Linear System Theory (Fall 2011): Homework 1. Solutions
|
|
- Blake Bryant
- 5 years ago
- Views:
Transcription
1 Linear System Theory (Fall 20): Homework Soltions De Sep. 29, 20 Exercise (C.T. Chen: Ex.3-8). Consider a linear system with inpt and otpt y. Three experiments are performed on this system sing the inpts (t), 2 (t) and 3 (t) for t In each case, the initial state at t = 0, x(0) is the same. The corresponding observed otpts are y (t), y 2 (t) and y 3 (t). Which of the following three predictions are tre if x(0) 0? a. If 3 = 2, then y 3 = y y 2. b. If 3 = 2 ( 2 ), then y 3 = 2 (y y 2 ). c. If 3 = 2, then y 3 = y y 2. Which of the above are tre if x(0) = 0? For each answer no, give a simple conterexample (nmerical or on paper). Soltion. The best soltion was given by Tsai-lin. If H is a linear system, it has state eqations sch as: ẋ = Ax B y = Cx The Laplace transform of the above by taking into accont the initial condition is (check yo basic control system corse notes): y(s) = C(sI A) B(s) C(sI A) x(0) Then, exploiting linearity, the following can be dedced:
2 Case. x(0) 0: a. Qestion: H( 2 ) =? H H 2? y = C(sI A) B C(sI A) x(0) y 2 = C(sI A) B 2 C(sI A) x(0) y 3 = C(sI A) B( 2 ) C(sI A) x(0) Hence, y 3 y y 3 if x(0) Answer: no. b. Qestion: H ( 2 2 ) =? H 2 H 2 2? y 3 = C(sI A) B 2 ( 2 ) C(sI A) x(0) = 2 C(sI A) B( 2 ) C(sI A) x(0) 2 y 2 y 2 = 2 [C(sI A) B C(sI A) x(0) Hence, y 3 = 2 (y y 2 ). Answer: yes! 2 [C(sI A) B 2 C(sI A) x(0) = 2 C(sI A) B[ C(sI A) x(0) c. Qestion: H( 2 )? = H H 2? The answer is the same as a. simply by replacing 2 with 2, i.e. no. Case 2. x(0) = 0: from the above, it is easy to see that the answer is yes for all a., b. and c. This is why the transfer-fnction representation mst assme that the system is initially relaxed.
3 Exercise 2 (C.T. Chen: Ex.3-4). The inpt and the otpt y of a system is described by: ÿ 2ẏ 3y = 2. What is the transfer fnction of the system? Find a state-space representation for the system. Soltion 2. Transfer fnction: y(s) (s) = 2s s 2s 3 Canonical controllable form: Canonical observable form: [ [ [ẋ 0 x = ẋ y = [ 2 [ x [ẋ = ẋ 2 [ y = [ 0 [ x [ x [ 0 [ 2 and there are other state-space representations.
4 Exercise 3 (C.T. Chen: Ex.3-23). Find the transfer fnction and a state-space representation of the network in the following figre. Simlate in MATLAB/Octave/Scilab the state-space representation for a constant A inpt. What happens to otpt and the internal state? Do yo think the transfer fnction is a good description of the system? V C2 i C2 F Z i L spply crrent i R Ω Ω F y = V C _ H V L _ Soltion 3. The circit eqations are (notice there are 3 differential eqations, so we expect a 3rd-order system): Choose the state variables Then i C i C2 =, i C2 i R =, i C = dv C dt, i C 2 = dv C 2 dt, V L i L = V C, V C2 = i R. x = V C = y, = V C2, x 3 = i L. ẋ = x 3, ẋ 2 =, ẋ 3 = x x 3. Ths, a state-space representation for the circit is: 0 0 ẋ = 0 0 x 0 0 y = [ 0 0 x The transfer fnction can be obtained from the state-space representation sing the formla H(s) = C(sI A) B, or by noticing that the relation between the inpt and
5 the otpt y is only dependent on the impedance Z of the parallel element across the otpt: Z = s //( s) = s s 2 s A simlation of the circit s response to a A step inpt at t =, with zero initial conditions at t = 0, is shown below. It can be seen that while the otpt voltage y = V C and the branch crrent i L are well-behaved, the crrent i C2 diverges. = V C2 x = V C = y x 3 = i L This means that by observing solely the otpt voltage, it is not possible to detect an exploding crrent (and voltage) across the inpt-side parallel element. Hence, the inptside element is not observable from the otpt. Electrically, this can be easily nderstood by noticing that the circit s transfer fnction is exactly the same as the one below, no matter what the inpt-side element is: spply crrent _ Ω H F y _
