Introduction Set invariance theory isteps sets Robust invariant sets. Set Invariance. D. Limon A. Ferramosca E.F. Camacho


 Briana York
 1 years ago
 Views:
Transcription
1 Set Invariance D. Limon A. Ferramosca E.F. Camacho Department of Automatic Control & Systems Engineering University of Seville HYCONEECI Graduate School on Control Limon, Ferramosca, Camacho Set Invariance Outline Some definitions One step set The reach set 4 Limon, Ferramosca, Camacho Set Invariance
2 Some definitions Set invariance Set invariance is a fundamental concept in design of controller for constrained systems. The reason: constraint satisfaction can be guaranteed for all time (and for all disturbances) if and only if the initial state is contained inside a (robust) control invariant set. The evolution of a constrained system is admissible if there exists an invariant set X, where X is the set where constraints on the state are fullfilled. Hence, if x X, then x k X, for all k. Limon, Ferramosca, Camacho Set Invariance Some definitions Set invariance Set invariance is strictly connected with stability. Lyapunov theory states that, if there exists a Lyapunov function V (x) such that: ΔV (x) then, for all x X, any set defined as: ={x R n : V (x) α} X is an invariant set for the system, and hence for any initial state x, the system fulfills the constraints and remains inside. Limon, Ferramosca, Camacho Set Invariance 4
3 Some definitions Positive invariant set Consider an autonomous system: x k+ = f (x k ), x k X Definition (Positive invariant set) R n is a positive invariant set if x X, x k, for all k. If the system reaches a positive invariant set, its future evolution remains inside this set. The maximum invariant set, max X, is the smallest positive invariant set that contains all the positive invariant sets contained in X. Limon, Ferramosca, Camacho Set Invariance 5 Some definitions Positive control invariant set Consider the system: x k R n and u k R m. x k+ = f (x k, uk), x k X, u k U Definition (Positive control invariant set) R n is a positive control invariant set if x X, there exists a control law u k = h(x k ) such that x k, for all k, and u k = h(x k ) U. Limon, Ferramosca, Camacho Set Invariance 6
4 Onestepset The reach set Onestepset Consider the system: x k+ = f (x k, uk), x k X, u k U x k R n and u k R m. Let f (, ) = be en equilibrium point. Definition (One step set) The set Q() is the set of states in R n for which an admissible control inputs exists which will guarantee that the system will be driven to in one step: Q() = {x k R n u k U : f (x k, u k ) } Limon, Ferramosca, Camacho Set Invariance 7 Onestepset The reach set Onestepset The previous definition is the same for a system controlled by a control law u = h(x): Q h () = {x k R n : f (x k, h(x k )) } Property: Monotonicity: consider sets, then: Q( ) Q( ) Limon, Ferramosca, Camacho Set Invariance 8
5 Onestepset The reach set Geometric invariance condition The one step set definition allow us to define a condition for guaranteing the invariance of a set. Geometric invariance condition: set is a control invariant set if and only if Q() Q().5.5 x.5 Q().5 x is not invariant is invariant Limon, Ferramosca, Camacho Set Invariance 9 The reach set Onestepset The reach set Definition (The reach set) The set R() is the set of states in R n to which the system will evolve at the next time step given any x k and admissible control input: R() = {z R n : x k, u k Us.t.z = f (x k, u k )} For closedloop systems, R h () is the set of states in R n to which the system will evolve at the next time step given any x k : R h () = {z R n : x k, s.t.z = f (x k, h(x k ))} Limon, Ferramosca, Camacho Set Invariance
6 Definition The K i (X, ) is the set of states for which exists an admissible control sequence such that the system reaches the set X in exactly i steps, with an admissible evolution. K i (X, ) = {x X : k =,..., i, u k Us.t.x k Xandx i } This set represents the set of all states that can reach a given set in i steps, with an admissible evolution and an admissible control sequence. Limon, Ferramosca, Camacho Set Invariance Properties K i+ (X, ) = Q(K i (X, )) X, withk (X, ) =. K i (X, ) K i+ (X, ) iff is invariant. The set K (X, ) is finitely determined if and only if i N such that K (X, ) = K i (X, ). The smallest element i N such that K (X, ) = K i (X, ) is called the determinedness index. If j N such that K i+ (X, ) = K i (X, ), i j, then K (X, ) is finitely determined. For closedloop systems: K h i (X, ) = {x X h : x k X h k =,..., i, andx i } where X h = {x X : h(x) U}. Limon, Ferramosca, Camacho Set Invariance
