doc: internals for MHE relaxation

Author: franckgaga

docs/src/internals/predictive_control.md

@@ -31,5 +31,5 @@ ModelPredictiveControl.relaxterminal
 ## Constraints
 
 ```@docs
-ModelPredictiveControl.linconstraint!
+ModelPredictiveControl.linconstraint!(::PredictiveController, ::LinModel)
 ```

docs/src/internals/state_estim.md

@@ -23,6 +23,21 @@ ModelPredictiveControl.init_predmat_mhe
 ModelPredictiveControl.init_matconstraint_mhe
 ```
 
+## Constraint Relaxation
+
+```@docs
+ModelPredictiveControl.relaxarrival
+ModelPredictiveControl.relaxX̂
+ModelPredictiveControl.relaxŴ
+ModelPredictiveControl.relaxV̂
+```
+
+## Constraints
+
+```@docs
+ModelPredictiveControl.linconstraint!(::MovingHorizonEstimator, ::LinModel)
+```
+
 ## Evaluate Estimated Output
 
 ```@docs

src/controller/construct.jl

@@ -672,8 +672,8 @@ constraints:
     \mathbf{A_{U_{max}}} 
 \end{bmatrix} \mathbf{ΔŨ} ≤
 \begin{bmatrix}
-    - \mathbf{U_{min}} + \mathbf{T} \mathbf{u}(k-1) \\
-    + \mathbf{U_{max}} - \mathbf{T} \mathbf{u}(k-1)
+    - \mathbf{U_{min} + T} \mathbf{u}(k-1) \\
+    + \mathbf{U_{max} - T} \mathbf{u}(k-1)
 \end{bmatrix}
 ```
 """
@@ -745,8 +745,8 @@ Denoting the input increments augmented with the slack variable
     \mathbf{A_{Y_{max}}}
 \end{bmatrix} \mathbf{ΔŨ} ≤
 \begin{bmatrix}
-    - \mathbf{Y_{min}} + \mathbf{F} \\
-    + \mathbf{Y_{max}} - \mathbf{F} 
+    - \mathbf{Y_{min} + F} \\
+    + \mathbf{Y_{max} - F} 
 \end{bmatrix}
 ```
 """
@@ -786,8 +786,8 @@ the inequality constraints:
     \mathbf{A_{x̂_{max}}}
 \end{bmatrix} \mathbf{ΔŨ} ≤
 \begin{bmatrix}
-    - \mathbf{x̂_{min}} + \mathbf{f_x̂} \\
-    + \mathbf{x̂_{max}} - \mathbf{f_x̂} 
+    - \mathbf{x̂_{min} + f_x̂} \\
+    + \mathbf{x̂_{max} - f_x̂}
 \end{bmatrix}
 ```
 """

src/estimator/mhe/construct.jl

@@ -633,7 +633,23 @@ end
         model::SimModel, C, c_x̂min, c_x̂max, x̂min, x̂max, ex̄
     ) -> A_x̃min, A_x̃max, x̃min, x̃max, ẽx̄
 
-TBW
+Augment arrival state constraints with slack variable ϵ for softening the MHE.
+
+Denoting the MHE decision variable augmented with the slack variable ``\mathbf{Z̃} = 
+[\begin{smallmatrix} ϵ \\ \mathbf{Z} \end{smallmatrix}]``, it returns the ``\mathbf{ẽ_x̄}``
+matrix that appears in the estimation error at arrival equation ``\mathbf{x̄} =
+\mathbf{ẽ_x̄ Z̃ + f_x̄}``. It also returns the augmented constraints ``\mathbf{x̃_{min}}`` and
+``\mathbf{x̃_{max}}``, and the ``\mathbf{A}`` matrices for the inequality constraints:
+```math
+\begin{bmatrix} 
+    \mathbf{A_{x̃_{min}}} \\ 
+    \mathbf{A_{x̃_{max}}}
+\end{bmatrix} \mathbf{Z̃} ≤
+\begin{bmatrix}
+    - \mathbf{x̃_{min} + f_x̄} \\
+    + \mathbf{x̃_{max} - f_x̄}
