Flexible Multibody System Analysis for Control Purposes
Lecturer:

Prof. dr. ir. J.B. Jonker, University of Twente

Dr. ir. R.G.K.M. Aarts, University of Twente

Dr. ir. J. van Dijk, University of Twente
Objectives:
For control design of mechatronic systems it is essential to make use of simple prototype models with a few degrees of freedom that capture only the relevant system dynamics. For this purpose, the multibody system approach is a wellsuited method to model the dynamical behaviour of the mechanical part of such systems. In this approach the mechanical components are considered as rigid or flexible bodies that interact with each other through a variety of connections such as hinges and flexible coupling elements like trusses and beams. The method is applicable for flexible multibody systems as well as for flexible structures in which the system members experience only small displacement motions and elastic deformations with respect to an equilibrium position. A mathematical description of these models is represented by the equations of motion derived from the multibody systems approach.
For control synthesis a linearized statespace formulation is required in which an arbitrary combination of positions, velocities, accelerations and forces can be taken both as input variables and as output variables, according to the control problem being solved. In this course basic concepts of flexible multibody system dynamics are presented using a nonlinear finite element method. This formulation accounts for geometric nonlinear effects of flexible elements due to axial and transverse displacements. The approach offers many possibilities for analysis, simulation and prototype modeling of mechatronic systems. This will be illustrated through a variety of design cases.
Contents:

Scope of flexible multibody kinematics and dynamics. Multibody versus finite element formulations. Description of angular orientation: Euler angles, Quarternions.

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.

Linearized equations for control system analysis. Stationary and equilibrium solutions. Linearized statespace equations.

Illustrative design examples e.g. a programmable focus system, a multi axes vibration isolation mount and a multi axes micro stage.
Lecture notes:
Will be distributed during the course.
Prerequisites:
Basic background in systems modeling 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 certain 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 setup by tearing and zooming. Thus the model will 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:
We will also introduce some important system properties as controllability and observability in this setting. In the first part of the course, we will review the main representations, their interrelations, and their basic properties. In the context of control, we will 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 will extensively discuss the notion of implementability as a concept to characterize the limits of performance of a plant to be controlled. We will discuss how the problems of poleplacement and stabilization look like in this setting.
Contents:

General ideas. Mathematical models of systems. Dynamical systems. Examples from physics and economics. Linear timeinvariant systems. Differential equations. Polynomial matrices.

Minimal and full row rank representation. Autonomous systems. Inputs and outputs. Equivalence of representations.

Differential systems with latent variables. State space models. I/S/O models.

Controllability. Controllable part. Observability.

Elimination of latent variables. Elimination of state variables.

From I/O to I/S/O models. Image representations.

Interconnection. Control in a behavioral setting. Implementability.

Stability. Stabilization and pole placement.
Prerequisites:
The course is pretty much selfcontained. 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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Stability and Control of Timedelay Systems
Lecturer:
Objective:
Timedelay systems represent a class of infinitedimensional systems largely used to describe transport and propagation phenomena. Roughly speaking, the reaction of real world systems to exogenous signals is never instantaneously, which can be translated into a mathematical language by including delay terms in the corresponding models.
The study of the delay effects on the stability and control of dynamical systems is a problem of recurring interest since the delay presence may induce complex behaviour (instability, oscillations, chaotic behaviour,…). In the control area, a lot of results point out that delays in feedback loops typically lead to a performance degradation, an increased sensitivity to uncertainty, and instability. In particular cases, however, timedelays have a stabilizing effect and are used precisely as controller or design parameters. All these aspects motivate a quantitative and qualitative study of the influence of timedelays.
The aim of the course is to present a range of methods and techniques for the analysis and control of dynamical systems with delays. Particular attention will be paid to the recently developed eigenvalue based approach for the analysis and control design of linear timedelay systems and on numerical methods. Examples from various application areas complete the presentation.
Throughout the course timedelay systems are represented by continuoustime functional differential equations. The lectures are supported by computer demonstrations.
Contents:

Introduction, basic notions, illustrative examples Basic notions and definitions ( types of delays, system representation, notion of state, definition of solutions), spectral properties of linear timedelay systems, motivating examples

Stability analysis Analytical and graphical stability tests, computation of characteristic roots, robustness of stability, computation of stability regions in parameter spaces

Stabilization and control Stabilizability, limitations induced by delays in feedback loops, classification and properties of controllers, controller design tools, the other way around: using delays as controller parameters

Software & applications Overview of software for simulation, stability analysis and control, applications from engineering and biosciences (e.g. highspeed communication networks, machine tool vibrations, transportation systems, human immune dynamics models)
Prerequisites:
Basic undergraduate courses in systems and control. Familiarity with MATLAB is useful for the homework exercises.
Lecture Notes:
The lecture notes are available electronically. The participants can download the slides as a portable document format file (PDF), as well as other supporting material.
Homework assignments:
Three homework assignments will be given during the course lectures. The deadline is always the next lecture. The assignments will be graded and the average grade will be the final grade for the course.
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Design Methods for Control Systems
Lecturers:

Prof.dr. ir. M. Steinbuch, Eindhoven University of Technology

Dr.ir. G. Meinsma, University of Twente
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 wellknown classical singleloop 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 inputoutput pairing. Further LQ, LQG and some of their extensions are reviewed. Their potential for single and multiloop design is examined. After a thorough presentation of structured and unstructured uncertainty, model design methods based on Hinfinityoptimization (in particular, the mixed sensitivity problem and McFarlaneGlover's loopshaping problem) and musynthesis are presented.
Contents

Introduction to feedback theory. Basic feedback theory, closedloop stability, stability robustness, loop shaping, limits of performance.

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.

Multivariable Control Multivariable poles and zeros, interaction, interaction measures, decoupling, inputoutput pairing, servo compensators.

