Experiment 2.1: 6
5
4
3
2
Ques (1) Given that : P1=s +7s +2s +9s +10s +12s+15; 6
5
4
3
2
P2=s +9s +8s +9s +12s +15s+20 Use Matlab to find: P3=P1+P2,
P4=P1- P2
P5=P1* P2
Solution:
MATLAB Scripts: >> P1=[1 7 2 9 10 12 15]; % vector with coefficients of polynomial P1 >> P2=[1 9 8 9 12 15 20]; % vector with coefficients of polynomial polynomial P2 >> P3=P1+P2 P3=P1+P2 % find P3 ( the sum of P1 and P2) P3 = 2 16 10 18 22 27 35 >> P4=P1-P2 % calculate the value of P4 P4 P4 = 0 -2 -6 0 -2 -3 -3 -5 >> P5=conv(P1,P2) % P5 is the product of the polynomials P1 and and P2 P5 = 1 16 73 92 182 291 433 599 523 609 560 465 300
Results:
Therefore, 6 5 4 3 2 P3=2s +16s +10s +18s +22s +27s+35 5
4
2
P4=-2s -6s -2s -3s-5 12 11 10 9 8 7 6 5 4 3 2 P5=s +16s +73s +92s +182s +291s +433s +599s +523s +609s +560s +465s+300
Ques (2)
P6=(s+7)(s+8)(s+3)(s+5)(s+9)(s+10) Roots of P6 are: s=-7,-8,-3,-5,-9 s=-7,-8,-3,-5,-9 & -10 -10
Matlab script >> r=[-7,-8,-3,-5,-9,-10] ; % root vector of P6 >> P6=poly(r) % computes the coefficient of the polynomial from the roots P6 = Columns 1 through 6 1 42 718 6372 30817 76530 Column 7 75600
Result: 6 5 4 3 2 P6=s +42s +718s +6372s +30817s + 76530s+75600
Ques (3):
Use two Matlab commands to find the transfer function G1(s) given below as a polynomial divide by a polynomial:
Solution: Matlab Script >> s=tf('s'); % Define ‘s’ as a Linear Time Invariant (LTI) object in polynomial form >> G=20*(s+2)*(s+3)*(s+6)*(s+8)/(s*(s+7)*(s+9)*(s+10)*(s+15)); % Form G(s) as a LTI transfer function in polynomial form >> G Transfer function: 20 s^4 + 380 s^3 + 2480 s^2 + 6480 s + 5760 ------------------------------------------s^5 + 41 s^4 + 613 s^3 + 3975 s^2 + 9450 s
Ques (4)
Use two Matlab commands to find the transfer function G2(s) expressed as factor in numerator divide by factor in the denominator:
Solution:
Matlab script >> s=zpk('s'); % Define ‘s’ as LTI object in factored form >> G2=(s^4+17*s^3+99*s^2+223*s+140)/(s^5+32*s^4+363*s^3+2092*s^2+5052*s+4320) Warning: Accuracy may be poor in parts of the frequency range. Use the "prescale" command to maximize acc uracy in the range of interest. > In warning at 26 In xscale at 135 In @zpkdata\private\utSS2ZPK at 12 In zpkdata.plus>LocalAddSISO at 41 In zpkdata.plus at 17 In lti.plus at 55 Zero/pole/gain: (s+1) (s+4) (s+5) (s+7) -----------------------------------------------------(s+16.79) (s^2 + 4.097s + 4.468) (s^2 + 11.12s + 57.6)
Ques (5)
Using various combinations of G1(s) and G2(s), find: i)
G3(s)
ii)
G4(s)
iii)
G5(s).
Solution:
i)
G3(s) = G1(s) + G2(s)
Matlab script: a) Various combinations for calculation of G3(s): >> % Calculation of G3 with G1 expressed in factor form and G2 expressed in factor form. >> s=zpk('s'); >> G1=20*(s+2)*(s+3)*(s+6)*(s+8)/(s*(s+7)*(s+9)*(s+10)*(s+15)); >> G2=(s^4+17*s^3+99*s^2+223*s+140)/(s^5+32*s^4+363*s^3+2092*s^2+5052*s+4320); Warning: Accuracy may be poor in parts of the frequency range.
