oriented

More information

Homework 7 - Solutions

Homework 7 - Solutions Homework 7 - Solutions Note: This homework is worth a total of 48 points. 1. Compensators (9 points) For a unity feedback system given below, with G(s) = K s(s + 5)(s + 11) do the following: (c) Find the

More information

AMME3500: System Dynamics & Control

AMME3500: System Dynamics & Control Stefan B. Williams May, 211 AMME35: System Dynamics & Control Assignment 4 Note: This assignment contributes 15% towards your final mark. This assignment is due at 4pm on Monday, May 3 th during Week 13

More information

State Regulator. Advanced Control. design of controllers using pole placement and LQ design rules

State Regulator. Advanced Control. design of controllers using pole placement and LQ design rules Advanced Control State Regulator Scope design of controllers using pole placement and LQ design rules Keywords pole placement, optimal control, LQ regulator, weighting matrixes Prerequisites Contact state

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

Root Locus Techniques

Root Locus Techniques 4th Edition E I G H T Root Locus Techniques SOLUTIONS TO CASE STUDIES CHALLENGES Antenna Control: Transient Design via Gain a. From the Chapter 5 Case Study Challenge: 76.39K G(s) = s(s+50)(s+.32) Since

More information

DESIGN USING TRANSFORMATION TECHNIQUE CLASSICAL METHOD

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

More information

Damping Matrix. Donkey2Ft

Damping Matrix. Donkey2Ft 1 Damping Matrix DonkeyFt Damping in a single-degree-of-freedom (SDOF) system is well studied. Whether the system is under-damped, over-damped, or critically damped is well known. For an under-damped system,

More information

EECS C128/ ME C134 Final Thu. May 14, pm. Closed book. One page, 2 sides of formula sheets. No calculators.

EECS C128/ ME C134 Final Thu. May 14, pm. Closed book. One page, 2 sides of formula sheets. No calculators. Name: SID: EECS C28/ ME C34 Final Thu. May 4, 25 5-8 pm Closed book. One page, 2 sides of formula sheets. No calculators. There are 8 problems worth points total. Problem Points Score 4 2 4 3 6 4 8 5 3

More information

سایت آموزش مهندسی مکانیک

سایت آموزش مهندسی مکانیک http://www.drshokuhi.com سایت آموزش مهندسی مکانیک 1 Single-degree-of-freedom Systems 1.1 INTRODUCTION In this chapter the vibration of a single-degree-of-freedom system will be analyzed and reviewed. Analysis,

More information

Root locus Analysis. P.S. Gandhi Mechanical Engineering IIT Bombay. Acknowledgements: Mr Chaitanya, SYSCON 07

Root locus Analysis. P.S. Gandhi Mechanical Engineering IIT Bombay. Acknowledgements: Mr Chaitanya, SYSCON 07 Root locus Analysis P.S. Gandhi Mechanical Engineering IIT Bombay Acknowledgements: Mr Chaitanya, SYSCON 07 Recap R(t) + _ k p + k s d 1 s( s+ a) C(t) For the above system the closed loop transfer function

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

Control Systems. EC / EE / IN. For

Control Systems.   EC / EE / IN. For Control Systems For EC / EE / IN By www.thegateacademy.com Syllabus Syllabus for Control Systems Basic Control System Components; Block Diagrammatic Description, Reduction of Block Diagrams. Open Loop

More information

(Continued on next page)

(Continued on next page) (Continued on next page) 18.2 Roots of Stability Nyquist Criterion 87 e(s) 1 S(s) = =, r(s) 1 + P (s)c(s) where P (s) represents the plant transfer function, and C(s) the compensator. The closedloop characteristic

More information

Frequency domain analysis

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

More information

Trajectory Planning, Setpoint Generation and Feedforward for Motion Systems

Trajectory Planning, Setpoint Generation and Feedforward for Motion Systems 2 Trajectory Planning, Setpoint Generation and Feedforward for Motion Systems Paul Lambrechts Digital Motion Control (4K4), 23 Faculty of Mechanical Engineering, Control Systems Technology Group /42 2

More information

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

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

More information

FATIMA MICHAEL COLLEGE OF ENGINEERING & TECHNOLOGY

FATIMA MICHAEL COLLEGE OF ENGINEERING & TECHNOLOGY FATIMA MICHAEL COLLEGE OF ENGINEERING & TECHNOLOGY Senkottai Village, Madurai Sivagangai Main Road, Madurai - 625 020. An ISO 9001:2008 Certified Institution DEPARTMENT OF ELECTRONICS AND COMMUNICATION

