CDS 101: Lecture 4.1 Linear Systems
|
|
- Osborne Byrd
- 5 years ago
- Views:
Transcription
1 CDS : Lecture 4. Linear Systems Richard M. Murray 8 October 4 Goals: Describe linear system models: properties, eamples, and tools Characterize stability and performance of linear systems in terms of eigenvalues Compute linearization of a nonlinear systems around an equilibrium point Reading: Åström and Murray, Analysis and Design of Feedback Systems, Ch 4 Packard, Poola and Horowitz, Dynamic Systems and Feedback, Sections 9,, (available via course web page)
2 Lecture Review 3.: Stability from Last and Week Performance - -π π Key topics for this lecture Stability of equilibrium points Local versus global behavior Performance specification via step and frequency response.5 u y Oct 4 R. M. Murray, Caltech CDS
3 What is a Linear System? Linearity of functions: f : Zero at the origin: f () = n Addition: f ( + y) = f( ) + f( y) Scaling: f ( α ) = α f ( ) m f( α + β y) = Canonical eample: α f ( ) + β f( y) f ( ) = A Linearity of systems: sums of solutions Dynamical system = A Control system = A+ Bu y = C + Du () = t () = () t () = α + β () = t () = () t t () = α() t + β() t () =, u( t) = u ( t) y() t = y () t () =, u( t) = u ( t) y() t = y () t () =, u( t) = αu ( t) + βu ( t) y() t = αy () t + βy () t 8 Oct 4 R. M. Murray, Caltech CDS 3
4 Linear Systems u u = A+ Bu y = C + Du () = y y Input/output linearity at () = Linear systems are linear in initial condition and input need to use () = to add outputs together For different initial conditions, you need to be more careful (sounds like a good midterm question) u u + u y y + y Linear system step response and frequency response scale with input amplitude X input X output Allows us to use ratios and percen-tages in step/freq response. These are independent of input amplitude Limitation: input saturation only holds up to certain input amplitude 8 Oct 4 R. M. Murray, Caltech CDS 4
5 Why are Linear Systems Important? Many important eamples Electronic circuits Many important tools Frequency response, step response, etc Traditional tools of control theory Developed in 93 s at Bell Labs; intercontinental telecom Especially true after feedback Frequency response is key performance specification (think telephones) Many mechanical systems q k m q Quantum mechanics, Markov chains, u (t ) k m k 3 b Classical control design toolbo Nyquist plots, gain/phase margin Loop shaping Optimal control and estimators Linear quadratic regulators Kalman estimators Robust control design H control design μ analysis for structured uncertainty 8 Oct 4 R. M. Murray, Caltech CDS 5 CDS / a CDS b CDS b/
6 Solutions of Linear Systems: The Matri Eponential = A+ Bu y = C+ Du Scalar linear system, with no input yt ( ) =??? y = a = c () = t () at at = e yt () = ce Matri version, with no input = A () = y = C t () e At yt () = Ce At = initial(a,b,c,d,); Matri eponential Analog to the scalar case; defined by series epansion: e M I M M M! 3! 3 = L P = epm(m) 8 Oct 4 R. M. Murray, Caltech CDS 6
7 = A+ Bu y = C+ Du Stability of Linear Systems () t = e At Q: when is the system asymptotically stable? lim t ( ) = t Stability is determined by the eigenvalues of the matri A Simple case: diagonal system λ = O λn λt e t () = O λnt e More generally: transform to Jordan form & = T JT J J = O J k λi = O λi Form of eigenvalues determines system behavior Linear systems are automatically globally stable or unstable J i Stable if λ i Asy stable if λ i < Unstable if λ i > Asy stable if Re(λ i ) < Unstable if Re( λ i )> Indeterminate if Re( λ i )= 8 Oct 4 R. M. Murray, Caltech CDS 7
8 Eigenstructure of Linear Systems Real e-values Re(λ i ) < Real e-values Re(λ i ) < Re(λ j ) > Comple e-values Re(λ i ) = Comple e-values Re(λ i ) < Oct 4 R. M. Murray, Caltech CDS 8
