added: MovingHorizonEstimator support for direct=true

[franckgaga]

No description provided.

[franckgaga]

https://juliacontrol.github.io/ModelPredictiveControl.jl/previews/PR95

[franckgaga]

I understood an interesting and kinda weird aspect of the MHE in the current form yesterday! I did not found any paper that discuss about that. It seems to be ovelooked in the litterature. I only found http://facta.junis.ni.ac.rs/acar/acar201202/acar20120202.pdf that discuss a little bit more in detailed about the MHE in the current form.

Before the estimation window is filled, the MHE is in fact the full information estimator. The initial covariance at arrival in the objective function is necessarily an a priori estimate $\mathbf{P̂}{-1}(0)$ (since we did not used any measurements yet). But the current form estimates $\mathbf{x̂}_k(k)$, an a posteriori state estimate. Thus, necessarily, at $k=H_e$, the covariance at arrival will switch from the a priori $\mathbf{P̂}{-1}(0)$ estimate to the a posteriori $\mathbf{P̂}_0(0)$ version to match the nature of the state estimate in the arrival cost term. The following time steps will also use the a posteriori estimates $\mathbf{P̂}_1(1)$, $\mathbf{P̂}_2(2)$, and so on.

That's the reason why my notation is now a little bit more complicated, with the introduction of $M_k$ process noises, $N_k$ sensor noises, and the constant $p$.

It is still work in progress, I will notice you when it's ready for you comments!

[franckgaga]

Ok, I struggled a lot to rigorously handle the switch in the arrival estimation covariance $\hat{P}$ for the MHE current form.

The main issue is with linear plant model. The whole optimization problem changes a lot at $k=H_e$ if we do not neglect the switch from an a priori $\hat{P_{k-1}}(k)$ to an a posteriori $\hat{P_{k}}(k)$ arrival estimation covariance.

That's mainly because with $k \lt H_e$ the arrival state estimate is inside the measurement window i.e.: the arrival estimate is $\hat{x_{-1}}(0)$ and the first measurement in the window is $y(0)$. At $k = H_e$ and later on, the arrival state estimate is outside this window i.e.: the arrival estimate is $\hat{x_{0}}(0)$ and the first measurement in the window is $y(1)$. The whole optimization problem change so much at this point that it becomes super complicated to handle that rigorously (we would need to change the objective, the linear inequality constraints, etc. at this point in time)

I think I will finally opt for this simple workaround (perhaps less rigorous):

If we assume that the initial estimate provided by the user is in fact $\hat{x_{-1}}(-1)$ and $\hat{P_{-1}}(-1)$ (instead of $\hat{x_{-1}}(0)$ and $\hat{P_{-1}}(0)$ like the prediction form of the MHE and the other Kalman Filter in the package), the optimization problem do not change at $k=H_e$ since the the arrival estimate is always outside the measurement window. I would mention that small inconsistency in the docstring of the MHE. I think that's a good compromise since the initial estimate is always a guess in practice and I do not see any good reason why $\hat{x_{-1}}(-1)$ would be a drastically different guess than $\hat{x_{-1}}(0)$ for the user. Another advantage is the new $M_k$ variable is no longer necessary (only $N_k$, as before).

Note that this only applies for the current form. The prediction form is easier since the arrival estimate is always inside the measurement window.

[franckgaga]

Closing, will merge #96