Hybrid Control and Switched Systems. Lecture #3 What can go wrong? Trajectories of hybrid systems
|
|
- Everett Wells
- 5 years ago
- Views:
Transcription
1 Hybrid Conrol and Swiched Sysems Lecure #3 Wha can go wrong? Trajecories of hybrid sysems João P. Hespanha Universiy of California a Sana Barbara Summary 1. Trajecories of hybrid sysems: Soluion o a hybrid sysem Execuion of a hybrid sysem 2. Degeneracies Finie escape ime Chaering Zeno rajecories Non-coninuous dependency on iniial condiions 1
2 Hybrid Auomaon (deerminisic) Q se of discree saes R n coninuous sae-space f : Q R n R n vecor field ϕ : Q R n Q discree ransiion ρ : Q R n R n rese map x ú ρ(q 1,x ) mode q 2 ϕ(q 1,x ) = q 2? mode q 1 mode q 3 ϕ(q 1,x ) = q 3? x ú ρ(q 1,x ) Q R n f : Q R n R n Φ : Q R n Q R n Hybrid Auomaon se of discree saes coninuous sae-space vecor field discree ransiion (& rese map) x ú Φ 2 (q 1,x ) mode q 2 Φ 1 (q 1,x ) = q 2? mode q 1 mode q 3 Φ 1 (q 1,x ) = q 3? x ú Φ 2 (q 1,x ) Compac represenaion of a hybrid auomaon 2
3 Soluion o a hybrid auomaon x ú Φ 2 (q 1,x ) Φ 1 (q 1,x ) = q 2? mode q 2 mode q 1 Definiion: A soluion o he hybrid auomaon is a pair of righ-coninuous signals x : [0, ) R n q : [0, ) Q such ha 1. x is piecewise differeniable & q is piecewise consan 2. on any inerval ( 1, 2 ) on which q is consan and x coninuous coninuous evoluion 3. discree ransiions Example #4: Invered pendulum swing-up u [-1,1] θ Hybrid conroller: 1 s pump/remove energy ino/from he sysem by applying maximum force, unil E 0 2 nd wai unil pendulum is close o he uprigh posiion 3 h nex o uprigh posiion use feedback linearizaion conroller remove energy E [ ε,ε]? E < ε? wai sabilize E>ε? pump energy E [ ε,ε]? ω + θ δ? 3
4 Example #4: Invered pendulum swing-up pump energy wai sabilize Ε ε? ω + θ δ? Q = { pump, wai, sab } R 2 = coninuous sae-space (pump,x) q = pump,e < ² (wai,x) q = pump,e ² Φ(q,x)= (wai,x) q =wai, ω + θ > δ (sab,x) q =wai, ω + θ δ (sab,x) q =sab Example #4: Invered pendulum swing-up pump energy wai sabilize Ε ε? ω + θ δ? Q = { pump, wai, sab } R 2 = coninuous sae-space δ ω + θ ε E q = wai q = sab q = pump Wha if we already have ω + θ δ when E reaches ε? 4
5 Example #4: Invered pendulum swing-up pump energy wai sabilize Q = { pump, wai, sab } δ ε ω + θ E q = pump * Ε ε? q = wai ω + θ δ? R 2 = coninuous sae-space (pump,x) q =pump,e < ² (wai,x) q =pump,e ² Φ(q, x) = (wai,x) q =wai, ω + θ > ² (sab,x) q =wai, ω + θ ² (sab,x) q =sab q = sab q( * ) =? In general, we may need more han one value for he sae a each insan of ime Hybrid signals Definiion: A hybrid ime rajecory is a (finie or infinie) sequence of closed inervals τ = { [τ i,τ' i ] : τ i τ' i, τ' i = τ i+1, i =1,2, } (if τ is finie he las inerval may by open on he righ) T se of hybrid ime rajecories Definiion: For a given τ = { [τ i,τ' i ] : τ i τ' i, τ i+1 = τ' i, i =1,2, } T a hybrid signal defined on τ wih values on X is a sequence of funcions x = {x i : [τ i,τ' i ] X i =1,2, } x : τ X hybrid signal defined on τ wih values on X E.g., τ ú { [0,1], [1,1], [1,1], [1,+ ) }, x ú { /2, 1/4, 3/4, 1/ } /2 3/4 1/4 1/ [0,1] [1,1] [1,1] [1,+ ) A hybrid signal can ake muliple values for he same ime-insan 5
6 Execuion of a hybrid auomaon x ú Φ 2 (q 1,x ) Φ 1 (q 1,x ) = q 2? mode q 2 mode q 1 Definiion: An execuion of he hybrid auomaon is a pair of hybrid signals x : τ R n q : τ Q τ = { [τ i,τ' i ] : i =1,2, } T such ha 1. on any [τ i,τ' i ] τ, q i is consan and coninuous evoluion 2. discree ransiions Example #4: Invered pendulum swing-up pump energy wai sabilize Ε ε? ω + θ δ? Q = { pump, wai, sab } R 2 = coninuous sae-space δ ε ω + θ * q = sab τ ú { [0, * ], [ *, * ], [ *,+ ) } q ú { pump, wai, sab } x ú {,, } E q = pump q = wai 1. There are oher conceps of soluion ha also avoid his problem [Teel 2005] 2. Could one fix his hybrid sysem o sill work wih he usual noion of soluion? 6
