forked from RussTedrake/underactuated
-
Notifications
You must be signed in to change notification settings - Fork 0
/
dp.html
945 lines (754 loc) · 53.9 KB
/
dp.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
<!DOCTYPE html>
<html>
<head>
<title>Underactuated Robotics: Dynamic Programming</title>
<meta name="Underactuated Robotics: Dynamic Programming" content="text/html; charset=utf-8;" />
<link rel="canonical" href="http://underactuated.mit.edu/dp.html" />
<script src="https://hypothes.is/embed.js" async></script>
<script type="text/javascript" src="htmlbook/book.js"></script>
<script src="htmlbook/mathjax-config.js" defer></script>
<script type="text/javascript" id="MathJax-script" defer
src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js">
</script>
<script>window.MathJax || document.write('<script type="text/javascript" src="htmlbook/MathJax/es5/tex-chtml.js" defer><\/script>')</script>
<link rel="stylesheet" href="htmlbook/highlight/styles/default.css">
<script src="htmlbook/highlight/highlight.pack.js"></script> <!-- http://highlightjs.readthedocs.io/en/latest/css-classes-reference.html#language-names-and-aliases -->
<script>hljs.initHighlightingOnLoad();</script>
<link rel="stylesheet" type="text/css" href="htmlbook/book.css" />
</head>
<body onload="loadChapter('underactuated');">
<div data-type="titlepage">
<header>
<h1><a href="underactuated.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, 2020<br/>
<a href="tocite.html">How to cite these notes</a><br/>
</p>
</header>
</div>
<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>
<table style="width:100%;"><tr style="width:100%">
<td style="width:33%;text-align:left;"><a class="previous_chapter" href=stochastic.html>Previous Chapter</a></td>
<td style="width:33%;text-align:center;"><a href=underactuated.html>Table of contents</a></td>
<td style="width:33%;text-align:right;"><a class="next_chapter" href=lqr.html>Next Chapter</a></td>
</tr></table>
<!-- EVERYTHING ABOVE THIS LINE IS OVERWRITTEN BY THE INSTALL SCRIPT -->
<chapter style="counter-reset: chapter 6"><h1>Dynamic Programming</h1>
<p>In chapter 2, we spent some time thinking about the phase portrait of the
simple pendulum, and concluded with a challenge: can we design a nonlinear
controller to <em>reshape</em> the phase portrait, with a very modest amount
of actuation, so that the upright fixed point becomes globally stable? With
unbounded torque, feedback-cancellation solutions (e.g., invert gravity) can
work well, but can also require an unnecessarily large amount of control
effort. The energy-based swing-up control solutions presented for the acrobot
and cart-pole systems are considerably more appealing, but required some
cleverness and might not scale to more complicated systems. Here we
investigate another approach to the problem, using computational optimal
control to synthesize a feedback controller directly.</p>
<section><h1>Formulating control design as an optimization</h1>
<p>In this chapter, we will introduce optimal control - a control design
process using optimization. This approach is powerful for a number of
reasons. First and foremost, it is very general - allowing us to specify
the goal of control equally well for fully- or under-actuated, linear or
nonlinear, deterministic or stochastic, and continuous or discrete
systems. Second, it permits concise descriptions of potentially very
complex desired behaviours, specifying the goal of control as an scalar
objective (plus a list of constraints). Finally, and most importantly,
optimal control is very amenable to numerical solutions.
<elib>Bertsekas00a</elib> is a fantastic reference on this material for
those who want a somewhat rigorous treatment; <elib>Sutton18</elib> is an
excellent (free) reference for those who want something more
approachable.</p>
<p>The fundamental idea in optimal control is to formulate the goal of
control as the <em>long-term</em> optimization of a scalar cost function.
Let's introduce the basic concepts by considering a system that is even
simpler than the simple pendulum.</p>
<example><h1>Optimal Control Formulations for the Double Integrator</h1>
<p>Consider the double integrator system $$\ddot{q} = u, \quad |u| \le
1.$$ If you would like a mechanical analog of the system (I always do),
then you can think about this as a unit mass brick moving along the x-axis
on a frictionless surface, with a control input which provides a
horizontal force, $u$. The task is to design a control system, $u =
\pi(\bx,t)$, $\bx=[q,\dot{q}]^T$ to regulate this brick to $\bx =
[0,0]^T$.</p>
<figure>
<img width="70%" src="figures/double_integrator_brick.svg"/>
<figcaption>The double integrator as a unit-mass brick on a frictionless
surface</figcaption>
</figure>
<p>In order to formulate this control design problem using optimal
control, we must define a scalar objective which scores the long-term
performance of running each candidate control policy, $\pi(\bx,t)$, from
each initial condition, $(\bx_0,t_0)$, and a list of constraints that must
be satisfied. For the task of driving the double integrator to the origin,
one could imagine a number of optimal control formulations which would
accomplish the task, e.g.: <ul> <li> Minimum time: $\min_\pi t_f,$
subject to $\bx(t_0) = \bx_0,$ $\bx(t_f) = {\bf 0}.$ </li> <li> Quadratic
cost: $\min_\pi \int_{t_0}^{\infty} \left[ \bx^T(t) {\bf Q} \bx(t)
\right] dt,$ ${\bf Q}\succ0$.</li> </ul> where the first is a constrained
optimization formulation which optimizes time, and the second accrues a
penalty at every instance according to the distance that the state is away
from the origin (in a metric space defined by the matrix ${\bf Q}$), and
therefore encourages trajectories that go more directly towards the goal,
possibly at the expense of requiring a longer time to reach the goal (in
fact it will result in an exponential approach to the goal, where as the
minimum-time formulation will arrive at the goal in finite time). Note
that both optimization problems only have well-defined solutions if it is
possible for the system to actually reach the origin, otherwise the
minimum-time problem cannot satisfy the terminal constraint, and the
integral in the quadratic cost would not converge to a finite value as
time approaches infinity (fortunately the double integrator system is
controllable, and therefore can be driven to the goal in finite time).</p>
<p> Note that the input limits, $|u|\le1$ are also required to make this
problem well-posed; otherwise both optimizations would result in the
optimal policy using infinite control input to approach the goal
infinitely fast. Besides input limits, another common approach to limiting
the control effort is to add an additional quadratic cost on the input (or
"effort"), e.g. $\int \left[ \bu^T(t) {\bf R} \bu(t) \right] dt,$ ${\bf
R}\succ0$. This could be added to either formulation above. We will
examine many of these formulations in some details in the examples worked
out at the end of this chapter. </p>
</example>
<p>Optimal control has a long history in robotics. For instance, there has
been a great deal of work on the minimum-time problem for pick-and-place
robotic manipulators, and the linear quadratic regulator (LQR) and linear
quadratic regulator with Gaussian noise (LQG) have become essential tools
for any practicing controls engineer. With increasingly powerful computers
and algorithms, the popularity of numerical optimal control has grown at an
incredible pace over the last few years.</p>
<example id="minimum_time_double_integrator"><h1>The minimum time problem for the double
integrator</h1>
<p>For more intuition, let's do an informal derivation of the solution to
the minimum time problem for the double integrator with input constraints:
\begin{align*}
\minimize_{\pi} \quad & t_f\\
\subjto \quad & \bx(t_0) = \bx_0, \\
& \bx(t_f) = {\bf 0}, \\
& \ddot{q}(t) = u(t), \\
& |u| \le 1.