Use the "prescale" command to maximize acc uracy in the range of interest. > In warning at 26 In xscale at 135 In @zpkdata\private\utSS2ZPK at 12 In zpkdata.plus>LocalAddSISO at 41 In zpkdata.plus at 17 In lti.plus at 55 >> G3=G1+G2 Zero/pole/gain: 21 (s+16.76) (s+7.999) (s+6.003) (s^2 + 6.079s + 9.499) (s^2 + 3.033s + 2.607) (s^2 + 11.46s + 59.47) ----------------------------------------------------------------------------------------------------s (s+7) (s+9) (s+10) (s+15) (s+16.79) (s^2 + 4.097s + 4.468) (s^2 + 11.12s + 57.6)
>> % Calculation of G3 with G1 expressed in polynomial form and G2 expressed in factor form >> G1=20*(s+2)*(s+3)*(s+6)*(s+8)/(s*(s+7)*(s+9)*(s+10)*(s+15)); % Inputs G1 >> G1zpk=G1; % stores G1 in G1 zpk >> G1tf=tf(G1zpk); %converts G1 to polynomial form >> G2=(s^4+17*s^3+99*s^2+223*s+140)/(s^5+32*s^4+363*s^3+2092*s^2+5052*s+4320); %input G2 >> G2tf=G2; % G2 stored in polynomial form >> G2zpk=zpk(G2tf); %converts G2 to factor form >> G3=G1+G2
Transfer function: 21 s^9 + 1078 s^8 + 23309 s^7 + 284298 s^6 + 2.156e006 s^5 + 1.043e007 s^4 + 3.173e007 s^3 + 5.816e007 s^2 + 5.842e007 s + 2.488e007 --------------------------------------------------------------------------------------------------s^10 + 73 s^9 + 2288 s^8 + 40566 s^7 + 449993 s^6 + 3.239e006 s^5 + 1.502e007 s^4 + 4.25e007 s^3 + 6.491e007 s^2 + 4.082e007 s >> % Calculation of G3 with both G1 and G2 expressed in polynomial form >> s=tf('s'); >> G1=20*(s+2)*(s+3)*(s+6)*(s+8)/(s*(s+7)*(s+9)*(s+10)*(s+15)); >> G2=(s^4+17*s^3+99*s^2+223*s+140)/(s^5+32*s^4+363*s^3+2092*s^2+5052*s+4320); >> G3=G1+G2 Transfer function: 21 s^9 + 1078 s^8 + 23309 s^7 + 284298 s^6 + 2.156e006 s^5 + 1.043e007 s^4 + 3.173e007 s^3 + 5.816e007 s^2 + 5.842e007 s + 2.488e007 ---------------------------------------------------------------------------------------------------
s^10 + 73 s^9 + 2288 s^8 + 40566 s^7 + 449993 s^6 + 3.239e006 s^5 + 1.502e007 s^4 + 4 .25e007 s^3 + 6.491e007 s^2 + 4.082e007 s
>> % Calculation of G3 with G1 expressed in factor form and G2 expressed in polynomial form. >> s=zpk('s'); >> G1=20*(s+2)*(s+3)*(s+6)*(s+8)/(s*(s+7)*(s+9)*(s+10)*(s+15)); >> s=tf('s'); >> G2=(s^4+17*s^3+99*s^2+223*s+140)/(s^5+32*s^4+363*s^3+2092*s^2+5052*s+4320); >> G3=G1+G2 Zero/pole/gain: 21 (s+16.76) (s+7.999) (s+6.003) (s^2 + 6.079s + 9.499) (s^2 + 3.033s + 2.607) (s^2 + 11.46s + 59.47) ----------------------------------------------------------------------------------------------------s (s+7) (s+9) (s+10) (s+15) (s+16.79) (s^2 + 4.097s + 4.468) (s^2 + 11.12s + 57.6)