6 Exercise 4 (C.T. Chen: Ex.3-3 (modified)). Consider the below feedback system with sbsystems S and S 2 described by: (Sys ) [ẋ = ẋ 2 [ 2 0 y = [ 0 [ x [ [ [ x 4, 2 2 [ [ ; 2 and (Sys 2 ) [ [ẋ3 2 = 3, ẋ 4 [ [ [ y2 2 0 x3 =. y 3 x 4 [ r r 2 [ 2 _ Sys y [ y2 Sys 2 3 y 3 a. Withot sing any compter programme, write down the state-space representation of the feedback system, with the states (x,, x 3, x 4 ), the inpt (r, r 2 ) and the otpt y. b. Next, se MATLAB/Octave/Scilab to obtain the transfer fnctions of the sbsystems Sys and Sys 2, respectively. Then, calclate the transfer fnction of the feedback system (type help feedback at the command prompt). Print (or copy and paste) all the steps as yor answer. c. Explain why the state-space representation in qestion a. is of dimension 4, bt the transfer fnction in qestion b. is of dimension 2. What is missing? Soltion 4. a. A straight-forward way to obtain a state-space representation for the feedback system is to recycle all the state variables (x,, x 3, x 4 ) of the sbsystems, and se the formla given in class (or work it ot manally again) [ [ [ [Ẋ A B = C 2 X B R, () Ẋ 2 B 2 C A 2 B 2 D C 2 X 2 B 2 D Y = [ [ X C D C 2 D R, (2) X 2
7 specializing it to This yields:: ẋ ẋ 2 ẋ 3 ẋ 4 A 2 = [ = y = [ 0 x x 3 x 4 x x 3 x [ r r 2 b. The following are Scilab command history and reslts; similar reslts can be obtained in MATLAB: -->A=[-2 ; 0 -; B=[4 ; - 2; C=[0 ; D=[ -; -->A2=[0 0; 0 0; B2=[2;; C2=[2 0; -; D2=[0;0; ->sys=syslin( c,a,b,c,d) sys = sys() (state-space system:)!lss A B C D X0 dt! sys(2) = A matrix = sys(3) = B matrix = sys(4) = C matrix =. sys(5) = D matrix =. -. sys(6) = X0 (initial state) = sys(7) = Time domain =
8 c -->sys2=syslin( c,a2,b2,c2,d2) sys2 = sys2() (state-space system:)!lss A B C D X0 dt! sys2(2) = A matrix = 2.. sys2(3) = B matrix = sys2(4) = C matrix = sys2(5) = D matrix = sys2(6) = X0 (initial state) = c sys2(7) = Time domain = -->tf=ss2tf(sys) tf = s - s s s -->tf2=ss2tf(sys2) tf2 = 4 - s - s
9 It can be observed that we have transfer-fnctions matrices instead of scalar transfer fnctions, since Sys is 2-inpt--otpt, and Sys 2 is -inpt-2-otpt. The feedback system is therefor 2-inpt--otpt: ->tf/.tf2 ans = 2 2 s s - s s s 4s s The fact that the reslting transfer matrices of the feedback system are only second order, whereas the total nmber of states is 4, is de to the fact that both sbsystems have transfer matrices that are first order. This is becase both Sys and Sys 2 have redndant states. First, notice that y only depends on and (, 2 ), bt not on x, so the same inpt-otpt relation can be represented by the following first-order state-space eqations: ẋ 2 = [ 2 [, 2 (Sys a ) y = [ [. 2 It can be easily verified that Sys a has the same transfer matrix as Sys : -->sysa=syslin( c,[-,[- 2,[,[ -); -->tfa=ss2tf(sysa) tfa = s - s s s Next, Sys 2 can be redced to 2nd order by noticing that x 3 and x 4 are linearly dependent if they both have zero initial conditions (remember, we want to find the transfer matrix, which has to assme a relaxed system). In fact, if x 3 (0) = x 4 (0) = 0,
10 then x 3 (t) = x 4 (t) = t 0 t o 2(t)dt (t)dt x 3 (t) = 2x 4 (t), y 2 = 2x 3 = 4x 4, y 3 = x 3 x 4 = x 4. Ths, Sys 2 can be redced to the following first-order system (provided we have zero initial conditions!) ẋ 4 = 3, [ [ (Sys 2a ) y2 4 = x 4. It can be easily verified that Sys 2a has the same transfer matrix as Sys 2 : y 3 -->sys2a=syslin( c,[0,[,[4;,[0;0); -->tf2a=ss2tf(sys2a) tf2a = 4 - s - s In later lectres, we will see that Sys is not observable, and Sys 2 is not controllable.