\end{align*}
What behavior would you expect an optimal controller exhibit?</p>
<p> Your intuition might tell you that the best thing that the brick can
do, to reach the goal in minimum time with limited control input, is to
accelerate maximally towards the goal until reaching a critical point,
then hitting the brakes in order to come to a stop exactly at the goal.
This would be called a <em>bang-bang</em> control policy; these are often
optimal for systems with bounded input, and it is in fact optimal for the
double integrator, although we will not prove it until we have developed
more tools. <!-- leave the proof to the pontryagin notes --></p>
<p> Let's work out the details of this bang-bang policy. First, we can
figure out the states from which, when the brakes are fully applied, the
system comes to rest precisely at the origin. Let's start with the case
where $q(0) < 0$, and $\dot{q}(0)>0$, and "hitting the brakes" implies
that $u=-1$ . Integrating the equations, we have \begin{gather*}
\ddot{q}(t) = u = -1 \\\dot{q}(t) = \dot{q}(0) - t \\ q(t) = q(0) +
\dot{q}(0) t - \frac{1}{2} t^2. \end{gather*} Substituting $t =
\dot{q}(0) - \dot{q}$ into the solution give $\dot{q}$ reveals that the
system orbits are parabolic arcs: \[ q = -\frac{1}{2} \dot{q}^2 + c_{-},
\] with $c_{-} = q(0) + \frac{1}{2}\dot{q}^2(0)$.</p>
<figure>
<img width="80%" src="figures/double_integrator_orbits.svg"/>
<figcaption>Two solutions for the system with $u=-1$</figcaption>
</figure>
<!-- t = qdot - qdot0,
q = q0 + qdot0(qdot-qdot0) + 1/2(qdot-qdot0)^2
= q0 + qdot0*qdot - qdot0^2 + 1/2qdot^2 - qdot*qdot0 + 1/2qdot0^2
= 1/2qdot^2 + (q0 - 1/2 qdot0^2)
-->
<p>Similarly, the solutions for $u=1$ are $q = \frac{1}{2} \dot{q}^2 +
c_{+}$, with $c_{+}=q(0)-\frac{1}{2}\dot{q}^2(0)$.</p>
<p> Perhaps the most important of these orbits are the ones that pass
directly through the origin (e.g., $c_{-}=0$). Following our initial
logic, if the system is going slower than this $\dot{q}$ for any $q$, then
the optimal thing to do is to slam on the accelerator
($u=-\text{sgn}(q)$). If it's going faster than the $\dot{q}$ that we've
solved for, then still the best thing to do is to brake; but inevitably
the system will overshoot the origin and have to come back. We can
summarize this policy with: \[ u = \begin{cases} +1 & \text{if } (\dot{q}
< 0 \text{ and } q \le \frac{1}{2} \dot{q}^2) \text{ or } (\dot{q}\ge 0
\text{ and } q < -\frac{1}{2} \dot{q}^2) \\ 0 & \text{if } q=0 \text{ and
} \dot{q}=0 \\ -1 & \text{otherwise} \end{cases} \]</p> <!--This policy is
cartooned in Figure~\ref{fig:mintime_double_int}. %Trajectories of the
system %executing this policy are also included - the fundamental
%characteristic is that the system is accelerated as quickly as %possible
toward the switching surface, then rides the switching %surface in to the
origin. -->
<figure>
<img width="80%" src="figures/double_integrator_mintime_policy.svg"/>
<figcaption>Candidate optimal (bang-bang) policy for the minimum-time
double integrator problem.</figcaption>
</figure>
<p>and illustrate some of the optimal solution trajectories:</p>
<figure>
<img width="80%" src="figures/double_integrator_mintime_orbits.svg"/>
<figcaption>Solution trajectories of system using the optimal policy</figcaption>
</figure>
<p>And for completeness, we can compute the optimal time to the goal by
solving for the amount of time required to reach the switching surface
plus the amount of time spent moving along the switching surface to the
goal. With a little algebra, you will find that the time to goal,
$T(\bx)$, is given by \[ T(\bx) = \begin{cases}
2\sqrt{\frac{1}{2}\dot{q}^2-q} - \dot{q} & \text{for } u=+1 \text{
regime}, \\ 0 & \text{for } u=0, \\ \dot{q} +
2\sqrt{\frac{1}{2}\dot{q}^2+q} & \text{for } u=-1, \end{cases} \]<!-- call
t_m the time to the surface, then the time on switching surface =
|qdot(t_m)|
for u=1
q0,qdot0 => qm,qdotm with qm = -1/2 qdotm^2
-1/2 qdotm^2 = 1/2 qdotm^2 + c+
qdotm^2 = - c+ , qdotm = sqrt(-c+)
T = (qdotm-qdot0)+qdotm = 2*sqrt(-c+) - qdot0
for u=-1, qdotm^2 = c- , qdotm = -sqrt(c-)
T = (qdot0-qdotm)-qdotm = qdot0 + 2*sqrt(c-)
-->
plotted here:</p>
<figure>
<img width="80%" src="figures/double_integrator_mintime_cost_to_go.svg"/>
<figcaption>Time to the origin using the bang-bang policy</figcaption>
</figure>
<p> Notice that the function is continuous (though not smooth), even
though the policy is discontinuous.</p>
</example> <!-- end of example -->
<subsection><h1>Additive cost</h1>
<p> As we begin to develop theoretical and algorithmic tools for optimal
control, we will see that some formulations are much easier to deal with
than others. One important example is the dramatic simplification that
can come from formulating objective functions using <em>additive
cost</em>, because they often yield recursive solutions. In the additive
cost formulation, the long-term "score" for a trajectory can be written
as $$\int_0^T \ell(x(t),u(t)) dt,$$ where $\ell()$ is the instantaneous cost
(also referred to as the "running cost"), and $T$ can be either a finite
real number or $\infty$. We will call a problem specification with a
finite $T$ a "finite-horizon" problem, and $T=\infty$ an
"infinite-horizon" problem. Problems and solutions for infinite-horizon
problems tend to be more elegant, but care is required to make sure that
the integral converges for the optimal controller (typically by having an
achievable goal that allows the robot to accrue zero-cost).</p>
<p> The quadratic cost function suggested in the double integrator example
above is clearly written as an additive cost. At first glance, our
minimum-time problem formulation doesn't appear to be of this form, but we
actually can write it as an additive cost problem using an infinite
