# EL2520 Control Theory and Practice

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 norms, system gains and the small gain theorem The closed-loop system and the design problem Characterized by six transfer functions: need to look at all! Fundamental limitations and waterbed effect. Loop shaping to satisfy sensitivity function specifications. From now and on: MIMO Today s lecture Linear systems with multiple inputs and multiple outputs Basic properties of multivariable systems State-space theory, state feedback and observers Decentralized and decoupled control Robust loop shaping H 2 and H optimal control The final part of the course considers systems with constraints Basic properties of multivariable systems Transfer matrices Block diagram calculations Gains and directions The multivariable frequency response Poles and zeros Chapters 2-3 in the textbook.

2 Transfer matrices Quiz: transfer matrices The Laplace transform X(s) of a signal x(t) is defined by. What is the transfer matrix G(s) for the system Given a linear system and assuming u(t)=0 for t<0 and x(0)=0, How does G(s) change when 2. Input two also affects the first state: If system has multiple inputs and outputs, Y and U are vectorvalued and G(s) is a matrix (i.e. a matrix-valued function of s). 3. The second state also affects output one: 4. The second state influences the first:. Answer and observations G(s) = 0 0 Example: series connection Linear system viewed as interconnected multivariable systems Independent subsystems à (block)diagonal transfer matrix 2., 3. G(s) = 0 Couplings à nondiagonal G(s). Different A, B, C can give same G(s) 4. G(s) = () have the same poles (eigenvalues of A). Hard to see from G(s) We see that Y(s)=G Z (s)z(s)= G Z (s)g U (s)u(s). Note that G Z G U apple G U G Z order matters! 2

3 Block diagram calculations wu w r u Fr G z Quiz: the closed-loop MIMO system Determine the sensitivity and complementary sensitivity for the linear multivariable system n wu w r Fr u G z -Fy Have to be careful when manipulating block diagrams. n Example. Let w u,w,n=0 and derive transfer matrix from r to z -Fy so Recall: S is transfer matrix from wà z, T is transfer matrix from -nà z Additional question: What is the relation between S and T? Today s lecture Frequency response and system gain For a scalar linear system G(s) driven by u(t)=sin(ωt), Basic properties of multivariable systems Transfer matrices Block diagram calculations Gains and directions The multivariable frequency response Poles and zeros (after transients have died out). So The system gain (cf. Lecture ) is defined as Attained for sinusoidal input with frequency ω such that G(iω) = G Q: What are the corresponding results for multivariable systems? 3

4 Operator norm of linear mapping The singular value decomposition Consider the linear mapping Since (x, y, A complex-valued) A matrix A 2 C m r (with r<m, rank(a)=r), can be represented by its singular value decomposition (SVD) we have So where The positive scalars i are the singular values of A v i are the input singular vectors of A, V * V=I u i are the output singular vectors of A, U * U=I Where are called the singular values of A. SVD interpretation Example A U V Interpretation: linear mapping y=ax can be decomposed as compute coefficients of x along input directions v i scale coefficients by i reconstitute along output directions u i Since apple apple r an input in the v r direction is amplified the most. It generates an output in the direction of u r (typically different from v r ). x = v 4

5 The multivariable frequency response The system gain For a linear multivariable system Y(s)=G(s)U(s), we have As for scalar systems, we can use Parseval s theorem to find Since this is a linear mapping, where with equality if U(iω) parallell w. corresponding input singular vector. For example, Note: Worst-case input is sinusoidal at the frequency that attains the supremum, but its components are appropriately scaled and phase shifted (as specified by the input singular vector of ) only if U(iω) parallell with input singular vector corresponding to Note: the infinity norm computes the maximum amplifications across frequency (sup ω) and input directions ( ) Example: heat exchanger Singular values of heat exchanger Objective: control outlet temperatures T c, T H by manipulating the flows q C, q H Model: System gain: G =3 db (=37) 5

