Analysis and control of linear systems by Philippe de Larminat

By Philippe de Larminat

Automation of linear structures is a basic and crucial concept. This ebook bargains with the idea of continuous-state computerized structures.

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This will be presented in this chapter, along with a few fundamental properties such as stability, controllability and observability. Chapter written by Patrick BOUCHER and Patrick TURELLE. 1. 1. 1. Dynamic system The controls and outputs are generally multiple and their evolution is as a function of time. In the determinist case which we are dealing with here, knowing the controls u(t ) from −∞ makes it possible to know the outputs y (t ) at instant t. 1. Internal representation of determinist systems: the concept of state A system can be represented externally through the relations that link the inputs to the outputs, which are described in the vector form: y(t ) = h{u(τ )} τ ∈ [0 , ∞ [ Hence, linear and invariant systems (LIS) are traditionally represented, in the mono-variable case, by the convolution equation: ∞ y (t ) = h(t ) ∗ u (t ) = ∫ ∞ h(τ ) u (t − τ )dτ = −∞ ∫ h(t − τ ) u(τ ) dτ −∞ This representation is linked to the concept of transfer function by Laplacian transformation of the convolution equation: Y ( p) = H ( p) U ( p) However, we are soon limited at the level of these representations by the nonlinearities and non-stationarity of systems.

According to the matrix structure adopted, this modeling is also currently used during the synthesis of control laws, irrespective of the method chosen. This internal representation, which is richer and more global than the inputoutput representation, is enabling the representation, in the form of a matrix, of any system: invariant or non-invariant, linear or non-linear, mono-variable or multivariable, continuous or discrete. This will be presented in this chapter, along with a few fundamental properties such as stability, controllability and observability.

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 . 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 .

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