How to solve this nonlinear output regulation problem

[spaveofvivi]

Hello, everyone

There is a control theory problem that i would like to ask for help here.

Suppose a very simple nonlinear system as follows

x' = u
y = (2-cosx)*(2-cosx)

how to find a control law u that makes the output y can track a constant asymptotically,
of course the x should be stable too.

best wishes

 

[Mark44]

Hello, everyone

There is a control theory problem that i would like to ask for help here.

Suppose a very simple nonlinear system as follows

x' = u
y = (2-cosx)*(2-cosx)

how to find a control law u that makes the output y can track a constant asymptotically,
of course the x should be stable too.

best wishes

Here's what I would do.
Starting with your 2nd equation, expand the right side.
y = 4 - 4cosx + cos²x

This equation is quadratic in cosx. Use the quadratic formula to solve for cosx, and then use trig to solve for x.
Take the derivative to get u.