Dead-beat controller design

Size: px
Start display at page:

Download "Dead-beat controller design"

Transcription

1 J. Hetthéssy, A. Barta, R. Bars: Dead beat cntrller design Nvember, 4 Dead-beat cntrller design In sampled data cntrl systems the cntrller is realised by an intelligent device, typically by a PLC (Prgrammable Lgic Cntrller). The tas f the PLC is t realise the cntrller algrithm and t handle the signals used fr peratin f the cntrller (filtering, A/D and D/A cnverters, signal cnditining interfaces). As the cntrller algrithm is realised by sftware, there is a pssibility t apply different special, mre sphisticated cntrl algrithms. One f these algrithms ensures the accurate settling f the utput signal during a finite, small number f the sampling perids. In the literature this algrithm is referred as the dead-beat cntrller algrithm. T understand the essence f the methd - fr sae f simplicity - we cnsider a stable plant withut dead time. The reference signal is a unit step. (Let us remar that the dead-beat cntrller can be designed fr unstable plants with dead time cnsidering a reference signal different f unit step as well, in a bit mre cmplex way). In the sequel a design prcedure is derived in three steps. This slutin satisfies the practical requirements as well. In the first step the cntrller is designed fr the fastest behaviur when the utput signal is settled in ne sampling step. It will be seen that with this design with typical sampling times the cntrl signal culd be extremely high, mrever in mst cases scillatins ccur between the sampling pints. In the secnd step the design is mdified t avid intersampling scillatins. It will be shwn that cancellatin f ers utside f the unit circle is the reasn fr scillatins. Zers f the plant pulse transfer functin are separated fr cancellable and nn-cancellable nes and nly the cancellable ers will appear in the cntrller algrithm. This mdificatin f the algrithm increases the settling time. If the cntrl signal is still higher than allwed, the slutin can be refined using a s-called design plynmial. In this case the settling time is increased further (but still remains finite). The design is executed in the peratr dmain. Interesting feature f the design methd is the fact that it remves undesirable time dmain prperties (scillatins, t high values f the cntrl signal) by cnsideratins dne in the peratr dmain. The basic tas is the design f a sampled data cntrller. r[] - e[] u[] u(t) C () D / A P(s) U() y(t) Y(s) y[] Y() A/ D Y P ( ) = U The hybrid (cntinuus-discrete) prblem is cnverted first t a discrete prblem. Pulse transfer functin P() f the plant is determined which cnsiders the D/A cnverter and hld element tgether with the plant transfer functin P(s). Sampling time T s has als t be given. Then design the cntrller C() and chec the clsed lp system perfrmance analysing the cntinuus signals nt nly in the sampling pints, but als between them. r[] e[] u[] y[] C () P ( ) - Speaing abut cntrllers given by their transfer functins let us analyse realisatin aspects. Be the pulse transfer functin f the cntrller: b + b + b + b C ( )= + a + a + a In rder t use the shift peratr - let us divide bth the numeratr and the denminatr by third pwer f. b + b + b + b C ( )= + a + a + a

2 J. Hetthéssy, A. Barta, R. Bars: Dead beat cntrller design Nvember, 4 As { u [ ]} { e [ ]} Z Z with crss-multiplicatin we get u [ ] + au [ ] + au [ ] + au [ ] = be [ ] + be [ ] + be [ ] + be [ ] r u [ ] = be [ ] + be [ ] + be [ ] + be [ ] au [ ] au [ ] au [ ] It is seen that the pulse transfer functin f the cntrller can be transfrmed simply t a recursive difference equatin. MATLAB functin dlsim calculates the utput f a discrete element n the basis f its difference equatin fr a given input signal. The abve interpretatin f pulse transfer functin C() means that the cntrller in a sampled data cntrl system is implemented by a recursive algrithm Returning t the design f the dead-beat cntrller let us cnsider the fllwing plant given by its transfer functin Ps () = ( + 5s)( + s) The sampling time is T s = sec.» Ts=;» s=p('s');» =p('',ts);» Ps=/((+5*s)*(+*s)) In rder t get an impressin abut the system let us calculate its unit step respnse.» step(ps) The pulse transfer functin f the plant tgether with the er rder hld is btained» P=cd(Ps,Ts) B ( ) ( +.948) P ( ) = =.9559 ( -.887) ( -.948) First design the dead-beat cntrller ensuring settling prcess during ne sampling perid. The cnditin fr this is that the resulting transfer functin f the clsed lp between the utput signal and the input signal (suppsed t be a sampled unit step) be a ne step shift, namely the shift peratr -.. CP ( ) ( ) T = = + CP ( ) ( ) Hence the cntrller pulse transfer functin is expressed as T = P ( )( T ( )) P ( ) Express P() as a rati f tw plynmials: B ( ) P ( ) = Then B ( ) The discrete transfer functin f the cntrller is calculated in Matlab by» T=/» C=T/(P*(-T))» C=minreal(C) ( -.887) ( -.948).45 ( +.948) ( -) The clsed lp system behaviur can be visualised by the step cmmand,

3 J. Hetthéssy, A. Barta, R. Bars: Dead beat cntrller design Nvember, 4» step(c*p/(+c*p)) The u cntrl signal is displayed:» step(c/(+c*p)) The system shws ne time step delay. Mre accurate behaviur can be investigated by a Simulin mdel. Clc t time u cntrl Scpe y utput C Ps Step Input LTI System Zer-Order Hld LTI SystemPs Scpe The reasn f the scillatins is the fact that the cntrller cntains the ers f the plant as its ples, and sme f these ples result scillatins in cntrl signal u[. (see the appendix at the end) The ers f the plant (rts f B) ( ) appear in the cntrller as ples, B ( ) P ( ) has nly ne er, = Examine this in mre detail.» C=/(+.948)» step(c) This cmpnent causes the scillatin. Cnvert bac this ple t the cntinuus dmain. Since, s = ln / TS» p=lg(-.948) p = i S = e st, Here we just emphasise that typically ples f negative real value cause the scillatins. Let us separate the cancellable and nn-cancellable ers f the prcess pulse transfer functin accrding t + B ( ) = B( B ) + where B cntains the cmpensable and B the nn cmpensable rts. If a er is nt cmpensated it will appear in the clsed lp transfer functin. Design a cntrller that des nt cmpensate the nn cmpensable rts. T =

