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<!DOCTYPE html>
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<head>
<title>Underactuated Robotics:
Output Feedback (aka Pixels-to-Torques)</title>
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Output Feedback (aka Pixels-to-Torques)" content="text/html; charset=utf-8;" />
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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;">
© Russ Tedrake, 2021<br/>
Last modified <span id="last_modified"></span>.</br>
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<a href="misc.html">How to cite these notes, use annotations, and give feedback.</a><br/>
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<p><b>Note:</b> These are working notes used for <a
href="http://underactuated.csail.mit.edu/Spring2021/">a course being taught
at MIT</a>. They will be updated throughout the Spring 2021 semester. <a
href="https://www.youtube.com/channel/UChfUOAhz7ynELF-s_1LPpWg">Lecture videos are available on YouTube</a>.</p>
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<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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<div id="references"><section><h1>References</h1>
<ol>
<li id=Hespanha09>
<span class="author">Joao P. Hespanha</span>,
<span class="title">"Linear Systems Theory"</span>,Princeton Press
, <span class="year">2009</span>.
</li><br>
<li id=Astrom10>
<span class="author">{\AA}str{\"o}m, Karl Johan and Murray, Richard M</span>,
<span class="title">"Feedback systems: an introduction for scientists and engineers"</span>,Princeton university press
, <span class="year">2010</span>.
</li><br>
</ol>
</section><p/>
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