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Feedback control for computer systems

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Feedback Control
for Computer Systems

Philipp K. Janert

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Feedback Control for Computer Systems
by Philipp K. Janert
Copyright © 2014 Philipp K. Janert. All rights reserved.
Printed in the United States of America.
Published by O’Reilly Media, Inc., 1005 Gravenstein Highway North, Sebastopol, CA
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ISBN: 978-1-449-36169-3
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Table of Contents

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Part I.

Foundations

1. Why Feedback? An Invitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
A Hands-On Example
Hoping for the Best
Establishing Control
Adding It Up
Summary
Code to Play With

4
5
7
8
10
11

2. Feedback Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Systems and Signals
Tracking Error and Corrective Action
Stability, Performance, Accuracy
The Setpoint
Uncertainty and Change
Feedback and Feedforward
Feedback and Enterprise Systems
Code to Play With

16
18
19
21
22
23
24
24

3. System Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Lags and Delays
Forced Response and Free Response
Transient Response and Steady-State Response
Dynamics in the Physical World and in the Virtual World
Dynamics and Memory

27
28
29
31
33
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The Importance of Lags and Delays for Feedback Loops
Avoiding Delays
Theory and Practice
Code to Play With

34
34
35
36

4. Controllers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Block Diagrams
On/Off Control
Proportional Control
Why Proportional Control Is Not Enough
Integral Control
Integral Control Changes the Dynamics
Integral Control Can Generate a Constant Offset
Derivative Control
Problems with Derivative Control
The Three-Term or PID Controller
Code to Play With

40
41
42
42
43
45
45
46
47
47
49

5. Identifying Input and Output Signals. . . . . . . . . . . . . . . . . . . . . . . . . . 51
Control Input and Output
Directionality of the Input/Output Relation
Examples
Thermal Control 1: Heating
Item Cache
Server Scaling
Controlling Supply and Demand by Dynamic Pricing
Thermal Control 2: Cooling
Criteria for Selecting Control Signals
For Control Inputs
For Control Outputs
A Note on Multidimensional Systems

51
52
54
54
56
58
60
62
64
64
65
65

6. Review and Outlook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
The Feedback Idea
Iteration
Process Knowledge
Avoiding Instability
The Setpoint

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67
68
68
69

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Control, Not Optimization

Part II.

69

Practice

7. Theory Preview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Frequency Representation
The Transfer Function
Block-Diagram Algebra
PID Controllers
Poles of the Transfer Function
Process Models

73
74
74
75
76
77

8. Measuring the Transfer Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Static Input/Output Relation: The Process Characteristic
Practical Considerations
Dynamic Response to a Step Input:
The Process Reaction Curve
Practical Aspects
Process Models
Self-Regulating Process
Accumulating Process
Self-Regulating Process with Oscillation
Non-Minimum Phase System
Other Methods of System Identification

80
81

82
83
83
83
86
88
89
90

9. PID Tuning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Tuning Objectives
General Effect of Changes to Controller Parameters
Ziegler–Nichols Tuning
Semi-Analytical Tuning Methods
Practical Aspects
A Closer Look at Controller Tuning Formulas

92
94
95
96
98
99

10. Implementation Issues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Actuator Saturation and Integrator Windup
Preventing Integrator Windup
Setpoint Changes and Integrator Preloading
Smoothing the Derivative Term
Choosing a Sampling Interval
Variants of the PID Controller

101
102
103
103
104
105

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Incremental Form
Error Feedback Versus Output Feedback
The General Linear Digital Controller
Nonlinear Controllers
Error-Square and Gap Controllers
Simulating Floating-Point Output
Categorical Output

106
107
108
109
109
110
111

11. Common Feedback Architectures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Changing Operating Conditions: Gain Scheduling
Gain Scheduling for Mildly Nonlinear Systems
Large Disturbances: Feedforward
Fast and Slow Dynamics: Nested or “Cascade” Control
Systems Involving Delays: The Smith Predictor

Part III.