More information

Analysis of SISO Control Loops

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

More information

Control Systems I Lecture 10: System Specifications

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

More information

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

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

More information

Jerk derivative feedforward control for motion systems

Jerk derivative feedforward control for motion systems Jerk derivative feedforward control for motion systems Matthijs Boerlage Rob Tousain Maarten Steinbuch Abstract This work discusses reference trajectory relevant model based feedforward design. For motion

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

Systems Analysis and Control

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

More information

CHAPTER # 9 ROOT LOCUS ANALYSES

CHAPTER # 9 ROOT LOCUS ANALYSES F K א CHAPTER # 9 ROOT LOCUS ANALYSES 1. Introduction The basic characteristic of the transient response of a closed-loop system is closely related to the location of the closed-loop poles. If the system

More information

Systems Analysis and Control

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

More information

Automatic Control (TSRT15): Lecture 7

Automatic Control (TSRT15): Lecture 7 Automatic Control (TSRT15): Lecture 7 Tianshi Chen Division of Automatic Control Dept. of Electrical Engineering Email: tschen@isy.liu.se Phone: 13-282226 Office: B-house extrance 25-27 Outline 2 Feedforward

More information

ECEn 483 / ME 431 Case Studies. Randal W. Beard Brigham Young University

ECEn 483 / ME 431 Case Studies. Randal W. Beard Brigham Young University ECEn 483 / ME 431 Case Studies Randal W. Beard Brigham Young University Updated: December 2, 2014 ii Contents 1 Single Link Robot Arm 1 2 Pendulum on a Cart 9 3 Satellite Attitude Control 17 4 UUV Roll

More information

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 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

More information

Automatic Control (MSc in Mechanical Engineering) Lecturer: Andrea Zanchettin Date: Student ID number... Signature...

Automatic Control (MSc in Mechanical Engineering) Lecturer: Andrea Zanchettin Date: Student ID number... Signature... Automatic Control (MSc in Mechanical Engineering) Lecturer: Andrea Zanchettin Date: 29..23 Given and family names......................solutions...................... Student ID number..........................

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

CHAPTER 2 TRANSFER FUNCTION ANALYSIS

CHAPTER 2 TRANSFER FUNCTION ANALYSIS . Introduction CHAPTER TRANSFER FUNCTION ANALYSIS The purpose of this chapter is to illustrate how to derive equations of motion for Multi Degree of Freedom (mdof) systems and how to solve for their transfer

More information

Exam. 135 minutes, 15 minutes reading time

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:

More information

Lecture 6 Classical Control Overview IV. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

Lecture 6 Classical Control Overview IV. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Lecture 6 Classical Control Overview IV Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Lead Lag Compensator Design Dr. Radhakant Padhi Asst.

More information

University of Alberta ENGM 541: Modeling and Simulation of Engineering Systems Laboratory #7. M.G. Lipsett & M. Mashkournia 2011

University of Alberta ENGM 541: Modeling and Simulation of Engineering Systems Laboratory #7. M.G. Lipsett & M. Mashkournia 2011 ENG M 54 Laboratory #7 University of Alberta ENGM 54: Modeling and Simulation of Engineering Systems Laboratory #7 M.G. Lipsett & M. Mashkournia 2 Mixed Systems Modeling with MATLAB & SIMULINK Mixed systems

More information

Solutions to Skill-Assessment Exercises

Solutions to Skill-Assessment Exercises Solutions to Skill-Assessment Exercises To Accompany Control Systems Engineering 4 th Edition By Norman S. Nise John Wiley & Sons Copyright 2004 by John Wiley & Sons, Inc. All rights reserved. No part

More information

Richiami di Controlli Automatici

Richiami di Controlli Automatici Richiami di Controlli Automatici Gianmaria De Tommasi 1 1 Università degli Studi di Napoli Federico II detommas@unina.it Ottobre 2012 Corsi AnsaldoBreda G. De Tommasi (UNINA) Richiami di Controlli Automatici

More information

Theory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati

Theory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Theory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Module - 2 Simpul Rotors Lecture - 2 Jeffcott Rotor Model In the

More information

Mechanical Vibrations Chapter 6 Solution Methods for the Eigenvalue Problem

Mechanical Vibrations Chapter 6 Solution Methods for the Eigenvalue Problem Mechanical Vibrations Chapter 6 Solution Methods for the Eigenvalue Problem Introduction Equations of dynamic equilibrium eigenvalue problem K x = ω M x The eigensolutions of this problem are written in

More information

MAE 143B - Homework 8 Solutions

MAE 143B - Homework 8 Solutions MAE 43B - Homework 8 Solutions P6.4 b) With this system, the root locus simply starts at the pole and ends at the zero. Sketches by hand and matlab are in Figure. In matlab, use zpk to build the system

More information

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

Today (10/23/01) Today. Reading Assignment: 6.3. Gain/phase margin lead/lag compensator Ref. 6.4, 6.7, 6.10 Today Today (10/23/01) Gain/phase margin lead/lag compensator Ref. 6.4, 6.7, 6.10 Reading Assignment: 6.3 Last Time In the last lecture, we discussed control design through shaping of the loop gain GK:

More information

Robust Control 3 The Closed Loop

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

More information

THE subject of the analysis is system composed by

THE subject of the analysis is system composed by MECHANICAL VIBRATION ASSIGNEMENT 1 On 3 DOF system identification Diego Zenari, 182160, M.Sc Mechatronics engineering Abstract The present investigation carries out several analyses on a 3-DOF system.