horizon and the instantaneous cost $$\ell(x,u) = \begin{cases} 0 & \text{if
} x=0, \\ 1 & \text{otherwise.} \end{cases}$$</p>
<p> We will examine a number of approaches to solving optimal control
problems throughout the next few chapters. For the remainder of this
chapter, we will focus on additive-cost problems and their solution via
<em>dynamic programming</em>.</p>
</subsection> <!-- end of additive cost -->
</section> <!-- control design as an optimization -->
<section><h1>Optimal control as graph search</h1>
<p> For systems with continuous states and continuous actions, dynamic
programming is a set of theoretical ideas surrounding additive cost optimal
control problems. For systems with a finite, discrete set of states and a
finite, discrete set of actions, dynamic programming also represents a set
of very efficient numerical <em>algorithms</em> which can compute optimal
feedback controllers. Many of you will have learned it before as a tool for
graph search. </p>
<p>Imagine you have a directed graph with states (or nodes) $\{s_1,s_2,...\}
\in S$ and "actions" associated with edges labeled as $\{a_1,a_2,...\} \in
A$, as in the following trivial example:</p>
<figure><img width="70%" src="figures/graph_search.svg"/><figcaption>A
simple directed graph.</figcaption></figure>
<p>Let us also assume that each edge has an associate weight or cost, using
$\ell(s,a)$ to denote the cost of being in state $s$ and taking action $a$.
Furthermore we will denote the transition "dynamics" using \[ s[n+1] =
f(s[n],a[n]). \] For instance, in the graph above, $f(s_1,a_1) = s_2$.</p>
<p>There are many algorithms for finding (or approximating) the optimal
path from a start to a goal on directed graphs. In dynamic programming,
the key insight is that we can find the shortest path from every node by
solving recursively for the optimal <em>cost-to-go</em> (the cost that will
be accumulated when running the optimal controller) from every node to the
goal. One such algorithm starts by initializing an estimate $\hat{J}^*=0$
for all $s_i$, then proceeds with an iterative algorithm which sets
\begin{equation} \hat{J}^*(s_i) \Leftarrow \min_{a \in A} \left[ \ell(s_i,a) +
\hat{J}^*\left({f(s_i,a)}\right) \right]. \label{eq:value_update}
\end{equation} In software, $\hat{J}^*$ can be represented as a vector with
dimension equal to the number of discrete states. This algorithm,
appropriately known as <em>value iteration</em>, is guaranteed to converge
to the optimal cost-to-go up to a constant factor, $\hat{J}^* \rightarrow
J^* + c$ <elib>Bertsekas96</elib>, and in practice does so rapidly.
Typically the update is done in <em>batch</em> -- e.g. the estimate is
updated for all $i$ at once -- but the <em>asynchronous</em> version where
states are updated one at a time is also known to converge, so long as
every state is eventually updated infinitely often. Assuming the graph has
a goal state with a zero-cost self-transition, then this cost-to-go
function represents the weighted shortest distance to the goal. </p>
<p> Value iteration is an amazingly simple algorithm, but it accomplishes
something quite amazing: it efficiently computes the long-term cost of an
optimal policy from <i>every</i> state by iteratively evaluating the
one-step cost. If we know the optimal cost-to-go, then it's easy to extract
the optimal policy, $a = \pi^*(s)$: \begin{equation} \pi^*(s_i) = \argmin_a
\left[ \ell(s_i,a) + J^*\left( f(s_i,a) \right) \right].
\label{eq:policy_update} \end{equation} It's a simple algorithm, but playing
with an example can help our intuition.</p>
<example><h1>Grid World</h1>
<p>Imagine a robot living in a grid (finite state) world. Wants to get
to the goal location. Possibly has to negotiate cells with obstacles.
Actions are to move up, down, left, right, or do nothing.
<elib>Sutton98</elib></p>
<figure>
<a href="figures/gridworld_mintime.swf">
<img width="80%" src="figures/gridworld_mintime.svg" />
</a>
<figcaption>The one-step cost for the grid-world minimum-time problem.
The goal state has a cost of zero, the obstacles have a cost of 10, and
every other state has a cost of 1. <em>Click the image to watch the
value iteration algorithm in action.</em></figcaption>
</figure>
<jupyter>examples/grid_world.ipynb</jupyter>
</example> <!-- end grid world -->
<todo>figure/text for graph approximation of a continuous state
space.</todo>
<example><h1>Dynamic Programming for the Double Integrator</h1>
<p>You can run value iteration for the double integrator (using
barycentric interpolation to interpolate between nodes) in <drake></drake>
using: </p>
<jupyter no_colab>examples/double_integrator/value_iteration.ipynb</jupyter>
<p>Please do take some time to try different cost functions by
editing the code yourself.</p>
</example>
<p> Let's take a minute to appreciate how amazing this is. Our solution to
finding the optimal controller for the double integrator wasn't all that
hard, but it required some mechanical intuition and solutions to
differential equations. The resulting policy was non-trivial -- bang-bang
control with a parabolic switching surface. The value iteration algorithm
doesn't use any of this directly -- it's a simple algorithm for graph
search. But remarkably, it can generate effectively the same policy with
just a few moments of computation.</p>
<p>It's important to note that there <em>are</em> some differences between
the computed policy and the optimal policy that we derived, due to
discretization errors. We will ask you to explore these in the
problems.</p>
<p>The real value of this numerical solution, however, is unlike our
analytical solution for the double integrator, we can apply this same
algorithm to any number of dynamical systems virtually without modification.
