# Singular Value Decomposition Analysis

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 designs? Definition and properties of singular values Singular value decomposition (SVD) provides directional information 376_069 Multivariable feedback control V3 1 of 36

2 Singular Value Decomposition Analysis SISO sinusoidal steady-state (I) u(s) g(s) y(s) Assume: g(s) strictly stable Sinusoidal steady-state u(t) = ue jωt y(t) = ye jωt y = g(jω)u Note g(jω) is a complex scalar g(jω) = g(jω) e jφ(ω) Magnitude: g(jω) = g*(jω)g(jω) Phase: φ(ω) = arctan Im( g(j ω)) Re( g (j ω )) Above plotted in Bode plot defines frequency response of SISO plant g(s) g(jω) defines plant gain at frequency ω 376_069 Multivariable feedback control V3 2 of 36

3 Singular Value Decomposition Analysis SISO sinusoidal steady-state (II) Input with complex amplitude u(t) = ue jωt u = u e jψ u(t) = u ej(ωt + ψ) Interpretation Re{u(t)} = u cos(ωt + ψ) Im{u(t)} = u sin(ωt + ψ) Complex steady-state output y(t) = y e jωt y = g(jω) u = g(jω) e jφ(ω) u e jψ = g(jω) u ej(φ(ω) + ψ) y = g(jω) u Re{y(t)} = y cos(ωt + ψ + φ(ω)) Im{y(t)} = y sin(ωt + ψ + φ(ω)) 376_069 Multivariable feedback control V3 3 of 36

4 Singular Value Decomposition Analysis SISO Bode plot information Provides graphical "summary" of plant gain at different frequencies Concepts of "small" and "large" gain are clear g(jω) >> 1 g(jω) << 1 large gain small gain 376_069 Multivariable feedback control V3 4 of 36

5 Singular Value Decomposition Analysis MIMO sinusoidal steady-state u(s) g(s) y(s) Assume: G(s) strictly stable Sinusoidal inputs generate at steady-state sinusoidal outputs Sinusoidal steady-state u(t) = ue jωt ;u C m y(t) = ye jωt ;y C p y = G(jω)u G(jω) : p x m complex matrix Need notion of size of G(jω) vs. frequency want visualize MIMO gain on Bode plot 376_069 Multivariable feedback control V3 5 of 36

6 Singular Value Decomposition Analysis Issues Deal with complex vectors and complex matrices. How to quantify "large" and "small" Impact of directions y(s) = G(s) u(s) u(t) = ue jωt ;u C m y(t) = ye jωt ;y C p y = G(jω)u Direction and size of u and Plant properties at frequency ω yield Direction and size of y Singular value decomposition (SVD) provides the "tool" for analysis 376_069 Multivariable feedback control V3 6 of 36

7 Singular Value Decomposition Analysis Notes on complex vectors x is a complex vector; x C n x = x 1 x 2: x n x i = a i + jb i ; i = 1, 2,..., n Magnitude x = x 2 x H = [x* 1 x* 2... x* n ] x 2 = x H x Example 1 j + x = 2 3j ;x H = [1 - j 2 + j3] x H x = (1 - j)(1 + j) + (2 + j3)(2 - j3) = 15 x 2 = _069 Multivariable feedback control V3 7 of 36

8 Singular Value Decomposition Analysis Notes on complex matrices A is a n x m matrix with complex-valued elements: a ik = α ik + jβ ik Notation A H : complex-conjugate transpose of A A H is a m x n matrix Note: A H A m x m matrix AA H n x n matrix Fact: AA H and A H A have real non-negative eigenvalues Example 1 1+ j A = j 2+ j A H 1 = j 1 j 2 j det(λι - A H A) = λ 2-9λ + 5 ;λ 1 = 8.41 ;λ 2 = 0.59 det(λι - A H A) = det(λι - AA H ) In this example λ i [A H A] = λ i [AA H ] > 0 376_069 Multivariable feedback control V3 8 of 36

9 Singular Value Decomposition Analysis Definition of singular values A is a n x m complex matrix Suppose : rank(a) = k Notation σ i (A): singular value of A Definition: The strictly positive square roots of the nonzero eigenvalues of A H A ( and AA H equivalently), are the singular values of A σ i (A) = λ i [A H A] = λ i [AA H ] > 0 i = 1, 2,..., k 376_069 Multivariable feedback control V3 9 of 36

10 Singular Value Decomposition Analysis The singular value decomposition A is a n x m complex matrix: rank(a) = k σ 1 σ 2... σ k 0 : are singular values of A Σ = σ σ σ k Σ n x m real matrix SVD A = U Σ V H A = U H Σ V U : n x n unitary matrix (U H = U -1 ) V : m x m unitary matrix (V H = V -1 ) Column vectors of U and V are orthonormal 376_069 Multivariable feedback control V3 10 of 36