9 Step and Frequency Response = A+ Bu y = C+ Du ut () = () t ut () = Asin( ωt) Effect of eigenstructure on step response Comple eigenvalues with small real part lead to oscillatory response Frequency of oscillations ω i Effects of eigenstructure on frequency response Eigenvalues determine break points for frequency response Comple eigenvalues lead to peaks in response function near ω i Imag Ais Phase (deg); Magnitude (db) - Pole-zero map ω p Real Ais Amplitude Bode Diagrams ω p - Frequency (rad/sec) Step Response T π/ω p Time (sec.) 8 Oct 4 R. M. Murray, Caltech CDS 9
10 Computing Frequency Responses Technique #: plot input and output, measure relative amplitude and phase Use MATLAB or SIMULINK to generate response of system to sinusoidal output.5.5 A u A y u y Gain = A y /A u Phase = π ΔT/T Note: In general, gain and phase will depend on the input amplitude ΔT T Technique # (linear systems): use MATLAB bode command Assumes linear dynamics in state space form: = A+ Bu y = C+ Du Gain plotted on log-log scale db = log (gain) Phase plotted on linear-log scale Frequency (rad/sec) 8 Oct 4 R. M. Murray, Caltech CDS Phase (deg) Magnitude (db) bode(a,b,c,d).
11 Eample: Electrical Circuit C 3 v i ~ R R C v o Derivation based on Kirchoff s laws for electrical circuits (Ph ) Sum of currents at nodes = : dv v v v v3 dv ( 3 v) v3 v C = C = dt R R dt R Rewrite in terms of new states: v c =v, v c =v 3 v Bridged Tee Circuit d dt v + c C R v R CR c + C R R v v = c v + c V c CR CR i v o [ ] v v c = + c v i 8 Oct 4 R. M. Murray, Caltech CDS
12 Linear Control Systems and Convolution = A+ Bu y = C+ Du Impulse response, h(t) = Ce At B Response to input impulse Equivalent to Green s function yt () = Ce At () +??? homogeneous Linearity compose response to arbitrary u(t) using convolution Decompose input into sum of shifted impulse functions Compute impulse response for each Sum impulse response to find y(t) Complete solution: use integral instead of sum t () = At A( t τ ) () + ( τ) τ + () τ = yt Ce Ce Bu d Dut linear with respect to initial condition and input X input X output when () = 8 Oct 4 R. M. Murray, Caltech CDS
13 Matlab Tools for Linear Systems t At A( t τ ) () = () + ( τ) τ + () yt Ce Ce Bu d Dut τ = A = [- ; -]; B = [; ]; C = [ ]; D = []; = [;.5]; sys = ss(a,b,c,d); initial(sys, ); impulse(sys); Amplitude To: Y() Initial Condition Results Linear Simulation Results t = :.:; u =.*sin(5*t) + cos(*t); lsim(sys, u, t, ); Other MATLAB commands gensig, square, sawtooth produce signals of diff. types step, impulse, initial, lsim time domain analysis bode, freqresp, evalfr frequency domain analysis Amplitude To: Y() Time (sec.) ltiview linear time invariant system plots 8 Oct 4 R. M. Murray, Caltech CDS 3
14 Linearization Around an Equilibrium Point = f(, u) z& = Az+ Bv y = h(, u) w= Cz+ Dv Linearize around = e f( e, ue) = ye = h( e, ue) z = v = u u w= y y e e e f A= B = f u (, u ) (, u ) e e e e h C = D = h u (, u ) (, u ) e e e e Remarks In eamples, this is often equivalent to small angle approimations, etc Only works near to equilibrium point - -π π Full nonlinear model Linear model (honest!) 8 Oct 4 R. M. Murray, Caltech CDS 4
15 Local Stability of Nonlinear Systems Asymptotic stability of the linearization implies local asymptotic stability of equilibrium point Linearization around equilibrium point captures tangent dynamics = f( ) = A ( + o ( ) e ) e higher order terms If linearization is unstable, can conclude that nonlinear system is locally unstable If linearization is stable but not asymptotically stable, can t conclude anything about nonlinear system: 3 & = =± linearize & linearization is stable (but not asy stable) nonlinear system can be asy stable or unstable Local approimation particularly appropriate for control systems design Control often used to ensure system stays near desired equilibrium point If dynamics are well-approimated by linearization near equilibrium point, can use this to design the controller that keeps you there (!) 8 Oct 4 R. M. Murray, Caltech CDS 5
