Model Predictive Control Lecture 5

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Model Predictive Control Lecture 5"

Transcription

1 Model Predictive Control Lecture 5 Klaus Trangbæk Automation & Control Aalborg University Denmark. mpc5 p. 1

2 Exercise from last time 1) Assume that you know linear constraints on the states and process noise. Show that the moving horizon estimator is a QP. Assume that we choose w (on the finite horizon) and x a as decision variables. The first step is to set up a vectorised predictor in terms of these: X = Γx a + BW. Secondly express the performance function in terms of the decision variables. This is done by inserting V = Y m ΛX. The file MHEQP.m sets up the predictor and performance weights. 2) Have a look at the three tank simulation. Replace the saturating observer with an MHE using quadprog to solve. MHE3tanks.m A simpler example is found in MHEsimple.m mpc5 p. 2

3 Today Explicit MPC Nonlinear MPC Summary of the course mpc5 p. 3

4 Explicit MPC It can be shown (Bemporad et al. 2002) that LQ MPC can be formulated as a piece-wise affine state feedback. We then get the explicit formulation: u k = K i x k + d i, x k X i, which is much faster than online optimisation. The number regions can be gigantic. Locating the current region can then be time consuming in itself. (mpt_demo4.m) mpc5 p. 4

5 Nonlinear MPC Nonlinear models are usually formulated in continuous time: dx dt = f(x,u) The control signal must be parameterised by a finite number of decision variables, e.g. by discretising in time or using splines. Closing inner loops increases chance of success. mpc5 p. 5

6 Nonlinear MPC The simplest method is probably sequential QP: Linearise along trajectory -> gradient and Hessian Solve QP -> optimal control sequence Recompute trajectory Repeat until convergence A faster alternative: Linearise along trajectory (using tail) -> gradient and Hessian Solve QP -> optimal control sequence Apply first sample of control sequence mpc5 p. 6

7 Simple nonlinear example Discrete time x k+1 = tanh(0.9x 3 k + u k) Nonlex.m mpc5 p. 7

8 Nonlinear MPC example Continuous time dx dt = 2tanh((x + 1)u) x 3 Linearising by simulation of small changes to input. (Linalong.m) Exc5.m, Exc5c.m mpc5 p. 8

9 Nonlinear MPC Direct collocation: Instead of approximating the performance function, we can approximate the model by linear constraints -> sparse NLP. Multiple shooting: A mixture of sequential QP and direct collocation. The performance function is approximated on intervals. The intervals must be connected -> constraints. mpc5 p. 9

10 Literature Predictive Control with Constraints by J.M. Maciejowski 1. Introduction 2. A Basic Formulation of Predictive Control 3. Solving Predictive Control Problems (4. Step Response and Transfer Function Formulations) 5. Other Formulations of Predictive Control 6. Stability (7. Tuning) 8. Robust Predictive Control (9. Two Case Studies) 10. Perspectives mpc5 p. 10

11 Literature Predictive Control with Constraints by J.M. Maciejowski Supplementary: The MPT manual Rao 99 on moving horizon estimation. Kothare 96 on robust MPC Bemporad 02 on explicit MPC Boyd and Vandenberghe: Convex Optimization mpc5 p. 11

12 Basic idea of MPC Predict future plant response using a model over a prediction horizon of H p samples. Use a prediction of the reference and disturbances. Set up cost function over the horizon. Define constraints on inputs and states/outputs. Determine control inputs over a control horizon H u H p that minimise the cost function. Apply first sample of determined inputs. Shift the horizon and repeat the procedure (receding horizon). mpc5 p. 12

13 Motivation The achievable performance of control problems is often dominated by constraints rather than plant dynamics. Constraints can be a more natural way to specify requirements than through penalties on variations from set-point. Nonlinear behaviour of MPC can improve performance. mpc5 p. 13

14 Pros and cons Advantages: Constraints are respected. Natural requirement specification. Anti-windup follows automatically. Standard tools for solving optimisation. Fairly natural extensions to nonlinear/hybrid systems. Disadvantages: A finite (and sometimes short) horizon is necessary problems with stability and feasibility. Essentially state feedback uncertainty (disturbances, model errors) is problematic. Complexity. mpc5 p. 14

15 Quadratic program The optimisation problem: minimise U(k)G + U(k) T H U(k) subject to F ΓΘ W U(k) F 1 u(k 1) f Γ[Ψˆx(k k) + Υu(k 1)] g w This is a quadratic programming problem (QP): 1 min θ 2 θt Φθ + φ T θ, s.t. Ωθ ω mpc5 p. 15

16 Soft constraints Infeasibility causes the solver to yield useless results. For some constraints we can accept a violation for a short period. Soft constraints are added through the performance index: 1 min θ 2 θ T Φθ + φ T θ +ρ ǫ s.t. Ωθ ω +ǫ ǫ 0 Soft constraints increase the number of decision variables. mpc5 p. 16

17 Linear programs For some systems, a 1-norm cost: V (k) = H p m q j ẑ j (k +i k) r j (k +i) + H u p r j û j (k +1 k) i=1 j=1 i=1 j=1 is more natural. For instance, the actual cost of fuel consumption is linear. An -norm is also possible: V (k) = max i max j q j ẑ j (k + i k) r j (k + i) 1- and -norms result in linear programs. mpc5 p. 17