114
115
116
117
118

Case Studies

12. Exploring Control Systems Through Simulation. . . . . . . . . . . . . . . . 125
The Case Studies
Modeling Time
Control Time
Simulation Time
The Simulation Framework
Components
Plants and Systems
Controllers
Actuators and Filters
Convenience Functions for Standard Loops
Generating Graphical Output

126
127
127
128
129
130
130
130
132
133
135

13. Case Study: Cache Hit Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Defining Components
Cache Misses as Manufacturing Defects
Measuring System Characteristics
Controller Tuning
Simulation Code

137
140
141
143
146

14. Case Study: Ad Delivery. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
The Situation
Measuring System Characteristics

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149
150


Establishing Control
Improving Performance
Variations
Cumulative Goal
Gain Scheduling
Integrator Preloading
Weekend Effects
Simulation Code

151
153
156
156
157
158
158
158

15. Case Study: Scaling Server Instances. . . . . . . . . . . . . . . . . . . . . . . . . 161
The Situation
Measuring and Tuning
Reaching 100 Percent With a Nonstandard Controller
Dealing with Latency
Simulation Code

161
162
164
167
169

16. Case Study: Waiting-Queue Control. . . . . . . . . . . . . . . . . . . . . . . . . . 173
On the Nature of Queues and Buffers
The Architecture
Setup and Tuning
Derivative Control to the Rescue
Controller Alternatives
Simulation Code

173
175
175
179
181
181

17. Case Study: Cooling Fan Speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
The Situation
The Model
Tuning and Commissioning
Closed-Loop Performance
Simulation Code

185
186
188
189
190

18. Case Study: Controlling Memory Consumption in a Game Engine. 193
The Situation
Problem Analysis
Architecture Alternatives
A Nontraditional Loop Arrangement
A Traditional Loop with Logarithms
Results
Simulation Code

193
194
195
196
196
199
200

19. Case Study Wrap-Up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
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Simple Controllers, Simple Loops
Measuring and Tuning
Staying in Control
Dealing with Noise

Part IV.

203
204
204
205

Theory

20. The Transfer Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Differential Equations
Laplace Transforms
Properties of the Laplace Transform
Using the Laplace Transform to Solve Differential Equations
A Worked Example
The Transfer Function
Worked Example: Step Response
Worked Example: Ramp Input
The Harmonic Oscillator
What If the Differential Equation Is Not Known?

209
210
211
212
213
214
215
216
217
218

21. Block-Diagram Algebra and the Feedback Equation. . . . . . . . . . . . 219
Composite Systems
The Feedback Equation
An Alternative Derivation of the Feedback Equation
Block-Diagram Algebra
Limitations and Importance of Transfer Function Methods

219
221
223
223
224

22. PID Controllers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
The Transfer Function of the PID Controller
The Canonical Form of the PID Controller
The General Controller
Proportional Droop Revisited
A Worked Example

227
228
229
230
231

23. Poles and Zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Structure of a Transfer Function
Effect of Poles and Zeros
Special Cases and Additional Details
Pole Positions and Response Patterns
Dominant Poles
Pole Placement

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233
234
236
238
239
241


What to Do About Delays

242

24. Root Locus Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
Construction of Root Locus Diagrams
Root Locus or “Evans” Rules
Angle and Magnitude Criteria
Practical Issues
Examples
Simple Lag with a P Controller
Simple Lag with a PI Controller

246
247
249
250
251
251
253

25. Frequency Response and the Bode Plot. . . . . . . . . . . . . . . . . . . . . . . 257
Frequency Response
Frequency Response in the Physical World
Frequency Response for Transfer Functions
A Worked Example
The Bode Plot
A Criterion for Marginal Stability
Other Graphical Techniques

257
258
258
259
261
263
265

26. Topics Beyond This Book. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Discrete-Time Modeling and the z-Transform
State-Space Methods
Robust Control
Optimal Control
Mathematical Control Theory

Part V.

