changed: algebraic variable at left limit for predicted outputs in MPC and MHE

[franckgaga]

This get VERY confusing with zero order hold inputs, it's better to be as precise as possible in the notation, and to stick to this convention everywhere in the codebase:

Caution

The predicted output $\mathbf{\hat{y}}$, both for MPC and MHE, are computed from the $\mathbf{\hat{a}_0}$ vectors and the h! function, ALWAYS corresponding to the left limit of the sampling instants (with one single exception, see below).

Some details about this convention:

• This convention is intuitive since it's the holding of the inputs for $T_s$ seconds that changes the predicted outputs.
• This is the simplest convention for MPC, since it leads to exactly $H_p$ data points in $\mathbf{\hat{A}_0}$. The right limit convention leads to $H_p + 1$ samples as it requires one additional algebraic vector just after $k + H_p$.
• The variables in $\mathbf{\bar{A}}$ are strictly reserved for the dynamic function fq!.
• It does not mean that $\mathbf{\hat{a}_0}$ vectors are never used in the fq! function. A notorious exception is the MHE with p=1 : the last $\mathbf{\hat{a}_0}$ vector is used in the last trapezoid, for example.
• The algebraic variable at the right limit of the sampling instants are denoted with $\mathbf{a^+}$. They are required for the TrapezoidalCollocation transcription, bundled in the $\mathbf{\bar{A}}$ vector.
• The algebraic variable at the left limit of the sampling instants are denoted with $\mathbf{\hat{a}_0}$ (instead of $\mathbf{a^-}$) so it matches perfectly the notation of the states in $\mathbf{\hat{X}_0}$, since both are used to compute the predicted outputs.
• The notation and the convention is introduced in the extended help of TrapezoidalCollocation and OrthogonalCollocation docstrings.
• Strictly speaking, we should store the manipulated inputs and the measured disturbances at the left limit of the arrival time step, to rigorously estimate the arrival algebraic estimate $\mathbf{\hat{a}_0}(k - N_k + p)$ in the MHE. But it would lead to data windows with $N_k + 1$ data points for them. It would significantly complexify the MHE codebase, with no real gain. So we make an exception instead for the arrival algebraic vector estimate: we use the values at the right limit. This exception as no impact mathematically since $\mathbf{\hat{a}_0}(k - N_k + p)$ is only used to compute the predicted output at the arrival when direct=false (useless if direct=true). Only strictly proper plant models are supported in this package, thus the predicted output at the left limit and at the right limit are equals.

[codecov-commenter]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 97.21%. Comparing base (9cc2550) to head (92c286f).
⚠️ Report is 1 commits behind head on main.

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #466      +/-   ##
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+ Coverage   97.19%   97.21%   +0.02%     
==========================================
  Files          28       28              
  Lines        6879     6865      -14     
==========================================
- Hits         6686     6674      -12     
+ Misses        193      191       -2     

Flag	Coverage Δ
unittests	97.21% <100.00%> (+0.02%)	⬆️

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