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Examples
========

.. include:: /_generated/nav/control_systems.rst

This gallery walks through every public feature of
``engineeringkit.control_systems``, one breakthrough per script: feedback
and PID control, algebraic and Lyapunov stability tests, frequency-response
design, transient specifications, the root locus, sampled-data control,
state-space design and optimal control and estimation.

:doc:`Units, assumptions and numerical validation </guides/control_systems>`
gives the validity limits. Each script is self-contained and can be run
directly with ``python examples/control_systems/<section>/<script>.py``.

Sections
--------

- **feedback** -- Watt's governor, Minorsky's PID, Black's feedback amplifier and Ziegler-Nichols tuning.
- **stability** -- Maxwell's characteristic roots, Routh's array, Lyapunov's direct method and Hurwitz's determinants.
- **frequency** -- Nyquist's criterion and Bode's stability margins.
- **time_response** -- Hazen's second-order servomechanism.
- **root_locus** -- Evans's root locus.
- **sampled** -- Ragazzini and Zadeh's zero-order-hold discretisation.
- **state_space** -- Kalman's controllability, Luenberger's observer and Wonham's pole placement.
- **optimal** -- Bellman's dynamic programming, Kalman's LQR and the Kalman-Bucy filter.


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Feedback
--------

Governors, feedback amplifiers and PID control.


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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>
    </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>
    </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>
    </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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Frequency response
------------------

The Nyquist criterion, Bode plots and stability margins.


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    <div class="sphx-glr-thumbcontainer" tooltip="Long-distance telephone repeaters with feedback sometimes &quot;sang&quot;. Harry Nyquist (1932) showed how to predict this from the measured open-loop frequency response alone. Trace L(j\omega) for all frequencies (~engineeringkit.control_systems.nyquist): the closed loop has Z = N + P unstable poles, where N counts clockwise encirclements of -1 (~engineeringkit.control_systems.nyquist_encirclements) and P the unstable open-loop poles.">

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  .. image:: /api/gallery/control_systems/frequency/images/thumb/sphx_glr_plot_01_nyquist_stability_criterion_thumb.png
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  :doc:`/api/gallery/control_systems/frequency/plot_01_nyquist_stability_criterion`

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      <div class="sphx-glr-thumbnail-title">Nyquist's stability criterion</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Hendrik Bode (1940, 1945) turned Nyquist&#x27;s criterion into design numbers read off a logarithmic plot of gain and phase (~engineeringkit.control_systems.bode). The gain margin is how much the loop gain can grow before the phase-crossover point reaches -1; the phase margin is how much extra lag the loop tolerates at the gain crossover (~engineeringkit.control_systems.stability_margins). For L = 2/(s(s+1)(s+2)) the phase crosses -180^\circ at \omega = \sqrt2, where |L| = 1/3, so the gain margin is 3.">

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  .. image:: /api/gallery/control_systems/frequency/images/thumb/sphx_glr_plot_02_bode_stability_margins_thumb.png
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  :doc:`/api/gallery/control_systems/frequency/plot_02_bode_stability_margins`

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      <div class="sphx-glr-thumbnail-title">Bode's gain and phase margins</div>
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Optimal control and estimation
------------------------------

Dynamic programming, the linear-quadratic regulator and the Kalman-Bucy filter.


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Root locus
----------

Closed-loop poles as the loop gain varies.


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    <div class="sphx-glr-thumbcontainer" tooltip="Walter Evans at North American Aviation (1948) showed how to sketch the paths of the closed-loop poles as a single gain K rises from zero, without solving a polynomial for each gain (~engineeringkit.control_systems.root_locus). For K/(s(s+2)(s+4)) the branches leave the poles at 0, -2 and -4. Two of them meet and break away from the real axis where \tfrac{d}{ds}[s(s+2)(s+4)] = 0, at s = -2 + 2/\sqrt3. They cross the imaginary axis at \pm j\sqrt8 when K = 48.">

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  .. image:: /api/gallery/control_systems/root_locus/images/thumb/sphx_glr_plot_01_evans_root_locus_thumb.png
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  :doc:`/api/gallery/control_systems/root_locus/plot_01_evans_root_locus`

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      <div class="sphx-glr-thumbnail-title">Evans's root locus</div>
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Sampled-data control
--------------------

Computer control with a zero-order hold.


