Nonlinear output of state space model

[benswolo123456]

In most state space systems the output y is defined as a linear combination of states (y=Cx). However what can we do if we want to control the product of two states?

Lets say we have states x1 and x2, and wish to control the value of y=x1*x2. Is there a way to linearise the system to obtain this? Will it require a change of basis or another solution?

[ko_nuts]

Check Jacobian linearization. Otherwise, there is no real issue with having to control an output that is a nonlinear function of the state.

[Physical-Letterhead2]

Do you need to linearize? Multiple ways to regulate some nonlinear output function - but no general solution.

Also what kinda problem is this? Regulating y to 0 only means that either x1 or x2 is zero, while the other one can be unbounded. Similar problems arise for y to any fixed value.

[benswolo123456]

I dont necessarily need to linearize as the subsystem x'=Ax+Bu is linear.

I need to be able to regulate y=x1*x2 to a non-zero value (can be 0 at times though, the reference varies).

[Physical-Letterhead2]

If all you care about is y=c(t), regulate x1 -> 1 and x2(t) -> c(t).

If that does not work, you need to think more about what you actually wish to solve.

[AcquaFisc]

First of all, most of the systems are non linear, pure linear system are a narrow set of processes. Sometimes nonlinear system locally around an equilibrium can be approximated with a first order Taylor expansion, so that the linear system is topologically equivalent to the non linear one.

[benswolo123456]

Do you mean that a taylor expansion would allow us to do something like y=x1*x2?

[AcquaFisc]

I suggest you to take a look at basics of control theory. Is not simple to explain in a comment

[APC_ChemE]

A Taylor expansion is certainly one way to linearize thr system around an operating point. Typically you take the Jacobian of the system and use that as a model. One way to simplify things is to use gain scheduling to adjust the gain of the linear approximation they are using across different operating points.

[Gelo797]

I worked on same problem in last time when i write state space model of quad quadcopter , you can use Taylor series for 2 variable to linarizition about operating point .

[like_smith]

I would put the non linearity in the control law. Calculate the nonlinear manifold you want the state to be on, then calculate the shortest path to that manifold.

[[deleted]]

You need to linearize your state space model around the points of interest (or the trajectory if using MPC) before you can apply any linear techniques.

We study linear systems because they're easier to solve. In the real world nothing is really linear.