267
268
272
273
274

Appendices

A. Glossary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
B. Creating Graphs with Gnuplot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
C. Complex Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
D. Further Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

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Preface

This is a book about feedback control—not a topic that programmers
(among others) tend to know much about. This is a pity, because feed‐
back control was originally devised to solve a problem that should be
all too familiar to software engineers, especially those who are working
on enterprise systems. Feedback control is a way to make sure that
large, complicated systems run reliably, even when subject to external
disturbances, and to make efficient use of constrained resources.
If you are looking for a system that can spin up some additional servers
when traffic in your data center spikes and take them down again when
the rush is over, then you have come to the right place.

What Is Feedback?
Feedback works by constantly comparing the actual behavior of a sys‐
tem to its desired behavior. If the actual behavior differs from the de‐
sired one, a corrective action is applied to counteract the deviation and
drive the system back to its target. This process is repeated constantly,
as long as the system is running.
One appealing feature of feedback control is that it requires relatively
little knowledge about the controlled system. As long as one knows
which direction to “nudge” the system when it has gotten off course,
one can build a feedback loop. For this reason, feedback is an attractive
technique for controlling large, complex, and opaque systems.
Moreover, feedback systems are self-correcting even in the presence
of external disturbances. Because the system’s behavior is monitored
and adjusted all the time, a feedback system naturally and automati‐
cally responds to changes in operating conditions. No need to make
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special provisions to activate additional servers for rush hour: the
feedback controller will notice the increase in load and spin up further
instances until the desired quality of service is met. It will also take the
instances down again, once the rush has passed, in proportion to the
lightening load.
One needs to be careful, though, since control actions that are either
too large or improperly timed can “overcorrect” a disturbance. Instead
of reducing the difference between the desired and the actual behavior,
such control actions replace a deviation in one direction with another
deviation in the opposite direction. In the worst case, the amplitude
of these deviations grows with each step until the system becomes
dysfunctional or, often literally, blows up!
To avoid this outcome, control theory has developed specific experi‐
ments for analyzing a system’s behavior. The results from these ex‐
periments can then be used to design and tune control loops that are
safe to operate and that will track a reference value accurately and
reliably.

Why This Book?
Feedback control has a long and successful history in applications to
electronic circuits, industrial processes, and vehicular engineering.
But it can do more. Feedback is self-correcting, so it can keep a system
on target even if conditions change unpredictably. Feedback requires
only limited knowledge about the process to be controlled; hence it
can be applied to situations involving complex and opaque systems,
such as those typical of enterprise environments.
In this book we will study the application of feedback principles to
several software engineering problems, such as the optimal sizing of
a cache, the management of a server farm, the control of waiting
queues or buffers, and more. In all these examples, feedback will help
us to make efficient use of scarce or expensive resources and to do so
in the face of changing conditions.
But applying feedback principles to computer systems raises different
questions than one typically encounters in the traditional application
areas. The laws describing the behavior of computer systems are much
less constrained than those for systems in the physical world; so we
will rely more on experimental measurements and phenomenological
descriptions than on theoretical analysis. (This is similar to the situa‐

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tion one finds in the application of feedback methods to industrial
processes.) At the same time, computer systems offer a greater variety
of control signals than most physical assemblies do; we therefore have
greater freedom in choosing the best signal to use and hence must be
aware of the trade-offs involved. We will pay particular attention to
such questions.
Feedback control has been barely explored as a design paradigm for
software systems. I hope to convince you that feedback control has
much to offer in this regard and is, in fact, the correct solution to many
problems that software engineers commonly face.

How to Read This Book
It can be difficult for an outsider to learn about feedback. Textbooks
and articles use specialized terminology and examples from existing
application areas, which sometimes obscures the underlying concepts.
The problems that arise in the classical application areas are not nec‐
essarily the same problems that are of greatest interest to program‐
mers. One also needs to be aware that many textbooks on feedback
control are primarily concerned with the mathematical manipulations
that underlie control theory and so give less attention to conceptual
development or practical implementation questions.
This book takes a different approach. Theoretical development, im‐
portant and beautiful as it may be, has been relegated to Part IV in the
back of the book. An introductory Part I introduces the concepts of
feedback control, system dynamics, and controller design. Part II de‐
scribes a variety of practical techniques for the implementation and
tuning of controllers, and it also discusses some examples of “design
patterns” for feedback loops. Part III consists of a collection of case
studies: specific problems involving computer systems that are solved
using feedback methods. For each case study, a number of different
approaches and their trade-offs are discussed in some detail.
The case studies are the heart of the book. I suggest beginning with
the introductory Part I in order to become familiar with the basic
feedback concepts. Then tackle the case studies, diving into Part II
(Practice) or Part IV (Theory) as the need for additional information
arises. (The sequence of case studies is arranged roughly in order of
increasing complexity.)

