Engineering First order non linear to state space equations

[Maniac_XOX]

Homework Statement

I keep seeing examples using second order where the x1 replaces the y' and x2 replaces y'' giving two differentials x2 = x1' and x2' = f(x,u) where u is the input. What if have two non-linear first order differentials?

Relevant Equations

Started off from non linear equations:
1) $$y' = \frac{c_1 u - c_2uy + c_3}{z}$$
2)$$z' = c_4u - c_5 \sqrt{z}$$
After linearizing (although i've read that state space can use non-linear systems)
1) $$y' = -k_1y-k_2z+k_3$$
2) $$ z' = -k_4 z +k_5 u - k_6$$

where c and k are constants

How to represent this system in state space form?
where:
$$ x' = Ax + Bu \text{ and
}y = Cx + Du$$
I am trying to create a state space model based on these equations on simulink, need to find A, B, C and D but like I mentioned, i cannot find the solution when the differentials are not of second order.

 

[DaveE]

Note that the dynamics of state z depend only on the input u. So, conceptually, this can be solved before the other state equation. This result then becomes another input to the solution of the y state dynamics. So this is more like a cascade of two single state systems than one system with two mutually coupled states. As such, I'm not sure the matrix notation is necessary.

However, based on your other post on this subject, I think you should make sure you are very familiar with LTI systems before you try to deal with non-linear systems. While nearly anything can be linearized around a small neighborhood and then solved with standard LTI methods, there are very complex issues with defining what "small-signal" means and how truly non-linear systems work. There are seldom easy answers for truly non-linear dynamic systems.

This text may help you with the basics you will need. Note that every good treatment of modern controls assumes you know about linear DEs and Laplace transforms. We switch into the s domain ASAP in the linear world, LOL.

 

[Maniac_XOX]

Note that the dynamics of state z depend only on the input u. So, conceptually, this can be solved before the other state equation. This result then becomes another input to the solution of the y state dynamics. So this is more like a cascade of two single state systems than one system with two mutually coupled states. As such, I'm not sure the matrix notation is necessary.

However, based on your other post on this subject, I think you should make sure you are very familiar with LTI systems before you try to deal with non-linear systems. While nearly anything can be linearized around a small neighborhood and then solved with standard LTI methods, there are very complex issues with defining what "small-signal" means and how truly non-linear systems work. There are seldom easy answers for truly non-linear dynamic systems.

This text may help you with the basics you will need. Note that every good treatment of modern controls assumes you know about linear DEs and Laplace transforms. We switch into the s domain ASAP in the linear world, LOL.

I have taken a look at example 2 of what you sent as it most relates, and I have tried working that example myself which i did successfully without much effort. but when I do it on my case, it becomes a much complex task..

In the example for first-order differentials, i had to reverse the process as the source started from state-space equations to find the differentials, while I am doing the opposite. The steps i have taken are:

1) Laplace transform of the linear differential z', as it is only in terms of z output and x input.
$z' = -az +bx-c$ turns into $z_s = \frac{sbx_s -c}{s(s+a)}$
2) Solved that for Hs, which gives the output equation
3) Used the laplace transform of the output state space and brought in equation of step 2:
$$ Zs = cXs + dUs = \frac{sbU_s -c}{s(s+a)}$$
To avoid confusion between state variable and input, i have changed the input x from previous steps to U to keep the state space equation conventional.
From here I figured this would be a reasonable way to find the state variable. Is this a reasonable approach? Although I believe i should change the name of c on either sides as they are probably not equal.
The next steps would basically be reversing the steps of the image i attached from the source you sent, but it gives me a second order differential with X'' and V'' in the equation after the inverse laplace, and I don't know if any of it makes sense at this point.

If i can have deeper guidance that is more related to my specific problem it would really help out further