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Studies in Systems, Decision and Control 139

Bijnan Bandyopadhyay
Abhisek K. Behera

Event-Triggered
Sliding Mode
Control
A New Approach to Control System
Design
Studies in Systems, Decision and Control

Volume 139

Series editor
Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland
e-mail: kacprzyk@ibspan.waw.pl
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Bijnan Bandyopadhyay Abhisek K. Behera
•

Event-Triggered Sliding
Mode Control
A New Approach to Control System Design

123
Bijnan Bandyopadhyay Abhisek K. Behera
Systems and Control Engineering Systems and Control Engineering
Indian Institute of Technology Bombay Indian Institute of Technology Bombay
Mumbai, Maharashtra Mumbai, Maharashtra
India India

ISSN 2198-4182 ISSN 2198-4190 (electronic)

Studies in Systems, Decision and Control
ISBN 978-3-319-74218-2 ISBN 978-3-319-74219-9 (eBook)
https://doi.org/10.1007/978-3-319-74219-9
Library of Congress Control Number: 2017963863

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Dedicated to my Ph.D. guide, late Dr. S. S.
Lamba, former Professor, IIT Delhi
Bijnan Bandyopadhyay

Dedicated to my parents and gurus

Abhisek K. Behera
Preface

Variable structure systems using sliding mode control (SMC) was originated in
USSR in the late fifties to stabilize the uncertain dynamical systems with relay as a
feedback control law. It gained popularity outside USSR only after the late sev-
enties due to a survey paper in English by Prof. Vadim I. Utkin. Since then, it has
become now a well-established robust control technique to deal with the uncer-
tainties in the plant, and achieving the system stability. A vast number of scientific
publications and the practical applications of this control technique have made
SMC as an important area in the control literatures.
The attention on design of SMC in discrete-time domain was paid by many
researchers soon after the importance of microprocessor and computer/processors
are realized in control applications in the early eighties. The first and important
observation in the discrete-time design is that no exact sliding mode is achieved as
in the continuous-time counterpart. A new notion of sliding mode is introduced
which is known a quasi-sliding mode (QSM). This has led to the development of
discrete-time SMC as an important area in SMC due to its practical importance.
Many design approaches have been proposed to improve the performance of SMC
for the sampled-data system.
In this monograph, a new approach to design SMC is presented using a novel
implementation strategy, namely event-triggering. In this strategy, the control is
updated whenever a certain stabilizing condition is violated, and hence, the system
stability is always maintained. Due to the need-based control strategy, it finds a
major application in spatially distributed control systems to reduce the communi-
cation among different sensor and actuator ends. So, the resources of the systems
are optimally used. The event-triggering-based design of SMC not only gives the
robust performance but also ensures minimal use of resources in the control system.
This monograph presents the recent results on event-triggered SMC for robust
stabilization of dynamical systems. In the first part of the monograph, the prelim-
inaries on sampled-data systems with an introduction to event-triggered control are
presented to familiarize the readers the event-triggering-based design of control
law. In addition to this, a brief introduction to SMC and its design are also dis-
cussed. Then, the design of event-triggered SMC for both linear and nonlinear

vii
viii Preface

systems is given in Chaps. 2 and 3. The event-triggered SMC is presented for linear
systems guaranteeing the semi-global and global stability in Chap. 2. However,
only local stabilization result is discussed for the nonlinear systems, which is
presented in detail in Chap. 3. In the event-triggered control, the state trajectory is
continuously measured to evaluate the triggering condition which may not be
practical in all applications. So, Chaps. 4 and 5 present few variants of
event-triggered control, namely self-triggered and discrete event-triggered control,
respectively. In the self-triggering strategy, the triggering strategy is developed
without using the continuous state measurements. On the other hand, the periodic
state measurements are used in discrete event-triggered control and the control is
updated when it is violated at some periodic instants. In recent time, there has been
a considerable amount of interest in stability of quantized control system. The final
chapter presents the design of event-triggered SMC with quantized state mea-
surements. It is our belief that the material of the monograph would serve its
purpose and explore the new challenges on the topic.
This work would have not been completed without the support and encour-
agement from many of our friends and colleagues. The authors would like to thank
Indian Institute of Technology Bombay for providing the conducive environment to
carry out the research reported in the monograph. Finally, we extend our gratitude
to our family for their love, support and understanding throughout the process of
this endeavour.

Mumbai, India Bijnan Bandyopadhyay

December 2017 Abhisek K. Behera
Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Computer-Controlled Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Basic Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Design Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Event-Triggered Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Preliminary Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Stability of Event-Triggered Systems . . . . . . . . . . . . . . . . 8
1.2.3 Need of Event-Triggered Control . . . . . . . . . . . . . . . . . . . 11
1.3 Sliding Mode: An Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 Dynamics During Sliding Mode . . . . . . . . . . . . . . . . . . . . 14
1.3.2 Design of Sliding Mode Control . . . . . . . . . . . . . . . . . . . . 17
1.4 Discrete-Time Sliding Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4.1 Switching-Based Reaching Law . . . . . . . . . . . . . . . . . . . . 22
1.4.2 Switching-Free Reaching Law . . . . . . . . . . . . . . . . . . . . . 23
1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Event-Triggered Sliding Mode Control for Linear Systems . . . . . . . 27
2.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Event-Triggered Sliding Mode Control . . . . . . . . . . . . . . . . . . . . 28
2.2.1 Stability of Sliding Motion . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.2 Stability of Event-Triggered System . . . . . . . . . . . . . . . . . 33
2.2.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Global Event-Triggered Sliding Mode Control . . . . . . . . . . . . . . . 37
2.3.1 Global Event-Triggering Rule . . . . . . . . . . . . . . . . . . . . . . 37
2.3.2 Design of Sliding Mode Control . . . . . . . . . . . . . . . . . . . . 38
2.3.3 Global Stability of Event-Triggered System . . . . . . . . . . . . 41
2.3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