7 Definition The set C (X ) is the maximal control invariant set contained in X for system x k+ = f (x k, u k ) if and only if C (X) is a control invariant set and contains all the invariant sets Φ contained in X. Φ C (X) X This set is derived from the definition of the isteps admissible set, C i (X ), that is the set of states for which exists an admissible control sequence such that the evolution of the system remains in X during the next i steps. C i (X )={x X : k =,..., i, u k Us.t.x k+ X} Limon, Ferramosca, Camacho Set Invariance Properties C i (X )=K i (X, X ). C i+ (X ) C i (X ). If x XC i (X ), there not exists an admissible control law which will ensure that the evolution of the system is admissible for i steps. C (X ) is the set of all states for which there exists an admissible control law which ensures the fulfillment of the constraints for all time. C (X ) is finitely determined if and only if there exists an element i N, such that C i+ (X) =C i (X), i i. Hence, C i (X )=C (X). K i (X, ) C (X ), i and X. Limon, Ferramosca, Camacho Set Invariance 4
8 Definition The set S (X, ) is the maximal stabilisable invariant set contained in X for system x k+ = f (x k, u k ) if and only if S (X, ) is the union of all istep stabilisable sets contained in X. This set is derived from the definition of the isteps stabilisable set, S i (X, ), that is the set of states for which exists an admissible control sequence that drive the system to the invariant set in i steps with an admissible evolution. S i (X, ) = {x X : k =,..., i, u k Us.t.x k Xandx i } The only difference between S i (X, ) and K i (X, ) is the invariance condition for. Limon, Ferramosca, Camacho Set Invariance 5 Properties S i+ (X, ) = Q(S i (X, )) X, withs (X, ) =. S i (X, ) S i+ (X, ). Any S i (X, ) is a control invariant set. Consider and invariant sets, such that. Then S i (X, ) S i (X, ). S i (X, S j (X, )) = S i+j (X, ). S (X, ) is finitely determined if and only if there exists an element i N such that S i+ (X, ) = S i (X, ), for any i i. Furthermore, S (X, ) = S i (X, ), for any i i. For closedloop systems: S h i (X, ) = {x X h : x k X h k =,..., i, andx i } where X h = {x X : h(x) U}. Limon, Ferramosca, Camacho Set Invariance 6
9 isteps controllable and stabilizable sets not Invariant Invariant S x x K x x Controllable set K (X, ) = Q() X Stabilisable set S (X, ) = Q() X Limon, Ferramosca, Camacho Set Invariance 7 isteps controllable and stabilizable sets x x S S K K x x Controllable set K (X, ) = Q(K ) X Stabilisable set S (X, ) = Q(S ) X Limon, Ferramosca, Camacho Set Invariance 8
10 K K isteps controllable and stabilizable sets S S S x x K x x Controllable set K (X, ) = Q(K ) X Stabilisable set S (X, ) = Q(S ) X Limon, Ferramosca, Camacho Set Invariance 9 isteps controllable and stabilizable sets S 4 S S S x x K K K K x x Controllable set K i+ (X, ) = Q(K i ) X K i (X, ) K i+ X Stabilisable set S i+ (X, ) = Q(S i ) X S i (X, ) S i+ X Limon, Ferramosca, Camacho Set Invariance
11 isteps controllable and stabilizable sets S 9 = S = S x x K K K K x x Controllable set K i+ (X, ) = Q(K i ) X K i (X, ) K i+ X Stabilisable set S i+ (X, ) = Q(S i ) X S i (X, ) S i+ X S finitely determined iff S i (X, ) = S i+ X Limon, Ferramosca, Camacho Set Invariance Comments The difference between the controllable set and the stabilisable set is the fact that the set is an invariant set. This difference is very important in relation to the concept of stability: if x S i (X, ), then there exists a control sequence such that the system is driven to, and there exist a control law such that the system remains inside. If is not invariant, then the system might evolve outside, hence loosing stability. S (X, ) C (X). Hence, x C (X) \ S i (X, ), there exists an admissible control law such that the system fulfills the constraints, but there not exists a control law that drives the system to. Set {} is an invariant set. Then, the set of states that asymptotically stabilize the system at the origin in i steps is given by S i (X, {}). Limon, Ferramosca, Camacho Set Invariance
12 Robust positive control invariant set Consider the system: x k+ = f (x k, uk, w k ), x k X, u k U, w k W x k R n, u k R m, w k R q. Definition (Robust positive control invariant set) R n is a robust positive control invariant set if x X, there exists a control law u k = h(x k ) such that x k, for all k and w k W, and u k = h(x k ) U. Limon, Ferramosca, Camacho Set Invariance Robust one step set Definition (Robust one step set) The set Q() is the set of states in R n for which an admissible control inputs exists which will guarantee that the system will be driven to in one step, for any w W: Q() = {x k R n u k U : f (x k, u k, w k ) w k W} For closedloop systems: Q h () = {x k R n : f (x k, h(x k ), w k ) w k W} Limon, Ferramosca, Camacho Set Invariance 4
13 Properties Monotonicity: consider sets, then: Q( ) Q( ) The one step set definition allow us to define a condition for guaranteing the invariance of a set. Geometric robust invariance condition: set is a control invariant set if and only if Q() Limon, Ferramosca, Camacho Set Invariance 5 Robust reach set Definition (Robust reach set) The set R() is the set of states in R n to which the system will evolve at the next time step given any x k, anyw k Wand admissible control input: R() = {z R n : x k, u k U, w k Ws.t.z = f (x k, u k, w k )} For closedloop systems, R h () is the set of states in R n to which the system will evolve at the next time step given any x k and any w k W: R h () = {z R n : x k, w k Ws.t.z = f (x k, h(x k ), w k )} Limon, Ferramosca, Camacho Set Invariance 6
