16.30/31, Fall 2010 Recitation # 1

Size: px
Start display at page:

Download "16.30/31, Fall 2010 Recitation # 1"

Transcription

1 6./, Fall Recitation # September, In this recitation we consiere the following problem. Given a plant with open-loop transfer function.569s +.5 G p (s) = s +.7s +.97, esign a feeback control system such that the close-loop ominant poles have unampe natural frequency ω n = ra/s an amping ratio ζ =. (This problem was mentione uring lecture, see the Topic notes, as a stability augmentation system for a B77.) r + - e G c (s) u G p (s) y Figure : The stanar block iagram for a unit-feeback loop. Since we are working with the root locus metho, it is convenient to rewrite the transfer function in the form z Zeroes(G (s z) p) s G p (s) = K p p Poles(G (s p) =.569 p) s +.7s Note that the root-locus gain of the plant s transfer function K p =.569 is negative. This is a fact that is easy to overlook, an may generate confusion if not recognize an hanle properly. More on this later. The pole-zero map for the system is shown in Figure, where: z =.969 is the zero (root of the numerator) of G p. p, =.75 ±.8888 are the poles (roots of the enominator) of G p. p, =.8 ±.j are the esire close-loop poles. (We know that the magnitue of the esire poles is equal to ω n =, an the real part is ζω n =.8.) Q. Can we achieve the control objective using proportional feeback? A first question one may want to answer is whether a simple proportional feeback (i.e., G c (s) = K, for some K) woul o the trick. You may convince yourself that this is not likely to be the case, e.g., by sketching the root locus. However, it is possible to check rigorously whether a point is on the root locus (i.e., whether the root locus goes through a given point), using the angle conition: (s z) (s p) = l8, l =, ±, ±,.... z Zeroes(G) p Poles(G) (In particular, the point will be on the positive-gain root locus for l o, an on the negativegain root locus for l even.)

2 Im(s) Re(s) Figure : The open loop poles (blue crosses) are in an unesirable locations: the response is too slow (ω n ra/s) an too lightly ampe (ζ.). The objective is to esign a control system such that the close loop system has its ominant poles in the location marke with re squares. Recall that, e.g., (s z) can be thought of as a vector going from z to s. The angle (s z) can be compute as, the arctangent of Im(s z)/re(s z). (You may want to use the atan function on your calculator or in Matlab to make sure you get the quarant right.) In our case, let us check where the esire poles are on the root locus. Due to symmetry, it is sufficient to check for one of them, i.e., for p =.8 +.j. We get p z =.5 +.j, hence (p z ) = atan(.,.5) =.6. p p = j, hence (p p ) = atan(.5,.95) =.. p p = j, hence (p p ) = atan(.888,.95) =.5. Hence, (p z ) (p p ) (p p ) =.8 = l8, l =, ±, ±,... an the point p is not on the root locus (for proportional feeback). Q. Can we achieve the control objective using a ynamic compensator? To make a long story short: yes. There are many ways one can achieve the objective. Let us consier one, we will iscuss others shortly. Let us use a ynamic compensator of the form G c (s) = K p s z c. s p c Let us choose p c = z, i.e., let us cancel the open-loop zero with a compensator pole. We still have two egrees of freeom remaining: the location of the zero z c an the gain K c. The angle conition in this case woul be (recall that the open-loop zero is cancele with the compensator pole): (p z c ) (p p ) (p p ) = l8, l =, ±, ±,...,

3 Root Locus Step Response Imaginary Axis Amplitue Real Axis Time (sec) Figure : Root locus an close-loop step response using the lag compensator propose in the text. that is (recall that we have alreay compute the angles of p p, ), (p z c ) = l8, l =, ±, ±,... In other wors, if we choose z c in such a way that (p z c ) = 66.9, we satisfy the angle conition, with l =. Since l is o, the esire poles will be on the positive-gain root locus for the propose feeback system. In orer to fin z c, just set (p z c ) = atan(.,.8 z c ) = 66.9, i.e.,. z c =.8 =.8. tan 66.9 Notice that given the location of the compensator pole an zero, this is a lag compensator (the pole is closer to the origin). The magnitue ratio between the pole an the zero is within the stanar guielines of a factor between 5 an. The last step in the esign of the compensator is the choice of the gain. The magnitue conition can be written as: z Zeroes(G) s z =, s p K z Poles(G) where G = G c G p an K = K c K p. Doing the calculations (remember we can ignore the cancele pole/zero), we get: K =.859, an ultimately K c = K / K p =.7. Since K p < as alreay mentione, we can conclue K c =.7, an s +.8 G c =.7. s For verification purposes, we can compute the characteristic equation of the close-loop system: s +.7s (s +.8) s +.6s G c G p = =,......

