Mathematical Modelling

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To describe natural phenomena and processes, mathematical models are widely used. The focus in this course shall be on dynamical models (i.e., where time plays a role) in particular those that have interaction with the environment through inputs and outputs. Mathematical systems theory provides the framework to deal with such models in a systematic and useful way. First we consider some general aspects of mathematical modeling. Then we briefly address dynamical systems without inputs and outputs - but which may show nonlinear behavior. We study basic properties such as equilibrium points, linearization, and stability. We then switch to linear dynamical models with inputs and outputs. They are used in many different areas of the natural sciences and in engineering disciplines. We discuss the following topics and concepts. Linear difference and differential equations, Laplace transforms, transfer functions of linear systems; controllability, observability, minimality; system representations with an emphasis on state-space representations and canonical forms; stability; the interconnection of linear systems including feedback; frequency domain analysis and the relationship with filter theory, Fourier analysis, and time series analysis. To demonstrate the applicability of the techniques and concepts, many examples from science and engineering are mentioned and briefly discussed.