Nonlinear systems as linear time-varying systems?

Is it possible to represent nonlinear systems as linear time-varying systems? I found little papers and no books covering this subject.
Does anyone have some references on that?
I'm interested in both methods for approximating nonlinear models as LTV models and identification of LTV systems from empirical data.

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I don't think you can.
How can you represent the nonlinear system
\dot{x} = -x^3
as a linear time varying system?
any linear system you pick will not truly represent this non linear system.
If you are trying to approximate the function then linearize the nonlinear system.

Maybe but would that be useful? You'd just be pushing off the non-linearity to the parameters.

as far as i recall you can approximate them by having a finite set of linear time varying systems which approximate the real system around certain points.

like you have a tensor or something of linear time varying systems that are used to do that.

>How can you represent the nonlinear system
>\dot{x} = -x^3
>as a linear time varying system?

well as the solution of the ODE has to be continous you can approximate it pointwise by linear functions.

That is what I said at the bottom, you can just linearize the system.

Yes and it is really useful if you think about it for a while.

Show an article or subject related to this.

Nah, it would just be rejected if I as much as tried. I'm gonna sit here and drink piwo instead.

...

Here, for your enlightenment.

I think it would be really useful for control systems design, since LTV systems are easier to analyze and would require less computational power for real time applications. Also, although I'm not sure about this, I think the identification task for LTV systems would be simpler than for nonlinear systems.

>That is what I said at the bottom, you can just linearize the system.
true, my bad

Are you also interested in dicks? Because you sound like a faggot.

>LTV systems are easier to analyze and would require less computational power for real time applications
Sure, but analyzing the evolution of the system will surely require nonlinearity. Either you don't know the state of a system at a given time or it somehow changes perfectly by itself. I might be missing something here, so let me know.

At least in the context of non-linear hyperbolic systems there are genereally analogues between a steady case and the unsteady case in one less dimension.

Linearising a non-linear hyperbolic system allows it to be transformed to an elliptic system through affine transformations in which case unsteadiness is dealt with quite simply when using numerical methods

Yeah, you could represent a parabolic arc as a line whos slope changes with time.

If you wanted to consider how the slope changes, it is going to be with t^2. Substitute that in and you're going to find out the form of the whole motion is at^2 + bt +c, a non-linear function.

non linear functions simplify complex behavior, it's like asking "can I solve triangles without trig?" yes, but you're an idiot.

Thinking about this a little more, it would be extremely simple to get the gradient numerically. However, I don't think there's any way of getting around calculating rate of change wrt time nonlinearly.
this

Yes, search for linear parameter varying (LPV) systems.

You're full of shit.

permalink.lanl.gov/object/tr?what=info:lanl-repo/lareport/LA-UR-80-5007

The fluid equations are
linearized about the static solution, and
the stability of the perturbation is
studied. To date, only the first instability
has been computed analytically. Once
we know the parameter value (for example,
the Rayleigh number) for the onset
of this first time-varying instability, we
must determine the correct form of the
solution after the perturbation has grown
large beyond the linear regime. To this
solution we add a new time-dependent
perturbative mode, again linearized (now
about a time-varying, nonanalytically
available solution) to discover the new
instability. To date, the second step of
the analysis has been performed only
numerically. This process, in principle,
can be repeated again and again until a
suitably turbulent flow has been obtained.
At each successive stage, the
computation grows successively more
intractable.
However, it is just at this point that
the universality theory solves the
problem; it works only after enough instabilities have entered to reach the
asymptotic regime. Since just two such
instabilities already serve as a good approximate
starting point, we need only a
few parameters for each flow to empower
the theory to complete the hard
part of the infinite cascade of more complex
instabilities.

No, not in any useful capacity. You could look at a single trajectory and fit it to a LTV system, but it won't accurately represent any significantly nonlinear system.