4 Root Locus Step Response Imaginary Axis Amplitue Real Axis Time (sec) Figure : Root locus an close-loop step response using the simple compensator with one pole iscusse uring the recitation. which shows that the close-loop poleas are inee at the esire locations. A similar lag-compensation approach woul have worke, without the nee for pole-zero cancellation, but the algebra woul have been a bit more complicate. The root locus an the (close-loop) step response using the propose compensator are shown in Figure. Other approaches Simple lag One approach propose in class was consier a ynamic compensator with only one pole p c. The angle conition can be written as (p z ) (p p ) (p p ) (p p c ) =.8 (p p c ) = l8, l =, ±, ±,..., i.e., we can choose (p p c ) = 55.6, satisfying the angle conition with l =. Since l is o, the esire close loop poles will be on the positive-gain root locus. Proceeing as in the previous case, we can compute p c =.8./ tan 55.6 =.7. The magnitue conition woul yiel K c = Inee, two of the close-loop poles will be at the esire location. However, a new pole has been introuce in the system, an will actually play a ominant role in the response, which is not esirable see Figure. Also note that the steay-state error is quite large.

5 MIT OpenCourseWare 6. / 6. Feeback Control Systems Fall For information about citing these materials or our Terms of Use, visit:

6.003 Homework #7 Solutions

6.003 Homework #7 Solutions 6.003 Homework #7 Solutions Problems. Secon-orer systems The impulse response of a secon-orer CT system has the form h(t) = e σt cos(ω t + φ)u(t) where the parameters σ, ω, an φ are relate to the parameters

More information

EE 370L Controls Laboratory. Laboratory Exercise #7 Root Locus. Department of Electrical and Computer Engineering University of Nevada, at Las Vegas

EE 370L Controls Laboratory. Laboratory Exercise #7 Root Locus. Department of Electrical and Computer Engineering University of Nevada, at Las Vegas EE 370L Controls Laboratory Laboratory Exercise #7 Root Locus Department of Electrical an Computer Engineering University of Nevaa, at Las Vegas 1. Learning Objectives To emonstrate the concept of error

More information

and from it produce the action integral whose variation we set to zero:

and from it produce the action integral whose variation we set to zero: Lagrange Multipliers Monay, 6 September 01 Sometimes it is convenient to use reunant coorinates, an to effect the variation of the action consistent with the constraints via the metho of Lagrange unetermine

More information

16.30/31, Fall 2010 Recitation # 2

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

More information

Root Locus. 1 Review of related mathematics. Ang Man Shun. October 30, Complex Algebra in Polar Form. 1.2 Roots of a equation

Root Locus. 1 Review of related mathematics. Ang Man Shun. October 30, Complex Algebra in Polar Form. 1.2 Roots of a equation Root Locus Ang Man Shun October 3, 212 1 Review of relate mathematics 1.1 Complex Algebra in Polar Form For a complex number z, it can be expresse in polar form as z = re jθ 1 Im z Where r = z, θ = tan.

More information

First Order Linear Differential Equations

First Order Linear Differential Equations LECTURE 6 First Orer Linear Differential Equations A linear first orer orinary ifferential equation is a ifferential equation of the form ( a(xy + b(xy = c(x. Here y represents the unknown function, y

More information

2.004 Dynamics and Control II Spring 2008

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

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.04A Systems and Controls Spring 2013

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.04A Systems and Controls Spring 2013 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.04A Systems and Controls Spring 2013 Problem Set #4 Posted: Thursday, Mar. 7, 13 Due: Thursday, Mar. 14, 13 1. Sketch the Root

More information

Chapter 31: RLC Circuits. PHY2049: Chapter 31 1

Chapter 31: RLC Circuits. PHY2049: Chapter 31 1 Chapter 31: RLC Circuits PHY049: Chapter 31 1 LC Oscillations Conservation of energy Topics Dampe oscillations in RLC circuits Energy loss AC current RMS quantities Force oscillations Resistance, reactance,

More information

'HVLJQ &RQVLGHUDWLRQ LQ 0DWHULDO 6HOHFWLRQ 'HVLJQ 6HQVLWLYLW\,1752'8&7,21

'HVLJQ &RQVLGHUDWLRQ LQ 0DWHULDO 6HOHFWLRQ 'HVLJQ 6HQVLWLYLW\,1752'8&7,21 Large amping in a structural material may be either esirable or unesirable, epening on the engineering application at han. For example, amping is a esirable property to the esigner concerne with limiting

More information

G j dq i + G j. q i. = a jt. and

G j dq i + G j. q i. = a jt. and Lagrange Multipliers Wenesay, 8 September 011 Sometimes it is convenient to use reunant coorinates, an to effect the variation of the action consistent with the constraints via the metho of Lagrange unetermine

More information

19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control

19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control 19 Eigenvalues, Eigenvectors, Orinary Differential Equations, an Control This section introuces eigenvalues an eigenvectors of a matrix, an iscusses the role of the eigenvalues in etermining the behavior