b) Various combinations for calculation of G4(s):
Matlab Script: >> % Calculation of G4 with both G1 and G2 expressed in factor form >> s=zpk('s'); >> G1=20*(s+2)*(s+3)*(s+6)*(s+8)/(s*(s+7)*(s+9)*(s+10)*(s+15)); >> G2=(s^4+17*s^3+99*s^2+223*s+140)/(s^5+32*s^4+363*s^3+2092*s^2+5052*s+4320); Warning: Accuracy may be poor in parts of the frequency range. Use the "prescale" command to maximize accuracy in the range of interest. > In warning at 26 In xscale at 135 In @zpkdata\private\utSS2ZPK at 12 In zpkdata.plus>LocalAddSISO at 41 In zpkdata.plus at 17 In lti.plus at 55 >> G4=G1-G2 Zero/pole/gain: 19 (s+16.81) (s+8.001) (s+5.997) (s+3.252) (s+1.496) (s^2 + 4.335s + 6.018) (s^2 + 10.74s + 55.47) -------------------------------------------------------------------------------------------------s (s+7) (s+9) (s+10) (s+15) (s+16.79) (s^2 + 4.097s + 4.468) (s^2 + 11.12s + 57.6) >> % Calculation of G4 with both G1 and G2 expressed in polynomial form >> s=tf('s'); >> G1=20*(s+2)*(s+3)*(s+6)*(s+8)/(s*(s+7)*(s+9)*(s+10)*(s+15)); >> G2=(s^4+17*s^3+99*s^2+223*s+140)/(s^5+32*s^4+363*s^3+2092*s^2+5052*s+4320); >> G4=G1-G2
Transfer function: 19 s^9 + 962 s^8 + 20491 s^7 + 246942 s^6 + 1.862e006 s^5 + 9.034e006 s^4 + 2.791e007 s^3 + 5.284e007 s^2 + 5.577e007 s + 2.488e007 ---------------------------------------------------------------------------------------------------s^10 + 73 s^9 + 2288 s^8 + 40566 s^7 + 449993 s^6 + 3.239e006 s^5 + 1.502e007 s^4 + 4 .25e007 s^3 + 6.491e007 s^2 + 4.082e007 s
>> % Calculation of G3 with G1 expressed in factor form and G2 expressed in polynomial form. >> s=zpk('s'); >> G1=20*(s+2)*(s+3)*(s+6)*(s+8)/(s*(s+7)*(s+9)*(s+10)*(s+15)); >> s=tf('s'); >> G2=(s^4+17*s^3+99*s^2+223*s+140)/(s^5+32*s^4+363*s^3+2092*s^2+5052*s+4320); >> G4=G1-G2 Zero/pole/gain: 19 (s+16.81) (s+8.001) (s+5.997) (s+3.252) (s+1.496) (s^2 + 4.335s + 6.018) (s^2 + 10.74s + 55.47) -------------------------------------------------------------------------------------------------s (s+7) (s+9) (s+10) (s+15) (s+16.79) (s^2 + 4.097s + 4.468) (s^2 + 11.12s + 57.6)
>> % Calculation of G4 with G1 expressed in polynomial form and G2 expressed in factor form >> G1=20*(s+2)*(s+3)*(s+6)*(s+8)/(s*(s+7)*(s+9)*(s+10)*(s+15)); % Inputs G1 >> G1zpk=G1; % stores G1 in G1 zpk >> G1tf=tf(G1zpk); %converts G1 to polynomial form >> G2=(s^4+17*s^3+99*s^2+223*s+140)/(s^5+32*s^4+363*s^3+2092*s^2+5052*s+4320); %input G2 >> G2tf=G2; % G2 stored in polynomial form >> G2zpk=zpk(G2tf); %converts G2 to factor form >> G4=G1-G2
