Control System Toolbox (Part-III) imtiaz.hussain@faculty.muet.edu.pk

Outline Frequency Domain Analysis of Control Systems Bode Plot Nyquist Plot Nichol’s Chart Gain & Phase Margins

Bode Plots To obtain the bode plot use following MATLAB code num =10*[1 10]; den1= [1 0]; den2=[1 2]; den3=[1 5]; den12=conv(den1,den2); den=conv(den12,den3); bode(num,den) grid on K =10; zero=-10; pole=[0 -2 -5]; sys=zpk(k, zero, pole); bode(sys) grid on 5/13/2019

Bode Plots

Bode Plots (for specific Frequency range) To obtain the bode plot for specific frequency range w=0.1:0.1:100; num =10*[1 10]; den1= [1 0]; den2=[1 2]; den3=[1 5]; den12=conv(den1,den2); den=conv(den12,den3); bode(num,den,w) grid on 5/13/2019

Bode Plots (for specific Frequency range)

[mag, pha, w] = bode(num, den, w) Bode Plots (with left hand arguments) When invoked with left-hand arguments, such as [mag, pha, w] = bode(num, den, w) bode returns the magnitude (mag), phase (pha) and frequency (w) of the system in matrices.

Bode Plots Exercise-1: - Obtain the bode plot of the second order system ωn = 0.1 rad / sec ζ = 0.1, 0.5, 1, 1.5 G(S) = (S2 + 0.02S+0.01) 0.01 G(S) = (S2 + 0.01S+0.01) 0.01 G(S) = (S2 + 0.02S+0.01) 0.01 G(S) = (S2 + 0.03S+0.01) 0.01 5/13/2019

Bode Plots Exercise-2: - Calculate the magnitude and phase of the following system at w (rad/sec)=0.1, 0.5, 1, 10, 100.

Polar plots or Nyquist plot To obtain the Nyquist plot use the following MATLAB code num =2500; den1= [1 0]; den2=[1 5]; den3=[1 50]; den12=conv(den1,den2); den=conv(den12,den3); nyquist(num,den) 5/13/2019

Polar plots or Nyquist plot -w w 5/13/2019

Polar plots or Nyquist plot To adjust the default axes of Nyquist plot use axis command num =2500; den1= [1 0]; den2=[1 5]; den3=[1 50]; den12=conv(den1,den2); den=conv(den12,den3); nyquist(num,den) axis([-2.5 0 -2 2]) grid on 5/13/2019

Polar plots or Nyquist plot 5/13/2019

Nyquist plot (Open-loop & Closed Loop Frequency Response )

Polar plots or Nyquist plot Exercise-3: - consider following transfer function Obtain the Nyquist plot of the following system (when w>0). Determine the open-loop & closed-loop magnitude responses when w=2.5 rad/sec 5/13/2019

Magnitude-Phase plot or Nichols Chart To obtain the Nichols plot use following MATLAB code num =2500; den1= [1 0]; den2=[1 5]; den3=[1 50]; den12=conv(den1,den2); den=conv(den12,den3); nichols(num,den) ngrid 5/13/2019

Magnitude-Phase plot or Nichols Chart

Magnitude-Phase plot or Nichols Chart To obtain the Nichols plot use following MATLAB code num =2500; den1= [1 0]; den2=[1 5]; den3=[1 50]; den12=conv(den1,den2); den=conv(den12,den3); nichols(num,den) ngrid; axis([-220 -90 -40 40]) 5/13/2019

Magnitude-Phase plot or Nichols Chart

Magnitude-Phase plot or Nichols Chart

Exercise-4: - For the following Transfer Function (i) Obtain the Nichols Chart (ii) Determine the open-loop as well as closed-loop magnitude and phase when w=1.39 rad/sec. 5/13/2019

Phase & Gain Margins To obtain the gain margin and phase use the following mat lab code. num =2500; den1= [1 0]; den2=[1 5]; den3=[1 50]; den12=conv(den1,den2); den=conv(den12,den3) [GM, PM, wp, wg]=margin(num,den)

Phase & Gain Margins GM = 5.5000 PM = 31.7124 num =2500; den1= [1 0]; To obtain the gain margin and phase use the following mat lab code. GM = 5.5000 PM = 31.7124 wp = 15.8114 wg = 6.2184 num =2500; den1= [1 0]; den2=[1 5]; den3=[1 50]; den12=conv(den1,den2); den=conv(den12,den3) [GM, PM, wp, wg]=margin(num,den)

Phase & Gain Margins To obtain the gain margin and phase use the following mat lab code. num =2500; den1= [1 0]; den2=[1 5]; den3=[1 50]; den12=conv(den1,den2); den=conv(den12,den3) margin(num,den)

Phase & Gain Margins

System Characteristics (Bode Plot)

System Characteristics (Nyquist Plot)

System Characteristics (Nichol’s Chart)

Exercise#5 Obtain the phase and gain margins of the system shown in following figure for the two cases where K=10 and K=100. Also obtain the Bode, Nyquist and Nichol's plots for either cases.

Exercise#5

Exercise#6 Consider the system shown in following figure. Obtain Bode diagram for the closed-loop transfer function. Obtain also the resonant peak, resonant frequency, and bandwidth.

End of Tutorial You can Download this tutorial from http://imtiazhussainkalwar.weebly.com/ End of Tutorial 5/13/2019