MPC and constrained systems

A general methodology

Model predictive control is a form of control in which the current control action is obtained by solving, at each sampling instant, a finite horizon open-loop optimal control problem, using the current state of the plant as the initial state; the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant. An important advantage of this type of control is its ability to cope with hard constraints on controls and states. It has, therefore, been widely applied in petro-chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls.

Its general formulation allows the application of the MPC control strategy to linear, nonlinear and hybrid systems, although the optimization problems to be solved on-line become increasingly complex (QP/LP, nonlinear optimization, and MIQP/MILP, respectively). Especially for hybrid systems, many fundamental properties of MPC have to be carefully re-examined. In a STW project on "MPC for hybrid systems" we studied these issues in detail, e.g. questions related to how to setup MPC schemes that guarantee attractivity, Lyapunov stability, robust stability (ISS), etc., for hybrid systems have been of interest.

Interestingly, for some classes of MPC strategies (linear systems with linear/quadratic costs and piecewise affine systems with linear costs) the optimal control law can also be expressed as piecewise affine functions of the state. For this so-called "explicit MPC" one no longer has to solve the optimization problems online. All this is appealing for various applications (e.g., for certification), the overall piecewise affine control law might be rather complex. Therefore, there recently was a major effort on approximate/suboptimal (explicit) MPC to obtain PWA control laws close to optimal, but much simpler in its implementation. We worked on this in the EU-FP7 project MOBYDIC for instance.

Our ideas have, amongst others, been applied to the design of adaptive cruise controllers and power converters.

Related Publications

2018

  • Robust Event-triggered MPC With Guaranteed Asymptotic Bound and Average Sampling Rate

    2018
    F.D. Brunner, W.P.M.H. Heemels, F. Allgower
    IEEE Transactions on Automatic Control

2017

  • Resource-aware MPC for constrained linear systems: Two rollout approaches

    2017
    T.M.P. Gommans, D. Antunes, W.P.M.H. Heemels
    Journal of Process Control 51, p. 68–83

  • Optimal Control for Integrated Emission Management in Diesel Engines

    2017
    M.C.F. Donkers, J. van Schijndel, W.P.M.H. Heemels, F.P.T. Willems
    Control Engineering Practice 61, p. 206–216

  • Optimal control of a wave energy converter

    August, 2017
    B Hendrikx, J. Leth, P. Andersen, W.P.M.H. Heemels
    Conference on Control Technology and Applications (CCTA) 2017, Hawaii

2016

  • Model predictive control algorithm for large-area regional hyperthermia

    2016
    L. Sebeke, Xi Luo, B. de Jager, W.P.M.H. Heemels, E. Heijman, H. Gruell
    Journal of Therapeutic Ultrasound 4, p. A173

  • Robust Self-Triggered MPC for Constrained Linear Systems: A Tube-Based Approach

    2016
    F.D. Brunner, W.P.M.H. Heemels, F. Allgower
    Automatica 72, p. 73-83

  • Numerical Evaluation of a Robust Self-Triggered MPC Algorithm

    2016
    F.D. Brunner, W.P.M.H. Heemels, F. Allgower
    6th IFAC Workshop on Distributed Estimation and Control in Networked Systems (NECSYS) 2016, Tokio, Japan

2015

  • Resource-aware MPC for Constrained Nonlinear Systems: A Self-Triggered Control Approach

    May, 2015
    T.M.P. Gommans and W.P.M.H. Heemels
    Systems and Control Letters 79, p. 59-67

  • Communication Scheduling in Robust Self-Triggered MPC for Linear Discrete-Time Systems

    2015
    F.B. Brunner, T.M.P. Gommans, W.P.M.H. Heemels, F. Allgower
    IFAC Workshop on Distributed Estimation and Control in Networked Systems (NECSYS) 2015, Philadelphia, USA "(IFAC-PapersOnLine, Volume 48, Issue 22, 2015, Pages 132-137)", p. 132-137

  • Set-Based MPC with an Application to Enhanced Local Hyperthermia for Cancer Treatment

    2015
    Xi Luo, B. de Jager, E. Heijman, W.P.M.H. Heemels
    IFAC Conference on Nonlinear Model Predictive Control (NMPC) 2015, Seville, Spain "(IFAC-PapersOnLine, Volume 48, Issue 23, 2015, Pages 477-482)", p. 477-482

