Control System lab Experiments using Compose (VTU University Syllabus)

Transcription

1 Control System lab Experiments using Compose (VTU University Syllabus) Author: Sijo George Reviewer: Sreeram Mohan Altair Engineering, Bangalore Version: 1 Date: 09/11/2016 1

2 CONTENTS EXP. 1: STEP RESPONSE OF A SECOND ORDER SYSTEM... 3 EXP.1.B STEP RESPONSE OF A SECOND ORDER SYSTEM FOR VARIOUS VALUES OF ZETA EXP.2 EFFECT OF P, PD, PI AND PID CONTROLLER EXP.3 ROOT LOCUS, BODE AND NYQUIST PLOT EXP.4 RELATION BETWEEN FREQUENCY RESPONSE AND TRANSIENT RESPONSE TABLE OF FIGURES FIGURE 1.a: STEP RESPONSE AND ADDITION OF ZERO TO THE GIVEN SYSTEM FIGURE 1.b: STEP RESPONSE FOR DIFFERENT DAMPING RATIO OF A SECOND ORDER SYSTEM FIGURE 2.a: EFFECT P, PD, PI AND PID CONTROLLERS FIGURE 3.a: ROOT LOCUS AND EFFECT OF ADDING POLES AND ZEROS 165 FIGURE 3.b: BODE PLOT OF THE GIVEN SYSTEM FIGURE 3.c: NYQUIST PLOT OF THE GIVEN SYSTEM...28 FIGURE 4.a: BODE PLOT FOR TRANSIENT AND FREQUENCY RESPONSE...31 FIGURE 4.b: RESONANT PEAK, MAX.OVERSHOOT AND DAMPING RATIO FIGURE 4.c: BANDWIDTH AND DAMPING RATIO

3 EXP. 1: STEP RESPONSE OF A SECOND ORDER SYSTEM Aim & Description: (1)To determine the step response of a second order system and evaluation of time domain specifications. (2) To evaluate the effect of additional zeros on time response of second order system. Step response is the time behavior of the outputs of a general system when its inputs change from zero to one in a very short time. G ( s) 3 s 1 10s 20 Software code: clc; clear; close all; sys.num = [1]; sys.den = [ ]; sys1=tf(sys); kp=dcgain(sys) ess=1/(1+kp) w = sqrt (sys.den(3)) zeta = sys.den(2) / (2*w) TD=(1+0.7*zeta)/w TS = 4/ (zeta*w) % Numerator % Denominator % Finding transfer function % gain when frequency is zero(s=0) % Steady state error % Finding natural frequency % Damping ratio % Delay time % Settling time TP = pi/ (w*sqrt(1-zeta^2)) % Peak Time TR=(pi-atan((sqrt(1-zeta^2))/zeta))/(w*sqrt(1-zeta^2)) Percentovershoot= exp(-zeta*pi/ sqrt(1-zeta^2))*100 % Rise Time % Overshoot subplot(2,2,1) [z,t]=step(sys); % Step response plot(t,z); title('step response'); xlabel('time (sec)'); 3

4 ylabel('amplitude');grid on; subplot(2,2,2) pzmap(sys.num,sys.den); % Pole zero plot title('pole zero map of the given system') grid on; %% Adding Zero to the given system sys.num = conv(sys.num,[1 1]); sys2=tf(sys) [f,t]=step(sys); subplot(2,2,3) plot(t,f); title('step response after addition of zero') xlabel('time (sec)'); ylabel('amplitude');grid on; subplot(2,2,4) pzmap(sys.num,sys.den) % Pole zero plot when zero is added title('pole zero map when zero is added') grid on; % Function created to display transfer function using string concept function [sys]=tf(varargin) if length(varargin)==1 sys=varargin{1}; b=sys.num; a=sys.den; if length(varargin)==2 b=varargin{1}; a=varargin{2}; 4