Let's apply it now to the simple pendulum, which was intractable
analytically.</p>
<example><h1>Dynamic Programming for the
Simple Pendulum</h1>
<p>You can run value iteration for the simple pendulum (using barycentric
interpolation to interpolate between nodes) in <drake></drake> using:</p>
<jupyter no_colab>examples/pendulum/value_iteration.ipynb</jupyter>
<p>Again, you can easily try different cost functions by
editing the code yourself.</p>
</example>
</section> <!-- end of graph search -->
<section><h1>Continuous dynamic programming</h1>
<p> I find the graph search algorithm extremely satisfying as a first step,
but also become quickly frustrated by the limitations of the discretization
required to use it. In many cases, we can do better; coming up with
algorithms which work more natively on continuous dynamical systems. We'll
explore those extensions in this section.</p>
<subsection><h1>The Hamilton-Jacobi-Bellman Equation</h1>
<p> It's important to understand that the value iteration equations,
equations (\ref{eq:value_update}) and (\ref{eq:policy_update}), are more
than just an algorithm. They are also sufficient conditions for
optimality: if we can produce a $J^*$ and $\pi^*$ which satisfy these
equations, then $\pi^*$ must be an optimal controller. There are an
analogous set of conditions for the continuous systems. For a system
$\dot{\bx} = f(\bx,\bu)$ and an infinite-horizon additive cost
$\int_0^\infty \ell(\bx,\bu)dt$, we have: \begin{gather} 0 = \min_\bu \left[
\ell(\bx,\bu) + \pd{J^*}{\bx}f(\bx,\bu) \right], \label{eq:HJB} \\
\pi^*(\bx) = \argmin_\bu \left[ \ell(\bx,\bu) + \pd{J^*}{\bx}f(\bx,\bu)
\right]. \end{gather} Equation \ref{eq:HJB} is known as the
<em>Hamilton-Jacobi-Bellman</em> (HJB) equation. <elib>Bertsekas05</elib>
<sidenote>Technically, a Hamilton-Jacobi equation is a PDE whose <em>time
derivative</em> depends on the first-order partial derivatives over
state, which we get in the finite-time deriviation; Eq \ref{eq:HJB} is
the steady-state solution of the Hamilton-Jacobi equation.</sidenote>
<!-- chapter 3 --> gives an informal derivation of these equations as the
limit of a discrete-time approximation of the dynamics, and also gives
the following sufficiency theorem:</p>
<theorem><h1>HJB Sufficiency Theorem</h1>
<p>The following statement is adapted from Proposition 3.2.1 of
<elib>Bertsekas05</elib>. Consider a system $\dot{\bx}=f(\bx,\bu)$ and
an infinite-horizon additive cost $\int_0^\infty \ell(\bx,\bu)dt$. Suppose
$J(\bx)$ is a solution to the HJB equation; that is $J$ is continuously
differentiable in $\bx$ and is such that \[ 0 = \min_{\bu \in U} \left[
\ell(\bx,\bu) + \pd{J}{\bx}f(\bx,\bu) \right],\quad \text{for all } \bx. \]
Suppose also that $\pi^*(\bx)$ attains the minimum in the equation for
all $\bx$. Further assume that the differential equation described by
$f$ has a unique solution starting from any state $\bx$, that the
control input trajectory, $\bu^*(t)$, obtained by evaluating $\pi^*$
along any solution is piecewise continuous as a function of $t$, and
that there exists at least one fixed point, $\bx_{fp}$, with
$f(\bx_{fp}, \pi^*(\bx_{fp})) = 0$ and $J^*(\bx_{fp}) = 0$. Then $J$ is
equal to the optimal cost-to-go function, $J(\bx)=J^*(\bx)$ for all
$\bx$, and the control trajectories $\bu^*(t)$ are optimal.</p>
</theorem>
<p>As a tool for verifying optimality, the HJB equations are actually
surprisingly easy to work with: we can verify optimality for an
infinite-horizon objective without doing any integration; we simply have
to check a derivative condition on the optimal cost-to-go function $J^*$.
Let's see this play out on the double integrator example.</p>
<example id="hjb_double_integrator"><h1>HJB for the Double
Integrator</h1>
<p>Consider the problem of regulating the double integrator (this time
without input limits) to the origin using a quadratic cost: $$
\ell(\bx,\bu) = q^2 + \dot{q}^2 + u^2. $$ I claim (without derivation)
that the optimal controller for this objective is $$\pi(\bx) = -q -
\sqrt{3}\dot{q}.$$ To convince you that this is indeed optimal, I
have produced the following cost-to-go function: $$J(\bx) = \sqrt{3}
q^2 + 2 q \dot{q} + \sqrt{3} \dot{q}^2.$$</p>
<p>Taking \begin{gather*} \pd{J}{q} = 2\sqrt{3} q + 2\dot{q}, \qquad
\pd{J}{\dot{q}} = 2q + 2\sqrt{3}\dot{q}, \end{gather*} we can write
\begin{align*} \ell(\bx,\bu) + \pd{J}{\bx}f(\bx,\bu) &= q^2 + \dot{q}^2 +
u^2 + (2\sqrt{3} q + 2\dot{q}) \dot{q} + (2q + 2\sqrt{3}\dot{q}) u
\end{align*} This is a convex quadratic function in $u$, so we can
find the minimum with respect to $u$ by finding where the gradient with
respect to $u$ evaluates to zero. \[ \pd{}{u} \left[ \ell(\bx,\bu) +
\pd{J}{\bx} f(\bx,\bu) \right] = 2u + 2q + 2\sqrt{3}\dot{q}. \] Setting
this equal to $0$ and solving for $u$ yields: $$u^* = -q - \sqrt{3}
\dot{q},$$ thereby confirming that our policy $\pi$ is in fact the
minimizer. Substituting $u^*$ back into the HJB reveals that the right
side does in fact simplify to zero. I hope you are convinced!</p>
</example>
<p>Note that evaluating the HJB for the time-to-go of the minimum-time
problem for the double integrator will also reveal that the HJB is
satisfied wherever that gradient is well-defined. This is certainly
mounting evidence in support of our bang-bang policy being optimal, but
since $\pd{J}{\bx}$ is not defined everywhere, it does not actually
satisfy the requirements of the sufficiency theorem as stated above.</p>
</subsection> <!-- end HJB -->
<subsection id="hjb_minimizing_control"><h1>Solving for the minimizing control</h1>
<p>We still face a few barriers to actually using the HJB in an algorithm.