Chapter 3 MATHEMATICAL MODELING OF DYNAMIC SYSTEMS
Chapter 3 MATHEMATICAL MODELING OF DYNAMIC SYSTEMS 3. System Modeling Mathematical Modeling In designing control systems we mst be able to model engineered system dynamics. The model of a dynamic system
More information1. State-Space Linear Systems 2. Block Diagrams 3. Exercises
LECTURE 1 State-Space Linear Sstems This lectre introdces state-space linear sstems, which are the main focs of this book. Contents 1. State-Space Linear Sstems 2. Block Diagrams 3. Exercises 1.1 State-Space
More informationVNVe 2017/ Final project
VNVe 2017/2018 - Final project Athor s name Febrary 21, 2018 Solve all examples and send yor final soltion (pdf file) and all sorce codes (LaTex, MATLAB,, ++, etc.) to e-mail address satek@fit.vtbr.cz
More informationBLOOM S TAXONOMY. Following Bloom s Taxonomy to Assess Students
BLOOM S TAXONOMY Topic Following Bloom s Taonomy to Assess Stdents Smmary A handot for stdents to eplain Bloom s taonomy that is sed for item writing and test constrction to test stdents to see if they
More informationFEA Solution Procedure
EA Soltion Procedre (demonstrated with a -D bar element problem) EA Procedre for Static Analysis. Prepare the E model a. discretize (mesh) the strctre b. prescribe loads c. prescribe spports. Perform calclations
More informationControl Systems
6.5 Control Systems Last Time: Introdction Motivation Corse Overview Project Math. Descriptions of Systems ~ Review Classification of Systems Linear Systems LTI Systems The notion of state and state variables
More informationSimulation investigation of the Z-source NPC inverter
octoral school of energy- and geo-technology Janary 5 20, 2007. Kressaare, Estonia Simlation investigation of the Z-sorce NPC inverter Ryszard Strzelecki, Natalia Strzelecka Gdynia Maritime University,
More informationPartial Differential Equations with Applications
Universit of Leeds MATH 33 Partial Differential Eqations with Applications Eamples to spplement Chapter on First Order PDEs Eample (Simple linear eqation, k + = 0, (, 0) = ϕ(), k a constant.) The characteristic
More informationSTEP Support Programme. STEP III Hyperbolic Functions: Solutions
STEP Spport Programme STEP III Hyperbolic Fnctions: Soltions Start by sing the sbstittion t cosh x. This gives: sinh x cosh a cosh x cosh a sinh x t sinh x dt t dt t + ln t ln t + ln cosh a ln ln cosh
More informationState Space Models Basic Concepts
Chapter 2 State Space Models Basic Concepts Related reading in Bay: Chapter Section Sbsection 1 (Models of Linear Systems) 1.1 1.1.1 1.1.2 1.1.3 1.1.5 1.2 1.2.1 1.2.2 1.3 In this Chapter we provide some
More informationLecture 3. (2) Last time: 3D space. The dot product. Dan Nichols January 30, 2018
Lectre 3 The dot prodct Dan Nichols nichols@math.mass.ed MATH 33, Spring 018 Uniersity of Massachsetts Janary 30, 018 () Last time: 3D space Right-hand rle, the three coordinate planes 3D coordinate system:
More informationUNCERTAINTY FOCUSED STRENGTH ANALYSIS MODEL
8th International DAAAM Baltic Conference "INDUSTRIAL ENGINEERING - 19-1 April 01, Tallinn, Estonia UNCERTAINTY FOCUSED STRENGTH ANALYSIS MODEL Põdra, P. & Laaneots, R. Abstract: Strength analysis is a
More informationLinear and Nonlinear Model Predictive Control of Quadruple Tank Process
Linear and Nonlinear Model Predictive Control of Qadrple Tank Process P.Srinivasarao Research scholar Dr.M.G.R.University Chennai, India P.Sbbaiah, PhD. Prof of Dhanalaxmi college of Engineering Thambaram
More information1. Tractable and Intractable Computational Problems So far in the course we have seen many problems that have polynomial-time solutions; that is, on
. Tractable and Intractable Comptational Problems So far in the corse we have seen many problems that have polynomial-time soltions; that is, on a problem instance of size n, the rnning time T (n) = O(n
More informationUncertainties of measurement
Uncertainties of measrement Laboratory tas A temperatre sensor is connected as a voltage divider according to the schematic diagram on Fig.. The temperatre sensor is a thermistor type B5764K [] with nominal
More informationLecture Notes: Finite Element Analysis, J.E. Akin, Rice University
9. TRUSS ANALYSIS... 1 9.1 PLANAR TRUSS... 1 9. SPACE TRUSS... 11 9.3 SUMMARY... 1 9.4 EXERCISES... 15 9. Trss analysis 9.1 Planar trss: The differential eqation for the eqilibrim of an elastic bar (above)
More informationThe Linear Quadratic Regulator
10 The Linear Qadratic Reglator 10.1 Problem formlation This chapter concerns optimal control of dynamical systems. Most of this development concerns linear models with a particlarly simple notion of optimality.