  • Robust Event-Triggered MPC for Constrained Linear Discrete-Time Systems with Guaranteed Average Sampling Rate

    2015
    F.D. Brunner, W.P.M.H. Heemels, F. Allgower
    IFAC Conference on Nonlinear Model Predictive Control (NMPC) 2015, Seville, Spain "(IFAC-PapersOnLine, Volume 48, Issue 23, 2015, Pages 117-122)", p. 117-122

  • Modeling, observer design and robust control of the particle density profile in tokamak plasmas

    2015
    T.C. Blanken, F. Felici, M. De Baar, W.P.M.H. Heemels
    IEEE Conference on Decision and Control (CDC) 2015, Osaka, Japan, p. 7628 - 7635

  • Robust Self-Triggered Model Predictive Control for Constrained Discrete-Time LTI Systems based on Homothetic Tubes

    2015
    E. Aydiner, F.D. Brunner, W P.M.H. Heemels, F. Allgower
    European Control Conference (ECC), 2015, Linz, Austria, p. 1587-1593

2014

  • Stabilizing Dynamic Control Design for Hybrid Systems: A Hybrid Control Lyapunov Function Approach

    May, 2014
    S. Di Cairano, W.P.M.H. Heemels, M. Lazar, A. Bemporad
    IEEE Transactions on Automatic Control 59(10), p. 2629-2643

  • Optimal cost-NOx trade-off in Diesel Engines by Integrated Emission Management

    June, 2014
    J. van Schijndel, M.C.F. Donkers, F.P.T. Willems, W.P.M.H. Heemels
    FISITA 2014, Maastricht

  • Dynamic Programming for Integrated Emission Management in Diesel Engines

    August, 2014
    J.C.A. van Schijndel, M.C.F. Donkers, F. Willems and W.P.M.H. Heemels
    IFAC World Congress 2014, Cape Town, South Africa, p. 11860-11865

  • Robust Self-Triggered MPC for Constrained Linear Systems

    June, 2014
    F.D. Brunner, W.P.M.H. Heemels and F. Allgower
    European Control Conference (ECC) 2014, Strasbourg, France, p. 472-477

2013

  • Further Input-to-State Stability Subtleties for Discrete-time Systems

    June, 2013
    M. Lazar, W.P.M.H. Heemels, and A.R. Teel
    IEEE Transactions on Automatic Control 58(6), p. 1609-1613

  • Hybrid Control Lyapunov Functions for Stabilization of Hybrid Systems

    April, 2013
    S. Di Cairano, W.P.M.H. Heemels, M. Lazar and A. Bemporad
    Hybrid Systems: Computation and Control (HSCC) 2013, Philadelphia, USA, p. 73-82

2012

  • FPGA implementations of piecewise affine functions based on multi-resolution hyper-rectangular partitions

    December, 2012
    F. Comaschi, B.A.G. Genuit, A. Oliveri, W.P.M.H. Heemels and M. Storace
    IEEE Transactions on Circuits and Systems 59(12), p. 2920-2933

  • Approximation of Explicit MPC Using Regular PWA Functions: An ISS Approach

    May, 2012
    B.A.G. Genuit, W.P.M.H. Heemels and L. Lu
    IET Control Theory & Applications "Special Issue: Implementation of Feedback Controllers on Special-Purpose Hardware" 6(8), p. 1015-1028

  • Linear Control of Time Domain Constrained Systems

    May, 2012
    W. Aangenent, W.P.M.H. Heemels, M.J.G. van de Molengraft, D. Henrion, and M. Steinbuch
    Automatica 48(5), p. 736-746

  • A Unified Numerical Scheme for Linear-Quadratic Optimal Control Problems with Joint Control and State Constraints

    September, 2012
    L. Han, M.K. Camlibel, J.-S. Pang and W.P.M.H. Heemels
    Optimization Methods & Software "Special Issue" 27(4-5), p. 761-799

  • MOBY-DIC: A MATLAB Toolbox for circuit-oriented design of explicit MPC

    August, 2012
    A. Oliveri, D. Barcelli, A. Bemporad, B.A.G. Genuit, W.P.M.H. Heemels, T. Poggi, M. Rubagotti and M. Storace
    IFAC Conference on Nonlinear Model Predictive Control 2012, p. 218-225

