Modern Control Engineering

530 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method Re Im Re Re Im Im 1 – 1 G Plane G Plane G ...

Example Problems and Solutions 531 Re Im 4 3 v= 2.45 v= 2 v= 1.5 –1 2 1 1 123 v= 1 6 v=^0 (^98) 10 v= 0.5 2.65 1 +jv 2.65e– 0. ...

532 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method A–7–11. Consider a unity-feedback system wi ...

Example Problems and Solutions 533 Real Axis – 3 – 2 – (^10123) Imag Axis 3 5 1 1 4 2 0 Nyquist Plot of G(s)= 20(s^2 +s+0.5) ...

534 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method MATLAB Program 7–19 num = [20 20 10]; den = ...

Example Problems and Solutions 535 MATLAB Program 7–20 num = [-1 -4 -6]; den = [1 5 4]; nyquist(num,den); grid title('Nyquist Pl ...

536 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method Note that the Nyquist plot for the positive ...

Example Problems and Solutions 537 Real Axis 2 – 1.5 – 1 – 0.5 0 0.5 1 1.5 2 Imag Axis 0.4 0.2 1 1 0.2 0.4 0.6 0.8 0 0 ...

538 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method Figure 7–132 Polar plot of K/G(jv). theK/G( ...

Example Problems and Solutions 539 Hence, we set from which we obtain Since the phase curve never crosses the –180° line, the ga ...

540 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 0 ...

Example Problems and Solutions 541 40 20 20 dB 0 40 60 80 0.1 0.2 0.4 0.6 1 2 4 6 10 20 4060 100 270 ° 180 ° 90 ° 0 ° v in ...

542 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method Consider the system shown in Figure 7–136. ...

Example Problems and Solutions 543 Solution.The angle of Gc(jv)is given by At we have Since or and also we have Thus, the angle ...

544 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method MATLAB Program 7–22 num = [10 1]; den = [1 ...

Example Problems and Solutions 545 If we use the command polar(theta,r) MATLAB will produce a plot in the polar coordinates. Sub ...

circle, and the phase angle of this point is –120°. Hence, the phase margin is 60°. The fact that pointG(j1.15)is on the unit ci ...

Example Problems and Solutions 547 Figure 7–143 shows the Bode diagram of the closed-loop transfer function. The magnitude curve ...

Clearly, the denominator of the closed-loop transfer function is a product of two lightly damped second-order terms (the damping ...

Example Problems and Solutions 549 MATLAB Program 7–24 produces the Bode diagram shown in Figure 7–145. From this plot, the phas ...