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All case studies are realized as computer simulations, and the code is
available from the book’s website. The code is intentionally simple and
straightforward so that it can be easily extended and modified. Ex‐
perimenting with simulations is an excellent way to build intuition for
the sometimes surprising behavior of closed-loop systems—and to
build the necessary confidence that this feedback stuff really works!

Conventions Used in This Book
The following typographical conventions are used in this book:
Italic
Indicates new terms, URLs, email addresses, filenames, and file
extensions.
Constant width

Used for program listings, as well as within paragraphs to refer to
program elements such as variable or function names, databases,
data types, environment variables, statements, and keywords.
Constant width bold

Shows commands or other text that should be typed literally by
the user.
Constant width italic

Shows text that should be replaced with user-supplied values or
by values determined by context.
This icon signifies a tip, suggestion, or general note.

This icon indicates a warning or caution.

Using Code Examples
This book is here to help you get your job done. In general, if this book
includes code examples, you may use the code in your programs and
documentation. You do not need to contact us for permission unless
you’re reproducing a significant portion of the code. For example,
writing a program that uses several chunks of code from this book
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does not require permission. Selling or distributing a CD-ROM of
examples from O’Reilly books does require permission. Answering a
question by citing this book and quoting example code does not re‐
quire permission. Incorporating a significant amount of example code
from this book into your product’s documentation does require per‐
mission.
We appreciate, but do not require, attribution. An attribution usually
includes the title, author, publisher, and ISBN. For example: “Feedback
Control for Computer Systems by Philipp K. Janert (O’Reilly). Copy‐
right 2014 Philipp K. Janert, 978-1-449-36169-3.”
If you feel your use of code examples falls outside fair use or the per‐
mission given above, feel free to contact us at permissions@oreilly.com.

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Acknowledgments
It was a pleasure working on this project with a familiar group of
friends and coworkers. Mike Loukides guided this project with his
familiar gentle touch. Matt Darnell again did a tremendous job of
copyediting the manuscript. The production team at O’Reilly was
most accommodating when it came to my special requests regarding
math typesetting and graphics.
Ben Peirce read the entire manuscript and provided valuable com‐
ments. I also acknowledge useful conversations with Austin King,
Chris Nauroth, and Joe Adler.
I am especially indebted to Richard Kreckel, who carefully read several
drafts of this manuscript and made many exceptionally valuable and
insightful suggestions. I owe him big-time.
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This book was written on Linux while running IceWM, tcsh, and
XEmacs. The manuscript was prepared using LaTeX together with the
AMS-LaTeX packages; the LaTeX manuscript was then transformed
into the publisher’s internal format for production. The graphs were
drawn using gnuplot and pic; some calculations for the root locus di‐
agrams in Chapter 24 were performed with Scilab. The simulations
were implemented in Python.

