Transfer function between states in state space

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Simon Detmer 2020-7-10

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https://ww2.mathworks.cn/matlabcentral/answers/563078-transfer-function-between-states-in-state-space

编辑： Christine Li 2020-7-17

Hello,

I'm working on controllers for a quadrocopter and I'm using the following state space in matlab

where r denotes the Position, r_dot the velocity ... Phi the Rotation in Euler angles, omega the angular velocity and so on.

u is a 4x1 vector containing the individual thrusts of the propellers.

Now I want to design a cascaded controller for which I need the transfer function between the acceleration r_ddot and the Euler angles Phi.

How can I get this transfer function from the state space representation?

Thanks in advance

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Paul 2020-7-16

Because the states are internal to the realization of your system, there really isn't a transfer function between any one state and another. So whether or not you can get what you want depends on what exactly you mean.

Are you trying to develop a relationship between two states in your system when the the system is stimulated by an external input? What about initial conditions on the states?

Maybe you want the transfer function from an external input that is a perturbing a state to the output defined by another state?

Maybe you're trying to accomplish something else?

Christine Li 2020-7-16

编辑：Christine Li 2020-7-17

Agree with Paul.

No matter you use state space matrix [A B C D] or transfer function, they all used to describe your system, and can be transfered by ss2tf in matlab or use

For your system, you have 4 states, which are internal. Suppose your 4 stataes are [

]. Now, you want to extract x2 and x3 and try to find a transfer function to describe the dynamics of x2 and x3. But if you write down your eqaution for x2 and x3, which will be:

You can find out that the dynamics of x2 and x3 are influenced by x1 and x4 as well. So if you want to get the transfer function for your system that only has two states: x2 and x3, then you need to check your equations to see if you can treat x1 and x2 as your system disturbances or inputs and extract them out of your original A matrix. If you can make your equaitons be in a format of