4 J. Hetthéssy, A. Barta, R. Bars: Dead beat cntrller design Nvember, 4 where = +deg ( B B ), =. B () The cmpnent is necassary t get a realisable system (the degree f the denminatr is higher than the degree f the numeratr). Anther requriment is that the static gain is, that is B is nrmalied. + B + B ( ) = B( B ) () = ( B ) n. B () B () = +.948, = = and = pb ()=.7 + [ ] and the MATLAB prgram» Bm =(+.948)» Bpn=P.*dcgain(Bm)» Bmn=Bm/dcgain(Bm)» T=Bmn/(^)» C=T/(P*(-T))»(C=minreal(C,.)) It is seen that there are n scillatins and the verexcitatin in the cntrl signal is als less than befre. The system became slwer, nw the utput signal reaches the steady state during tw sampling steps. As the maximum value f the cntrl signal is still t high, its value f abut 5 wuld exceed the pssibilities f a usual actuatr. S we have t find a mdificatin f the design which wuld decrease the value f the verexcitatin eeping the prperty f a finite settling time. Let us cmplete the cntrl algrithm with a design plynmial, which will lead the finite time settling prcess. Fr example chsing design plynmial + + F = + + = its smthing effect is shwn by its unit step respnse.» F=(^++)/(*^)» step(f) The cntrl equatin with the design plynmial: T = F and the cntrller algrithm: AF ( ) ( ) + B [ B F] Let us bserve that F()=, s the design plynmial des nt affect the er static errr.» T=F*Bmn/(^)» C=T/(P*(-T))» C=minreal(C) n The dead-beat cntrller can be designed als in cases when the plant cntains dead time. Let us suppse that the cntinuus dead time T d is a multiple integer f the sampling time T s. Be this rati d=t d /T s. B ( ) d P ( ) = n

5 J. Hetthéssy, A. Barta, R. Bars: Dead beat cntrller design Nvember, 4 Suppsing that the cntinuus plant is stable and the reference input is a unit step, the clsed lp transfer functin is written as: T = but nw the value f has t be increased by d steps. = + deg ( B ) + d Cnsequently the pen lp pulse transfer functin is CP ( ) ( ) =, d A + [ ] In the previus example a T d = sec dead time is added t the cntinuus prcess (d=). Calculate the cntrller and simulate the behaviur f the cntrl system. The cntrller:» Td=» d=td/ts» T=Bmn/(^(+d))» C=T/(P*(-T))» C=minreal(C) The delay can be simulated in Simulin with an added delay blc (Simulin >Cntinuus >Transprt Delay). Similarly t the previus discussin the design can be mdified with a design plynmial. Appendix: Let us analyse the cntur f a cnjugate cmplex pair with a given damping factr in the dmain. In the s dmain the cnstant ζ lines are straight lines s=σ+jω ging thrugh the rig where fr a given σ value ω = σ ς ζ st = e t. These cmplex s values are transfrmed t the dmain by relatinship s curves f heart shape. As a demnstratin be» sigma=:.:.6;» eta=.4;» Ts=;» =exp(ts*(-sigma+j*sqrt(-eta*eta)*sigma/eta));» plt(real(),imag(),real(),-imag()),grid; Thse rts f plynmial B() which belng t plynmial B are inside f the clsed curve (where the damping factr is higher than n the cntur). Thse rts f plynmial B() which belng t plynmial Bare n the cntur r utside f it. 5

Determining the Accuracy of Modal Parameter Estimation Methods

Determining the Accuracy of Modal Parameter Estimation Methods Determining the Accuracy f Mdal Parameter Estimatin Methds by Michael Lee Ph.D., P.E. & Mar Richardsn Ph.D. Structural Measurement Systems Milpitas, CA Abstract The mst cmmn type f mdal testing system

More information

Cambridge Assessment International Education Cambridge Ordinary Level. Published

Cambridge Assessment International Education Cambridge Ordinary Level. Published Cambridge Assessment Internatinal Educatin Cambridge Ordinary Level ADDITIONAL MATHEMATICS 4037/1 Paper 1 Octber/Nvember 017 MARK SCHEME Maximum Mark: 80 Published This mark scheme is published as an aid

More information

Lead/Lag Compensator Frequency Domain Properties and Design Methods

Lead/Lag Compensator Frequency Domain Properties and Design Methods Lectures 6 and 7 Lead/Lag Cmpensatr Frequency Dmain Prperties and Design Methds Definitin Cnsider the cmpensatr (ie cntrller Fr, it is called a lag cmpensatr s K Fr s, it is called a lead cmpensatr Ntatin

More information

Design and Simulation of Dc-Dc Voltage Converters Using Matlab/Simulink

Design and Simulation of Dc-Dc Voltage Converters Using Matlab/Simulink American Jurnal f Engineering Research (AJER) 016 American Jurnal f Engineering Research (AJER) e-issn: 30-0847 p-issn : 30-0936 Vlume-5, Issue-, pp-9-36 www.ajer.rg Research Paper Open Access Design and

More information

Part a: Writing the nodal equations and solving for v o gives the magnitude and phase response: tan ( 0.25 )

Part a: Writing the nodal equations and solving for v o gives the magnitude and phase response: tan ( 0.25 ) + - Hmewrk 0 Slutin ) In the circuit belw: a. Find the magnitude and phase respnse. b. What kind f filter is it? c. At what frequency is the respnse 0.707 if the generatr has a ltage f? d. What is the

More information

Review Problems 3. Four FIR Filter Types

Review Problems 3. Four FIR Filter Types Review Prblems 3 Fur FIR Filter Types Fur types f FIR linear phase digital filters have cefficients h(n fr 0 n M. They are defined as fllws: Type I: h(n = h(m-n and M even. Type II: h(n = h(m-n and M dd.