14 isteps robust controllable set Definition The isteps robust controllable set K i (X, ) is the set of states for which exists an admissible control sequence such that the system reaches the set X in exactly i steps, with an admissible evolution, for any w k W. K i (X, ) = {x X : k =,..., i, u k Us.t.x k Xandx i w k W} This set represents the set of all states that can reach a given set in i steps, with an admissible evolution and an admissible control sequence. Limon, Ferramosca, Camacho Set Invariance 7 Maximal robust control invariant set Definition The set C (X ) is the maximal control invariant set contained in X for system x k+ = f (x k, u k, w k ) if and only if C (X) is a robust control invariant set and contains all the robust invariant sets Φ contained in X. Φ C (X) X This set is derived from the definition of the isteps robust admissible set, C i (X ), that is the set of states for which exists an admissible control sequence such that the evolution of the system remains in X during the next i steps, for any w k W. C i (X) ={x X : u k Us.t.x k+ X, w k W, k =,..., i } Limon, Ferramosca, Camacho Set Invariance 8
15 Maximal robust stabilisable set Definition The set S (X, ) is the maximal robust stabilisable invariant set contained in X for system x k+ = f (x k, u k, w k ) if and only if S (X, ) is the union of all istep stabilisable sets contained in X, for any w k W. This set is derived from the definition of the isteps robust stabilisable set, S i (X, ), that is the set of states for which exists an admissible control sequence that drive the system to the invariant set in i steps with an admissible evolution. S i (X, ) = {x X : k =,..., i, u k Us.t.x k Xandx i w k W} All the properties of the nominal invariant sets are applicable to the robust case. Limon, Ferramosca, Camacho Set Invariance 9 Bibliography F. Blanchini. Set invariance in control. Automatica. 5: , 999. E. Kerrigan. Robust Constrained Satisfaction: Invariant Sets and Predictive Control. PhD Dissertation. Limon, Ferramosca, Camacho Set Invariance
15 Limit sets. Lyapunov functions
15 Limit sets. Lyapunov functions At this point, considering the solutions to ẋ = f(x), x U R 2, (1) we were most interested in the behavior of solutions when t (sometimes, this is called asymptotic behavior
More informationExample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x
Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written
More informationPredictive Control Algorithms: Stability despite Shortened Optimization Horizons
Predictive Control Algorithms: Stability despite Shortened Optimization Horizons Philipp Braun Jürgen Pannek Karl Worthmann University of Bayreuth, 9544 Bayreuth, Germany University of the Federal Armed
More information6.231 Dynamic Programming Midterm, Fall 2008. Instructions
6.231 Dynamic Programming Midterm, Fall 2008 Instructions The midterm comprises three problems. Problem 1 is worth 60 points, problem 2 is worth 40 points, and problem 3 is worth 40 points. Your grade
More informationUsing the Theory of Reals in. Analyzing Continuous and Hybrid Systems
Using the Theory of Reals in Analyzing Continuous and Hybrid Systems Ashish Tiwari Computer Science Laboratory (CSL) SRI International (SRI) Menlo Park, CA 94025 Email: ashish.tiwari@sri.com Ashish Tiwari
More informationModel Predictive Control Lecture 5
Model Predictive Control Lecture 5 Klaus Trangbæk ktr@es.aau.dk Automation & Control Aalborg University Denmark. http://www.es.aau.dk/staff/ktr/mpckursus/mpckursus.html mpc5 p. 1 Exercise from last time
More informationPassive control. Carles Batlle. II EURON/GEOPLEX Summer School on Modeling and Control of Complex Dynamical Systems Bertinoro, Italy, July 1822 2005
Passive control theory I Carles Batlle II EURON/GEOPLEX Summer School on Modeling and Control of Complex Dynamical Systems Bertinoro, Italy, July 1822 25 Contents of this lecture Change of paradigm in
More informationPID Controller Design for Nonlinear Systems Using DiscreteTime Local Model Networks
PID Controller Design for Nonlinear Systems Using DiscreteTime Local Model Networks 4. Workshop für Modellbasierte Kalibriermethoden Nikolaus EulerRolle, Christoph Hametner, Stefan Jakubek Christian
More informationElgersburg Workshop 2010, 1.4. März 2010 1. PathFollowing for Nonlinear Systems Subject to Constraints Timm Faulwasser
#96230155 2010 Photos.com, ein Unternehmensbereich von Getty Images. Alle Rechte vorbehalten. Steering a Car as a Control Problem PathFollowing for Nonlinear Systems Subject to Constraints Chair for Systems
More informationThe MaxDistance Network Creation Game on General Host Graphs
The MaxDistance Network Creation Game on General Host Graphs 13 Luglio 2012 Introduction Network Creation Games are games that model the formation of largescale networks governed by autonomous agents.