18 Inner loop state feedback Adding a state feedback in an inner loop has several purposes: Stabilising predictions. Dual mode schemes. Constraint handling as perturbations to optimal control. Make behaviour after control horizon more realistic. mpc5 p. 18

19 Stability Stability problems can occur because We must use a finite horizon to get a finite dimensional problem. The finite horizon causes prediction mismatch. The optimisation is solved online analysis is difficult. In summary, the main problem is that we ignore what happens after the horizon. Stability can be ensured by ensuring that the terminal state is forced into a sufficiently small set and penalised. mpc5 p. 19

20 Terminal constraints A terminal constraint specifies that the state must be in a given set by the end of the prediction horizon. The the terminal constraint set should be chosen so that the state trajectories, after entering, stay inside, i.e an invariant set. mpc5 p. 20

21 Infinite horizons Optimising over an infinite horizon yields stability. On an infinite horizon, the tail will be the first part of the optimal solution the cost function serves as a Lyapunov function. We can write the infinite horizon cost as V (k) = = i=1 H u i=1 + H u x(k + i k) 2 Q + i=1 u(k + i 1 k) 2 R x(k + i k) 2 Q + x(k + H u + 1) T Qx(k + Hu + 1) H u i=1 u(k + i 1 k) 2 R mpc5 p. 21

22 Robustness Robust control can be used for dealing with uncertainty and nonlinearities. Assume that the system matrices will belong to a convex set [A(k),B(k)] Ω. We would like to minimise the worst case performance: min U(k) max V (k) A(k+j),B(k+j) We approximate this by using the performance function as a Lyapunov function and minimising at each sample: V (x(k + j + 1)) V (x(k + j)) x(k + j) 2 Q 1 u(k + j) 2 R and V (x) = x T Px, P > 0. mpc5 p. 22

23 Robustness We now assume u(k + j) = K k x(k + j) and want to find K k that minimises V while respecting the constraints. With standard manipulations, this can be as an LMI in the auxiliary variables Q = γp 1 and Y = K k Q. The LMIs must be fulfilled for the infinitely many [A(k),B(k)] Ω. The LMIs are affine in A and B, so if Ω is a convex polytope, we only need to satisfy the LMIs at the vertices of the polytope, which is a finite-dimensional problem. mpc5 p. 23

24 Adding constraints The above method makes it fairly easy to add constraints. To ensure feasibility on an infinite horizon, we often employ the concept of invariant sets: For an autonomous system x k+1 = Fx k, S is an invariant set if x k S x k+1 = Fx k S. So if the constraints are satisfied for any x S, and S is invariant, then the constraints are satisfied forever. With some conservatism, constraints can be cast as LMIs. mpc5 p. 24

25 Observers The prediction requires an estimate ˆx(k k) of the state. If the entire state vector is not measured then an observer must be used. The optimisation problem is the same as before, but sometimes uncertainty must be taken into account. State/output constraints can lead to feasibility problems when the state is uncertain. mpc5 p. 25

26 Observers Because of the constraints, we no longer have perfect separation of observer and state feedback. Methods: Separation, e.g. by high gain observers. Direct: considering the state uncertainty in the control. I/O-models. mpc5 p. 26

27 Constrained estimation One type of observer is a constrained estimator: Past values of states and inputs (disturbances) are known to respect constraints. Constraints provide extra information. Often, noise distributions are very uncertain but some limits are known. A finite horizon is necessary. Linear constrained estimation is a QP. mpc5 p. 27

28 Constrained estimation x k+1 = Ax k + Gw k, y k = Cx k + v k where w, v have covariances Q, R. We can formulate the problem min x N M,{w k } N 1 k=n M x N M ˆx N M N M 1 2 Π 1 N M + N 1 k=n M ( v k 2 R 1 + s.t. the system equations and x X, w W. This is a QP, i.e. constrained estimation is a dual of constrained control. Online optimisation -> extension to nonlinear systems. mpc5 p. 28

29 Online optimisation Online optimisation gives a lot of flexibility: the model can be changed online. This allows for adaptive/fault tolerant control. The model does not have to be constant over the horizon. This allows for time varying and even nonlinear models. mpc5 p. 29

30 Nonlinear MPC Nonlinear models are usually formulated in continuous time: dx dt = f(x,u) The control signal must be parameterised by a finite number of decision variables, e.g. by discretising in time or using splines. Closing inner loops increases chance of success. mpc5 p. 30

31 Nonlinear MPC The simplest method is probably sequential QP: Linearise along trajectory -> gradient and Hessian Solve QP -> optimal control sequence Recompute trajectory Repeat until convergence A faster alternative: Linearise along trajectory (using tail) -> gradient and Hessian Solve QP -> optimal control sequence Apply first sample of control sequence mpc5 p. 31

32 Nonlinear MPC Direct collocation: Instead of approximating the performance function, we can approximate the model by linear constraints -> sparse NLP. Multiple shooting: A mixture of sequential QP and direct collocation. The performance function is approximated on intervals. The intervals must be connected -> constraints. mpc5 p. 32

33 Using MPT MPT (or the MPC Toolbox) converts the plant and problem description into a quadratic program, which can then be solved by e.g. quadprog. MPT can handle many things beyond linear MPC, e.g. PWA, MILP, nonlinear (polynomials). The your MPC feature adds even more flexibility. However, setting up the QP by hand gives more control over details. mpc5 p. 33