5 [fopen_fileid] = fopen('testfile.txt', 'w'); %open a text file l=length(b); m=length(a); % Number of numerator coefficients % Number of denominator coefficients %%%% Numerator if length(b)==1 % Check the numerator has only one coefficient fprintf(fopen_fileid, ' %g\n',b(1)); % Print numerator fprintf(fopen_fileid,' %s',' '); % to draw the separation b/w numerator and denominator if length(b)==2 % Check the numerator has two coefficients c{1}=sprintf('%g%s',b(1),'s'); % store first numerator coefficient in a cell 'c{1}' c{2}=b(); % store second numerator coefficient in a cell 'c{2}' fprintf(fopen_fileid, '%s',c{1}); % print numerator if b()>0 fprintf(fopen_fileid, '%s','+'); % to print + sign before the last coefficient if it is positive fprintf(fopen_fileid, '%g\n',c{}); fprintf(fopen_fileid,' %s',' '); % to draw the separation b/w numerator and denominator if length(b)>2 i=l-1; for k=1:l-2 coefficient % Check the numerator has more than two coefficients % Initializing value of i to print the exponential term % Loop defined for the numerator coefficients excluding the last and second last h=sprintf('%g%s%g',b(k),'s^',i); % Print numerator coefficient and storing to the variable 'h' c{k}=h; i=i-1; %store the numerator coefficient in a cell 'c' % Decrementing the power of i for printing the exponential terms in the correct way c{k+1}=sprintf('%g%s',b(k+1),'s'); % store the second last coefficient in 'c' c{k+2}=b(); % store last coefficient in 'c' c; 5

6 for t=1:length(c)-1 fprintf(fopen_fileid, '%s',c{t}); transfer function % Print the numerator coefficients except the last coefficient of the R=cell2mat(c(t+1)); T=strcmp(R(1),'-'); %returns 1 if the comparison matches if (t~=(length(c)-1)) if T==0 fprintf(fopen_fileid, '%s','+'); %to print + sign if the next coefficient is positive except the last digit if b()>0 fprintf(fopen_fileid, '%s','+'); %To print + sign before it when the last coefficient is positive fprintf(fopen_fileid,'%g\n',c{}); % Print the last numerator coefficient of the transfer function fprintf(fopen_fileid,' %s',' '); % Denominator if length(a)==1 % Check if the denominator has only one coefficient fprintf(fopen_fileid, '\n %g\n',a(1)); if length(a)==2 f{1}=sprintf('\n % Check if the denominator has two coefficients %g%s',a(1),'s'); f{2}=a(); fprintf(fopen_fileid, '%s',f{1}); % print denominator if a()>0 fprintf(fopen_fileid, '%s','+'); %to print + sign before the last coefficient it is positive fprintf(fopen_fileid, '%g\n',f{}); %print the last coefficient 6

7 if length(a)>2 %Check if the length of denominator has more than 2 coefficients j=m-1; for g=1:m-2 e=sprintf('%g%s%g',a(g),'s^',j); % storing s terms and its coefficient power in a variable f{g}=e; %assigning to a cell j=j-1; f{g+1}=sprintf('%g%s',a(g+1),'s'); f{g+2}=a(); fprintf(fopen_fileid, '\n term %s',f{1}); %to print the first coefficient of denominator along with the s R2=cell2mat(f(2)); T2=strcmp(R2(1),'-'); %returns 1 if the comparison matches if T2==0 fprintf(fopen_fileid, '%s','+'); %To print + sign before second term if it is a positive for u=2:length(f)-1 fprintf(fopen_fileid, '%s',f{u}); % to print the coefficients from second to last R1=cell2mat(f(u+1)); %converting cell to matrix T1=strcmp(R1(1),'-'); % gives 1 as output if the comparison matches if (u~=(length(f)-1)) if T1==0 fprintf(fopen_fileid, '%s','+'); %to print the + sign if the next coefficient is positive if a()>0 fprintf(fopen_fileid, '%s','+'); %to print + sign if the last coefficient is positive 7

8 fprintf(fopen_fileid, '%g\n',f{}); %to print the last coefficient fclose(fopen_fileid); content = type('testfile.txt'); sys=content{1,1} ; % Function for finding step response function [z,t]= step(varargin) if length(varargin)==1 % Check whether the input variable is of length 1 sys=varargin{1}; % store the incoming structure to variable sys num=sys.num; den=sys.den; t=0:0.1:25; % store numerator coefficients in variable num % store denominator coefficients in variable den % Plotting time length if length(varargin)==2 % Check whether the input variable is of length 1 num=varargin{1}; % store the numerator to variable num den=varargin{2}; % store the numerator to variable num t=0:0.1:25; % Plotting time length if length(varargin)==3 num=varargin{1}; den=varargin{2}; t=varargin{3}; % User defined time length [A,B,C,D]=tf2ss(sys.num,sys.den); % Finding State space matrices f=length(a); U=1; %Size of the A matrix % Step input 8

