about approximate linearization


hello all,
I'm a little bit unexperienced in the area and wanna learn probably a
basic key issue about approximate linearization of nonlinear models
for control.
For an example if a process model is like "x1dot=square(x2) +..", then
I know that approximate linearization of the model is extracted via
taylor expansion,
but if the nonlinear model is like "x1dot=x2.x3 +..", how should I
realize the linearization?
Thank you in advance...
Reply to
melda
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Explaining it by Taylor's expansion is one way.
What you're really trying to do is find the derivative around the operating point for the whole state vector. So your
x1dot = x2 * x3
would linearize as
x1dot = x2(0) *
x3 + x3(0) * x2,
where the x2(0) and x3(0) are the values of those two states at the nominal operating point.
More formally (but still from memory, so I may not be quoting textbooks exactly), if x is a vector, and if your system is described as
dx -- = f(x, u, t) dt
then your linearization would be
dx del | -- = f(x(0), u, t) + ----- f(x, u, t) | , dt del x | x = x(0)
where the "del / del x" operation takes the Jordanian of the system function.
This still isn't a perfectly linear system because you're adding in the value of the system function at the operating point, but it's affine, and pretty easy to turn into a linear system by replacing that expression with some extra integrators, appropriately initialized.
Reply to
Tim Wescott
Small correction: designating the derivative/Jacobian of f(x,u,t) with respect to x as f'(x0,u,t) the linearization would be f(x0,u,t)+f'(x0,u,t)(x-x0) Where the second term yields a linear correction; the Jacobian matrix times a variation vector x0. Ray
Reply to
RRogers
        | x =
Thank you, yes.
Reply to
Tim Wescott

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