doc: additional details

Author: franckgaga

src/controller/construct.jl

@@ -536,9 +536,9 @@ The matrix appear in the quadratic general form:
 ```
 The Hessian matrix is constant if the model and weights are linear and time invariant (LTI): 
 ```math
-    \mathbf{H̃} = 2 (   \mathbf{Ẽ}'      \mathbf{M}_{H_p} \mathbf{Ẽ} 
-                     + \mathbf{P̃_{Δu}}' \mathbf{Ñ}_{H_c} \mathbf{P̃_{Δu}} 
-                     + \mathbf{P̃_{u}}'  \mathbf{L}_{H_p} \mathbf{P̃_{u}}     )
+    \mathbf{H̃} = 2 (   \mathbf{Ẽ'}      \mathbf{M}_{H_p} \mathbf{Ẽ} 
+                     + \mathbf{P̃_{Δu}'} \mathbf{Ñ}_{H_c} \mathbf{P̃_{Δu}} 
+                     + \mathbf{P̃_{u}'}  \mathbf{L}_{H_p} \mathbf{P̃_{u}}     )
 ```
 The vector ``\mathbf{q̃}`` and scalar ``r`` need recalculation each control period ``k``, see
 [`initpred!`](@ref). ``r`` does not impact the minima position. It is thus useless at
@@ -641,7 +641,7 @@ Augment manipulated inputs constraints with slack variable ϵ for softening.
 
 Denoting the decision variables augmented with the slack variable
 ``\mathbf{Z̃} = [\begin{smallmatrix} \mathbf{Z} \\ ϵ \end{smallmatrix}]``, it returns the
-augmented conversion matrix ``\mathbf{P̃u}``, similar to the one described at
+augmented conversion matrix ``\mathbf{P̃_u}``, similar to the one described at
 [`init_ZtoU`](@ref). It also returns the ``\mathbf{A}`` matrices for the inequality
 constraints:
 ```math

src/controller/execute.jl

@@ -183,8 +183,8 @@ They are computed with these equations using in-place operations:
     \mathbf{C_u}     &= \mathbf{T}\mathbf{u}(k-1)    - \mathbf{R̂_u}                     \\
     \mathbf{q̃}       &= 2[    (\mathbf{M}_{H_p} \mathbf{Ẽ})'   \mathbf{C_y} 
                             + (\mathbf{L}_{H_p} \mathbf{P̃_U})' \mathbf{C_u}   ]         \\
-    r                &=     \mathbf{C_y}' \mathbf{M}_{H_p} \mathbf{C_y} 
-                          + \mathbf{C_u}' \mathbf{L}_{H_p} \mathbf{C_u}
+    r                &=     \mathbf{C_y'}  \mathbf{M}_{H_p} \mathbf{C_y} 
+                          + \mathbf{C_u'} \mathbf{L}_{H_p} \mathbf{C_u}
 \end{aligned}
 ```
 """

src/controller/transcription.jl

@@ -44,8 +44,9 @@ operating point ``\mathbf{x̂_{op}}`` (see [`augment_model`](@ref)):
 where ``\mathbf{x̂}_i(k+j)`` is the state prediction for time ``k+j``, estimated by the
 observer at time ``i=k`` or ``i=k-1`` depending on its `direct` flag. This transcription
 method is generally more efficient for large control horizon ``H_c``, unstable or highly
-nonlinear plant models/constraints. Sparse optimizers like `OSQP` or `Ipopt` are recommended
-for large-scale problems.
+nonlinear plant models/constraints. 
+
+Sparse optimizers like `OSQP` or `Ipopt` are recommended for this method.
 """
 struct MultipleShooting <: TranscriptionMethod end
 

src/estimator/construct.jl

@@ -154,9 +154,9 @@ returns the augmented matrices `Â`, `B̂u`, `Ĉ`, `B̂d` and `D̂d`:
 \end{aligned}
 ```
 An error is thrown if the augmented model is not observable and `verify_obsv == true`. The
-augmented operating points `x̂op` and `f̂op` are simply ``\mathbf{x_{op}}`` and
-``\mathbf{f_{op}}`` vectors appended with zeros (see [`setop!`](@ref)). See Extended Help
-for a detailed definition of the augmented matrices.
+augmented operating points ``\mathbf{x̂_{op}}`` and ``\mathbf{f̂_{op}}`` are simply 
+``\mathbf{x_{op}}`` and ``\mathbf{f_{op}}`` vectors appended with zeros (see [`setop!`](@ref)). 
+See Extended Help for a detailed definition of the augmented matrices and vectors.
 
 # Extended Help
 !!! details "Extended Help"
@@ -179,6 +179,13 @@ for a detailed definition of the augmented matrices.
     \mathbf{D̂_d} &= \mathbf{D_d}
     \end{aligned}
     ```
+    and the operating points of the augmented model are:
+    ```math
+    \begin{aligned}
+        \mathbf{x̂_{op}} &= \begin{bmatrix} \mathbf{x_{op}} \\ \mathbf{0} \end{bmatrix} \\
+        \mathbf{f̂_{op}} &= \begin{bmatrix} \mathbf{f_{op}} \\ \mathbf{0} \end{bmatrix}
+    \end{aligned}
+    ```
 """
 function augment_model(model::LinModel{NT}, As, Cs_u, Cs_y; verify_obsv=true) where NT<:Real
     nu, nx, nd = model.nu, model.nx, model.nd