9 K=[1]; % Weighting factor for i=1:length(t) h{i}=c*(a^-1)*(e^(a*t(i))-eye(f))*b*k +D*K*U; % finding step response values z=cell2mat(h); % Conversion of cell to matrix % Function for finding pole zero map function pzmap(varargin) if length(varargin)==1 sys=varargin{1}; % assigning structure to variable sys num=sys.num; %Numerator den=sys.den; % Denominator if length(varargin)==2 num=varargin{1}; den=varargin{2}; % Numerator % Denominator z=roots(num); p=roots(den); % Finding zeros % Finding poles %%%.For Plotting poles for i=1:length(p) R=isreal(p(i)); % Check whether the poles are real or not switch (R) case 1 % case when Poles are real p1=[p(i),0]; p2=[p(i),0]; % Defining first point for plotting poles % Defining second point for plotting poles theta = atan2( p2(2) - p1(2), p2(1) - p1(1)); % Finding slope r = sqrt( (p2(1) - p1(1))^2 + (p2(2) - p1(2))^2); % Finding the length between the first and second point line = 0:0.01: r; 9

10 x = p1(1) + line*cos(theta); % set of x coordinates to plot y = p1(2) + line*sin(theta); % set of y coordinates to plot plot(x,y,'x'); % Plotting poles hold on; case 0 % Case when poles are imaginary. p1=[real(p(i)),imag(p(i))]; p2=[real(p(i)),imag(p(i))]; % Since both points(p1 &p2) are equal, it plots only the specified point theta = atan2( p2(2) - p1(2), p2(1) - p1(1)); r = sqrt( (p2(1) - p1(1))^2 + (p2(2) - p1(2))^2); line = 0:0.01: r; x = p1(1) + line*cos(theta); y = p1(2) + line*sin(theta); plot(x,y,'x'); hold on; %%%% Zeros for i=1:length(z) S=isreal(z(i)) % Check whether the zeros are real or not switch (S) case 1 % case when Poles are real q1=[z(i),0]; q2=[z(i),0]; theta = atan2( q2(2) - q1(2), q2(1) - q1(1)); r = sqrt( (q2(1) - q1(1))^2 + (q2(2) - q1(2))^2); line = 0:0.01: r; a = q1(1) + line*cos(theta); b = q1(2) + line*sin(theta); plot(a,b,'o'); 10

11 hold on; case 0 % Case when zeros are imaginary. q1=[real(z(i)),imag(z(i))]; q2=[real(z(i)),imag(z(i))]; theta = atan2( q2(2) - q1(2), q2(1) - q1(1)); r = sqrt( (q2(1) - q1(1))^2 + (q2(2) - q1(2))^2); line = 0:0.01: r; a = q1(1) + line*cos(theta); b = q1(2) + line*sin(theta); plot(a,b,'o'); % plotting zeros hold on; % Function for finding DC gain function kp=dcgain(varargin) if length(varargin)==1 % Check whether the incoming variable has length 1 sys=varargin{1}; num=sys.num; den=sys.den; % Assigning the incoming structure to sys % Numerator % Denominator if length(varargin)==2 % Check whether the incoming variable has length 1 num=varargin{1}; den=varargin{2}; % Numerator % Denominator kp=num()/den(); % DC gain 11

12 %Function for finding state space matrices in controllable canonical form from transfer function function[a,b,c,d]=tf2ss(b,a) d=length(a); %Length of denominator if a(1)~=1 %to make first coefficient of denominator equal to 1 a=a/a(1); m=d-1; %%%%%%%%----A MATRIX for j=1:m A(m,j)=-a(d); d=d-1; for k=m:-1:2 A(k-1,k)=1; B(m,1)=1; %B MATRIX %%%%%%%-----C MATRIX g=abs(length(a)-length(b)); if length(a)>length(b) for i=length(b):-1:1 b(i+g)=b(i); for l=1:g b(l)=0; if length(b)>length(a) for i=length(a):-1:1 a(i+g)=a(i); 12

