• 1. Scope of flexible multibody kinematics and dynamics. Multibody versus finite element formulations. Description of angular orientation: Euler angles, Quarternions.
• 2. Finite element representation of flexible multibody systems. Kinematical analysis: the concept of constraints, degrees of freedom and geometric transfer functions. Dynamic analysis: lumped mass formulation, consistent mass formulation, stiffness matrices, equations of motion, equations of reaction.
• 3. Linearized equations for control system analysis. Stationary and equilibrium solutions. Linearized state-space equations.
• 4. Illustrative design examples e.g. an active encoder head and a multi axes vibration isolation system.

lecture notes

Will be distributed during the course.

prerequisites

Basic background in systems modelling and control theory.

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Mathematical Models of Systems

lecturers

Dr. J.W. Polderman, University of Twente
Prof. dr. H.L. Trentelman, University of Groningen

objective

The purpose of this course is to discuss the ideas and principles behind modeling using the behavioral approach, and to apply these ideas to control system design. In the behavioral approach, dynamical models are specified in a different way than is customary in transfer function or state space models. The main difference is that it does not start with an input/output representation. Instead, models are simply viewed as relations among variables. The collection of all time trajectories which the dynamical model allows is called the behavior of the system. Specification of the behavior is the outcome of a modelling process. Models obtained from first principles are usually set-up by tearing and zooming. Thus the model consist of the laws of the subsystems on the one hand, and the interconnection laws on the other. In such a situation it is natural to distinguish between two types of variables: the manifest variables which are the variables which the model aims at and the latent variables which are auxiliary variables introduced in the modelling process. Behavioral models easily accommodate static relations in addition to the dynamic ones. A number of system representation questions occur in this framework, among others:
• the elimination of latent variables
• input/output structures
• state space representations
We also introduce some important system properties as controllability and observability in this setting. In the first part of the course, we review the main representations, their interrelations, and their basic properties. In the context of control, we view interconnection as the basic principle of design. In the to–be–controlled plant there are certain control terminals and the controller imposes additional laws on these terminal variables. Thus the controlled system has to obey the laws of both the plant and the controller. Control design procedures thus consist of algorithms which associate with a specification of the plant (for example, a kernel, an image, or a hybrid representation involving latent variables) a specification of the controller, thus passing directly from the plant model to the controller. We extensively discuss the notion of implementability as a concept to characterize the limits of performance of a plant to be controlled. We discuss how the problems of pole-placement and stabilization look like in this setting.

contents

1. General ideas. Mathematical models of systems. Dynamical systems. Examples from physics and economics. Linear time-invariant systems. Differential equations. Polynomial matrices.
2. Minimal and full row rank representation. Autonomous systems. Inputs and outputs. Equivalence of representations.
3. Differential systems with latent variables. State space models. I/S/O models.
4. Controllability. Controllable part. Observability.
5. Elimination of latent variables. Elimination of state variables.
6. From I/O to I/S/O models. Image representations.
7. Interconnection. Control in a behavioral setting. Implementability.
8. Stability. Stabilization and pole placement.

prerequisites

The course is pretty much self-contained. Basic linear algebra and calculus should suffice.

course material

The main reference is Introduction to Mathematical Systems Theory: A Behavioral Approach by J.W. Polderman and J.C. Willems (Springer 1998). The last lecture is based on a paper by M.N. Belur and H.L. Trentelman

examination

Four sets of homework exercises will be handed out during the course. The average grade of these four assignments determines the final grade. There is no final exam.