More informationAssignment Fall 2014
Assignment 5.086 Fall 04 De: Wednesday, 0 December at 5 PM. Upload yor soltion to corse website as a zip file YOURNAME_ASSIGNMENT_5 which incldes the script for each qestion as well as all Matlab fnctions
More informationIJSER. =η (3) = 1 INTRODUCTION DESCRIPTION OF THE DRIVE
International Jornal of Scientific & Engineering Research, Volme 5, Isse 4, April-014 8 Low Cost Speed Sensor less PWM Inverter Fed Intion Motor Drive C.Saravanan 1, Dr.M.A.Panneerselvam Sr.Assistant Professor
More informationReflections on a mismatched transmission line Reflections.doc (4/1/00) Introduction The transmission line equations are given by
Reflections on a mismatched transmission line Reflections.doc (4/1/00) Introdction The transmission line eqations are given by, I z, t V z t l z t I z, t V z, t c z t (1) (2) Where, c is the per-nit-length
More informationSection 7.4: Integration of Rational Functions by Partial Fractions
Section 7.4: Integration of Rational Fnctions by Partial Fractions This is abot as complicated as it gets. The Method of Partial Fractions Ecept for a few very special cases, crrently we have no way to
More informationReview of course Nonlinear control Lecture 5. Lyapunov based design Torkel Glad Lyapunov design Control Lyapunov functions Control Lyapunov function
Review of corse Nonlinear control Lectre 5 Lyapnov based design Reglerteknik, ISY, Linköpings Universitet Geometric control theory inpt-otpt linearization controller canonical form observer canonical form
More informationSystem identification of buildings equipped with closed-loop control devices
System identification of bildings eqipped with closed-loop control devices Akira Mita a, Masako Kamibayashi b a Keio University, 3-14-1 Hiyoshi, Kohok-k, Yokohama 223-8522, Japan b East Japan Railway Company
More informationClassify by number of ports and examine the possible structures that result. Using only one-port elements, no more than two elements can be assembled.
Jnction elements in network models. Classify by nmber of ports and examine the possible strctres that reslt. Using only one-port elements, no more than two elements can be assembled. Combining two two-ports
More informationImage and Multidimensional Signal Processing
Image and Mltidimensional Signal Processing Professor William Hoff Dept of Electrical Engineering &Compter Science http://inside.mines.ed/~whoff/ Forier Transform Part : D discrete transforms 2 Overview
More informationFRÉCHET KERNELS AND THE ADJOINT METHOD
PART II FRÉCHET KERNES AND THE ADJOINT METHOD 1. Setp of the tomographic problem: Why gradients? 2. The adjoint method 3. Practical 4. Special topics (sorce imaging and time reversal) Setp of the tomographic
More informationMean Value Formulae for Laplace and Heat Equation
Mean Vale Formlae for Laplace and Heat Eqation Abhinav Parihar December 7, 03 Abstract Here I discss a method to constrct the mean vale theorem for the heat eqation. To constrct sch a formla ab initio,
More informationMethods of Design-Oriented Analysis The GFT: A Final Solution for Feedback Systems
http://www.ardem.com/free_downloads.asp v.1, 5/11/05 Methods of Design-Oriented Analysis The GFT: A Final Soltion for Feedback Systems R. David Middlebrook Are yo an analog or mixed signal design engineer
More informationControl Systems Design
ELEC4410 Control Systems Design Lectre 16: Controllability and Observability Canonical Decompositions Jlio H. Braslavsky jlio@ee.newcastle.ed.a School of Electrical Engineering and Compter Science Lectre
More informationINPUT-OUTPUT APPROACH NUMERICAL EXAMPLES
INPUT-OUTPUT APPROACH NUMERICAL EXAMPLES EXERCISE s consider the linear dnamical sstem of order 2 with transfer fnction with Determine the gain 2 (H) of the inpt-otpt operator H associated with this sstem.
