Multivariable state space where forcing function has derivative terms

[nandlald]

Homework Statement

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

Homework 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.

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.

 

[mighty2000]

Hi there,

Thank you for reaching out for help with your problem. I understand the frustration of not being able to find a solution in existing literature. However, I believe I can offer some insight that may help you in your state space representation.

Firstly, it is important to note that the derivative terms in the forcing function do not necessarily mean you have to lose an input to the system. In fact, these terms can be rewritten using the chain rule as:

d(du1)/dt = d(u1)/dt + d(u2)/dt

This means that you can still represent the system with both inputs, u1 and u2, while also accounting for the derivative terms in the forcing function.

Secondly, as you mentioned, the SISO solution may not be directly applicable in a multivariable context. However, you can still use the principles of state space representation to solve your problem. One approach you could take is to define new state variables that incorporate both the input and its derivative, such as:

x1 = u1
x2 = d(u1)/dt

Then, using these new state variables, you can rewrite your equations as:

dx1/dt = x2
dx2/dt = 3*x2 + 2*u2 - 4*u1 + 4*du1/dt

This approach may require some additional manipulation and substitution, but it should ultimately lead to a state space representation for your system.

I hope this helps you in finding a solution for your problem. If you have any further questions or need clarification, please don't hesitate to ask. Best of luck!