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March 9th, 2018, 10:23 AM   #1
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Identification of unknown function in a DAE system

I have a system of 1st order odes given by

\dot{x_1}(t) = \alpha_1 f_1(x_1,t) + \beta_1 u(t) \\
\dot{x_2}(t) = \alpha_2 f_2(x_2,t) + \beta_2 u(t)

They are constrained by an algebraic equation

$$ x_1(t) + x_2(t) = k $$

where $\left( \alpha_1,\alpha_2, \beta_1,\beta_2 , k \right) \in \mathbb{R}$ are known constants (i.e. parameters). $f_1(t)$ and $f_2(t)$ are both unknown.

Starting from a rich set of input-output **noise-free** data available from simulating a complex proxy system, what would be the best procedure to identify (even a subset of repeatable/characteristic properties) the unknown **_possibly time-varying_** functions $f_1(x_1,t)$ and $f_2(x_2,t)?$ I am almost certain that $f_1(x_1,t)$ and $f_2(x_2,t)$ are both linear.

I am looking for a grey-box system-id approach that shall work well to arbitrary excitations in all future simulations. (NOT merely a curve-fitting procedure to match a specific excitation input-output dataset)
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March 9th, 2018, 08:47 PM   #2
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Interesting question. I have a few observations which might help get you started.

Initially, lets focus on the question of determining if the equation is linear. If this is the case, then note that it is a completely uncoupled linear equation. Each of your functions has the form $f_i(x_i,t) = c_i(t)x_i(t)$ and therefore, the ODE is given by writing
\[\dot x = A(t)x(t) + \mu(t) \] where $A$ is amatrix and $\mu$ is the vectorized inhomogenous term. Most importantly, the observation that it is uncoupled means that $A$ is a diagonal matrix.

This leads to a fundamental matrix solution defined by the usual matrix exponential:
\[x \mapsto \exp(tA(0))\cdot x \]
and applying the variation of constants formula gives the fundamental solution operator (flow)
\[\Phi(x,t) = \exp(tA(0)) \cdot x + \int_0^t \exp((t-s)A(s)) \mu(s) \ ds \]

Now, you claim you have the ability to query this flow for a bunch of initial data. In other words, you can pick a few values, $x_1,x_2$ and compute the trajectory $x_1(t) = \Phi(x_1,t), x_2(t) = \Phi(x_2,t)$ and from this data and the fact that $A$ is diagonal, hence easily invertible, you can obtain explicit values for $c_1(t),c_2(t)$.

Now, compute the flow for a "bunch" of other initial data by querying the black box, and also compute those trajectories in the linear ODE defined by choosing $c_1,c_2$ to satisfy the first two trajectories, and check if these trajectories match. If they don't, then $f_1,f_2$ are not linear, and if they do, then they are linear, and moreover, you know explicitly what they are.

2. The conserved quantity $x_1 + x_2 = k$ means that a particular linear combination of solutions for the ODE will conserve this quantity. Hence, if the equations are linear, then this solution must correspond to an eigenvalue of 0 for $A$ so you only have a 1-dimensional subspace of nontrivial solutions. Thus, you actually expect to be able to uniquely determine $c_1,c_2$ using only a single initial condition.

3. In the event it turns out to be nonlinear, then how much you can learn depends whether you are allow to query $f_1,f_2$ directly, rather than just the solution operator. If so, you can still uncover a great deal of information about the phase space structure. I can provide more details if this is relevant.

I'll think about this question some more since it is interesting. I hope my comments so far are helpful.
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March 16th, 2018, 11:56 AM   #3
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Try for $\displaystyle x_1 ' (t)+ x_2 ' (t)=k'=0$
Now plug to system expression above
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