The first barrier is the minimization over $u$. When the action set was
discrete, as in the graph search version, we could evaluate the one-step
cost plus cost-to-go for every possible action, and then simply take the
best. For continuous action spaces, in general we cannot rely on the
strategy of evaluating a finite number of possible $\bu$'s to find the
minimizer.</p>
<p>All is not lost. In the quadratic cost double integrator example
above, we were able to solve explicitly for the minimizing $\bu$ in
terms of the cost-to-go. It turns out that this strategy will actually
work for a number of the problems we're interested in, even when the
system (which we are given) or cost function (which we are free to pick,
but which should be expressive) gets more complicated.</p>
<p>Recall that I've already tried to convinced you that a majority of the
systems of interest are <em>control affine</em>, e.g. I can write \[
f(\bx,\bu) = f_1(\bx) + f_2(\bx)\bu. \] We can make another dramatic
simplification by restricting ourselves to instantaneous cost functions of
the form \[ \ell(\bx,\bu) = \ell_1(\bx) + \bu^T {\bf R} \bu, \qquad {\bf R}={\bf
R}^T \succ 0. \] In my view, this is not very restrictive - many of the
cost functions that I find myself choosing to write down can be expressed
in this form. Given these assumptions, we can write the HJB as \[ 0 =
\min_{\bu} \left[ \ell_1(\bx) + \bu^T {\bf R} \bu + \pd{J}{\bx} \left[
f_1(\bx) + f_2(\bx)\bu \right]\right]. \] Since this is a positive
quadratic function in $\bu$, if the system does not have any constraints
on $\bu$, then we can solve in closed-form for the minimizing $\bu$ by
taking the gradient of the right-hand side: \[ \pd{}{\bu} = 2\bu^T {\bf R}
+ \pd{J}{\bx} f_2(\bx) = 0, \] and setting it equal to zero to obtain \[
\bu^* = -\frac{1}{2}{\bf R}^{-1}f_2^T(\bx) \pd{J}{\bx}^T.\] If there are
linear constraints on the input, such as torque limits, then more
generally this could be solved (at any particular $\bx$) as a quadratic
program.</p>
<p> What happens in the case where our system is not control affine or if
we really do need to specify an instantaneous cost function on $\bu$
that is not simply quadratic? If the goal is to produce an iterative
algorithm, like value iteration, then one common approach is to make a
(positive-definite) quadratic approximation in $\bu$ of the HJB, and
updating that approximation on every iteration of the algorithm. This
broad approach is often referred to as <em>differential dynamic
programming</em> (c.f. <elib>Jacobson70</elib>). </p>
</subsection> <!-- end solve for u -->
<subsection><h1>Numerical solutions for $J^*$</h1>
<p> The other major barrier to using the HJB in a value iteration
algorithm is that the estimated optimal cost-to-go function, $\hat{J}^*$,
must somehow be represented with a finite set of numbers, but we don't yet
know anything about the potential form it must take. In fact, knowing the
time-to-goal solution for minimum-time problem with the double integrator,
we see that this function might need to be non-smooth for even very simple
dynamics and objectives.</p>
<p>One natural way to parameterize $\hat{J}^*$ -- a scalar
valued-function defined over the state space -- is to define the values
on a mesh. This approach then admits algorithms with close ties to the
relatively very advanced numerical methods used to solve other partial
differential equations (PDEs), such as the ones that appear in finite
element modeling or fluid dynamics. One important difference, however, is
that our PDE lives in the dimension of the state space, while many of the
<a href="https://en.wikipedia.org/wiki/Types_of_mesh">mesh
representations</a> from the other disciplines are optimized for two or
three dimensional space. Also, our PDE may have discontinuities (or at
least discontinuous gradients) at locations in the state space which are
not known apriori.</p>
<p>A slightly more general view of the problem would describe the mesh
(and the associated interpolation functions) as just one form of
representations for <a
href="https://en.wikipedia.org/wiki/Function_approximation">function
approximation</a>. Using a <a
href="https://en.wikipedia.org/wiki/Deep_learning">
neural network</a> to represent the cost-to-go also falls under the
domain of function approximation, perhaps representing the other extreme
in terms of complexity; using deep networks in approximate dynamic
programming is common in <a
href="http://rail.eecs.berkeley.edu/deeprlcourse/">deep reinforcement
learning</a>, which we will discuss more later in the book.</p>
<todo> (see Appendix C for a brief background on function approximation)
</todo>
<subsubsection><h1>Value iteration with function
approximation</h1>
<p>If we approximate $J^*$ with a finitely-parameterized function
$\hat{J}_\balpha^*$, with parameter vector $\balpha$, then this
immediately raises many important questions:
<ul>
<li>What if the true cost-to-go function does not live in the
prescribed function class; e.g., there does not exist an $\balpha$ which
satisfies the sufficiency conditions for all $\bx$? (This seems very
likely to be the case.)</li>
<li>What update should we apply in the iterative algorithm?</li>
<li>What can we say about it's convergence?</li>
</ul>
Let us start by considering updates given by a least-squares approximation of the value iteration update.</p>
<p>Using the least squares solution in a value iteration update is
sometimes referred to as <i>fitted value iteration</i>, where $\bx_k$
are some number of samples taken from the continuous space and for
discrete-time systems the iterative approximate solution to
\begin{gather*} J^*(\bx_0) = \min_{u[\cdot]} \sum_{n=0}^\infty
\ell(\bx[n],\bu[n]), \\ \text{ s.t. } \bx[n+1] = f(\bx[n], \bu[n]),
\bx[0] = \bx_0\end{gather*} becomes \begin{gather} J^d_k = \min_\bu
\left[ \ell(\bx_k,\bu) + \hat{J}^*_\alpha\left({f(\bx_k,\bu)}\right)
\right], \\ \balpha \Leftarrow \argmin_\balpha \sum_k
\left(\hat{J}^*_\balpha(\bx_k) - J^d_k \right)^2.