7 Wha can go wrong? 1. Problems in he coninuous evoluion : exisence uniqueness finie escape 2. Problems in he hybrid execuion: Chaering Zeno 3. Non-coninuous dependency on iniial condiions Exisence f(x) disconinuous x There is no soluion o his differenial equaion ha sars wih x(0) = 0 Why? one any inerval [0,ε) x canno: remain zero, become posiive, or become negaive. (x = 0 would make some sense) Theorem [Exisence of soluion] If f : R n R n is coninuous, hen x 0 R n here exiss a leas one soluion wih x(0) = x 0, defined on some inerval [0,ε) 7
8 Uniqueness f(x) derivaive a 0 There are muliple soluions ha sar a x(0) = 0, e.g., x x Definiions: A funcion f : R n R n is Lipschiz coninuous if in any bounded subse of S of R n here exiss a consan c such ha (f is differeniable almos everywhere and he derivaive is bounded on any bounded se) Theorem [Uniqueness of soluion] If f : R n R n is Lipschiz coninuous, hen x 0 R n here a single soluion wih x(0) = x 0, defined on some inerval [0,ε) Finie escape ime x Any soluion ha does no sar a x(0) = 0 is of he form unbounded derivaive T -1 T finie escape soluion x ends o in finie ime Definiions: A funcion f : R n R n is globally Lipschiz coninuous if here exiss a consan c such ha f grows no faser han linearly Theorem [Uniqueness of soluion] If f : R n R n is globally Lipschiz coninuous, hen x 0 R n here a single soluion wih x(0) = x 0, defined on [0, ) 8
9 Degenerae execuions due o problems in he coninuous evoluion x 1? q = 1 q = 2 τ = { [0,1], [1,2] } q = {q 1 () = 1, q 2 () = 2 } x = {x 1 () =, () = 2 } x x 1 () () 1 2 τ = { [0,1], [1,2) } q = {q 1 () = 1, q 2 () = 2 } x = {x 1 () =, () = 1/(2 ) } x 1? q = 1 q = 2 x x 1 () () 1 2 Chaering There is no soluion pas x = 0 x () 1 q = 1 x 0? q = 2 bu here is an execuion x < 0? q 1 () q 2k+1 () k 1 τ = { [0,1], [1], [1], [1], } q = {q 1 () = 2, q 2 () = 1, q 1 () = 2, } x = {x 1 () = 1, () = 0, () = 0, } x 1 () q 2k () k 1 x k () k 2 1 Chaering execuion τ is infinie bu afer some ime, all inervals are singleons 9
10 Example #3: Semi-auomaic ransmission v() { up, down, keep } drivers inpu (discree) v = up or ω ω 2? v = up or ω ω 3? v = up or ω ω 4? g = 1 g = 2 g = 3 g = 4 v = down or ω ϖ 1? v = down or ω ϖ 2? v = down or ω ϖ 3? ω 2 g = 1 ϖ 1 g = 2 ω 3 ϖ 2 g = 3 ϖ 3 g = 4 ω 4 If he driver ses v() = up * and ω( ) ϖ 1 one ges chaering. For ever? Example #1: Bouncing ball (Zeno execuion) y g Free fall Collision c [0,1) energy absorbed a impac x 1 = 0 & <0? ú c 10
11 Zeno soluion x 1 = 0 & <0? ú c Zeno soluion x 1 = 0 & = 0? x 1 = 0 & <0? ú c 11
12 Zeno execuion x 1 = 0 & <0? ú c Zeno execuion τ is infinie bu he execuion does no exend o = + Zeno ime τ ú sup s τ s Example #9: Waer ank sysem (regularizaion) inflow λ goal preven boh anks from becoming empying h 2? q = 1 q = 2 x 1 h 1 h 2 ouflow μ 1 ouflow μ 2 x 1 h 1? x 1 h 1 = h 2 q = 1 q = 2 q = 1 12
13 inflow λ Temporal regularizaion Example #9: Waer ank sysem h 2? q = 2 x 1 h 1 h 2 x 1 h 1? ouflow μ 1 ouflow μ 2 only jump afer a minimum ime has elapsed h 2, τ > ε? q = 1 τ ú 0 q = 2 τ ú 0 x 1 h 1, τ > ε? inflow λ Spaial regularizaion Example #9: Waer ank sysem h 2? q = 2 x 1 ouflow μ 1 h 1 h 2 ouflow μ 2 only jump afer a minimum change in he coninuous sae has occurred h 2, x 1 x 3 + x 4 > ε? q = 1 x 1 h 1? x 3 ú x 1, x 4 ú q = 2 x 3 ú x 1, x 4 ú x 1 h 1, x 1 x 3 + x 4 > ε? 13
14 Coninuiy wih respec o iniial condiions Theorem [Uniqueness & coninuiy of soluion] If f : R n R n is Lipschiz coninuous, hen x 0 R n here a single soluion wih x(0) = x 0, defined on some inerval [0,ε) Moreover, given any T <, and wo soluions x 1, ha exis on [0,T]: ε > 0 δ > 0 : x 1 (0) (0) δ x 1 () () ε [0,T] value of he soluion on he inerval [0,T] is coninuous wih respec o he iniial condiions Disconinuiy wih respec o iniial condiions x 1 0 & > 0? mode q 1 ú 1 mode q 2 disconinuiy of he rese map x 1 0 & 0? ú 2 2 ε 1 x no maer how close o zero (0) = ε > 0 is, (1) = 1 x 1 if (0) = 0 hen (1) = 2 14
15 Disconinuiy wih respec o iniial condiions x 1 0 & > 0? mode q 1 x 1 0 & 0? ẋ 1 =1 ẋ 2 =0 mode q 2 mode q 3 ẋ 1 =0 ẋ 1 =1 ẋ 2 = 1 ẋ 2 =1 ε 1 x 1 ε 1 x 1 no maer how close o zero (0) = ε > 0 is, (2) = ε 1 if (0) = 0 hen (2) = 1 problem arises from disconinuiy of he ransiion funcion Nex class 1. Numerical simulaion of hybrid auomaa simulaions of ODEs zero-crossing deecion 2. Simulaors Simulink Saeflow SHIFT Modelica Follow-up homework Find condiions for he exisence of soluion o a hybrid sysem 15