More information

Lecture 6: Phase Space and Damped Oscillations

Lecture 6: Phase Space and Damped Oscillations Lecture 6: Phase Space and Damped Oscillatins Oscillatins in Multiple Dimensins The preius discussin was fine fr scillatin in a single dimensin In general, thugh, we want t deal with the situatin where:

More information

Sections 15.1 to 15.12, 16.1 and 16.2 of the textbook (Robbins-Miller) cover the materials required for this topic.

Sections 15.1 to 15.12, 16.1 and 16.2 of the textbook (Robbins-Miller) cover the materials required for this topic. Tpic : AC Fundamentals, Sinusidal Wavefrm, and Phasrs Sectins 5. t 5., 6. and 6. f the textbk (Rbbins-Miller) cver the materials required fr this tpic.. Wavefrms in electrical systems are current r vltage

More information

Revision: August 19, E Main Suite D Pullman, WA (509) Voice and Fax

Revision: August 19, E Main Suite D Pullman, WA (509) Voice and Fax .7.4: Direct frequency dmain circuit analysis Revisin: August 9, 00 5 E Main Suite D Pullman, WA 9963 (509) 334 6306 ice and Fax Overview n chapter.7., we determined the steadystate respnse f electrical

More information

Lecture 20a. Circuit Topologies and Techniques: Opamps

Lecture 20a. Circuit Topologies and Techniques: Opamps Lecture a Circuit Tplgies and Techniques: Opamps In this lecture yu will learn: Sme circuit tplgies and techniques Intrductin t peratinal amplifiers Differential mplifier IBIS1 I BIS M VI1 vi1 Vi vi I

More information

EEO 401 Digital Signal Processing Prof. Mark Fowler

EEO 401 Digital Signal Processing Prof. Mark Fowler EEO 4 Digital Signal Prcessing Pr. ar Fwler DT Filters te Set #2 Reading Assignment: Sect. 5.4 Prais & anlais /29 Ideal LP Filter Put in the signal we want passed. Suppse that ( ) [, ] X π xn [ ] y[ n]

More information

Section I5: Feedback in Operational Amplifiers

Section I5: Feedback in Operational Amplifiers Sectin I5: eedback in Operatinal mplifiers s discussed earlier, practical p-amps hae a high gain under dc (zer frequency) cnditins and the gain decreases as frequency increases. This frequency dependence

More information

Dataflow Analysis and Abstract Interpretation

Dataflow Analysis and Abstract Interpretation Dataflw Analysis and Abstract Interpretatin Cmputer Science and Artificial Intelligence Labratry MIT Nvember 9, 2015 Recap Last time we develped frm first principles an algrithm t derive invariants. Key

More information

ENSC Discrete Time Systems. Project Outline. Semester

ENSC Discrete Time Systems. Project Outline. Semester ENSC 49 - iscrete Time Systems Prject Outline Semester 006-1. Objectives The gal f the prject is t design a channel fading simulatr. Upn successful cmpletin f the prject, yu will reinfrce yur understanding

More information

ECE 2100 Circuit Analysis

ECE 2100 Circuit Analysis ECE 00 Circuit Analysis Lessn 6 Chapter 4 Sec 4., 4.5, 4.7 Series LC Circuit C Lw Pass Filter Daniel M. Litynski, Ph.D. http://hmepages.wmich.edu/~dlitynsk/ ECE 00 Circuit Analysis Lessn 5 Chapter 9 &

More information

Thermodynamics Partial Outline of Topics

Thermodynamics Partial Outline of Topics Thermdynamics Partial Outline f Tpics I. The secnd law f thermdynamics addresses the issue f spntaneity and invlves a functin called entrpy (S): If a prcess is spntaneus, then Suniverse > 0 (2 nd Law!)

More information

Lyapunov Stability Stability of Equilibrium Points

Lyapunov Stability Stability of Equilibrium Points Lyapunv Stability Stability f Equilibrium Pints 1. Stability f Equilibrium Pints - Definitins In this sectin we cnsider n-th rder nnlinear time varying cntinuus time (C) systems f the frm x = f ( t, x),

More information

2.161 Signal Processing: Continuous and Discrete Fall 2008

2.161 Signal Processing: Continuous and Discrete Fall 2008 MIT OpenCurseWare http://cw.mit.edu 2.161 Signal Prcessing: Cntinuus and Discrete Fall 2008 Fr infrmatin abut citing these materials r ur Terms f Use, visit: http://cw.mit.edu/terms. Massachusetts Institute

More information

Homology groups of disks with holes

Homology groups of disks with holes Hmlgy grups f disks with hles THEOREM. Let p 1,, p k } be a sequence f distinct pints in the interir unit disk D n where n 2, and suppse that fr all j the sets E j Int D n are clsed, pairwise disjint subdisks.

More information

, which yields. where z1. and z2

, which yields. where z1. and z2 The Gaussian r Nrmal PDF, Page 1 The Gaussian r Nrmal Prbability Density Functin Authr: Jhn M Cimbala, Penn State University Latest revisin: 11 September 13 The Gaussian r Nrmal Prbability Density Functin

More information

NOTE ON APPELL POLYNOMIALS

NOTE ON APPELL POLYNOMIALS NOTE ON APPELL POLYNOMIALS I. M. SHEFFER An interesting characterizatin f Appell plynmials by means f a Stieltjes integral has recently been given by Thrne. 1 We prpse t give a secnd such representatin,

More information

Lecture 7: Damped and Driven Oscillations

Lecture 7: Damped and Driven Oscillations Lecture 7: Damped and Driven Oscillatins Last time, we fund fr underdamped scillatrs: βt x t = e A1 + A csω1t + i A1 A sinω1t A 1 and A are cmplex numbers, but ur answer must be real Implies that A 1 and

More information

Department of Electrical Engineering, University of Waterloo. Introduction

Department of Electrical Engineering, University of Waterloo. Introduction Sectin 4: Sequential Circuits Majr Tpics Types f sequential circuits Flip-flps Analysis f clcked sequential circuits Mre and Mealy machines Design f clcked sequential circuits State transitin design methd

