What is the relationship between matrix dimensions in state space control?

[kidsasd987]

Hello, I want to verify this question.

In short,
"Where did input matrix Bu arise from?"

I was wondering why the state equation has to be in the form of x_dot=Ax+Bu. I got to the point that the highest order terms can be expressed in the form of linear superposition of lower degree terms.

If that is the case, we can find dim(x)=n. (state vector in R^n)
Also because d/dt is a linear operator, dim(x_dot)=dim(x) (because we need n terms to uniquely determine x_dot).

And this gives a conclusion that dim(A)+dim(B)=dim(x). which means, Bu is compensating dimension for the 2nd order differential terms (since x_dot's 2nd order terms are linear superposition of lower differential terms)

Professor told me that, x_dot doesn't need to be in the same dimension in the case that if A has a nullity greater than 1 (A doesn't have full rank)

but considering the canonical solution, wouldn't they have to be in the same dimension?

 

[donpacino]

https://en.wikibooks.org/wiki/Control_Systems/State-Space_Equations

go down the section heading
Matrix Dimensions

Per my understanding A will always be a square matrix. This is due to the fact that every state in the X vector must be multiplies by every other state. Now if A is not full rank, there will be a section of the matrix that will have multiple or infinite possibilities. Now this might be something you don't care about for certain states, or something that doesn't matter. I suppose when representing your state equations you can show only part of the Xdot values, as many of the others simply do not matter. However in that case they would still 'exist.' they simply would not be written down on paper.

dim(A)+dim(B)=dim(x)

follow the link. B and A will have different dimensions. X and Xdot are and will always be vectors of the states and their derivatives. A and B are simply sized based on the number of states and the number of inputs. They are the mathematical representations of how the states relate to one another, and how they are affected by the inputs.