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Fuzzy and Neural Control

lecturers

Prof.dr. R. Babuska, Delft University of Technology
Prof.dr.ir. J. Hellendoorn, Siemens Nederland and Delft Center for Systems and Control, Delft University of Technology

objective

This course provides basic knowledge of fuzzy and neural (also called "intelligent") methods for the modeling and control of nonlinear systems. An overview of typical applications is given. The participants will have the opportunity to gain hands-on experience by solving Matlab/Simulink oriented assignments. While traditional control engineering methods are based on differential and difference equations, intelligent techniques employ alternative representation schemes such as fuzzy logic rules, which can incorporate human knowledge and deductive processes, or artificial neural networks to realize learning and adaptation capabilities. These techniques can be used for black box and gray-box modeling, knowledge-based as well as model-based control and decision support.

contents

1. Introduction and motivation. Essentials of fuzzy sets and artificial neural networks.
2. Knowledge-based design, direct and supervisory control.
3. Data-driven neural and fuzzy modeling, model-based control using fuzzy and neural models.
4. Overview of industrial applications.

lecture notes

The lecture notes are available electronically. The participants can download the lecture notes as a zipped postscript file (ZIP) or as a portable document format file (PDF). Transparencies and MATLAB/Simulink demos shown at lectures can be downloaded as well.

prerequisites

Linear algebra and analysis. Basics of linear systems, control and identification.

homework assignments

Three homework assignments will be given during the course lectures. The deadline is always the next following lecture. The assignments will be graded and the average grade will be the final grade for the course.

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Model Predictive Control

lecturers

Dr.ir. A.J.J. van den Boom, Delft University of Technology
Prof.dr. A.A. Stoorvogel, University of Twente

objective

The model predictive control (MPC) strategy yields the optimization of a performance index with respect to some future control sequence, using predictions of the output signal based on a process model, coping with amplitude constraints on inputs, outputs and states. The course presents an overview of the most important predictive control strategies, the theoretical aspects as well as the practical implications, which makes model predictive control so successful in many areas of industry, such as petro-chemical industry and chemical process industry. Hands-on experience is obtained by MATLAB exercises.
Aims of the course:
- Introduction to the basic concepts of model predictive control.
- Theoretical foundation as well as the practical issues in MPC.
- Overview of current research and future directions for MPC.

contents

1. General introduction. Different type of models and model-structures, advantages and limitations. Signal constraints in control.
2. Standard predictive control scheme. Relation standard form with GPC, LQPC and other predictive control schemes. Finite/infinite horizon MPC. Solution of the standard predictive control problem.
3. Stability and the role of endpoint penalties. The effects of model uncertainty and robustness analysis.
4. The effects of noise on prediction and constraints.
5. Limitations in MPC: complexity, feasibility, computational requirements, real-time implementation.

lecture notes

The lecture notes of the course will be made available electronically.

prerequisites

Calculus and linear algebra. Basics of linear system and control (Sections §1.13, §1.15, §1.16, §3.2.C, §3.3.C, §6.10 and appendix A.3 from the book Linear systems by P.J. Antsaklis and A.N. Michel, McGraw-Hill 1998).

homework assignments

Two homework assignments will be handed out. The assignments will be graded, and the average will be your final grade for this course.

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Filtering algorithms for Large-scale Systems

lecturers

Prof.dr.ir. A.W. Heemink, Delft University of Technology
Dr. R.G. Hanea, Delft University of Technology

objective

Real-time filtering and prediction of physical phenomena is of great interest and relies on accurate and fast dynamical models. The modeling process, based on first principle models that involve dynamic conservation laws, normally leads to models consisting of many thousands of differential equations. Numerical weather prediction is an example of a very challenging large-scale filtering problem. Efficient reduction techniques are indispensable to simplify the filtering problem to obtain a computationally feasible solution method. The solution of the reduced problem however still have to capture the essential properties of the underlying physical system. This course aims to address various issues of filtering problems of large-scale systems that lead to computationally attractive methods for estimation and prediction purposes. Attention will be concentrated on models that are based upon partial differential equations and that have to be approximated numerically in order to obtain a discrete state space representation of the physical system.