More informationDimension Theory for Ordinary Differential Equations
Vladimir A. Boichenko, Gennadij A. Leonov, Volker Reitmann Dimension Theory for Ordinary Differential Equations Teubner Contents Singular values, exterior calculus and Lozinskiinorms 15 1 Singular values
More informationFormulations of Model Predictive Control. Dipartimento di Elettronica e Informazione
Formulations of Model Predictive Control Riccardo Scattolini Riccardo Scattolini Dipartimento di Elettronica e Informazione Impulse and step response models 2 At the beginning of the 80, the early formulations
More informationMaximization versus environmental compliance
Maximization versus environmental compliance Increase use of alternative fuels with no risk for quality and environment Reprint from World Cement March 2005 Dr. Eduardo Gallestey, ABB, Switzerland, discusses
More informationA Robust Optimization Approach to Supply Chain Management
A Robust Optimization Approach to Supply Chain Management Dimitris Bertsimas and Aurélie Thiele Massachusetts Institute of Technology, Cambridge MA 0139, dbertsim@mit.edu, aurelie@mit.edu Abstract. We
More informationINPUTTOSTATE STABILITY FOR DISCRETETIME NONLINEAR SYSTEMS
INPUTTOSTATE STABILITY FOR DISCRETETIME NONLINEAR SYSTEMS ZhongPing Jiang Eduardo Sontag,1 Yuan Wang,2 Department of Electrical Engineering, Polytechnic University, Six Metrotech Center, Brooklyn,
More informationChapter 3 Nonlinear Model Predictive Control
Chapter 3 Nonlinear Model Predictive Control In this chapter, we introduce the nonlinear model predictive control algorithm in a rigorous way. We start by defining a basic NMPC algorithm for constant reference
More informationLecture 13 Linear quadratic Lyapunov theory
EE363 Winter 289 Lecture 13 Linear quadratic Lyapunov theory the Lyapunov equation Lyapunov stability conditions the Lyapunov operator and integral evaluating quadratic integrals analysis of ARE discretetime
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationNonlinear Systems and Control Lecture # 15 Positive Real Transfer Functions & Connection with Lyapunov Stability. p. 1/?
Nonlinear Systems and Control Lecture # 15 Positive Real Transfer Functions & Connection with Lyapunov Stability p. 1/? p. 2/? Definition: A p p proper rational transfer function matrix G(s) is positive
More informationA Passivity Measure Of Systems In Cascade Based On Passivity Indices
49th IEEE Conference on Decision and Control December 57, Hilton Atlanta Hotel, Atlanta, GA, USA A Passivity Measure Of Systems In Cascade Based On Passivity Indices Han Yu and Panos J Antsaklis Abstract
More informationStateDriven Testing of Distributed Systems: Appendix
StateDriven Testing of Distributed Systems: Appendix Domenico Cotroneo, Roberto Natella, Stefano Russo, Fabio Scippacercola Università degli Studi di Napoli Federico II {cotroneo,roberto.natella,sterusso,fabio.scippacercola}@unina.it
More informationNotes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand
Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such
More informationChange Management in Enterprise IT Systems: Process Modeling and Capacityoptimal Scheduling
Change Management in Enterprise IT Systems: Process Modeling and Capacityoptimal Scheduling Praveen K. Muthusamy, Koushik Kar, Sambit Sahu, Prashant Pradhan and Saswati Sarkar Rensselaer Polytechnic Institute
More informationAn Asymptotically Optimal Scheme for P2P File Sharing
An Asymptotically Optimal Scheme for P2P File Sharing Panayotis Antoniadis Costas Courcoubetis Richard Weber Athens University of Economics and Business Athens University of Economics and Business Centre
More informationRELATIONSHIPS BETWEEN AFFINE FEEDBACK POLICIES FOR ROBUST CONTROL WITH CONSTRAINTS. Paul J. Goulart Eric C. Kerrigan
RELATIOSHIPS BETWEE AFFIE FEEDBACK POLICIES FOR ROBUST COTROL WITH COSTRAITS Paul J Goulart Eric C Kerrigan Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
More information4 Lyapunov Stability Theory
4 Lyapunov Stability Theory In this section we review the tools of Lyapunov stability theory. These tools will be used in the next section to analyze the stability properties of a robot controller. We
More informationStrengthening International Courts and the Early Settlement of Disputes
Strengthening International Courts and the Early Settlement of Disputes Michael Gilligan, Leslie Johns, and B. Peter Rosendorff November 18, 2008 Technical Appendix Definitions σ(ŝ) {π [0, 1] s(π) = ŝ}
