control system toolbox - state space model containing constant vectors

I am trying to create a state space model using ss. My system has the following format:

dx/dt = Ax + Bu + f

y = Cx + Du

(x,y and u are states, outputs and inputs resp)

f is a constant real valued vector.

How can I create such models in control toolbox?

After this, I would like to obtain the discrete time equivalent.

Please help.

Abhishek

Answer 1

[Rajiv Singh]

One way is to add a new state to the system whose derivative is zero.

Let X = [x; x2] where x2 is a single variable if f is a scalar.

dX/dt = [A 1; 0 0] X + [B; 0] u
y = [C 0] X + Du
X(0) = [x(0); f]

[Abhishek Murthy]

Thanks for the reply!

These linear systems are actually "pieces" of a PWA non-linear system. So the initially condition one piece is the final condition of the previous "piece".

I will revert back once I understand how your suggestion will affect this process.

Abhishek

Answer 2

[Fangjun Jiang]

That means your input has an offset. For example Define New_u=u+Offset, so u=New_u-Offset and put it in your equation.

dx/dt=Ax+B(New_u-Offset)+f;
y=Cx+D(New_u-Offset)

And solve: B*Offset=f for the value of Offset. All shall be proper size matrix.

And you don't need to worry about the constant term at y=Cx+D*New_u-D*Offset because it is just a constant term at your output.

[Abhishek Murthy]

Thanks for the reply. I want to make sure, I understand your answer completely:

Are you suggesting that the new system will look this?

dx/dt = Ax + B(New_u)

y = Cx + D(New_u)

After the designing the controller with the above state space equations, during operation,

the actual input u to the system will be New_u-offset and output from the system will be be shifted by D*offset ?

Thanks again.

[Fangjun Jiang]

Yes. That's what I remembered. After using ss() to solve your problem, your original input and output are just off by a constant term.