How can be the state-space representation rewritten into

Coradossi7xod

Coradossi7xod

Answered question

2022-03-25

How can be the state-space representation rewritten into differential equation?
x˙1=x1+x2
x˙2=x1+u
A(1110)xB(01)u

Answer & Explanation

Lesly Fernandez

Lesly Fernandez

Beginner2022-03-26Added 16 answers

Step 1
For this specific example,
x˙1 =x1+x2x˙2 =x1+u
  x¨1 =x˙1+x˙2x˙2 =x1+u
x¨1=x˙1+ux1x¨1+x˙1+x1=u
where we substituted x˙2 (the underlined equation) into x¨1. In general, if you have a single input single output system of the form,
x˙=Ax+Bu
y=Cx+Du
You can compute the transfer function
G(s)=C(sIA)1B+D
Let G(s)=P(s)Q(s)
is a polynomial of s in the numerator and Q(s) is a polynomial of s in the denominator of G(s).
Then from Y(s)=G(s)U(s)=P(s)Q(s)U(s) we can get Q(s)Y(s)=P(s)U(s) and by taking the inverse Laplace transform we can obtain the differential equation, i.e.,
L1(Q(s)Y(s))=L1(P(s)U(s)).
For your example,
A=[1110] B=[01] C=[10] D=0
Therefore,
C(sI-A)-1B+D=10s+1-11s-101=1s2+s+1
and from the transfer function we can obtain,
Y(s)=1s2+s+1U(s)
(s2+s+1)Y(s)=U(s)
L1y¨+y˙+y=u
Since y=x1 we can write this as x¨1+x˙1+x1=u (same as above).

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