Question

Please explain every step as clearly and detailed as possible.

B Frequency Response Modeling Frequency response modeling of a linear system is based on the premise that the dynamics of a linear system can be recovered from a knowledge of how the system responds to sinusoidal inputs. (This will be made mathematically precise in Theorem 13.) In other words, to determine (or iden- tify) a linear system, all one has to do is observe how the system reacts to sinusoidal inputs Let's assume that we have a linear system governed by where p and q are real constants. The function g(t) is called the forcing function or input function. When g(t) is a sinusoid, the particular solution to (10) obtained by the method of undetermined coefficients is the steady-state solution or output function yss (t) corresponding to g(t). We can think of a linear system as a compartment or block into which goes an input function g and out of which comes the output function yss (see Figure 7.32). To identify a linear system means to determine the coefficients p and q in equation (10). Linear system Input g (t) Output Figure 7.32 Block diagram depicting a linear system It will be convenient for us to work with complex variables. A complex number z is usually expressed in the form z-α + ia with α, β real numbers and i denoting V-1. We can also express z in polar form, z-re". where r-r + F and tan θ-β/α. Here r (20) is called the magnitude and θ the phase angle of z. The following theorem gives the relationship between the linear system and its response to sinusoidal inputs in terms of the transfer function H(s) [see Project A, equation (2), page 417]. Steady-State Solutions to Sinusoidal Inputs Theorem 13. Let H(s) be the transfer function for equation (10). If H(s) is finite at S-lu, with ω real, then the steady-state solution to (10) for g(t)-zu is Prove Theorem 13. [ Hint: Guess y,s(t) -Azu and show that A -H ( io). ] (a) (b) Use Theorem 13 to show that if g()sin ot, then the steady-state solution to (10) is M() sin ot +N()]. where H(ia) M()i the polar form for H(io) (c) Solve for M() and N() in terms of p and q (d) Experimental results for modeling done by frequency response methods are usually pre- sented in frequency response or Bode plots. There are two types of Bode plots. The first is the log of the magnitude M(a) of H (m) versus the angular frequency ω using a log scale for ω. The second is a plot of the phase angle or argument N(a) of H(io) versus the angular frequency using a log scale for. The Bode plots for the transfer function H(s)0.2s +) are given in Figure 7.33 1.0 0.5 E -20 3-40 -0.5 6 -80 -100 -120 -140 -160 -180 0.1 0.2 04 0.71.0247 10 Angular frequeney w 0.1 0.2 04 0.7 1.0247 10 Angular frequency Figure 7.33 Bode plots for H(w)-[ 1 + 0.2(ίω) + (m)2r Sketch the Bode plots of the linear system governed by equation (10) with p-04 and q-1.0. Use ω-0.3, 0.6, 0.9, 1.2, and 1.5 for the plot of M (ω) and ω-05. 0.8, 1, 2, and 5 for the plot of N() (e) Assume we know that q-1. When we input a sine wave with ω-2, the system settles into a steady-state sinusoidal output with magnitude M(2)0.333. Find p and thus iden- tify the linear system. (f) Suppose a sine wave input with2 produces a steady-state sinusoidal output with magnitude M(2) identify the system. 0.5 and that when ω-4, then M (4)-0.1. Find p and q and thus Frequency response curves are also discussed in Section 4.10.

Answer 1

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