EXERCISES FOR SECTION 1.5
1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler
More informationHamilton- J acobi Equation: Explicit Formulas In this lecture we try to apply the method of characteristics to the Hamilton-Jacobi equation: u t
M ah 5 2 7 Fall 2 0 0 9 L ecure 1 0 O c. 7, 2 0 0 9 Hamilon- J acobi Equaion: Explici Formulas In his lecure we ry o apply he mehod of characerisics o he Hamilon-Jacobi equaion: u + H D u, x = 0 in R n
More informationLecture 20: Riccati Equations and Least Squares Feedback Control
34-5 LINEAR SYSTEMS Lecure : Riccai Equaions and Leas Squares Feedback Conrol 5.6.4 Sae Feedback via Riccai Equaions A recursive approach in generaing he marix-valued funcion W ( ) equaion for i for he
More informationChapter 1 Fundamental Concepts
Chaper 1 Fundamenal Conceps 1 Signals A signal is a paern of variaion of a physical quaniy, ofen as a funcion of ime (bu also space, disance, posiion, ec). These quaniies are usually he independen variables
More informationHamilton- J acobi Equation: Weak S olution We continue the study of the Hamilton-Jacobi equation:
M ah 5 7 Fall 9 L ecure O c. 4, 9 ) Hamilon- J acobi Equaion: Weak S oluion We coninue he sudy of he Hamilon-Jacobi equaion: We have shown ha u + H D u) = R n, ) ; u = g R n { = }. ). In general we canno
More informationIntermediate Differential Equations Review and Basic Ideas
Inermediae Differenial Equaions Review and Basic Ideas John A. Burns Cener for Opimal Design And Conrol Inerdisciplinary Cener forappliedmahemaics Virginia Polyechnic Insiue and Sae Universiy Blacksburg,
More informationCrossing the Bridge between Similar Games
Crossing he Bridge beween Similar Games Jan-David Quesel, Marin Fränzle, and Werner Damm Universiy of Oldenburg, Deparmen of Compuing Science, Germany CMACS Seminar CMU, Pisburgh, PA, USA 2nd December
More informationPredator - Prey Model Trajectories and the nonlinear conservation law
Predaor - Prey Model Trajecories and he nonlinear conservaion law James K. Peerson Deparmen of Biological Sciences and Deparmen of Mahemaical Sciences Clemson Universiy Ocober 28, 213 Ouline Drawing Trajecories
More informationFrom Particles to Rigid Bodies
Rigid Body Dynamics From Paricles o Rigid Bodies Paricles No roaions Linear velociy v only Rigid bodies Body roaions Linear velociy v Angular velociy ω Rigid Bodies Rigid bodies have boh a posiion and
More information15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
More information(1) (2) Differentiation of (1) and then substitution of (3) leads to. Therefore, we will simply consider the second-order linear system given by (4)
Phase Plane Analysis of Linear Sysems Adaped from Applied Nonlinear Conrol by Sloine and Li The general form of a linear second-order sysem is a c b d From and b bc d a Differeniaion of and hen subsiuion
More informationSolutions to Assignment 1
MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we
More informationOrdinary Differential Equations
Lecure 22 Ordinary Differenial Equaions Course Coordinaor: Dr. Suresh A. Karha, Associae Professor, Deparmen of Civil Engineering, IIT Guwahai. In naure, mos of he phenomena ha can be mahemaically described
More informationt dt t SCLP Bellman (1953) CLP (Dantzig, Tyndall, Grinold, Perold, Anstreicher 60's-80's) Anderson (1978) SCLP
Coninuous Linear Programming. Separaed Coninuous Linear Programming Bellman (1953) max c () u() d H () u () + Gsusds (,) () a () u (), < < CLP (Danzig, yndall, Grinold, Perold, Ansreicher 6's-8's) Anderson
More informationFour Generations of Higher Order Sliding Mode Controllers. L. Fridman Universidad Nacional Autonoma de Mexico Aussois, June, 10th, 2015
Four Generaions of Higher Order Sliding Mode Conrollers L. Fridman Universidad Nacional Auonoma de Mexico Aussois, June, 1h, 215 Ouline 1 Generaion 1:Two Main Conceps of Sliding Mode Conrol 2 Generaion
More informationMath Week 14 April 16-20: sections first order systems of linear differential equations; 7.4 mass-spring systems.