\label{eq:fitted_value_iteration} \end{gather} Since the desired values $J^d_k$ are only an initial guess of the cost-to-go, we will apply this algorithm iteratively until (hopefully) we achieve some numerical convergence.</p>
<p>Note that the update in \eqref{eq:fitted_value_iteration} is not
<i>quite</i> the same as doing least-squares optimization of $$\sum_k
\left(\hat{J}^*_\balpha(\bx_k) - \min_\bu \left[ \ell(\bx_k,\bu) +
\hat{J}^*_\alpha\left({f(\bx_k,\bu)}\right) \right] \right)^2,$$ because
in this equation $\alpha$ has an effect on both occurences of
$\hat{J}^*$. In \eqref{eq:fitted_value_iteration}, we cut that
dependence by taking $J_k^d$ as fixed desired values; this version
performs better in practice. Like many things, this is an old idea that
has been given a new name in the deep reinforcement learning literature
-- people think of the $\hat{J}^*_\alpha$ on the right hand side as
being the output from a fixed "target network". For nonlinear function
approximators, the update to $\alpha$ in
\eqref{eq:fitted_value_iteration} is often replaced with just one step
of gradient descent.</p>
<example><h1>Neural Value Iteration</h1>
<p>Let us try reproducing our double-integrator value iteration examples using neural networks in <a href="https://pytorch.org/tutorials/">PyTorch</a>:</p>
<jupyter>examples/double_integrator/neural_value_iteration.ipynb</jupyter>
<!--
<p>In this example, you'll see that we use two different approaches to
minimizing over $\bu$. First, we use discrete actions where we simply
evaluate all possible actions and keep the best. Second, we explore
the case which exploits the quadratic cost and the control-affine
dynamics -- but note that we also have to go to continuous time to
have the closed-form solution. (Can you understand why?)</p>
-->
</example>
<p>In general, the convergence and accuracy guarantees for value
iteration with generic function approximators are quite weak. But we do
have some results for the special case of <em>linear function
approximators</em>. A linear function approximator takes the form: \[
\hat{J}^*_\balpha(\bx) = \sum_i \alpha_i \psi_i(\bx) = \bpsi^T(\bx)
\balpha, \] where $\bpsi(\bx)$ is a vector of potentially nonlinear
features. Common examples of features include polynomials, radial basis
functions, or most interpolation schemes used on a mesh. The
distinguishing feature of a linear function approximator is the ability
to exactly solve for $\balpha$ in order to represent a desired function
optimally, in a least-squares sense. For linear function approximators,
this is simply: \begin{gather*} \balpha \Leftarrow \begin{bmatrix}
\bpsi^T(\bx_1) \\ \vdots \\ \bpsi^T(\bx_K)\end{bmatrix}^+
\begin{bmatrix} J^d_1 \\ \vdots \\ J^d_K \end{bmatrix}, \end{gather*}
where the $^+$ notation refers to a Moore-Penrose pseudoinverse.
Remarkably, for linear function approximators, this update is still
known to converge to the globally optimal $\balpha^*$.</p>
<todo> add citations for convergence results </todo>
</subsubsection>
<subsubsection><h1>Value iteration on a mesh</h1>
<p>Imagine that we use a mesh to approximate the cost-to-go function
over that state space with $K$ mesh points $\bx_k$. We would like to
perform the value iteration update: \begin{equation} \forall k,
\hat{J}^*(\bx_k) \Leftarrow \min_\bu \left[ \ell(\bx_k,\bu) +
\hat{J}^*\left({f(\bx_k,\bu)}\right) \right],
\label{eq:mesh_value_iteration} \end{equation} but must deal with the
fact that $f(\bx_k,\bu)$ might not result in a next state that is
directly at a mesh point. Most interpolation schemes for a mesh can be
written as some weighted combination of the values at nearby mesh
points, e.g. \[ \hat{J}^*(\bx) = \sum_i \beta_i(\bx) \hat{J}^*(\bx_i),
\quad \sum_i \beta_i = 1 \] with $\beta_i$ the relative weight of the
$i$th mesh point. In <drake></drake> we have implemented barycentric
interpolation<elib>Munos98</elib>. Taking $\alpha_i = \hat{J}^*(\bx_i)$,
the cost-to-go estimate at mesh point $i$, we can see that this is
precisely an important special case of fitted value iteration with
linear function approximation. Furthermore, assuming $\beta_i(\bx_i) =
1,$ (e.g., the only point contributing to the interpolation <i>at a mesh
point</i> is the value at the mesh point itself), the update in Eq.
(\ref{eq:mesh_value_iteration}) is precisely the least-squares update
(and it achieves zero error). This is the representation used in the value iteration examples that you've already experimented with above.</p>
</subsubsection>
<todo>
<subsubsection><h1>Representing the cost-to-go in a deep network</h1>
</subsubsection>
</todo>
<subsubsection><h1>Continuous-time systems</h1>
<p>For solutions to systems with continuous-time dynamics, I have to
uncover one of the details that I've so far been hiding to keep the
notation simpler. Let us consider a problem with a finite-horizon:
\begin{gather*} \min_{\bu[\cdot]} \sum_{n=0}^N \ell(\bx[n],\bu[n]), \\
\text{ s.t. } \bx[n+1] = f(\bx[n], \bu[n]), \bx[0] = \bx_0\end{gather*}
In fact, the way that we compute this is by solving the <em>time-varying
cost-to-go function</em> backwards in time: \begin{gather*}J^*(\bx,N) =
\min_\bu \ell(\bx, \bu) \\ J^*(\bx,n-1) = \min_\bu \left[ \ell(\bx, \bu) +
J^*(f(\bx,\bu), n) \right]. \end{gather*} The convergence of the value
iteration update is equivalent to solving this time-varying cost-to-go
backwards in time until it reaches a steady-state solution (the
infinite-horizon solution). Which explains why value iteration only
converges if the optimal cost-to-go is bounded.</p>
<p>Now let's consider the continuous-time version. Again, we have a
time-varying cost-to-go, $J^*(\bx,t)$. Now $$\frac{dJ^*}{dt} =
\pd{J^*}{\bx}f(\bx,\bu) + \pd{J^*}{t},$$ and our sufficiency condition
is $$0 = \min_\bu \left[\ell(\bx, \bu) + \pd{J^*}{\bx}f(\bx,\bu) +
\pd{J^*}{t} \right].$$ But since $\pd{J^*}{t}$ doesn't depend on $\bu$,
we can pull it out of the $\min$ and write the (true) HJB:
$$-\pd{J^*}{t} = \min_\bu \left[\ell(\bx, \bu) + \pd{J^*}{\bx}f(\bx,\bu)
\right].$$ The standard numerical recipe <elib>Osher03 </elib> for
solving this is to approximate $\hat{J}^*(\bx,t)$ on a mesh and then
integrate the equations backwards in time (until convergence, if the
goal is to find the infinite-horizon solution). If, for mesh point
$\bx_i$ we have $\alpha_i(t) = \hat{J}^*(\bx_i, t)$, then:
$$-\dot\alpha_i(t) = \min_\bu \left[\ell(\bx_i, \bu) + \pd{J^*(\bx_i,
t)}{\bx}f(\bx_i,\bu) \right],$$ where the partial derivative is
estimated with a suitable finite-difference approximation on the mesh
and often some "viscosity" terms are added to the right-hand side to
provide additional numerical robustness; see the Lax-Friedrichs scheme
<elib>Osher03 </elib> (section 5.3.1) for an example. </p>
<p>Probably most visible and consistent campaign using numerical HJB
solutions in applied control (at least in robotics) has come from <a
href="http://hybrid.eecs.berkeley.edu/index.html">Claire Tomlin's group
at Berkeley</a>. Their work leverages <a
href="https://www.cs.ubc.ca/~mitchell/ToolboxLS/">Ian Mitchell's Level
Set Toolbox</a>, which solves the Hamilton-Jacobi PDEs on a Cartesian
mesh using the technique cartooned above, and even includes the
minimum-time problem for the double integrator as a tutorial
example<elib>Mitchell05</elib>.</p>
</subsubsection>
</subsection> <!-- end function approx -->
</section> <!-- end continuous time DP -->
<todo> try just running snopt on quartic objective </todo>
<!--
<subsection><h1>A continuous policy iteration algorithm</h1>
<todo> simple (e.g. one-d example) </todo>
</section> -->
<!-- end policy iteration -->
<!--
<subsection><h1>How far can we take this? (Performance and Scaling)</h1>
<todo>
Errors in bang-bang for double integrator due to discretization.