11 Singular Value Decomposition Analysis More on SVD SVD A = U Σ V H A = U H Σ V U : n x n unitary matrix (U H = U -1 ) U = [u 1 u 2... u n ] ;u H i u j = δ ij u i : Left singular vectors of A (is right eigenvector of AA H associated with λ i [AA H ] ) V : m x m unitary matrix (V H = V -1 ) V = [v 1 v 2... v m ] ;v H i v j = δ ij v i : Right singular vectors of A (is right eigenvector of A H A associated with λ i [A H A] ) 376_069 Multivariable feedback control V3 11 of 36

12 Singular Value Decomposition Analysis Geometric interpretation A complex n x n matrix A -1 exists λ i (A) 0 Consider linear transformation y = Ax ;x, y R n x A y Euclidean norm x 2 = x'x ; y 2 = y'y Spectral norm of matrix A A 2 = max x 0 Ax 2 x 2 = max Ax x = Singular value relations σ max ( A) = max x 2 =1 Ax 2 = A 2 σ min ( A) = min x 2 =1 Ax 2 = 1 A _069 Multivariable feedback control V3 12 of 36

13 Singular Value Decomposition Analysis Graphical visualization Real case : y = Ax ; y, x R n, A R nxn ; n=2 SVD A = U Σ V H Σ = σ 1 = σ max 0 0 σ 2 =σ min U = [u 1 u 2 ] V = [v 1 v 2 ] Vector OD = v 1 ; Vector OD" = u 1 Vector OE = v 2 ; Vector OE" = u 2 Length of vector OD' = σ max = σ 1 Length of vector OE' = σ min = σ 2 376_069 Multivariable feedback control V3 13 of 36

14 Singular Value Decomposition Analysis MIMO frequency response u(t) = ue jωt G(s) y(t) = ye jωt Restrict u to unit (complex) sphere, i.e. u 2 = 1 u i (t) is complex sinusoid, e.g. u i (t) = u i e jψ ie jωt Re{u i (t)} = u i cos(ωt + ψ i ) Output response y(s) = G(s)u(s) Singular values σ max (G(jω)) = max G(j ω)u u = = y max (ω) 2 σ min (G(jω)) = min G(j ω)u u = = y min (ω) 2 Maximum and minimum singular values of G(jω) define max and min amplification of unit sinusoidal input at frequency ω 376_069 Multivariable feedback control V3 14 of 36

15 Singular Value Decomposition Analysis Bode plot visualization 376_069 Multivariable feedback control V3 15 of 36

16 Singular Value Decomposition Analysis Discussion The concept of singular values will be heavily exploited in analysis and design of MIMO feedback systems Correct interpretation of singular value plot hinges on units of physical variables (scaling) Singular value results assume "roundness" (convexity) of input signal space u(t) = ue jωt G(s) y(t) = ye jωt Input Space Output Space 376_069 Multivariable feedback control V3 16 of 36

17 Feedback Performance Specifications in the Frequency Domain Feedback Performance Specifications in the Frequency Domain Introduction Use singular values to establish nature of mimo performance specs in frequency domain Performance attributes - command following - disturbance rejection - insensitivity to sensor noise Stability-robustness to be addressed later 376_069 Multivariable feedback control V3 17 of 36

18 Feedback Performance Specifications in the Frequency Domain Fundamental relations True tracking error: e(s) = r(s) - y(s) Loop TFM: L(s) L(s) = G(s)K(s) Sensitivity TFM: S(s) S(s) = [Ι + L(s)] -1 Closed-loop TFM: T(s) T(s) = [Ι + L(s)] -1 L(s) Performance equation e(s) = S(s)[r(s) - d(s)] + T(s)n(s) Constraint: T(s) + S(s) = Ι 376_069 Multivariable feedback control V3 18 of 36

19 Feedback Performance Specifications in the Frequency Domain Command following (I) Sinusoidal command yields sinusoidal error r(t) = re jωt e(t) = ee jωt Relation: e = S(jω)r e 2 σ max [S(jω)] r 2 Ω r range of frequencies where command input has energy Prescription for good command following make σ max [S(jω)] << 1 for all ω Ω r 376_069 Multivariable feedback control V3 19 of 36

20 Feedback Performance Specifications in the Frequency Domain Command following (II) Interpretation for unit command sinusoid r(t) = re jωt ; r 2 = 1 Worst error at frequency ω e(t) = ee jωt e 2 = σ max [S(jω)] Attained when r points along right singular vector associated with σ max Best error at frequency ω e 2 = σ min [S(jω)] Attained when r points along right singular vector associated with σ min In general σ min [S(jω)] e 2 σ max [S(jω)] 376_069 Multivariable feedback control V3 20 of 36

