Modern Control Engineering

430 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method Im Re v =` v = 0 Resonant peak vn vr 0 Figu ...

Section 7–3 / Polar Plots 431 Im Re v = 0 01 v ` Figure 7–30 Polar plot of 1 + 2 zaj forz>0. v vn b+ aj v vn b 2 Next, consid ...

432 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method The general shape of the polar plot of G(jv ...

Section 7–3 / Polar Plots 433 Im (^0) Re 0 0 v = 0 v v v v v ` Type 2 system Type 1 system Type 0 system Figure 7–33 Polar plo ...

434 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method denominator polynomial of G(jv)is greater t ...

Section 7–3 / Polar Plots 435 v Im Re Im (^0) Re 0 0 Im Re Im (^00) Re Im Re Im (^0) Re Im Im Im Im Re Re Re Re 0 0 0 0 1 0 v v ...

436 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method The command nyquist(num,den) draws the Nyqu ...

Section 7–3 / Polar Plots 437 MATLAB Program 7–5 num = [1]; den = [1 0.8 1]; nyquist(num,den) grid title('Nyquist Plot of G(s) = ...

438 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method Caution. In drawing a Nyquist plot, where a ...

Section 7–3 / Polar Plots 439 Real Axis − 212 −1.5 − 10 −0.5 0.5 1.5 Imaginary Axis − 2 1 − 5 5 − 1 2 − 3 − 4 0 3 4 Nyquist Plot ...

440 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method Drawing Nyquist Plots of a System Defined i ...

Section 7–3 / Polar Plots 441 MATLAB Program 7–9 A = [0 1;-25 -4]; B = [0;25]; C = [1 0]; D = [0]; nyquist(A,B,C,D) grid title(' ...

442 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method EXAMPLE 7–13 Consider the system defined by ...

Section 7–4 / Log-Magnitude-versus-Phase Plots 443 7–4 Log-Magnitude-versus-Phase Plots Another approach to graphically portrayi ...

444 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method and Figure 7–43 compares frequency-response ...

Section 7–5 / Nyquist Stability Criterion 445 |G | in dB 20 10 0 –10 –20 –180° 0 ° 180 ° G |G | in dB 20 10 0 –10 –20 –180° 0 ° ...

446 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method For stability, all roots of the characteris ...

Section 7–5 / Nyquist Stability Criterion 447 s Plane 3 F(s) Plane 2 220 34 2 3 Re Im jv v=– 2 s=– 1 s=– 2 s= 1 s v= 0 v= 2 ...

448 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method jv j 2 s Plane j 1 0 j 1 j 2 2 – 121 3 A ...

Section 7–5 / Nyquist Stability Criterion 449 We shall not present a formal proof of this theorem here, but leave the proof to P ...