How to achieve the unification of addition and multiplication in state space models

for example,sys=1/(s+1),I have a test,sys1+sys1=2s/(s^2+2s+1),but 2*sys=2/(s+1),so which I should choose? and now I want to calculate 1/2 *sys,maybe a fraction 1.5,hope to get your help :)

[kang]

I am sorry that I make a mistake that sys1+sys2=(2s+2)/(s^2+2s+1).it seems that it's same as 2*sys,but its (sys1+sys2).a is 2*2，it means that (2s+2)/(s^2+2s+1) is different from 2/s+1 with system.A .

Answer 1

[Mr. Pavl M.]

clc
clear all
close all

%Algebraic help:
%sys = 1/(s+1)
%2s/(s^2+2*s+1) = 2*s/(s+1)(s+1) = s/(s+1)(s+1) + s/(s+1)s+1)
%(2*s+2)/(s^2+2*s+1) = 2*(s+1)/(s+1)(s+1) = 2/(s+1)
%You need to show that (2*s+2)/(s^2+2*s+1) = 2*sys


num1 = 1;
den1 = [1,2];
sys1 = tf(num1,den1)

sys1 =
 
    1
  -----
  s + 2
 
Continuous-time transfer function.

sys1 = tf(num1, den1)

sys1 =
 
    1
  -----
  s + 2
 
Continuous-time transfer function.

sys2 = 2*sys1

sys2 =
 
    2
  -----
  s + 2
 
Continuous-time transfer function.

sys2 = tf(2*num1,den1)

sys2 =
 
    2
  -----
  s + 2
 
Continuous-time transfer function.

num3 = [2 2]

num3 = 1×2

2 2

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den3 = [1 2 1]

den3 = 1×3

1 2 1

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sys3 = tf(num3,den3)

sys3 =
 
     2 s + 2
  -------------
  s^2 + 2 s + 1
 
Continuous-time transfer function.

minreal(sys3)

ans =
 
    2
  -----
  s + 1
 
Continuous-time transfer function.

s = tf( 's' )

s =
 
  s
 
Continuous-time transfer function.

sys4 = 2/(s+1)

sys4 =
 
    2
  -----
  s + 1
 
Continuous-time transfer function.

sys5 = (2*s+2)/(s^2+2*s+1)

sys5 =
 
     2 s + 2
  -------------
  s^2 + 2 s + 1
 
Continuous-time transfer function.

sys4.numerator

ans = 1x1 cell array
    {[0 2]}

sys4.denominator

ans = 1x1 cell array
    {[1 1]}

sys6 = minreal(sys4)

sys6 =
 
    2
  -----
  s + 1
 
Continuous-time transfer function.

if norm(sys4 - sys6,inf) == 0
    display('Transfer functions are equal')
else
    
tf(zpk(sys4)) - tf(zpk(sys6))
   
sys4/sys6 

disp('They differ')
end

Transfer functions are equal

 

%if all(cell2mat(sys6.numerator) == cell2mat(sys4.numerator)) && all(cell2mat(sys6.denominator) == cellmat(sys4.denominator))
%    display('OK')
%end

%Constructed by
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%kindly accept my the answer-solution.