ix
x Contents

2.4 Event-Triggered Sliding Mode Control for Multivariable

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4.1 Event-Triggered Design of SMC . . . . . . . . . . . . . . . . . . . 45
2.4.2 Event-Triggering Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3 Event-Triggered Sliding Mode Control for Nonlinear Systems . . . . . 55
3.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1.1 Design of Sliding Mode Control . . . . . . . . . . . . . . . . . . . . 56
3.2 Event-Triggered Sliding Mode Control . . . . . . . . . . . . . . . . . . . . 57
3.2.1 Stability of Sliding Motion . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Event-Triggering Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3.1 Design of Event-Triggering Scheme with Constraints . . . . 63
3.4 Event-Triggering with Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5.1 Without Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.5.2 With Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.7 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4 Self-Triggered Sliding Mode Control for Linear Systems . . . . . . . . . 77
4.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Self-Triggering Scheme Without Delay . . . . . . . . . . . . . . . . . . . . 78
4.2.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3 Self-Triggering Scheme with Delay . . . . . . . . . . . . . . . . . . . . . . . 81
4.3.1 Design of Self-Triggered Sliding Mode Control . . . . . . . . . 85
4.3.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5 Discrete Event-Triggered Sliding Mode Control for Linear
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Discrete-Time Sliding Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2.1 Bartoszewicz’s Reaching Law . . . . . . . . . . . . . . . . . . . . . 96
5.2.2 Design of Discrete-Time Sliding Mode Control . . . . . . . . . 97
5.3 Discrete Event-Triggered Sliding Mode Control: State
Feedback Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... 98
5.3.1 Event-Triggered Bartoszewicz’s Reaching Law . . ....... 99
Contents xi

5.3.2 Event-Triggering Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.3.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4 Discrete Event-Triggered Sliding Mode Control: Output
Feedback Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.4.1 Multirate Output Feedback Technique . . . . . . . . . . . . . . . . 103
5.4.2 Multirate-Based Event-Triggered Discrete-Time
Sliding Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6 Event-Triggered Sliding Mode Control with Quantized State
Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.1.1 Quantizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.2 Design of Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3 Design of Event-Triggered Sliding Mode Control . . . . . . . . . . . . . 117
6.3.1 Design of Event-Triggering Rule . . . . . . . . . . . . . . . . . . . 119
6.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Acronyms

A-D Analog-to-digital
D-A Digital-to-analog
DTSM Discrete-time sliding mode
ETCS Event-triggered control systems
FOS Fast output sampling
ISS Input-to-state stable
LTI Linear time-invariant
MIMO Multiple-input multiple-output
MROF Multirate output feedback
QSM Quasi-sliding mode
SISO Single-input single-output
SMC Sliding mode control
VSS Variable structure system

xiii
Symbols

R Set of real numbers

R
0 Set of nonnegative real numbers
Z Set of integers
Z
0 Set of nonnegative integers
Rn n-dimensional vector space over R
ða  bÞðxÞ Composition of the two functions a and b
xy Dot product of any two vectors x and y in Rn
P [ ð
Þ0 Positive (semi)-definite matrix P
Q⊤ Transpose of any matrix Q
jj Absolute value of a scalar variable ‘’
k: k1 1-norm of a finite-dimensional vector ‘’
k: k Euclidean (2-) norm of a finite-dimensional vector ‘’ or a matrix ‘’
of appropriate dimension
kmaxðminÞ fg Largest (smallest) eigenvalue of a square matrix ‘’
rf ðxÞ Gradient of a real-valued function f ðxÞ
sup (inf) Supremum or least upper bound (infimum or greatest lower bound)
b xc Floor function that returns largest integer less than or equal to x
ln Natural logarithm with base e (¼ 2:71828)
K Set of strictly increasing and continuous real-valued functions
defined on the nonnegative interval with zero at zero
K1 Set of unbounded class-K functions
sign Signum function
FðxÞ Set-valued map of the vector field f ðxÞ at the point of discontinuity x
in Filippov’s inclusion
co Convex closure
Bd ðxÞ Open ball of radius d centred at x
lðÞ Lebesgue measure of a set ‘’
v Sensitivity of the quantizer
M Saturation level of the quantizer

xv
xvi Symbols

Ti Inter-event time/time interval between two consecutive triggering

instants
s Constant sampling period for the discrete-time systems
N An integer greater than or equal to the observability index of the
system
Chapter 1
Introduction