6 Heat exchanger steady-state Heat exchanger step responses Singular value decomposition reveals Maximum effect input (input singular vector corresponding to ) Minimum effect input (corresponding to ) Input direction has dramatic effect! (agrees with physical intuition) Note: large difference in time-scales! Singular values and bandwidths Today s lecture What is the bandwidth of the system? Basic properties of multivariable systems Transfer matrices Block diagram calculations Gains and directions The multivariable frequency response Poles and zeros No single value, but a range. Depends on input directions. 6

7 Poles Poles cont d Definition. The poles of a linear systems are the eigenvalues of the system matrix in a minimal state-space realization. Since the transfer matrix is given by Definition. The pole polynomial is the characteristic polynomial of the A matrix, (s) = det(si-a). Alternatively, the poles of a linear system are the zeros of the pole polynomial, i.e., the values p i such that (p i )=0 where r(s) is a polynomial in s (see book for precise expression), the pole polynomial must be at least the least common denominator of the the elements of the transfer matrix. Example: The system must (at least) have poles in s=- and s=-2. Poles of multivariable systems Zeros Theorem. The pole polynomial of a system with transfer matrix G(s) is the common denominator of all minors of G(s) Recall: a minor of a matrix M is the determinant of a (smaller) square matrix obtained by deleting some rows and columns of M Example: The minors of Theorem. The zero polynomial of G(s) is the greatest common divisor of the maximal minors of G(s), normed so that they have the pole polynomial of G(s) as denominator. The zeros of G(s) are the roots of its zero polynomial. Example: The maximal minor of are Thus, the system has poles in s=- (a double pole) and s=-2. is Thus, G(s) has a zero at s=. (already normed!). 7

8 Quiz: multivariable poles and zeros What are the poles and zeros of the multivariable system Notes on poles and zeros For scalar system G(s) with poles p i and zeros z i, For a multivariable system, directions matter! For a system with pole p, there exist vectors u p, v p : Similarly, a zero at z i implies the existence of vectors u z, v z : As for scalar systems, a zero at s=z implies that there exists a signal on the form u(t)=v z e -zt for t 0, and u(t)=0 for t<0, and initial values x(0)=x z so that y(t)=0 for t 0 Summary An introduction to multivariable linear systems: Block diagram manipulations (order matters!) System gain (directions matter!) Poles and zeros More next week! 8

### EL2520 Control Theory and Practice

EL2520 Control Theory and Practice Lecture 8: Linear quadratic control Mikael Johansson School of Electrical Engineering KTH, Stockholm, Sweden Linear quadratic control Allows to compute the controller

### Singular Value Decomposition Analysis

Singular Value Decomposition Analysis Singular Value Decomposition Analysis Introduction Introduce a linear algebra tool: singular values of a matrix Motivation Why do we need singular values in MIMO control

### Multivariable Control. Lecture 05. Multivariable Poles and Zeros. John T. Wen. September 14, 2006

Multivariable Control Lecture 05 Multivariable Poles and Zeros John T. Wen September 4, 2006 SISO poles/zeros SISO transfer function: G(s) = n(s) d(s) (no common factors between n(s) and d(s)). Poles:

### Lecture plan: Control Systems II, IDSC, 2017

Control Systems II MAVT, IDSC, Lecture 8 28/04/2017 G. Ducard Lecture plan: Control Systems II, IDSC, 2017 SISO Control Design 24.02 Lecture 1 Recalls, Introductory case study 03.03 Lecture 2 Cascaded

### 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

### Chapter Robust Performance and Introduction to the Structured Singular Value Function Introduction As discussed in Lecture 0, a process is better desc

Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology c Chapter Robust

### 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski. Solutions to Problem Set 1 1. Massachusetts Institute of Technology

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Solutions to Problem Set 1 1 Problem 1.1T Consider the