21 Feedback Performance Specifications in the Frequency Domain Command following (III) Objective: express prescription for good command following in terms of loop TFM L(s) = G(s)K(s) Singular value facts σ max [A -1 ] = σ min 1 [ A] σ min [A] - 1 σ min [Ι + A] σ min [A] + 1 Recall: Need σ max [S(jω)] << 1 ;ω Ω r But S(jω) = [Ι + L(jω)] -1 σ max [S(jω)] = 1 << 1 σ min I +L(jω ) σ min [Ι + L(jω)] >> 1 ;ω Ω r But σ min [Ι + L(jω)] < σ min [L(jω)] + 1 Need σ min [L(jω)] >> 1 ;ω Ω r For good command following make σ min [G(s)K(s)] >> 1 for all ω Ω r 376_069 Multivariable feedback control V3 21 of 36

22 Feedback Performance Specifications in the Frequency Domain Disturbance rejection (I) Sinusoidal disturbance yields sinusoidal error d(t) = de jωt e(t) = ee jωt Relation e = S(jω)d e 2 σ max [S(jω)] d 2 Ω d range of frequencies where disturbance inputs has energy Prescription for good disturbance rejection make σ max [S(jω)] << 1 for all ω Ω d or make σ min [G(s)K(s)] >> 1 for all ω Ω d 376_069 Multivariable feedback control V3 22 of 36

23 Feedback Performance Specifications in the Frequency Domain Disturbance rejection (II) Interpretation for unit disturbance sinusoid d(t) = de jωt ; d 2 = 1 Worst error at frequency ω e 2 = σ max [S(jω)] Attained when d points along right singular vector associated with σ max Best error at frequency ω e 2 = σ min [S(jω)] Attained when d points along right singular vector associated with σ min In general σ min [S(jω)] e 2 σ max [S(jω)] 376_069 Multivariable feedback control V3 23 of 36

24 Feedback Performance Specifications in the Frequency Domain Quantitative relations Loop TFM: L(jω) = G(jω)K(jω) Sensitivity TFM: S(jω) = [Ι + L(jω)] -1 Closed-loop TFM: T(jω) = [Ι + L(jω)] -1 L(jω) Ω p = Ω r Ω d Key relations Let 0 < δ << 1 If σ max [S(jω)] δ 1 ; all ω Ω p Then 1 << 1 δ δ σ min [L(jω)] and ; all ω Ω p 1 - δ σ min [T(jω)] σ max [T(jω)] 1 + δ ; all ω Ω p T(jω) Ι - Proofs: not in this course 376_069 Multivariable feedback control V3 24 of 36

25 Feedback Performance Specifications in the Frequency Domain Comment Good command following and good disturbance rejection served by similar requirements ω Ω p = Ω r Ω d Large loop gain σ min [L(jω)] >> 1 Small sensitivity σ max [S(jω)] << 1 Flat closed-loop response σ min [T(jω)] σ max [T(jω)] 1 376_069 Multivariable feedback control V3 25 of 36

26 Feedback Performance Specifications in the Frequency Domain Insensitivity to sensor noise Sinusoidal noise yields sinusoidal error n(t) = ne jωt e(t) = ee jωt Relation e = T(jω)n e 2 σ max [T(jω)] n 2 Ω n range of frequencies where noise has significant energy Prescription for good sensor noise rejection make σ max [T(jω)]<<1 for all ω Ω n 376_069 Multivariable feedback control V3 26 of 36

27 Feedback Performance Specifications in the Frequency Domain Conflict with performance Let 0 < γ << 1 Suppose that Then σ max [T(jω)] γ for all ω Ω n σ min [L(jω)] σ max [L(jω)] γ 1-γ γ ω Ω n and low loop gain for all ω Ω n γ σ min [S(jω)] σ max [S(jω)] large sensitivity for all ω Ω n Bad command following and disturbance rejection in frequency range ω Ω n Consequence of constraint S(s) + T(s) = Ι 376_069 Multivariable feedback control V3 27 of 36

28 Feedback Performance Specifications in the Frequency Domain Design implications Need wide frequency separation between sets Ω p = Ω r Ω d and Ω n Cannot do good command following and disturbance rejection with noisy sensors that make low frequency errors (drift, bias, etc.) Stability-robustness to unmodelled highfrequency dynamics, far-away nonminimum phase zeros, and neglected small time-delays impact design in the same way as region Ω n 376_069 Multivariable feedback control V3 28 of 36

29 Feedback Performance Specifications in the Frequency Domain 376_069 Multivariable feedback control V3 29 of 36

30 Directional Information in Singular Value Plots Directional Information in Singular Value Plots Introduction Plots of min and max singular values vs. Frequency provide valuable insight into frequency domain properties of mimo systems Singular value decomposition provides directional information left singular vectors right singular vectors Need to understand and exploit directional information 376_069 Multivariable feedback control V3 30 of 36