More information

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

EE C128 / ME C134 Fall 2014 HW 6.2 Solutions. HW 6.2 Solutions EE C28 / ME C34 Fall 24 HW 6.2 Solutions. PI Controller For the system G = K (s+)(s+3)(s+8) HW 6.2 Solutions in negative feedback operating at a damping ratio of., we are going to design a PI controller

More information

d dx But have you ever seen a derivation of these results? We ll prove the first result below. cos h 1

d dx But have you ever seen a derivation of these results? We ll prove the first result below. cos h 1 Lecture 5 Some ifferentiation rules Trigonometric functions (Relevant section from Stewart, Seventh Eition: Section 3.3) You all know that sin = cos cos = sin. () But have you ever seen a erivation of

More information

Linear First-Order Equations

Linear First-Order Equations 5 Linear First-Orer Equations Linear first-orer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory)

More information

New Simple Controller Tuning Rules for Integrating and Stable or Unstable First Order plus Dead-Time Processes

New Simple Controller Tuning Rules for Integrating and Stable or Unstable First Order plus Dead-Time Processes Proceeings of the 3th WSEAS nternational Conference on SYSTEMS New Simple Controller Tuning Rules for ntegrating an Stable or Unstable First Orer plus Dea-Time Processes.G.ARVANTS Department of Natural

More information

Determine Power Transfer Limits of An SMIB System through Linear System Analysis with Nonlinear Simulation Validation

Determine Power Transfer Limits of An SMIB System through Linear System Analysis with Nonlinear Simulation Validation Determine Power Transfer Limits of An SMIB System through Linear System Analysis with Nonlinear Simulation Valiation Yin Li, Stuent Member, IEEE, Lingling Fan, Senior Member, IEEE Abstract This paper extens

More information

LINEAR DIFFERENTIAL EQUATIONS OF ORDER 1. where a(x) and b(x) are functions. Observe that this class of equations includes equations of the form

LINEAR DIFFERENTIAL EQUATIONS OF ORDER 1. where a(x) and b(x) are functions. Observe that this class of equations includes equations of the form LINEAR DIFFERENTIAL EQUATIONS OF ORDER 1 We consier ifferential equations of the form y + a()y = b(), (1) y( 0 ) = y 0, where a() an b() are functions. Observe that this class of equations inclues equations

More information

Mathematics 116 HWK 25a Solutions 8.6 p610

Mathematics 116 HWK 25a Solutions 8.6 p610 Mathematics 6 HWK 5a Solutions 8.6 p6 Problem, 8.6, p6 Fin a power series representation for the function f() = etermine the interval of convergence. an Solution. Begin with the geometric series = + +

More information

G4003 Advanced Mechanics 1. We already saw that if q is a cyclic variable, the associated conjugate momentum is conserved, L = const.

G4003 Advanced Mechanics 1. We already saw that if q is a cyclic variable, the associated conjugate momentum is conserved, L = const. G4003 Avance Mechanics 1 The Noether theorem We alreay saw that if q is a cyclic variable, the associate conjugate momentum is conserve, q = 0 p q = const. (1) This is the simplest incarnation of Noether

More information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 2: Drawing Bode Plots, Part 2 Overview In this Lecture, you will learn: Simple Plots Real Zeros Real Poles Complex

More information

Review of Differentiation and Integration for Ordinary Differential Equations

Review of Differentiation and Integration for Ordinary Differential Equations Schreyer Fall 208 Review of Differentiation an Integration for Orinary Differential Equations In this course you will be expecte to be able to ifferentiate an integrate quickly an accurately. Many stuents

More information

Compensator Design to Improve Transient Performance Using Root Locus

Compensator Design to Improve Transient Performance Using Root Locus 1 Compensator Design to Improve Transient Performance Using Root Locus Prof. Guy Beale Electrical and Computer Engineering Department George Mason University Fairfax, Virginia Correspondence concerning

More information

APPPHYS 217 Thursday 8 April 2010

APPPHYS 217 Thursday 8 April 2010 APPPHYS 7 Thursay 8 April A&M example 6: The ouble integrator Consier the motion of a point particle in D with the applie force as a control input This is simply Newton s equation F ma with F u : t q q

More information

Unit #6 - Families of Functions, Taylor Polynomials, l Hopital s Rule

Unit #6 - Families of Functions, Taylor Polynomials, l Hopital s Rule Unit # - Families of Functions, Taylor Polynomials, l Hopital s Rule Some problems an solutions selecte or aapte from Hughes-Hallett Calculus. Critical Points. Consier the function f) = 54 +. b) a) Fin

More information

6 General properties of an autonomous system of two first order ODE

6 General properties of an autonomous system of two first order ODE 6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x

