61 lines
1.4 KiB
ReStructuredText
61 lines
1.4 KiB
ReStructuredText
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Matrix Riccati Solver
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=====================
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The module :code:`optimal_control::matrix_riccati` integrates backward in time
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the matrix Riccati differential equation
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.. math::
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\frac{dA(t)}{dt} = -2\,A(t)\,M\,A(t) + Q,
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\qquad A(T) = A_T,
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together with the affine and constant components
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.. math::
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\frac{dB(t)}{dt} = -2\,A(t)\,M\,B(t),
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\qquad B(T) = B_T,
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.. math::
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\frac{dC(t)}{dt} = -B(t)^\top\,M\,B(t),
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\qquad C(T) = C_T.
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Discretisation
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--------------
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The grid :math:`\{t_n = T - n\,\Delta t\}_{n=0}^{N}` with
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:math:`\Delta t = T / N` is traversed backward and a classical RK4 step is
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applied to the joint vector field :math:`(A, B, C)`. Each macro step is
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optionally subdivided into :math:`s` sub-steps for stability on stiff
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problems.
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Validation
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----------
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In the scalar case :math:`A, M, Q \in \mathbb{R}` with :math:`A(T) = 0`,
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.. math::
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A(t) \;=\; -\sqrt{\frac{Q}{2M}}\;\tanh\!\Big(\sqrt{2QM}\,(T - t)\Big),
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a closed form used by the unit test :code:`scalar_riccati_matches_analytic`
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to certify :math:`L^\infty` convergence below :math:`10^{-5}` on
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:math:`[0, T]`.
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API
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---
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.. code-block:: rust
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pub fn solve_matrix_riccati(
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m_matrix: ArrayView2<f64>,
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q: ArrayView2<f64>,
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n: ArrayView2<f64>,
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a_terminal: ArrayView2<f64>,
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b_terminal: ArrayView1<f64>,
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c_terminal: f64,
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t_horizon: f64,
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config: RiccatiConfig,
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) -> Result<RiccatiResult>;
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