34 Conclusions The main feature of MPC is constraint handling. Uncertainty (noise, model) cause problems (just as with most nonlinear methods). State constraints cause feasibility problems. It is possible to use standard solvers. Closing inner loops is usually a good idea. mpc5 p. 34

Model Predictive Control Lecture 1

Model Predictive Control Lecture 1 Model Predictive Control Lecture 1 (Palle Andersen) pa@es.aau.dk Automation & Control Aalborg University Denmark mpc1 p. 1/32 Book Predictive Control with Constraints by J.M. Maciejowski 1. Introduction

More information

C21 Model Predictive Control

C21 Model Predictive Control C21 Model Predictive Control Mark Cannon 4 lectures Hilary Term 216-1 Lecture 1 Introduction 1-2 Organisation 4 lectures: week 3 week 4 { Monday 1-11 am LR5 Thursday 1-11 am LR5 { Monday 1-11 am LR5 Thursday

More information

Optimal Control. Palle Andersen. Aalborg University. Opt lecture 1 p. 1/2

Optimal Control. Palle Andersen. Aalborg University. Opt lecture 1 p. 1/2 Opt lecture 1 p. 1/2 Optimal Control Palle Andersen pa@control.aau.dk Aalborg University Opt lecture 1 p. 2/2 Optimal Control, course outline 1st lecture: Introduction to optimal control and quadratic

More information

TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA 2015 School of Information Technology and Electrical Engineering at the University of Queensland TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA Schedule Week Date

More information

Lecture 7: Finding Lyapunov Functions 1

Lecture 7: Finding Lyapunov Functions 1 Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions 1

More information

Nonlinear Optimization: Algorithms 3: Interior-point methods

Nonlinear Optimization: Algorithms 3: Interior-point methods Nonlinear Optimization: Algorithms 3: Interior-point methods INSEAD, Spring 2006 Jean-Philippe Vert Ecole des Mines de Paris Jean-Philippe.Vert@mines.org Nonlinear optimization c 2006 Jean-Philippe Vert,

More information

Adaptive cruise controller design: a comparative assessment for PWA systems

Adaptive cruise controller design: a comparative assessment for PWA systems Adaptive cruise controller design: a comparative assessment for PWA systems Hybrid control problems for vehicle dynamics and engine control. Cagliari, PhD Course on discrete event and hybrid systems Daniele

More information

Nonlinear Model Predictive Control: From Theory to Application

Nonlinear Model Predictive Control: From Theory to Application J. Chin. Inst. Chem. Engrs., Vol. 35, No. 3, 299-315, 2004 Nonlinear Model Predictive Control: From Theory to Application Frank Allgöwer [1], Rolf Findeisen, and Zoltan K. Nagy Institute for Systems Theory

More information

MPC in Column control

MPC in Column control MPC in Column control Inge N. Åkesson, Senior Control and Application Specialist ABB, BA ATCF Control and Force Measurement, SE-721 57 Västerås, Sweden e-mail address: inge.akesson@se.abb.com Abstract:

More information

Chapter 3 Nonlinear Model Predictive Control

Chapter 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 information

Support Vector Machine (SVM)

Support Vector Machine (SVM) Support Vector Machine (SVM) CE-725: Statistical Pattern Recognition Sharif University of Technology Spring 2013 Soleymani Outline Margin concept Hard-Margin SVM Soft-Margin SVM Dual Problems of Hard-Margin

More information

Robust Path Planning and Feedback Design under Stochastic Uncertainty

Robust Path Planning and Feedback Design under Stochastic Uncertainty Robust Path Planning and Feedback Design under Stochastic Uncertainty Lars Blackmore Autonomous vehicles require optimal path planning algorithms to achieve mission goals while avoiding obstacles and being

More information

Some Optimization Fundamentals

Some Optimization Fundamentals ISyE 3133B Engineering Optimization Some Optimization Fundamentals Shabbir Ahmed E-mail: sahmed@isye.gatech.edu Homepage: www.isye.gatech.edu/~sahmed Basic Building Blocks min or max s.t. objective as

More information

054414 PROCESS CONTROL SYSTEM DESIGN. 054414 Process Control System Design. LECTURE 6: SIMO and MISO CONTROL

054414 PROCESS CONTROL SYSTEM DESIGN. 054414 Process Control System Design. LECTURE 6: SIMO and MISO CONTROL 05444 Process Control System Design LECTURE 6: SIMO and MISO CONTROL Daniel R. Lewin Department of Chemical Engineering Technion, Haifa, Israel 6 - Introduction This part of the course explores opportunities

More information

Optimal Control. Lecture 2. Palle Andersen, Aalborg University. Opt lecture 2 p. 1/44

Optimal 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 information

Dual Methods for Total Variation-Based Image Restoration

Dual Methods for Total Variation-Based Image Restoration Dual Methods for Total Variation-Based Image Restoration Jamylle Carter Institute for Mathematics and its Applications University of Minnesota, Twin Cities Ph.D. (Mathematics), UCLA, 2001 Advisor: Tony

More information

Parameter Estimation for Bingham Models

Parameter Estimation for Bingham Models Dr. Volker Schulz, Dmitriy Logashenko Parameter Estimation for Bingham Models supported by BMBF Parameter Estimation for Bingham Models Industrial application of ceramic pastes Material laws for Bingham

More information

Summer course on Convex Optimization. Fifth Lecture Interior-Point Methods (1) Michel Baes, K.U.Leuven Bharath Rangarajan, U.