### SMITH MCMILLAN FORMS

Appendix B SMITH MCMILLAN FORMS B. Introduction Smith McMillan forms correspond to the underlying structures of natural MIMO transfer-function matrices. The key ideas are summarized below. B.2 Polynomial

### Introduction to MVC. least common denominator of all non-identical-zero minors of all order of G(s). Example: The minor of order 2: 1 2 ( s 1)

Introduction to MVC Definition---Proerness and strictly roerness A system G(s) is roer if all its elements { gij ( s)} are roer, and strictly roer if all its elements are strictly roer. Definition---Causal

### 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

### Solution via Laplace transform and matrix exponential

EE263 Autumn 2015 S. Boyd and S. Lall Solution via Laplace transform and matrix exponential Laplace transform solving ẋ = Ax via Laplace transform state transition matrix matrix exponential qualitative

### 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

### Lecture 5 Input output stability

Lecture 5 Input output stability or How to make a circle out of the point 1+0i, and different ways to stay away from it... k 2 yf(y) r G(s) y k 1 y y 1 k 1 1 k 2 f( ) G(iω) Course Outline Lecture 1-3 Lecture

### Zeros and zero dynamics

CHAPTER 4 Zeros and zero dynamics 41 Zero dynamics for SISO systems Consider a linear system defined by a strictly proper scalar transfer function that does not have any common zero and pole: g(s) =α p(s)

### MIMO analysis: loop-at-a-time

MIMO robustness MIMO analysis: loop-at-a-time y 1 y 2 P (s) + + K 2 (s) r 1 r 2 K 1 (s) Plant: P (s) = 1 s 2 + α 2 s α 2 α(s + 1) α(s + 1) s α 2. (take α = 10 in the following numerical analysis) Controller:

### Full-State Feedback Design for a Multi-Input System

Full-State Feedback Design for a Multi-Input System A. Introduction The open-loop system is described by the following state space model. x(t) = Ax(t)+Bu(t), y(t) =Cx(t)+Du(t) () 4 8.5 A =, B =.5.5, C

### 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

### 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

### Autonomous Mobile Robot Design

Autonomous Mobile Robot Design Topic: Guidance and Control Introduction and PID Loops Dr. Kostas Alexis (CSE) Autonomous Robot Challenges How do I control where to go? Autonomous Mobile Robot Design Topic:

### HANKEL-NORM BASED INTERACTION MEASURE FOR INPUT-OUTPUT PAIRING

Copyright 2002 IFAC 15th Triennial World Congress, Barcelona, Spain HANKEL-NORM BASED INTERACTION MEASURE FOR INPUT-OUTPUT PAIRING Björn Wittenmark Department of Automatic Control Lund Institute of Technology

### Introduction to Model Order Reduction

KTH ROYAL INSTITUTE OF TECHNOLOGY Introduction to Model Order Reduction Lecture 1: Introduction and overview Henrik Sandberg, Bart Besselink, Madhu N. Belur Overview of Today s Lecture What is model (order)

### FRTN 15 Predictive Control

Department of AUTOMATIC CONTROL FRTN 5 Predictive Control Final Exam March 4, 27, 8am - 3pm General Instructions This is an open book exam. You may use any book you want, including the slides from the

### CONTROL OF DIGITAL SYSTEMS

AUTOMATIC CONTROL AND SYSTEM THEORY CONTROL OF DIGITAL SYSTEMS Gianluca Palli Dipartimento di Ingegneria dell Energia Elettrica e dell Informazione (DEI) Università di Bologna Email: gianluca.palli@unibo.it

### 10 Transfer Matrix Models

MIT EECS 6.241 (FALL 26) LECTURE NOTES BY A. MEGRETSKI 1 Transfer Matrix Models So far, transfer matrices were introduced for finite order state space LTI models, in which case they serve as an important

### EL 625 Lecture 10. Pole Placement and Observer Design. ẋ = Ax (1)

EL 625 Lecture 0 EL 625 Lecture 0 Pole Placement and Observer Design Pole Placement Consider the system ẋ Ax () The solution to this system is x(t) e At x(0) (2) If the eigenvalues of A all lie in the