More informationPredictive Control Algorithms for Nonlinear Systems
Predictive Control Algorithms for Nonlinear Systems DOCTORAL THESIS for receiving the doctoral degree from the Gh. Asachi Technical University of Iaşi, România The Defense will take place on 15 September
More informationA New Natureinspired Algorithm for Load Balancing
A New Natureinspired Algorithm for Load Balancing Xiang Feng East China University of Science and Technology Shanghai, China 200237 Email: xfeng{@ecusteducn, @cshkuhk} Francis CM Lau The University of
More information6.231 Dynamic Programming and Stochastic Control Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 6.231 Dynamic Programming and Stochastic Control Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 6.231
More informationProposal for Communication Ph.D. at UC Davis 8
Proposal for Communication Ph.D. at UC Davis 8 6. Administration of the Program The program will be administered by the Department of Communication, which is currently administering the Communication M.A.
More informationProbability Generating Functions
page 39 Chapter 3 Probability Generating Functions 3 Preamble: Generating Functions Generating functions are widely used in mathematics, and play an important role in probability theory Consider a sequence
More informationDynamically Managing the Realtime Fabric of a Wireless SensorActuator Network
Dynamically Managing the Realtime Fabric of a Wireless SensorActuator Network Award No: CNS0931195 Duration: Sept. 1 2009  Aug. 31 2012 M.D. Lemmon, Univ. of Notre Dame (PI) S.X. Hu, Univ. of Notre
More informationHow will the programme be delivered (e.g. interinstitutional, summerschools, lectures, placement, rotations, online etc.):
Titles of Programme: Hamilton Hamilton Institute Institute Structured PhD Structured PhD Minimum 30 credits. 15 of Programme which must be obtained from Generic/Transferable skills modules and 15 from
More information20 Selfish Load Balancing
20 Selfish Load Balancing Berthold Vöcking Abstract Suppose that a set of weighted tasks shall be assigned to a set of machines with possibly different speeds such that the load is distributed evenly among
More informationControl Systems with Actuator Saturation
Control Systems with Actuator Saturation Analysis and Design Tingshu Hu Zongli Lin With 67 Figures Birkhauser Boston Basel Berlin Preface xiii 1 Introduction 1 1.1 Linear Systems with Actuator Saturation
More informationLOOP TRANSFER RECOVERY FOR SAMPLEDDATA SYSTEMS 1
LOOP TRANSFER RECOVERY FOR SAMPLEDDATA SYSTEMS 1 Henrik Niemann Jakob Stoustrup Mike Lind Rank Bahram Shafai Dept. of Automation, Technical University of Denmark, Building 326, DK2800 Lyngby, Denmark
More informationChapter 5. Linear Inequalities and Linear Programming. Linear Programming in Two Dimensions: A Geometric Approach
Chapter 5 Linear Programming in Two Dimensions: A Geometric Approach Linear Inequalities and Linear Programming Section 3 Linear Programming gin Two Dimensions: A Geometric Approach In this section, we
More informationUNIVERSITY OF VIENNA
WORKING PAPERS Egbert Dierker Hildegard Dierker Birgit Grodal Nonexistence of Constrained Efficient Equilibria when Markets are Incomplete August 2001 Working Paper No: 0111 DEPARTMENT OF ECONOMICS UNIVERSITY
More informationInvestigación Operativa. The uniform rule in the division problem
Boletín de Estadística e Investigación Operativa Vol. 27, No. 2, Junio 2011, pp. 102112 Investigación Operativa The uniform rule in the division problem Gustavo Bergantiños Cid Dept. de Estadística e
More informationProgramme Specification MSc/PGDip/PGCert Sustainable Building: Performance and Design
Programme Specification MSc/PGDip/PGCert Sustainable Building: Performance and Design Valid from: September 2012 Faculty of Technology, Design and Environment Oxford Brookes University SECTION 1: GENERAL
More informationTesting on proportions
Testing on proportions Textbook Section 5.4 April 7, 2011 Example 1. X 1,, X n Bernolli(p). Wish to test H 0 : p p 0 H 1 : p > p 0 (1) Consider a related problem The likelihood ratio test is where c is
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationGenerating Direct Powers
Generating Direct Powers Nik Ruškuc nik@mcs.stand.ac.uk School of Mathematics and Statistics, Novi Sad, 16 March 2012 Algebraic structures Classical: groups, rings, modules, algebras, Lie algebras. Semigroups.