Summer course on Convex Optimization. Fifth Lecture Interior-Point Methods (1) Michel Baes, K.U.Leuven Bharath Rangarajan, U. Summer course on Convex Optimization Fifth Lecture Interior-Point Methods (1) Michel Baes, K.U.Leuven Bharath Rangarajan, U.Minnesota Interior-Point Methods: the rebirth of an old idea Suppose that f is

More information

Lecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method

Lecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method Lecture 3 3B1B Optimization Michaelmas 2015 A. Zisserman Linear Programming Extreme solutions Simplex method Interior point method Integer programming and relaxation The Optimization Tree Linear Programming

More information

Neuro-Dynamic Programming An Overview

Neuro-Dynamic Programming An Overview 1 Neuro-Dynamic Programming An Overview Dimitri Bertsekas Dept. of Electrical Engineering and Computer Science M.I.T. September 2006 2 BELLMAN AND THE DUAL CURSES Dynamic Programming (DP) is very broadly

More information

Optimization of Design. Lecturer:Dung-An Wang Lecture 12

Optimization of Design. Lecturer:Dung-An Wang Lecture 12 Optimization of Design Lecturer:Dung-An Wang Lecture 12 Lecture outline Reading: Ch12 of text Today s lecture 2 Constrained nonlinear programming problem Find x=(x1,..., xn), a design variable vector of

More information

Formulations of Model Predictive Control. Dipartimento di Elettronica e Informazione

Formulations 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 information

Introduction to Support Vector Machines. Colin Campbell, Bristol University

Introduction to Support Vector Machines. Colin Campbell, Bristol University Introduction to Support Vector Machines Colin Campbell, Bristol University 1 Outline of talk. Part 1. An Introduction to SVMs 1.1. SVMs for binary classification. 1.2. Soft margins and multi-class classification.

More information

Scientific Computing: An Introductory Survey

Scientific Computing: An Introductory Survey Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign

More information

Machine Learning and Pattern Recognition Logistic Regression

Machine Learning and Pattern Recognition Logistic Regression Machine Learning and Pattern Recognition Logistic Regression Course Lecturer:Amos J Storkey Institute for Adaptive and Neural Computation School of Informatics University of Edinburgh Crichton Street,

More information

PID Controller Design for Nonlinear Systems Using Discrete-Time Local Model Networks

PID Controller Design for Nonlinear Systems Using Discrete-Time Local Model Networks PID Controller Design for Nonlinear Systems Using Discrete-Time Local Model Networks 4. Workshop für Modellbasierte Kalibriermethoden Nikolaus Euler-Rolle, Christoph Hametner, Stefan Jakubek Christian

More information

19 LINEAR QUADRATIC REGULATOR

19 LINEAR QUADRATIC REGULATOR 19 LINEAR QUADRATIC REGULATOR 19.1 Introduction The simple form of loopshaping in scalar systems does not extend directly to multivariable (MIMO) plants, which are characterized by transfer matrices instead

More information

Conic optimization: examples and software

Conic optimization: examples and software Conic optimization: examples and software Etienne de Klerk Tilburg University, The Netherlands Etienne de Klerk (Tilburg University) Conic optimization: examples and software 1 / 16 Outline Conic optimization

More information

Duality in General Programs. Ryan Tibshirani Convex Optimization 10-725/36-725

Duality in General Programs. Ryan Tibshirani Convex Optimization 10-725/36-725 Duality in General Programs Ryan Tibshirani Convex Optimization 10-725/36-725 1 Last time: duality in linear programs Given c R n, A R m n, b R m, G R r n, h R r : min x R n c T x max u R m, v R r b T

More information

Lecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 10

Lecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 10 Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 10 Boundary Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction

More information

Towards Dual MPC. Tor Aksel N. Heirung B. Erik Ydstie Bjarne Foss

Towards Dual MPC. Tor Aksel N. Heirung B. Erik Ydstie Bjarne Foss 4th IFAC Nonlinear Model Predictive Control Conference International Federation of Automatic Control Towards Dual MPC Tor Aksel N. Heirung B. Erik Ydstie Bjarne Foss Department of Engineering Cybernetics,

More information

constraint. Let us penalize ourselves for making the constraint too big. We end up with a

constraint. Let us penalize ourselves for making the constraint too big. We end up with a Chapter 4 Constrained Optimization 4.1 Equality Constraints (Lagrangians) Suppose we have a problem: Maximize 5, (x 1, 2) 2, 2(x 2, 1) 2 subject to x 1 +4x 2 =3 If we ignore the constraint, we get the

More information

Cyber-Security Analysis of State Estimators in Power Systems

Cyber-Security Analysis of State Estimators in Power Systems Cyber-Security 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, KTH-Royal Institute

More information

Linear Models for Classification

Linear Models for Classification Linear Models for Classification Sumeet Agarwal, EEL709 (Most figures from Bishop, PRML) Approaches to classification Discriminant function: Directly assigns each data point x to a particular class Ci