More information

EEO 401 Digital Signal Processing Prof. Mark Fowler

EEO 401 Digital Signal Processing Prof. Mark Fowler EEO 401 Digital Signal Prcessing Prf. Mark Fwler Intrductin Nte Set #1 ading Assignment: Ch. 1 f Prakis & Manlakis 1/13 Mdern systems generally DSP Scenari get a cntinuus-time signal frm a sensr a cnt.-time

More information

Keysight Technologies Understanding the Kramers-Kronig Relation Using A Pictorial Proof

Keysight Technologies Understanding the Kramers-Kronig Relation Using A Pictorial Proof Keysight Technlgies Understanding the Kramers-Krnig Relatin Using A Pictrial Prf By Clin Warwick, Signal Integrity Prduct Manager, Keysight EEsf EDA White Paper Intrductin In principle, applicatin f the

More information

Pattern Recognition 2014 Support Vector Machines

Pattern Recognition 2014 Support Vector Machines Pattern Recgnitin 2014 Supprt Vectr Machines Ad Feelders Universiteit Utrecht Ad Feelders ( Universiteit Utrecht ) Pattern Recgnitin 1 / 55 Overview 1 Separable Case 2 Kernel Functins 3 Allwing Errrs (Sft

More information

NUMBERS, MATHEMATICS AND EQUATIONS

NUMBERS, MATHEMATICS AND EQUATIONS AUSTRALIAN CURRICULUM PHYSICS GETTING STARTED WITH PHYSICS NUMBERS, MATHEMATICS AND EQUATIONS An integral part t the understanding f ur physical wrld is the use f mathematical mdels which can be used t

More information

Lab 11 LRC Circuits, Damped Forced Harmonic Motion

Lab 11 LRC Circuits, Damped Forced Harmonic Motion Physics 6 ab ab 11 ircuits, Damped Frced Harmnic Mtin What Yu Need T Knw: The Physics OK this is basically a recap f what yu ve dne s far with circuits and circuits. Nw we get t put everything tgether

More information

THERMAL-VACUUM VERSUS THERMAL- ATMOSPHERIC TESTS OF ELECTRONIC ASSEMBLIES

THERMAL-VACUUM VERSUS THERMAL- ATMOSPHERIC TESTS OF ELECTRONIC ASSEMBLIES PREFERRED RELIABILITY PAGE 1 OF 5 PRACTICES PRACTICE NO. PT-TE-1409 THERMAL-VACUUM VERSUS THERMAL- ATMOSPHERIC Practice: Perfrm all thermal envirnmental tests n electrnic spaceflight hardware in a flight-like

More information

Synchronous Motor V-Curves

Synchronous Motor V-Curves Synchrnus Mtr V-Curves 1 Synchrnus Mtr V-Curves Intrductin Synchrnus mtrs are used in applicatins such as textile mills where cnstant speed peratin is critical. Mst small synchrnus mtrs cntain squirrel

More information

4th Indian Institute of Astrophysics - PennState Astrostatistics School July, 2013 Vainu Bappu Observatory, Kavalur. Correlation and Regression

4th Indian Institute of Astrophysics - PennState Astrostatistics School July, 2013 Vainu Bappu Observatory, Kavalur. Correlation and Regression 4th Indian Institute f Astrphysics - PennState Astrstatistics Schl July, 2013 Vainu Bappu Observatry, Kavalur Crrelatin and Regressin Rahul Ry Indian Statistical Institute, Delhi. Crrelatin Cnsider a tw

More information

OTHER USES OF THE ICRH COUPL ING CO IL. November 1975

OTHER USES OF THE ICRH COUPL ING CO IL. November 1975 OTHER USES OF THE ICRH COUPL ING CO IL J. C. Sprtt Nvember 1975 -I,," PLP 663 Plasma Studies University f Wiscnsin These PLP Reprts are infrmal and preliminary and as such may cntain errrs nt yet eliminated.

More information

ENG2410 Digital Design Sequential Circuits: Part A

ENG2410 Digital Design Sequential Circuits: Part A ENG2410 Digital Design Sequential Circuits: Part A Fall 2017 S. Areibi Schl f Engineering University f Guelph Week #6 Tpics Sequential Circuit Definitins Latches Flip-Flps Delays in Sequential Circuits

More information

ECE 5318/6352 Antenna Engineering. Spring 2006 Dr. Stuart Long. Chapter 6. Part 7 Schelkunoff s Polynomial

ECE 5318/6352 Antenna Engineering. Spring 2006 Dr. Stuart Long. Chapter 6. Part 7 Schelkunoff s Polynomial ECE 538/635 Antenna Engineering Spring 006 Dr. Stuart Lng Chapter 6 Part 7 Schelkunff s Plynmial 7 Schelkunff s Plynmial Representatin (fr discrete arrays) AF( ψ ) N n 0 A n e jnψ N number f elements in

More information

B. Definition of an exponential

B. Definition of an exponential Expnents and Lgarithms Chapter IV - Expnents and Lgarithms A. Intrductin Starting with additin and defining the ntatins fr subtractin, multiplicatin and divisin, we discvered negative numbers and fractins.

More information

We can see from the graph above that the intersection is, i.e., [ ).

We can see from the graph above that the intersection is, i.e., [ ). MTH 111 Cllege Algebra Lecture Ntes July 2, 2014 Functin Arithmetic: With nt t much difficulty, we ntice that inputs f functins are numbers, and utputs f functins are numbers. S whatever we can d with

More information

ECE 2100 Circuit Analysis

ECE 2100 Circuit Analysis ECE 2100 Circuit Analysis Lessn 25 Chapter 9 & App B: Passive circuit elements in the phasr representatin Daniel M. Litynski, Ph.D. http://hmepages.wmich.edu/~dlitynsk/ ECE 2100 Circuit Analysis Lessn

More information

Math Foundations 10 Work Plan

Math Foundations 10 Work Plan Math Fundatins 10 Wrk Plan Units / Tpics 10.1 Demnstrate understanding f factrs f whle numbers by: Prime factrs Greatest Cmmn Factrs (GCF) Least Cmmn Multiple (LCM) Principal square rt Cube rt Time Frame

More information

Building to Transformations on Coordinate Axis Grade 5: Geometry Graph points on the coordinate plane to solve real-world and mathematical problems.