More information3.3 Operations With Vectors, Linear Combinations
Operations With Vectors, Linear Combinations Performance Criteria: (d) Mltiply ectors by scalars and add ectors, algebraically Find linear combinations of ectors algebraically (e) Illstrate the parallelogram
More informationFRTN10 Exercise 12. Synthesis by Convex Optimization
FRTN Exercise 2. 2. We want to design a controller C for the stable SISO process P as shown in Figre 2. sing the Yola parametrization and convex optimization. To do this, the control loop mst first be
More informationLecture 6 : Linear Fractional Transformations (LFTs) Dr.-Ing. Sudchai Boonto
Lectre 6 : (LFTs) Dr-Ing Sdchai Boonto Department of Control System and Instrmentation Engineering King Mongkts Unniversity of Technology Thonbri Thailand Feedback Strctre d i d r e y z K G g n The standard
More information1 The space of linear transformations from R n to R m :
Math 540 Spring 20 Notes #4 Higher deriaties, Taylor s theorem The space of linear transformations from R n to R m We hae discssed linear transformations mapping R n to R m We can add sch linear transformations
More informationOptimization via the Hamilton-Jacobi-Bellman Method: Theory and Applications
Optimization via the Hamilton-Jacobi-Bellman Method: Theory and Applications Navin Khaneja lectre notes taken by Christiane Koch Jne 24, 29 1 Variation yields a classical Hamiltonian system Sppose that
More informationLecture Notes On THEORY OF COMPUTATION MODULE - 2 UNIT - 2
BIJU PATNAIK UNIVERSITY OF TECHNOLOGY, ODISHA Lectre Notes On THEORY OF COMPUTATION MODULE - 2 UNIT - 2 Prepared by, Dr. Sbhend Kmar Rath, BPUT, Odisha. Tring Machine- Miscellany UNIT 2 TURING MACHINE
More informationOn the circuit complexity of the standard and the Karatsuba methods of multiplying integers
On the circit complexity of the standard and the Karatsba methods of mltiplying integers arxiv:1602.02362v1 [cs.ds] 7 Feb 2016 Igor S. Sergeev The goal of the present paper is to obtain accrate estimates
More informationSafe Manual Control of the Furuta Pendulum
Safe Manal Control of the Frta Pendlm Johan Åkesson, Karl Johan Åström Department of Atomatic Control, Lnd Institte of Technology (LTH) Box 8, Lnd, Sweden PSfrag {jakesson,kja}@control.lth.se replacements
More informationIntegration of Basic Functions. Session 7 : 9/23 1
Integration o Basic Fnctions Session 7 : 9/3 Antiderivation Integration Deinition: Taking the antiderivative, or integral, o some nction F(), reslts in the nction () i ()F() Pt simply: i yo take the integral
More informationSecond-Order Wave Equation
Second-Order Wave Eqation A. Salih Department of Aerospace Engineering Indian Institte of Space Science and Technology, Thirvananthapram 3 December 016 1 Introdction The classical wave eqation is a second-order
More informationWe automate the bivariate change-of-variables technique for bivariate continuous random variables with
INFORMS Jornal on Compting Vol. 4, No., Winter 0, pp. 9 ISSN 09-9856 (print) ISSN 56-558 (online) http://dx.doi.org/0.87/ijoc.046 0 INFORMS Atomating Biariate Transformations Jeff X. Yang, John H. Drew,
More informationMATH2715: Statistical Methods
MATH275: Statistical Methods Exercises VI (based on lectre, work week 7, hand in lectre Mon 4 Nov) ALL qestions cont towards the continos assessment for this modle. Q. The random variable X has a discrete
More informationReduction of over-determined systems of differential equations
Redction of oer-determined systems of differential eqations Maim Zaytse 1) 1, ) and Vyachesla Akkerman 1) Nclear Safety Institte, Rssian Academy of Sciences, Moscow, 115191 Rssia ) Department of Mechanical
More informationA New Method for Calculating of Electric Fields Around or Inside Any Arbitrary Shape Electrode Configuration
Proceedings of the 5th WSEAS Int. Conf. on Power Systems and Electromagnetic Compatibility, Corf, Greece, Agst 3-5, 005 (pp43-48) A New Method for Calclating of Electric Fields Arond or Inside Any Arbitrary
More informationAPPENDIX B MATRIX NOTATION. The Definition of Matrix Notation is the Definition of Matrix Multiplication B.1 INTRODUCTION
APPENDIX B MAIX NOAION he Deinition o Matrix Notation is the Deinition o Matrix Mltiplication B. INODUCION { XE "Matrix Mltiplication" }{ XE "Matrix Notation" }he se o matrix notations is not necessary
More informationIII. Demonstration of a seismometer response with amplitude and phase responses at:
GG5330, Spring semester 006 Assignment #1, Seismometry and Grond Motions De 30 Janary 006. 1. Calibration Of A Seismometer Using Java: A really nifty se of Java is now available for demonstrating the seismic
More informationCFD-Simulation thermoakustischer Resonanzeffekte zur Bestimmung der Flammentransferfunktion
CFD-Simlation thermoakstischer Resonanzeffekte zr Bestimmng der Flammentransferfnktion Ator: Dennis Paschke Technische Universität Berlin Institt für Strömngsmechanik nd Technische Akstik FG Experimentelle
More informationElements of Coordinate System Transformations
B Elements of Coordinate System Transformations Coordinate system transformation is a powerfl tool for solving many geometrical and kinematic problems that pertain to the design of gear ctting tools and
More informationIntroduction to Quantum Information Processing
Introdction to Qantm Information Processing Lectre 5 Richard Cleve Overview of Lectre 5 Review of some introdctory material: qantm states, operations, and simple qantm circits Commnication tasks: one qbit
More information2E1252 Control Theory and Practice
2E1252 Control Theory and Practice Lectre 11: Actator satration and anti wind-p Learning aims After this lectre, yo shold nderstand how satration can case controller states to wind p know how to modify
More informationVectors in Rn un. This definition of norm is an extension of the Pythagorean Theorem. Consider the vector u = (5, 8) in R 2
MATH 307 Vectors in Rn Dr. Neal, WKU Matrices of dimension 1 n can be thoght of as coordinates, or ectors, in n- dimensional space R n. We can perform special calclations on these ectors. In particlar,
More informationPhysicsAndMathsTutor.com
. Two smooth niform spheres S and T have eqal radii. The mass of S is 0. kg and the mass of T is 0.6 kg. The spheres are moving on a smooth horizontal plane and collide obliqely. Immediately before the
More informationStudy of the diffusion operator by the SPH method
IOSR Jornal of Mechanical and Civil Engineering (IOSR-JMCE) e-issn: 2278-684,p-ISSN: 2320-334X, Volme, Isse 5 Ver. I (Sep- Oct. 204), PP 96-0 Stdy of the diffsion operator by the SPH method Abdelabbar.Nait
More informationMaterial. Lecture 8 Backlash and Quantization. Linear and Angular Backlash. Example: Parallel Kinematic Robot. Backlash.
