By Philippe de Larminat
Automation of linear structures is a basic and crucial concept. This ebook offers with the speculation of continuous-state automatic structures.
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Additional resources for Analysis and control of linear systems
The idea is to add to the pure imaginary argument 2π j f a real part σ which is chosen in order to converge the integral considered: +∞ ∫ x(t)e −∞ −(σ + 2π jf ) t dt Transfer Functions and Spectral Models 35 By determining that p = σ + 2πjf , we define a function of a complex variable, called the Laplace transform of x(t ) , defined into a vertical band of the complex plane, which is determined by the conditions on σ ensuring the convergence of the integral: X ( p) = +∞ − pt ∫ x(t )e dt = TL(x(t )) −∞ The instability phenomena that can interfere in a linear system are characterized by exponential divergence signals; hence, we perceive the interest in the complex variable transformations for the analysis and synthesis of linear systems.
4. Laplace transforms of ordinary causal signals x(t) X(p) σ0 convergence δ (0) 1 −∞ δ ((0n)) pn −∞ U (t ) = 1 t ≥ 0 1 p 0 1 p+a −a U (t ) = 0 t < 0 U (t )e − at U (t )t n U (t )t n e − at U (t ) sin(ωt ) U (t ) cos(ωt ) U (t )e − at sin(ωt ) U (t )e − at cos(ωt ) n! p n +1 n! 5. 7. , Linear Control System Analysis and Design, McGraw-Hill, 1988 (3rd edition). , Cours de mathématiques, vol. III, Masson, Paris, 1971. , Automatique, Hermès, Paris, 1996 (2nd edition). , Distributions et Transformation de Fourier, Ediscience, 1971.
6. Causality, stability and transfer function We have seen that the necessary and sufficient condition of stability of an SLI is for its impulse response to be absolutely integrable: ∫− ∞ h(θ ) dθ < +∞ . +∞ The consequence of the hypothesis of causality modifies this condition because we thus integrate from 0 to +∞ . On the other hand, if we seek a necessary and sufficient condition of stability for the expression of transfer functions, the hypothesis of causality is determining. Since the impulse response h(θ ) is a causal function, the transfer function H ( p) is holomorphic (defined, continuous, derivable with respect to the complex number p) in a right half-plane defined by Re ( p) > σ o .