Abstract
The first part of this chapter constitutes a short introduction to systems theory. The basic notions of input, output and state variables are presented in abstract terms, as well as the notion of input-output operator. We discuss the main properties involved in the investigation of a system, and illustrate how distinct systems can be combined to give rise to a new system. In this framework, we also introduce the main concern of this book: how to exploit the input channel in order to control the evolution of a system. The exposition becomes more concrete in the remaining part of the chapter, where we explain how a system can be represented by certain mathematical models: impulse response, state space equations. The role of initial conditions is emphasized, in a deterministic philosophy, in connection with the notion of state variable.
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Notes
- 1.
This may require some restrictions on the nature of the system and the set of admissible inputs.
- 2.
Alternatively, we may agree that the admissible inputs are restricted to functions \(u(\cdot )\in \mathcal{F}(\mathbf{R},\mathbf{R}^m)\) vanishing for \(t<t_0\).
- 3.
Provided that the input is defined for each \(t\in \mathbf{R}\), existence and uniqueness of solutions is actually guaranteed on \((-\infty ,+\infty )\).
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Bacciotti, A. (2019). Introduction. In: Stability and Control of Linear Systems. Studies in Systems, Decision and Control, vol 185. Springer, Cham. https://doi.org/10.1007/978-3-030-02405-5_1
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DOI: https://doi.org/10.1007/978-3-030-02405-5_1
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Publisher Name: Springer, Cham
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Online ISBN: 978-3-030-02405-5
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