Digital Control Systems [MEE 4003] - Kckong.info

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2.2. LAPLACE TRANSFORMS AND TRANSFER FUNCTIONS 43 which is called the characteristic equation. Also remind that a(s) = det{sI −F} and the eigenvalues of the matrixF are calculated from det{λI −F} = 0 Therefore, the poles ofG(s) are equivalent to the eigenvalues of the state matrix F . A system is asymptotically stable if • all the poles have strictly negative real parts (in the Laplace domain), or • all the eigenvalues have strictly negative real parts (in the state space). Digital Control Systems, Sogang University Kyoungchul Kong
2.3. FREQUENCY RESPONSE ANALYSIS 44 2.3 Frequency Response Analysis 2.3.1 Fourier transform versus Laplace transform There are several common conventions for defining the Fourier transform F(jω) of an integrable functionf(t). A common expression is F(jω) = ∫ ∞ 0 f(t)e −jωt dt where ω ∈ R + is a positive scalar. Notice that the Fourier transform is related to the Laplace transform F(s) = ∫ ∞ 0 f(t)e −st dt byjω = s. Therefore, Fourier-transformed functions can be obtained by replacingjω for s. [Example 2-15] The following functions (signals) are equivalent: Description Time-domain Laplace-domain Frequency-domain (t ∈ R + ) (s ∈ C) (ω ∈ R + ) Unit impulse δ(t) 1 1 Delayed impulse δ(t−τ) e −τs e −jωτ Unit step u(t) 1 s e Delayed unit step u(t−τ) −τs s 1 Ramp t n th power for an integern t n n! Exponential decay e −αt 1 Sine sin(ω 0 t) 1 jω e −jωτ jω − 1 s 2 ω 2 1 1 s n+1 s+α ω 0 s 2 +ω0 2 (jω) n+1 1 jω+α ω 0 ω0 2−ω2 Notice that the unit impulse signal includes all frequency components. In the Laplace domain, all the coefficients are real numbers but the Laplace operator, s, is a complex number. In the frequency domain, however, the coefficients are complex numbers while the frequency term,ω, is a positive real number. As the input and output signals can be transformed into the frequency domain by taking the Fourier transform, its relationship can also be represented in the frequency domain. The frequency-domain input-output relationship, which is usually represented by a Bode plot, is obtained by replacing jω forsof a transfer function. Digital Control Systems, Sogang University Kyoungchul Kong
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