More information

Experimental Robustness Study of a Second-Order Sliding Mode Controller

Experimental Robustness Study of a Second-Order Sliding Mode Controller Experimental Robustness Stuy of a Secon-Orer Sliing Moe Controller Anré Blom, Bram e Jager Einhoven University of Technology Department of Mechanical Engineering P.O. Box 513, 5600 MB Einhoven, The Netherlans

More information

1 Lecture 20: Implicit differentiation

1 Lecture 20: Implicit differentiation Lecture 20: Implicit ifferentiation. Outline The technique of implicit ifferentiation Tangent lines to a circle Derivatives of inverse functions by implicit ifferentiation Examples.2 Implicit ifferentiation

More information

Design A Robust Power System Stabilizer on SMIB Using Lyapunov Theory

Design A Robust Power System Stabilizer on SMIB Using Lyapunov Theory Design A Robust Power System Stabilizer on SMIB Using Lyapunov Theory Yin Li, Stuent Member, IEEE, Lingling Fan, Senior Member, IEEE Abstract This paper proposes a robust power system stabilizer (PSS)

More information

1(b) Compensation Example S 0 L U T I 0 N S

1(b) Compensation Example S 0 L U T I 0 N S S 0 L U T I 0 N S Compensation Example I 1U Note: All references to Figures and Equations whose numbers are not preceded by an "S"refer to the textbook. (a) The solution of this problem is outlined in

More information

Consider for simplicity a 3rd-order IIR filter with a transfer function. where

Consider for simplicity a 3rd-order IIR filter with a transfer function. where Basic IIR Digital Filter The causal IIR igital filters we are concerne with in this course are characterie by a real rational transfer function of or, equivalently by a constant coefficient ifference equation

More information

ME 375 FINAL EXAM Friday, May 6, 2005

ME 375 FINAL EXAM Friday, May 6, 2005 ME 375 FINAL EXAM Friay, May 6, 005 Divisio: Kig 11:30 / Cuigham :30 (circle oe) Name: Istructios (1) This is a close book examiatio, but you are allowe three 8.5 11 crib sheets. () You have two hours

More information

Iterated Point-Line Configurations Grow Doubly-Exponentially

Iterated Point-Line Configurations Grow Doubly-Exponentially Iterate Point-Line Configurations Grow Doubly-Exponentially Joshua Cooper an Mark Walters July 9, 008 Abstract Begin with a set of four points in the real plane in general position. A to this collection

More information

Basic IIR Digital Filter Structures

Basic IIR Digital Filter Structures Basic IIR Digital Filter Structures The causal IIR igital filters we are concerne with in this course are characterie by a real rational transfer function of or, equivalently by a constant coefficient

More information

PD Controller for Car-Following Models Based on Real Data

PD Controller for Car-Following Models Based on Real Data PD Controller for Car-Following Moels Base on Real Data Xiaopeng Fang, Hung A. Pham an Minoru Kobayashi Department of Mechanical Engineering Iowa State University, Ames, IA 5 Hona R&D The car following

More information

Proof by Mathematical Induction.

Proof by Mathematical Induction. Proof by Mathematical Inuction. Mathematicians have very peculiar characteristics. They like proving things or mathematical statements. Two of the most important techniques of mathematical proof are proof

More information

2.004 Dynamics and Control II Spring 2008

2.004 Dynamics and Control II Spring 2008 MIT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 8 I * * Massachusetts

More information

Unit 8: Part 2: PD, PID, and Feedback Compensation

Unit 8: Part 2: PD, PID, and Feedback Compensation Ideal Derivative Compensation (PD) Lead Compensation PID Controller Design Feedback Compensation Physical Realization of Compensation Unit 8: Part 2: PD, PID, and Feedback Compensation Engineering 5821:

More information

Advanced Partial Differential Equations with Applications

Advanced Partial Differential Equations with Applications MIT OpenCourseWare http://ocw.mit.eu 18.306 Avance Partial Differential Equations with Applications Fall 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.eu/terms.

More information

Antiderivatives. Definition (Antiderivative) If F (x) = f (x) we call F an antiderivative of f. Alan H. SteinUniversity of Connecticut

Antiderivatives. Definition (Antiderivative) If F (x) = f (x) we call F an antiderivative of f. Alan H. SteinUniversity of Connecticut Antierivatives Definition (Antierivative) If F (x) = f (x) we call F an antierivative of f. Antierivatives Definition (Antierivative) If F (x) = f (x) we call F an antierivative of f. Definition (Inefinite

More information

First Order Linear Differential Equations

First Order Linear Differential Equations LECTURE 8 First Orer Linear Differential Equations We now turn our attention to the problem of constructing analytic solutions of ifferential equations; that is to say,solutions that can be epresse in

