state space model with constant term

If i linearize a nonlinear system model of the form

 dx/dt=f(x,u) so that it can be written in the form

dx/dt=Ax+Bu+constant

What is the best way to handle the constant if I want to create an lqr regulator or an observer? For example the system below?

A=[0 1 0;0 0 1;0 0 0] B=[0;0;1] Constant=[0;1;0]

[Shashank Prasanna]

This is not technically a state space representation. Could you give some information on what the constant is? is it an initial condition? Or atleast share what the f(x,u) looks like.

[Nikolaj Goodger]

f(x,u) could be a nonlinear model of the form dx1/dt=x2 dx2/dt=x3^2 dx3/dt=u

If I then want to linearize the system at the point t0_x1,t0_x2,t0_x3 I end up with

dx1/dt=x2

dx2/dt=(x3-t0_x3)*2*t0_x3+t0_x3^2

dx3/dt=u

So then I could write in the form

dx/dt=Ax+Bu+constant where

A=[0 1 0;0 0 2*t0_x3;0 0 0] B=[0;0;1] constant=[0 t0_x3^2-t0_x3 0]

but as Azzi said i can consider the constant as a disturbance.

Answer 1

[Azzi Abdelmalek]

You can consider a constant as a disturbance signal. You can calculate your lqr regulator without this constant, but you should add to your system an integrator which will eliminate the effect of any disturbance signal. (m integrators for m outputs)

[Nikolaj Goodger]

Thankyou. This put me on the right track and I was able to get a solution :-)

Answer 2

[Hesam Mazaheri]

hello, regarding your answer, how can i model this disturbance as an constant to the system in m-file by defining matrices A,B,C,D as ss, how can I add this constant term. after linearing a nonlinear system, my state space is like this: Xdot=Ax+Bu+G1(Constant term) y=Cx+Du+G2(another constant term)

[Asser Mohamed]

I have the same question. hope you can give us an answer