### Chapter 7 - Solved Problems

Chapter 7 - Solved Problems Solved Problem 7.1. A continuous time system has transfer function G o (s) given by G o (s) = B o(s) A o (s) = 2 (s 1)(s + 2) = 2 s 2 + s 2 (1) Find a controller of minimal

### 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

### Robust Performance Example #1

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

### Problem Set 5 Solutions 1

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Problem Set 5 Solutions The problem set deals with Hankel

### 1 Steady State Error (30 pts)

Professor Fearing EECS C28/ME C34 Problem Set Fall 2 Steady State Error (3 pts) Given the following continuous time (CT) system ] ẋ = A x + B u = x + 2 7 ] u(t), y = ] x () a) Given error e(t) = r(t) y(t)

### Essence of the Root Locus Technique

Essence of the Root Locus Technique In this chapter we study a method for finding locations of system poles. The method is presented for a very general set-up, namely for the case when the closed-loop

### Systems Engineering/Process Control L4

1 / 24 Systems Engineering/Process Control L4 Input-output models Laplace transform Transfer functions Block diagram algebra Reading: Systems Engineering and Process Control: 4.1 4.4 2 / 24 Laplace transform

### 6.241 Dynamic Systems and Control

6.41 Dynamic Systems and Control Lecture 5: H Synthesis Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology May 11, 011 E. Frazzoli (MIT) Lecture 5: H Synthesis May 11, 011

### One-Sided Laplace Transform and Differential Equations

One-Sided Laplace Transform and Differential Equations As in the dcrete-time case, the one-sided transform allows us to take initial conditions into account. Preliminaries The one-sided Laplace transform

### Course Outline. Higher Order Poles: Example. Higher Order Poles. Amme 3500 : System Dynamics & Control. State Space Design. 1 G(s) = s(s + 2)(s +10)

Amme 35 : System Dynamics Control State Space Design Course Outline Week Date Content Assignment Notes 1 1 Mar Introduction 2 8 Mar Frequency Domain Modelling 3 15 Mar Transient Performance and the s-plane

### Transfer Functions. Chapter Introduction. 6.2 The Transfer Function

Chapter 6 Transfer Functions As a matter of idle curiosity, I once counted to find out what the order of the set of equations in an amplifier I had just designed would have been, if I had worked with the

### Multistage Amplifier Frequency Response

Multistage Amplifier Frequency Response * Summary of frequency response of single-stages: CE/CS: suffers from Miller effect CC/CD: wideband -- see Section 0.5 CB/CG: wideband -- see Section 0.6 (wideband

### 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)

### Singular Value Decomposition of the frequency response operator

Measures of input-output amplification (across frequency) Largest singular value Hilbert-Schmidt norm (power spectral density) Systems with one spatial variable Two point boundary value problems EE 8235:

### Control Systems II. ETH, MAVT, IDSC, Lecture 4 17/03/2017. G. Ducard

Control Systems II ETH, MAVT, IDSC, Lecture 4 17/03/2017 Lecture plan: Control Systems II, IDSC, 2017 SISO Control Design 24.02 Lecture 1 Recalls, Introductory case study 03.03 Lecture 2 Cascaded Control

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

Prüfung Regelungstechnik I (Control Systems I) Prof. Dr. Lino Guzzella 3.. 24 Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Do not mark up this translation aid -

### Raktim Bhattacharya. . AERO 632: Design of Advance Flight Control System. Norms for Signals and Systems

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

### 2-DOF BLOCK POLE PLACEMENT CONTROL APPLICATION TO: HAVE-DASH-II BTT MISSILE

2-DOF BLOCK POLE PLACEMENT CONTROL APPLICATION TO: HAVE-DASH-II BTT MISSILE BEKHITI Belkacem 1 DAHIMENE Abdelhakim 1 NAIL Bachir 2 and HARICHE Kamel 1 1 Electronics and Electrotechnics Institute, University

### Chapter 5. Standard LTI Feedback Optimization Setup. 5.1 The Canonical Setup

Chapter 5 Standard LTI Feedback Optimization Setup Efficient LTI feedback optimization algorithms comprise a major component of modern feedback design approach: application problems involving complex models

### 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

### 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

### 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

### 8 sin 3 V. For the circuit given, determine the voltage v for all time t. Assume that no energy is stored in the circuit before t = 0.