This chapter briefly introduces the readers to the preliminary ideas on design and
analysis of computer-controlled systems and then sliding mode control (SMC). In
general, computer-controlled systems consists of both continuous and discrete-time
systems that interact among themselves through the feedback channel to achieve cer-
tain objectives. Different available classical techniques, namely emulation, discrete-
time and hybrid approaches, are summarized here with their own advantages and
disadvantages. In almost all these techniques, the periodic sampling interval is often
used to design and analyse the sampled-data systems for its simplicity and easier in
design. On the other hand, aperiodic control implementation is desired in sampled-
data systems to reduce the periodic computational burden and cost associated with
the implementation. However, this introduces few difficulties in analysing closed
loop system stability. A novel sampling strategy known as event-triggered control
is introduced here where the control is updated whenever it is demanded. In this
technique, the time instants for updating the control signal is determined using some
rule that ensures the stability of the system. So, this strategy maintains the system
stability while reducing extra burden on the system.
The design of SMC is also presented in this chapter to familiarize the readers. This
is a robust controller that stabilizes the plant in the presence of external disturbances.
The sliding motion and SMC are briefly elaborated to understand sliding mode
with discontinuous control action. This is followed by the design of SMC for linear
systems. The discrete realization of SMC, unfortunately, does not yield sliding motion
exactly due to discrete nature of control is also discussed. Some control design
techniques are reviewed for discrete-time sliding mode.

© Springer International Publishing AG, part of Springer Nature 2018 1

B. Bandyopadhyay and A. K. Behera, Event-Triggered Sliding Mode Control, Studies
in Systems, Decision and Control 139, https://doi.org/10.1007/978-3-319-74219-9_1
2 1 Introduction

1.1 Computer-Controlled Systems

Computer-controlled systems, in general, are broadly defined as a control system that

allows interaction of both analog system and digital controller through computer
or any other digital platform. Such systems are ubiquitous in almost all fields of
engineering due to rapid advancement of digital technology. The plant represents
the continuous-time system, while the control signal is a discrete in nature that is
applied to the plant. The control signal is updated at the discrete instants only and,
however, is held constant in between two consecutive sampling instants. This is why,
it is also called as sampled-data system. The plant dynamics evolves in open loop
manner between two discrete instants as there is no control variable. So, the interest
is mainly focused on the plant behaviour at the discrete instants only leading to
the so-called discrete-time systems. In other words, the system dynamics evolves
at discrete instants with the control input applied at these instants. This is not only
easy to analyse but also one of the simplest ways of implementing the control law to
achieve a certain objective compared to its analog counterparts.
In actual practice, computer-controlled systems have facilitated the control design
problems to a great extent by introducing the digital control signal. However, it should
not be thought of as an ideal control design and implementation scenario for a physi-
cal plant. Rather, it is a limiting approach of the analog control system (both plant and
control evolve in continuous-time) and all the analyses are carried out for the analog
control system in sampled-data fashion. The control law is designed for any plant
using system dynamics which may be defined on continuous- and/or discrete-time
domain. However, in the present case only, we focus on continuous-time plant. So,
the control is designed from the continuous time plant which achieves desired perfor-
mance of the system when it is implemented in continuous manner. But, in discrete
implementation, the continuous-time performance may also be achieved if it is imple-
mented at a faster sampling rate. So, the computer-controlled system yet provides an
alternative to analog control implementation subject to some desired performance.
In spite of this, there are numerous challenges associated with computer-controlled
systems that make the researchers to rethink possible new techniques for control
implementation and design philosophy.

1.1.1 Basic Architecture

The basic architecture of a computer-controlled system is shown in Fig. 1.1. The

plant, which evolves in continuous-time, interacts with the controller only at discrete
instants. The plant states are sampled at discrete instants and converted into a digital
signals. The whole process of converting analog-to-digital conversion is represented
by the analog-to-digital (A-D) converter. These digital signals are then processed
through a control algorithm to generate a new set of digital signals, known as discrete
control signals. We see that there is same number of discrete control signals as the
1.1 Computer-Controlled Systems 3

Fig. 1.1 Basic architecture

D-A Analog Plant A-D
of a computer-controlled
systems

Clock

Control Law/
Algorithm

samples of plant states. Now, the discrete signal is processed by digital-to-analog (D-
A) converter to generate a continuous-time signal between two consecutive sampling
instants. But, this continuous signal is an approximation of analog control signal. The
overall process from A-D to D-A is coordinated by a clock that synchronizes all the
tasks carried out at A-D, control synthesis and D-A converter. Here, the control signal
may be designed in the continuous-time frame work or in discrete domain depending
on the design requirements.
Both the converters, often, operate in periodic time interval in the sampled-data
system. The information is sampled at time instants known as sampling instant, and
the interval between successive sampling instants is called as sampling period. For
every sampling instant, the control signal is computed and the same is applied to the
plant by its approximate analog signal through D-A converter. The control signal is
held constant in every sampling interval making the system as open loop, and hence,
the system evolves in open loop manner. This is one of the differences that makes
the computer-controlled systems different from other feedback systems. There has
been many techniques available to explore the analysis and design of such system.
But, yet the full potential of this system needs to be investigated for effective use of
computer-controlled systems. For instance, the effect of sampling interval on system
performance, the absence of clock that synchronizes both A-D and D-A converters,
etc. are need to be investigated. Of all these, sampling interval variation is one of
the most important and challenging problems in computer-controlled system. Too
fast sampling is sometime unnecessary, while the slow sampling may deteriorate the
system performance. So, it is always desirable to have an optimal sampling interval
that stabilizes the computer-controlled system.