Mah 2250-004 Week 4 April 6-20 secions 7.-7.3 firs order sysems of linear differenial equaions; 7.4 mass-spring sysems. Mon Apr 6 7.-7.2 Sysems of differenial equaions (7.), and he vecor Calculus we need
More informationHybrid Control and Switched Systems. Lecture #1 Hybrid systems are everywhere: Examples
Hybrid Control and Switched Systems Lecture #1 Hybrid systems are everywhere: Examples João P. Hespanha University of California at Santa Barbara Summary Examples of hybrid systems 1. Bouncing ball 2.
More informationLAPLACE TRANSFORM AND TRANSFER FUNCTION
CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dep. of Chemical and Biological Engineering 5-1 Road Map of he Lecure V Laplace Transform and Transfer funcions
More informationBasic Circuit Elements Professor J R Lucas November 2001
Basic Circui Elemens - J ucas An elecrical circui is an inerconnecion of circui elemens. These circui elemens can be caegorised ino wo ypes, namely acive and passive elemens. Some Definiions/explanaions
More informationAn Introduction to Malliavin calculus and its applications
An Inroducion o Malliavin calculus and is applicaions Lecure 5: Smoohness of he densiy and Hörmander s heorem David Nualar Deparmen of Mahemaics Kansas Universiy Universiy of Wyoming Summer School 214
More information4.6 One Dimensional Kinematics and Integration
4.6 One Dimensional Kinemaics and Inegraion When he acceleraion a( of an objec is a non-consan funcion of ime, we would like o deermine he ime dependence of he posiion funcion x( and he x -componen of
More informationSimulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010
Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid
More information6. Stochastic calculus with jump processes
A) Trading sraegies (1/3) Marke wih d asses S = (S 1,, S d ) A rading sraegy can be modelled wih a vecor φ describing he quaniies invesed in each asse a each insan : φ = (φ 1,, φ d ) The value a of a porfolio
More informationLecture 4 Notes (Little s Theorem)
Lecure 4 Noes (Lile s Theorem) This lecure concerns one of he mos imporan (and simples) heorems in Queuing Theory, Lile s Theorem. More informaion can be found in he course book, Bersekas & Gallagher,
More informationLecture 1 Overview. course mechanics. outline & topics. what is a linear dynamical system? why study linear systems? some examples
EE263 Auumn 27-8 Sephen Boyd Lecure 1 Overview course mechanics ouline & opics wha is a linear dynamical sysem? why sudy linear sysems? some examples 1 1 Course mechanics all class info, lecures, homeworks,
More informationChapter 7 Response of First-order RL and RC Circuits
Chaper 7 Response of Firs-order RL and RC Circuis 7.- The Naural Response of RL and RC Circuis 7.3 The Sep Response of RL and RC Circuis 7.4 A General Soluion for Sep and Naural Responses 7.5 Sequenial
More information2. Nonlinear Conservation Law Equations
. Nonlinear Conservaion Law Equaions One of he clear lessons learned over recen years in sudying nonlinear parial differenial equaions is ha i is generally no wise o ry o aack a general class of nonlinear
More informationPhysics 221 Fall 2008 Homework #2 Solutions Ch. 2 Due Tues, Sept 9, 2008
Physics 221 Fall 28 Homework #2 Soluions Ch. 2 Due Tues, Sep 9, 28 2.1 A paricle moving along he x-axis moves direcly from posiion x =. m a ime =. s o posiion x = 1. m by ime = 1. s, and hen moves direcly
More informationHamilton Jacobi equations
Hamilon Jacobi equaions Inoducion o PDE The rigorous suff from Evans, mosly. We discuss firs u + H( u = 0, (1 where H(p is convex, and superlinear a infiniy, H(p lim p p = + This by comes by inegraion
More informationf t te e = possesses a Laplace transform. Exercises for Module-III (Transform Calculus)
Exercises for Module-III (Transform Calculus) ) Discuss he piecewise coninuiy of he following funcions: =,, +, > c) e,, = d) sin,, = ) Show ha he funcion ( ) sin ( ) f e e = possesses a Laplace ransform.
