Abstract
The subject of the last chapter is the relationship between two possible approaches to the analysis of a system: the time domain approach (developed in the previous chapters) and the more traditional frequency domain approach based on the Laplace transform. We study in particular the realization problem. For the case of SISO systems, we also give a different solution to the stabilization problem by output feedback. Finally, we illustrate the decoupling problem, whose solution takes advantages of both approaches.
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Notes
- 1.
Alternatively, performing a decomposition to partial fractions, we can rewrite (8.19) as
$${{1}\over {s_2-s_1}} \left( {{U(s)}\over {s-s_2}} - {{U(s)}\over {s-s_1}} \right) \ .$$Taking the inverse transform we obtain
$$\begin{aligned}&{{1}\over {s_2-s_1}} \left( \int _0^t e^{s_2(t-\tau )} u(\tau )\, d\tau - \int _0^t e^{s_1(t-\tau )} u(\tau )\, d\tau \right) \\&= {{1}\over {s_2-s_1}} \left( \int _0^t [ e^{s_2(t-\tau )} - e^{s_1(t-\tau )}] u(\tau )\, d\tau \right) \end{aligned}$$ - 2.
If T(s) is the transfer function of a system, its zeros give useful information about the behavior of the system: the interested reader is referred to [8].
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Bacciotti, A. (2019). Frequency Domain Approach. 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_8
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DOI: https://doi.org/10.1007/978-3-030-02405-5_8
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Publisher Name: Springer, Cham
Print ISBN: 978-3-030-02404-8
Online ISBN: 978-3-030-02405-5
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