1.1.2 Design Techniques

Many techniques have been used to analyse and design of computer-controlled

systems. The closed loop system is hybrid in nature involving both continuous
and discrete dynamics; however, the standard available technique deals with either
continuous-time design or discrete-time design. That means design the control either
4 1 Introduction

using continuous-time or discrete-time model and apply it to the continuous-time

plant. In the same manner, the stability analysis of the system is carried out using
either of these models depending on control signal. Broadly speaking, there are three
methods available for analysing the computer-controller system, namely emulation-
based approach, discrete-time approach and hybrid system approach. Also, in most
applications the constant sampling period is used for implementing the discrete con-
troller.

1.1.2.1 Emulation-Based Approach

It may also be seen as continuous-time approach design to sampled-data system. Here,

the performance specifications are in continuous-time domain since the plant and
controller both are in continuous-time domain. The basic idea is first to ignore the A-D
and D-A converters in Fig. 1.1 and follow the design steps of any stabilizing controller.
The continuous-time controller is then approximated by replacing derivatives with
finite-differences and continuous-time signals with sampled values at that instants.
Though it is an approximation, the satisfactory performance of the sampled-data
system is still achieved by this approximation. Here, the controller designed from
continuous-time plant emulates the behaviour of continuous-time plant in spite of
discrete implementations, and hence, this is known as emulation-based approach.
However, the main issue in this approach is that the sampling period is not taken
into account in the design of controller. It is very natural that the emulated controller
gives the stability of sampled-data system for a range of sampling period only while
it destabilizes for other values of sampling period.

1.1.2.2 Discrete-Time Approach

This approach is simple and is based on the discrete-time model of the plant. The
continuous plant interacts with digital controller through A-D and D-A converters.
Thus, the controller sees the plant as discrete-time model through these converters.
The discrete-time representation of the plant is obtained by combining the plant with
A-D and D-A converters. There are numerous approaches available for discretizing
the plant for a given sampling period, and the popular among them are Euler dis-
cretization, zero-order hold (ZOH) discretization, etc. The closed loop response is
analysed only at discrete instants since it ignores the inter-sampling behaviour of
the plant. On the other hand, the stability of the system is analysed by ignoring the
dynamics between two sampling instants. The wide application of this technique is
found in many slow processes where it is enough to study the system behavior only
at some periodic intervals. However, the well-known shortcomings of this technique
are that no inter-sampling dynamics of the plant can be analysed and selection of a
suitable sampling interval that captures the undesirable phenomenon in the system.
1.1 Computer-Controlled Systems 5

1.1.2.3 Hybrid Approach

As the name suggests, in this technique, both the continuous and discrete behaviours
are analysed for the sampled-data system without representing the system by some
approximated system dynamics. This is why, it is also known as direct design
approach to sampled-data system. Due to this hybrid nature of the system, the design
and stability methods are complicated than earlier two methods.
It is to be noted that variable sampling period may also be used for designing
the controller for sampled-data systems. However, it involves many design issues
for analysing the stability due to restricted mathematical tools and/or time-varying
nature of system dynamics. Nevertheless, many attempts have been made for sta-
bilizing the sampled-data systems with aperiodic control sampling process. Event-
triggering strategy is one of such techniques that generates nonuniform sampling
instants while ensuring system stability. In this book, only event-triggered technique
will be emphasized for computer-controlled systems.

1.2 Event-Triggered Control

Event-triggering strategy is a control implementation technique that is motivated

from the Lebesgue sampling. In this case, sampling period is not constant but is
determined by the evolution of system trajectory satisfying some stability condi-
tion. To understand the concept of event-triggered control, first Lebesgue sampling
technique is discussed.
In classical sampled-data system, generally constant sampling period is chosen
and the control is implemented once this constant time period is elapsed. This is
generally referred as Reimann sampling as shown in Fig. 1.2a. For any constant hR >
0, the sampling of the continuous-time signal ψ(t) takes place at every hR intervals
of time. It does not monitor the evolution of signal ψ(t). As a result of this, the
important concern in Reimann sampling is the proper selection of sampling period
to capture the transient behaviour of the plant. On the other hand, the signal may
also be sampled at the time instants when state evolution from its immediate past
sampled value crosses a certain threshold value (say, hL ) as the time does in Reimann
sampling. This is known as Lebesgue sampling which is shown in Fig. 1.2b. Though
both the sampling techniques are different, these have similarity in the sampling
mechanism. In the former, the time is measured while state is monitored in the latter
case. However, in doing so many advantages are obtained in the case of Lebesgue
sampling. For example, it is not necessary most of the time to update the control
signal frequently at periodic interval. So, Lebesgue sampling gives sampling instant
whenever it is needed subject to some satisfactory system performance.
In case the function ψ(t) is a finite-dimensional vector for any fixed t, the sampling
instant may be decided by observing the individual state evolution of the vector-
valued function. Then, for any given constants hLi > 0 with i = 1, 2, . . . , n, the
individual state evolution, denoted by hLi (t), is observed for generating sampling
6 1 Introduction

Fig. 1.2 Comparison of ψ (t)