16 Eample: Inverted Pendulum on a Cart m θ f M State:, θ,, & θ Input: u = F Output: y = Linearize according to previous formula around θ = π 8 Oct 4 R. M. Murray, Caltech CDS 6
17 Summary: Linear Systems u = A+ Bu y = C+ Du () = y t () = At A( t τ ) () + ( τ) τ + () τ = yt Ce Ce Bu d Dut Properties of linear systems Linearity with respect to initial condition and inputs Stability characterized by eigenvalues Many applications and tools available Provide local description for nonlinear systems 8 Oct 4 R. M. Murray, Caltech CDS 7
CDS 101/110: Lecture 3.1 Linear Systems
CDS /: Lecture 3. Linear Systems Goals for Today: Describe and motivate linear system models: Summarize properties, examples, and tools Joel Burdick (substituting for Richard Murray) jwb@robotics.caltech.edu,
More informationCDS 101/110: Lecture 3.1 Linear Systems
CDS /: Lecture 3. Linear Systems Goals for Today: Revist and motivate linear time-invariant system models: Summarize properties, examples, and tools Convolution equation describing solution in response
More informationCDS 101/110a: Lecture 2.1 Dynamic Behavior
CDS 11/11a: Lecture 2.1 Dynamic Behavior Richard M. Murray 6 October 28 Goals: Learn to use phase portraits to visualize behavior of dynamical systems Understand different types of stability for an equilibrium
More informationCDS 101/110a: Lecture 2.1 Dynamic Behavior
CDS 11/11a: Lecture.1 Dynamic Behavior Richard M. Murray 6 October 8 Goals: Learn to use phase portraits to visualize behavior of dynamical systems Understand different types of stability for an equilibrium
More informationCDS 101: Lecture 5-1 Reachability and State Space Feedback
CDS 11: Lecture 5-1 Reachability and State Space Feedback Richard M. Murray 23 October 26 Goals: Define reachability of a control system Give tests for reachability of linear systems and apply to examples
More informationCALIFORNIA INSTITUTE OF TECHNOLOGY Control and Dynamical Systems
CDS 101 1. For each of the following linear systems, determine whether the origin is asymptotically stable and, if so, plot the step response and frequency response for the system. If there are multiple
More informationControl 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 informationCDS 101/110a: Lecture 10-2 Control Systems Implementation
CDS 101/110a: Lecture 10-2 Control Systems Implementation Richard M. Murray 5 December 2012 Goals Provide an overview of the key principles, concepts and tools from control theory - Classical control -
More informationCDS 110: Lecture 2-2 Modeling Using Differential Equations
CDS 110: Lecture 2-2 Modeling Using Differential Equations Richard M. Murray and Hideo Mabuchi 4 October 2006 Goals: Provide a more detailed description of the use of ODEs for modeling Provide examples
More informationCDS 101/110a: Lecture 8-1 Frequency Domain Design
CDS 11/11a: Lecture 8-1 Frequency Domain Design Richard M. Murray 17 November 28 Goals: Describe canonical control design problem and standard performance measures Show how to use loop shaping to achieve
More informationControl 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 informationControl 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 informationCDS 101/110a: Lecture 10-1 Robust Performance
CDS 11/11a: Lecture 1-1 Robust Performance Richard M. Murray 1 December 28 Goals: Describe how to represent uncertainty in process dynamics Describe how to analyze a system in the presence of uncertainty
More informationLinear Systems. Chapter Basic Definitions
Chapter 5 Linear Systems Few physical elements display truly linear characteristics. For example the relation between force on a spring and displacement of the spring is always nonlinear to some degree.