More information

Dynamic Real-time Optimization with Direct Transcription and NLP Sensitivity

Dynamic Real-time Optimization with Direct Transcription and NLP Sensitivity Dynamic Real-time Optimization with Direct Transcription and NLP Sensitivity L. T. Biegler, R. Huang, R. Lopez Negrete, V. Zavala Chemical Engineering Department Carnegie Mellon University Pittsburgh,

More information

Discrete Optimization

Discrete Optimization Discrete Optimization [Chen, Batson, Dang: Applied integer Programming] Chapter 3 and 4.1-4.3 by Johan Högdahl and Victoria Svedberg Seminar 2, 2015-03-31 Todays presentation Chapter 3 Transforms using

More information

Discrete mechanics, optimal control and formation flying spacecraft

Discrete mechanics, optimal control and formation flying spacecraft Discrete mechanics, optimal control and formation flying spacecraft Oliver Junge Center for Mathematics Munich University of Technology joint work with Jerrold E. Marsden and Sina Ober-Blöbaum partially

More information

Introduction to the Finite Element Method (FEM)

Introduction to the Finite Element Method (FEM) Introduction to the Finite Element Method (FEM) ecture First and Second Order One Dimensional Shape Functions Dr. J. Dean Discretisation Consider the temperature distribution along the one-dimensional

More information

Model Predictive Control: Basic Concepts by A. Bemporad Controllo di Processo e dei Sistemi di Produzione A.a. 2008/09 1/94

Model Predictive Control: Basic Concepts by A. Bemporad Controllo di Processo e dei Sistemi di Produzione A.a. 2008/09 1/94 Model Predictive Control: Basic Concepts 2009 by A. Bemporad Controllo di Processo e dei Sistemi di Produzione A.a. 2008/09 1/94 Model Predictive Control (MPC) model based optimizer process reference input

More information

CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XVI - Fault Accomodation Using Model Predictive Methods - Jovan D. Bošković and Raman K.

CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XVI - Fault Accomodation Using Model Predictive Methods - Jovan D. Bošković and Raman K. FAULT ACCOMMODATION USING MODEL PREDICTIVE METHODS Scientific Systems Company, Inc., Woburn, Massachusetts, USA. Keywords: Fault accommodation, Model Predictive Control (MPC), Failure Detection, Identification

More information

Robust Optimization for Unit Commitment and Dispatch in Energy Markets

Robust Optimization for Unit Commitment and Dispatch in Energy Markets 1/41 Robust Optimization for Unit Commitment and Dispatch in Energy Markets Marco Zugno, Juan Miguel Morales, Henrik Madsen (DTU Compute) and Antonio Conejo (Ohio State University) email: mazu@dtu.dk Modeling

More information

Optimal Control. Lecture Notes. February 13, 2009. Preliminary Edition

Optimal Control. Lecture Notes. February 13, 2009. Preliminary Edition Lecture Notes February 13, 29 Preliminary Edition Department of Control Engineering, Institute of Electronic Systems Aalborg University, Fredrik Bajers Vej 7, DK-922 Aalborg Ø, Denmark Page II of IV Preamble

More information

Support Vector Machines for Classification and Regression

Support Vector Machines for Classification and Regression UNIVERSITY OF SOUTHAMPTON Support Vector Machines for Classification and Regression by Steve R. Gunn Technical Report Faculty of Engineering, Science and Mathematics School of Electronics and Computer

More information

2.3 Convex Constrained Optimization Problems

2.3 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 information

An optimisation framework for determination of capacity in railway networks

An optimisation framework for determination of capacity in railway networks CASPT 2015 An optimisation framework for determination of capacity in railway networks Lars Wittrup Jensen Abstract Within the railway industry, high quality estimates on railway capacity is crucial information,

More information

DELFT UNIVERSITY OF TECHNOLOGY

DELFT UNIVERSITY OF TECHNOLOGY DELFT UNIVERSITY OF TECHNOLOGY REPORT 12-06 ADJOINT SENSITIVITY IN ELECTRICAL IMPEDANCE TOMOGRAPHY USING COMSOL MULTIPHYSICS W. MULCKHUYSE, D. LAHAYE, A. BELITSKAYA ISSN 1389-6520 Reports of the Department

More information

Optimization Modeling for Mining Engineers

Optimization Modeling for Mining Engineers Optimization Modeling for Mining Engineers Alexandra M. Newman Division of Economics and Business Slide 1 Colorado School of Mines Seminar Outline Linear Programming Integer Linear Programming Slide 2

More information

Nonlinear Programming Methods.S2 Quadratic Programming

Nonlinear Programming Methods.S2 Quadratic Programming Nonlinear Programming Methods.S2 Quadratic Programming Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard A linearly constrained optimization problem with a quadratic objective

More information

Lecture notes for the course Advanced Control of Industrial Processes. Morten Hovd Institutt for Teknisk Kybernetikk, NTNU

Lecture notes for the course Advanced Control of Industrial Processes. Morten Hovd Institutt for Teknisk Kybernetikk, NTNU Lecture notes for the course Advanced Control of Industrial Processes Morten Hovd Institutt for Teknisk Kybernetikk, NTNU November 3, 2009 2 Contents 1 Introduction 9 1.1 Scope of note..............................

More information

Linear Optimal Control. How does this guy remain upright?