Reimann and Lebesgue
samplings

hR t

(a) Reimann sampling

ψ (t)

hL

t
(b) Lebesgue sampling

instant of the corresponding state. However, it is very complicated and difficult to

analyse the stability of closed loop system. Another school of thought is to sample all
the states simultaneously whenever certain condition is violated. Thus, this strategy
does not necessitate individual sampling of the state at different time instants. Due to
this, the latter is more convenient for implementing practically than the earlier one.
1.2 Event-Triggered Control 7

1.2.1 Preliminary Idea

Event-triggered control is one of such techniques that generates the sampling instant
(also called as triggering instant) for sampling and updating the control signal. To pro-
vide a preliminary idea on event-triggered control, we consider a nonlinear dynamical
system

ẋ = f (x, u), x(0) = x0 ∈ Rn (1.1)

where the function f (·, ·) is Lipschitz with respect to both the arguments u ∈ Rm . Let
there exists a continuous feedback control law u(x) = π(x) such that the dynamics

ẋ = f (x, π(x))

is asymptotically stable. It is assumed that the control is implemented digitally to

the plant. So, the control signal π(x) is computed for every sampling instant and is
applied to the plant at these discrete instants. Then, the system becomes open loop
between two consecutive sampling instants. However, due to this, the discrete error
is introduced in the plant, defined by e(t) = x(ti ) − x(t) with e(ti ) = x(ti ) − x(ti ) = 0
where t ∈ [ti , ti+1 ). This error appears in the plant due to discrete implementation of
continuous-time control, but its value is zero if the control is continuously updated
as in analog implementation.
Further, we assume that the system (1.1) is input-to-state stable (ISS) with respect
to the error e(t). That means there exists a continuously differentiable Lyapunov
function V : Rn → R≥0 such that

a(x) ≤ V (x) ≤ a(x) (1.2)

∇V (x) · f (x, π(x + e)) < −a(x) + γ (e) (1.3)

for some class-K∞ 1 functions a, a, a, and class-K function γ . Here, the notation
‘·’ denotes inner (scalar) product. Event-triggering strategy is developed for deter-
mining the sampling instants such that desired stability is achieved. In this case, the
asymptotic stability of the system is desired with the discrete implementation of the
control law. So, the obvious condition for which this holds is γ (e) < σ a(x) for
some σ ∈ (0, 1). This can be simplified further, by assuming a−1 and γ are Lipschitz
on some compacts, as σ x > Le e, where Le is an appropriate constant. Thus, the
triggering instant may be generated by executing the following,

ti+1 = inf {t > ti : σ x(t) ≤ Le e(t)} . (1.4)

1 Any function a is said to be class-K if it is continuous, strictly increasing, zero at zero. Again, it
is said to be class-K∞ if it belongs to class-K and is unbounded. Clearly, class-K∞ functions are
subsets of class-K functions.
8 1 Introduction

This is known as triggering rule for event-triggered control π(x). It is seen that
this ensures Le e < σ x which implies that γ (e) < σ a(x) also holds. This
implies from (1.2) and (1.3) that

V̇ < −(1 − σ )a(x)

 
≤ −(1 − σ ) a ◦ a−1 (V )
= −(1 − σ )a (V )
<0

where K∞ a := a ◦ a−1 which is the composition of two functions a and a−1 .

This shows that closed loop system is asymptotically stable with the control applied
at discrete instants. Moreover, in the event-triggering technique the inter-sampling
behaviour is considered for stability of the closed loop system.
It is worth mentioning here that the event-triggered control scheme not only guar-
antees system stability but also ensures the convergence of inter-sampling behaviour.
From the above, it is observed that the Lyapunov function decreases continuously
and goes to zero as the time tends to infinity. This is one of the important properties
of the event-triggered control system (ETCS).

1.2.2 Stability of Event-Triggered Systems

In this technique, the triggering instants are generated by the triggering rule at which
the control signal is updated and this results the closed loop system stability. However,
it might happen that the control signal is not updated at the triggering instant when
triggering instants are too close to each other. This demands fast execution of control
tasks, or even in worst-case continuous-time like execution which is not possible
by digital processor. It may be noted here that in periodic execution such situation
does not arise due to each sampling/triggering instant that occurs after every constant
sampling period.
Let {ti }i∈Z≥0 be sequence of triggering instants generated by some stabilizing
triggering rule. We define Ti = ti+1 − ti as the inter-event/execution time for any
given triggering sequence {ti }i∈Z≥0 . For stability of the event-triggered system, the
inter-event time must be strictly greater than zero, i.e. Ti > aT for all i ∈ Z≥0 and
some positive constant aT . This guarantees the Zeno-free execution of triggering
sequence. The positive inter-event time ensures control is updated after every finite-
time interval only. This is essential for the processor to execute the control task and
update the control signal. In other words, it can be said that {ti }i∈Z≥0 is an increasing
sequence, i.e. t0 < t1 < t2 < · · · such that ti+1 > ti + aT . Such a triggering sequence
is feasible for implementing the control practically to ensure the stability of closed
loop system. The triggering instants generated by some triggering rule that is not
necessarily satisfying the above property would make the event-triggered system
unstable.
1.2 Event-Triggered Control 9

Example 1.1 Consider a scalar nonlinear control system as

ẋ = x2 + u

where x ∈ {[−c, c] : c ∈ R>0 } which is a compact set. Any stabilizing controller can
be designed for the above system to ensure the asymptotic stability. Let u = −x2 −kx
be a feedback control which ensures the asymptotic stability of the system with k > 0.
This control is applied to the above system at discrete instants only such that closed
loop system is stable. So, the discrete-time control is given as

u(t) = −x2 (ti ) − kx(ti ), t ∈ [ti , ti+1 ), i ∈ Z≥0 .