Transfer function: 19 s^9 + 962 s^8 + 20491 s^7 + 246942 s^6 + 1.862e006 s^5 + 9.034e006 s^4 + 2.791e007 s^3 + 5.284e007 s^2 + 5.577e007 s + 2.488e007 --------------------------------------------------------------------------------------------------s^10 + 73 s^9 + 2288 s^8 + 40566 s^7 + 449993 s^6 + 3.239e006 s^5 + 1.502e007 s^4 + 4 .25e007 s^3 + 6.491e007 s^2 + 4.082e007 s
c) Various combinations for calculation of G5(s):
Matlab script:
>> % Calculation of G5 with both G1 and G2 expressed in factor form >> G1=20*(s+2)*(s+3)*(s+6)*(s+8)/(s*(s+7)*(s+9)*(s+10)*(s+15)); >> G2=(s^4+17*s^3+99*s^2+223*s+140)/(s^5+32*s^4+363*s^3+2092*s^2+5052*s+4320); %input G2 >> G2tf=G2; % G2 stored in polynomial form >> G2zpk=zpk(G2tf); %converts G2 to factor form >> G5=G1*G2zpk % calculates G5 with G1 and G2 expressed in factor form Zero/pole/gain: 20 (s+8) (s+7) (s+6) (s+5) (s+4) (s+3) (s+2) (s+1) ---------------------------------------------------------------------------------s (s+15) (s+16.79) (s+10) (s+9) (s+7) (s^2 + 4.097s + 4.468) (s^2 + 11.12s + 57.6)
>> % Calculation of G5 with both G1 and G2 expressed in polynomial form >> G1=20*(s+2)*(s+3)*(s+6)*(s+8)/(s*(s+7)*(s+9)*(s+10)*(s+15)); % Inputs G1 >> G1zpk=G1; % stores G1 in G1 zpk >> G1tf=tf(G1zpk); %converts G1 to polynomial form >> G2=(s^4+17*s^3+99*s^2+223*s+140)/(s^5+32*s^4+363*s^3+2092*s^2+5052*s+4320); %input G2 >> G5=G1tf*G2 % calculates G5 with G1 and G2 expressed in polynomial form
Transfer function: 20 s^8 + 720 s^7 + 10920 s^6 + 90720 s^5 + 448980 s^4 + 1.346e006 s^3 + 2.362e006 s^2 + 2.192e006 s + 806400 ---------------------------------------------------------------------------------------------------s^10 + 73 s^9 + 2288 s^8 + 40566 s^7 + 449993 s^6 + 3.239e006 s^5 + 1.502e007 s^4 + 4.25e007 s^3 + 6.491e007 s^2 + 4.082e007 s >> % Calculation of G5 with G1 expressed in factor form and G2 expressed in polynomial form >> G1=20*(s+2)*(s+3)*(s+6)*(s+8)/(s*(s+7)*(s+9)*(s+10)*(s+15)); % Inputs G1 >> G2=(s^4+17*s^3+99*s^2+223*s+140)/(s^5+32*s^4+363*s^3+2092*s^2+5052*s+4320); %input G2 >> G5=G1*G2 % calculates G5 with G1 expressed in factor form and G2 expressed in polynomial form
Transfer function: 20 s^8 + 720 s^7 + 10920 s^6 + 90720 s^5 + 448980 s^4 + 1.346e006 s^3 + 2 .362e006 s^2 + 2.192e006 s + 806400 ---------------------------------------------------------------------------------------------------s^10 + 73 s^9 + 2288 s^8 + 40566 s^7 + 449993 s^6 + 3.239e006 s^5 + 1.502e007 s^4 + 4.25e007 s^3 + 6.491e007 s^2 + 4.082e007 s
>> % Calculation of G5 with G1 expressed in polynomial form and G2 expressed in factor form >> G1=20*(s+2)*(s+3)*(s+6)*(s+8)/(s*(s+7)*(s+9)*(s+10)*(s+15)); % Inputs G1
