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    <h1><a href="index.html" style="text-decoration:none;">Underactuated Robotics</a></h1>
    <p data-type="subtitle">Algorithms for Walking, Running, Swimming, Flying, and Manipulation</p>
    <p style="font-size: 18px;"><a href="http://people.csail.mit.edu/russt/">Russ Tedrake</a></p>
    <p style="font-size: 14px; text-align: right;">
      &copy; Russ Tedrake, 2020<br/>
      <a href="tocite.html">How to cite these notes</a> &nbsp; | &nbsp;
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<p><b>Note:</b> These are working notes used for <a
href="http://underactuated.csail.mit.edu/Spring2020/">a course being taught
at MIT</a>. They will be updated throughout the Spring 2020 semester.  <a
href="https://www.youtube.com/channel/UChfUOAhz7ynELF-s_1LPpWg">Lecture  videos are available on YouTube</a>.</p>

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<!-- EVERYTHING ABOVE THIS LINE IS OVERWRITTEN BY THE INSTALL SCRIPT -->
<chapter style="counter-reset: chapter 14"><h1>
Output Feedback (aka Pixels-to-Torques)</h1>

  <p>In this chapter we will start considering systems of the form:
  \begin{gather*} \bx[n+1] = {\bf f}(\bx[n], \bu[n], \bw[n], n) \\ \by[n] = {\bf
  g}(\bx[n], \bu[n], \bv[n], n).\end{gather*}  In other words, we'll finally
  start addressing the fact that we have to make decisions based on sensor
  measurements -- most of our discussions until now have tacitly assumed that we
  have access to the true state of the system for use in our feedback
  controllers (and that's already been a hard problem).
  </p>

  <p>In some cases, we will see that the assumption of "full-state feedback" is
  not so bad -- we do have good tools for state estimation from raw sensor data.
  But even our best state estimation algorithms do add some dynamics to the
  system in order to filter out noisy measurements; if the time constants of
  these filters is near the time constant of our dynamics, then it becomes
  important that we include the dynamics of the estimator in our analysis of the
  closed-loop system.</p>

  <p>In other cases, it's entirely too optimistic to design a controller
  assuming that we will have an estimate of the full state of the system.  Some
  state variables might be completely unobservable, others might require
  specific "information-gathering" actions on the part of the controller.</p>

  <p>For me, the problem of robot manipulation is the application domain where
  more direct approaches to output feedback become critically important. Imagine
  you are trying to design a controller for a robot that needs to button the
  buttons on your dress shirt.  If step one is to estimate the state of the
  shirt (how many degrees of freedom does my shirt have?), then it feels like
  we're not going to be successful.  Or if you want to program a robot to make a
  salad -- what's the state of the salad?  Do I really need to know the
  positions and velocities of every piece of lettuce in order to be
  successful?</p>

  <section><h1>The Classical Perspective</h1>

    <p>To some extent, this idea of calling out "output feedback" as a special,
    advanced topic is a new problem.  Before state space and optimization-based
    approaches to control ushered in "modern control", we had "classical
    control".  Classical control focused predominantly (though not exclusively)
    on linear time-invariant (LTI) systems, and made very heavy use of
    frequency-domain analysis (e.g. via the Fourier Transform/Laplace
    Transform).  There are many excellent books on the subject;
    <elib>Hespanha09+Astrom10</elib> are nice examples of modern treatments that
    start with state-space representations but also treat the frequency-domain
    perspective.</p>

    <p>What's important for us to acknowledge here is that in classical control,
    basically everything was built around the idea of output feedback.  The
    fundamental concept is the transfer function of a system, which is a
    input-to-output map (in frequency domain) that can completely characterize
    an LTI system.  Core concepts like pole placement and loop shaping were
    fundamentally addressing the challenge of output feedback that we are
    discussing here.  Sometimes I feel that, despite all of the things we've
    gain with modern, optimization-based control, I worry that we've lost
    something in terms of considering rich characterizations of closed-loop
    performance (rise time, dwell time, overshoot, ...) and perhaps even in
    practical robustness of our systems to unmodeled errors.</p>

    <todo>Add a few examples here that capture it.</todo>

  </section>

  <section><h1>Observer-based Feedback</h1>

    <subsection><h1>Luenberger Observer</h1></subsection>

    <subsection><h1>Linear Quadratic Regulator w/ Gaussian Noise
    (LQG)</h1></subsection>

    <subsection><h1>Partially-observable Markov Decision Processes</h1>

    </subsection>

    <todo>Defer the rest of the discussion to the state estimation
    chapter</todo>

  </section>

  <section><h1>Static Output Feedback</h1>

    <subsection><h1>For Linear Systems</h1></subsection>

    <todo>Bilinear alternations with SOS, Policy search with SGD</todo>

  </section>

  <section><h1>Disturbance-based feedback</h1>

    <todo>State-space models.  ARX Models.</todo>
    <subsection><h1>System-Level Synthesis</h1></subsection>

  </section>

  <todo>Task-relevant variables</todo>

<!--  <section><h1>Feedback from Pixels</h1></section> -->

</chapter>
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