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15 lines
706 B
TeX
15 lines
706 B
TeX
\subsection{Linear Quadratic Regulator (LQR)}
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For lin. cont.-time dyn. $\dot{\vec{x}}(t) = \vec{F}_c \vec{x}(t) + \vec{G}_c \vec{u}(t)$.
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We try to minimise cost functional (for $\vec{u}(t) = -\mat{K}\vec{x}(t)$)
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\[
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J = \int_{t}^{\8} \vec{x}(\tau)^\top \vec{Q}\vec{x}(\tau) + \vec{u}(\tau)^\top \vec{R} \vec{u}(\tau) \dx \tau
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\]
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Solution: $\mat{K} = \mat{R}^{-1} (\mat{G}_c^\top \mat{P})$, with $\mat{P}$ found from
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\[
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\mat{F}_c^\top \mat{P} + \mat{PF}_c - (\mat{PG}_c) \mat{R}^{-1} + \mat{Q} = \mat{0}
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\]
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Finding $\mat{K}$ is expensive, but \textit{offline}, at runtime only $\vec{u}(t)$.
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\bi{Non-Lin}: Approx, $\delta\dot{\vec{x}}(t) = \vec{F}_c \delta \vec{x}(t) + \mat{G}_c \delta \vec{u}(t)$
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