More information

Lecture 2 Lagrangian formulation of classical mechanics Mechanics

Lecture 2 Lagrangian formulation of classical mechanics Mechanics Lecture Lagrangian formulation of classical mechanics 70.00 Mechanics Principle of stationary action MATH-GA To specify a motion uniquely in classical mechanics, it suffices to give, at some time t 0,

More information

Trigonometric Functions

Trigonometric Functions 72 Chapter 4 Trigonometric Functions 4 Trigonometric Functions To efine the raian measurement system, we consier the unit circle in the y-plane: (cos,) A y (,0) B So far we have use only algebraic functions

More information

Control Systems I Lecture 10: System Specifications

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

More information

Math Implicit Differentiation. We have discovered (and proved) formulas for finding derivatives of functions like

Math Implicit Differentiation. We have discovered (and proved) formulas for finding derivatives of functions like Math 400 3.5 Implicit Differentiation Name We have iscovere (an prove) formulas for fining erivatives of functions like f x x 3x 4x. 3 This amounts to fining y for 3 y x 3x 4x. Notice that in this case,

More information

Harmonic Modelling of Thyristor Bridges using a Simplified Time Domain Method

Harmonic Modelling of Thyristor Bridges using a Simplified Time Domain Method 1 Harmonic Moelling of Thyristor Briges using a Simplifie Time Domain Metho P. W. Lehn, Senior Member IEEE, an G. Ebner Abstract The paper presents time omain methos for harmonic analysis of a 6-pulse

More information

Design of a Lead Compensator

Design of a Lead Compensator Design of a Lead Compensator Dr. Bishakh Bhattacharya Professor, Department of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc - Funded by MHRD The Lecture Contains Standard Forms of

More information

Optimum design of tuned mass damper systems for seismic structures

Optimum design of tuned mass damper systems for seismic structures Earthquake Resistant Engineering Structures VII 175 Optimum esign of tune mass amper systems for seismic structures I. Abulsalam, M. Al-Janabi & M. G. Al-Taweel Department of Civil Engineering, Faculty

More information

State observers and recursive filters in classical feedback control theory

State observers and recursive filters in classical feedback control theory State observers an recursive filters in classical feeback control theory State-feeback control example: secon-orer system Consier the riven secon-orer system q q q u x q x q x x x x Here u coul represent

More information

Implicit Differentiation

Implicit Differentiation Implicit Differentiation Thus far, the functions we have been concerne with have been efine explicitly. A function is efine explicitly if the output is given irectly in terms of the input. For instance,

More information

Bohr Model of the Hydrogen Atom

Bohr Model of the Hydrogen Atom Class 2 page 1 Bohr Moel of the Hyrogen Atom The Bohr Moel of the hyrogen atom assumes that the atom consists of one electron orbiting a positively charge nucleus. Although it oes NOT o a goo job of escribing

More information

Calculus of Variations

Calculus of Variations 16.323 Lecture 5 Calculus of Variations Calculus of Variations Most books cover this material well, but Kirk Chapter 4 oes a particularly nice job. x(t) x* x*+ αδx (1) x*- αδx (1) αδx (1) αδx (1) t f t

More information

Review: transient and steady-state response; DC gain and the FVT Today s topic: system-modeling diagrams; prototype 2nd-order system

Review: transient and steady-state response; DC gain and the FVT Today s topic: system-modeling diagrams; prototype 2nd-order system Plan of the Lecture Review: transient and steady-state response; DC gain and the FVT Today s topic: system-modeling diagrams; prototype 2nd-order system Plan of the Lecture Review: transient and steady-state

More information

Compensation 8. f4 that separate these regions of stability and instability. The characteristic S 0 L U T I 0 N S

Compensation 8. f4 that separate these regions of stability and instability. The characteristic S 0 L U T I 0 N S S 0 L U T I 0 N S Compensation 8 Note: All references to Figures and Equations whose numbers are not preceded by an "S"refer to the textbook. As suggested in Lecture 8, to perform a Nyquist analysis, we

More information

Total Energy Shaping of a Class of Underactuated Port-Hamiltonian Systems using a New Set of Closed-Loop Potential Shape Variables*

Total Energy Shaping of a Class of Underactuated Port-Hamiltonian Systems using a New Set of Closed-Loop Potential Shape Variables* 51st IEEE Conference on Decision an Control December 1-13 212. Maui Hawaii USA Total Energy Shaping of a Class of Uneractuate Port-Hamiltonian Systems using a New Set of Close-Loop Potential Shape Variables*

More information

2.004 Dynamics and Control II Spring 2008

2.004 Dynamics and Control II Spring 2008 MT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control Spring 2008 or information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Reading: ise: Chapter 8 Massachusetts

More information

ensembles When working with density operators, we can use this connection to define a generalized Bloch vector: v x Tr x, v y Tr y

ensembles When working with density operators, we can use this connection to define a generalized Bloch vector: v x Tr x, v y Tr y Ph195a lecture notes, 1/3/01 Density operators for spin- 1 ensembles So far in our iscussion of spin- 1 systems, we have restricte our attention to the case of pure states an Hamiltonian evolution. Toay