13 for l=1:g a(l)=0; b f=m; for p=2:1:m+1 C(1,f)=b(p)-a(p)*b(1); f=f-1; D=b(1); %D MATRIX Simulation Results: sys1 = s^2+10s+20 sys2 = 1s s^2+10s+20 kp = 0.05 ( DC gain) ess = ( Steady state error) w = (Natural frequency) zeta = (Damping ratio) TD = (Delay time) TS = 0.8 (Settling Time) TP = i (Peak Time) TR = i (Rise Time) Percent overshoot = i (Percent Overshoot) 13

14 Figure 1.a: Step Response and addition of zero to the given system 14

15 EXP.1.B STEP RESPONSE OF A SECOND ORDER SYSTEM FOR VARIOUS VALUES OF ZETA Description: Damping ratio is a dimensionless measure describing how oscillations in a system decay after a disturbance. Many systems exhibit oscillatory behavior when they are disturbed from their position of static equilibrium. Software Code: clc; close all; clear zeta=[ ]; % Various damping ratio t=0:0.2:10; num=[0 0 25]; % Plotting time length % Numerator den1=[1 10*zeta(1) 25]; den2=[1 10*zeta(2) 25]; den3=[1 10*zeta(3) 25]; den4=[1 10*zeta(4) 25]; figure(1) [f1,t]=step(num,den1,t);plot(t,f1);hold on; % Step Response [f2,t]=step(num,den2,t);plot(t,f2);hold on; [f3,t]=step(num,den3,t);plot(t,f3);hold on; [f4,t]=step(num,den4,t);plot(t,f4);hold on; leg('zeta=0.0,system is undamped','zeta=0.6,under Damped','zeta=1.0,Critically damped','zeta=1.5,over damped') title('step response for various values of zeta') 15

16 Simulation Results: Figure 2.a: Step Response of the given system for different damping ratio 16

17 EXP.2 EFFECT OF P, PD, PI AND PID CONTROLLER Aim & Description: To obtain step response of the given system and evaluate the effect P,PD,PI and PID controllers. A controller is a device, historically using mechanical, hydraulic, pneumatic or electronic techniques often in combination, but more recently in the form of a microprocessor or computer, which monitors and physically alters the operating conditions of a given dynamical system. G s) 0.5s ( 3 Software Code: 1 1s 4 clc; clear; close all; sys.num=[1]; % Numerator of the system is defined as a structure sys.den=[ ]; % Denominator of the system is defined as a structure sys1=tf(sys) sys5.num=[1]; sys5.den=[1]; % Print transfer function and store it in sys1. %Numerator of unity feedback system %Denominator of unity feedback system [f,g]=feedback(sys,sys5) [z,t]=step(g); % Gives transfer function of the system with unity feedback % Step response of the system subplot(2,3,1);plot(t,z);grid on; title('step response of given system'); %Proportional controller kp=10; sys.num=kp*sys.num; % Numerator augmented with P controller sys2=tf(sys); [f,g]=feedback(sys,sys5) [z,t]=step(g); subplot(2,3,2); plot(t,z);grid on; title('proportional contol Kp=10') k=dcgain(sys) % DC gain of the system ( s=0) 17

18 essp=1/(1+k) % % PD controler Kd=10; numc=[kd*kp kp]; %Transfer function of a PD controller is [ Kp+Kd*s] sys.num=conv(numc,sys.num); % Numerator augmented with PD controller sys3=tf(sys); [f,g]=feedback(sys,sys5) [m,t]=step(g); subplot(2,3,3);plot(t,m);grid on; title('pd control Kp=10 and Kd=10') %PI controller ki=10; sys.num=[kp ki*kp]; % %Transfer function of a PI controller is [ Kp+ ki*kp/s] deni=[1 0]; sys.den=conv(deni,sys.den); sys4=tf(sys) [f,g]=feedback(sys,sys5); [m,t]=step(g); subplot(2,3,4);plot(t,m);grid; k=dcgain(g) esspi=1/(1+k) title('pi control Kp=10 and Ki=10') % %PID controller sys.num=conv(numc,[1 ki]); % Numerator augmented with PID controller [Kp+Ki/s+Kd*s] sys3=tf(sys); [f,g]=feedback(sys,sys5); [m,t]=step(g); subplot(2,3,5);plot(t,m);grid; k=dcgain(g) 18