More informationMath 2214 Solution Test 1A Spring 2016
Mah 14 Soluion Tes 1A Spring 016 sec Problem 1: Wha is he larges -inerval for which ( 4) = has a guaraneed + unique soluion for iniial value (-1) = 3 according o he Exisence Uniqueness Theorem? Soluion
More informationarxiv: v1 [math.ca] 15 Nov 2016
arxiv:6.599v [mah.ca] 5 Nov 26 Counerexamples on Jumarie s hree basic fracional calculus formulae for non-differeniable coninuous funcions Cheng-shi Liu Deparmen of Mahemaics Norheas Peroleum Universiy
More informationExistence of multiple positive periodic solutions for functional differential equations
J. Mah. Anal. Appl. 325 (27) 1378 1389 www.elsevier.com/locae/jmaa Exisence of muliple posiive periodic soluions for funcional differenial equaions Zhijun Zeng a,b,,libi a, Meng Fan a a School of Mahemaics
More informationOn-line Adaptive Optimal Timing Control of Switched Systems
On-line Adapive Opimal Timing Conrol of Swiched Sysems X.C. Ding, Y. Wardi and M. Egersed Absrac In his paper we consider he problem of opimizing over he swiching imes for a muli-modal dynamic sysem when
More informationMath 333 Problem Set #2 Solution 14 February 2003
Mah 333 Problem Se #2 Soluion 14 February 2003 A1. Solve he iniial value problem dy dx = x2 + e 3x ; 2y 4 y(0) = 1. Soluion: This is separable; we wrie 2y 4 dy = x 2 + e x dx and inegrae o ge The iniial
More information4 Sequences of measurable functions
4 Sequences of measurable funcions 1. Le (Ω, A, µ) be a measure space (complee, afer a possible applicaion of he compleion heorem). In his chaper we invesigae relaions beween various (nonequivalen) convergences
More informationEssential Microeconomics : OPTIMAL CONTROL 1. Consider the following class of optimization problems
Essenial Microeconomics -- 6.5: OPIMAL CONROL Consider he following class of opimizaion problems Max{ U( k, x) + U+ ( k+ ) k+ k F( k, x)}. { x, k+ } = In he language of conrol heory, he vecor k is he vecor
More informationSTABILITY OF RESET SWITCHING SYSTEMS
STABILITY OF RESET SWITCHING SYSTEMS J.P. Paxman,G.Vinnicombe Cambridge Universiy Engineering Deparmen, UK, jpp7@cam.ac.uk Keywords: Sabiliy, swiching sysems, bumpless ransfer, conroller realizaion. Absrac
More informationGuest Lectures for Dr. MacFarlane s EE3350 Part Deux
Gues Lecures for Dr. MacFarlane s EE3350 Par Deux Michael Plane Mon., 08-30-2010 Wrie name in corner. Poin ou his is a review, so I will go faser. Remind hem o go lisen o online lecure abou geing an A
More informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More informationE β t log (C t ) + M t M t 1. = Y t + B t 1 P t. B t 0 (3) v t = P tc t M t Question 1. Find the FOC s for an optimum in the agent s problem.
Noes, M. Krause.. Problem Se 9: Exercise on FTPL Same model as in paper and lecure, only ha one-period govenmen bonds are replaced by consols, which are bonds ha pay one dollar forever. I has curren marke
More informationChapter 6. Systems of First Order Linear Differential Equations
Chaper 6 Sysems of Firs Order Linear Differenial Equaions We will only discuss firs order sysems However higher order sysems may be made ino firs order sysems by a rick shown below We will have a sligh
More informationCONTRIBUTION TO IMPULSIVE EQUATIONS
European Scienific Journal Sepember 214 /SPECIAL/ ediion Vol.3 ISSN: 1857 7881 (Prin) e - ISSN 1857-7431 CONTRIBUTION TO IMPULSIVE EQUATIONS Berrabah Faima Zohra, MA Universiy of sidi bel abbes/ Algeria
More informationMath 527 Lecture 6: Hamilton-Jacobi Equation: Explicit Formulas
Mah 527 Lecure 6: Hamilon-Jacobi Equaion: Explici Formulas Sep. 23, 2 Mehod of characerisics. We r o appl he mehod of characerisics o he Hamilon-Jacobi equaion: u +Hx, Du = in R n, u = g on R n =. 2 To
More information4.5 Constant Acceleration
4.5 Consan Acceleraion v() v() = v 0 + a a() a a() = a v 0 Area = a (a) (b) Figure 4.8 Consan acceleraion: (a) velociy, (b) acceleraion When he x -componen of he velociy is a linear funcion (Figure 4.8(a)),
More information10. State Space Methods
. Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he
More informationEE 315 Notes. Gürdal Arslan CLASS 1. (Sections ) What is a signal?
EE 35 Noes Gürdal Arslan CLASS (Secions.-.2) Wha is a signal? In his class, a signal is some funcion of ime and i represens how some physical quaniy changes over some window of ime. Examples: velociy of
More informationdt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.
Mah Rahman Exam Review Soluions () Consider he IVP: ( 4)y 3y + 4y = ; y(3) = 0, y (3) =. (a) Please deermine he longes inerval for which he IVP is guaraneed o have a unique soluion. Soluion: The disconinuiies
More informationThe Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales
Advances in Dynamical Sysems and Applicaions. ISSN 0973-5321 Volume 1 Number 1 (2006, pp. 103 112 c Research India Publicaions hp://www.ripublicaion.com/adsa.hm The Asympoic Behavior of Nonoscillaory Soluions
More information3.6 Derivatives as Rates of Change
3.6 Derivaives as Raes of Change Problem 1 John is walking along a sraigh pah. His posiion a he ime >0 is given by s = f(). He sars a =0from his house (f(0) = 0) and he graph of f is given below. (a) Describe
More informationMath 334 Test 1 KEY Spring 2010 Section: 001. Instructor: Scott Glasgow Dates: May 10 and 11.
