Control System Toolbox imtiaz.hussain@faculty.muet.edu.pk

Outline Transfer Function Models Pole-Zero maps Simplification of Block Diagrams Time Domain Analysis of 1st and 2nd Order Systems Frequency Domain Analysis of Control Systems Root Locus of LTI models Excercises

Transfer Function Model Using Numerator & Denominator Coefficients This transfer function can be stored into the MATLAB num = 100; den = [1 14 10]; sys=tf(num,den) To check your entry you can use the command printsys as shown below: printsys(num,den); 11/21/2018

Transfer Function Model Using Zeros, Poles and Gain (ZPK model) This transfer function can be stored into the MATLAB Zeros=-3; Poles= [-1 -2]; K=100; sys=zpk(Zeros,Poles,K) 11/21/2018

Poles & Zeros To plot the poles and zeros of any transfer function there is a built in function pzmap in the MATLAB pzmap(num,den) 11/21/2018

Series Blocks Blocks in series can be simplified by using series command S 9S + 17 9(S+3) 2S2 + 9s + 27 num1 = [1 0]; den1 = [9 17]; num2 = 9*[1 3]; den2 = [2 9 27]; [num12, den12] = series (num1,den1,num2,den2); printsys(num12,den12); 11/21/2018

Contd… Series Blocks Blocks in series can also be simplified by using conv command S 9S + 17 9(S+3) 2S2 + 9s + 27 num1 = [1 0]; den1 = [9 17]; num2 = 9*[1 3]; den2 = [2 9 27]; num12 =conv(num1,num2); den12 = conv(,den1,den2); printsys(num12,den12); 11/21/2018

Parallel Block Blocks in parallel can be simplified by using parallel command num1 = [1 2]; den1 = [1 2 3]; num2 = [1 3]; den2 = [1 -4 1]; [num, den]=parallel(num1,den1,num2,den2); printsys(num,den); 11/21/2018

Closed-loop transfer function The block in feedback can be reduced using feedback command. C(S) R(S) - 1 S + 1 2 S num1 = 1; den1 = [1 1]; num2 = 2; den2 = [1 0]; [numcl,dencl] = feedback(num1,den1,num2,den2,-1); printsys(numcl,dencl) 11/21/2018

Time Response of 1st order Systems The first order system has only one pole. Where K is the D.C gain and T is the time constant of the system. Time constant is a measure of how quickly a 1st order system responds to a unit step input. D.C Gain of the system is ratio between the input signal and the steady state value of output.

Step Response To obtain the step response of the system num = 10; den = [3 1]; step(num,den) or num = 10; den = [3 1]; sys=tf(num, den) step(sys)

Impulse Response To find the step response of the system num = 10; den = [3 1]; impulse(num,den) 11/21/2018

Ramp Response To find the ramp response of the system t = 0:0.01:10 r = t; num = 10; den = [2 1]; lsim(num,den,r,t) 11/21/2018

Time Response of 2nd Order Systems A general second-order system is characterized by the following transfer function.

Undamped Natural Frequency & Damping ratio Determine the un-damped natural frequency and damping ratio of the following second order system. To find the undamped natural frequency and damping ratio in matlab num=4 den=[1 2 4] sys=tf(num, den) damp(sys)

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 11/21/2018

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 11/21/2018

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.

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) 11/21/2018

Polar plots or Nyquist plot -w w 11/21/2018

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 11/21/2018

Polar plots or Nyquist plot 11/21/2018

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

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 11/21/2018

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]) 11/21/2018

Magnitude-Phase plot or Nichols Chart

Magnitude-Phase plot or Nichols Chart

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)

Root Locus of 1st Order System Consider the following unity feedback system Matlab Code num=1; den=[1 0]; G=tf(num,den); rlocus(G) sgrid

Root Locus of 1st Order System Consider the following unity feedback system num=[1 0]; den=[1 1]; G=tf(num,den); rlocus(G) sgrid Continued…..

Root Locus 2nd order systems Consider the following unity feedback system num=1; den1=[1 0]; den2=[1 3]; den=conv(den1,den2); G=tf(num,den); rlocus(G) sgrid Determine the location of closed loop poles that will modify the damping ratio to 0.8 and natural undapmed frequency to 1.7 r/sec. Also determine the gain K at that point.