Building to Transformations on Coordinate Axis Grade 5: Geometry Graph points on the coordinate plane to solve real-world and mathematical problems. Building t Transfrmatins n Crdinate Axis Grade 5: Gemetry Graph pints n the crdinate plane t slve real-wrld and mathematical prblems. 5.G.1. Use a pair f perpendicular number lines, called axes, t define

More information

Verification of Quality Parameters of a Solar Panel and Modification in Formulae of its Series Resistance

Verification of Quality Parameters of a Solar Panel and Modification in Formulae of its Series Resistance Verificatin f Quality Parameters f a Slar Panel and Mdificatin in Frmulae f its Series Resistance Sanika Gawhane Pune-411037-India Onkar Hule Pune-411037- India Chinmy Kulkarni Pune-411037-India Ojas Pandav

More information

Lecture 17: Free Energy of Multi-phase Solutions at Equilibrium

Lecture 17: Free Energy of Multi-phase Solutions at Equilibrium Lecture 17: 11.07.05 Free Energy f Multi-phase Slutins at Equilibrium Tday: LAST TIME...2 FREE ENERGY DIAGRAMS OF MULTI-PHASE SOLUTIONS 1...3 The cmmn tangent cnstructin and the lever rule...3 Practical

More information

Preparation work for A2 Mathematics [2017]

Preparation work for A2 Mathematics [2017] Preparatin wrk fr A2 Mathematics [2017] The wrk studied in Y12 after the return frm study leave is frm the Cre 3 mdule f the A2 Mathematics curse. This wrk will nly be reviewed during Year 13, it will

More information

CS 477/677 Analysis of Algorithms Fall 2007 Dr. George Bebis Course Project Due Date: 11/29/2007

CS 477/677 Analysis of Algorithms Fall 2007 Dr. George Bebis Course Project Due Date: 11/29/2007 CS 477/677 Analysis f Algrithms Fall 2007 Dr. Gerge Bebis Curse Prject Due Date: 11/29/2007 Part1: Cmparisn f Srting Algrithms (70% f the prject grade) The bjective f the first part f the assignment is

More information

Medium Scale Integrated (MSI) devices [Sections 2.9 and 2.10]

Medium Scale Integrated (MSI) devices [Sections 2.9 and 2.10] EECS 270, Winter 2017, Lecture 3 Page 1 f 6 Medium Scale Integrated (MSI) devices [Sectins 2.9 and 2.10] As we ve seen, it s smetimes nt reasnable t d all the design wrk at the gate-level smetimes we just

More information

SIGNALS AND SYSTEMS 15. Z TRANSFORM SOLUTION OF BACKWARD DE S FROM DEQ S WITH INITIAL CONDITIONS

SIGNALS AND SYSTEMS 15. Z TRANSFORM SOLUTION OF BACKWARD DE S FROM DEQ S WITH INITIAL CONDITIONS 73 SIGNALS AND SYSTEMS 5. Z TRANSFRM SLUTIN F BACKWARD DE S FRM DEQ S WITH INITIAL CNDITINS Intrductin In the previus handut, The Z Transfrm Slutin f Difference Equatins, the difference equatins were btained

More information

Relationship Between Amplifier Settling Time and Pole-Zero Placements for Second-Order Systems *

Relationship Between Amplifier Settling Time and Pole-Zero Placements for Second-Order Systems * Relatinship Between Amplifier Settling Time and Ple-Zer Placements fr Secnd-Order Systems * Mark E. Schlarmann and Randall L. Geiger Iwa State University Electrical and Cmputer Engineering Department Ames,

More information

CHARACTERIZATION OF A PERIODIC SURFACE PROFILE BY POLE-ZERO PARAMETERIZATION OF ELASTODYNAMIC PULSE REFLECTIONS * R.A. Roberts and J.D.

CHARACTERIZATION OF A PERIODIC SURFACE PROFILE BY POLE-ZERO PARAMETERIZATION OF ELASTODYNAMIC PULSE REFLECTIONS * R.A. Roberts and J.D. CHARACTERZATON OF A PERODC SURFACE PROFLE BY POLE-ZERO PARAMETERZATON OF ELASTODYNAMC PULSE REFLECTONS * R.A. Rberts and J.D. Achenbach The Technlgical nstitute Nrthwestern University Evanstn, L. 60201

More information

Bootstrap Method > # Purpose: understand how bootstrap method works > obs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(obs) >

Bootstrap Method > # Purpose: understand how bootstrap method works > obs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(obs) > Btstrap Methd > # Purpse: understand hw btstrap methd wrks > bs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(bs) > mean(bs) [1] 21.64625 > # estimate f lambda > lambda = 1/mean(bs);

More information

Lecture 02 CSE 40547/60547 Computing at the Nanoscale

Lecture 02 CSE 40547/60547 Computing at the Nanoscale PN Junctin Ntes: Lecture 02 CSE 40547/60547 Cmputing at the Nanscale Letʼs start with a (very) shrt review f semi-cnducting materials: - N-type material: Obtained by adding impurity with 5 valence elements

More information

Public Key Cryptography. Tim van der Horst & Kent Seamons

Public Key Cryptography. Tim van der Horst & Kent Seamons Public Key Cryptgraphy Tim van der Hrst & Kent Seamns Last Updated: Oct 5, 2017 Asymmetric Encryptin Why Public Key Crypt is Cl Has a linear slutin t the key distributin prblem Symmetric crypt has an expnential

More information

MATHEMATICS SYLLABUS SECONDARY 5th YEAR

MATHEMATICS SYLLABUS SECONDARY 5th YEAR Eurpean Schls Office f the Secretary-General Pedaggical Develpment Unit Ref. : 011-01-D-8-en- Orig. : EN MATHEMATICS SYLLABUS SECONDARY 5th YEAR 6 perid/week curse APPROVED BY THE JOINT TEACHING COMMITTEE

More information

AN INTERMITTENTLY USED SYSTEM WITH PREVENTIVE MAINTENANCE

AN INTERMITTENTLY USED SYSTEM WITH PREVENTIVE MAINTENANCE J. Operatins Research Sc. f Japan V!. 15, N. 2, June 1972. 1972 The Operatins Research Sciety f Japan AN INTERMITTENTLY USED SYSTEM WITH PREVENTIVE MAINTENANCE SHUNJI OSAKI University f Suthern Califrnia

More information

ZVS Boost Converter. (a) (b) Fig 6.29 (a) Quasi-resonant boost converter with M-type switch. (b) Equivalent circuit.