More informationChapter 7: Termination Detection
Chapter 7: Termination Detection Ajay Kshemkalyani and Mukesh Singhal Distributed Computing: Principles, Algorithms, and Systems Cambridge University Press A. Kshemkalyani and M. Singhal (Distributed Computing)
More informationLecture 2: Consumer Theory
Lecture 2: Consumer Theory Preferences and Utility Utility Maximization (the primal problem) Expenditure Minimization (the dual) First we explore how consumers preferences give rise to a utility fct which
More informationLECTURE 15: AMERICAN OPTIONS
LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These
More informationExam Introduction Mathematical Finance and Insurance
Exam Introduction Mathematical Finance and Insurance Date: January 8, 2013. Duration: 3 hours. This is a closedbook exam. The exam does not use scrap cards. Simple calculators are allowed. The questions
More informationProximal mapping via network optimization
L. Vandenberghe EE236C (Spring 234) Proximal mapping via network optimization minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping Introduction this lecture:
More informationLas Vegas and Monte Carlo Randomized Algorithms for Systems and Control
Las Vegas and Monte Carlo Randomized Algorithms for Systems and Control Roberto Tempo IEIITCNR Politecnico di Torino roberto.tempo@polito.it BasarFest,, Urbana RT 2006 1 CSL UIUC Six months at CSL in
More informationNetwork Traffic Modelling
University of York Dissertation submitted for the MSc in Mathematics with Modern Applications, Department of Mathematics, University of York, UK. August 009 Network Traffic Modelling Author: David Slade
More informationChapter 7 Robust Stabilization and Disturbance Attenuation of Switched Linear ParameterVarying Systems in Discrete Time
Chapter 7 Robust Stabilization and Disturbance Attenuation of Switched Linear ParameterVarying Systems in Discrete Time JiWoong Lee and Geir E. Dullerud Abstract Nonconservative analysis of discretetime
More informationSpanish Regional Accounts. Base 2010. Regional Gross Domestic Product. Year 2014 Income accounts of the household sector.
27 March 2015 Spanish Regional Accounts. Base 2010 Regional Gross Domestic Product. Year 2014 Income accounts of the household sector. 20102012 series Main results Regional Gross Domestic Product. Year
More informationChapter 7. Sealedbid Auctions
Chapter 7 Sealedbid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)
More informationDecentralized Utilitybased Sensor Network Design
Decentralized Utilitybased Sensor Network Design Narayanan Sadagopan and Bhaskar Krishnamachari University of Southern California, Los Angeles, CA 900890781, USA narayans@cs.usc.edu, bkrishna@usc.edu
More informationHackingproofness and Stability in a Model of Information Security Networks
Hackingproofness and Stability in a Model of Information Security Networks Sunghoon Hong Preliminary draft, not for citation. March 1, 2008 Abstract We introduce a model of information security networks.
More informationLoad balancing of temporary tasks in the l p norm
Load balancing of temporary tasks in the l p norm Yossi Azar a,1, Amir Epstein a,2, Leah Epstein b,3 a School of Computer Science, Tel Aviv University, Tel Aviv, Israel. b School of Computer Science, The
More informationChapter 7 ELECTRICITY PRICES IN A GAME THEORY CONTEXT. 1. Introduction. Mireille Bossy. Geert Jan Olsder Odile Pourtallier Etienne Tanré
Chapter 7 ELECTRICITY PRICES IN A GAME THEORY CONTEXT Mireille Bossy Nadia Maïzi Geert Jan Olsder Odile Pourtallier Etienne Tanré Abstract We consider a model of an electricity market in which S suppliers
More informationRoots of Polynomials
Roots of Polynomials (Com S 477/577 Notes) YanBin Jia Sep 24, 2015 A direct corollary of the fundamental theorem of algebra is that p(x) can be factorized over the complex domain into a product a n (x
More information4/1/2017. PS. Sequences and Series FROM 9.2 AND 9.3 IN THE BOOK AS WELL AS FROM OTHER SOURCES. TODAY IS NATIONAL MANATEE APPRECIATION DAY