31 Directional Information in Singular Value Plots SVD and linear equations y = Ax x A y SVD A = U Σ V H y = U Σ V H x Suppose : x = v i (right singular vector) y = U Σ V H v i Note (since v H i v j = δ ij ) V H v i = 0 : 1 : 0 (1 in row i) ; ΣV H v i = 0 : σ i : 0 (σ i in row i) then y i = σ i u i (u i left singular vector) Visualization 376_069 Multivariable feedback control V3 31 of 36

32 Directional Information in Singular Value Plots SVD directional information (I) Maximum singular value σ 1 (A) = σ max (A) - Associated max right singular vector v max = v 1 ; v max 2 = 1 - Associated max left singular vector u max = u 1 ; u max 2 = 1 Max amplification direction Let If Then y = Ax x = v max y = σ max u max y 2 = σ max (A) ;max amplification Visualization y Vmax = x Umax 376_069 Multivariable feedback control V3 32 of 36

33 Directional Information in Singular Value Plots SVD directional information (II) Minimum singular value: Rank(A)= m σ m (A) = σ min (A) - Associated min right singular vector v min = v m ; v min 2 = 1 - Associated min left singular vector u min = u m ; u min 2 = 1 Min amplification direction Let If Then y = Ax x = v min y = σ min u min y 2 = σ min (A) ;min amplification Visualization Vmin = x Umin y 376_069 Multivariable feedback control V3 33 of 36

34 Directional Information in Singular Value Plots Utilizing SVD directional information System: y(s) = G(s)u(s) ;G(s) m x m matrix Pick ω, calculate G(jω), do SVD G(jω) = U(jω) Σ(jω) V H (jω) Maximum direction analysis - Find σ max (ω), v max (ω), u max (ω) - Write [v max (ω)] i = a i e jψ i - Write [u max (ω)] i = b i e jφ i - Apply input u i (t) = a i sin(ωt + ψ i ) with u(t) = (u 1 (t); ;u i (t); u m (t)) then y i (t) = σ max b i sin(ωt + φ i ) Minimum directional analysis similar Complex directions of right and left singular vectors correspond to sinusoidal vectors with relative phase-shift among their elements 376_069 Multivariable feedback control V3 34 of 36

35 Directional Information in Singular Value Plots DC gain matrix analysis SVD-based direction analysis easiest at DC because plant is real (ω = 0) Plant model (strictly stable) dx(t)/dt = Ax(t) + Bu(t) y(t) = Cx(t) G(s) = C(sΙ - A) -1 B Plant DC gain matrix s = jω = 0 G(0) = - CA -1 B At steady-state u(t) = u = real constant vector y(t ) = y = real constant vector y = G(0)u or y = - CA -1 Bu 376_069 Multivariable feedback control V3 35 of 36

36 Directional Information in Singular Value Plots Steady-state analysis SVD at G(0) SVD at DC y = G(0)u G(0) = - CA -1 B = real G(0) = UΣV H (U, Σ, V = real) Max amplification direction If u = v max ; u 2 = 1 Then y = σ max u max Min amplification direction If u = v min ; u 2 = 1 Then y = σ min u min Above provides valuable insight upon MIMO plant characteristics at DC 376_069 Multivariable feedback control V3 36 of 36

### Control Systems 2. Lecture 4: Sensitivity function limits. Roy Smith

Control Systems 2 Lecture 4: Sensitivity function limits Roy Smith 2017-3-14 4.1 Input-output controllability Control design questions: 1. How well can the plant be controlled? 2. What control structure

### 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

### 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

### 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 =

### (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax =

. (5 points) (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? dim N(A), since rank(a) 3. (b) If we also know that Ax = has no solution, what do we know about the rank of A? C(A)

### RELAY CONTROL WITH PARALLEL COMPENSATOR FOR NONMINIMUM PHASE PLANTS. Ryszard Gessing

RELAY CONTROL WITH PARALLEL COMPENSATOR FOR NONMINIMUM PHASE PLANTS Ryszard Gessing Politechnika Śl aska Instytut Automatyki, ul. Akademicka 16, 44-101 Gliwice, Poland, fax: +4832 372127, email: gessing@ia.gliwice.edu.pl

### 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

### 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

### 2. Review of Linear Algebra

2. Review of Linear Algebra ECE 83, Spring 217 In this course we will represent signals as vectors and operators (e.g., filters, transforms, etc) as matrices. This lecture reviews basic concepts from linear

### 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

### 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

### 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

### 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

### 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.

### A NEW APPROACH TO MIXED H 2 /H OPTIMAL PI/PID CONTROLLER DESIGN

Copyright 2002 IFAC 15th Triennial World Congress, Barcelona, Spain A NEW APPROACH TO MIXED H 2 /H OPTIMAL PI/PID CONTROLLER DESIGN Chyi Hwang,1 Chun-Yen Hsiao Department of Chemical Engineering National

