Titlebook: Introduction to Mathematical Systems Theory; Linear Systems, Iden Christiaan Heij,André Ran,Freek Schagen Textbook 20071st edition Birkh?us

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Input-Output Systems,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.

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Stability,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.

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Filtering and Prediction,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.

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Cycles and Trends,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.

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State Space Models,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 r

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Optimal Control,e 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 co

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