More information

4.2 First Differentiation Rules; Leibniz Notation

4.2 First Differentiation Rules; Leibniz Notation .. FIRST DIFFERENTIATION RULES; LEIBNIZ NOTATION 307. First Differentiation Rules; Leibniz Notation In this section we erive rules which let us quickly compute the erivative function f (x) for any polynomial

More information

Nested Saturation with Guaranteed Real Poles 1

Nested Saturation with Guaranteed Real Poles 1 Neste Saturation with Guarantee Real Poles Eric N Johnson 2 an Suresh K Kannan 3 School of Aerospace Engineering Georgia Institute of Technology, Atlanta, GA 3332 Abstract The global stabilization of asymptotically

More information

The Principle of Least Action

The Principle of Least Action Chapter 7. The Principle of Least Action 7.1 Force Methos vs. Energy Methos We have so far stuie two istinct ways of analyzing physics problems: force methos, basically consisting of the application of

More information

AP CALCULUS AB Summer Work. The following are guidelines for completing the summer work packet

AP CALCULUS AB Summer Work. The following are guidelines for completing the summer work packet Name: Perio: AP CALCULUS AB Summer Work For stuents to successfully complete the objectives of the AP Calculus curriculum, the stuent must emonstrate a high level of inepenence, capability, eication, an

More information

A Comparison between a Conventional Power System Stabilizer (PSS) and Novel PSS Based on Feedback Linearization Technique

A Comparison between a Conventional Power System Stabilizer (PSS) and Novel PSS Based on Feedback Linearization Technique J. Basic. Appl. Sci. Res., ()9-99,, TextRoa Publication ISSN 9-434 Journal of Basic an Applie Scientific Research www.textroa.com A Comparison between a Conventional Power System Stabilizer (PSS) an Novel

More information

Exam 2 Review Solutions

Exam 2 Review Solutions Exam Review Solutions 1. True or False, an explain: (a) There exists a function f with continuous secon partial erivatives such that f x (x, y) = x + y f y = x y False. If the function has continuous secon

More information

Experiment I Electric Force

Experiment I Electric Force Experiment I Electric Force Twenty-five hunre years ago, the Greek philosopher Thales foun that amber, the harene sap from a tree, attracte light objects when rubbe. Only twenty-four hunre years later,

More information

Pure Further Mathematics 1. Revision Notes

Pure Further Mathematics 1. Revision Notes Pure Further Mathematics Revision Notes June 20 2 FP JUNE 20 SDB Further Pure Complex Numbers... 3 Definitions an arithmetical operations... 3 Complex conjugate... 3 Properties... 3 Complex number plane,

More information

f(x + h) f(x) f (x) = lim

f(x + h) f(x) f (x) = lim Introuction 4.3 Some Very Basic Differentiation Formulas If a ifferentiable function f is quite simple, ten it is possible to fin f by using te efinition of erivative irectly: f () 0 f( + ) f() However,

More information

Prep 1. Oregon State University PH 213 Spring Term Suggested finish date: Monday, April 9

Prep 1. Oregon State University PH 213 Spring Term Suggested finish date: Monday, April 9 Oregon State University PH 213 Spring Term 2018 Prep 1 Suggeste finish ate: Monay, April 9 The formats (type, length, scope) of these Prep problems have been purposely create to closely parallel those

More information

Transformations of Random Variables

Transformations of Random Variables Transformations of Ranom Variables September, 2009 We begin with a ranom variable an we want to start looking at the ranom variable Y = g() = g where the function g : R R. The inverse image of a set A,

More information

Year 11 Matrices Semester 2. Yuk

Year 11 Matrices Semester 2. Yuk Year 11 Matrices Semester 2 Chapter 5A input/output Yuk 1 Chapter 5B Gaussian Elimination an Systems of Linear Equations This is an extension of solving simultaneous equations. What oes a System of Linear

More information

An inductance lookup table application for analysis of reluctance stepper motor model

An inductance lookup table application for analysis of reluctance stepper motor model ARCHIVES OF ELECTRICAL ENGINEERING VOL. 60(), pp. 5- (0) DOI 0.478/ v07-0-000-y An inuctance lookup table application for analysis of reluctance stepper motor moel JAKUB BERNAT, JAKUB KOŁOTA, SŁAWOMIR

More information

Systems Analysis and Control

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

More information

BEYOND THE CONSTRUCTION OF OPTIMAL SWITCHING SURFACES FOR AUTONOMOUS HYBRID SYSTEMS. Mauro Boccadoro Magnus Egerstedt Paolo Valigi Yorai Wardi