  • Self-triggered MPC for constrained linear systems and quadratic costs

    August, 2012
    J.D.J. Barradas Berglind, T.M.P. Gommans and W.P.M.H. Heemels
    IFAC Conference on Nonlinear Model Predictive Control 2012, p. 342-348

2011

  • Approximation of PWA Control Laws Using Regular Partitions: An ISS Approach

    September, 2011
    B.A.G. Genuit, L. Lu and W.P.M.H. Heemels
    IFAC Word Conference 2011, Milano, Italy, p. 4540-4545

  • Synthesis of low-complexity stabilizing piecewise affine controllers: A control-Lyapunov function approach

    December, 2011
    L. Lu, W.P.M.H. Heemels and A. Bemporad
    IEEE Conference on Decision and Control (CDC) 2011, Orlando, USA, p. 1227-1232

  • Low-complexity Approximations of PWA Functions: a Case Study on Adaptive Cruise Control

    August, 2011
    A. Oliveri, G.J.L. Naus, M. Storace and W.P.M.H. Heemels
    European conference on circuit theory and design (ECCTD) 2011, Linköping, Sweden, p. 694-697

2010

  • Design and implementation of parameterized adaptive cruise control: An explicit model predictive control approach

    April, 2010
    G.J.L. Naus, J. Ploeg, M.J.G. van de Molengraft, W.P.M.H. Heemels and M. Steinbuch
    Control Engineering Practice 18(8), p. 882-892

  • Reference Governors for Controlled Belt Restraint Systems

    2010
    E.P. van der Laan, W.P.M.H. Heemels, H. Luijten, F.E. Veldpaus and M. Steinbuch
    Vehicle System Dynamics 48(7), p. 831-850

  • On Integration of Event-based Estimation and Robust MPC in a Feedback Loop

    April, 2010
    J. Sijs, M. Lazar and W.P.M.H. Heemels
    Hybrid Systems: Computation and Control (HSCC) 2010, Stockholm, Sweden, p. 31-41

  • Convergence of Discrete-time Approximations of Constrained Linear-Quadratic Optimal Control Problems

    December, 2010
    L. Han, M.K. Camlibel, J.-S. Pang and W.P.M.H. Heemels
    IEEE Conference on Decision and Control (CDC) 2010, Atlanta, USA, p. 5210-5215

  • A Model Predictive Control Approach for Stochastic Networked Control Systems

    September, 2010
    D. Bernardini, M.C.F. Donkers, A. Bemporad and W.P.M.H. Heemels
    IFAC Workshop on Distributed Estimation and Control in Networked Systems (NECSYS), Annecy, France, p. 7-12

  • A Model Predictive Control Approach to Design a Parameterized Adaptive Cruise Control

    2010
    G.J.L. Naus, J. Ploeg, M.J.G. van de Molengraft, W.P.M.H. Heemels and M. Steinbuch
    in "Automotive Model Predictive Control Models: Methods and Applications", Springer 17, p. 273-284

2009

  • Predictive control of hybrid systems: Input-to-state stability results for sub-optimal solutions

    January, 2009
    M. Lazar and W.P.M.H. Heemels
    Automatica 45(1), p. 180-185

  • Linear Control of Time-Domain Constrained Systems

    December, 2009
    W.H.T.M. Aangenent, W.P.M.H. Heemels, M.J.G. van de Molengraft and M. Steinbuch
    IEEE Conference on Decision and Control (CDC) 2009, Shanghi, China, p. 5339-5344

  • Further results on "Robust MPC using Linear Matrix Inequalities"

    2009
    M. Lazar, W.P.M.H. Heemels, D. Munoz de la Pena and T. Alamo
    in "Assessment and Future Directions of Nonlinear Model Predictive Control, Lecture Notes in Control and Information Sciences", Springer 384, p. 89-98

  • Self-optimizing Robust Nonlinear Model Predictive Control

    2009
    M. Lazar, W.P.M.H. Heemels and A. Jokic
    in "Assessment and Future Directions of Nonlinear Model Predictive Control, Lecture Notes in Control and Information Sciences", Springer 384, p. 27-40

