doc: additional comments MHE

Author: franckgaga

src/estimator/internal_model.jl

@@ -99,12 +99,9 @@ function InternalModel(
     return InternalModel{NT, SM}(model, i_ym, stoch_ym.A, stoch_ym.B, stoch_ym.C, stoch_ym.D)
 end
 
-"Validate if `model` is asymptotically stable for [`LinModel`](@ref)."
-function validate_internalmodel(model::LinModel, nym, Csm, Dsm)
-    poles = eigvals(model.A)
-    if any(abs.(poles) .≥ 1) 
-        error("InternalModel does not support integrating or unstable model")
-    end
+"Validate if deterministic `model` and stochastic model `Csm, Dsm` for `InternalModel`s."
+function validate_internalmodel(model::SimModel, nym, Csm, Dsm)
+    validate_poles(model)
     if size(Csm,1) ≠ nym || size(Dsm,1) ≠ nym
         error("Stochastic model output quantity ($(size(Csm,1))) is different from "*
               "measured output quantity ($nym)")
@@ -115,18 +112,15 @@ function validate_internalmodel(model::LinModel, nym, Csm, Dsm)
     return nothing
 end
 
-"Only validate stochastic model size is `model` is not a [`LinModel`](@ref)."
-function validate_internalmodel(::SimModel, nym, Csm, Dsm)
-    if size(Csm,1) ≠ nym || size(Dsm,1) ≠ nym
-        error("Stochastic model output quantity ($(size(Csm,1))) is different from "*
-              "measured output quantity ($nym)")
-    end
-    if iszero(Dsm)
-        error("Stochastic model requires a nonzero direct transmission matrix D")
+"Validate if `model` is asymptotically stable for `LinModel`s."
+function validate_poles(model::LinModel)
+    poles = eigvals(model.A)
+    if any(abs.(poles) .≥ 1) 
+        error("InternalModel does not support integrating or unstable model")
     end
     return nothing
 end
-
+validate_poles(::SimModel) = nothing
 
 @doc raw"""
     matrices_internalmodel(model::LinModel)

src/estimator/mhe/construct.jl

@@ -186,9 +186,9 @@ N_k =                     \begin{cases}
 ```
 The vectors ``\mathbf{Ŵ}`` and ``\mathbf{V̂}`` encompass the estimated process noise
 ``\mathbf{ŵ}(k-j)`` and sensor noise ``\mathbf{v̂}(k-j)`` from ``j=N_k-1`` to ``0``. The 
-Extended Help defines the two vectors. See [`UnscentedKalmanFilter`](@ref) for details on 
-the augmented process model and ``\mathbf{R̂}, \mathbf{Q̂}`` covariances. The matrix 
-``\mathbf{P̂}_{k-N_k}(k-N_k+1)`` is estimated with an [`ExtendedKalmanFilter`](@ref).
+Extended Help defines the two vectors and the scalar ``ϵ``. See [`UnscentedKalmanFilter`](@ref)
+for details on he augmented process model and ``\mathbf{R̂}, \mathbf{Q̂}`` covariances. The
+matrix ``\mathbf{P̂}_{k-N_k}(k-N_k+1)`` is estimated with an [`ExtendedKalmanFilter`](@ref).
 
 !!! warning
     See the Extended Help if you get an error like:    
@@ -244,6 +244,10 @@ MovingHorizonEstimator estimator with a sample time Ts = 10.0 s, Ipopt optimizer
         \mathbf{x̂}_k(k-j+1) &= \mathbf{f̂}\Big(\mathbf{x̂}_k(k-j), \mathbf{u}(k-j), \mathbf{d}(k-j)\Big) + \mathbf{ŵ}(k-j)
     \end{aligned}
     ```
+    The slack variable ``ϵ`` relaxes the constraints if enabled, see [`setconstraint!`](@ref). 
+    It is disabled by default for the MHE (from `Cwt=Inf`) but it should be activated for
+    problems with two or more types of bounds, to ensure feasibility (e.g. on the estimated
+    state and sensor noise).
 
     For [`LinModel`](@ref), the optimization is treated as a quadratic program with a
     time-varying Hessian, which is generally cheaper than nonlinear programming. For 
@@ -317,10 +321,10 @@ noise ``\mathbf{ŵ}`` and sensor noise ``\mathbf{v̂}``:
 and also ``ϵ ≥ 0``. All the constraint parameters are vector. Use `±Inf` values when there
 is no bound. The constraint softness parameters ``\mathbf{c}``, also called equal concern
 for relaxation, are non-negative values that specify the softness of the associated bound.
-Use `0.0` values for hard constraints. The process and sensor noise constraints are all soft
-by default. Notice that constraining the estimated sensor noises is equivalent to bounding 
-the innovation term, since ``\mathbf{v̂}(k) = \mathbf{y^m}(k) - \mathbf{ŷ^m}(k)``. See 
-Extended Help for details on model augmentation and time-varying constraints.
+Use `0.0` values for hard constraints (default for all of them). Notice that constraining
+the estimated sensor noises is equivalent to bounding the innovation term, since 
+``\mathbf{v̂}(k) = \mathbf{y^m}(k) - \mathbf{ŷ^m}(k)``. See Extended Help for details on 
+model augmentation and time-varying constraints.
 
 # Arguments
 !!! info