Shifting state space variables by constant value

Hi,

I have a discrete time state space model for which I want to shift the state variables by a constant value. The actual model is as follows,

x(k+1) = Ax(k) + Bu(k)

It is a 4-state system. The goal is to change state variables and input to the following variable.

z(k) = x(k) - c

v(k) = u(k) - c

Where c is a constant. My question here is, how shall this change of variable be implemented to change the state space equation accordingly. Is the following change going to work? I could not see similarity transform working here.

z(k+1) = Az(k)+ Ac + Bv(k) + Bc

Am I missing anything here? Please advise of your thoughts.

Thank you.

Answer 1

[Azzi Abdelmalek]

You gave an incomplete state space equation. What about the output?

y[k]=e.x[k]+d u[k] ,

The output should also change

y[k]=e.(z[k]+c])+d.(v[k]+c)

The states change but not the output

[Salman]

I am so sorry about it.

Output is simply

Y(k) = x(k)

[Azzi Abdelmalek]

Ok, then y[k] will be

y[k]=z(k)+c

[Salman]

Thank you for your feedback. But what about the x(k+1)? It is a discrete time system. For continuous time system, dx/dt = dz/dt as the derivative of constant c = 0. But what will happen to x(k+1) in discrete time?

[Azzi Abdelmalek]

Nothing happens to x[k+1],

x(k+1) = Ax(k) + Bu(k)

In your case you have considered another state z[k], you have the choice to work with x or z, just do not forget to change the equations