More informationFrequency 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 informationD(s) G(s) A control system design definition
R E Compensation D(s) U Plant G(s) Y Figure 7. A control system design definition x x x 2 x 2 U 2 s s 7 2 Y Figure 7.2 A block diagram representing Eq. (7.) in control form z U 2 s z Y 4 z 2 s z 2 3 Figure
More informationControl Systems I. Lecture 2: Modeling and Linearization. Suggested Readings: Åström & Murray Ch Jacopo Tani
Control Systems I Lecture 2: Modeling and Linearization Suggested Readings: Åström & Murray Ch. 2-3 Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich September 28, 2018 J. Tani, E.
More informationIntroduction to Modern Control MT 2016
CDT Autonomous and Intelligent Machines & Systems Introduction to Modern Control MT 2016 Alessandro Abate Lecture 2 First-order ordinary differential equations (ODE) Solution of a linear ODE Hints to nonlinear
More informationCDS 101: Lecture 5.1 Controllability and State Space Feedback
CDS, Lecture 5. CDS : Lecture 5. Cotrollability ad State Space Feedback Richard M. Murray 8 October Goals: Deie cotrollability o a cotrol system Give tests or cotrollability o liear systems ad apply to
More informationOutline. 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 informationExercise Sheet 1 with solutions
ecture Energy Systems: Hardware and Control - Control Part University of Freiburg Winterterm 27/28 Eercise Sheet with solutions Prof. Dr. Moritz Diehl, Dr. Gianluca Frison and Benjamin Stickan For questions
More informationEET 3212 Control Systems. Control Systems Engineering, 6th Edition, Norman S. Nise December 2010, A. Goykadosh and M.
NEW YORK CITY COLLEGE OF TECHNOLOGY The City University of New York 300 Jay Street Brooklyn, NY 11201-2983 Department of Electrical and Telecommunications Engineering Technology TEL (718) 260-5300 - FAX:
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering Dynamics and Control II Fall 2007
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering.4 Dynamics and Control II Fall 7 Problem Set #9 Solution Posted: Sunday, Dec., 7. The.4 Tower system. The system parameters are
More informationControl Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Jacopo Tani. Institute for Dynamic Systems and Control D-MAVT ETH Zürich
Control Systems I Lecture 7: Feedback and the Root Locus method Readings: Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 2, 2018 J. Tani, E. Frazzoli (ETH) Lecture 7:
More informationDr. Ian R. Manchester
Dr Ian R. Manchester Week Content Notes 1 Introduction 2 Frequency Domain Modelling 3 Transient Performance and the s-plane 4 Block Diagrams 5 Feedback System Characteristics Assign 1 Due 6 Root Locus
More informationEECS 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 informationÜbersetzungshilfe / 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 informationState 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 informationRichiami 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 informationTransfer function and linearization
Transfer function and linearization Daniele Carnevale Dipartimento di Ing. Civile ed Ing. Informatica (DICII), University of Rome Tor Vergata Corso di Controlli Automatici, A.A. 24-25 Testo del corso:
More informationEC Control Engineering Quiz II IIT Madras
EC34 - Control Engineering Quiz II IIT Madras Linear algebra Find the eigenvalues and eigenvectors of A, A, A and A + 4I Find the eigenvalues and eigenvectors of the following matrices: (a) cos θ sin θ
More informationẋ n = f n (x 1,...,x n,u 1,...,u m ) (5) y 1 = g 1 (x 1,...,x n,u 1,...,u m ) (6) y p = g p (x 1,...,x n,u 1,...,u m ) (7)
EEE582 Topical Outline A.A. Rodriguez Fall 2007 GWC 352, 965-3712 The following represents a detailed topical outline of the course. It attempts to highlight most of the key concepts to be covered and