BEYOND THE CONSTRUCTION OF OPTIMAL SWITCHING SURFACES FOR AUTONOMOUS HYBRID SYSTEMS. Mauro Boccadoro Magnus Egerstedt Paolo Valigi Yorai Wardi BEYOND THE CONSTRUCTION OF OPTIMAL SWITCHING SURFACES FOR AUTONOMOUS HYBRID SYSTEMS Mauro Boccaoro Magnus Egerstet Paolo Valigi Yorai Wari {boccaoro,valigi}@iei.unipg.it Dipartimento i Ingegneria Elettronica

More information

DYNAMIC PERFORMANCE OF RELUCTANCE SYNCHRONOUS MACHINES

DYNAMIC PERFORMANCE OF RELUCTANCE SYNCHRONOUS MACHINES Annals of the University of Craiova, Electrical Engineering series, No 33, 9; ISSN 184-485 7 TH INTERNATIONAL CONFERENCE ON ELECTROMECHANICAL AN POWER SYSTEMS October 8-9, 9 - Iaşi, Romania YNAMIC PERFORMANCE

More information

A Review of Feedforward Control Approaches in Nanopositioning for High-Speed SPM

A Review of Feedforward Control Approaches in Nanopositioning for High-Speed SPM Garrett M. Clayton Department of Mechanical Engineering, Villanova University, Villanova, PA 1985 e-mail: garrett.clayton@villanova.eu Szuchi Tien Department of Mechanical Engineering, National Cheng Kung

More information

ay (t) + by (t) + cy(t) = 0 (2)

ay (t) + by (t) + cy(t) = 0 (2) Solving ay + by + cy = 0 Without Characteristic Equations, Complex Numbers, or Hats John Tolle Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213-3890 Some calculus courses

More information

Experiment 2, Physics 2BL

Experiment 2, Physics 2BL Experiment 2, Physics 2BL Deuction of Mass Distributions. Last Upate: 2009-05-03 Preparation Before this experiment, we recommen you review or familiarize yourself with the following: Chapters 4-6 in Taylor

More information

EXERCISES FOR SECTION 6.3

EXERCISES FOR SECTION 6.3 y 6. Secon-Orer Equation 499.58 4 t EXERCISES FOR SECTION 6.. We ue integration by part twice to compute Lin ωt Firt, letting u in ωt an v e t t,weget Lin ωt in ωt e t e t lim b in ωt e t t. in ωt ω e

More information

3.7 Implicit Differentiation -- A Brief Introduction -- Student Notes

3.7 Implicit Differentiation -- A Brief Introduction -- Student Notes Fin these erivatives of these functions: y.7 Implicit Differentiation -- A Brief Introuction -- Stuent Notes tan y sin tan = sin y e = e = Write the inverses of these functions: y tan y sin How woul we

More information

2 ODEs Integrating Factors and Homogeneous Equations

2 ODEs Integrating Factors and Homogeneous Equations 2 ODEs Integrating Factors an Homogeneous Equations We begin with a slightly ifferent type of equation: 2.1 Exact Equations These are ODEs whose general solution can be obtaine by simply integrating both

More information

Homework 6 Solutions and Rubric

Homework 6 Solutions and Rubric Homework 6 Solutions and Rubric EE 140/40A 1. K-W Tube Amplifier b) Load Resistor e) Common-cathode a) Input Diff Pair f) Cathode-Follower h) Positive Feedback c) Tail Resistor g) Cc d) Av,cm = 1/ Figure

More information

Final Exam: Sat 12 Dec 2009, 09:00-12:00

Final Exam: Sat 12 Dec 2009, 09:00-12:00 MATH 1013 SECTIONS A: Professor Szeptycki APPLIED CALCULUS I, FALL 009 B: Professor Toms C: Professor Szeto NAME: STUDENT #: SECTION: No ai (e.g. calculator, written notes) is allowe. Final Exam: Sat 1

More information

MAE 143B - Homework 8 Solutions

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

More information

The Exact Form and General Integrating Factors

The Exact Form and General Integrating Factors 7 The Exact Form an General Integrating Factors In the previous chapters, we ve seen how separable an linear ifferential equations can be solve using methos for converting them to forms that can be easily

More information

Multirate Feedforward Control with State Trajectory Generation based on Time Axis Reversal for Plant with Continuous Time Unstable Zeros

Multirate Feedforward Control with State Trajectory Generation based on Time Axis Reversal for Plant with Continuous Time Unstable Zeros Multirate Feeforwar Control with State Trajectory Generation base on Time Axis Reversal for with Continuous Time Unstable Zeros Wataru Ohnishi, Hiroshi Fujimoto Abstract with unstable zeros is known as

More information

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

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

More information

Optimization Notes. Note: Any material in red you will need to have memorized verbatim (more or less) for tests, quizzes, and the final exam.

