Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top

2021 | Book

Dynamical Systems Read chapter Read first chapter

Introduction to Mathematical Systems Theory

Discrete Time Linear Systems, Control and Identification

Authors: Christiaan Heij, André C.M. Ran, Frederik van Schagen

Publisher: Springer International Publishing

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Login to get access
insite
SEARCH

About this book

This book provides an introduction to the theory of linear systems and control for students in business mathematics, econometrics, computer science, and engineering. The focus is on discrete time systems, which are the most relevant in business applications, as opposed to continuous time systems, requiring less mathematical preliminaries. The subjects treated are among the central topics of deterministic linear system theory: controllability, observability, realization theory, stability and stabilization by feedback, LQ-optimal control theory. Kalman filtering and LQC-control of stochastic systems are also discussed, as are modeling, time series analysis and model specification, along with model validation.
This second edition has been updated and slightly expanded. In addition, supplementary material containing the exercises is now available on the Springer Link's book website.

Table of Contents

Frontmatter
1. Dynamical Systems
Abstract
Many phenomena investigated in such diverse areas as physics, biology, engineering, and economics show a dynamical evolution over time. Examples are thermodynamics and electromagnetism in physics, chemical processes and adaptation in biology, control systems in engineering, and decision making in macro economics, finance, and business economics.
Christiaan Heij, André C.M. Ran, Frederik van Schagen
2. Input-Output Systems
Abstract
In this chapter we consider input-output systems. Such systems can be described in the time domain, in terms of the impulse response, and in the frequency domain, by the transfer function. For rational transfer functions the system can be represented by a finite dimensional state space model.
Christiaan Heij, André C.M. Ran, Frederik van Schagen
3. State Space Models
Abstract
This chapter discusses structure theory of state space systems. The central concepts are observability (the possibility to reconstruct the state from inputs and outputs) and controllability (the possibility to influence the state by manipulating the inputs). Minimal realizations are observable and controllable, and the converse is also true. We characterize all non-minimal realizations, and give an algorithm to compute the matrices in a minimal realization from the impulse response of the system.
Christiaan Heij, André C.M. Ran, Frederik van Schagen
4. Stability
Abstract
Input-output systems are applied in control, where the inputs are chosen in such a way that the system shows satisfactory performance. Stability is an important objective, that is, disturbances have a limited effect on the system. Systems can be stabilized by feedback, where past performance is used to choose the input variables.
Christiaan Heij, André C.M. Ran, Frederik van Schagen
5. Optimal Control
Abstract
In this chapter we consider quantitative control objectives for rather general systems. The inputs are chosen to minimize a function that expresses the costs associated with the system evolution. This can be solved by dynamic programming. We pay special attention to the so-called LQ problem, where the system is linear and the cost function is quadratic. In this case the optimal control is given by state feedback, and the feedback matrix can be computed by solving certain matrix equations (so called Riccati equations).
Christiaan Heij, André C.M. Ran, Frederik van Schagen
6. Stochastic Systems
Abstract
In stochastic systems, the outputs are (partly) driven by unobserved random inputs. This chapter is concerned with stationary processes and their approximation with finite dimensional linear stochastic systems. Similar to the results for deterministic input-output systems there is an equivalence between finite dimensional stochastic state space models, polynomial (ARMA) representations, and rational spectra (in the frequency domain), which are the analogue of the transfer function.
Christiaan Heij, André C.M. Ran, Frederik van Schagen
7. Filtering and Prediction
Abstract
Stochastic systems can be applied for forecasting purposes. The classical solution for filtering, smoothing and prediction of linear systems was proposed by Wiener and Kolmogorov in terms of spectral representations. The Kalman filter is a much more efficient, recursive solution in terms of state space models.
Christiaan Heij, André C.M. Ran, Frederik van Schagen
8. Stochastic Control
Abstract
Stochastic optimal control problems can in principle be solved by stochastic dynamic programming. We pay special attention to the LQG problem where the system is linear, the cost function is quadratic, and the random variables have Gaussian distributions. The optimal controller is given by the LQG feedback law where the unobserved state is replaced by the Kalman filter estimate.
Christiaan Heij, André C.M. Ran, Frederik van Schagen
9. System Identification
Abstract
System identification is concerned with the estimation of a system on the basis of observed data. This involves specification of the model structure, estimation of the unknown model parameters, and validation of the resulting model. Least squares and maximum likelihood methods are discussed, for stationary processes (without inputs) and for input-output systems.
Christiaan Heij, André C.M. Ran, Frederik van Schagen
10. Cycles and Trends
Abstract
For many time series, trends and cyclical fluctuations dominate the stationary part. The main cyclical components can be identified by spectral analysis. Trends and seasonals can either be incorporated explicitly in the model or they can be removed by filtering the data.
Christiaan Heij, André C.M. Ran, Frederik van Schagen
11. Further Developments
Abstract
We close this book with a few sections that provide glimpses of further developments in the area of systems and control theory. In all sections we shall give a pointer to further literature on the subject.
Christiaan Heij, André C.M. Ran, Frederik van Schagen
Backmatter
Metadata
Title
Introduction to Mathematical Systems Theory
Authors
Christiaan Heij
André C.M. Ran
Frederik van Schagen
Copyright Year
2021
Electronic ISBN
978-3-030-59654-5
Print ISBN
978-3-030-59652-1
DOI
https://doi.org/10.1007/978-3-030-59654-5