More informationControl Systems Design
ELEC4410 Control Systems Design Lecture 13: Stability Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 13: Stability p.1/20 Outline Input-Output
More information2. Higher-order Linear ODE s
2. Higher-order Linear ODE s 2A. Second-order Linear ODE s: General Properties 2A-1. On the right below is an abbreviated form of the ODE on the left: (*) y + p()y + q()y = r() Ly = r() ; where L is the
More informationCDS 101: Lecture 2.1 System Modeling
CDS 101: Lecture 2.1 System Modeling Richard M. Murray 4 October 2004 Goals: Define what a model is and its use in answering questions about a system Introduce the concepts of state, dynamics, inputs and
More informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 21: Stability Margins and Closing the Loop Overview In this Lecture, you will learn: Closing the Loop Effect on Bode Plot Effect
More informationLinear System Theory. Wonhee Kim Lecture 1. March 7, 2018
Linear System Theory Wonhee Kim Lecture 1 March 7, 2018 1 / 22 Overview Course Information Prerequisites Course Outline What is Control Engineering? Examples of Control Systems Structure of Control Systems
More informationSome solutions of the written exam of January 27th, 2014
TEORIA DEI SISTEMI Systems Theory) Prof. C. Manes, Prof. A. Germani Some solutions of the written exam of January 7th, 0 Problem. Consider a feedback control system with unit feedback gain, with the following
More informationAppendix 3B MATLAB Functions for Modeling and Time-domain analysis
Appendix 3B MATLAB Functions for Modeling and Time-domain analysis MATLAB control system Toolbox contain the following functions for the time-domain response step impulse initial lsim gensig damp ltiview
More informationEEE582 Homework Problems
EEE582 Homework Problems HW. Write a state-space realization of the linearized model for the cruise control system around speeds v = 4 (Section.3, http://tsakalis.faculty.asu.edu/notes/models.pdf). Use
More informationHomogeneous Constant Matrix Systems, Part II
4 Homogeneous Constant Matri Systems, Part II Let us now epand our discussions begun in the previous chapter, and consider homogeneous constant matri systems whose matrices either have comple eigenvalues
More informationControl 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
More informationAPPLICATIONS FOR ROBOTICS
Version: 1 CONTROL APPLICATIONS FOR ROBOTICS TEX d: Feb. 17, 214 PREVIEW We show that the transfer function and conditions of stability for linear systems can be studied using Laplace transforms. Table
More informationSystems 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 informationSystems 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 informationDr 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 informationReturn Difference Function and Closed-Loop Roots Single-Input/Single-Output Control Systems
Spectral Properties of Linear- Quadratic Regulators Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 2018! Stability margins of single-input/singleoutput (SISO) systems! Characterizations
More informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 22: The Nyquist Criterion Overview In this Lecture, you will learn: Complex Analysis The Argument Principle The Contour
More informationModule 09 From s-domain to time-domain From ODEs, TFs to State-Space Modern Control
Module 09 From s-domain to time-domain From ODEs, TFs to State-Space Modern Control Ahmad F. Taha EE 3413: Analysis and Desgin of Control Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/
More informationMAE 143B - Homework 9
MAE 143B - Homework 9 7.1 a) We have stable first-order poles at p 1 = 1 and p 2 = 1. For small values of ω, we recover the DC gain K = lim ω G(jω) = 1 1 = 2dB. Having this finite limit, our straight-line
More informationECEN 420 LINEAR CONTROL SYSTEMS. Lecture 6 Mathematical Representation of Physical Systems II 1/67