19 esspid=1/(1+k) title('pid control Kp=10,Ki=10 & kd=10') % Function defined for feedback function [f,g]=feedback(sys1,sys2) num1=sys1.num; %Numerator of the first system den1=sys1.den; %Denominator of the first system num2=sys2.num; %Numerator of the second system den2=sys2.den; %Denominator of the second system [num1,den2]=equal_length(num1,den2); % to make numerator and denominator equal length [den1,den2]=equal_length(den1,den2); [num1,num2]=equal_length(num1,num2); num_n=conv([num1],[den2]); %Numerator of the feedback system d1=conv([den1],[den2]); d2=conv([num1],[num2]); [d1 d2]=equal_length(d1,d2); den_n=d1+d2; % denominator of the feedback system G.num=num_n; G.den=den_n; f=tf(g); s1=find(g.num); s2=find(g.den); if s1(1)>1 G.num=[G.num(s1(1):length(G.num))]; % to avoid initial unwanted zeros in the numerator if s2(1)>1 G.den=[G.den(s2(1):length(G.den))]; % to avoid initial unwanted zeros in the denominator %Function defined to make numerator and denominator equal length function [num1,den1]=equal_length(num1,den1) g=abs(length(den1)-length(num1)); %Difference in the length of numerator and denominator % To make numerator equal length with denominator if length(den1)>length(num1) for i=length(num1):-1:1 num1(i+g)=num1(i); for l=1:g num1(l)=0; 19

20 % To make denominator equal length with numerator if length(num1)>length(den1) for i=length(den1):-1:1 den1(i+g)=den1(i); for l=1:g den1(l)=0; num1; den1; Simulation Results: Figure 2.a: Effect of P, PD, PI and PID controllers 20

21 EXP.3 ROOT LOCUS, BODE AND NYQUIST PLOT Aim & Description: 1) To sketch the root locus for the given transfer function 2) To sketch the nyquist plots for the given transfer function 3) To sketch the bode plots for the given transfer function Root locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly a gain within a feedback system. Nyquist plot is a representation of the vector response of a feedback system (especially an amplifier) as a complex graphical plot showing the relationship between feedback and gain. Bode plot is a graph of the frequency response of a system. It is usually a combination of a Bode magnitude plot, expressing the magnitude (usually in decibels) of the frequency response, and a Bode phase plot, expressing the phase shift. Both quantities are plotted against a horizontal axis proportional to the logarithm of frequency. Software Code: 1 G ( s) 3 2 s 5s 6s 0 clc; clear; close all; sys.num=[ ]; % Numerator of the system sys.den=[ ]; % Denominator of the system sys1=tf(sys); % Printing transfer function subplot(2,2,1) rlocus(sys); % root locus of the system title('root locus'); ylabel('imaginary Axis') % %Effect of adding poles sys.den=conv([1 10],sys.den); % Adding poles to the denominator sys2=tf(sys); subplot(2,2,2) rlocus(sys); % root locus of the system when extra poles are added title(' Effect of adding poles');ylabel('imaginary Axis') % %Effect of adding zeros 21

22 sys.num=conv([1 1],sys.num); % Adding zeros to the denominator sys.den=[ ]; sys3=tf(sys); subplot(2,2,3) rlocus(sys); % root locus of the system when poles are added title(' Effect of adding zeros in Root locus');ylabel('imaginary Axis') % % effect of k on the transient response k=10; t=0:0.01:20; G.num=[0 0 0 k] G.den=[1 5 6 k]; [z,t]=step(g); % Finding step response of the system subplot(2,2,4);xlabel('time');ylabel('amplitude') plot(t,z); grid on; title('effect of k on transient response'); % Function defined for rlocus function rlocus(sys) num=sys.num; den=sys.den; %% To make numerator and denominator equal length g=abs(length(den)-length(num)); if length(den)>length(num) % If denominator length is greater than numerator length for i=length(num):-1:1 num(i+g)=num(i); % shifting numerator coefficients to the right for l=1:g num(l)=0; % Apping zeros to the index where numerator coefficients originally existed 22