Limited in number of dimensions. Function approximation, but lacks convergence results.
</todo>
</section> --><!-- end scaling -->
<section><h1>Extensions</h1>
<todo> add section on extensions. discounting. finite-horizon / time-varying dynamics or cost. call-out to the stochastic case in the future chapter.</todo>
<p>There are many many nice extensions to the basic formulations that we've presented so far. I'll try to list a few of the most important ones here. I've also had a number of students in this course explore very interesting extensions; for example <elib>Yang20</elib> looked a imposing a low-rank structure on the (discrete-state) value function using ideas from matrix completion to obtain good estimates despite updating only a small fraction of the states.</p>
<subsection><h1>Linear Programming Approach</h1>
<p>For discrete MDPs, value iteration is a magical algorithm because it is
simple, but known to converge to the global optimal, $J^*$. However,
other important algorithms are known; one of the most important is a
solution approach using linear programming. This formulation provides an
alternative view, but may also be more generalizable and even more
efficient for some instances of the problem.</p>
<todo>I've moved the stochastic dp equations, but this text depended on them.</todo>
<p>Recall the optimality conditions from Eq.
(\ref{eq:stochastic_dp_optimality_cond}). If we describe the cost-to-go
function as a vector, $J_i = J(s_i)$, then these optimality conditions can
be rewritten in vector form as \begin{equation} \bJ = \min_a \left[ {\bf \ell}(a)
+ \bT(a) \bJ \right], \label{eq:vector_stochastic_dp} \end{equation} where
$\ell_i(a) = \ell(s_i,a)$ is the cost vector, and $T_{i,j}(a) = \Pr(s_j|s_i,a)$
is the transition probability matrix. Let us take $\bJ$ as the vector of
decision variables, and replace Eq. (\ref{eq:vector_stochastic_dp}) with
the constraints: \begin{equation} \forall a, \bJ \le {\bf \ell}(a) + \bT(a)
\bJ.\end{equation} Clearly, for finite $a$, this is finite list of linear
constraints, and for any vector $\bJ$ satisfying these constraints we have
$\bJ \le \bJ^*$ (elementwise). Now write the linear program:
\begin{gather*} \maximize_\bJ \quad \bc^T \bJ, \\ \subjto \quad \forall a,
\bJ \le
{\bf \ell}(a) + \bT(a) \bJ, \end{gather*} where $c$ is any positive vector. The
solution to this problem is $\bJ = \bJ^*$.</p>
<p>Perhaps even more interesting is that this approach can be generalized
to linear function approximators. Taking a vector form of my notation
above, now we have a matrix of features with $\bPsi_{i,j} =
\psi^T_j(\bx_i)$ and we can write the LP \begin{gather} \maximize_\balpha
\quad \bc^T \bPsi \balpha, \\ \subjto \quad \forall a, \bPsi \balpha \le
{\bf \ell}(a) + \bT(a) \bPsi \balpha. \end{gather} This time the solution is not
necessarily optimal, because $\bPsi \balpha^*$ only approximates $\bJ^*$,
and the relative values of the elements of $\bc$ (called the
"state-relevance weights") can determine the relative tightness of the
approximation for different features <elib>Farias02</elib>.</p>
</subsection>
</section> <!-- end of extensions -->
<section><h1>Exercises</h1>
<exercise><h1>Choosing a Cost Function</h1>
<figure>
<img width="50%" src="figures/exercises/choosing_cost.svg"/>
<figcaption>Autonomous car moving at velocity $v$ on a straight road.</figcaption>
</figure>
<p>The figure above shows an autonomous car moving at constant velocity $v>0$ on a straight road. Let $x$ be the (longitudinal) position of the car along the road, $y$ its (transversal) distance from the centerline, and $\theta$ the angle between the centerline and the direction of motion. The only control action is the steering velocity $u$, which is constrained in the interval $[u_{\text{min}}, u_{\text{max}}]$ (where $u_{\text{min}}<0$ and $u_{\text{max}}>0$). We describe the car dynamics with the simple kinematic model \begin{align*}\dot x &= v \cos \theta, \\ \dot y &= v \sin \theta, \\ \dot \theta &= u.\end{align*} Let $\bx = [x, y, \theta]^T$ be the state vector. To optimize the car trajectory we consider a quadratic objective function $$J = \int_{0}^{\infty} [\bx^T(t) \bQ \bx(t) + R u^2(t)] dt,$$ where $\bQ$ is a constant positive-semidefinite (hence symmetric) matrix and $R$ is a constant nonnegative scalar (note that $R=0$ is allowed here).</p>
<ol type="a">
<li>Suppose our only goal is to keep the distance $y$ between the car and the centerline as small as possible, without worrying about anything else. What would be your choice for $\bQ$ and $R$?</li>
<li>How would the behavior of the car change if you were to multiply the weights $\bQ$ and $R$ from the previous point by an arbitrary positive coefficient $\alpha$?</li>
<li>The cost function from point (a) might easily lead to excessively sharp turns. Which entry of $\bQ$ or $R$ would you increase to mitigate this issue?</li>
<li>Country roads are more slippery on the sides than in the center. Is this class of objective functions rich enough to include a penalty on sharp turns that increases with the distance of the car from the centerline?</li>
<li>With this dynamic model and this objective function, would you ever choose a weight matrix $\bQ$ which is strictly positive definite (independent of the task you want the car to accomplish)? Why?</li>
</ol>
</exercise>
<exercise><h1>Ill-Posed Optimal Control Problem</h1>
