Analysis and control of linear systems by Philippe de Larminat

By Philippe de Larminat

Automation of linear platforms is a primary and crucial conception. This e-book offers with the idea of continuous-state automatic platforms.

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34 Analysis and Control of Linear Systems The energy exchanged between x (t ) and the harmonic signal of frequency f o ( e2π jfot ) can be finite. X ( f o ) is then finite, or infinite if x(t ) is also a harmonic signal and X ( f ) is then characterized by a Dirac impulse δ ( fo ) ( f ) . According to the nature of the signal considered, by using various mathematical theories concerning the convergence of indefinite integrals, we can define the Fourier transform in the following cases: – positively integrable signal: ∫ x(t) dt ≤ ∞ .

Finally, for ξ = 0 , the system becomes a pure oscillator with a infinite module in ω o and a real phase mistaken for the asymptotic phase. 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.

We note that in this particular case, the impulse response of the system is a function that tends infinitely toward 0. 7. 1. Harmonic analysis Let us consider a stable LTI whose impulse response h(θ) is canceled after a period of time t R . For the models of physical systems, this period of time t R is in fact rejected infinitely; however, for reasons of clarity, let us suppose t R as finite, corresponding to the response time to 1% of the system. When this system is subject to a harmonic excitation x ( t ) = Ae2π jf0t from t = 0 , we obtain: y (t ) = t ∫0 h (θ ) Ae 2π jf0 ( t −θ ) dθ = Ae2π jf0t t ∫0 h (θ ) Ae −2π jf0θ dθ For t > t R , the impulse response being zero, we have: t ∫0 h(θ ) e −2π jf0θ dθ =H ( f0 ) = +∞ ∫0 h(θ ) e−2π jf0θ dθ = H ( f0 ) e jΦ ( f0 ) and hence for t > t R , we obtain y(t ) = AH ( fo ) e2π jf0 t = A H ( f0 ) e j (2π f0 t +Φ ( f0 )) .

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