Lectre 8 Backlash and Qantization Material Toda s Goal: To know models and compensation methods for backlash Lectre slides Be able to analze the effect of qantization errors Note: We are sing analsis methods
More informationA Survey of the Implementation of Numerical Schemes for Linear Advection Equation
Advances in Pre Mathematics, 4, 4, 467-479 Pblished Online Agst 4 in SciRes. http://www.scirp.org/jornal/apm http://dx.doi.org/.436/apm.4.485 A Srvey of the Implementation of Nmerical Schemes for Linear
More informationMath 116 First Midterm October 14, 2009
Math 116 First Midterm October 14, 9 Name: EXAM SOLUTIONS Instrctor: Section: 1. Do not open this exam ntil yo are told to do so.. This exam has 1 pages inclding this cover. There are 9 problems. Note
More informationLecture 17 Errors in Matlab s Turbulence PSD and Shaping Filter Expressions
Lectre 7 Errors in Matlab s Trblence PSD and Shaping Filter Expressions b Peter J Sherman /7/7 [prepared for AERE 355 class] In this brief note we will show that the trblence power spectral densities (psds)
More informationGraphs and Networks Lecture 5. PageRank. Lecturer: Daniel A. Spielman September 20, 2007
Graphs and Networks Lectre 5 PageRank Lectrer: Daniel A. Spielman September 20, 2007 5.1 Intro to PageRank PageRank, the algorithm reportedly sed by Google, assigns a nmerical rank to eery web page. More
More informationSimplified Identification Scheme for Structures on a Flexible Base
Simplified Identification Scheme for Strctres on a Flexible Base L.M. Star California State University, Long Beach G. Mylonais University of Patras, Greece J.P. Stewart University of California, Los Angeles
More information5. The Bernoulli Equation
5. The Bernolli Eqation [This material relates predominantly to modles ELP034, ELP035] 5. Work and Energy 5. Bernolli s Eqation 5.3 An example of the se of Bernolli s eqation 5.4 Pressre head, velocity
More informationAMS 212B Perturbation Methods Lecture 05 Copyright by Hongyun Wang, UCSC
AMS B Pertrbation Methods Lectre 5 Copright b Hongn Wang, UCSC Recap: we discssed bondar laer of ODE Oter epansion Inner epansion Matching: ) Prandtl s matching ) Matching b an intermediate variable (Skip
More informationNonlinear predictive control of dynamic systems represented by Wiener Hammerstein models
Nonlinear Dn (26) 86:93 24 DOI.7/s7-6-2957- ORIGINAL PAPER Nonlinear predictive control of dnamic sstems represented b Wiener Hammerstein models Maciej Ławrńcz Received: 7 December 25 / Accepted: 2 Jl
More informationAffine Invariant Total Variation Models
Affine Invariant Total Variation Models Helen Balinsky, Alexander Balinsky Media Technologies aboratory HP aboratories Bristol HP-7-94 Jne 6, 7* Total Variation, affine restoration, Sobolev ineqality,
More informationA Note on Irreducible Polynomials and Identity Testing
A Note on Irrecible Polynomials an Ientity Testing Chanan Saha Department of Compter Science an Engineering Inian Institte of Technology Kanpr Abstract We show that, given a finite fiel F q an an integer
More informationState variable feedback
State variable feedbak We have previosly disssed systems desribed by the linear state-spae eqations Ax B y Cx n m with xt () R the internal state, t () R the ontrol inpt, and yt () R the measred otpt.
More informationSystems in equilibrium. Graph models and Kirchoff s laws.