1/67 ECEN 420 LINEAR CONTROL SYSTEMS Lecture 6 Mathematical Representation of Physical Systems II State Variable Models for Dynamic Systems u 1 u 2 u ṙ. Internal Variables x 1, x 2 x n y 1 y 2. y m Figure
More information1 Continuous-time Systems
Observability Completely controllable systems can be restructured by means of state feedback to have many desirable properties. But what if the state is not available for feedback? What if only the output
More informationSTABILITY OF CLOSED-LOOP CONTOL SYSTEMS
CHBE320 LECTURE X STABILITY OF CLOSED-LOOP CONTOL SYSTEMS Professor Dae Ryook Yang Spring 2018 Dept. of Chemical and Biological Engineering 10-1 Road Map of the Lecture X Stability of closed-loop control
More informationLecture 9 Nonlinear Control Design. Course Outline. Exact linearization: example [one-link robot] Exact Feedback Linearization
Lecture 9 Nonlinear Control Design Course Outline Eact-linearization Lyapunov-based design Lab Adaptive control Sliding modes control Literature: [Khalil, ch.s 13, 14.1,14.] and [Glad-Ljung,ch.17] Lecture
More informationState will have dimension 5. One possible choice is given by y and its derivatives up to y (4)
A Exercise State will have dimension 5. One possible choice is given by y and its derivatives up to y (4 x T (t [ y(t y ( (t y (2 (t y (3 (t y (4 (t ] T With this choice we obtain A B C [ ] D 2 3 4 To
More informationSAMPLE 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 informationÜbersetzungshilfe / 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 -
More informationPattern formation and Turing instability
Pattern formation and Turing instability. Gurarie Topics: - Pattern formation through symmetry breaing and loss of stability - Activator-inhibitor systems with diffusion Turing proposed a mechanism for
More informationEE 380. Linear Control Systems. Lecture 10
EE 380 Linear Control Systems Lecture 10 Professor Jeffrey Schiano Department of Electrical Engineering Lecture 10. 1 Lecture 10 Topics Stability Definitions Methods for Determining Stability Lecture 10.
More informationTopic # 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
More informationES.1803 Topic 19 Notes Jeremy Orloff. 19 Variation of parameters; exponential inputs; Euler s method
ES83 Topic 9 Notes Jeremy Orloff 9 Variation of parameters; eponential inputs; Euler s method 9 Goals Be able to derive and apply the eponential response formula for constant coefficient linear systems
More information21 Linear State-Space Representations
ME 132, Spring 25, UC Berkeley, A Packard 187 21 Linear State-Space Representations First, let s describe the most general type of dynamic system that we will consider/encounter in this class Systems may
More informationECE 3793 Matlab Project 3 Solution
ECE 3793 Matlab Project 3 Solution Spring 27 Dr. Havlicek. (a) In text problem 9.22(d), we are given X(s) = s + 2 s 2 + 7s + 2 4 < Re {s} < 3. The following Matlab statements determine the partial fraction
More informationDigital Control Systems
Digital Control Systems Lecture Summary #4 This summary discussed some graphical methods their use to determine the stability the stability margins of closed loop systems. A. Nyquist criterion Nyquist
More informationTopic # Feedback Control Systems
Topic #19 16.31 Feedback Control Systems Stengel Chapter 6 Question: how well do the large gain and phase margins discussed for LQR map over to DOFB using LQR and LQE (called LQG)? Fall 2010 16.30/31 19
More informationExam in Systems Engineering/Process Control
Department of AUTOMATIC CONTROL Exam in Systems Engineering/Process Control 27-6-2 Points and grading All answers must include a clear motivation. Answers may be given in English or Swedish. The total
More information1 (30 pts) Dominant Pole