More information

The loop shaping paradigm. Lecture 7. Loop analysis of feedback systems (2) Essential specifications (2)

The loop shaping paradigm. Lecture 7. Loop analysis of feedback systems (2) Essential specifications (2) Lecture 7. Loop analysis of feedback systems (2). Loop shaping 2. Performance limitations The loop shaping paradigm. Estimate performance and robustness of the feedback system from the loop transfer L(jω)

More information

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

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

More information

Linear Systems Theory

Linear Systems Theory ME 3253 Linear Systems Theory Review Class Overview and Introduction 1. How to build dynamic system model for physical system? 2. How to analyze the dynamic system? -- Time domain -- Frequency domain (Laplace

More information

Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam!

Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Prüfung Regelungstechnik I (Control Systems I) Prof. Dr. Lino Guzzella 9. 8. 2 Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Do not mark up this translation aid -

More information

Lecture 3: The Root Locus Method

Lecture 3: The Root Locus Method Lecture 3: The Root Locus Method Venkata Sonti Department of Mechanical Engineering Indian Institute of Science Bangalore, India, 56001 This draft: March 1, 008 1 The Root Locus method The Root Locus method,

More information

2C9 Design for seismic and climate changes. Jiří Máca

2C9 Design for seismic and climate changes. Jiří Máca 2C9 Design for seismic and climate changes Jiří Máca List of lectures 1. Elements of seismology and seismicity I 2. Elements of seismology and seismicity II 3. Dynamic analysis of single-degree-of-freedom

More information

Unit 11 - Week 7: Quantitative feedback theory (Part 1/2)

Unit 11 - Week 7: Quantitative feedback theory (Part 1/2) X reviewer3@nptel.iitm.ac.in Courses» Control System Design Announcements Course Ask a Question Progress Mentor FAQ Unit 11 - Week 7: Quantitative feedback theory (Part 1/2) Course outline How to access

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

ELECTRONICS & COMMUNICATIONS DEP. 3rd YEAR, 2010/2011 CONTROL ENGINEERING SHEET 5 Lead-Lag Compensation Techniques

ELECTRONICS & COMMUNICATIONS DEP. 3rd YEAR, 2010/2011 CONTROL ENGINEERING SHEET 5 Lead-Lag Compensation Techniques CAIRO UNIVERSITY FACULTY OF ENGINEERING ELECTRONICS & COMMUNICATIONS DEP. 3rd YEAR, 00/0 CONTROL ENGINEERING SHEET 5 Lead-Lag Compensation Techniques [] For the following system, Design a compensator such

More information

Frequency Response Techniques

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

More information

AA242B: MECHANICAL VIBRATIONS

AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS 1 / 50 AA242B: MECHANICAL VIBRATIONS Undamped Vibrations of n-dof Systems These slides are based on the recommended textbook: M. Géradin and D. Rixen, Mechanical Vibrations:

More information

Problem Set 4 Solution 1

Problem Set 4 Solution 1 Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Problem Set 4 Solution Problem 4. For the SISO feedback

More information

KINGS COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING

KINGS COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING KINGS COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING QUESTION BANK SUB.NAME : CONTROL SYSTEMS BRANCH : ECE YEAR : II SEMESTER: IV 1. What is control system? 2. Define open

More information

Control Systems Engineering ( Chapter 8. Root Locus Techniques ) Prof. Kwang-Chun Ho Tel: Fax:

Control Systems Engineering ( Chapter 8. Root Locus Techniques ) Prof. Kwang-Chun Ho Tel: Fax: Control Systems Engineering ( Chapter 8. Root Locus Techniques ) Prof. Kwang-Chun Ho kwangho@hansung.ac.kr Tel: 02-760-4253 Fax:02-760-4435 Introduction In this lesson, you will learn the following : The

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

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 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)

More information

Methods for analysis and control of. Lecture 4: The root locus design method

Methods for analysis and control of. Lecture 4: The root locus design method Methods for analysis and control of Lecture 4: The root locus design method O. Sename 1 1 Gipsa-lab, CNRS-INPG, FRANCE Olivier.Sename@gipsa-lab.inpg.fr www.lag.ensieg.inpg.fr/sename Lead Lag 17th March

More information