2008

  • On input-to-state stability of Min-max Nonlinear Model Predictive Control

    January, 2008
    M. Lazar, D. Munoz de la Pena, W.P.M.H. Heemels and T. Alamo
    Systems and Control Letters 57(1), p. 39-48

  • Input-to-state stabilizing sub-optimal NMPC with an application to DC-DC converters

    April, 2008
    M. Lazar, W.P.M.H. Heemels, B.J.P. Roset, H. Nijmeijer and P.P.J. v.d. Bosch
    International Journal of Robust and Nonlinear Control 18(8), p. 890-904

  • On robustness of constrained discrete-time systems to state measurement errors

    April, 2008
    B.J.P. Roset, W.P.M.H. Heemels, M. Lazar and H. Nijmeijer
    Automatica 44(4), p. 1161-1165

  • Design of ISS-Lyapunov functions for discrete-time linear uncertain systems

    July, 2008
    T. Alamo, D. Munoz de la Pena, M. Lazar and W.P.M.H. Heemels
    IFAC World Congress 2008, Seoul, Korea

  • Further results on robust MPC using linear matrix inequalities

    2008
    M. Lazar, W.P.M.H Heemels, D. Munoz de la Pena and T. Alamo
    International Workshop on Assessment and Future Directions of Nonlinear MPC 2008, Pavia, Italy

  • Optimized input-to-state stabilization of discrete-time nonlinear systems with bounded inputs

    June, 2008
    M. Lazar and W.P.M.H. Heemels
    American Control Conference (ACC) 2008, Seattle, USA, p. 2310-2315

  • Reference governors for controlled belt restraint systems

    September, 2008
    E.P. van der Laan, H.J.C. Luijten, F.E. Veldpaus, W.P.M.H. Heemels and M. Steinbuch
    International Conference on Vehicular Electronics and Safety (ICVES) in Columbus, Ohio, USA, p. 114-119

  • Null controllability of discrete-time linear systems with input and state constraints

    December, 2008
    W.P.M.H. Heemels and M.K. Camlibel
    IEEE Conference on Decision and Control (CDC) 2008, Cancun, Mexico, p. 3487-3492

  • Predictive control of hybrid systems: Stability results for suboptimal solutions

    July, 2008
    M. Lazar and W.P.M.H. Heemels
    IFAC World Congress 2008, Seoul, Korea, p. 11196-1120

  • Self-optimizing robust nonlinear model predictive control

    2008
    M. Lazar, W.P.M.H Heemels and A. Jokic
    International Workshop on Assessment and Future Directions of Nonlinear MPC 2008, Pavia, Italy

  • A Control Lyapunov Approach to Predictive Control of Hybrid Systems

    2008
    S. Di Cairano, M. Lazar, A. Bemporad and W.P.M.H. Heemels
    in "Lecture Notes in Computer Science. 11th International Workshop on Hybrid Systems: Computation and Control 2008, St. Louis, USA", Springer, p. 130-143

  • Nonlinear Control of Linear Motion Systems: An Exploratory Study

    2008
    W.H.T.M. Aangenent
    Eindhoven University of Technology "PhD Thesis"

2007

  • Squaring the circle: An algorithm for generating polyhedral invariant sets from ellipsoidal ones

    December, 2007
    A. Alessio, M. Lazar, A. Bemporad and W.P.M.H. Heemels
    Automatica 43(12), p. 2096-2103

  • New methods for computing the terminal cost for min-max model predictive control

    June, 2007
    M. Lazar, D. Munoz de la Pena, W.P.M.H. Heemels and T. Alamo
    American Control Conference (ACC) 2007, New York, USA, p. 4476-4481

  • H2 performance analysis of reset control systems

    December, 2007
    G. Witvoet, W.H.T.M. Aangenent, W.P.M.H. Heemels, M.J.G. van de Molengraft and M. Steinbuch
    IEEE Conference on Decision and Control (CDC) 2007, New Orleans, USA, p. 3278-3284

  • Assessment of nonlinear predictive control techniques for DC-DC converters

    August, 2007
    A.G. Beccuti, M. Lazar, W.P.M.H. Heemels, G. Papafotiou and M. Morari
    NOLCOS 2007, Pretoria, South-Africa

  • Controllability of linear systems with input and state constraints

    December, 2007
    W.P.M.H. Heemels and M.K. Camlibel
    IEEE Conference on Decision and Control (CDC) 2007, New Orleans, USA, p. 536-541

