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doc: minor modification
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src/estimator/mhe/construct.jl

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@@ -318,10 +318,10 @@ MovingHorizonEstimator estimator with a sample time Ts = 10.0 s, Ipopt optimizer
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``\mathbf{x̂_i}`` can be manually specified with [`setstate!`](@ref), or automatically
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with [`initstate!`](@ref) for [`LinModel`](@ref). Note the MHE with ``p=0`` is slightly
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inconsistent with all the other estimators here. It interprets the initial values as
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``\mathbf{x̂}_{-1}(-1) = \mathbf{x̂_i}`` and ``\mathbf{P̂}_{-1}(-1) = \mathbf{P̂_i}``, that
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``\mathbf{x̂_i} = \mathbf{x̂}_{-1}(-1)`` and ``\mathbf{P̂_i} = \mathbf{P̂}_{-1}(-1)``, that
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is, an *a posteriori* estimate[^2] from the last time step. The MHE with ``p=1`` is
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consistent, interpreting them as ``\mathbf{x̂}_{-1}(0) = \mathbf{x̂_i}`` and
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``\mathbf{P̂}_{-1}(0) = \mathbf{P̂_i}``.
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consistent, interpreting them as ``\mathbf{x̂_i} = \mathbf{x̂}_{-1}(0)`` and
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``\mathbf{P̂_i} = \mathbf{P̂}_{-1}(0)``.
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[^2]: M. Hovd (2012), "A Note On The Smoothing Formulation Of Moving Horizon Estimation",
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*Facta Universitatis*, Vol. 11 №2.

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