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System Identification for Control
Lecturers
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Dr.ir. X.J.A. Bombois, Delft University of Technology
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Prof.dr.ir. P.M.J. Van den Hof, Delft University of Technology
Objective
System Identification concerns the modeling of dynamical systems on the basis of observed data. Control Design concerns the design of a controller using a model description of dynamical systems. The objective of this course to present methodologies for system identification with a particular emphasis on the question how to obtain models that are suited to serve as a basis for control design. In the first part of the course, different methods of system identification are presented with a particular focus on prediction error methods. In the second part of the course attention is given to the question how to identify models that are control-relevant, i.e. that lead to properly designed model-based controllers. This includes issues like closed-loop identification and model uncertainty.
Contents
- 1. Introduction; concepts; discrete-time signal and system analysis.
- 2. Parametric (prediction error) identification methods - model sets, identification criterion, bias and variance.
- 3. Parametric (prediction error) identification methods - model validation, experiment design and approximate modelling
- 4. Extension on model structures and identification methods
- 5. Closed-loop identification
- 6. Control-relevant models; iterative performance enhancement
- 7. Handling model uncertainty
- 8. Experiment-based controller validation
Prerequisites
Calculus and linear algebra. Some knowledge of statistics and linear systems theory and/or time series analysis is helpful, but not required. The lecture notes contain useful summaries of the important notions used during the course.
Lecture notes
Lecture notes will be distributed during the course.
Course assessment
The assessment of this course will be in the form of three homework assignments.
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Mathematical Models of Systems
Lecturers
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Dr. J.W. Polderman, University of Twente
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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 set-up 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 pole-placement and stabilization look like in this setting.
Contents
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General ideas. Mathematical models of systems. Dynamical systems. Examples from physics and economics. Linear time-invariant systems. Differential equations. Polynomial matrices.
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Minimal and full row rank representation. Autonomous systems. Inputs and outputs. Equivalence of representations.
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Differential systems with latent variables. State space models. I/S/O models.
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Controllability. Controllable part. Observability.
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Elimination of latent variables. Elimination of state variables.
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From I/O to I/S/O models. Image representations.
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Interconnection. Control in a behavioral setting. Implementability.
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Stability. Stabilization and pole placement.
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.
Prerequisites
The course is pretty much self-contained. Basic linear algebra and calculus should suffice.
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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Modeling and Control of Hybrid Systems
Lecturers
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Prof.dr.ir. Bart De Schutter, Delft University of Technology
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Dr.ir. Maurice Heemels, Eindhoven University of Technology
Framework
Recent technological innovations have caused a considerable interest in the study of dynamical processes of a mixed continuous and discrete nature. Such processes are called hybrid systems and are characterized by the interaction of time-continuous models (governed by differential or difference equations) on the one hand, and logic rules and discrete-event systems (described by, e.g., automata, finite state machines, etc.) on the other. In practice a hybrid system arises when continuous physical processes are controlled via embedded software that intrinsically has a finite number of states only (e.g., on/off control).
Objectives of the course
This course will offer a brief overview of the field of hybrid systems ranging from modeling, over analysis and simulation, to verification and control. We will particularly focus on modeling, analysis, and control of tractable classes of hybrid systems.
Contents
1. General introduction. Examples of hybrid systems & motivation. Modeling frameworks (automata, hybrid automata,
piecewise-affine systems, complementarity systems, mixed logic dynamical systems, ...)
2. Properties and analysis of hybrid systems (well-posedness, Zeno behavior, stability, liveness, safety, ....).
3. Control of hybrid systems (switching controllers, model predictive control, ...)
4. Control of hybrid systems (continued). Verification. Tools.
Prerequisites
Basic undergraduate courses in systems and control. Basic programming skills (Matlab).
Lecture notes
The lecture notes will be made available electronically.
Homework assignments
Four homework assignments will be handed out. The assignments will be graded and the average grade will be the final grade for this course.
Course website
http://www.dcsc.tudelft.nl/~disc_hs/course/
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Design Methods for Control Systems
Lecturers
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Prof.dr. ir. M. Steinbuch, Eindhoven University of Technology
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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 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 structured and unstructured uncertainty, model design methods based on H-infinity-optimization (in particular, the mixed sensitivity problem and McFarlane-Glover's loopshaping problem) and mu-synthesis are presented.
Contents
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Introduction to feedback theory. Basic feedback theory, closed-loop stability, stability robustness, loop shaping, limits of performance.
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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.
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Multivariable Control Multivariable poles and zeros, interaction, interaction measures, decoupling, input-output pairing, servo compensators.
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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
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Uncertainty models and robustness Parametric robustness analysis, the basic perturbation model, the small-gain theorem, stability robustness of the basic perturbation model, stability robustness of feedback systems, numerator-denominator perturbations, structured singular value robustness analysis, combined performance and stability robustness.
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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.