  • Design of stabilizing output feedback nonlinear model predictive controllers with an application to DC-DC converters

    July, 2007
    B.J.P. Roset, M. Lazar, W.P.M.H. Heemels and H. Nijmeijer
    European Control Conference (ECC) 2007, Kos, Greece, p. 2999-3004

  • Discrete-time non-smooth nonlinear MPC: Stability and input-to-state stability

    2007
    M. Lazar, W.P.M.H. Heemels, A. Bemporad and S. Weiland
    in "Lecture Notes in Control and Information Sciences, Assessment and Future Directions of Nonlinear Model Predictive Control", Springer, p. 93-109

2006

  • Stabilizing Model Predictive Control of Hybrid Systems

    November, 2006
    M. Lazar, W.P.M.H. Heemels, S. Weiland and A. Bemporad
    IEEE Transactions on Automatic Control 51(11), p. 1813-1818

  • A hybrid MPC approach to the design of a Smart adaptive cruise controller

    October, 2006
    D. Corona, M. Lazar, B. De Schutter and W.P.M.H. Heemels
    International Conference on Control Applications (CCA) 2006, Munich, Germany, p. 231-236

  • Polyhedral invariant sets for closed-loop linear MPC systems

    December, 2006
    A. Alessio, A. Bemporad, M. Lazar and W.P.M.H. Heemels
    IEEE Conference on Decision and Control (CDC) 2006, San Diego, USA, p. 4532-4537

  • ISS sub-optimal nonlinear MPC algorithms with an application to DC-DC converters

    2006
    M. Lazar, B.J.P. Roset, W.P.M.H. Heemels, H. Nijmeijer and P.P.J. v.d. Bosch
    IFAC Workshop on Nonlinear Model Predictive Control for Fast systems 2006, Genoble, France

  • A new dual-mode hybrid MPC algorithm with a robust stability guarantee

    June, 2006
    M. Lazar and W.P.M.H. Heemels
    IFAC Conference on Analysis and Design of Hybrid Systems (ADHS) 2006, Alghero, Italy, p. 321-328

  • Min-max nonlinear model predictive control with guaranteed input-to-state stability

    October, 2006
    M. Lazar, D. Munoz de la Pena, W.P.M.H. Heemels and T. Alamo
    International Symposium on Mathematical Theory of Networks and Systems (MTNS) 2006, Kyoto, Japan

  • Stabilizing output feedback nonlinear model predictive control: An extended observer approach

    2006
    B.J.P. Roset, M. Lazar, H. Nijmeijer and W.P.M.H. Heemels
    International Symposium on Mathematical Theory of Networks and Systems (MTNS) 2006, Kyoto, Japan 2006, p. 771-781

  • A stabilizing output-based nonlinear model predictive control scheme

    December, 2006
    B.J.P. Roset, M. Lazar, W.P.M.H. Heemels and H. Nijmeijer
    IEEE Conference on Decision and Control (CDC) 2006, San Diego, USA, p. 4627-4632

  • Squaring the circle: An algorithm for generating polyhedral invariant sets from ellipsoidal ones

    June, 2006
    M. Lazar, A. Alessio, A., Bemporad and W.P.M.H. Heemels
    American Control Conference (ACC) 2006, Minneapolis, USA, p. 3007-3012

  • Model Predictive Control of Hybrid Systems: Stability and Robustness

    2006
    M. Lazar
    Eindhoven University of Technology "PhD Thesis"

2005

  • Energy Management Strategies for Vehicular Electric Power Systems

    May, 2005
    M.W.T. Koot, J.T.B.A. Kessels, A.G. de Jager, W.P.M.H. Heemels, P.P.J. van den Bosch and M. Steinbuch
    IEEE Transactions on Vehicular Technology 54(3), p. 771-782

  • On the stability and robustness of non-smooth nonlinear model predictive control

    2005
    M. Lazar, W.P.M.H. Heemels, S. Weiland and A. Bemporad
    International Workshop on Assessment and Future Directions of Nonlinear MPC 2005, Freudenstadt-Lauterbad, Germany, p. 3-10