For the circuit given, determine the voltage v for all time t. Assume that no energy is stored in the circuit before t = 0. Spring 2015, Exam #5, Problem #1 4t Answer: e tut 8 sin 3 V 1 For the circuit

### 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

### AN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES

AN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES JOEL A. TROPP Abstract. We present an elementary proof that the spectral radius of a matrix A may be obtained using the formula ρ(a) lim

### An Internal Stability Example

An Internal Stability Example Roy Smith 26 April 2015 To illustrate the concept of internal stability we will look at an example where there are several pole-zero cancellations between the controller and

### 16.30/31, Fall 2010 Recitation # 2

16.30/31, Fall 2010 Recitation # 2 September 22, 2010 In this recitation, we will consider two problems from Chapter 8 of the Van de Vegte book. R + - E G c (s) G(s) C Figure 1: The standard block diagram

### CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. VII Multivariable Poles and Zeros - Karcanias, Nicos

MULTIVARIABLE POLES AND ZEROS Karcanias, Nicos Control Engineering Research Centre, City University, London, UK Keywords: Systems, Representations, State Space, Transfer Functions, Matrix Fraction Descriptions,

### Singular Value Decomposition

Singular Value Decomposition Motivatation The diagonalization theorem play a part in many interesting applications. Unfortunately not all matrices can be factored as A = PDP However a factorization A =

### PARAMETERIZATION OF STATE FEEDBACK GAINS FOR POLE PLACEMENT

PARAMETERIZATION OF STATE FEEDBACK GAINS FOR POLE PLACEMENT Hans Norlander Systems and Control, Department of Information Technology Uppsala University P O Box 337 SE 75105 UPPSALA, Sweden HansNorlander@ituuse

### Chapter 12. Feedback Control Characteristics of Feedback Systems

Chapter 1 Feedbac Control Feedbac control allows a system dynamic response to be modified without changing any system components. Below, we show an open-loop system (a system without feedbac) and a closed-loop

### Differential equations

Differential equations Math 7 Spring Practice problems for April Exam Problem Use the method of elimination to find the x-component of the general solution of x y = 6x 9x + y = x 6y 9y Soln: The system

### Topic # Feedback Control. State-Space Systems Closed-loop control using estimators and regulators. Dynamics output feedback

Topic #17 16.31 Feedback Control State-Space Systems Closed-loop control using estimators and regulators. Dynamics output feedback Back to reality Copyright 21 by Jonathan How. All Rights reserved 1 Fall

### The Relation Between the 3-D Bode Diagram and the Root Locus. Insights into the connection between these classical methods. By Panagiotis Tsiotras

F E A T U R E The Relation Between the -D Bode Diagram and the Root Locus Insights into the connection between these classical methods Bode diagrams and root locus plots have been the cornerstone of control

### Lecture 4 and 5 Controllability and Observability: Kalman decompositions

1 Lecture 4 and 5 Controllability and Observability: Kalman decompositions Spring 2013 - EE 194, Advanced Control (Prof. Khan) January 30 (Wed.) and Feb. 04 (Mon.), 2013 I. OBSERVABILITY OF DT LTI SYSTEMS

### Principal Component Analysis CS498

Principal Component Analysis CS498 Today s lecture Adaptive Feature Extraction Principal Component Analysis How, why, when, which A dual goal Find a good representation The features part Reduce redundancy

