By Philippe de Larminat
Automation of linear structures is a basic and crucial thought. This booklet offers with the idea of continuous-state automatic structures.
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Additional resources for Analysis and control of linear systems
26 illustrate the diagrams presenting the aspect of the frequency response for a second order system with different values of ξ . 23. 24. 25. 26. 6. 1. Fourier transform Any signal has a reality in time and frequency domains. Our ear is sensitive to amplitude (sound level) and frequency of a sound (low or high-pitched tone). These time and frequency domains, which are characterized by variables that are opposite to one another, are taken in the broad sense: if a magnitude evolves according to a distance (atmospheric pressure according to altitude), the concept of frequency will be homogenous, contrary to a length.
The integral definition of the TF converges in absolute value. X ( f ) is a function that tends toward 0 infinitely; – integrable square signal or finite energy signal ∫ x(t )2 dt ≤ ∞ . The integral definition of the TF exists in the sense of the convergence in root mean square: +A lim A→∞ ∫ X( f ) − ∫ x(t) e 2 −2π jft dt df = 0 −A k – slightly ascending signal: ∃A and k t > A ⇒ x (t ) < t . The Fourier transform exists in the distribution sense. 2. Laplace transform When the signal considered has an exponential divergence, irrespectively of the mathematical theory considered, we cannot attribute any sense to the integral definition of the Fourier transform.
23. 24. 25. 26. 6. 1. Fourier transform Any signal has a reality in time and frequency domains. Our ear is sensitive to amplitude (sound level) and frequency of a sound (low or high-pitched tone). These time and frequency domains, which are characterized by variables that are opposite to one another, are taken in the broad sense: if a magnitude evolves according to a distance (atmospheric pressure according to altitude), the concept of frequency will be homogenous, contrary to a length. The Fourier transform is the mathematical tool that makes it possible to link these two domains.