It can be shown that the closed loop system with the above discrete control is
ISS with respect to the error. Choose V (x) = 21 x2 . Then, with some calculation, we
arrive at
 
V̇ (x(t)) = x(t) x2 (t) − x2 (ti ) − kx(ti )
= −x(t) (x(t) + x(ti )) e(t) − kx(t)x(ti )
≤ 2c|e(t)||x(t)| − kx(t)(e(t) + x(t))
= 2c|e(t)||x(t)| − kx(t)e(t) − kx2 (t)
≤ 2c|e(t)||x(t)| + k|x(t)||e(t)| − kx2 (t)
≤ (2c + k)|e(t)||x(t)| − kx2 (t).

Here, we use the fact |x| ≤ c and x(ti ) = e(t) + x(t). Now applying Young’s
inequality2 to the first term (ε = 2c+k
k
), we obtain

k (2c + k)2
V̇ (t) ≤ − |x(t)|2 + |e(t)|2
2 2k
= −a(|x(t)|) + γ (|e(t)|)

where a(r) and γ (r) are given as

k 2 (2c + k)2 2
a(r) = r and γ (r) = r .
2 2k
Thus, the triggering rule is designed according to (1.4) which stabilizes the system
and is given by

2 Young’s inequality for exponent two states that for any nonnegative real numbers p, q and every

ε > 0, the following holds

p2 εq2
pq ≤ + .
2ε 2
10 1 Introduction
 
k
ti+1 = inf t > ti : σ |x(t)| ≤ |e(t)|
2c + k

for some σ ∈ (0, 1). This triggering rule ensures |e(t)| < σ 2c+k k
|x(t)| for all time
and thus implies that V̇ < 0 for all time. Hence, the closed loop system becomes
asymptotically stable even if the control is applied at the discrete instants generated
by the triggering sequence {ti }i∈Z≥0 . It can also be established that the triggering rule
does not have a Zeno triggering sequence. For x ∈ {[−c, c] : c ∈ R>0 }\{0}, one can
write
 
d |e(t)| 1 d d
= 2 |e(t)| |x(t)| − |x(t)| |e(t)|
dt |x(t)| x (t) dt dt
  
1  
≤ 2  d e(t) |x(t)| − d |x(t)| |e(t)|
x (t)  dt  dt
    
1    
≤ 2  d x(t) |x(t)| +  d x(t) |e(t)|
x (t)  dt   dt 
 
1 d 
= 2 (|x(t)| + |e(t)|)  x(t) .
x (t) dt

Now, using the fact |x| ≤ c and the control law u(t) = −x2 (ti ) − kx(ti ) for all
t ∈ [ti , ti+1 ) and i ∈ Z≥0 , one can easily obtain that
 
d   
 x(t) = x2 (t) − x2 (ti ) − kx(ti )
 dt 
= |−(x(t) + x(ti )) e(t) − k (e(t) + x(t))|
= |x(t) + x(ti )| |e(t)| + k |e(t) + x(t)|
≤ 2c|e(t)| + k|e(t)| + k|x(t)|
< (2c + k)|e(t)| + (2c + k)|x(t)|
= (2c + k)(|e(t)| + |x(t)|).

Using this in the following, it can be written as

d |e(t)| 1
< 2 (|x(t)| + |e(t)|)2 (2c + k)
dt |x(t)| x (t)
 2
|e(t)|
= (2c + k) +1 .
|x(t)|

The solution to the above differential inequality can be obtained using comparison
Lemma. Then, the solution to the above differential can be obtained as

|e(t)|
≤ μ(t), t ∈ [ti , ti+1 )
|x(t)|
1.2 Event-Triggered Control 11

where μ(t) satisfies the differential equation μ̇ = (2c + k)(1 + μ)2 with the initial
|e(ti )|
condition |x(ti )|
= μ(ti ) = 0. Then, corresponding to triggering mechanism for this
system, the lower bound of the inter-event time is obtained as

σk
Ti > > 0.
(2c + (1 + σ )k)(2c + k)

This shows that the inter-event time is lower bounded by a positive quantity which
is strictly greater than zero. Indeed, it is necessary to eliminate the Zeno execution
of triggering sequence and ultimately to guarantee the system stability.
In the numerical simulation, the values of c, k and σ are selected as 5, 1 and
0.85, respectively. The initial condition is taken as x(0) = 4. The response of the
system is shown in Fig. 1.3. It is seen that state trajectory goes to zero as time tends
to infinity. The varying sampling interval or inter-event time generated by executing
the triggering rule is shown in Fig. 1.4. It is seen that the inter-event time is lower
bounded from zero which is given by 0.0065 and it increases to a value as high as
0.072. The plot of control signal is also shown in Fig. 1.5. As the sampling interval
increases, the control signal also remains constant until the next sampling instant is
generated. The plot of Lyapunov function is given in Fig. 1.6. It is seen that event-
triggered control implementation achieves asymptotic stability of the closed loop
system with guaranteed convergence of inter-sampling behaviour of the system.