Linear Optimal Control. How does this guy remain upright? Linear Optimal Control How does this guy remain upright? Overview 1. expressing a linear system in state space form 2. discrete time linear optimal control (LQR) 3. linearizing around an operating point

More information

Lecture 3: Linear methods for classification

Lecture 3: Linear methods for classification Lecture 3: Linear methods for classification Rafael A. Irizarry and Hector Corrada Bravo February, 2010 Today we describe four specific algorithms useful for classification problems: linear regression,

More information

Linear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc.

Linear 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 information

Statistical Machine Learning

Statistical Machine Learning Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes

More information

Las Vegas and Monte Carlo Randomized Algorithms for Systems and Control

Las Vegas and Monte Carlo Randomized Algorithms for Systems and Control Las Vegas and Monte Carlo Randomized Algorithms for Systems and Control Roberto Tempo IEIIT-CNR Politecnico di Torino roberto.tempo@polito.it BasarFest,, Urbana RT 2006 1 CSL UIUC Six months at CSL in

More information

Two-Stage Stochastic Linear Programs

Two-Stage Stochastic Linear Programs Two-Stage Stochastic Linear Programs Operations Research Anthony Papavasiliou 1 / 27 Two-Stage Stochastic Linear Programs 1 Short Reviews Probability Spaces and Random Variables Convex Analysis 2 Deterministic

More information

Uncertainty modeling revisited: What if you don t know the probability distribution?

Uncertainty modeling revisited: What if you don t know the probability distribution? : What if you don t know the probability distribution? Hans Schjær-Jacobsen Technical University of Denmark 15 Lautrupvang, 2750 Ballerup, Denmark hschj@dtu.dk Uncertain input variables Uncertain system

More information

A Simple Introduction to Support Vector Machines

A Simple Introduction to Support Vector Machines A Simple Introduction to Support Vector Machines Martin Law Lecture for CSE 802 Department of Computer Science and Engineering Michigan State University Outline A brief history of SVM Large-margin linear

More information

An Overview Of Software For Convex Optimization. Brian Borchers Department of Mathematics New Mexico Tech Socorro, NM 87801 borchers@nmt.

An Overview Of Software For Convex Optimization. Brian Borchers Department of Mathematics New Mexico Tech Socorro, NM 87801 borchers@nmt. An Overview Of Software For Convex Optimization Brian Borchers Department of Mathematics New Mexico Tech Socorro, NM 87801 borchers@nmt.edu In fact, the great watershed in optimization isn t between linearity

More information

Summary of specified general model for CHP system

Summary of specified general model for CHP system Fakulteta za Elektrotehniko Eva Thorin, Heike Brand, Christoph Weber Summary of specified general model for CHP system OSCOGEN Deliverable D1.4 Contract No. ENK5-CT-2000-00094 Project co-funded by the

More information

Regression Using Support Vector Machines: Basic Foundations

Regression Using Support Vector Machines: Basic Foundations Regression Using Support Vector Machines: Basic Foundations Technical Report December 2004 Aly Farag and Refaat M Mohamed Computer Vision and Image Processing Laboratory Electrical and Computer Engineering

More information

AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS

AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS Revised Edition James Epperson Mathematical Reviews BICENTENNIAL 0, 1 8 0 7 z ewiley wu 2007 r71 BICENTENNIAL WILEY-INTERSCIENCE A John Wiley & Sons, Inc.,

More information

Tetris: Experiments with the LP Approach to Approximate DP

Tetris: Experiments with the LP Approach to Approximate DP Tetris: Experiments with the LP Approach to Approximate DP Vivek F. Farias Electrical Engineering Stanford University Stanford, CA 94403 vivekf@stanford.edu Benjamin Van Roy Management Science and Engineering

More information

Cheng Soon Ong & Christfried Webers. Canberra February June 2016

Cheng Soon Ong & Christfried Webers. Canberra February June 2016 c Cheng Soon Ong & Christfried Webers Research Group and College of Engineering and Computer Science Canberra February June (Many figures from C. M. Bishop, "Pattern Recognition and ") 1of 31 c Part I

More information

Clustering and scheduling maintenance tasks over time

Clustering and scheduling maintenance tasks over time Clustering and scheduling maintenance tasks over time Per Kreuger 2008-04-29 SICS Technical Report T2008:09 Abstract We report results on a maintenance scheduling problem. The problem consists of allocating

More information

Lecture 13 Linear quadratic Lyapunov theory

Lecture 13 Linear quadratic Lyapunov theory EE363 Winter 28-9 Lecture 13 Linear quadratic Lyapunov theory the Lyapunov equation Lyapunov stability conditions the Lyapunov operator and integral evaluating quadratic integrals analysis of ARE discrete-time

More information

Optimization of warehousing and transportation costs, in a multiproduct multi-level supply chain system, under a stochastic demand

Optimization of warehousing and transportation costs, in a multiproduct multi-level supply chain system, under a stochastic demand Int. J. Simul. Multidisci. Des. Optim. 4, 1-5 (2010) c ASMDO 2010 DOI: 10.1051/ijsmdo / 2010001 Available online at: http://www.ijsmdo.org Optimization of warehousing and transportation costs, in a multiproduct

More information

Stochastic control of HVAC systems: a learning-based approach. Damiano Varagnolo

Stochastic control of HVAC systems: a learning-based approach. Damiano Varagnolo Stochastic control of HVAC systems: a learning-based approach Damiano Varagnolo Something about me 2 Something about me Post-Doc at KTH Post-Doc at U. Padova Visiting Scholar at UC Berkeley Ph.D. Student