PS. Sequences and Series FROM 9.2 AND 9.3 IN THE BOOK AS WELL AS FROM OTHER SOURCES. TODAY IS NATIONAL MANATEE APPRECIATION DAY 1 Oh the things you should learn How to recognize and write arithmetic sequences
More informationA Note on Best Response Dynamics
Games and Economic Behavior 29, 138 150 (1999) Article ID game.1997.0636, available online at http://www.idealibrary.com on A Note on Best Response Dynamics Ed Hopkins Department of Economics, University
More informationLinear Programming Notes V Problem Transformations
Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material
More informationSOLUTIONS TO ASSIGNMENT 1 MATH 576
SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts
More informationCPC/CPA Hybrid Bidding in a Second Price Auction
CPC/CPA Hybrid Bidding in a Second Price Auction Benjamin Edelman Hoan Soo Lee Working Paper 09074 Copyright 2008 by Benjamin Edelman and Hoan Soo Lee Working papers are in draft form. This working paper
More informationScheduling Realtime Tasks: Algorithms and Complexity
Scheduling Realtime Tasks: Algorithms and Complexity Sanjoy Baruah The University of North Carolina at Chapel Hill Email: baruah@cs.unc.edu Joël Goossens Université Libre de Bruxelles Email: joel.goossens@ulb.ac.be
More informationEvolution Feature Oriented Model Driven Product Line Engineering Approach for Synergistic and Dynamic Service Evolution in Clouds
Evolution Feature Oriented Model Driven Product Line Engineering Approach for Synergistic and Dynamic Service Evolution in Clouds Zhe Wang, Xiaodong Liu, Kevin Chalmers School of Computing Edinburgh Napier
More informationSOFTWARE ENGINEERING OVERVIEW
SOFTWARE ENGINEERING OVERVIEW http://www.tutorialspoint.com/software_engineering/software_engineering_overview.htm Copyright tutorialspoint.com Let us first understand what software engineering stands
More informationScheduling Algorithm with Optimization of Employee Satisfaction
Washington University in St. Louis Scheduling Algorithm with Optimization of Employee Satisfaction by Philip I. Thomas Senior Design Project http : //students.cec.wustl.edu/ pit1/ Advised By Associate
More informationLinearQuadratic Optimal Controller 10.3 Optimal Linear Control Systems
LinearQuadratic Optimal Controller 10.3 Optimal Linear Control Systems In Chapters 8 and 9 of this book we have designed dynamic controllers such that the closedloop systems display the desired transient
More information10.2 Series and Convergence
10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and
More informationDevelopment of dynamically evolving and selfadaptive software. 1. Background
Development of dynamically evolving and selfadaptive software 1. Background LASER 2013 Isola d Elba, September 2013 Carlo Ghezzi Politecnico di Milano DeepSE Group @ DEIB 1 Requirements Functional requirements
More informationStationary random graphs on Z with prescribed iid degrees and finite mean connections
Stationary random graphs on Z with prescribed iid degrees and finite mean connections Maria Deijfen Johan Jonasson February 2006 Abstract Let F be a probability distribution with support on the nonnegative
More informationOPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN. E.V. Grigorieva. E.N. Khailov
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Supplement Volume 2005 pp. 345 354 OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN E.V. Grigorieva Department of Mathematics
More informationDYNAMICAL NETWORKS: structural analysis and synthesis
A social networks synchronization DYNAMICAL NETWORKS: structural analysis and synthesis traffic management B C (bio)chemical processes water distribution networks biological systems, ecosystems production
More informationTraining on Quality Assurance in PhD. Dissertation and mentorship: what standards should there be? by Mariangela SCIANDRA
Training on Quality Assurance in PhD Dissertation and mentorship: what standards should there be? by Mariangela SCIANDRA Guidelines and Principles for Accreditation Aim: to illustrate the main Principles
More informationModèle géométrique de plissement constraint par la rhéologie et l équilibre mécanique: Application possibe en régime extensif?