1 Mah 334 Tes 1 KEY Spring 21 Secion: 1 Insrucor: Sco Glasgow Daes: Ma 1 and 11. Do NOT wrie on his problem saemen bookle, excep for our indicaion of following he honor code jus below. No credi will be
More informationt 2 B F x,t n dsdt t u x,t dxdt
Evoluion Equaions For 0, fixed, le U U0, where U denoes a bounded open se in R n.suppose ha U is filled wih a maerial in which a conaminan is being ranspored by various means including diffusion and convecion.
More informationMATH 4330/5330, Fourier Analysis Section 6, Proof of Fourier s Theorem for Pointwise Convergence
MATH 433/533, Fourier Analysis Secion 6, Proof of Fourier s Theorem for Poinwise Convergence Firs, some commens abou inegraing periodic funcions. If g is a periodic funcion, g(x + ) g(x) for all real x,
More informationVariational Iteration Method for Solving System of Fractional Order Ordinary Differential Equations
IOSR Journal of Mahemaics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 1, Issue 6 Ver. II (Nov - Dec. 214), PP 48-54 Variaional Ieraion Mehod for Solving Sysem of Fracional Order Ordinary Differenial
More informationClass Meeting # 10: Introduction to the Wave Equation
MATH 8.5 COURSE NOTES - CLASS MEETING # 0 8.5 Inroducion o PDEs, Fall 0 Professor: Jared Speck Class Meeing # 0: Inroducion o he Wave Equaion. Wha is he wave equaion? The sandard wave equaion for a funcion
More informationMath 334 Fall 2011 Homework 11 Solutions
Dec. 2, 2 Mah 334 Fall 2 Homework Soluions Basic Problem. Transform he following iniial value problem ino an iniial value problem for a sysem: u + p()u + q() u g(), u() u, u () v. () Soluion. Le v u. Then
More informationBasic definitions and relations
Basic definiions and relaions Lecurer: Dmiri A. Molchanov E-mail: molchan@cs.u.fi hp://www.cs.u.fi/kurssi/tlt-2716/ Kendall s noaion for queuing sysems: Arrival processes; Service ime disribuions; Examples.
More information= ( ) ) or a system of differential equations with continuous parametrization (T = R
XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of
More informationTHE 2-BODY PROBLEM. FIGURE 1. A pair of ellipses sharing a common focus. (c,b) c+a ROBERT J. VANDERBEI
THE 2-BODY PROBLEM ROBERT J. VANDERBEI ABSTRACT. In his shor noe, we show ha a pair of ellipses wih a common focus is a soluion o he 2-body problem. INTRODUCTION. Solving he 2-body problem from scrach
More informationProperties Of Solutions To A Generalized Liénard Equation With Forcing Term
Applied Mahemaics E-Noes, 8(28), 4-44 c ISSN 67-25 Available free a mirror sies of hp://www.mah.nhu.edu.w/ amen/ Properies Of Soluions To A Generalized Liénard Equaion Wih Forcing Term Allan Kroopnick
More informationdi Bernardo, M. (1995). A purely adaptive controller to synchronize and control chaotic systems.
di ernardo, M. (995). A purely adapive conroller o synchronize and conrol chaoic sysems. hps://doi.org/.6/375-96(96)8-x Early version, also known as pre-prin Link o published version (if available):.6/375-96(96)8-x
More informationSolution: b All the terms must have the dimension of acceleration. We see that, indeed, each term has the units of acceleration
PHYS 54 Tes Pracice Soluions Spring 8 Q: [4] Knowing ha in he ne epression a is acceleraion, v is speed, is posiion and is ime, from a dimensional v poin of view, he equaion a is a) incorrec b) correc
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More information2.1: What is physics? Ch02: Motion along a straight line. 2.2: Motion. 2.3: Position, Displacement, Distance
Ch: Moion along a sraigh line Moion Posiion and Displacemen Average Velociy and Average Speed Insananeous Velociy and Speed Acceleraion Consan Acceleraion: A Special Case Anoher Look a Consan Acceleraion
More informationODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004
ODEs II, Lecure : Homogeneous Linear Sysems - I Mike Raugh March 8, 4 Inroducion. In he firs lecure we discussed a sysem of linear ODEs for modeling he excreion of lead from he human body, saw how o ransform
More informationModule 3: The Damped Oscillator-II Lecture 3: The Damped Oscillator-II
Module 3: The Damped Oscillaor-II Lecure 3: The Damped Oscillaor-II 3. Over-damped Oscillaions. This refers o he siuaion where β > ω (3.) The wo roos are and α = β + α 2 = β β 2 ω 2 = (3.2) β 2 ω 2 = 2
More information( ) ( ) if t = t. It must satisfy the identity. So, bulkiness of the unit impulse (hyper)function is equal to 1. The defining characteristic is
UNIT IMPULSE RESPONSE, UNIT STEP RESPONSE, STABILITY. Uni impulse funcion (Dirac dela funcion, dela funcion) rigorously defined is no sricly a funcion, bu disribuion (or measure), precise reamen requires
More informationChapter 3 Boundary Value Problem
Chaper 3 Boundary Value Problem A boundary value problem (BVP) is a problem, ypically an ODE or a PDE, which has values assigned on he physical boundary of he domain in which he problem is specified. Le
More informationPOSITIVE SOLUTIONS OF NEUTRAL DELAY DIFFERENTIAL EQUATION
Novi Sad J. Mah. Vol. 32, No. 2, 2002, 95-108 95 POSITIVE SOLUTIONS OF NEUTRAL DELAY DIFFERENTIAL EQUATION Hajnalka Péics 1, János Karsai 2 Absrac. We consider he scalar nonauonomous neural delay differenial
More informationMA 214 Calculus IV (Spring 2016) Section 2. Homework Assignment 1 Solutions
MA 14 Calculus IV (Spring 016) Secion Homework Assignmen 1 Soluions 1 Boyce and DiPrima, p 40, Problem 10 (c) Soluion: In sandard form he given firs-order linear ODE is: An inegraing facor is given by
More informationTHE WAVE EQUATION. part hand-in for week 9 b. Any dilation v(x, t) = u(λx, λt) of u(x, t) is also a solution (where λ is constant).