### 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

### 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

### Linear System Theory

Linear System Theory Wonhee Kim Lecture 3 Mar. 21, 2017 1 / 38 Overview Recap Nonlinear systems: existence and uniqueness of a solution of differential equations Preliminaries Fields and Vector Spaces

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

Prüfung Regelungstechnik I (Control Systems I) Prof. Dr. Lino Guzzella 3. 8. 24 Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Do not mark up this translation aid

### 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

### DESCRIPTOR FRACTIONAL LINEAR SYSTEMS WITH REGULAR PENCILS

Int. J. Appl. Math. Comput. Sci., 3, Vol. 3, No., 39 35 DOI:.478/amcs-3-3 DESCRIPTOR FRACTIONAL LINEAR SYSTEMS WITH REGULAR PENCILS TADEUSZ KACZOREK Faculty of Electrical Engineering Białystok Technical

### 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

### DS-GA 1002 Lecture notes 10 November 23, Linear models

DS-GA 2 Lecture notes November 23, 2 Linear functions Linear models A linear model encodes the assumption that two quantities are linearly related. Mathematically, this is characterized using linear functions.

### x(t) = x(t h), x(t) 2 R ), where is the time delay, the transfer function for such a e s Figure 1: Simple Time Delay Block Diagram e i! =1 \e i!t =!

1 Time-Delay Systems 1.1 Introduction Recitation Notes: Time Delays and Nyquist Plots Review In control systems a challenging area is operating in the presence of delays. Delays can be attributed to acquiring

### 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

### 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

### Pole Placement Design

Department of Automatic Control LTH, Lund University 1 Introduction 2 Simple Examples 3 Polynomial Design 4 State Space Design 5 Robustness and Design Rules 6 Model Reduction 7 Oscillatory Systems 8 Summary

### 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.

### Reglerteknik Allmän Kurs. Del 2. Lösningar till Exempelsamling. Läsår 2015/16

Reglerteknik Allmän Kurs Del Lösningar till Exempelsamling Läsår 5/6 Avdelningen för Reglerteknik, KTH, SE 44 Stockholm, SWEDEN AUTOMATIC CONTROL COMMUNICATION SYSTEMS LINKÖPINGS UNIVERSITET Reglerteknik

### Robust control of MIMO systems

Robust control of MIMO systems Olivier Sename Grenoble INP / GIPSA-lab September 2017 Olivier Sename (Grenoble INP / GIPSA-lab) Robust control September 2017 1 / 162 O. Sename [GIPSA-lab] 2/162 1. Some

### Fundamental of Control Systems Steady State Error Lecturer: Dr. Wahidin Wahab M.Sc. Aries Subiantoro, ST. MSc.

Fundamental of Control Systems Steady State Error Lecturer: Dr. Wahidin Wahab M.Sc. Aries Subiantoro, ST. MSc. Electrical Engineering Department University of Indonesia 2 Steady State Error How well can

### 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

### 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)

### INTERACTIVE APPLICATIONS IN A MANDATORY CONTROL COURSE

INTERACTIVE APPLICATIONS IN A MANDATORY CONTROL COURSE Yves Piguet Roland Longchamp Calerga Sàrl, av. de la Chablière 35, 4 Lausanne, Switzerland. E-mail: yves.piguet@calerga.com Laboratoire d automatique,

### U.C. Berkeley CS294: Spectral Methods and Expanders Handout 11 Luca Trevisan February 29, 2016

U.C. Berkeley CS294: Spectral Methods and Expanders Handout Luca Trevisan February 29, 206 Lecture : ARV In which we introduce semi-definite programming and a semi-definite programming relaxation of sparsest

### Simple Learning Control Made Practical by Zero-Phase Filtering: Applications to Robotics

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL 49, NO 6, JUNE 2002 753 Simple Learning Control Made Practical by Zero-Phase Filtering: Applications to Robotics Haluk