>> G1zpk=G1; % stores G1 in G1 zpk >> G1tf=tf(G1zpk); %converts G1 to polynomial form >> G2=(s^4+17*s^3+99*s^2+223*s+140)/(s^5+32*s^4+363*s^3+2092*s^2+5052*s+4320); %input G2 >> G2tf=G2; % G2 stored in polynomial form >> G2zpk=zpk(G2tf); %converts G2 to factor form >> G5=G1tf*G2zpk % calculates G5 with G1 expressed in polynomial form and G2 expressed in factor form Transfer function: 20 s^8 + 720 s^7 + 10920 s^6 + 90720 s^5 + 448980 s^4 + 1.346e006 s^3 + 2 .362e006 s^2 + 2.192e006 s + 806400 ---------------------------------------------------------------------------------------------------s^10 + 73 s^9 + 2288 s^8 + 40566 s^7 + 449993 s^6 + 3.239e006 s^5 + 1.502e007 s^4 + 4.25e007 s^3 + 6.491e007 s^2 + 4.082e007 s
Ques (4)
Use Matlab to evaluate the partial fraction expansions below:
Matlab Script (a) >> % Evaluate the partial fraction expansion of G6 >> b=[5,10]; % b is the coefficient of the polynomial in the numerator >> a=[1,8,15,0]; %a is the coefficient of the polynomial in the denominator >> [r,p,k]=residue(b,a)
r= -1.5000 0.8333
0.6667
p= -5 -3 0
k= []
Result: G6= -1.5/(s+5) + 0.833/(s+3) + 0.6667/s (b) >> % Evaluate the partial fraction expansion of G7 >> b=[5,10]; % b is the coefficient of the polynomial in the numerator >> a=[1,6,9,0]; %a is the coefficient of the polynomial in the denominator >> [r,p,k]=residue(b,a) r= -1.1111 1.6667 1.1111 p= -3 -3 0 k= [] 2
Result: G7= -1.1/(s+3) + 1.7/(s+3) + 1.1/s
c) >> % Evaluate the partial fraction expansion of G8 >> b=[5,10]; % b is the coefficient of the polynomial in the numerator >> a=[1,6,34,0]; %a is the coefficient of the polynomial in the denominator
>> [r,p,k]=residue(b,a) r= -0.1471 - 0.4118i -0.1471 + 0.4118i 0.2941 p= -3.0000 + 5.0000i -3.0000 - 5.0000i 0 k= []
Result: G8= ( -0.1471-j0.4118)/(s+3-j5) + (-0.1471+j0.4118)/(s+3+j5) + 0.2941/s
Experiment 2.2 (LAB): Ques (1a) Use MATLAB and the Symbolic Math Toolbox to generate symbolically the
time function f(t) given as:
Matlab script: >> %Generate a symbolically the time function f(t) >> f=0.0075-0.00034*exp(-2.5*t)*cos(22*t)+0.087*exp(-2.5*t)*sin(22*t)-0.0072*exp(-8*t); >> %Generate a symbolically the time function f(t) >> syms t; % define symbolic variable >> ft=0.0075-0.00034*exp(-2.5*t)*cos(22*t)+0.087*exp(-2.5*t)*sin(22*t)-0.0072*exp(-8*t); >> ft f t= (87*sin(22*t))/(1000*exp((5*t)/2)) - (17*cos(22*t))/(50000*exp((5*t)/2)) - 9/(1250*exp(8*t)) + 3/400
1b) Generate symbolically F(s) below and obtain result symbolically in both factored and
polynomial forms.