1.2.3 Need of Event-Triggered Control

Event-triggered control is one of the aperiodic control implementation strategies in

digital platform that ensures closed loop system stability. Unlike periodic sampling,

Fig. 1.3 Response of the

system
Another random document with
no related content on Scribd:
1 lb. 3 oz.
The dimensions of a young male shot in autumn were as follows:
To end of tail 24 inches, to end of wings 24, to end of claws 29;
extent of wings 26; wing from flexure 10 1/2. Weight 1 lb. 1 1/2 oz.

In dissecting this bird, the extreme compression of the body strikes

one with surprise, its greatest breadth being scarcely an inch and a
half, although it is capable of being much dilated. The great length
and thickness of the neck are also remarkable; but these
circumstances are not peculiar to the present species, being equally
observed in many other Herons. On the roof of the mouth are three
longitudinal ridges; the aperture of the posterior nares is linear, with
an oblique flap on each side; the lower mandible is deeply concave,
its crura elastic and expansile; the tongue 2 1/12 inches long,
sagittate at the base with a single very slender papilla on each side,
trigonal, tapering, flattened above; the width of the mouth is 10
twelfths; but the pharynx is much wider. The œsophagus, a b c,
which is fifteen inches long, is very wide, having at its upper part,
when inflated, a diameter of 2 inches, but gradually contracting to 1/2
inch at its entrance into the thorax, and again expanding to 1 inch. Its
walls are extremely thin, and when contracted, its mucous coat
forms strongly marked longitudinal plaits. The proventriculus is very
wide, its glandules oblong and arranged in a belt 10 twelfths in
breadth. The stomach, e, is of moderate size, membranous, that is
with its muscular coat very thin, and not forming lateral muscles; its
tendinous spaces large and round, its inner coat smooth and soft; its
greatest diameter 1 inch. There is a small roundish pyloric lobe, as in
other Herons. Both lobes of the liver lie on the right side of the
proventriculus; one, i, being 1 inch 10 twelfths, the other, j, 1 inch 2
twelfths long; the gall-bladder large, 11 twelfths long. The intestine is
long and very slender, measuring 4 feet 7 inches, with a diameter of
only 2 twelfths at its upper part, and 1 1/2 twelfth at the lower, when
inflated; the rectum 4 inches long, and 4 twelfths in diameter, its
anterior extremity rounded, and having a minute papilliform
termination, only 1 twelfth long.

The trachea, which is 12 1/2 inches long, differs from that of ordinary
Herons in being much compressed, especially at its upper and lower
extremities; the middle part being less so. It is also proportionally
wider, and its rings are narrower. At the top its diameter is 5 twelfths,
at the middle 4 1/4 twelfths, towards the lower part 4 3/4 twelfths, at
the end 4 1/4 twelfths. The rings are osseous, in number 180; the five
lower divided in front and behind, and much arched, the last
measuring half an inch in a direct line between its extremities. The
bronchi are in consequence very broad at their commencement, but
gradually taper, and are composed of about 18 half rings. The
contractor muscles are inconspicuous, the sterno-tracheal slender;
and there is a single pair of inferior laryngeal, going to the first
bronchial ring. The aperture of the glottis is 8 twelfths long, without
any papillæ, but with a deep groove behind, and two thin-edged
flaps.
In the digestive organs of this bird, there is nothing remarkably
different from that of other Herons. The stomach contained remains
of fishes and large coleopterous insects. The examination of the
trachea, bronchi, and lungs, would not lead us to suppose that its cry
is of the curious character represented, although it certainly would
induce us to believe it different from that of ordinary Herons, which
have the trachea narrower, round, and with broader and more bony
rings.
Although in external appearance and habits it exhibits some affinity
to the Rails, its digestive organs have no resemblance to theirs.
An egg presented by Dr Brewer of Boston measures two inches in
length by one inch and a half, and is of a broadly oval shape, rather
pointed at the smaller end, and of a uniform dull olivaceous tint.
 BREWER’S DUCK.

Anas Breweri.
PLATE CCCXXXVIII. Male.