contents

1. General Introduction. Linear filtering problem (Kalman filtering), nonlinear extensions. State space representation of numerical models based on PDE's.
2. Computational issues in large-scale filtering problems, Square root filtering, Potter algorithm
3. Ensemble Kalman filtering algorithms (Ensemble Kalman filter, Reduced-Rank square root filtering, Hybrid algorithms, Symmetric versions of the algorithms).
4. Euler-Lagrange equations. Representer method for linear state estimation. Relation with Kalman filtering. Examples and real-life applications

prerequisites

Basic undergraduate courses in systems and control, basic knowledge of pde's, and basic programming skills.

lecture notes

Will be distributed during the course.

home work assignments

A take-home exam will be handed out during the last lecture.

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Linear Quadratic Differential Games

lecturers

Dr. J.C. Engwerda, Tilburg University

objective

Interaction between processes at various levels takes place everywhere around us. Either in industry, economics, ecology or on a social level, in many places processes influence each other. Particularly in those case where subjects can affect the outcome of a process, the question arises how subjects come to a certain action. To get more grip on this question within mathematics the paradigms of optimal control theory and game theory evolved. As a merge of both strands of this literature dynamic game theory resulted. This theory brings together the issues of: optimizing behavior, presence of multiple agents, enduring consequences of decisions, and robustness with respect to variability in the environment. Within this field, linear quadratic differential games developed and plays an important role for three reasons: first, many applications of differential game theory fall into this category, second, there are many analytical results available and, third, efficient numerical solution techniques can be used to solve these games. The aim of this course is to give an introduction into the theory of differential games. For the above mentioned reasons we will mainly focus on the linear quadratic case. In the linear quadratic case it is assumed that there are several agents which can influence the evolution of the state of a system, described by a linear differential equation. Each agent has his own goals which he likes to achieve. These goals are assumed to be described by a quadratic function of the state of the system, the control efforts of the involved agent and (sometimes) the control efforts used by the other agents. In particular, by viewing ”nature” as a separate player in the game who can choose an input function that works against the other player(s), one can model worst-case scenarios and, consequently, analyze the robustness of the ”undisturbed” solution. We start by analyzing the one-player case. The obtained results are used later on to analyze the multi-player case. After this case, we consider the so-called cooperative case. That is, the case where all agents agree to reach their goals by coordinating their control efforts. In that case the outcome of the game depends strongly on the bargaining concept used. Some bargaining concepts and its numerical calculation will be discussed. In case the agents decide not to coordinate their actions, the information the different agents have about the game turns out to be an important feature in the analysis of the multi-player case. We will consider two information structures. The so-called open-loop and the feedback information case. For both information structures we will derive the individually rational (Nash) outcomes of the game and present numerical algorithms to calculate outcomes. Finally, the consequences of model uncertainty will be discussed for the pursued actions of the agents. The presented theory will be illustrated in a number of examples.

contents

1. Some main results on regular linear quadratic optimal control.
2. Cooperative games. Necessary and sufficient conditions for existence of Pareto solutions. Bargaining theory. Numerical solutions.
3. Non-Cooperative Open-Loop information games. Nash equilibrium concept. Necessary and sufficient conditions for existence of a unique Nash equilibrium. Some main results on the linear quadratic case.
4. Non-Cooperative Feedback information games. Characterization and existence results. Planning horizon convergence issues.
5. Uncertain Non-Cooperative Feedback information games. Stochastic Approach and Deterministic Approach.

prerequisites

Some familiarity with differential equations and linear algebra.

course material

J.C. Engwerda, Linear Quadratic Dynamic Optimization and Differential Games, ISBN: 0-470-01524-1, Wiley, 2005, Chapters 5-9.

homework assignments

Every week a set of homework exercises has to be handed in. There is no final exam.