MATLAB Script >> %Generate symbolically the function F(s) >> syms s % Define symbolic variable >> F=2*(s+3)*(s+5)*(s+7)/(s*(s+8)*(s^2+10*s+100)); >> F F= ((2*s + 6)*(s + 5)*(s + 7))/(s*(s + 8)*(s^2 + 10*s + 100)) >> % Obtain F in factored form >> s=zpk('s'); >> Fzpk=2*(s+3)*(s+5)*(s+7)/(s*(s+8)*(s^2+10*s+100)); % Fzpk is the factored form of F >> Fzpk Zero/pole/gain: 2 (s+3) (s+5) (s+7) ------------------------s (s+8) (s^2 + 10s + 100) >> % Obtain F in polynomial form >> s=tf('s'); >> Ftf=2*(s+3)*(s+5)*(s+7)/(s*(s+8)*(s^2+10*s+100)); % Ftf is the polynomial form of F >> Ftf Transfer function: 2 s^3 + 30 s^2 + 142 s + 210 -----------------------------s^4 + 18 s^3 + 180 s^2 + 800 s
1c) Find the Laplace transform of f(t) in (1a)
MATLAB script
>> % Calculate the Laplace transform of ft >> syms t s >> ft=0.0075-0.00034*exp(-2.5*t)*cos(22*t)+0.087*exp(-2.5*t)*sin(22*t)-0.0072*exp(-8*t); >> laplace(ft,t,s) %computes the laplace transmform of ft ans = 3/(400*s) - 9/(1250*(s + 8)) - (17*(s + 5 /2))/(50000*((s + 5/2)^2 + 484)) + 957/(500*((s + 5/2)^2 + 484))
1d) Find the inverse Laplace transform of F(s) in (1b)
MATLAB Script >> % Calculate the inverse Laplace transform of F(s) >> syms t s % Define symbolic variable >> FS=2*(s+3)*(s+5)*(s+7)/(s*(s+8)*(s^2+10*s+100)); >> IFS=ilaplace(FS,s,t) %Computes the inverse laplace transform of FS Result: IFS = 5/(112*exp(8*t)) + (237*(cos(5*3^(1/2)*t) + (3^(1/2)*sin(5*3^(1/2)*t))/9))/(140*exp(5*t)) + 21/80
1f) Solve for loop currents in diagram below:
MATLAB Script >> syms s I1 I2 I3 V
>> z=[7+s+(5/s) -(2+s) -5; -(2+s) (4+2*s+3/s) -(2+s); -5 -(2+s) 8+s+4/s]; >> I=[I1; I2; I3]; >> VS=[V; 0; 0]; >> I=inv(z)*VS; >> pretty(I) +-
-+
| |
4
3
2
|
V s (s + 16 s + 39 s + 40 s + 12)
|
| ----------------------------------------- | | 5
4
3
2
|
| s + 26 s + 205 s + 396 s + 284 s + 60 | | | |
| 2 3
2
|
V s (s + 15 s + 30 s + 8)
|
| ----------------------------------------- | | 5
4
3
2
|
| s + 26 s + 205 s + 396 s + 284 s + 60 | | | |
| 2 3
2
|
V s (s + 14 s + 24 s + 15)
|
| ----------------------------------------- | | 5
4
3
2
|
| s + 26 s + 205 s + 396 s + 284 s + 60 | Interpretation of Result
( ) ( ) ( )
Experiment 2.3
LAB USING MATLAB 2) Generate the polynomial operations stated in P1 and P2 given below: 6
5
4
3
2
6
5
4
3
2
P1=s +7s +2s +9s +10s +12s+15; P2=s +9s +8s +9s +12s +15s+20 MATLAB Script: >> % Generate the polynomial P1 and P2 >> P1=s^6+7*s^5+2*s^4+9*s^3+10*s^2+12*s+15; >> s=tf('s'); >> P1=s^6+7*s^5+2*s^4+9*s^3+10*s^2+12*s+15 %P1 in polynomial form Transfer function: s^6 + 7 s^5 + 2 s^4 + 9 s^3 + 10 s^2 + 12 s + 15 >> s=tf('s'); %define 's'as an LTI object in polynomial form >> P2=s^6+9*s^5+8*s^4+9*s^3+12*s^2+15*s+20 %P2 defined in polynomial form Transfer function: s^6 + 9 s^5 + 8 s^4 + 9 s^3 + 12 s^2 + 15 s + 20