ZVS Boost Converter. (a) (b) Fig 6.29 (a) Quasi-resonant boost converter with M-type switch. (b) Equivalent circuit. EEL6246 Pwer Electrnics II Chapter 6 Lecture 6 Dr. Sam Abdel-Rahman ZVS Bst Cnverter The quasi-resnant bst cnverter by using the M-type switch as shwn in Fig. 6.29(a) with its simplified circuit shwn in

More information

Flipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System

Flipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal Mass-Spring System A Hrizntal Mass-Spring System is where a mass is attached t a spring, riented hrizntally, and then placed

More information

ELECTRON CYCLOTRON HEATING OF AN ANISOTROPIC PLASMA. December 4, PLP No. 322

ELECTRON CYCLOTRON HEATING OF AN ANISOTROPIC PLASMA. December 4, PLP No. 322 ELECTRON CYCLOTRON HEATING OF AN ANISOTROPIC PLASMA by J. C. SPROTT December 4, 1969 PLP N. 3 These PLP Reprts are infrmal and preliminary and as such may cntain errrs nt yet eliminated. They are fr private

More information

k-nearest Neighbor How to choose k Average of k points more reliable when: Large k: noise in attributes +o o noise in class labels

k-nearest Neighbor How to choose k Average of k points more reliable when: Large k: noise in attributes +o o noise in class labels Mtivating Example Memry-Based Learning Instance-Based Learning K-earest eighbr Inductive Assumptin Similar inputs map t similar utputs If nt true => learning is impssible If true => learning reduces t

More information

Chapter Summary. Mathematical Induction Strong Induction Recursive Definitions Structural Induction Recursive Algorithms

Chapter Summary. Mathematical Induction Strong Induction Recursive Definitions Structural Induction Recursive Algorithms Chapter 5 1 Chapter Summary Mathematical Inductin Strng Inductin Recursive Definitins Structural Inductin Recursive Algrithms Sectin 5.1 3 Sectin Summary Mathematical Inductin Examples f Prf by Mathematical

More information

Least Squares Optimal Filtering with Multirate Observations

Least Squares Optimal Filtering with Multirate Observations Prc. 36th Asilmar Cnf. n Signals, Systems, and Cmputers, Pacific Grve, CA, Nvember 2002 Least Squares Optimal Filtering with Multirate Observatins Charles W. herrien and Anthny H. Hawes Department f Electrical

More information

Interference is when two (or more) sets of waves meet and combine to produce a new pattern.

Interference is when two (or more) sets of waves meet and combine to produce a new pattern. Interference Interference is when tw (r mre) sets f waves meet and cmbine t prduce a new pattern. This pattern can vary depending n the riginal wave directin, wavelength, amplitude, etc. The tw mst extreme

More information

Distributions, spatial statistics and a Bayesian perspective

Distributions, spatial statistics and a Bayesian perspective Distributins, spatial statistics and a Bayesian perspective Dug Nychka Natinal Center fr Atmspheric Research Distributins and densities Cnditinal distributins and Bayes Thm Bivariate nrmal Spatial statistics

More information

CHAPTER 3 INEQUALITIES. Copyright -The Institute of Chartered Accountants of India

CHAPTER 3 INEQUALITIES. Copyright -The Institute of Chartered Accountants of India CHAPTER 3 INEQUALITIES Cpyright -The Institute f Chartered Accuntants f India INEQUALITIES LEARNING OBJECTIVES One f the widely used decisin making prblems, nwadays, is t decide n the ptimal mix f scarce

More information

SAMPLING DYNAMICAL SYSTEMS

SAMPLING DYNAMICAL SYSTEMS SAMPLING DYNAMICAL SYSTEMS Melvin J. Hinich Applied Research Labratries The University f Texas at Austin Austin, TX 78713-8029, USA (512) 835-3278 (Vice) 835-3259 (Fax) hinich@mail.la.utexas.edu ABSTRACT

More information

Introduction to Smith Charts

Introduction to Smith Charts Intrductin t Smith Charts Dr. Russell P. Jedlicka Klipsch Schl f Electrical and Cmputer Engineering New Mexic State University as Cruces, NM 88003 September 2002 EE521 ecture 3 08/22/02 Smith Chart Summary

More information

Pre-Calculus Individual Test 2017 February Regional

Pre-Calculus Individual Test 2017 February Regional The abbreviatin NOTA means Nne f the Abve answers and shuld be chsen if chices A, B, C and D are nt crrect. N calculatr is allwed n this test. Arcfunctins (such as y = Arcsin( ) ) have traditinal restricted

More information

Chapter 30. Inductance

Chapter 30. Inductance Chapter 30 nductance 30. Self-nductance Cnsider a lp f wire at rest. f we establish a current arund the lp, it will prduce a magnetic field. Sme f the magnetic field lines pass thrugh the lp. et! be the

More information

CONSTRUCTING STATECHART DIAGRAMS

CONSTRUCTING STATECHART DIAGRAMS CONSTRUCTING STATECHART DIAGRAMS The fllwing checklist shws the necessary steps fr cnstructing the statechart diagrams f a class. Subsequently, we will explain the individual steps further. Checklist 4.6

More information

Support-Vector Machines

Support-Vector Machines Supprt-Vectr Machines Intrductin Supprt vectr machine is a linear machine with sme very nice prperties. Haykin chapter 6. See Alpaydin chapter 13 fr similar cntent. Nte: Part f this lecture drew material

