forked from mirrors/org-mode
31 lines
1.1 KiB
Org Mode
31 lines
1.1 KiB
Org Mode
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#+latex_header: \usepackage{amsmath}
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#+latex_header: \usepackage{amssymb}
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* Inline fragments and LaTeX environments
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:PROPERTIES:
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:ID: 0b3807b3-69af-40cb-a27a-b380d54879cc
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:END:
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The LQR problem for a time-periodic system of the form
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\begin{align}
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\dot{x} = A(t) x + B(t) u, \quad t \in [0, \infty), \quad x(0) = x_i \label{eq:time-varying-system}\\
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A(t+T) = A(t),\ B(t + T) = B(t) \nonumber
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\end{align}
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is as follows. With a quadratic form defined on \( (x,u) \) pairs
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\begin{align}
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\label{eq:quadratic-form}
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\mathbf{q}(x, u) := \lim_{t_f \to \infty} \int_{0}^{t_f} \begin{bmatrix} x \\ u \end{bmatrix}^{\star} \begin{bmatrix}
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Q & 0 \\
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0 & r
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\end{bmatrix} \begin{bmatrix} x \\ u \end{bmatrix} =: \lim_{t_f \to \infty} \int_{0}^{t_f} q(x,u) dt
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\end{align}
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with \( q \ge 0 \) and \( r \ge 0 \), find the infimum of the quadratic form \( \mathbf{q} \) subject to the dynamics:
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\[
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\inf_{x,u} \mathbf{q}(x,u).
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\]
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\begin{align}
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\label{eq:lqr-inf-via-duality}
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\inf_{x, u} \mathbf{q}(x, u) = x_i^{\star} \bar{\lambda}(0) x_i,
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\end{align}
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where \( \bar{\lambda} \) is the maximal solution of the differential linear matrix inequality over \( [0, t] \).
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