  • Control of PWA systems using a stable receding horizon method

    July, 2005
    I. Necoara, B. De Schutter, W.P.M.H. Heemels, S. Weiland, M.. Lazar and T.T.J. van den Boom
    IFAC World Congress on Automatic Control 2006, Prague

  • On the stability of quadratic forms based model predictive control of constrained PWA systems

    June, 2005
    M. Lazar, W.P.M.H. Heemels, S. Weiland and A. Bemporad
    American Control Conference (ACC) 2005, Portland, USA, p. 575-580

  • Infinity norms as Lyapunov functions for model predictive control of constrained PWA systems

    2005
    M. Lazar, W.P.M.H. Heemels, S. Weiland, A. Bemporad and O. Pastravanu
    in "Hybrid Systems: Computation and Control 2005, Lecture Notes in Computer Science", Springer 3414, p. 417-432

2004

  • Energy management strategies for vehicle power nets

    June, 2004
    M. Koot, B. de Jager, J.T.B.A. Kessels, W.P.M.H. Heemels and P.P.J. van den Bosch
    American Control Conference (ACC) 2004, Boston, USA, p. 4072-4077

  • Stabilization conditions for model predictive control of constrained PWA systems

    December, 2004
    M. Lazar, W.P.M.H. Heemels, S. Weiland and A. Bemporad
    IEEE Conference on Decision and Control (CDC) 2004, Bahamas, p. 4595-4600

  • Stabilizing receding horizon control of piecewise linear systems

    2004
    M. Lazar, W.P.M.H. Heemels, S. Weiland and A. Bemporad
    International Symposium on Mathematical Theory of Networks and Systems (MTNS) 2004, Leuven, Belgium

2003

  • A semi-explicit MPC set-up for constrained piecewise affine systems

    2003
    M. Lazar and W.P.M.H. Heemels
    European Control Conference (ECC) 2003, Cambridge, U.K.

  • Advanced energy management strategies for vehicle power nets

    2003
    E.H.J.A. Nuijten, M.W.T. Koot, J.T.B.A. Kessels, M. Eifert, A.G. de Jager, W.P.M.H. Heemels and P.P.J. van den Bosch
    International congress on “European automotive industry driving global changes” 2003, Paris, France

2002

  • Positive feedback stabilization of centrifugal compressor surge

    February, 2002
    F.P.T. Willems, W.P.M.H. Heemels, A.G. de Jager and A.A. Stoorvogel
    Automatica 38(2), p. 311-318

  • On hybrid systems and closed-loop MPC systems

    May, 2002
    A. Bemporad, W.P.M.H. Heemels and B. De Schutter
    IEEE Transactions on Automatic Control 47(5), p. 863-869

  • Max-plus-algebraic problems and the extended linear complementarity problem — algorithmic aspects

    2002
    B. De Schutter, W.P.M.H. Heemels and A. Bemporad
    IFAC World Congress 2002, Barcelona, Spain

2001

  • On hybrid systems and closed-loop MPC systems

    December, 2001
    A. Bemporad, W.P.M.H. Heemels and B. De Schutter
    IEEE Conference on Decision and Control (CDC) 2001, Orlando, USA, p. 1645-1649

1999

  • Positive feedback stabilization of surge in a centrifugal compressor

    December, 1999
    F. Willems, W.P.M.H. Heemels, B. de Jager and A.A. Stoorvogel
    IEEE Conference on Decision and Control (CDC) 1999, Phoenix, USA, p. 3295-3300

1998

  • Linear Quadratic Regulator Problem with Positive Controls

    1998
    W.P.M.H. Heemels, S.J.L. van Eijndhoven and A.A. Stoorvogel
    International Journal of Control 70(4), p. 551-578

1997

  • Linear quadratic regulator problem with positive controls

    1997
    W.P.M.H. Heemels, S.J.L. van Eijndhoven and A.A. Stoorvogel
    European Control Conference (ECC) 1997, Brussels, Belgium

1996

  • Optimal control of a linear control system with positive inputs

    1996
    W.P.M.H. Heemels, S.J.L. van Eijndhoven and A.A. Stoorvogel
    International Symposium on Mathematical Theory of Networks and Systems (MTNS) 1996, St. Louis, USA

1995

  • Optimal positive control and optimal control with positive state entries

    1995
    W.P.M.H. Heemels
    Eindhoven University of Technology "M.S.c. thesis"