EECS C8/ME C34 Fall Problem Set 9 Solutions (3 pts) Dominant Pole For the following transfer function: Y (s) U(s) = (s + )(s + ) a) Give state space description of the system in parallel form (ẋ = Ax +
More informationControl 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 informationCALIFORNIA INSTITUTE OF TECHNOLOGY Control and Dynamical Systems
CDS 101 1. Åström and Murray, Exercise 1.3 2. Åström and Murray, Exercise 1.4 3. Åström and Murray, Exercise 2.6, parts (a) and (b) CDS 110a 1. Åström and Murray, Exercise 1.4 2. Åström and Murray, Exercise
More informationVALLIAMMAI ENGINEERING COLLEGE
VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur 6 DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING QESTION BANK ME-Power Systems Engineering I st Year SEMESTER I IN55- SYSTEM THEORY Regulation
More informationECE557 Systems Control
ECE557 Systems Control Bruce Francis Course notes, Version.0, September 008 Preface This is the second Engineering Science course on control. It assumes ECE56 as a prerequisite. If you didn t take ECE56,
More informationx(n + 1) = Ax(n) and y(n) = Cx(n) + 2v(n) and C = x(0) = ξ 1 ξ 2 Ex(0)x(0) = I
A-AE 567 Final Homework Spring 213 You will need Matlab and Simulink. You work must be neat and easy to read. Clearly, identify your answers in a box. You will loose points for poorly written work. You
More informationIntroduction to Feedback Control
Introduction to Feedback Control Control System Design Why Control? Open-Loop vs Closed-Loop (Feedback) Why Use Feedback Control? Closed-Loop Control System Structure Elements of a Feedback Control System
More informationLab 6a: Pole Placement for the Inverted Pendulum
Lab 6a: Pole Placement for the Inverted Pendulum Idiot. Above her head was the only stable place in the cosmos, the only refuge from the damnation of the Panta Rei, and she guessed it was the Pendulum
More informationAnalog Signals and Systems and their properties
Analog Signals and Systems and their properties Main Course Objective: Recall course objectives Understand the fundamentals of systems/signals interaction (know how systems can transform or filter signals)
More informationEE 16B Midterm 2, March 21, Name: SID #: Discussion Section and TA: Lab Section and TA: Name of left neighbor: Name of right neighbor:
EE 16B Midterm 2, March 21, 2017 Name: SID #: Discussion Section and TA: Lab Section and TA: Name of left neighbor: Name of right neighbor: Important Instructions: Show your work. An answer without explanation
More informationModule 03 Linear Systems Theory: Necessary Background
Module 03 Linear Systems Theory: Necessary Background Ahmad F. Taha EE 5243: Introduction to Cyber-Physical Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha/index.html September
More informationExam. 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
More informationIntroduction to Nonlinear Control Lecture # 4 Passivity
p. 1/6 Introduction to Nonlinear Control Lecture # 4 Passivity È p. 2/6 Memoryless Functions ¹ y È Ý Ù È È È È u (b) µ power inflow = uy Resistor is passive if uy 0 p. 3/6 y y y u u u (a) (b) (c) Passive
More informationProblem Weight Score Total 100
EE 350 EXAM IV 15 December 2010 Last Name (Print): First Name (Print): ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Problem Weight Score 1 25 2 25 3 25 4 25 Total
More informationAutomatic Control 2. Loop shaping. Prof. Alberto Bemporad. University of Trento. Academic year
Automatic Control 2 Loop shaping Prof. Alberto Bemporad University of Trento Academic year 21-211 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 21-211 1 / 39 Feedback
More informationagree w/input bond => + sign disagree w/input bond => - sign
1 ME 344 REVIEW FOR FINAL EXAM LOCATION: CPE 2.204 M. D. BRYANT DATE: Wednesday, May 7, 2008 9-noon Finals week office hours: May 6, 4-7 pm Permitted at final exam: 1 sheet of formulas & calculator I.