23 if length(num)>length(den) % If numerator length is greater than denominator length for i=length(den):-1:1 den(i+g)=den(i); % shifting denominator coefficients to the right for l=1:g den(l)=0; % Apping zeros to the index where denominator coefficients originally existed %%% d=roots(den); % Finding poles n=roots(num); % Finding zeros P=length(d); % Number of poles Z=length(n); % Number of zeros pzmap(num,den); % P-Z map of the given system grid on;hold on; i = 1; for k=0:0.1:30 r3=den+k*num; r2{i}=roots(r3); % Denominator of unity feedback system for various gain values % poles of the feedback system i = i+1; if P==4 && Z==0 % Special case for no zeros and 4 poles for k=0:5:1000 r3=den+k*num; r2{i}=roots(r3); i = i+1; z2=cell2mat(r2); % Conversion of cell to matrix 23

24 % Extraction of values from the matrix j=1; for k=1:length(d) for i=p+k:length(d):numel(z2) b(j)=z2(i); j=j+1; %------Plotting the extracted values for i=1:numel(b) p1=[real(b(i)),imag(b(i))]; p2=[real(b(i)),imag(b(i))]; % Define the first point to plot the root locus % Define the second point to plot the root locus theta = atan2( p2(2) - p1(2), p2(1) - p1(1)); %Define slope r = sqrt( (p2(1) - p1(1))^2 + (p2(2) - p1(2))^2); % Distance between first and second point line = 0:0.01: r; x = p1(1) + line*cos(theta); y = p1(2) + line*sin(theta); % set of x coordinates to plot root locus % set of y coordinates to plot root locus if k==1 plot(x,y,'cyan.'); % plotting different branches with different color hold on; if k==2 plot(x,y,'g.'); % plotting different branches with different color hold on; if k==3 plot(x,y,'r.'); % plotting different branches with different color hold on; if k==4 24

25 plot(x,y,'b.'); % plotting different branches with different color hold on; j=1; ; Simulation Results: Figure 3.a: Root locus & effect of adding zeros and poles % Bode Plot sys.num=[ ]; sys.den=[ ]; % Numerator of the system % denominator of the system w=logspace(-2,2,100); % Define frequency magnitude=1./(w.*sqrt(w.^2+3^2).*sqrt(w.^2+2^2)); % Magnitude expression of the given system mag_db=20*log10(magnitude); % Magnitude in db phase_rad=-atan(w/0)-atan(w/3)-atan(w/2); %Phase angle phase_deg=phase_rad*180/pi; %Phase angle in degree figure(2) subplot(2,1,1) 25

26 semilogx(w,mag_db) % Magnitude plot in semilog graph xlabel('frequency [rad/s]'),ylabel('magnitude [db]'),grid on; subplot(2,1,2) semilogx(w,phase_deg) % Phase plot xlabel('frequency [rad/s]'),ylabel('phase Angle [deg]'),grid on; %% Gain Margin [wgc,fval] = fsolve(@func,5) % gain crossover frequency gm=1./(wgc.*sqrt(wgc.^2+3^2).*sqrt(wgc.^2+2^2)) % Gain Margin % Phase margin j=1; for i=1:length(mag_db) if mag_db(i)>-0.2 && mag_db(i)<0.2 % setting a limit for finding wpm b(j)=(i); j=j+1; wpm=min(w(b)) % phase crossover frequency pm=(-atan(wpm/0)-atan(wpm/3)-atan(wpm/2))*180/pi+180 % Phase margin % Function defined for solving gain crossover frequency function w= func(w) f=-atan(w/0)-atan(w/3)-atan(w/2)+180; 26