3) Generate the polynomial operations P3=P1+P2, P4=P1-P2, P5=P1*P2
MATLAB Script Transfer function: >> % Generate the polynomial P3=P1+P2 >> P1=s^6+7*s^5+2*s^4+9*s^3+10*s^2+12*s+15; >> P2=s^6+9*s^5+8*s^4+9*s^3+12*s^2+15*s+20;
>> P3=P1+P2 Transfer function: 2 s^6 + 16 s^5 + 10 s^4 + 18 s^3 + 22 s^2 + 27 s + 35 >> % Generate the polynomial P4=P1-P2 >> P1=s^6+7*s^5+2*s^4+9*s^3+10*s^2+12*s+15; >> P2=s^6+9*s^5+8*s^4+9*s^3+12*s^2+15*s+20; >> P4=P1-P2
Transfer function: -2 s^5 - 6 s^4 - 2 s^2 - 3 s - 5 >> % Generate the polynomial P5=P1*P2 >> P1=s^6+7*s^5+2*s^4+9*s^3+10*s^2+12*s+15; >> P2=s^6+9*s^5+8*s^4+9*s^3+12*s^2+15*s+20; >> P5=P1*P2
Transfer function: s^12 + 16 s^11 + 73 s^10 + 92 s^9 + 182 s^8 + 291 s^7 + 433 s^6 + 599 s^5 + 523 s^4 + 609 s^3 + 560 s^2 + 465 s + 300
4) Generate the polynomial whose roots are: -7, -8, -3, -5, -9, 10
MATLAB Script >> %Generate a polynomial from given roots >> r=[-7 -8 -3 -5 -9 -10]; %roots of a polynomial >> p=poly(r) % Find the coefficient of the polynomial whose roots are given p= 1
42
718
6372
30817
76530
75600
Result:
5) Generate the partial fraction expansion of the transfer function G(s)
MATLAB Script
>> %Perform the partial fraction expansion of G(s) >> num=[5,10]; % Coefficient of numerator's polynomial >> den=[1,8,15,0]; %Coefficient of denomerator's polynomial >> [r,p,k]=residue(num,den) r= -1.5000 0.8333 0.6667 p= -5 -3 0
k= []
Result: ()
6. Construct the two transfer functions enumerated in question (5)
MATLAB Script >> G1=tf([1],[1 1 2]) G1 = 1 ----------s^2 + s + 2 Continuous-time transfer function. >> G2=tf([1 1],[1 4 3]) G2 = s+1 ------------s^2 + 4 s + 3 Continuous-time transfer function. >> F=G1+G2 F= s^3 + 3 s^2 + 7 s + 5 -----------------------------s^4 + 5 s^3 + 9 s^2 + 11 s + 6
Continuous-time transfer function. >> P=G1-G2 P= -s^3 - s^2 + s + 1 -----------------------------s^4 + 5 s^3 + 9 s^2 + 11 s + 6 Continuous-time transfer function. >> T=G1*G2 T= s+1 -----------------------------s^4 + 5 s^3 + 9 s^2 + 11 s + 6 Continuous-time transfer function.
7.
For G1(s) + G2(s)
MATLAB Script >>num=[1 3 7 5]; >> den=[1 5 9 11 6]; >> [r,p,k]=residue(num,den) r= 1.0000 0.0000 - 0.3780i 0.0000 + 0.3780i -0.0000 p= -3.0000 -0.5000 + 1.3229i -0.5000 - 1.3229i -1.0000 k= []
Result () () G1(s) – G2(s) MATLAB Script >>num=[-1 -1 1 1];
>> den=[1 5 9 11 6]; >> [r,p,k]=residue(num,den) r= -1.0000 -0.0000 - 0.3780i -0.0000 + 0.3780i 0 p= -3.0000 -0.5000 + 1.3229i -0.5000 - 1.3229i -1.0000 k= []
INTERPRETATION
() ()
G1(s)G2(s)
MATLAB Script num=[1 1]; >> den=[1 5 9 11 6]; >> [r,p,k]=residue(num,den) r= 0.1250 -0.0625 - 0.1181i -0.0625 + 0.1181i -0.0000 p= -3.0000 -0.5000 + 1.3229i -0.5000 - 1.3229i -1.0000 k= []
INTERPRETATION
() ()
