Grey-box system ID for a set of DAEs

[Krishnakumar Gopalakrishnan]

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 $\{ \alpha_1,\alpha_2, \beta_1,\beta_2 , k \} \in \mathbb{R}$  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)

Can sysquake be used for this?

[Yves Piguet]

I'm not sure to understand your problem. Are x1 and x2 your outputs? Do they already satisfy x1(t)+x2(t)=k? Otherwise, a solution is constant x1 and x2; then their derivatives are zero and you can solve for f1 and f2. But I guess it isn't that.

In general, Sysquake doesn't include a DAE solver, but you could write one.

Yves

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