LQ, LQG and Control System Design LQ basic theory, some extensions of LQ theory, design by LQ theory, LQG basic theory, asymptotic analysis, design by LQG theory, optimization, examples and applications

Uncertainty models and robustness Parametric robustness analysis, the basic perturbation model, the smallgain theorem, stability robustness of the basic perturbation model, stability robustness of feedback systems, numeratordenominator perturbations, structured singular value robustness analysis, combined performance and stability robustness.

Hinfinity optimization and musynthesis The mixed sensitivity problem, loop shaping, the standard Hinfinity control problem, state space solution, optimal and suboptimal solutions, integral control and HF rolloff, musynthesis, application of musynthesis.

Appendix on Matrices

Appendix on norms of signals and systems
Lecture notes:
A full set of lecture notes will be made available on the course website.
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 the grade 1. The final grade for the course is the average of the grades for the four homework sets.
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Model Predictive Control
Lecturers:

Dr.ir. Ton van den Boom, Delft University of Technology,

Prof.dr. Anton 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 petrochemical industry and chemical process industry. Handson 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 industrial MPC.
Contents:

General introduction. Different type of models and modelstructures, advantages and limitations. Signal constraints in control.

Standard predictive control scheme. Relation standard form with GPC, QPC and other predictive control schemes. Finite/infinite horizon MPC. Solution of the standard predictive control problem.

Stability and the role of endpoint penalties. The effects of model uncertainty and robustness analysis.

The effects of noise on prediction and constraints

MPC in industry based on a case study. Limitations in MPC: complexity, feasibility, computational requirements, realtime implementation.
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, McGrawHill 1998).
Lecture Notes:
The lecture notes of the course will be made available electronically at the website: http://www.dcsc.tudelft.nl/~discmpc/index.html
Homework assignments:
Four homework assignments will be handed out. The assignments will be graded, and the average grade (over four assignments) will be your final grade for this course.
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Control and System Theory of Stochastic Systems
Lecturers:
Prof.dr. A. Bagchi (University of Twente)
Prof.dr.ir. J.H. van Schuppen (Vrije Universiteit/Centrum voor Wiskunde en Informatica).
Objective:
The purpose of the course is to present the concept of a stochastic system as a probabilistic model for signals and to present synthesis methods for control and filtering of stochastic systems. The course is directed jointly to engineers and to mathematicians. Because of the intended audience, the mathematical content of the course is not that deep (measure theory is used but lightly). Only discretetime stochastic systems will be treated. Although several classes of stochastic systems will be discussed most attention will be given to the class of Gaussian systems.
Contents:
Probability and stochastic processes. Introduction to the course. Probability distribution functions, probability densities. Gaussian random variables. Examples of probability distribution functions arising in engineering. Expectation. Independence and conditioning. Stochastic processes, Gaussian processes.
Stochastic systems. Stochastic system, Gaussian system, finite stochastic system. Properties of Gaussian systems. Examples from control, communication, mechanical engineering, and other areas of engineering.
Stochastic realization. Stochastic realization of stationary Gaussian processes. Existence and classification of minimal realizations. Special stochastic realizations. Subspace version of stochastic realization algorithm.
Kalman filters. Filtering problem for Gaussian processes. Kalman filter. Examples. Asymptotic Kalman filter. Design of Kalman filters. Prediction and smoothing of Gaussian systems.
Filter theory and filter techniques. Examples. Extended Kalman filter. Filters for finite stochastic systems. Particle filtering.
Stochastic control problems. Examples. Stochastic control system and stochastic controllability. Control objectives. Control synthesis methods and design.
Stochastic control with complete observations. Example. Dynamic programming. Gaussian systems and quadratic criteria (LQG and LEQG). Stochastic control of finite stochastic systems. Infinitehorizon stochastic control.
Stochastic control with partial observations. Example. Filtering in the presence of an input process. Dynamic programming in case of partial observations. Gaussian systems with quadratic criteria (LQG and LEQG). Separation property and principle. Asymptotic controller. Perspective on control and system theory of stochastic systems.
Lecture notes:
Lecture notes are under preparation and will be distributed during the presentations of the course.
Grading:
Homeworksets and a takehome exam.
Prerequisites:
The participants are assumed to have knowledge as provided by a basic undergraduate course in control of linear systems; and by an undergraduate course in probability and elementary stochastic processes. The minimum level is indicated by the books: (1) for probability theory the book Sheldon Ross, A first course in probability, (5th Ed.), PrenticeHall, Upper Saddle River, 1998; and (2) for control and system theory the book G.J. Olsder, J.W. van der Woude, Mathematical systems theory, Delft University Press, Delft, 2005.
A recommended prerequisite is knowledge of elementary measure theory and measure theoretic probability as provided by the book J. Jacod and Ph. Protter, Probability essentials, Springer, Berlin, 1999 of which there exists also a more recent edition. Students will benefit from having studied this book prior to the start of the course but this is not a requirement. Measure theory at this level is usually provided by a course of a mathematics department of a university.
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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 Twente
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 wellmotivated examples, for instance from the area of robotic manipulators and mobile robots.
Contents:
Introduction. What is a nonlinear system? Characteristic examples. Limitations of linearization. Nonlinear inputoutput maps.
Controllability and observability. Lie brackets; rank conditions, relations with controllability and observability of linearized systems, examples.
State space transformations and feedback. State feedback, feedback linearization, computed torque control of robot manipulators, observer design, and examples.
Decoupling problems. Disturbance and inputoutput decoupling, tracking, geometrical formulation and controlled invariant distributions, examples.
Stability and stabilization. Stabilization and linearization, stabilization of noncontrollable critical eigenvalues, zero dynamics and decoupling problems with stability, passivitybased 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 takehome 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.