### Lecture 3: Review of Linear Algebra

ECE 83 Fall 2 Statistical Signal Processing instructor: R Nowak, scribe: R Nowak Lecture 3: Review of Linear Algebra Very often in this course we will represent signals as vectors and operators (eg, filters,

### 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

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

### Singular Value Decomposition

Chapter 6 Singular Value Decomposition In Chapter 5, we derived a number of algorithms for computing the eigenvalues and eigenvectors of matrices A R n n. Having developed this machinery, we complete our

### Basic Properties of Feedback

4 Basic Properties of Feedback A Perspective on the Properties of Feedback A major goal of control design is to use the tools available to keep the error small for any input and in the face of expected

### Properties of Matrices and Operations on Matrices

Properties of Matrices and Operations on Matrices A common data structure for statistical analysis is a rectangular array or matris. Rows represent individual observational units, or just observations,

### 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

### PID controllers, part I

Faculty of Mechanical and Power Engineering Dr inŝ. JANUSZ LICHOTA CONTROL SYSTEMS PID controllers, part I Wrocław 2007 CONTENTS Controller s classification PID controller what is it? Typical controller

### 18.06 Problem Set 8 - Solutions Due Wednesday, 14 November 2007 at 4 pm in

806 Problem Set 8 - Solutions Due Wednesday, 4 November 2007 at 4 pm in 2-06 08 03 Problem : 205+5+5+5 Consider the matrix A 02 07 a Check that A is a positive Markov matrix, and find its steady state

### Parallel Singular Value Decomposition. Jiaxing Tan

Parallel Singular Value Decomposition Jiaxing Tan Outline What is SVD? How to calculate SVD? How to parallelize SVD? Future Work What is SVD? Matrix Decomposition Eigen Decomposition A (non-zero) vector

### 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

### Review of Linear Algebra

Review of Linear Algebra Dr Gerhard Roth COMP 40A Winter 05 Version Linear algebra Is an important area of mathematics It is the basis of computer vision Is very widely taught, and there are many resources

### Modelling and Mathematical Methods in Process and Chemical Engineering

FS 07 February 0, 07 Modelling and Mathematical Methods in Process and Chemical Engineering Solution Series. Systems of linear algebraic equations: adjoint, determinant, inverse The adjoint of a (square)

### 100 (s + 10) (s + 100) e 0.5s. s 100 (s + 10) (s + 100). G(s) =

1 AME 3315; Spring 215; Midterm 2 Review (not graded) Problems: 9.3 9.8 9.9 9.12 except parts 5 and 6. 9.13 except parts 4 and 5 9.28 9.34 You are given the transfer function: G(s) = 1) Plot the bode plot

### 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

### Closed-Form Solution Of Absolute Orientation Using Unit Quaternions

Closed-Form Solution Of Absolute Orientation Using Unit Berthold K. P. Horn Department of Computer and Information Sciences November 11, 2004 Outline 1 Introduction 2 3 The Problem Given: two sets of corresponding

### Image Registration Lecture 2: Vectors and Matrices

Image Registration Lecture 2: Vectors and Matrices Prof. Charlene Tsai Lecture Overview Vectors Matrices Basics Orthogonal matrices Singular Value Decomposition (SVD) 2 1 Preliminary Comments Some of this

### Introduction - Motivation. Many phenomena (physical, chemical, biological, etc.) are model by differential equations. f f(x + h) f(x) (x) = lim

Introduction - Motivation Many phenomena (physical, chemical, biological, etc.) are model by differential equations. Recall the definition of the derivative of f(x) f f(x + h) f(x) (x) = lim. h 0 h Its

### 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

### Homework 5 EE235, Summer 2013 Solution

Homework 5 EE235, Summer 23 Solution. Fourier Series. Determine w and the non-zero Fourier series coefficients for the following functions: (a f(t 2 cos(3πt + sin(πt + π 3 w π f(t e j3πt + e j3πt + j2

### CIRCUIT ANALYSIS II. (AC Circuits)

Will Moore MT & MT CIRCUIT ANALYSIS II (AC Circuits) Syllabus Complex impedance, power factor, frequency response of AC networks including Bode diagrams, second-order and resonant circuits, damping and

### Order Reduction for Large Scale Finite Element Models: a Systems Perspective

Order Reduction for Large Scale Finite Element Models: a Systems Perspective William Gressick, John T. Wen, Jacob Fish ABSTRACT Large scale finite element models are routinely used in design and optimization

### Computational Methods CMSC/AMSC/MAPL 460. EigenValue decomposition Singular Value Decomposition. Ramani Duraiswami, Dept. of Computer Science

Computational Methods CMSC/AMSC/MAPL 460 EigenValue decomposition Singular Value Decomposition Ramani Duraiswami, Dept. of Computer Science Hermitian Matrices A square matrix for which A = A H is said

