14 Problem A.5.10 Ogata 4edition (partial-fraction expansion)

Laplace.T	Root-locus	Transient response	Frecuency response

14 Problem A.5.10 Ogata 4edition (partial-fraction expansion)

Lets do the partial-fraction expansion by Scilab of the following system:

$\begin{displaymath}\frac{C(s)}{R(s)}=\frac{5\cdot (s+20)}{s \cdot (s+4.59)\cdot (s^{2}+3.14\cdot s+16.35)}\end{displaymath}$

Program in Scilab:

s=%s;

num=5*(s+20);

den=s*(s+4.59)*(s^2+3.41*s+16.35);

g=syslin('c',num/den);

cr=g/. 1;

c=cr*(1/s);

cs=tf2ss(c);

fs=pfss(cs)

for k=1:3

clean(8*fs(k))

end;

Result:

  ans  =
 
    - 13.959394 + 3.0016262s     
    --------------------------   
                              2  
    10.011621 + 2.0004717s + s   
 ans  =
 
    - 46.040505 - 11.001626s     
    --------------------------   
                              2  
    9.9883925 + 5.9995283s + s   
 ans  =
 
    8   
    -   
    s  
    
     
-->c
 c  =
 
                 100 + 5s                 
    ----------------------------------    
                   2          3    4   5  
    100s + 80.0465s + 32.0019s + 8s + s

With these equations do partial-fraction expansion of close-loop system to unit-step input.

$\begin{displaymath}C(s)=\frac{5\cdot s+100}{(s^{5}+8\cdot s^{4}+32\cdot s^{3}+80\cdot s^{2}+100\cdot s)}=\end{displaymath}$

$\begin{displaymath}=\frac{1}{s}+\frac{1}{8}\cdot \frac{(3\cdot s-14)}{ s^{2}+2\c... ...10}+\frac{1}{8}\cdot \frac{-(46+11\cdot s)}{s^{2}+6\cdot s+10}=\end{displaymath}$

$\begin{displaymath}=\frac{1}{s}+\frac{1}{8}\cdot \frac{(3\cdot s+3-3-14)}{ ((s+1... ...}+\frac{1}{8}\cdot \frac{-(46+33-33+11\cdot s)}{((s+3)^{2}+1)}=\end{displaymath}$

$\begin{displaymath}=\frac{1}{s}+\frac{1}{8}\cdot \frac{(3\cdot (s+1)-17)}{ ((s+1... ...2})}+\frac{1}{8}\cdot \frac{-(11\cdot(s+3)+13)}{((s+3)^{2}+1)}=\end{displaymath}$

$\begin{displaymath}=\frac{1}{s}+\frac{3}{8}\cdot \frac{(s+1)}{ ((s+1)^{2}+3^{2})... ...s+3)}{((s+3)^{2}+1)}-\frac{1}{8}\cdot \frac{13}{((s+3)^{2}+1)}=\end{displaymath}$

$\begin{displaymath}=1+\frac{3}{8}\cdot e^{-t}\cdot cos(3\cdot t)-\frac{17}{24}\c... ...cdot t}\cdot cos(t)-\frac{13}{8}\cdot e^{-3\cdot t}\cdot sin(t)\end{displaymath}$

Let's plot the tranference function and obtained equation of partial-fraction expansion.

Program in Scilab:

s=%s;

num=5*(s+20);

den=100+80*s+32*s^2+8*s^3+s^4;

g=syslin('c',num/den);

t=0:0.01:5;

gs=csim('step',t,g);

y=1+(3/8)*exp(-t).*cos(3*t)-(17/24)*exp(-t).*sin(3*t)-(11/8)*exp(-3*t).*cos(t)
-(13/8)*exp(-3*t).*sin(t);

clf;

subplot(2,1,1);

xgrid;

xtitle('Reponse to unit-step of y=1+(3/8)*exp(-t)*cos(3*t)
-(17/24)*exp(-t)*sin(3*t)-(11/8)*exp(-3*t)*cos(t)-(13/8)*exp(-3*t)*sin(t)',
't(seg)','Amplitude');

plot2d(t,y,3);

subplot(2,1,2);

plot2d(t,gs);

xgrid;

xtitle('Reponse to unit-step','t(seg)','Amplitude')