# Homework Help: Multivariable state space where forcing function has derivative terms

1. Mar 14, 2009

### nandlald

1. The problem statement, all variables and given/known data
Hi,

I have two linear differential equations describing some multivariable dynamic system, and I need to represent the system in a state space representation.

This would be normally very easy if the forcing functions on the RHS did not contain derivative terms e.g.
dy/dt + y + z = u1 + u2
dz/dt + y + z = u2 + 2*u1

where u1 and u2 are the system inputs.

The problem is that I have a derivative term in the forcing function i.e. du(1,2)/dt

2. Relevant equations

The equations describing the system are:

d2y/dt + dy/dt +4y + 2z = 3*du1/dt + 2*u2 ... (1)

dz/dt + 3z + 2*dy/dt + 4y = 4*u2 + u1 ... (2)

where u1 and u2 are the system inputs.

3. The attempt at a solution

I can describe the solution for a SISO system but I don't think that is relevant in the multivariable context. I've considered rearranging (2) to make u2 the independant variable and substituting into (1) but that would mean I lose an input to the system. But that still doesn't help because i'll still have two independant variables (y and z) and I cannot apply the SISO solution.

I tried many textbooks including Maciejowski, and Skogestad and Posthlewaite who are authorities on multivariable control and none of them address this situation. The usual example don't involve derivatives of the inputs.