THE WAVE EQUATION 43. (S) Le u(x, ) be a soluion of he wave equaion u u xx = 0. Show ha Q43(a) (c) is a. Any ranslaion v(x, ) = u(x + x 0, + 0 ) of u(x, ) is also a soluion (where x 0, 0 are consans).
More information2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes
Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion
More informationMath Final Exam Solutions
Mah 246 - Final Exam Soluions Friday, July h, 204 () Find explici soluions and give he inerval of definiion o he following iniial value problems (a) ( + 2 )y + 2y = e, y(0) = 0 Soluion: In normal form,
More informationCHAPTER 12 DIRECT CURRENT CIRCUITS
CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As
More informationInventory Analysis and Management. Multi-Period Stochastic Models: Optimality of (s, S) Policy for K-Convex Objective Functions
Muli-Period Sochasic Models: Opimali of (s, S) Polic for -Convex Objecive Funcions Consider a seing similar o he N-sage newsvendor problem excep ha now here is a fixed re-ordering cos (> 0) for each (re-)order.
More information1 Subdivide the optimization horizon [t 0,t f ] into n s 1 control stages,
Opimal Conrol Formulaion Opimal Conrol Lecures 19-2: Direc Soluion Mehods Benoî Chachua Deparmen of Chemical Engineering Spring 29 We are concerned wih numerical soluion procedures for
More informationAP CALCULUS AB 2003 SCORING GUIDELINES (Form B)
SCING GUIDELINES (Form B) Quesion 4 A paricle moves along he x-axis wih velociy a ime given by v( ) = 1 + e1. (a) Find he acceleraion of he paricle a ime =. (b) Is he speed of he paricle increasing a ime
More informationarxiv: v1 [math.fa] 9 Dec 2018
AN INVERSE FUNCTION THEOREM CONVERSE arxiv:1812.03561v1 [mah.fa] 9 Dec 2018 JIMMIE LAWSON Absrac. We esablish he following converse of he well-known inverse funcion heorem. Le g : U V and f : V U be inverse
More informationME 391 Mechanical Engineering Analysis
Fall 04 ME 39 Mechanical Engineering Analsis Eam # Soluions Direcions: Open noes (including course web posings). No books, compuers, or phones. An calculaor is fair game. Problem Deermine he posiion of
More information5. Stochastic processes (1)
Lec05.pp S-38.45 - Inroducion o Teleraffic Theory Spring 2005 Conens Basic conceps Poisson process 2 Sochasic processes () Consider some quaniy in a eleraffic (or any) sysem I ypically evolves in ime randomly
More informationRepresenting a Signal. Continuous-Time Fourier Methods. Linearity and Superposition. Real and Complex Sinusoids. Jean Baptiste Joseph Fourier
Represening a Signal Coninuous-ime ourier Mehods he convoluion mehod for finding he response of a sysem o an exciaion aes advanage of he lineariy and imeinvariance of he sysem and represens he exciaion
More informationProjection-Based Optimal Mode Scheduling
Projecion-Based Opimal Mode Scheduling T. M. Caldwell and T. D. Murphey Absrac This paper develops an ieraive opimizaion echnique ha can be applied o mode scheduling. The algorihm provides boh a mode schedule
More informationDirect Current Circuits. February 19, 2014 Physics for Scientists & Engineers 2, Chapter 26 1
Direc Curren Circuis February 19, 2014 Physics for Scieniss & Engineers 2, Chaper 26 1 Ammeers and Volmeers! A device used o measure curren is called an ammeer! A device used o measure poenial difference
More informationQuestion 1: Question 2: Topology Exercise Sheet 3
Topology Exercise Shee 3 Prof. Dr. Alessandro Siso Due o 14 March Quesions 1 and 6 are more concepual and should have prioriy. Quesions 4 and 5 admi a relaively shor soluion. Quesion 7 is harder, and you
More informationKEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
1 KEY Mah 4 Miderm I Fall 8 secions 1 and Insrucor: Sco Glasgow Please do NOT wrie on his eam. No credi will be given for such work. Raher wrie in a blue book, or on our own paper, preferabl engineering
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationu(x) = e x 2 y + 2 ) Integrate and solve for x (1 + x)y + y = cos x Answer: Divide both sides by 1 + x and solve for y. y = x y + cos x
. 1 Mah 211 Homework #3 February 2, 2001 2.4.3. y + (2/x)y = (cos x)/x 2 Answer: Compare y + (2/x) y = (cos x)/x 2 wih y = a(x)x + f(x)and noe ha a(x) = 2/x. Consequenly, an inegraing facor is found wih
More informationi L = VT L (16.34) 918a i D v OUT i L v C V - S 1 FIGURE A switched power supply circuit with diode and a switch.