+\end{bmatrix}
+```
 """
 function relaxarrival(::SimModel{NT}, C, c_x̂min, c_x̂max, x̂min, x̂max, ex̄) where {NT<:Real}
     ex̂ = -ex̄
@@ -655,7 +671,22 @@ end
 @doc raw"""
     relaxX̂(model::SimModel, C, C_x̂min, C_x̂max, Ex̂) -> A_X̂min, A_X̂max, Ẽx̂
 
-TBW
+Augment estimated state constraints with slack variable ϵ for softening the MHE.
+
+Denoting the MHE decision variable augmented with the slack variable ``\mathbf{Z̃} = 
+[\begin{smallmatrix} ϵ \\ \mathbf{Z} \end{smallmatrix}]``, it returns the ``\mathbf{Ẽ_x̂}``
+matrix that appears in estimated states equation ``\mathbf{X̂} = \mathbf{Ẽ_x̂ Z̃ + F_x̂}``. It
+also returns the ``\mathbf{A}`` matrices for the inequality constraints:
+```math
+\begin{bmatrix} 
+    \mathbf{A_{X̂_{min}}} \\ 
+    \mathbf{A_{X̂_{max}}}
+\end{bmatrix} \mathbf{Z̃} ≤
+\begin{bmatrix}
+    - \mathbf{X̂_{min} + F_x̂} \\
+    + \mathbf{X̂_{max} - F_x̂}
+\end{bmatrix}
+```
 """
 function relaxX̂(::LinModel{NT}, C, C_x̂min, C_x̂max, Ex̂) where {NT<:Real}
     if !isinf(C) # Z̃ = [ϵ; Z]
@@ -680,7 +711,21 @@ end
 @doc raw"""
     relaxŴ(model::SimModel, C, C_ŵmin, C_ŵmax, nx̂) -> A_Ŵmin, A_Ŵmax
 
-TBW
+Augment estimated process noise constraints with slack variable ϵ for softening the MHE.
+
+Denoting the MHE decision variable augmented with the slack variable ``\mathbf{Z̃} = 
+[\begin{smallmatrix} ϵ \\ \mathbf{Z} \end{smallmatrix}]``, it returns the ``\mathbf{A}`` 
+matrices for the inequality constraints:
+```math
+\begin{bmatrix} 
+    \mathbf{A_{Ŵ_{min}}} \\ 
+    \mathbf{A_{Ŵ_{max}}}
+\end{bmatrix} \mathbf{Z̃} ≤
+\begin{bmatrix}
+    - \mathbf{Ŵ_{min}} \\
+    + \mathbf{Ŵ_{max}}
+\end{bmatrix}
+```
 """
 function relaxŴ(::SimModel{NT}, C, C_ŵmin, C_ŵmax, nx̂) where {NT<:Real}
     A = [zeros(NT, length(C_ŵmin), nx̂) I]
@@ -695,7 +740,22 @@ end
 @doc raw"""
     relaxV̂(model::SimModel, C, C_v̂min, C_v̂max, E) -> A_V̂min, A_V̂max, Ẽ
 
-TBW
+Augment estimated sensor noise constraints with slack variable ϵ for softening the MHE.
+
+Denoting the MHE decision variable augmented with the slack variable ``\mathbf{Z̃} = 
+[\begin{smallmatrix} ϵ \\ \mathbf{Z} \end{smallmatrix}]``, it returns the ``\mathbf{Ẽ}``
+matrix that appears in estimated sensor noise equation ``\mathbf{V̂} = \mathbf{Ẽ Z̃ + F}``. It
+also returns the ``\mathbf{A}`` matrices for the inequality constraints:
+```math
+\begin{bmatrix} 
+    \mathbf{A_{V̂_{min}}} \\ 
+    \mathbf{A_{V̂_{max}}}
+\end{bmatrix} \mathbf{Z̃} ≤
+\begin{bmatrix}
+    - \mathbf{V̂_{min} + F} \\
+    + \mathbf{V̂_{max} - F}
+\end{bmatrix}
+```
 """
 function relaxV̂(::LinModel{NT}, C, C_v̂min, C_v̂max, E) where {NT<:Real}
     if !isinf(C) # Z̃ = [ϵ; Z]