Transformation of state space model

Hi,

I have a state space model with 23 states, 5 inputs and 9 outputs. So the matrices A,B,C and D are known.

I am looking for a way to calculate the state space model for a different set of inputs and outputs, by transforming the initial matrices A,B,C and D?

Example: Let's say that the initial system is:

X' = A.X + B.U with U = [u1 ; u2 ; u3 ; u4 ; u5] and X is the state vector.

Y = C.X + D.U with Y = [y1 ; y2 ; y3 ; y4 ; y5 ; y6 ; y7 ; y8 ; y9]

and A, B, C and D are known.

How can I transform this model into a new model with another set of inputs and outputs like:

U_new = [y1 ; y2 ; y3 ; y4 ; y5]

Y_new = [y6 ; y7 ; y8 ; y9]

which are basically two subsets of the original output vector Y.

Is it possile without transforming the State Space Model into Transfer Functions?

I appreciate your help.

Best regards. A1ireza

Answer 1

[Sam Chak]

Hi @Alireza Aghdaei

System 1

System 2

If I understand your description correctly, the dynamics of the system does not change. Only the output matrix changes from 9 outputs to 4 outputs.

[Alireza Aghdaei]

Thank you @Sam Chak. Do you mean that I only need to take the first 5 rows of the matrix B and use them as a new output matrix?

[Sam Chak]

Hi @Alireza Aghdaei, it is more important to clarify what condition that triggers the switching from to .

Since there are 23 states and 5 inputs, your input matrix should look like this: , where each is a column vector of size .

Initially the input term looks like this:

At some point, something triggers the switch and it becomes

As you can see, there is no change to your input matrix if the inputs are injected at the same original input points of .

If I interpreted your description incorrectly, where are injected at different locations, then your input matrix should change as well: .

Answer 2

[Paul]

It sounds like the Question is about forming a feedback loop. In this case use feedback.

[Alireza Aghdaei]

Thank you @Paul for your answer but I can not figure it out how to convert my original state space system (A,B,C,D) to a closed loop system in such a way that it recieves U_new and generates Y_new?

[Paul]

Did you try to use feedback? Please show what you tried.