### 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

### Robust Control with Classical Methods QFT

Robust Control with Classical Methods QT Per-Olof utman Review of the classical Bode-Nichols control problem QT in the basic Single nput Single Output (SSO) case undamental Design Limitations dentification

### d n 1 f dt n 1 + K+ a 0f = C cos(ωt + φ)

Tutorial TUTOR: THE PHASOR TRANSFORM All voltages currents in linear circuits with sinusoidal sources are described by constant-coefficient linear differential equations of the form (1) a n d n f dt n

### 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

### 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

### 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

### BUMPLESS SWITCHING CONTROLLERS. William A. Wolovich and Alan B. Arehart 1. December 27, Abstract

BUMPLESS SWITCHING CONTROLLERS William A. Wolovich and Alan B. Arehart 1 December 7, 1995 Abstract This paper outlines the design of bumpless switching controllers that can be used to stabilize MIMO plants

### EE/ACM Applications of Convex Optimization in Signal Processing and Communications Lecture 2

EE/ACM 150 - Applications of Convex Optimization in Signal Processing and Communications Lecture 2 Andre Tkacenko Signal Processing Research Group Jet Propulsion Laboratory April 5, 2012 Andre Tkacenko

### 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

### Reflections in Hilbert Space III: Eigen-decomposition of Szegedy s operator

Reflections in Hilbert Space III: Eigen-decomposition of Szegedy s operator James Daniel Whitfield March 30, 01 By three methods we may learn wisdom: First, by reflection, which is the noblest; second,

### Ph.D. Katarína Bellová Page 1 Mathematics 2 (10-PHY-BIPMA2) EXAM - Solutions, 20 July 2017, 10:00 12:00 All answers to be justified.

PhD Katarína Bellová Page 1 Mathematics 2 (10-PHY-BIPMA2 EXAM - Solutions, 20 July 2017, 10:00 12:00 All answers to be justified Problem 1 [ points]: For which parameters λ R does the following system

### Lecture 1 Introduction

L. Vandenberghe EE236A (Fall 2013-14) Lecture 1 Introduction course overview linear optimization examples history approximate syllabus basic definitions linear optimization in vector and matrix notation

### Lecture: Sampling. Automatic Control 2. Sampling. Prof. Alberto Bemporad. University of Trento. Academic year

Automatic Control 2 Sampling Prof. Alberto Bemporad University of rento Academic year 2010-2011 Prof. Alberto Bemporad (University of rento) Automatic Control 2 Academic year 2010-2011 1 / 31 ime-discretization

### Principal Components Analysis (PCA) and Singular Value Decomposition (SVD) with applications to Microarrays

Principal Components Analysis (PCA) and Singular Value Decomposition (SVD) with applications to Microarrays Prof. Tesler Math 283 Fall 2015 Prof. Tesler Principal Components Analysis Math 283 / Fall 2015

### Transfer func+ons, block diagram algebra, and Bode plots. by Ania- Ariadna Bae+ca CDS Caltech 11/05/15

Transfer func+ons, block diagram algebra, and Bode plots by Ania- Ariadna Bae+ca CDS Caltech 11/05/15 Going back and forth between the +me and the frequency domain (1) Transfer func+ons exist only for

### Astro 250 Crash course on Control Systems Part I, March 3, 2003 Andy Packard, Zachary Jarvis-Wloszek, Weehong Tan, Eric Wemhoff

Astro 25 Crash course on Control Systems Part I, March 3, 23 Andy Packard, Zachary Jarvis-Wloszek, Weehong Tan, Eric Wemhoff pack@me.berkeley.edu 1 Feedback Systems Motivation Process to be controlled

### Boolean Inner-Product Spaces and Boolean Matrices

Boolean Inner-Product Spaces and Boolean Matrices Stan Gudder Department of Mathematics, University of Denver, Denver CO 80208 Frédéric Latrémolière Department of Mathematics, University of Denver, Denver

### 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

### c c c c c c c c c c a 3x3 matrix C= has a determinant determined by

Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant.

### Minimum Time Control of A Second-Order System

49th IEEE Conference on Decision and Control December 5-7, 2 Hilton Atlanta Hotel, Atlanta, GA, USA Minimum Time Control of A Second-Order System Zhaolong Shen and Sean B. Andersson Department of Mechanical

### Overview of Bode Plots Transfer function review Piece-wise linear approximations First-order terms Second-order terms (complex poles & zeros)

Overview of Bode Plots Transfer function review Piece-wise linear approximations First-order terms Second-order terms (complex poles & zeros) J. McNames Portland State University ECE 222 Bode Plots Ver.

