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17 lines
823 B
TeX
17 lines
823 B
TeX
\subsection{MPC}
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\bi{Cost function} ($p(\vec{x}_N)$ \textit{terminal cost}, sum the \textit{stage cost})
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\[
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J_{0 \rightarrow N}(\vec{x}_0, \vec{u}_0, \ldots, \vec{u_{N - 1}}) = p(\vec{x}_N) + \sum_{k = 0}^{N - 1} q(\vec{x}_k, \vec{u}_k)
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\]
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We minimize the above s.t. for $k \in \{ 0, \ldots, N - 1 \}$ we have $\vec{x}_{k + 1} = \vec{f}(\vec{x}_k, \vec{u}_k)$, $\vec{g}(\vec{x}_k, \vec{u}_k) \leq \vec{0}$
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and $\vec{x}_N \in \cX_f$ and $\vec{x}_0 = \vec{x}(0)$
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\bi{Finite-Horizon Lin-Quad Control}
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Quad. Cost:
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\[
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J_{0 \rightarrow N}(\vec{x}_0, \vec{u}_0, \ldots) =
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\vec{x}_N^\top \vec{P} \vec{x}_N + \sum_{k = 0}^{N - 1} \vec{x}_k^\top \mat{Q} \vec{x}_k + \vec{u}_k^\top \mat{R} \vec{u}_k
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\]
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without constraints, \bi{State Feedback Law} $\vec{u}^*_0 = -\mat{K}\vec{x}(0)$.
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With constraints, minimize as above.
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