Course content

The course gives a basic introduction to system theory for continuous-time dynamical systems, with emphasis on modelling, analysis, and interpretation of linear systems. The main purpose of the course is to help students understand dynamical systems through their mathematical models and through their behaviour in time and frequency domains.

The course covers mathematical representations of dynamical systems, including differential equations, state-space models, transfer functions, block diagrams, time responses, and frequency responses. Particular emphasis is placed on how these representations are related, and on how they can be used to understand the qualitative and quantitative behaviour of a system.

Central topics include:

• continuous-time dynamical systems and signals;
• modelling with differential equations;
• state-space models and vector differential equations;
• linearization of nonlinear models around operating points;
• eigenvalues, eigenvectors, modes, and their relation to time responses;
• generalized eigenvectors, Jordan forms, and their relation to time responses;
• stability of linear systems;
• impulse responses and step responses;
• transfer functions, poles, zeros, and block diagrams;
• Laplace and Fourier transforms;
• frequency response and Bode plots;
• phase-plane descriptions;
• practical phenomena such as delays and non-minimum phase behaviour.

The course combines theory with practical examples, exercises, numerical calculations, simulations, visualizations, and workshops where we will build together some simple electronics systems exhibiting the dynamics and properties studied in the more theory-oriented parts of the course. These activities are used to connect mathematical results to the behaviour of dynamical systems in engineering and other application areas.

Learning outcome

After completing the course, the student should have distinct learning outcomes, as categorized below.

Knowledge

After successfully completing the course, the student should be able to:

• explain the main mathematical representations of continuous-time dynamical systems, including differential equations, state-space models, transfer functions, block diagrams, time responses, and frequency responses;
• describe the role of inputs, outputs, states, initial conditions, parameters, and signals in dynamical system models;
• explain the relationship between state-space models and transfer-function models for linear time-invariant systems;
• explain how eigenvalues, eigenvectors, generalized eigenvectors, and modes influence the qualitative behaviour of linear autonomous systems;
• describe central concepts such as stability, poles, zeros, impulse response, step response, frequency response, gain, phase, and resonance;
• explain the basic role of Laplace and Fourier transforms in the analysis of signals and systems;
• describe what linearization is, why it is useful, and what its limitations are;
• explain how idealized mathematical models can support, but also limit, reasoning about practical systems.

Skills

After successfully completing the course, the student should be able to:

• formulate simple continuous-time dynamical models using differential equations, state-space representations, and transfer functions;
• transform between different representations of linear systems when the required assumptions are satisfied;
• compute and interpret time responses of linear systems, including free responses, forced responses, impulse responses, and step responses;
• analyse the stability of linear systems using eigenvalues, poles, and related mathematical criteria;
• compute and interpret frequency responses and Bode plots for linear systems;
• use analytical calculations and numerical tools to simulate and analyse dynamical systems;
• linearize simple nonlinear models around operating points and interpret the resulting linear approximation;
• interpret plots, simulations, and numerical results in relation to the underlying system model;
• identify typical modelling and analysis limitations when applying system theory to practical examples.

General competence

After successfully completing the course, the student should be able to:

• use system-theoretic thinking to analyse dynamical phenomena in technical and non-technical contexts;
• communicate system-theoretic reasoning using equations, diagrams, plots, simulations, and verbal explanations;
• connect mathematical properties of models to qualitative system behaviour;
• reflect on the choice of model representation and on how this choice affects analysis and interpretation;
• use computational tools as support for analysis while still being able to explain the mathematical meaning of the computations;
• recognize how the concepts of the course support later studies in automatic control, estimation, signal processing, system identification, robotics, and cyber-physical systems.

Learning methods and activities

The course combines lectures, exercises, discussion, student-active learning, hands-on workshops and computational exploration. Teaching sessions normally combine short lecture sequences with examples, concept questions, problem solving, and discussion.

The lectures are organized so that theoretical concepts are introduced gradually and connected to concrete examples, calculations, plots, and simulations. In many sessions, the teaching will alternate between explanation, short activities, and collective discussion, rather than consisting only of long uninterrupted lectures.

Learning activities may include:

• lectures and worked examples;
• hands-on workshops on systems based on low-voltage electronics;
• individual and group exercises;
• concept questions and peer instruction;
• collective coding and simulation activities;
• use of Jupyter notebooks, MATLAB, Python, or similar tools;
• interpretation of plots and numerical results;
• self-assessment quizzes and digital question sets;
• discussion of typical misconceptions and alternative solution strategies.

Canvas is used as the main channel for course information, announcements, learning resources, links, practical information, and, when relevant, recordings or information about digital or hybrid teaching activities.

Self-assessment resources and digital question sets may be used to help students evaluate their own understanding of central topics in the course. These activities are intended as formative support for learning and are not used directly in the final grading, unless explicitly stated in Canvas.

Generative AI tools may be used as learning support, for example to ask for alternative explanations, generate practice questions, obtain hints, or debug code. Students are responsible for checking the correctness of AI-generated material and must be able to explain all concepts, calculations, and solutions independently.

Recommended previous knowledge

Students should have knowledge corresponding to mathematics from an engineering bachelor’s programme, including linear algebra, calculus, ordinary differential equations, complex numbers, and basic numerical calculations.

It is useful, but not strictly required, to have previous experience with basic automatic control, physics-based modelling, and simple programming or numerical computation in MATLAB, Python, or similar environments.

Students are encouraged to revise the following topics before or during the first part of the course:

• matrix algebra, rank, inverse matrices, eigenvalues, and eigenvectors;
• elementary ordinary differential equations;
• complex numbers and complex exponentials;
• basic plotting and numerical simulation;
• basic concepts from signals and systems or automatic control, if previously encountered.

Required previous knowledge

Mathematics corresponding to an engineering bachelor’s programme.

Course materials

Course materials will be announced in Canvas before and during the semester.

The material may include:

• textbook chapters;
• lecture notes;
• slides;
• exercises;
• Jupyter notebooks;
• MATLAB, Python, or similar computational examples;
• self-assessment questions;
• learning-flow maps;
• recorded videos, when available;
• supplementary online resources.

Canvas will indicate which materials are required, recommended, or optional.

Credit reductions