Why is the Identity Matrix essential in Multivariable Control Theory?

[MikeSv]

Hello everyone.

Iam working on a course in multivariable control theory and I stumbled over the Identity Matrix.

I understand what the identity matrix is, though the use of it is a mistery...

I was reading about going from state space to transfer functions and I found this expressions:

Known:
X'=AX+BU

Taking Laplace transform (with zero initial conditions)
sX(s)=AX(s)+BU(s)

The state equation can be write in the form
(sI−A)X(s)= BU(s)

Now Iam wondering why I would need an Identity Matrix when bringing A to the left sided of the equation?

Thanks in advance for any help,

Cheers,

Michael

 

[fresh_42]

Hello everyone.

Iam working on a course in multivariable control theory and I stumbled over the Identity Matrix.

I understand what the identity matrix is, though the use of it is a mistery...

I was reading about going from state space to transfer functions and I found this expressions:

Known:
X'=AX+BU

Taking Laplace transform (with zero initial conditions)
sX(s)=AX(s)+BU(s)

The state equation can be write in the form
(sI−A)X(s)= BU(s)

Now Iam wondering why I would need an Identity Matrix when bringing A to the left sided of the equation?

Thanks in advance for any help,

Cheers,

Michael

I'm not sure I know what you mean. If you have $sX(s)- AX(s)=BU(s)$ and pull the common factor $X(s)$ out via the distributive law, you get $(s-A(s))X(s)=BU(s)$. However, $s$ minus $A(s)$ isn't defined, it isn't even in the same space (in general; don't know where your objects are from). Thus we have to write $sX(s)=s\cdot I \cdot X(s)$ first and are then left with $(s\cdot I)\cdot X(s)$.

 

[mfb]

You cannot subtract a matrix from a real number. What would the result be? A matrix or a real number?
If you have sX(s) - AX(s), you cannot directly factor out X(s) due to this issue.
You know that X(s) = IX(s), and if you write sX(s) - AX(s) = sIX(s) - AX(s), then you can factor out X(s), as both sI and A are matrices.

edit: fresh was faster.

 

[MikeSv]

Hi and thanks for all replies.

That makes sense :-)

So the identity matrix is needed to convert my real number s into a Matrice that has the same size as A to be able to do the subtraction, right?

Cheers,

Mike

 

[mfb]

Right.

 

[MikeSv]

Great

Thanks again for the quick reply and and help!

/Mike

 

[WWGD]

Hi and thanks for all replies.

That makes sense :-)

So the identity matrix is needed to convert my real number s into a Matrice that has the same size as A to be able to do the subtraction, right?

Cheers,

Mike

Just to note, the term I think is most common for singular of matrices is matrix; just in case you run into it ( figuratively, I hope ; ) )..EDIT: It makes sense to go from either matrix to matrixes within the English language as well as from matrices to matrice, but it does not work out this way in this case.

 

[Mark44]

Just to note, the term I think is most common for singular of matrices is matrix

Yes, that's correct. This is something left over from Latin. In some cases, the suffix "rix" is used for feminine agents; e.g., aviatrix (fem. counterpart of aviator) and dominatrix. The plurals change the "rix" to "rices".