Modèle géométrique de plissement constraint par la rhéologie et l équilibre mécanique: Application possibe en régime extensif? Yves M. Leroy (Laboratoire de Géologie, ENS), Bertrand Maillot (University
More informationCyberSecurity Analysis of State Estimators in Power Systems
CyberSecurity Analysis of State Estimators in Electric Power Systems André Teixeira 1, Saurabh Amin 2, Henrik Sandberg 1, Karl H. Johansson 1, and Shankar Sastry 2 ACCESS Linnaeus Centre, KTHRoyal Institute
More informationTitle: Integrating Management of Truck and Rail Systems in LA. INTERIM REPORT August 2015
Title: Integrating Management of Truck and Rail Systems in LA Project Number: 3.1a Year: 20132017 INTERIM REPORT August 2015 Principal Investigator Maged Dessouky Researcher Lunce Fu MetroFreight Center
More informationRobust output feedbackstabilization via risksensitive control
Automatica 38 22) 945 955 www.elsevier.com/locate/automatica Robust output feedbackstabilization via risksensitive control Valery A. Ugrinovskii, Ian R. Petersen School of Electrical Engineering, Australian
More informationWhen Promotions Meet Operations: CrossSelling and Its Effect on CallCenter Performance
When Promotions Meet Operations: CrossSelling and Its Effect on CallCenter Performance Mor Armony 1 Itay Gurvich 2 Submitted July 28, 2006; Revised August 31, 2007 Abstract We study crossselling operations
More informationStudent Project Allocation Using Integer Programming
IEEE TRANSACTIONS ON EDUCATION, VOL. 46, NO. 3, AUGUST 2003 359 Student Project Allocation Using Integer Programming A. A. Anwar and A. S. Bahaj, Member, IEEE Abstract The allocation of projects to students
More informationDevelopment of global specification for dynamically adaptive software
Development of global specification for dynamically adaptive software Yongwang Zhao School of Computer Science & Engineering Beihang University zhaoyw@act.buaa.edu.cn 22/02/2013 1 2 About me Assistant
More informationOptimal Control. Lecture 2. Palle Andersen, Aalborg University. Opt lecture 2 p. 1/44
Optimal Control Lecture 2 pa@control.aau.dk Palle Andersen, Aalborg University Opt lecture 2 p. 1/44 Summary of LQ for DT systems Linear discrete time, dynamical system x(k +1) = Φx(k)+Γu(k) y(k) = Hx(k)
More informationBrief Paper. of discretetime linear systems. www.ietdl.org
Published in IET Control Theory and Applications Received on 28th August 2012 Accepted on 26th October 2012 Brief Paper ISSN 17518644 Temporal and onestep stabilisability and detectability of discretetime
More information24. The Branch and Bound Method
24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NPcomplete. Then one can conclude according to the present state of science that no
More informationFaculty of Computer Science
Faculty of Computer Science PhD programme in COMPUTER SCIENCE Website: http://www.unibz.it/en/inf/progs/phdcs/default.html Duration: 3 years Academic year: 2015/2016 Start date: 01/11/2015 Official programme
More informationAnalysis of an Artificial Hormone System (Extended abstract)
c 2013. This is the author s version of the work. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purpose or for creating
More informationReview of Basic Options Concepts and Terminology
Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some
More informationSpecific amendments to the Capacity Allocation and Congestion Management Network Code
Annex: Specific amendments to the Capacity Allocation and Congestion Management Network Code I. Amendments with respect to entry into force and application The Network Code defines deadlines for several
More informationOptimization models for congestion control with multipath routing in TCP/IP networks
Optimization models for congestion control with multipath routing in TCP/IP networks Roberto Cominetti Cristóbal Guzmán Departamento de Ingeniería Industrial Universidad de Chile Workshop on Optimization,
More informationAnalysis of Approximation Algorithms for kset Cover using FactorRevealing Linear Programs
Analysis of Approximation Algorithms for kset Cover using FactorRevealing Linear Programs Stavros Athanassopoulos, Ioannis Caragiannis, and Christos Kaklamanis Research Academic Computer Technology Institute
More informationTail inequalities for order statistics of logconcave vectors and applications
Tail inequalities for order statistics of logconcave vectors and applications Rafał Latała Based in part on a joint work with R.Adamczak, A.E.Litvak, A.Pajor and N.TomczakJaegermann Banff, May 2011 Basic
More informationManagement and optimization of multiple supply chains
Management and optimization of multiple supply chains J. Dorn Technische Universität Wien, Institut für Informationssysteme Paniglgasse 16, A1040 Wien, Austria Phone ++4315880118426, Fax ++4315880118494
More informationThe Performance Management Process How to establish goals, objectives and KPI s
Performance Management Part 3 The Performance Management Process How to establish goals, objectives and KPI s Agenda Review of what is Performance Management? Developing measures Goals, Objectives & KPI
More informationLinear Programming I
Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins
More informationMSc International Banking and Financial Services For students entering in 2006
MSc International Banking and Financial Services For students entering in 2006 Awarding Institution Teaching Institution Faculty of Economic and Social Sciences Date of specification: October 2006 Programme
More informationLinear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc.
1. Introduction Linear Programming for Optimization Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1.1 Definition Linear programming is the name of a branch of applied mathematics that
More informationApplied mathematics and mathematical statistics
Applied mathematics and mathematical statistics The graduate school is organised within the Department of Mathematical Sciences.. Deputy head of department: Aila Särkkä Director of Graduate Studies: Marija
More informationLecture 11: Sponsored search
Computational Learning Theory Spring Semester, 2009/10 Lecture 11: Sponsored search Lecturer: Yishay Mansour Scribe: Ben Pere, Jonathan Heimann, Alon Levin 11.1 Sponsored Search 11.1.1 Introduction Search
More information