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Appendix on Matrices
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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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Linear Matrix Inequalities for Control
Lecturers
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Prof.dr. C.W. Scherer, Delft University of Technology,
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Dr. S. Weiland, Eindhoven University of Technology
Objective
Linear matrix inequalities (LMI's) have emerged as a powerful tool to approach control problems that appear hard
if not impossible to solve in an analytic fashion. Although the history of LMI's goes back to the fourties with a major emphasis of their role in control in the sixties (Kalman, Yakubovich, Popov, Willems), only recently powerful numerical
interior point techniques have been developed to solve LMI's in a practically efficient manner (Nesterov, Nemirovskii 1994).
Several Matlab software packages are available that allow a simple coding of general LMI problems and of those that
arise in typical control problems. In particular, the former LMI toolbox has been integrated in the Matlab robust control
toolbox.
Boosted by the availability of fast LMI solvers, research in robust control has experienced a paradigm shift - instead of
arriving at an analytical solution the intention is to reformulate a given problem to verifying whether an LMI is solvable or to optimizing functionals over LMI constraints.
The main emphasis of the course is
• to reveal the basic principles of formulating desired properties of a control system in the form of LMI's
• to demonstrate the techniques how to reduce the corresponding controller synthesis problem to an LMI problem
• to get familiar with the use of software packages for performance analysis and controller synthesis using LMI tools.
The power of this approach is illustrated by several fundamental robustness and performance problems in analysis and design of linear control systems.
Topics
1. Some facts from convex analysis. Linear Matrix Inequalities: Introduction. History. Algorithms for their solution.
2. The role of Lyapunov functions to ensure invariance, stability, performance, robust performance.
Considered criteria: Dissipativity, integral quadratic constraints, H2-norm, H∞ norm, upper bound of peak-to-peak norm.
LMI stability regions.
3. Frequency domain techniques for the robustness analysis of a control system. Integral Quadratic Constraints.
Multipliers. Relations to classical tests and to µ-theory.
4. A general technique to proceed from LMI analysis to LMI synthesis. State-feedback and output-feedback synthesis
algorithms for robust stability, nominal performance and robust performance using general scalings.
5. Extensions to mixed control problems and to linear parametrically-varying controller design.
Material
The main reference material for the course will be lectures notes and
[1] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM
studies in Applied Mathematics, Philadelphia, 1994.
[2] L. El Ghaoui and S.I.Niculescu (Editors), Advances in Linear Matrix Inequality Methods in Control, SIAM, Philadelphia,
2000.
[3] A. Ben-Tal, A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications,
SIAM-MPS Series in Optimizaton, SIAM, Philadelphia, 2001.
[4] G. Balas, R. Chiang, et al. (2006). Robust Control Toobox (Version 3.1), The MathWorks Inc.
[5] J. L¨ofberg, YALMIP, http://control.ee.ethz.ch/˜joloef/yalmip.php.
Prerequisites
Linear algebra, calculus, basic system theory, MATLAB.
Examination
Full credit is received for successfully solving at least 50% of the assigned take-home exams.
Course website
http://w3.ele.tue.nl/nl/cs/education/courses/DISClmi/
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Distributed Parameter Systems
Lecturers
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Dr. B. Jacob, Delft University of Technology
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Dr. H.J. Zwart, University of Twente
Objective
Modeling of dynamical systems with a spatial component leads to lumped parameter systems, when the spatial component may be denied, and to distributed parameter systems otherwise. The mathematical model of distributed parameter systems will be a partial differential equation. Examples of dynamical sytems with a spatial component are, among others, temperature distribution of metal slabs or plates, and the vibration of aircraft wings.
This course provides an introduction to distributed parameter systems. In particular, we will study distributed parameter port Hamiltonian systems. This class contains the above mentioned examples. The norm of such a system is given by the energy (Hamiltonian) of the system. This fact enables us to show relatively easy the existence and stability of solutions. Further, it is possible to determine which boundary variables are suitable as inputs and outputs, and how the system can be stabilized via the boundary.
At the end of the course the students should be able to model distributed parameter systems as distributed parameter port Hamiltonian system, and should be able to apply known concepts from system and control theory such as controllability, observability, stability, stabilizability and transfer functions to these systems.
Contents
1.) Distributed parameter port Hamiltonian system
2.) Wellposedness of distributed parameter port Hamiltonian system
3.) Control and observation at the boundary
4.) Transfer functions
5.) Wellposedness of control and observation operators
6.) Stability, stabilizability, controllability and observability
7.) Equations with diffusion
8.) Extensions
Prerequisites
Basic undergraduate courses in systems and control.
Lecture notes
Lecture notes are under preparation and will be distributed during the presentations of the course.
Homework assignments
Four homework assignments will be given during the course lectures. The assignments will be graded and the average grade will be the final grade for the course.
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System and Control Theory of Nonlinear Systems
Lecturers
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Prof.dr. H. Nijmeijer, Eindhoven University of Technology
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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 well-motivated 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 input-output 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 input-output decoupling, tracking, geometrical formulation and controlled invariant distributions, examples.
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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