Optimization Notes. Note: Any material in red you will need to have memorized verbatim (more or less) for tests, quizzes, and the final exam. MATH 2250 Calculus I Date: October 5, 2017 Eric Perkerson Optimization Notes 1 Chapter 4 Note: Any material in re you will nee to have memorize verbatim (more or less) for tests, quizzes, an the final

More information

Differentiability, Computing Derivatives, Trig Review

Differentiability, Computing Derivatives, Trig Review Unit #3 : Differentiability, Computing Derivatives, Trig Review Goals: Determine when a function is ifferentiable at a point Relate the erivative graph to the the graph of an original function Compute

More information

Optimal LQR Control of Structures using Linear Modal Model

Optimal LQR Control of Structures using Linear Modal Model Optimal LQR Control of Structures using Linear Moal Moel I. Halperin,2, G. Agranovich an Y. Ribakov 2 Department of Electrical an Electronics Engineering 2 Department of Civil Engineering Faculty of Engineering,

More information

Applications of First Order Equations

Applications of First Order Equations Applications of First Orer Equations Viscous Friction Consier a small mass that has been roppe into a thin vertical tube of viscous flui lie oil. The mass falls, ue to the force of gravity, but falls more

More information

Introduction to Systems with Dynamics

Introduction to Systems with Dynamics S 0 L U T I 0 N S Introduction to Systems with Dynamics I- 3 Note: All references to Figures and Equations whose numbers are not preceded by an "S"refer to the textbook. From Figure 3.6 on page 79 of the

More information

Table of Common Derivatives By David Abraham

Table of Common Derivatives By David Abraham Prouct an Quotient Rules: Table of Common Derivatives By Davi Abraham [ f ( g( ] = [ f ( ] g( + f ( [ g( ] f ( = g( [ f ( ] g( g( f ( [ g( ] Trigonometric Functions: sin( = cos( cos( = sin( tan( = sec

More information

Physics 115C Homework 4

Physics 115C Homework 4 Physics 115C Homework 4 Problem 1 a In the Heisenberg picture, the ynamical equation is the Heisenberg equation of motion: for any operator Q H, we have Q H = 1 t i [Q H,H]+ Q H t where the partial erivative

More information

SELF-ERECTING, ROTARY MOTION INVERTED PENDULUM Quanser Consulting Inc.

SELF-ERECTING, ROTARY MOTION INVERTED PENDULUM Quanser Consulting Inc. SELF-ERECTING, ROTARY MOTION INVERTED PENDULUM Quanser Consulting Inc. 1.0 SYSTEM DESCRIPTION The sel-erecting rotary motion inverte penulum consists of a rotary servo motor system (SRV-02) which rives

More information

Sliding mode approach to congestion control in connection-oriented communication networks

Sliding mode approach to congestion control in connection-oriented communication networks JOURNAL OF APPLIED COMPUTER SCIENCE Vol. xx. No xx (200x), pp. xx-xx Sliing moe approach to congestion control in connection-oriente communication networks Anrzej Bartoszewicz, Justyna Żuk Technical University

More information

MEEN 363. EXAMPLE of ANALYSIS (1 DOF) Luis San Andrés

MEEN 363. EXAMPLE of ANALYSIS (1 DOF) Luis San Andrés MEEN 363. EAMPLE of ANALYSIS (1 DOF) Luis San Anrés Objectives: a) To erive EOM for a 1-DOF (one egree of freeom) system b) To unerstan concept of static equilibrium c) To learn the correct usage of physical

More information

Interpolated Rigid-Body Motions and Robotics

Interpolated Rigid-Body Motions and Robotics Interpolate Rigi-Boy Motions an Robotics J.M. Selig Faculty of Business, Computing an Info. Management. Lonon South Bank University, Lonon SE AA, U.K. seligjm@lsbu.ac.uk Yaunquing Wu Dept. Mechanical Engineering.

More information

Lecture 6: Control of Three-Phase Inverters

Lecture 6: Control of Three-Phase Inverters Yoash Levron The Anrew an Erna Viterbi Faculty of Electrical Engineering, Technion Israel Institute of Technology, Haifa 323, Israel yoashl@ee.technion.ac.il Juri Belikov Department of Computer Systems,

More information

Recitation 11: Time delays

Recitation 11: Time delays Recitation : Time delays Emilio Frazzoli Laboratory for Information and Decision Systems Massachusetts Institute of Technology November, 00. Introduction and motivation. Delays are incurred when the controller

More information

Topic # Feedback Control Systems

Topic # 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 information

Differentiability, Computing Derivatives, Trig Review. Goals:

Differentiability, Computing Derivatives, Trig Review. Goals: Secants vs. Derivatives - Unit #3 : Goals: Differentiability, Computing Derivatives, Trig Review Determine when a function is ifferentiable at a point Relate the erivative graph to the the graph of an

More information