Systems in eqilibrim Graph models and Kirchoff s laws nna-karin ornberg Mathematical Models, nalysis and Simlation Fall semester, 2011 system in eqilibrim [Material from Strang, sections 21 and 22] Consider
More informationPredictive Control- Exercise Session 4 Adaptive Control: Self Tuning Regulators and Model Reference Adaptive Systems
Predictive Control- Exercise Session 4 Adaptive Control: Self Tning Reglators and Model Reference Adaptive Systems 1. Indirect Self Tning Reglator: Consider the system where G(s)=G 1 (s)g 2 (s) G 1 (s)=
More informationThe Cross Product of Two Vectors in Space DEFINITION. Cross Product. u * v = s ƒ u ƒƒv ƒ sin ud n
12.4 The Cross Prodct 873 12.4 The Cross Prodct In stdying lines in the plane, when we needed to describe how a line was tilting, we sed the notions of slope and angle of inclination. In space, we want
More informationVisualisations of Gussian and Mean Curvatures by Using Mathematica and webmathematica
Visalisations of Gssian and Mean Cratres by Using Mathematica and webmathematica Vladimir Benić, B. sc., (benic@grad.hr), Sonja Gorjanc, Ph. D., (sgorjanc@grad.hr) Faclty of Ciil Engineering, Kačićea 6,
More informationsin u 5 opp } cos u 5 adj } hyp opposite csc u 5 hyp } sec u 5 hyp } opp Using Inverse Trigonometric Functions
13 Big Idea 1 CHAPTER SUMMARY BIG IDEAS Using Trigonometric Fnctions Algebra classzone.com Electronic Fnction Library For Yor Notebook hypotense acent osite sine cosine tangent sin 5 hyp cos 5 hyp tan
More informationMath 263 Assignment #3 Solutions. 1. A function z = f(x, y) is called harmonic if it satisfies Laplace s equation:
Math 263 Assignment #3 Soltions 1. A fnction z f(x, ) is called harmonic if it satisfies Laplace s eqation: 2 + 2 z 2 0 Determine whether or not the following are harmonic. (a) z x 2 + 2. We se the one-variable
More informationAdaptive partial state feedback control of the DC-to-DC Ćuk converter
5 American Control Conference Jne 8-, 5. Portlan, OR, USA FrC7.4 Aaptive partial state feeback control of the DC-to-DC Ćk converter Hgo Rorígez, Romeo Ortega an Alessanro Astolfi Abstract The problem of
More informationPassivity-based Control of NPC Three-level Inverter
International Form on Mechanical, Control and Atomation (IFMCA 06) Passivity-based Control of NPC hree-level Inverter Shilan Shen, a, Qi Hong,b and Xiaofan Zh,c Gangzho Power Spply, Gangzho, Gangdong province,
More informationMaterial Transport with Air Jet
Material Transport with Air Jet Dr. István Patkó Bdapest Tech Doberdó út 6, H-1034 Bdapest, Hngary patko@bmf.h Abstract: In the field of indstry, there are only a very few examples of material transport
More informationConceptual Questions. Problems. 852 CHAPTER 29 Magnetic Fields
852 CHAPTER 29 Magnetic Fields magnitde crrent, and the niform magnetic field points in the positive direction. Rank the loops by the magnitde of the torqe eerted on them by the field from largest to smallest.
More informationThe Oscillatory Stable Regime of Nonlinear Systems, with two time constants
6th WSES International Conference on CIRCUITS SYSTEMS ELECTRONICSCONTROL & SIGNL PROCESSING Cairo Egpt Dec 9-3 7 5 The Oscillator Stable Regime of Nonlinear Sstems with two time constants NUŢU VSILE *
More informationFormal Methods for Deriving Element Equations
Formal Methods for Deriving Element Eqations And the importance of Shape Fnctions Formal Methods In previos lectres we obtained a bar element s stiffness eqations sing the Direct Method to obtain eact
More informationEssentials of optimal control theory in ECON 4140
Essentials of optimal control theory in ECON 4140 Things yo need to know (and a detail yo need not care abot). A few words abot dynamic optimization in general. Dynamic optimization can be thoght of as
More informationDepartment of Industrial Engineering Statistical Quality Control presented by Dr. Eng. Abed Schokry
Department of Indstrial Engineering Statistical Qality Control presented by Dr. Eng. Abed Schokry Department of Indstrial Engineering Statistical Qality Control C and U Chart presented by Dr. Eng. Abed
More information3. Several Random Variables
. Several Random Variables. To Random Variables. Conditional Probabilit--Revisited. Statistical Independence.4 Correlation beteen Random Variables Standardied (or ero mean normalied) random variables.5
More information4 Exact laminar boundary layer solutions
4 Eact laminar bondary layer soltions 4.1 Bondary layer on a flat plate (Blasis 1908 In Sec. 3, we derived the bondary layer eqations for 2D incompressible flow of constant viscosity past a weakly crved
More informationA Single Species in One Spatial Dimension