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Design Methods for Control Systems

lecturers

Prof.dr.ir. M. Steinbuch, Eindhoven University of Technology
Dr. S. Weiland, Eindhoven University of Technology

objective

The course presents "classical," "modern" and "post modern" notions about linear control system design. First the basic principles, potentials, advantages, pitfalls and limitations of feedback control are presented. An effort is made to explain the fundamental design aspects of stability, performance and robustness.
Next, various well-known classical single-loop control system design methods, including Quantitative Feedback Theory, are reviewed and their strengths and weaknesses are analyzed. The course includes a survey of design aspects that are characteristic for multivariable systems, such as interaction, decoupling and input-output pairing. Further LQ, LQG and some of their extensions are reviewed. Their potential for single- and multi-loop design is examined.
After a thorough presentation of the generalized plant framework and the notions of structured and unstructured uncertainties, design methods based on H-infinity-optimization and mu-synthesis are presented.

contents

1. Introduction to feedback theory. Basic feedback theory, closed-loop stability, stability robustness, loop shaping, limits of performance.
2. Classical control system design. Design goals and classical performance criteria, integral control, frequency response analysis, compensator design, classical methods for compensator design. Quantitative Feedback Theory.
3. Multivariable Control Multivariable poles and zeros, interaction, interaction measures, decoupling, input-output pairing, servo compensators.
4. LQ, LQG and Control System DesignLQ basic theory, some extensions of LQ theory, design by LQ theory, LQG basic theory, asymptotic analysis, design by LQG theory, optimization, examples and applications
5. The generalized plant framework, parametric and dynamic uncertainty models, the small-gain theorem, stability robustness of feedback systems, structured singular value robustness analysis, combined stability and performance robustness.
6. H-infinity optimization and mu-synthesis, the mixed sensitivity problem, loop shaping; the standard H-infinity control problem, state space solution, optimal and suboptimal solutions, integral control and HF roll-off, mu-synthesis, application of mu-synthesis.
a. Appendix on Matrices
b. Appendix on norms of signals and systems

lecture notes

Will be distributed during the course.

prerequisites

Basic undergraduate courses in systems and control. Some familiarity with MATLAB is helpful for doing the homework exercises.

homework assignments

Four homework sets will be distributed via the course website. Homework is graded on a scale from 1 to 10. Missing sets receive grade 0. The final grade for the course is the average of the grades for the four homework sets.

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System and Control Theory of Nonlinear Systems

lecturers

Prof.dr. H. Nijmeijer, Eindhoven University of Technology
Prof.dr. A.J. van der Schaft, University of Groningen

objective

This course provides an introduction to the use of modern mathematical techniques in nonlinear system and control theory. The intrinsic difficulties of the control of nonlinear systems, as well as the effectiveness of the newly developed mathematical theory, are illustrated throughout the course by some apparently simple and physically well-motivated examples, for instance from the area of robotic manipulators and mobile robots.

contents

1. Introduction. What is a nonlinear system? Characteristic examples. Limitations of linearization. Nonlinear input-output maps.
2. Controllability and observability. Lie brackets; rank conditions, relations with controllability and observability of linearized systems, examples.
3. State space transformations and feedback. State feedback, feedback linearization, computed torque control of robot manipulators, observer design, and examples.
4. Decoupling problems. Disturbance and input-output decoupling, tracking, geometrical formulation and controlled invariant distributions, examples.
5. Stability and stabilization. Stabilization and linearization, stabilization of non-controllable critical eigenvalues, zero dynamics and decoupling problems with stability, passivity-based control, DISContinuous feedback, examples.

lecture notes

Nonlinear Dynamical Control Systems, by H. Nijmeijer and A.J. van der Schaft, Springer Verlag, New York, 1990 (fourth printing 1999). Some additional lecture notes are distributed during the course.

grading

The evaluation will be done on the basis of three take-home exams that will be handed out during the course, and which need to be made individually.

prerequisites

An undergraduate course in state space methods for linear control systems. Also a course covering basic knowledge on ordinary differential equations is needed. Likewise, it is highly recommended to have attended a course on linear algebra.

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