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PART I

Foundations

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CHAPTER 1

Why Feedback? An Invitation

Workflow, order processing, ad delivery, supply chain management—
enterprise systems are often built to maintain the flow of certain items
through various processing steps. For instance, at a well-known online
retailer, one of our systems was responsible for managing the flow of
packages through the facilities. Our primary control mechanism was
the number of pending orders we would release to the warehouses at
any one time. Over time, these orders would turn into shipments and
be ready to be loaded onto trucks. The big problem was to throttle the
flow of pending orders just right so that the warehouses were never
idle, but without overflowing them (quite literally) either.
Later I encountered exactly the same problem, but in an entirely dif‐
ferent context, at a large publisher of Internet display ads. In this case,
the flow consisted of ad impressions.1 Again, the primary “knob” that
we could adjust was the number of ads released to the web servers, but
the constraint was a different one. Overflowing the servers was not a
concern, but it was essential to achieve an even delivery of ads from
various campaigns over the course of the month. Because the intensity
of web traffic changes from hour to hour and from day to day, we were
constantly struggling to accomplish this goal.
As these two examples demonstrate, maintaining an even flow of items
or work units, while neither overwhelming nor starving downstream
processing steps, is a common objective when building enterprise sys‐
1. Every time an advertisement is shown on a website, this event is counted as an im‐
pression. The concept is important in the advertising industry, since advertisers often
buy a certain number of such impressions.

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tems. However, the changes and uncertainties that are present in all
real-world processes frequently make it difficult, if not impossible, to
achieve this goal. Conveyors run slower than expected and web traffic
suddenly spikes, disrupting all carefully made plans. To succeed, we
therefore require systems that can detect changes in the environment
and respond to them.
In this book, we will study a particular strategy that has proven its
effectiveness many times in all forms of engineering, but that has rarely
been exploited in software development: feedback control. The essen‐
tial ingredient is that we base the operations of our system specifically
on the system’s output, rather than on other, more general environ‐
mental factors. (For example, instead of monitoring the ups and downs
of web traffic directly, we will base our delivery plan only on the actual
rate at which ads are being served.) By taking the actual output into
account (that’s what “feedback” means), we establish a firm and reliable
control over the system’s behavior. At the same time, feedback intro‐
duces complexity and the risk of instability, which occurs when inap‐
propriate control actions reinforce each other, and much of our at‐
tention will be devoted to techniques that prevent this problem. Once
properly implemented, however, feedback control leads to systems
that exhibit reliable behavior, even when subject to uncertainty and
change.

A Hands-On Example
As we have seen, flow control is a common objective in enterprise
systems. Unfortunately, things often seem rigged to make this objec‐
tive difficult to attain. Here is a typical scenario (see Figure 1-1).
1. We are in charge of a system that releases items to a downstream
processing step.
2. The downstream system maintains a buffer of items.
3. At each time step, the downstream system completes work on
some number of items from its buffer. Completed items are re‐
moved from the buffer (and presumably kicked down to the next
processing step).
4. We cannot put items directly into the downstream buffer. Instead,
we can only release items into a “ready pool,” from which they will
eventually transfer into the downstream buffer.

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5. Once we have placed items into the ready pool, we can no longer
influence their fate: they will move into the downstream buffer
owing to factors beyond our control.
6. The number of items that are completed by the downstream sys‐
tem (step 3) or that move from the ready pool to the downstream
buffer (step 5) fluctuates randomly.
7. At each time step, we need to decide how many items to release
into the ready pool in order to keep the downstream buffer filled
without overflowing it. In fact, the owners of the downstream
system would like us to keep the number of items in their buffer
constant at all times.

Figure 1-1. Block diagram of a workflow system. Items are being re‐
leased into the “ready pool,” from which they are transferred to the
downstream buffer.
It is somewhat natural at this point to say: this is unfair! We are sup‐
posed to control a quantity (the number of units in the downstream
buffer) that we can’t even manipulate directly. How are we supposed
to do that—in particular, given that the downstream people can’t even
keep constant the number of items they complete at each time step?
Unfortunately, life isn’t always fair.

Hoping for the Best
What are we to do? One way of approaching this problem is to realize
that, in the steady state, the number of units flowing into the buffer
must equal the number of units flowing out. We can therefore measure
the average number of units leaving the buffer at each time step and
then make sure we release the same number of units into the ready
pool. In the long run, things should just work out. Right?
Figure 1-2 (top) shows what happens when we do this. The number
of units in the buffer (the queue length) fluctuates wildly—sometimes
exceeding 100 units and other times dropping down to zero. If the
space in the buffer is limited (which may well be the case if we are
dealing with a physical processing plant), then we may frequently be
Hoping for the Best

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