Lectre 6 A Single Species in One Spatial Dimension Reading: Material similar to that in this section of the corse appears in Sections 1. and 13.5 of James D. Mrray (), Mathematical Biology I: An introction,
More informationNEURAL CONTROL OF NONLINEAR SYSTEMS: A REFERENCE GOVERNOR APPROACH
NEURAL CONTROL OF NONLINEAR SYSTEMS: A REFERENCE GOVERNOR APPROACH L. Schnitman Institto Tecnológico e Aeronática.8-900 - S.J. os Campos, SP Brazil leizer@ele.ita.cta.br J.A.M. Felippe e Soza Universiae
More informationModelling by Differential Equations from Properties of Phenomenon to its Investigation
Modelling by Differential Eqations from Properties of Phenomenon to its Investigation V. Kleiza and O. Prvinis Kanas University of Technology, Lithania Abstract The Panevezys camps of Kanas University
More informationSetting The K Value And Polarization Mode Of The Delta Undulator
LCLS-TN-4- Setting The Vale And Polarization Mode Of The Delta Undlator Zachary Wolf, Heinz-Dieter Nhn SLAC September 4, 04 Abstract This note provides the details for setting the longitdinal positions
More informationECON3120/4120 Mathematics 2, spring 2009
University of Oslo Department of Economics Arne Strøm ECON3/4 Mathematics, spring 9 Problem soltions for Seminar 4, 6 Febrary 9 (For practical reasons some of the soltions may inclde problem parts that
More informationModel Predictive Control Lecture VIa: Impulse Response Models
Moel Preictive Control Lectre VIa: Implse Response Moels Niet S. Kaisare Department of Chemical Engineering Inian Institte of Technolog Maras Ingreients of Moel Preictive Control Dnamic Moel Ftre preictions
More informationLecture: Corporate Income Tax
Lectre: Corporate Income Tax Ltz Krschwitz & Andreas Löffler Disconted Cash Flow, Section 2.1, Otline 2.1 Unlevered firms Similar companies Notation 2.1.1 Valation eqation 2.1.2 Weak atoregressive cash
More informationDecision making is the process of selecting
Jornal of Advances in Compter Engineering and Technology, (4) 06 A New Mlti-Criteria Decision Making Based on Fzzy- Topsis Theory Leila Yahyaie Dizaji, Sohrab khanmohammadi Received (05-09-) Accepted (06--)
More informationPulses on a Struck String
8.03 at ESG Spplemental Notes Plses on a Strck String These notes investigate specific eamples of transverse motion on a stretched string in cases where the string is at some time ndisplaced, bt with a
More informationOPTIMUM EXPRESSION FOR COMPUTATION OF THE GRAVITY FIELD OF A POLYHEDRAL BODY WITH LINEARLY INCREASING DENSITY 1
OPTIMUM EXPRESSION FOR COMPUTATION OF THE GRAVITY FIEL OF A POLYHERAL BOY WITH LINEARLY INCREASING ENSITY 1 V. POHÁNKA2 Abstract The formla for the comptation of the gravity field of a polyhedral body
More informationTheoretical and Experimental Implementation of DC Motor Nonlinear Controllers
Theoretical and Experimental Implementation of DC Motor Nonlinear Controllers D.R. Espinoza-Trejo and D.U. Campos-Delgado Facltad de Ingeniería, CIEP, UASLP, espinoza trejo dr@aslp.mx Facltad de Ciencias,
More informationThe Real Stabilizability Radius of the Multi-Link Inverted Pendulum
Proceedings of the 26 American Control Conference Minneapolis, Minnesota, USA, Jne 14-16, 26 WeC123 The Real Stabilizability Radis of the Mlti-Link Inerted Pendlm Simon Lam and Edward J Daison Abstract
More informationEXERCISES WAVE EQUATION. In Problems 1 and 2 solve the heat equation (1) subject to the given conditions. Assume a rod of length L.
.4 WAVE EQUATION 445 EXERCISES.3 In Problems and solve the heat eqation () sbject to the given conditions. Assme a rod of length.. (, t), (, t) (, ),, > >. (, t), (, t) (, ) ( ) 3. Find the temperatre
More informationSpring, 2008 CIS 610. Advanced Geometric Methods in Computer Science Jean Gallier Homework 1, Corrected Version
Spring, 008 CIS 610 Adanced Geometric Methods in Compter Science Jean Gallier Homework 1, Corrected Version Febrary 18, 008; De March 5, 008 A problems are for practice only, and shold not be trned in.
More informationConcepts Introduced. Digital Electronics. Logic Blocks. Truth Tables
Concepts Introdced Digital Electronics trth tables, logic eqations, and gates combinational logic seqential logic Digital electronics operate at either high or low voltage. Compters se a binary representation
More informationDesigning Single-Cycle MIPS Processor
CSE 32: Introdction to Compter Architectre Designing Single-Cycle IPS Processor Presentation G Stdy:.-. Gojko Babić 2/9/28 Introdction We're now ready to look at an implementation of the system that incldes
More informationA Model-Free Adaptive Control of Pulsed GTAW
A Model-Free Adaptive Control of Plsed GTAW F.L. Lv 1, S.B. Chen 1, and S.W. Dai 1 Institte of Welding Technology, Shanghai Jiao Tong University, Shanghai 00030, P.R. China Department of Atomatic Control,
More information