.. _sphx_glr_api_gallery_control_systems_feedback:

Feedback
--------

Governors, feedback amplifiers and PID control.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbcontainer" tooltip="James Watt&#x27;s flyball governor (1788) closed the steam valve as the engine sped up: the throttle moved in proportion to the speed error. Model the engine as J\dot\omega = \tau - b\omega - \tau_{load} and the governor as \tau = K_p(\omega_{ref} - \omega), a proportional PID. A proportional governor holds speed far better than an open throttle, but it always leaves a steady droop, \Delta\omega = \tau_{load}/(b + K_p), which shrinks with gain but never vanishes.">

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  .. image:: /api/gallery/control_systems/feedback/images/thumb/sphx_glr_plot_01_watt_centrifugal_governor_thumb.png
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  :doc:`/api/gallery/control_systems/feedback/plot_01_watt_centrifugal_governor`

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      <div class="sphx-glr-thumbnail-title">Watt's centrifugal governor and proportional droop</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Nicolas Minorsky (1922) watched helmsmen on the battleship New Mexico and noticed they reacted to the heading error, to how fast it was changing, and to any error that persisted. His three-term controller, \delta = K_p e + K_i \int e\,dt + K_d \dot e, is the PID. Using Nomoto&#x27;s ship model \psi/\delta = K/(s(Ts + 1)) and a steady wind moment, proportional plus derivative action leaves a constant heading offset d/K_p; adding integral action removes it.">

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  .. image:: /api/gallery/control_systems/feedback/images/thumb/sphx_glr_plot_02_minorsky_pid_ship_steering_thumb.png
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  :doc:`/api/gallery/control_systems/feedback/plot_02_minorsky_pid_ship_steering`

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      <div class="sphx-glr-thumbnail-title">Minorsky's PID controller for ship steering</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Harold Black (1927), commuting on the Lackawanna ferry, sketched an amplifier that fed a fraction \beta of its output back out of phase. With a huge open-loop gain A, the closed-loop gain A/(1 + A\beta) (~engineeringkit.control_systems.feedback) is almost exactly 1/\beta, set by a passive divider rather than by drifting vacuum tubes. Distortion and gain variations shrink by the factor 1 + A\beta, and the bandwidth grows by the same factor.">

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  .. image:: /api/gallery/control_systems/feedback/images/thumb/sphx_glr_plot_03_black_negative_feedback_amplifier_thumb.png
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  :doc:`/api/gallery/control_systems/feedback/plot_03_black_negative_feedback_amplifier`

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      <div class="sphx-glr-thumbnail-title">Black's negative-feedback amplifier</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="John Ziegler and Nathaniel Nichols of Taylor Instruments (1942) gave plant engineers a recipe: turn up a proportional gain until the loop oscillates steadily, note that ultimate gain K_u and period T_u, and read P, PI or PID settings from a table (~engineeringkit.control_systems.ziegler_nichols). The ultimate point is exactly the gain margin and phase-crossover frequency of the loop (~engineeringkit.control_systems.stability_margins). For 1/(s+1)^3, K_u = 8 and \omega_u = \sqrt3 rad/s.">

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  .. image:: /api/gallery/control_systems/feedback/images/thumb/sphx_glr_plot_04_ziegler_nichols_tuning_thumb.png
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  :doc:`/api/gallery/control_systems/feedback/plot_04_ziegler_nichols_tuning`

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      <div class="sphx-glr-thumbnail-title">Ziegler and Nichols tune a PID from the edge of instability</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/control_systems/feedback/plot_01_watt_centrifugal_governor
   /api/gallery/control_systems/feedback/plot_02_minorsky_pid_ship_steering
   /api/gallery/control_systems/feedback/plot_03_black_negative_feedback_amplifier
   /api/gallery/control_systems/feedback/plot_04_ziegler_nichols_tuning