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    <div class="sphx-glr-thumbcontainer" tooltip="When a digital computer closes the loop it reads the plant only at sampling instants and holds each command constant until the next. John Ragazzini and Lotfi Zadeh (1952) developed the z-transform analysis of such loops. For a zero-order hold the continuous model is replaced exactly by x_{k+1} = e^{AT}x_k + \int_0^T e^{A\tau}d\tau\,B\,u_k (~engineeringkit.control_systems.discretize_zoh). The naive Euler step x_{k+1} = (I + AT)x_k + BTu_k is only an approximation.">

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  .. image:: /api/gallery/control_systems/sampled/images/thumb/sphx_glr_plot_01_ragazzini_zadeh_zero_order_hold_thumb.png
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  :doc:`/api/gallery/control_systems/sampled/plot_01_ragazzini_zadeh_zero_order_hold`

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      <div class="sphx-glr-thumbnail-title">Ragazzini and Zadeh's sampled-data systems</div>
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Stability
---------

Characteristic roots, the Routh and Hurwitz tests and Lyapunov functions.


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    <div class="sphx-glr-thumbcontainer" tooltip="Governors fitted with integral action to remove droop sometimes &quot;hunted&quot;, oscillating with growing amplitude. James Clerk Maxwell (1868) linearised the engine-governor equations and showed that motion is stable exactly when every root of the characteristic polynomial has a negative real part (~engineeringkit.control_systems.poles). For the cubic s^3 + a_1 s^2 + a_2 s + a_3 he found the condition a_1 a_2 &gt; a_3: too much integral gain destabilises the governor.">

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  .. image:: /api/gallery/control_systems/stability/images/thumb/sphx_glr_plot_01_maxwell_on_governors_thumb.png
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  :doc:`/api/gallery/control_systems/stability/plot_01_maxwell_on_governors`

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      <div class="sphx-glr-thumbnail-title">Maxwell's "On Governors": stability from characteristic roots</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Maxwell could not state a test for polynomials above fourth order. Edward Routh&#x27;s Adams Prize essay (1877) solved the problem with a table built from the coefficients alone: the number of sign changes down its first column (~engineeringkit.control_systems.routh_first_column) equals the number of roots in the right half-plane. For the closed loop of K/(s(s+1)(s+2)), s^3 + 3s^2 + 2s + K, the table shows the loop is stable only for 0 &lt; K &lt; 6.">

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  .. image:: /api/gallery/control_systems/stability/images/thumb/sphx_glr_plot_02_routh_stability_criterion_thumb.png
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  :doc:`/api/gallery/control_systems/stability/plot_02_routh_stability_criterion`

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      <div class="sphx-glr-thumbnail-title">Routh's stability criterion</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Aleksandr Lyapunov&#x27;s doctoral thesis (1892) proved stability without solving the equations of motion. Find an energy-like function V(x) &gt; 0 that decreases along every trajectory, and the equilibrium is stable. For linear dynamics \dot x = Ax, the quadratic V = x^\mathsf{T} P x works exactly when the Lyapunov equation A^\mathsf{T}P + PA = -Q has a positive definite solution (~engineeringkit.control_systems.lyapunov_equation, lyapunov_stable).">

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  :doc:`/api/gallery/control_systems/stability/plot_03_lyapunov_direct_method`

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      <div class="sphx-glr-thumbnail-title">Lyapunov's direct method</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Aurel Stodola, designing turbine regulators in Zürich, asked Adolf Hurwitz for a practical stability condition. Hurwitz (1895), unaware of Routh&#x27;s work, gave it as determinants: build the matrix H_{ij} = a_{2j-i} from the coefficients, and every root lies in the left half-plane exactly when all leading minors are positive (~engineeringkit.control_systems.hurwitz_determinants). At the boundary \Delta_{n-1} passes through zero as a pair of roots crosses the imaginary axis.">

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State space
-----------

Controllability, pole placement and observers.


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Time response
-------------

Transient specifications of second-order servomechanisms.


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    <div class="sphx-glr-thumbcontainer" tooltip="Harold Hazen at MIT (1934) coined servomechanism and analysed position-following systems as second-order dynamics, \omega_n^2/(s^2 + 2\zeta\omega_n s + \omega_n^2). The damping ratio \zeta alone fixes the overshoot, and \zeta with the natural frequency \omega_n sets peak and settling times (~engineeringkit.control_systems.second_order_metrics). The simulated step and impulse responses (~engineeringkit.control_systems.step_response, impulse_response) confirm the formulas.">

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  :doc:`/api/gallery/control_systems/time_response/plot_01_hazen_servomechanism`

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      <div class="sphx-glr-thumbnail-title">Hazen's theory of servomechanisms</div>
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.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/control_systems/feedback/index.rst
   /api/gallery/control_systems/frequency/index.rst
   /api/gallery/control_systems/optimal/index.rst
   /api/gallery/control_systems/root_locus/index.rst
   /api/gallery/control_systems/sampled/index.rst
   /api/gallery/control_systems/stability/index.rst
   /api/gallery/control_systems/state_space/index.rst
   /api/gallery/control_systems/time_response/index.rst


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    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download all examples in Jupyter notebooks: control_systems_jupyter.zip </api/gallery/control_systems/control_systems_jupyter.zip>`


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    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_