27 Simulation results: Figure 3.b: Bode plot for the given system % Nyquist Plot clc; clear; close all; sys.num=[ ]; % Numerator of the given system sys.den=[ ]; % Denominator of the given system w=0:-0.01:-10; % Defining frequency for negative values for i=1:length(w) real_part(i)=-5*w(i)^2/(25*w(i)^4+(6*w(i)-w(i)^3)^2); % Real part of the given system for i=1:length(w) imag_part(i)=(w(i)^3-6*w(i))/(25*w(i)^4+(6*w(i)-w(i)^3)^2); % Imaginary part of the given system plot(real_part,imag_part,'b'); hold on; axis([ ]); % Define Axis fill(real_part(4),1.5, '^'); % To mark the Nyquist plot direction w=0:0.01:10; % Defining frequency for positive values 27

28 for i=1:length(w) real_part(i)=-5*w(i)^2/(25*w(i)^4+(6*w(i)-w(i)^3)^2); for i=1:length(w) imag_part(i)=(w(i)^3-6*w(i))/(25*w(i)^4+(6*w(i)-w(i)^3)^2); plot(real_part,imag_part,'b'); hold on; axis([ ]); % Define Axis xlabel('real Axis'); ylabel('imaginary Axis'); fill(real_part(4),-1.5, '^'); grid on; Figure 3.c: Nyquist plot for the given system 28

29 EXP.4 RELATION BETWEEN FREQUENCY RESPONSE AND TRANSIENT RESPONSE Aim & Description: To obtain the relation between frequency response and transient response Transient response or natural response is the response of a system to a change from an equilibrium or a steady state. Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input. Software Code: clc; clear; close all; zeta=0:0.01:0.8; num=[1]; % Different damping ratio values % Numerator den=conv([1 0],[0.1 1]) % Denominator g=tf(num,den); w=logspace(-1,3,100); % Frequency range in log scale magnitude=10./(w.*sqrt(w.^2+10^2)); mag_db=20*log10(magnitude); % Magnitude expression % Magnitude expression in decibel phase_rad=-atan(w/0)-atan(w/10); % Phase angle expression in radian phase_deg=phase_rad*180/pi; % Phase angle expression in degree figure(1) subplot(2,1,1) semilogx(w,mag_db) % Magnitude plot in semilog graph xlabel('frequency [rad/s]'),ylabel('magnitude [db]'),grid on; subplot(2,1,2) semilogx(w,phase_deg) % Phase plot in semi log graph xlabel('frequency [rad/s]'),ylabel('phase Angle [deg]'),grid on; %% Gain Margin [wgc,fval] = fsolve(@gain_margin,5) gm=1./(wgc.*sqrt(wgc.^2+10^2)) %---Gain margin 29

30 % Phase margin j=1; for i=1:length(mag_db) if mag_db(i)>-0.25 && mag_db(i)<0.25 % Selection criteria for finding the index of corresponding 0db frequency b(j)=(i); j=j+1; wpm=min(w(b)) % phase crossover frequency pm=(-atan(wpm/0)-atan(wpm/10))*180/pi+180 %---Phase margin for i=1:length(zeta) resonant_peak(i)=1/(2*zeta(i)*sqrt(1-zeta(i)^2)); % expression for finding resonant peak for i=1:length(zeta) max_overshoot(i)=exp(-pi*zeta(i)/sqrt(1-zeta(i)^2)); % expression for finding maximum overshoot for i=1:length(zeta) bandwidth(i)=1*sqrt((1-2*zeta(i)^2)+sqrt(4*zeta(i)^4-4*zeta(i)^2+2)); %expression for finding bandwidth figure(2) plot(zeta,resonant_peak) axis([ ]); hold on plot(zeta,max_overshoot) hold off; grid on; xlabel('zeta') ylabel('resonant peak,max.overshoot') figure(3) 30

31 plot(zeta,bandwidth); grid on; axis([ ]); xlabel('zeta') ylabel('bandwidth') %%%----Function for finding gain margin function w= gain_margin(w) f=-atan(w/0)-atan(w/10)+180; Simulation results: Phase crossover frequency, wpm = Gain margin, gm = Inf Phase margin, pm = Gain crossover frequency, wgc = 0 Figure 4.a: Bode plot of the given system 31

32 Figure 4.b: Resonant Peak, Max Overshoot and Damping ratio Figure 4.c: Bandwidth and damping ratio 32