### 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

### Feedback Control of Linear SISO systems. Process Dynamics and Control

Feedback Control of Linear SISO systems Process Dynamics and Control 1 Open-Loop Process The study of dynamics was limited to open-loop systems Observe process behavior as a result of specific input signals

### 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

### Sub-Stiefel Procrustes problem. Krystyna Ziętak

Sub-Stiefel Procrustes problem (Problem Procrustesa z macierzami sub-stiefel) Krystyna Ziętak Wrocław April 12, 2016 Outline 1 Joao Cardoso 2 Orthogonal Procrustes problem 3 Unbalanced Stiefel Procrustes

### MA 575 Linear Models: Cedric E. Ginestet, Boston University Regularization: Ridge Regression and Lasso Week 14, Lecture 2

MA 575 Linear Models: Cedric E. Ginestet, Boston University Regularization: Ridge Regression and Lasso Week 14, Lecture 2 1 Ridge Regression Ridge regression and the Lasso are two forms of regularized

### Linear systems, small signals, and integrators

Linear systems, small signals, and integrators CNS WS05 Class Giacomo Indiveri Institute of Neuroinformatics University ETH Zurich Zurich, December 2005 Outline 1 Linear Systems Crash Course Linear Time-Invariant

### CONTROL SYSTEMS, ROBOTICS AND AUTOMATION - Vol. VIII - Design Techniques in the Frequency Domain - Edmunds, J.M. and Munro, N.

DESIGN TECHNIQUES IN THE FREQUENCY DOMAIN Edmunds, Control Systems Center, UMIST, UK Keywords: Multivariable control, frequency response design, Nyquist, scaling, diagonal dominance Contents 1. Frequency

### Transistor amplifiers: Biasing and Small Signal Model

Transistor amplifiers: iasing and Small Signal Model Transistor amplifiers utilizing JT or FT are similar in design and analysis. Accordingly we will discuss JT amplifiers thoroughly. Then, similar FT

### 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

### [POLS 8500] Review of Linear Algebra, Probability and Information Theory

[POLS 8500] Review of Linear Algebra, Probability and Information Theory Professor Jason Anastasopoulos ljanastas@uga.edu January 12, 2017 For today... Basic linear algebra. Basic probability. Programming

### Transform Representation of Signals

C H A P T E R 3 Transform Representation of Signals and LTI Systems As you have seen in your prior studies of signals and systems, and as emphasized in the review in Chapter 2, transforms play a central

### The Dirty MIMO Multiple-Access Channel

The Dirty MIMO Multiple-Access Channel Anatoly Khina, Caltech Joint work with: Yuval Kochman, Hebrew University Uri Erez, Tel-Aviv University ISIT 2016 Barcelona, Catalonia, Spain July 12, 2016 Motivation:

### ECE382/ME482 Spring 2005 Homework 7 Solution April 17, K(s + 0.2) s 2 (s + 2)(s + 5) G(s) =

ECE382/ME482 Spring 25 Homework 7 Solution April 17, 25 1 Solution to HW7 AP9.5 We are given a system with open loop transfer function G(s) = K(s +.2) s 2 (s + 2)(s + 5) (1) and unity negative feedback.

### HW #3 Solutions: M552 Spring (c)

HW #3 Solutions: M55 Spring 6. (4.-Trefethen & Bau: parts (a), (c) and (e)) Determine the SVDs of the following matrices (by hand calculation): (a) [ 3 (c) (e) [ ANS: In each case we seek A UΣV. The general

### Phasor Diagram. Figure 1: Phasor Diagram. A φ. Leading Direction. θ B. Lagging Direction. Imag. Axis Complex Plane. Real Axis

1 16.202: PHASORS Consider sinusoidal source i(t) = Acos(ωt + φ) Using Eulers Notation: Acos(ωt + φ) = Re[Ae j(ωt+φ) ] Phasor Representation of i(t): = Ae jφ = A φ f v(t) = Bsin(ωt + ψ) First convert the

### Disturbance Estimation and Rejection An equivalent input disturbance estimator approach

Disturbance Estimation and Rejection n equivalent input disturbance estimator approach Jin-Hua She, Hiroyuki Kobayashi, Yasuhiro Ohyama and Xin Xin bstract his paper presents a new method of improving

### CHAPTER 7 STEADY-STATE RESPONSE ANALYSES

CHAPTER 7 STEADY-STATE RESPONSE ANALYSES 1. Introduction The steady state error is a measure of system accuracy. These errors arise from the nature of the inputs, system type and from nonlinearities of

### Ensembles and incomplete information

p. 1/32 Ensembles and incomplete information So far in this course, we have described quantum systems by states that are normalized vectors in a complex Hilbert space. This works so long as (a) the system

### Pitch Rate CAS Design Project

Pitch Rate CAS Design Project Washington University in St. Louis MAE 433 Control Systems Bob Rowe 4.4.7 Design Project Part 2 This is the second part of an ongoing project to design a control and stability