Root Locus of Higher Order Systems Consider the following unity feedback system num=1; den1=[1 0]; den2=[1 1]; den3=[1 2]; den12=conv(den1,den2); den=conv(den12,den3); G=tf(num,den); rlocus(G) sgrid Determine the closed loop gain that would make the system marginally stable.

Root Locus of a Zero-Pole-Gain Model k=2; z=-5; p=[0 -1 -2]; G=zpk(z,p,k); rlocus(G) sgrid

Root Locus of a State-Space Model B=[1;0]; C=[1 0]; D=0; sys=ss(A,B,C,D); rlocus(sys) sgrid

Choosing Desired Gain num=[2 3]; den=[3 4 1]; G=tf(num,den); [kd,poles]=rlocfind(G) sgrid

Please attach the hard copy of following exercises under the title solution to lab-6 in your practical book. Exercises

Exercise#1 Simplify the following block diagram and determine the following (Assume K=10). Closed loop transfer function (C/R) Poles Zeros Pole-zero-map

Exercise#2 Simplify the following block diagram and determine the following. Closed loop transfer function (C/R) Poles Zeros Pole-Zero-map + - R C

Exercise-3: Obtain the step and ramp responses of the following 1st order system for T=1, 2, 3 5, 10 seconds. 11/21/2018

Exercise-4: Obtain the step and ramp responses of the following 1st order system for K=1, 5, 10 and 15. 11/21/2018

Exercise-5: Obtain the step response of the following 1st order system with zero for K=1, α=4 and T=3 sec K=1, α=3 and T=4 sec K=1, α=3 and T=3 sec And compare the results with system w/o zero 11/21/2018

Exercise#6 Describe the nature of the second-order system response via the value of the damping ratio for the systems with transfer function

Exercise-7: Obtain the pole zero map and step response of the 2nd order system and determine the mode of damping in the system. If the system is underdamped obtain the time domain specifications. On the pole zero map show that corresponding damping ratio and natural undamped frequency of the poles. ωn=3 r/s and ζ=1 ωn=3 r/s and ζ=2 ωn=3 r/s and ζ=0.1 ωn=3 r/s and ζ=0.5 ωn=3 r/s and ζ=0

Exercise-8: Obtain the step response of the 2nd order system if ωn=0.1 r/s and ζ=0.5 ωn=0.3 r/s and ζ=0.5 ωn=0.6 r/s and ζ=0.5 ωn=1 r/s and ζ=0.5 ωn=1.5 r/s and ζ=0.5

Exercise#9 Consider the system shown in following figure, where damping ratio is 0.6 and natural undamped frequency is 5 rad/sec. Obtain the rise time tr, peak time tp, maximum overshoot Mp, and settling time 2% and 5% criterion ts when the system is subjected to a unit-step input.

Bode Plots Exercise-10: - 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 11/21/2018

Bode Plots Exercise-11: - 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 Exercise-12: - 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 11/21/2018

Exercise-13: - 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. 11/21/2018

Exercise#14 Obtain the phase and gain margins of the system shown in following figure for the two cases where K=10 and K=100.

Exercise#15 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.

Plot the root locus of following first order systems. Exercise#16 Plot the root locus of following first order systems.

Plot the root locus of following 2nd order systems. Exercise#17 Plot the root locus of following 2nd order systems. (1) (2)

Exercise#18 Plot the root locus of following systems. (1) (2)

Exercise#19 Plot the root Loci for the above ZPK model and find out the location of closed loop poles for =0.505 and n=8.04 r/sec. b=0.505; wn=8.04; sgrid(b, wn) axis equal Open File  New  M-File Save File as H:\ECE4115\Module1\Mod1_1.m

Exercise#20: Consider the following unity feedback system Plot the root Loci for the above transfer function Find the gain when both the roots are equal Also find the roots at that point Determine the settling of the system when two roots are equal. Open File  New  M-File Save File as H:\ECE4115\Module1\Mod1_1.m

Exercise#21 Consider the following velocity feedback system Plot the root Loci for the above system Determine the gain K at which the system produces sustained oscillations with frequency 8 rad/sec. Open File  New  M-File Save File as H:\ECE4115\Module1\Mod1_1.m

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