Analysis and control of linear systems by Philippe de Larminat

By Philippe de Larminat

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

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2. 1] – equation of measurement: y (t ) = C(t ) x(t ) + D(t ) u(t ) . 2] We will approach the resolution of the equation of state in two instances: – free state u(t ) = 0 ; – forced state u(t ) ≠ 0 . 1. Free state This refers to solving the equation x(t ) = A (t ) x(t ) from the initial condition x(t 0 ) = x 0 . Since the equation is linear, the solution x(t ) is expressed linearly according to x(t 0 ) as follows: x(t ) = Φ (t, t0 ) x(t0 ) The matrix Φ (t, t0 ) of size n × n is called a system transition matrix.

Frequency response H ( jω ) = K ⎛ ω2 ⎜1 − ⎜ ω2 0 ⎝ ⎞ ⎟ + 2 jξ ω ⎟ ω0 ⎠ ξ ≥ 1 : the system is a cascading of two systems of the first order H 1 and H 2 . 23). Transfer Functions and Spectral Models 31 ξ < 1 : the characteristics of the frequency response vary according to the value of ξ . Module and phase are obtained from the following expressions: H ( jω ) = K 2 φ (ω ) = − Arctg ( ⎛ ω2 ⎞ ω2 ⎜ 1 − 2 ⎟ + 4 ξ2 2 ⎜ ω ⎟ ω0 0 ⎠ ⎝ For ξ < 1 2 , the module reaches a maximum Amax = 2ξωωo ωo2 − ω 2 ) K 2ξ 1 − ξ 2 in an angular frequency called of resonance ω r = ω 0 1 − 2ξ 2 .

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.

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