New PDF release: Analysis and control of linear systems

By Philippe de Larminat

ISBN-10: 1905209355

ISBN-13: 9781905209354

Automation of linear platforms is a primary and crucial thought. This ebook offers with the speculation of continuous-state automatic platforms.

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Extra resources for Analysis and control of linear systems

Example text

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 . We note that the smaller ξ , the more significant this extremum and the more the phase follows its asymptotes to undertake a sudden transition along ω o .

4. 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.

Chapter 2 State Space Representation Control techniques based on spectral representation demonstrated their performances though numerous industrial implementations, but they also revealed their limitations for certain applications. The objective of this chapter is to provide the basis for a more general representation than the one adopted for the frequency approach and to offer the necessary elements to comprehend time control through the state approach. Time control, which is largely used in space and aeronautic applications but also in industrial applications such as servomechanisms, is based on the representation of systems through state variables.

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Analysis and control of linear systems by Philippe de Larminat


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