More informationCONTROL DESIGN FOR SET POINT TRACKING
Chapter 5 CONTROL DESIGN FOR SET POINT TRACKING In this chapter, we extend the pole placement, observer-based output feedback design to solve tracking problems. By tracking we mean that the output is commanded
More informationProfessor Fearing EE C128 / ME C134 Problem Set 7 Solution Fall 2010 Jansen Sheng and Wenjie Chen, UC Berkeley
Professor Fearing EE C8 / ME C34 Problem Set 7 Solution Fall Jansen Sheng and Wenjie Chen, UC Berkeley. 35 pts Lag compensation. For open loop plant Gs ss+5s+8 a Find compensator gain Ds k such that the
More informationEE16B, Spring 2018 UC Berkeley EECS. Maharbiz and Roychowdhury. Lectures 4B & 5A: Overview Slides. Linearization and Stability
EE16B, Spring 2018 UC Berkeley EECS Maharbiz and Roychowdhury Lectures 4B & 5A: Overview Slides Linearization and Stability Slide 1 Linearization Approximate a nonlinear system by a linear one (unless
More informationIntroduction. Performance and Robustness (Chapter 1) Advanced Control Systems Spring / 31
Introduction Classical Control Robust Control u(t) y(t) G u(t) G + y(t) G : nominal model G = G + : plant uncertainty Uncertainty sources : Structured : parametric uncertainty, multimodel uncertainty Unstructured
More informationDynamic 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 information6.1 Sketch the z-domain root locus and find the critical gain for the following systems K., the closed-loop characteristic equation is K + z 0.
6. Sketch the z-domain root locus and find the critical gain for the following systems K (i) Gz () z 4. (ii) Gz K () ( z+ 9. )( z 9. ) (iii) Gz () Kz ( z. )( z ) (iv) Gz () Kz ( + 9. ) ( z. )( z 8. ) (i)
More information2.010 Fall 2000 Solution of Homework Assignment 1
2. Fall 2 Solution of Homework Assignment. Compact Disk Player. This is essentially a reprise of Problems and 2 from the Fall 999 2.3 Homework Assignment 7. t is included here to encourage you to review
More informationECE504: Lecture 8. D. Richard Brown III. Worcester Polytechnic Institute. 28-Oct-2008
ECE504: Lecture 8 D. Richard Brown III Worcester Polytechnic Institute 28-Oct-2008 Worcester Polytechnic Institute D. Richard Brown III 28-Oct-2008 1 / 30 Lecture 8 Major Topics ECE504: Lecture 8 We are
More informationL2 gains and system approximation quality 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.242, Fall 24: MODEL REDUCTION L2 gains and system approximation quality 1 This lecture discusses the utility
More information= 0 otherwise. Eu(n) = 0 and Eu(n)u(m) = δ n m
A-AE 567 Final Homework Spring 212 You will need Matlab and Simulink. You work must be neat and easy to read. Clearly, identify your answers in a box. You will loose points for poorly written work. You
More information3. Fundamentals of Lyapunov Theory
Applied Nonlinear Control Nguyen an ien -.. Fundamentals of Lyapunov heory he objective of this chapter is to present Lyapunov stability theorem and illustrate its use in the analysis and the design of
More informationDO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED.
EE 537 Homewors Friedland Text Updated: Wednesday November 8 Some homewor assignments refer to Friedland s text For full credit show all wor. Some problems require hand calculations. In those cases do
More informationState Feedback and State Estimators Linear System Theory and Design, Chapter 8.
1 Linear System Theory and Design, http://zitompul.wordpress.com 2 0 1 4 State Estimator In previous section, we have discussed the state feedback, based on the assumption that all state variables are
More informationChap 4. State-Space Solutions and
Chap 4. State-Space Solutions and Realizations Outlines 1. Introduction 2. Solution of LTI State Equation 3. Equivalent State Equations 4. Realizations 5. Solution of Linear Time-Varying (LTV) Equations
More informationControl 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 information100 (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
More informationENGIN 211, Engineering Math. Laplace Transforms
ENGIN 211, Engineering Math Laplace Transforms 1 Why Laplace Transform? Laplace transform converts a function in the time domain to its frequency domain. It is a powerful, systematic method in solving
More informationClassical Dual-Inverted-Pendulum Control
PRESENTED AT THE 23 IEEE CONFERENCE ON DECISION AND CONTROL 4399 Classical Dual-Inverted-Pendulum Control Kent H. Lundberg James K. Roberge Department of Electrical Engineering and Computer Science Massachusetts
More information