More information

Advanced Lecture on Mathematical Science and Information Science I. Optimization in Finance

Advanced Lecture on Mathematical Science and Information Science I. Optimization in Finance Advanced Lecture on Mathematical Science and Information Science I Optimization in Finance Reha H. Tütüncü Visiting Associate Professor Dept. of Mathematical and Computing Sciences Tokyo Institute of Technology

More information

International Doctoral School Algorithmic Decision Theory: MCDA and MOO

International Doctoral School Algorithmic Decision Theory: MCDA and MOO International Doctoral School Algorithmic Decision Theory: MCDA and MOO Lecture 2: Multiobjective Linear Programming Department of Engineering Science, The University of Auckland, New Zealand Laboratoire

More information

CHAPTER 3 SECURITY CONSTRAINED OPTIMAL SHORT-TERM HYDROTHERMAL SCHEDULING

CHAPTER 3 SECURITY CONSTRAINED OPTIMAL SHORT-TERM HYDROTHERMAL SCHEDULING 60 CHAPTER 3 SECURITY CONSTRAINED OPTIMAL SHORT-TERM HYDROTHERMAL SCHEDULING 3.1 INTRODUCTION Optimal short-term hydrothermal scheduling of power systems aims at determining optimal hydro and thermal generations

More information

SIMS 2015 Plenary. Accomplishing Ground Moving Innovations through Modeling, Simulation, and Optimal Control

SIMS 2015 Plenary. Accomplishing Ground Moving Innovations through Modeling, Simulation, and Optimal Control SIMS 2015 Plenary Accomplishing Ground Moving Innovations through Modeling, Simulation, and Optimal Control Lars Eriksson lars.eriksson@liu.se Professor Division of Vehicular Systems Department of Electrical

More information

Numerical Analysis Introduction. Student Audience. Prerequisites. Technology.

Numerical Analysis Introduction. Student Audience. Prerequisites. Technology. Numerical Analysis Douglas Faires, Youngstown State University, (Chair, 2012-2013) Elizabeth Yanik, Emporia State University, (Chair, 2013-2015) Graeme Fairweather, Executive Editor, Mathematical Reviews,

More information

Real-Time Embedded Convex Optimization

Real-Time Embedded Convex Optimization Real-Time Embedded Convex Optimization Stephen Boyd joint work with Michael Grant, Jacob Mattingley, Yang Wang Electrical Engineering Department, Stanford University Spencer Schantz Lecture, Lehigh University,

More information

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 1

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 1 This article has been accepted for inclusion in a future issue of this journal Content is final as presented, with the exception of pagination IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 1 An Improved

More information

Inner Product Spaces

Inner Product Spaces Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

More information

17.3.1 Follow the Perturbed Leader

17.3.1 Follow the Perturbed Leader CS787: Advanced Algorithms Topic: Online Learning Presenters: David He, Chris Hopman 17.3.1 Follow the Perturbed Leader 17.3.1.1 Prediction Problem Recall the prediction problem that we discussed in class.

More information

Optimal Control. Lecture 3. Palle Andersen, Aalborg University. Opt lecture 3 p. 1/30

Optimal Control. Lecture 3. Palle Andersen, Aalborg University. Opt lecture 3 p. 1/30 Optimal Control Lecture 3 pa@control.aau.dk Palle Andersen, Aalborg University Opt lecture 3 p. 1/30 Stochastic Optimal Control In this lecture we are going to introduce disturbances which are modelled

More information

MODEL REFERENCE CONTROL IN INVENTORY AND SUPPLY CHAIN MANAGEMENT The implementation of a more suitable cost function

MODEL REFERENCE CONTROL IN INVENTORY AND SUPPLY CHAIN MANAGEMENT The implementation of a more suitable cost function MODEL REFERECE COTROL I IVETORY AD SUPPLY CHAI MAAGEMET The implementation of a more suitable cost function Heikki Rasku, Juuso Rantala, Hannu Koivisto Institute of Automation and Control, Tampere University

More information

Convex Optimization SVM s and Kernel Machines

Convex Optimization SVM s and Kernel Machines Convex Optimization SVM s and Kernel Machines S.V.N. Vishy Vishwanathan vishy@axiom.anu.edu.au National ICT of Australia and Australian National University Thanks to Alex Smola and Stéphane Canu S.V.N.

More information

Understanding and Applying Kalman Filtering

Understanding and Applying Kalman Filtering Understanding and Applying Kalman Filtering Lindsay Kleeman Department of Electrical and Computer Systems Engineering Monash University, Clayton 1 Introduction Objectives: 1. Provide a basic understanding

More information

Partial Fractions. (x 1)(x 2 + 1)

Partial Fractions. (x 1)(x 2 + 1) Partial Fractions Adding rational functions involves finding a common denominator, rewriting each fraction so that it has that denominator, then adding. For example, 3x x 1 3x(x 1) (x + 1)(x 1) + 1(x +

More information

1 Review of Least Squares Solutions to Overdetermined Systems

1 Review of Least Squares Solutions to Overdetermined Systems cs4: introduction to numerical analysis /9/0 Lecture 7: Rectangular Systems and Numerical Integration Instructor: Professor Amos Ron Scribes: Mark Cowlishaw, Nathanael Fillmore Review of Least Squares

More information

MOVING HORIZON ESTIMATION FOR AN INDUSTRIAL GAS PHASE POLYMERIZATION REACTOR. Jasmeer Ramlal Kenneth V. Allsford John D.

