ThotDjehuty
dd51156174
Each of the eight v2.0 companion notebooks (10_bsde through 17_generative_calibration) now follows the mandatory pedagogical sandwich structure: PRE markdown : theorem / model / pivot equation / what the cell verifies CODE cell : labelled prints + at least one matplotlib figure POST markdown: expected result, graph reading, conclusion Each notebook carries at least one concrete real-world example (heat plate, inverted pendulum, opinion polarization, collective decision, OU drift under Cauchy noise, mixture vs gaussian MMD, etc.) Generator script: scripts/enrich_v2_notebooks.py Doc plots refreshed via scripts/inject_doc_plots.py.
101 lines
2.9 KiB
ReStructuredText
101 lines
2.9 KiB
ReStructuredText
Quadratic-impact control — closed-form Riccati
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==============================================
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Closed-form Riccati feedback for a controlled 1-D SDE with quadratic running cost (`quadratic_impact_control_py`).
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.. note:: Companion executed notebook: `13_quadratic_impact.ipynb <../../examples/notebooks/13_quadratic_impact.ipynb>`_
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13 — Quadratic-impact controlled SDE
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====================================
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.. code-block:: python
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import numpy as np
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import matplotlib.pyplot as plt
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from optimizr import _core as opt
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plt.rcParams['figure.figsize'] = (7, 4)
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plt.rcParams['figure.dpi'] = 110
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Riccati fixed-point check
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-------------------------
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$h'(t) = h(t)^2/γ - φ$ with $h(T) = A$. When $γ = φ = A = 1$ the right-hand side is $h^2 - 1 = 0$ at $h = 1$, so `h ≡ 1`.
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.. code-block:: python
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res = opt.quadratic_impact_control_py(
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gamma=1.0, phi=1.0, a_terminal=1.0,
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t_horizon=0.5, n_steps=500,
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)
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tg = np.array(res['time_grid'])
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h = np.array(res['h']); k = np.array(res['feedback_gain'])
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print('h drift from 1:', float(np.max(np.abs(h - 1.0))))
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.. code-block:: python
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fig, ax = plt.subplots()
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ax.plot(tg, h, label='h(t)')
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ax.plot(tg, k, '--', label='k(t) = h(t)/γ')
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ax.axhline(1.0, color='k', alpha=0.3, ls=':', label='fixed point')
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ax.set_xlabel('t'); ax.legend(); ax.grid(alpha=0.3)
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ax.set_title('Riccati fixed point γ=φ=A=1')
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fig.tight_layout(); plt.show()
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.. AUTO-PLOT-BEGIN
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.. image:: ../_static/auto/algorithms__quadratic_impact_control/block_03_fig_01.png
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:align: center
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:width: 80%
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.. AUTO-PLOT-END
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.. image:: ../_static/v2/quadratic_impact_control/plot_01.png
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:align: center
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:width: 80%
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Sensitivity to the terminal weight
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----------------------------------
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Vary $A$, fix $γ = 1$, $φ = 0.25$, $T = 1$.
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.. code-block:: python
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fig, ax = plt.subplots()
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for A in [0.0, 0.25, 0.5, 1.0, 2.0, 5.0]:
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r = opt.quadratic_impact_control_py(1.0, 0.25, A, 1.0, 1000)
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ax.plot(r['time_grid'], r['h'], label=f'A = {A:g}')
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ax.set_xlabel('t'); ax.set_ylabel('h(t)'); ax.legend(); ax.grid(alpha=0.3)
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ax.set_title('Riccati sensitivity to terminal weight')
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fig.tight_layout(); plt.show()
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.. AUTO-PLOT-BEGIN
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.. image:: ../_static/auto/algorithms__quadratic_impact_control/block_04_fig_01.png
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:align: center
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:width: 80%
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.. AUTO-PLOT-END
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.. image:: ../_static/v2/quadratic_impact_control/plot_02.png
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:align: center
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:width: 80%
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**Verified:** `h ≡ 1` with `max|h - 1| < 1e-9` at the fixed point.
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API
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---
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.. code-block:: rust
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pub fn solve_quadratic_impact_control(cfg: &QuadraticImpactConfig) -> Result<QuadraticImpactResult>;
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pub struct QuadraticImpactConfig { pub gamma: f64, pub phi: f64, pub a_terminal: f64, pub t_horizon: f64, pub n_steps: usize }
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pub struct QuadraticImpactResult { pub time_grid: Array1<f64>, pub h: Array1<f64>, pub feedback_gain: Array1<f64> }
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