More information

Differentiation Applications 1: Related Rates

Differentiation Applications 1: Related Rates Differentiatin Applicatins 1: Related Rates 151 Differentiatin Applicatins 1: Related Rates Mdel 1: Sliding Ladder 10 ladder y 10 ladder 10 ladder A 10 ft ladder is leaning against a wall when the bttm

More information

3. Design of Channels General Definition of some terms CHAPTER THREE

3. Design of Channels General Definition of some terms CHAPTER THREE CHAPTER THREE. Design f Channels.. General The success f the irrigatin system depends n the design f the netwrk f canals. The canals may be excavated thrugh the difference types f sils such as alluvial

More information

ABSORPTION OF GAMMA RAYS

ABSORPTION OF GAMMA RAYS 6 Sep 11 Gamma.1 ABSORPTIO OF GAMMA RAYS Gamma rays is the name given t high energy electrmagnetic radiatin riginating frm nuclear energy level transitins. (Typical wavelength, frequency, and energy ranges

More information

Root locus ( )( ) The given TFs are: 1. Using Matlab: >> rlocus(g) >> Gp1=tf(1,poly([0-1 -2])) Transfer function: s^3 + 3 s^2 + 2 s

Root locus ( )( ) The given TFs are: 1. Using Matlab: >> rlocus(g) >> Gp1=tf(1,poly([0-1 -2])) Transfer function: s^3 + 3 s^2 + 2 s The given TFs are: 1 1() s = s s + 1 s + G p, () s ( )( ) >> Gp1=tf(1,ply([0-1 -])) Transfer functin: 1 ----------------- s^ + s^ + s Rt lcus G 1 = p ( s + 0.8 + j)( s + 0.8 j) >> Gp=tf(1,ply([-0.8-*i

More information

0606 ADDITIONAL MATHEMATICS

0606 ADDITIONAL MATHEMATICS PAPA CAMBRIDGE CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge Internatinal General Certificate f Secndary Educatin MARK SCHEME fr the Octber/Nvember 0 series 0606 ADDITIONAL MATHEMATICS 0606/ Paper, maimum

More information

MINIMIZATION OF ACTUATOR REPOSITIONING USING NEURAL NETWORKS WITH APPLICATION IN NONLINEAR HVAC 1 SYSTEMS

MINIMIZATION OF ACTUATOR REPOSITIONING USING NEURAL NETWORKS WITH APPLICATION IN NONLINEAR HVAC 1 SYSTEMS MINIMIZATION OF ACTUATOR REPOSITIONING USING NEURAL NETWORKS WITH APPLICATION IN NONLINEAR HVAC SYSTEMS M. J. Yazdanpanah *, E. Semsar, C. Lucas * yazdan@ut.ac.ir, semsar@chamran.ut.ac.ir, lucas@ipm.ir

More information

Level Control in Horizontal Tank by Fuzzy-PID Cascade Controller

Level Control in Horizontal Tank by Fuzzy-PID Cascade Controller Wrld Academy f Science, Engineering and Technlgy 5 007 Level Cntrl in Hrizntal Tank by Fuzzy-PID Cascade Cntrller Satean Tunyasrirut, and Santi Wangnipparnt Abstract The paper describes the Fuzzy PID cascade

More information

Flipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System

Flipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal Mass-Spring System A Hrizntal Mass-Spring System is where a mass is attached t a spring, riented hrizntally, and then placed

More information

Space Shuttle Ascent Mass vs. Time

Space Shuttle Ascent Mass vs. Time Space Shuttle Ascent Mass vs. Time Backgrund This prblem is part f a series that applies algebraic principles in NASA s human spaceflight. The Space Shuttle Missin Cntrl Center (MCC) and the Internatinal

More information

Drought damaged area

Drought damaged area ESTIMATE OF THE AMOUNT OF GRAVEL CO~TENT IN THE SOIL BY A I R B O'RN EMS S D A T A Y. GOMI, H. YAMAMOTO, AND S. SATO ASIA AIR SURVEY CO., l d. KANAGAWA,JAPAN S.ISHIGURO HOKKAIDO TOKACHI UBPREFECTRAl OffICE

More information

Frequency Response Analysis: A Review

Frequency Response Analysis: A Review Prcess Dynamics and Cntrl Subject: Frequency Respnse Analysis Prfessr Cstas Kiparissides Department f Chemical Engineering Aristtle University f Thessalniki December 6, 204 Frequency Respnse Analysis:

More information

[COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t o m a k e s u r e y o u a r e r e a d y )

[COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t o m a k e s u r e y o u a r e r e a d y ) (Abut the final) [COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t m a k e s u r e y u a r e r e a d y ) The department writes the final exam s I dn't really knw what's n it and I can't very well

More information

Lecture 13: Electrochemical Equilibria

Lecture 13: Electrochemical Equilibria 3.012 Fundamentals f Materials Science Fall 2005 Lecture 13: 10.21.05 Electrchemical Equilibria Tday: LAST TIME...2 An example calculatin...3 THE ELECTROCHEMICAL POTENTIAL...4 Electrstatic energy cntributins

More information

General Chemistry II, Unit II: Study Guide (part 1)

General Chemistry II, Unit II: Study Guide (part 1) General Chemistry II, Unit II: Study Guide (part 1) CDS Chapter 21: Reactin Equilibrium in the Gas Phase General Chemistry II Unit II Part 1 1 Intrductin Sme chemical reactins have a significant amunt

More information

COMP 551 Applied Machine Learning Lecture 5: Generative models for linear classification

COMP 551 Applied Machine Learning Lecture 5: Generative models for linear classification COMP 551 Applied Machine Learning Lecture 5: Generative mdels fr linear classificatin Instructr: Herke van Hf (herke.vanhf@mail.mcgill.ca) Slides mstly by: Jelle Pineau Class web page: www.cs.mcgill.ca/~hvanh2/cmp551

More information

3.4 Shrinkage Methods Prostate Cancer Data Example (Continued) Ridge Regression

3.4 Shrinkage Methods Prostate Cancer Data Example (Continued) Ridge Regression 3.3.4 Prstate Cancer Data Example (Cntinued) 3.4 Shrinkage Methds 61 Table 3.3 shws the cefficients frm a number f different selectin and shrinkage methds. They are best-subset selectin using an all-subsets