### Laboratory 11 Control Systems Laboratory ECE3557. State Feedback Controller for Position Control of a Flexible Joint

Laboratory 11 State Feedback Controller for Position Control of a Flexible Joint 11.1 Objective The objective of this laboratory is to design a full state feedback controller for endpoint position control

### Additional Closed-Loop Frequency Response Material (Second edition, Chapter 14)

Appendix J Additional Closed-Loop Frequency Response Material (Second edition, Chapter 4) APPENDIX CONTENTS J. Closed-Loop Behavior J.2 Bode Stability Criterion J.3 Nyquist Stability Criterion J.4 Gain

### 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

### Computation of Stabilizing PI and PID parameters for multivariable system with time delays

Computation of Stabilizing PI and PID parameters for multivariable system with time delays Nour El Houda Mansour, Sami Hafsi, Kaouther Laabidi Laboratoire d Analyse, Conception et Commande des Systèmes

### IDENTIFICATION FOR CONTROL

IDENTIFICATION FOR CONTROL Raymond A. de Callafon, University of California San Diego, USA Paul M.J. Van den Hof, Delft University of Technology, the Netherlands Keywords: Controller, Closed loop model,

### Dynamic System Response. Dynamic System Response K. Craig 1

Dynamic System Response Dynamic System Response K. Craig 1 Dynamic System Response LTI Behavior vs. Non-LTI Behavior Solution of Linear, Constant-Coefficient, Ordinary Differential Equations Classical

### HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION)

HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION) PROFESSOR STEVEN MILLER: BROWN UNIVERSITY: SPRING 2007 1. CHAPTER 1: MATRICES AND GAUSSIAN ELIMINATION Page 9, # 3: Describe

### A New Internal Model Control Method for MIMO Over-Actuated Systems

Vol. 7, No., 26 A New Internal Model Control Method for MIMO Over-Actuated Systems Ahmed Dhahri University of Tuins El Manar, National Engineering School of Tunis, Laboratory of research in Automatic Control,

### MATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)

MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m

### VI. Transistor amplifiers: Biasing and Small Signal Model

VI. Transistor amplifiers: iasing and Small Signal Model 6.1 Introduction Transistor amplifiers utilizing JT or FET are similar in design and analysis. Accordingly we will discuss JT amplifiers thoroughly.

### Lecture 7 Spectral methods

CSE 291: Unsupervised learning Spring 2008 Lecture 7 Spectral methods 7.1 Linear algebra review 7.1.1 Eigenvalues and eigenvectors Definition 1. A d d matrix M has eigenvalue λ if there is a d-dimensional

### On the Stability of Linear Systems

On the Stability of Linear Systems by Daniele Sasso * Abstract The criteria of stability defined in the standard theory of linear systems aren t exhaustive and show some inconsistencies. In this article

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

### 5.6. PSEUDOINVERSES 101. A H w.

5.6. PSEUDOINVERSES 0 Corollary 5.6.4. If A is a matrix such that A H A is invertible, then the least-squares solution to Av = w is v = A H A ) A H w. The matrix A H A ) A H is the left inverse of A and

### Chemical Process Dynamics and Control. Aisha Osman Mohamed Ahmed Department of Chemical Engineering Faculty of Engineering, Red Sea University

Chemical Process Dynamics and Control Aisha Osman Mohamed Ahmed Department of Chemical Engineering Faculty of Engineering, Red Sea University 1 Chapter 4 System Stability 2 Chapter Objectives End of this

### Math 443 Differential Geometry Spring Handout 3: Bilinear and Quadratic Forms This handout should be read just before Chapter 4 of the textbook.

Math 443 Differential Geometry Spring 2013 Handout 3: Bilinear and Quadratic Forms This handout should be read just before Chapter 4 of the textbook. Endomorphisms of a Vector Space This handout discusses

### FAST RECOVERY ALGORITHMS FOR TIME ENCODED BANDLIMITED SIGNALS. Ernő K. Simonyi

FAST RECOVERY ALGORITHS FOR TIE ENCODED BANDLIITED SIGNALS Aurel A Lazar Dept of Electrical Engineering Columbia University, New York, NY 27, USA e-mail: aurel@eecolumbiaedu Ernő K Simonyi National Council

### I = i 0,

Special Types of Matrices Certain matrices, such as the identity matrix 0 0 0 0 0 0 I = 0 0 0, 0 0 0 have a special shape, which endows the matrix with helpful properties The identity matrix is an example

### 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,

### Solving a RLC Circuit using Convolution with DERIVE for Windows

Solving a RLC Circuit using Convolution with DERIVE for Windows Michel Beaudin École de technologie supérieure, rue Notre-Dame Ouest Montréal (Québec) Canada, H3C K3 mbeaudin@seg.etsmtl.ca - Introduction