small doc correction

Author: franckgaga

docs/src/public/state_estim.md

@@ -18,7 +18,8 @@ error with closed-loop control (offset-free tracking).
 
 The estimators are all implemented in the predictor form (a.k.a. observer form), that is,
 they all estimates at each discrete time ``k`` the states of the next period
-``\mathbf{x̂}_k(k+1)``. In contrast, the filter form that estimates ``\mathbf{x̂}_k(k)`` is
+``\mathbf{x̂}_k(k+1)`` (also written ``\mathbf{x̂}_{k+1|k}`` [elsewhere](https://en.wikipedia.org/wiki/Kalman_filter)).
+In contrast, the filter form that estimates ``\mathbf{x̂}_k(k)`` is
 sometimes slightly more accurate.
 
 The predictor form comes in handy for control applications since the estimations come after

src/controller/linmpc.jl

@@ -109,7 +109,7 @@ and with the following nomenclature:
 
 The ``\mathbf{ΔU}`` vector includes the manipulated input increments ``\mathbf{Δu}(k+j) = 
 \mathbf{u}(k+j) - \mathbf{u}(k+j-1)`` from ``j=0`` to ``H_c-1``, the ``\mathbf{Ŷ}`` vector, 
-the output predictions ``\mathbf{ŷ(k+j)}`` from ``j=1`` to ``H_p``, and the ``\mathbf{U}`` 
+the output predictions ``\mathbf{ŷ}(k+j)`` from ``j=1`` to ``H_p``, and the ``\mathbf{U}`` 
 vector, the manipulated inputs ``\mathbf{u}(k+j)`` from ``j=0`` to ``H_p-1``. The 
 manipulated input setpoint predictions ``\mathbf{R̂_u}`` are constant at ``\mathbf{r_u}``.
 

src/estimator/kalman.jl

@@ -283,7 +283,7 @@ It implements the time-varying Kalman Filter in its predictor (observer) form :
 ```math
 \begin{aligned}
     \mathbf{M}(k)       &= \mathbf{P̂}_{k-1}(k)\mathbf{Ĉ^m}'
-                           [\mathbf{Ĉ^m P̂}_{k-1}(k)\mathbf{Ĉ^m + R̂}]^{-1}                 \\
+                           [\mathbf{Ĉ^m P̂}_{k-1}(k)\mathbf{Ĉ^m}' + \mathbf{R̂}]^{-1}       \\
     \mathbf{K}(k)       &= \mathbf{Â M(k)}                                                \\
     \mathbf{ŷ^m}(k)     &= \mathbf{Ĉ^m x̂}_{k-1}(k) + \mathbf{D̂_d^m d}(k)                  \\
     \mathbf{x̂}_{k}(k+1) &= \mathbf{Â x̂}_{k-1}(k) + \mathbf{B̂_u u}(k) + \mathbf{B̂_d d}(k)
@@ -643,7 +643,7 @@ substitutions ``\mathbf{Â = F̂}(k)`` and ``\mathbf{Ĉ^m = Ĥ^m}(k)``:
 ```math
 \begin{aligned}
     \mathbf{M}(k)       &= \mathbf{P̂}_{k-1}(k)\mathbf{Ĥ^m}'(k)
-                           [\mathbf{Ĥ^m}(k)\mathbf{P̂}_{k-1}(k)\mathbf{Ĥ^m}(k) + \mathbf{R̂}]^{-1}    \\
+                           [\mathbf{Ĥ^m}(k)\mathbf{P̂}_{k-1}(k)\mathbf{Ĥ^m}'(k) + \mathbf{R̂}]^{-1}    \\
     \mathbf{K}(k)       &= \mathbf{F̂}(k) \mathbf{M}(k)                                    \\
     \mathbf{ŷ^m}(k)     &= \mathbf{ĥ^m}\Big( \mathbf{x̂}_{k-1}(k), \mathbf{d}(k) \Big)     \\
     \mathbf{x̂}_{k}(k+1) &= \mathbf{f̂}\Big( \mathbf{x̂}_{k-1}(k), \mathbf{u}(k), \mathbf{d}(k) \Big)