<p>In this exercise we will see how seemingly simple cost functions can give surprising results. Consider the single-integrator system $\dot x = u$ with initial state $x(0)=0$. We would like to find the control signal $u(t)$ that minimizes the seemingly innocuous cost function $$J = \int_0^T x^2(t) + (u^2(t) - 1)^2 dt,$$ with $T$ finite. To this end, we consider a <a href="https://en.wikipedia.org/wiki/Square_wave">square-wave</a> control parameterized by $\tau>0$: $$u_\tau(t) = \begin{cases} 1 &\text{if} & t \in [0, \tau) \cup [3 \tau, 5 \tau) \cup [7 \tau, 9 \tau) \cup \cdots \\ -1 &\text{if} & t \in [\tau, 3 \tau) \cup [5 \tau, 7 \tau) \cup [9 \tau, 11 \tau) \cup \cdots \end{cases}.$$</p>
<ol type="a">
<li>What are the states $x$ and the inputs $u$ for which the running cost $$\ell(x, u) = x^2 + (u^2 - 1)^2$$ is equal to zero?</li>
<li>Consider now two control signals $u_{\tau_1}(t)$ and $u_{\tau_2}(t)$, with $\tau_1 = \tau_2 / 2$. Which one of the two incurs the lower cost $J$? (Hint: start by sketching how the system state $x(t)$ evolves under these two control signals.)</li>
<li>What happens to the cost $J$ when $\tau$ gets smaller and smaller? What does the optimal control signal look like? Could you implement it with a finite-bandwidth controller?</li>
</ol>
</exercise>
<exercise><h1>A Linear System with Quadratic Cost</h1>
<p>Consider the scalar control differential equation $$\dot{x} = x + u,$$ and the infinite horizon cost function $$J = \int_0^{\infty} [3x^2(t) + u^2(t)] dt.$$ As we will see in the <a href="lqr.html">chapter on linear-quadratic regulation</a>, the optimal cost-to-go for a problem of this kind assumes the form $J^* = S x^2$. It is not hard to see that this, in turn, implies that the optimal controller has the form $u^* = - K x$.</p>
<ol type="a">
<li>Imagine that you plugged the expression $J^* = S x^2$ in the HJB equations for this problem, you solved them for $S$, and you got $S \leq 0$. Would this result ring any alarm bells? Explain your answer.</li>
<li>Use the HJB sufficiency theorem to derive the optimal values of $S$ and $K$.</li>
<li>
Working with digital controllers, we typically have to sample the dynamics of our systems, and approximate them with discrete-time models. Let us introduce a time step $h > 0$ and discretize the dynamics as $$\frac{x[n+1] - x[n]}{h} = x[n] + u[n],$$ and the objective as $$h \sum_{n=0}^{\infty} (3 x^2[n] + u^2[n]).$$ One of the following expressions is the correct cost-to-go $J_h^*(x)$ for this discretized problem. Can you identify it without solving the discrete-time HJB equation? Explain your answer.
<ol ol type="i">
<li>$J_h^* (x)= S_h x^4$ with $S_h = 3 + h + \sqrt{6}h^2$.</li>
<li>$J_h^* (x)= S_h x^2$ with $S_h = 1 + 2h + 2 \sqrt{h^2 + h +1}$.</li>
<li>$J_h^* (x)= S_h x^2$ with $S_h = 3 + 2h^2 + \sqrt{h^2 + h + 2}$.</li>
</ol>
</li>
</ol>
</exercise>
<exercise><h1>Value Iteration for Minimum-Time Problems</h1>
<p>In this exercise we analyze the performance of the value-iteration algorithm, considering its application to the <a href="#minimum_time_double_integrator">minimum time problem for the double integrator</a>. <a href="https://colab.research.google.com/github/RussTedrake/underactuated/blob/master/exercises/dp/minimum_time/minimum_time.ipynb" target="_blank">In this python notebook</a>, you will find everything you need for this analysis. Take the time to go through the notebook and understand the code in it, then answer the following questions.</p>
<ol type="a">
<li>
At the end of the <a href="https://colab.research.google.com/github/RussTedrake/underactuated/blob/master/exercises/dp/minimum_time/minimum_time.ipynb" target="_blank">notebook</a> section "Performance of the Value-Iteration Policy", we plot the state trajectory of the double integrator in closed loop with the value-iteration policy, as well as the resulting control signal.
<ol ol type="i">
<li>Does the state-space trajectory follow the theoretically-optimal quadratic arcs we have seen in <a href="#minimum_time_double_integrator">the example</a>?</li>
<li>Is the control policy we get from value iteration a bang-bang policy? In other words, does the control signal take values in the set $\{-1, 0, 1\}$ exclusively?</li>
<li>Explain in a couple of sentences, what is the reason for this behavior.</li>
</ol>
</li>
<li>In the "Value Iteration Algorithm" section of the <a href="https://colab.research.google.com/github/RussTedrake/underactuated/blob/master/exercises/dp/minimum_time/minimum_time.ipynb" target="_blank">notebook</a>, increase the number of knot points on the $q$ and the $\dot q$ axes to refine the state-space mesh used in the value iteration. Does this fix the issues you have seen in point (a)?</li>
<li>In the final section of the <a href="https://colab.research.google.com/github/RussTedrake/underactuated/blob/master/exercises/dp/minimum_time/minimum_time.ipynb" target="_blank">notebook</a>, implement the theoretically-optimal control policy from <a href="#minimum_time_double_integrator">the example</a>, and use the plots to verify that the closed-loop system behaves as expected.</li>
</ol>
</exercise>
</section>
</chapter>
<!-- EVERYTHING BELOW THIS LINE IS OVERWRITTEN BY THE INSTALL SCRIPT -->
<table style="width:100%;"><tr style="width:100%">
<td style="width:33%;text-align:left;"><a class="previous_chapter" href=stochastic.html>Previous Chapter</a></td>
<td style="width:33%;text-align:center;"><a href=underactuated.html>Table of contents</a></td>
<td style="width:33%;text-align:right;"><a class="next_chapter" href=lqr.html>Next Chapter</a></td>
</tr></table>
<div id="footer">
<hr>
<table style="width:100%;">
<tr><td><em>Underactuated Robotics</em></td><td align="right">© Russ
Tedrake, 2020</td></tr>
</table>
</div>
</body>
</html>