16.4.3 A SWITHED POWER SUPPY USINGA DIODE In his example, we will analyze he behavior of he diodebased swiched power supply circui shown in Figure 16.15. Noice ha his circui is similar o ha in Figure 12.41,
More informationSliding Mode Extremum Seeking Control for Linear Quadratic Dynamic Game
Sliding Mode Exremum Seeking Conrol for Linear Quadraic Dynamic Game Yaodong Pan and Ümi Özgüner ITS Research Group, AIST Tsukuba Eas Namiki --, Tsukuba-shi,Ibaraki-ken 5-856, Japan e-mail: pan.yaodong@ais.go.jp
More informationDesigning Information Devices and Systems I Spring 2019 Lecture Notes Note 17
EES 16A Designing Informaion Devices and Sysems I Spring 019 Lecure Noes Noe 17 17.1 apaciive ouchscreen In he las noe, we saw ha a capacior consiss of wo pieces on conducive maerial separaed by a nonconducive
More informationMost Probable Phase Portraits of Stochastic Differential Equations and Its Numerical Simulation
Mos Probable Phase Porrais of Sochasic Differenial Equaions and Is Numerical Simulaion Bing Yang, Zhu Zeng and Ling Wang 3 School of Mahemaics and Saisics, Huazhong Universiy of Science and Technology,
More informationand v y . The changes occur, respectively, because of the acceleration components a x and a y
Week 3 Reciaion: Chaper3 : Problems: 1, 16, 9, 37, 41, 71. 1. A spacecraf is raveling wih a veloci of v0 = 5480 m/s along he + direcion. Two engines are urned on for a ime of 84 s. One engine gives he
More information( ) a system of differential equations with continuous parametrization ( T = R + These look like, respectively:
XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of
More informationCHE302 LECTURE IV MATHEMATICAL MODELING OF CHEMICAL PROCESS. Professor Dae Ryook Yang
CHE302 LECTURE IV MATHEMATICAL MODELING OF CHEMICAL PROCESS Professor Dae Ryook Yang Fall 200 Dep. of Chemical and Biological Engineering Korea Universiy CHE302 Process Dynamics and Conrol Korea Universiy
More information6.003 Homework 1. Problems. Due at the beginning of recitation on Wednesday, February 10, 2010.
6.003 Homework Due a he beginning of reciaion on Wednesday, February 0, 200. Problems. Independen and Dependen Variables Assume ha he heigh of a waer wave is given by g(x v) where x is disance, v is velociy,
More informationContinuous Time. Time-Domain System Analysis. Impulse Response. Impulse Response. Impulse Response. Impulse Response. ( t) + b 0.
Time-Domain Sysem Analysis Coninuous Time. J. Robers - All Righs Reserved. Edied by Dr. Rober Akl 1. J. Robers - All Righs Reserved. Edied by Dr. Rober Akl 2 Le a sysem be described by a 2 y ( ) + a 1
More informationNotes on Kalman Filtering
Noes on Kalman Filering Brian Borchers and Rick Aser November 7, Inroducion Daa Assimilaion is he problem of merging model predicions wih acual measuremens of a sysem o produce an opimal esimae of he curren
More informationUtility maximization in incomplete markets
Uiliy maximizaion in incomplee markes Marcel Ladkau 27.1.29 Conens 1 Inroducion and general seings 2 1.1 Marke model....................................... 2 1.2 Trading sraegy.....................................
More informationAnn. Funct. Anal. 2 (2011), no. 2, A nnals of F unctional A nalysis ISSN: (electronic) URL:
Ann. Func. Anal. 2 2011, no. 2, 34 41 A nnals of F uncional A nalysis ISSN: 2008-8752 elecronic URL: www.emis.de/journals/afa/ CLASSIFICAION OF POSIIVE SOLUIONS OF NONLINEAR SYSEMS OF VOLERRA INEGRAL EQUAIONS
More informationMEI Mechanics 1 General motion. Section 1: Using calculus
Soluions o Exercise MEI Mechanics General moion Secion : Using calculus. s 4 v a 6 4 4 When =, v 4 a 6 4 6. (i) When = 0, s = -, so he iniial displacemen = - m. s v 4 When = 0, v = so he iniial velociy
More information