MOVING HORIZON ESTIMATION FOR AN INDUSTRIAL GAS PHASE POLYMERIZATION REACTOR. Jasmeer Ramlal Kenneth V. Allsford John D. MOVING HORIZON ESTIMATION FOR AN INDUSTRIAL GAS PHASE POLYMERIZATION REACTOR Jasmeer Ramlal Kenneth V. Allsford John D. Hedengren, Sasol Polymers, 56 Grosvenor Road, Bryanston, Randburg, South Africa 225

More information

itesla Project Innovative Tools for Electrical System Security within Large Areas

itesla Project Innovative Tools for Electrical System Security within Large Areas itesla Project Innovative Tools for Electrical System Security within Large Areas Samir ISSAD RTE France samir.issad@rte-france.com PSCC 2014 Panel Session 22/08/2014 Advanced data-driven modeling techniques

More information

Minimize subject to. x S R

Minimize subject to. x S R Chapter 12 Lagrangian Relaxation This chapter is mostly inspired by Chapter 16 of [1]. In the previous chapters, we have succeeded to find efficient algorithms to solve several important problems such

More information

CONSTRAINED NONLINEAR PROGRAMMING

CONSTRAINED NONLINEAR PROGRAMMING 149 CONSTRAINED NONLINEAR PROGRAMMING We now turn to methods for general constrained nonlinear programming. These may be broadly classified into two categories: 1. TRANSFORMATION METHODS: In this approach

More information

Applications to Data Smoothing and Image Processing I

Applications to Data Smoothing and Image Processing I Applications to Data Smoothing and Image Processing I MA 348 Kurt Bryan Signals and Images Let t denote time and consider a signal a(t) on some time interval, say t. We ll assume that the signal a(t) is

More information

11. Nonlinear equations with one variable

11. Nonlinear equations with one variable EE103 (Fall 2011-12) 11. Nonlinear equations with one variable definition and examples bisection method Newton s method secant method 11-1 Definition and examples x is a zero (or root) of a function f

More information

Big Data - Lecture 1 Optimization reminders

Big Data - Lecture 1 Optimization reminders Big Data - Lecture 1 Optimization reminders S. Gadat Toulouse, Octobre 2014 Big Data - Lecture 1 Optimization reminders S. Gadat Toulouse, Octobre 2014 Schedule Introduction Major issues Examples Mathematics

More information

Math 5311 Gateaux differentials and Frechet derivatives

Math 5311 Gateaux differentials and Frechet derivatives Math 5311 Gateaux differentials and Frechet derivatives Kevin Long January 26, 2009 1 Differentiation in vector spaces Thus far, we ve developed the theory of minimization without reference to derivatives.

More information

Self-Tuning Memory Management of A Database System

Self-Tuning Memory Management of A Database System Self-Tuning Memory Management of A Database System Yixin Diao diao@us.ibm.com IM 2009 Tutorial: Recent Advances in the Application of Control Theory to Network and Service Management DB2 Self-Tuning Memory

More information

Mathematical finance and linear programming (optimization)

Mathematical finance and linear programming (optimization) Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may

More information

Decentralization and Private Information with Mutual Organizations

Decentralization and Private Information with Mutual Organizations Decentralization and Private Information with Mutual Organizations Edward C. Prescott and Adam Blandin Arizona State University 09 April 2014 1 Motivation Invisible hand works in standard environments

More information

Identification of Hybrid Systems

Identification of Hybrid Systems Identification of Hybrid Systems Alberto Bemporad Dip. di Ingegneria dell Informazione Università degli Studi di Siena bemporad@dii.unisi.it http://www.dii.unisi.it/~bemporad Goal Sometimes a hybrid model

More information

Principles of Scientific Computing Nonlinear Equations and Optimization

Principles of Scientific Computing Nonlinear Equations and Optimization Principles of Scientific Computing Nonlinear Equations and Optimization David Bindel and Jonathan Goodman last revised March 6, 2006, printed March 6, 2009 1 1 Introduction This chapter discusses two related

More information

Operation of Manufacturing Systems with Work-in-process Inventory and Production Control

Operation of Manufacturing Systems with Work-in-process Inventory and Production Control Operation of Manufacturing Systems with Work-in-process Inventory and Production Control Yuan-Hung (Kevin) Ma, Yoram Koren (1) NSF Engineering Research Center for Reconfigurable Manufacturing Systems,

More information

Introduction to Process Optimization

Introduction to Process Optimization Chapter 1 Introduction to Process Optimization Most things can be improved, so engineers and scientists optimize. While designing systems and products requires a deep understanding of influences that achieve

More information

24. The Branch and Bound Method

24. 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 NP-complete. Then one can conclude according to the present state of science that no

More information

The Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Line-search Method

The Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Line-search Method The Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Line-search Method Robert M. Freund February, 004 004 Massachusetts Institute of Technology. 1 1 The Algorithm The problem

More information

Two Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering

Two Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering Two Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering Department of Industrial Engineering and Management Sciences Northwestern University September 15th, 2014

More information