### Problem Set 8 Solutions 1

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6245: MULTIVARIABLE CONTROL SYSTEMS by A Megretski Problem Set 8 Solutions 1 The problem set deals with H-Infinity

### 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

### Review of Linear Time-Invariant Network Analysis

D1 APPENDIX D Review of Linear Time-Invariant Network Analysis Consider a network with input x(t) and output y(t) as shown in Figure D-1. If an input x 1 (t) produces an output y 1 (t), and an input x

### Relay feedback and multivariable control

Relay feedback and multivariable control Johansson, Karl Henrik Published: 1997-1-1 Link to publication Citation for published version (APA): Johansson, K. H. (1997). Relay feedback and multivariable control

### Kernel Method: Data Analysis with Positive Definite Kernels

Kernel Method: Data Analysis with Positive Definite Kernels 2. Positive Definite Kernel and Reproducing Kernel Hilbert Space Kenji Fukumizu The Institute of Statistical Mathematics. Graduate University

### 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

### Multi Degrees of Freedom Systems

Multi Degrees of Freedom Systems MDOF s http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano March 9, 07 Outline, a System

### Mathematical Relationships Between Representations of Structure in Linear Interconnected Dynamical Systems

American Control Conference on O'Farrell Street, San Francisco, CA, USA June 9 - July, Mathematical Relationships Between Representations of Structure in Linear Interconnected Dynamical Systems E. Yeung,

### 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

### DESIGN OF LINEAR STATE FEEDBACK CONTROL LAWS

7 DESIGN OF LINEAR STATE FEEDBACK CONTROL LAWS Previous chapters, by introducing fundamental state-space concepts and analysis tools, have now set the stage for our initial foray into statespace methods

### Modeling and Control Overview

Modeling and Control Overview D R. T A R E K A. T U T U N J I A D V A N C E D C O N T R O L S Y S T E M S M E C H A T R O N I C S E N G I N E E R I N G D E P A R T M E N T P H I L A D E L P H I A U N I

### Design of Decentralised PI Controller using Model Reference Adaptive Control for Quadruple Tank Process

Design of Decentralised PI Controller using Model Reference Adaptive Control for Quadruple Tank Process D.Angeline Vijula #, Dr.N.Devarajan * # Electronics and Instrumentation Engineering Sri Ramakrishna

### Frequency Response of Linear Time Invariant Systems

ME 328, Spring 203, Prof. Rajamani, University of Minnesota Frequency Response of Linear Time Invariant Systems Complex Numbers: Recall that every complex number has a magnitude and a phase. Example: z

### AN OPTIMIZATION-BASED APPROACH FOR QUASI-NONINTERACTING CONTROL. Jose M. Araujo, Alexandre C. Castro and Eduardo T. F. Santos

ICIC Express Letters ICIC International c 2008 ISSN 1881-803X Volume 2, Number 4, December 2008 pp. 395 399 AN OPTIMIZATION-BASED APPROACH FOR QUASI-NONINTERACTING CONTROL Jose M. Araujo, Alexandre C.

### Design and Analysis of a Novel L 1 Adaptive Controller, Part I: Control Signal and Asymptotic Stability

Proceedings of the 26 American Control Conference Minneapolis, Minnesota, USA, June 4-6, 26 ThB7.5 Design and Analysis of a Novel L Adaptive Controller, Part I: Control Signal and Asymptotic Stability

### Information Retrieval

Introduction to Information CS276: Information and Web Search Christopher Manning and Pandu Nayak Lecture 13: Latent Semantic Indexing Ch. 18 Today s topic Latent Semantic Indexing Term-document matrices

### Comparison of methods for parametric model order reduction of instationary problems

Max Planck Institute Magdeburg Preprints Ulrike Baur Peter Benner Bernard Haasdonk Christian Himpe Immanuel Maier Mario Ohlberger Comparison of methods for parametric model order reduction of instationary

### 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