doc: clarifying model augmentation for NonLinMPC in f̂! docstring

docs/src/internals/state_estim.md

@@ -4,13 +4,6 @@
 Pages = ["state_estim.md"]
 ```
 
-## Augmented Model
-
-```@docs
-ModelPredictiveControl.f̂!
-ModelPredictiveControl.ĥ!
-```
-
 ## Estimator Construction
 
 ```@docs
@@ -28,6 +21,13 @@ ModelPredictiveControl.init_matconstraint_mhe
 ModelPredictiveControl.get_optim_functions(::MovingHorizonEstimator, ::ModelPredictiveControl.GenericModel)
 ```
 
+## Augmented Model
+
+```@docs
+ModelPredictiveControl.f̂!
+ModelPredictiveControl.ĥ!
+```
+
 ## Update Quadratic Optimization
 
 ```@docs

src/estimator/construct.jl

@@ -236,8 +236,8 @@ end
 
 Augment [`LinModel`](@ref) state-space matrices with stochastic ones `As`, `Cs_u`, `Cs_y`.
 
-If ``\mathbf{x_0}`` are `model.x0` states, and ``\mathbf{x_s}``, the states defined at
-[`init_estimstoch`](@ref), we define an augmented state vector ``\mathbf{x̂} = 
+If ``\mathbf{x_0}`` is `model.x0` state, and ``\mathbf{x_s}``, the states defined at
+[`init_estimstoch`](@ref), we define an augmented state vector ``\mathbf{x̂_0} = 
 [ \begin{smallmatrix} \mathbf{x_0} \\ \mathbf{x_s} \end{smallmatrix} ]``. The method
 returns the augmented matrices `Â`, `B̂u`, `Ĉ`, `B̂d` and `D̂d`:
 ```math
@@ -253,9 +253,9 @@ See Extended Help for a detailed definition of the augmented matrices and vector
 
 # Extended Help
 !!! details "Extended Help"
-    Using the `As`, `Cs_u` and `Cs_y` matrices of the stochastic model provided in argument
-    and the `model.A`, `model.Bu`, `model.Bd`, `model.C`, `model.Dd` matrices, the 
-    state-space matrices of the augmented model are defined as follows:
+    Using the `As`, `Cs_u` and `Cs_y` matrices of the stochastic model constructed in
+    [`init_estimstoch`](@ref)), and `model.A`, `model.Bu`, `model.Bd`, `model.C`, `model.Dd`
+    matrices, the state-space matrices of the augmented model are defined as follows:
     ```math
     \begin{aligned}
     \mathbf{Â}   &=                                    \begin{bmatrix} 

src/estimator/execute.jl

@@ -18,19 +18,44 @@ end
 
 Mutating state function ``\mathbf{f̂}`` of the augmented model.
 
-By introducing an augmented state vector ``\mathbf{x̂_0}`` like in [`augment_model`](@ref), the
-function returns the next state of the augmented model, defined as:
+By introducing an augmented state vector ``\mathbf{x̂_0}`` like in [`augment_model`](@ref), 
+the function returns the next state of the augmented model, defined as:
 ```math
 \begin{aligned}
     \mathbf{x̂_0}(k+1) &= \mathbf{f̂}\Big(\mathbf{x̂_0}(k), \mathbf{u_0}(k), \mathbf{d_0}(k)\Big) \\
     \mathbf{ŷ_0}(k)   &= \mathbf{ĥ}\Big(\mathbf{x̂_0}(k), \mathbf{d_0}(k)\Big) 
 \end{aligned}
 ```
 where ``\mathbf{x̂_0}(k+1)`` is stored in `x̂0next` argument. The method mutates `x̂0next`, 
-`û0` and `k0` in place. The argument `û0` is the input vector of the augmented model, 
-computed by ``\mathbf{û_0 = u_0 + ŷ_{s_u}}``. The argument `k0` is used to store the
-intermediate stage values of `model.solver` (when applicable). The model parameter vector
-`model.p` is not included in the function signature for conciseness.
+`û0` and `k0` in place. The argument `û0` stores the disturbed input of the augmented model
+``\mathbf{û_0}``, and `k0`, the intermediate stage values of `model.solver`, when applicable.
+The model parameter `model.p` is not included in the function signature for conciseness. See
+Extended Help for details on ``\mathbf{û_0, f̂}`` and ``\mathbf{ĥ}`` implementations.
+
+# Extended Help
+!!! details "Extended Help"
+    Knowing that the augmented state vector is defined as
+    ``\mathbf{x̂_0} = [ \begin{smallmatrix} \mathbf{x_0} \\ \mathbf{x_s} \end{smallmatrix} ]``,
+    the augmented model functions are:
+    ```math
+    \begin{aligned}
+    \mathbf{f̂}\Big(\mathbf{x̂_0}(k), \mathbf{u_0}(k), \mathbf{d_0}(k)\Big)  &=               \begin{bmatrix}
+        \mathbf{f}\Big(\mathbf{x_0}(k), \mathbf{û_0}(k), \mathbf{d_0}(k), \mathbf{p}\Big)   \\
+        \mathbf{A_s} \mathbf{x_s}(k)                                                        \end{bmatrix} \\
+    \mathbf{ĥ}\Big(\mathbf{x̂_0}(k), \mathbf{d_0}(k)\Big)                   &=
+        \mathbf{h}\Big(\mathbf{x_0}(k), \mathbf{d_0}(k), \mathbf{p}\Big) + \mathbf{y_{s_y}}(k)
+    \end{aligned}
+    ```
+    in which:
+    ```math
+    \begin{aligned}
+    \mathbf{û_0}(k)     &= \mathbf{u_0}(k) + \mathbf{y_{s_u}}(k)                            \\
+    \mathbf{y_{s_u}}(k) &= \mathbf{C_{s_u} x_s}(k)                                          \\
+    \mathbf{y_{s_y}}(k) &= \mathbf{C_{s_y} x_s}(k)
+    \end{aligned}
+    ```
+    The ``\mathbf{f}`` and ``\mathbf{h}`` functions above are in fact the [`f!`](@ref) and 
+    [`h!`](@ref) methods, respectively.
 """
 function f̂!(x̂0next, û0, k0, estim::StateEstimator, model::SimModel, x̂0, u0, d0)
     return f̂!(x̂0next, û0, k0, model, estim.As, estim.Cs_u, x̂0, u0, d0)

src/estimator/kalman.jl

@@ -1125,7 +1125,7 @@ augmented process model:
 \end{aligned}
 ```
 The matrix ``\mathbf{Ĥ^m}`` is the rows of ``\mathbf{Ĥ}`` that are measured outputs. The
-Jacobians are computed with [`ForwardDiff`](@extref ForwardDiff) bu default. The correction
+Jacobians are computed with [`ForwardDiff`](@extref ForwardDiff) by default. The correction
 and prediction step equations are provided below. The correction step is skipped if 
 `estim.direct == true` since it's already done by the user.