More information

Activity Guide Loops and Random Numbers

Activity Guide Loops and Random Numbers Unit 3 Lessn 7 Name(s) Perid Date Activity Guide Lps and Randm Numbers CS Cntent Lps are a relatively straightfrward idea in prgramming - yu want a certain chunk f cde t run repeatedly - but it takes a

More information

LHS Mathematics Department Honors Pre-Calculus Final Exam 2002 Answers

LHS Mathematics Department Honors Pre-Calculus Final Exam 2002 Answers LHS Mathematics Department Hnrs Pre-alculus Final Eam nswers Part Shrt Prblems The table at the right gives the ppulatin f Massachusetts ver the past several decades Using an epnential mdel, predict the

More information

Preparation work for A2 Mathematics [2018]

Preparation work for A2 Mathematics [2018] Preparatin wrk fr A Mathematics [018] The wrk studied in Y1 will frm the fundatins n which will build upn in Year 13. It will nly be reviewed during Year 13, it will nt be retaught. This is t allw time

More information

A New Evaluation Measure. J. Joiner and L. Werner. The problems of evaluation and the needed criteria of evaluation

A New Evaluation Measure. J. Joiner and L. Werner. The problems of evaluation and the needed criteria of evaluation III-l III. A New Evaluatin Measure J. Jiner and L. Werner Abstract The prblems f evaluatin and the needed criteria f evaluatin measures in the SMART system f infrmatin retrieval are reviewed and discussed.

More information

Chapter 8. Root Locus Techniques

Chapter 8. Root Locus Techniques Chapter 8 Rt Lcu Technique Intrductin Sytem perfrmance and tability dt determined dby cled-lp l ple Typical cled-lp feedback cntrl ytem G Open-lp TF KG H Zer -, - Ple 0, -, -4 K 4 Lcatin f ple eaily fund

More information

ChE 471: LECTURE 4 Fall 2003

ChE 471: LECTURE 4 Fall 2003 ChE 47: LECTURE 4 Fall 003 IDEL RECTORS One f the key gals f chemical reactin engineering is t quantify the relatinship between prductin rate, reactr size, reactin kinetics and selected perating cnditins.

More information

Dispersion Ref Feynman Vol-I, Ch-31

Dispersion Ref Feynman Vol-I, Ch-31 Dispersin Ref Feynman Vl-I, Ch-31 n () = 1 + q N q /m 2 2 2 0 i ( b/m) We have learned that the index f refractin is nt just a simple number, but a quantity that varies with the frequency f the light.

More information

A Correlation of. to the. South Carolina Academic Standards for Mathematics Precalculus

A Correlation of. to the. South Carolina Academic Standards for Mathematics Precalculus A Crrelatin f Suth Carlina Academic Standards fr Mathematics Precalculus INTRODUCTION This dcument demnstrates hw Precalculus (Blitzer), 4 th Editin 010, meets the indicatrs f the. Crrelatin page references

More information

T(s) 1+ T(s) 2. Phase Margin Test for T(s) a. Unconditionally Stable φ m = 90 o for 1 pole T(s) b. Conditionally Stable Case 1.

T(s) 1+ T(s) 2. Phase Margin Test for T(s) a. Unconditionally Stable φ m = 90 o for 1 pole T(s) b. Conditionally Stable Case 1. Lecture 49 Danger f Instability/Oscillatin When Emplying Feedback In PWM Cnverters A. Guessing Clsed Lp Stability Frm Open Lp Frequency Respnse Data. T(s) versus T(s) + T(s) 2. Phase Margin Test fr T(s)

More information

Inference in the Multiple-Regression

Inference in the Multiple-Regression Sectin 5 Mdel Inference in the Multiple-Regressin Kinds f hypthesis tests in a multiple regressin There are several distinct kinds f hypthesis tests we can run in a multiple regressin. Suppse that amng

More information

Simulation of Line Outage Distribution Factors (L.O.D.F) Calculation for N-Buses System

Simulation of Line Outage Distribution Factors (L.O.D.F) Calculation for N-Buses System Simulatin f Line Outage Distributin Factrs (L.O.D.F) Calculatin fr N-Buses System Rashid H. AL-Rubayi Department f Electrical Engineering, University f Technlgy Afaneen A. Abd Department f Electrical Engineering,

More information

READING STATECHART DIAGRAMS

READING STATECHART DIAGRAMS READING STATECHART DIAGRAMS Figure 4.48 A Statechart diagram with events The diagram in Figure 4.48 shws all states that the bject plane can be in during the curse f its life. Furthermre, it shws the pssible

More information

This section is primarily focused on tools to aid us in finding roots/zeros/ -intercepts of polynomials. Essentially, our focus turns to solving.

This section is primarily focused on tools to aid us in finding roots/zeros/ -intercepts of polynomials. Essentially, our focus turns to solving. Sectin 3.2: Many f yu WILL need t watch the crrespnding vides fr this sectin n MyOpenMath! This sectin is primarily fcused n tls t aid us in finding rts/zers/ -intercepts f plynmials. Essentially, ur fcus

More information

Resampling Methods. Chapter 5. Chapter 5 1 / 52

Resampling Methods. Chapter 5. Chapter 5 1 / 52 Resampling Methds Chapter 5 Chapter 5 1 / 52 1 51 Validatin set apprach 2 52 Crss validatin 3 53 Btstrap Chapter 5 2 / 52 Abut Resampling An imprtant statistical tl Pretending the data as ppulatin and

More information

EDA Engineering Design & Analysis Ltd

EDA Engineering Design & Analysis Ltd EDA Engineering Design & Analysis Ltd THE FINITE ELEMENT METHOD A shrt tutrial giving an verview f the histry, thery and applicatin f the finite element methd. Intrductin Value f FEM Applicatins Elements

More information

x x

x x Mdeling the Dynamics f Life: Calculus and Prbability fr Life Scientists Frederick R. Adler cfrederick R. Adler, Department f Mathematics and Department f Bilgy, University f Utah, Salt Lake City, Utah

More information