The beautiful Duck from which I made the drawing copied on the
plate before you, was shot on Lake Barataria, in Louisiana, in
February 1822. It was in company with seven or eight Canvass-back
Ducks. No other individuals of the species were in sight at the time,
and all my efforts to procure another have been ineffectual.
You will see that this curious bird is named in the plate “Anas
glocitans,” the descriptions of that species having induced me to
consider it identical with this. But on comparing my drawing with
specimens in the Museum of the Zoological Society of London, I
found that the former represents a much larger bird, which, besides,
is differently coloured in some of its parts. The individual figured was
a male; but I have some doubts whether it had acquired the full
beauty of its mature plumage, and I considered it at the time as a
bird of the preceding season.
In form and proportions this bird is very nearly allied to the Mallard,
from which it differs in having the bill considerably narrower, in
wanting the recurved feathers of the tail, in having the feet dull
yellow in place of orange-red, the speculum more green and duller,
without the white bands of that bird, and in the large patch of light
red on the side of the head. It may possibly be an accidental variety,
or a hybrid between that bird and some other species, perhaps the
Gadwall, to which also it bears a great resemblance.
Bill nearly as long as the head, higher than broad at the base,
depressed and widened towards the end, rounded at the tip, the
lamellæ short and numerous, the unguis obovate, curved, the nasal
groove elliptical, the nostrils oblong.
Head of moderate size, oblong, compressed; neck rather long and
slender; body full, depressed. Feet short, stout, placed behind the
centre of the body; legs bare a little above the joint; tarsus short, a
little compressed, anteriorly with small scutella, laterally and behind
with reticulated angular scales. Hind toe very small, with a narrow
free membrane; third toe longest, fourth a little shorter; claws small,
arched, compressed, acute.
Plumage dense, soft, and elastic; of the hind head and neck short
and blended; of the other parts in general broad and rounded. Wings
of moderate length, acute; tail short, graduated.
Bill dull yellow, slightly tinged with green, dusky along the ridge. Iris
brown. Feet dull yellow, claws dusky, webs dull grey. Head and
upper part of the neck deep glossy green; but there is an elongated
patch of pale reddish-yellow, extending from the base of the bill over
the cheek to two inches and a quarter behind the eye, and meeting
that of the other side on the chin; the space immediately over and
behind the eye light dull purple. A narrow ring of pale yellowish-red
on the middle of the neck; the lower part of the neck dull brownish-
red, the feathers with a transverse band of dusky, and edged with
paler. The upper parts are dull greyish-brown, transversely undulated
with dusky; the smaller wing-coverts without undulations, but each
feather with a dusky bar behind another of light dull yellow; first row
of smaller coverts tipped with black; primaries and their coverts, light
brownish-grey; some of the outer secondaries similar, the next five or
six duck-green, the next light grey with a dusky patch toward the
end. The rump and upper tail-coverts black, as are the parts under
the tail, excepting two longitudinal white bands; tail-feathers light
brownish-grey, edged with whitish. All the rest of the lower parts are
greyish-white tinged with yellow, beautifully undulated with dusky
lines, on the middle of the breast these lines less numerous, and
each feather with a reddish-grey central streak.
Length to end of tail 23 inches, to end of claws 24; extent of wings
39; bill along the ridge 2 1/2, along the edge of lower mandible 2 1/8;
tarsus 1 1/8, middle toe 2, its claw 5/12; hind toe 3/8, its claw 1/8.
Weight 2 lb. 9 oz.
I have named this Duck after my friend Thomas M. Brewer of
Boston, as a mark of the estimation in which I hold him as an
accomplished ornithologist.
 LITTLE GUILLEMOT.

Uria Alle, Temm.

PLATE CCCXXXIX. Male and Female.

This interesting little bird sometimes makes its appearance on our

eastern coasts during very cold and stormy weather. It does not
proceed much farther southward than the shores of New Jersey,
where it is of very rare occurrence. Now and then some are caught
in a state of exhaustion, as I have known to be the case especially in
Passamaquody Bay near Eastport in Maine, and in the vicinity of
Boston and Salem in Massachusetts.
In the course of my voyages across the Atlantic, I have often
observed the Little Guillemots in small groups, rising and flying to
short distances at the approach of the ship, or diving close to the
bow and reappearing a little way behind. Now with expanded wings
they would flutter and run as it were on the surface of the deep;
again, they would seem to be busily engaged in procuring food,
which consisted apparently of shrimps, other crustacea, and
particles of sea-weeds, all of which I have found in their stomach. I
have often thought how easy it would be to catch these tiny
wanderers of the ocean with nets thrown expertly from the bow of a
boat, for they manifest very little apprehension of danger from the
proximity of one, insomuch that I have seen several killed with the
oars. Those which were caught alive and placed on the deck, would
at first rest a few minutes with their bodies flat, then rise upright and
run about briskly, or attempt to fly off, which they sometimes
accomplished, when they happened to go in a straight course the
whole length of the ship so as to rise easily over the bulwarks. On
effecting their escape they would alight on the water and
immediately disappear.
During my visit to Labrador and Newfoundland I met with none of
these birds, although the cod-fishers assured me that they frequently
breed there. I am informed by Dr Townsend that this species is
found near the mouth of the Columbia River.

Alca alle, Linn. Syst. Nat. vol. i. p. 211.—Lath. Ind. Ornith. vol. ii. p. 795.
Little Auk, Alca alle, Wils. Amer. Ornith. vol. ix. p. 94, pl. 74, fig. 5.
Uria alle, Ch. Bonaparte, Synopsis of Birds of United States, p. 425.
Little Guillemot, Uria alle, Richards. and Swains. Faun. Bor.-Amer. vol. ii.
p. 479.
Little Auk, or Sea Dove, Nuttall, Manual, vol. ii. p. 531.

Adult Male in summer. Plate CCCXXXIX.

Bill shorter than the head, stout, straightish, subpentagonal at the
base, compressed towards the end. Upper mandible with the dorsal
line convexo-declinate, the ridge convex, the sides sloping, the
edges sharp and overlapping, the tip rather obtuse. Nasal
depression short and broad; nostrils basal, oblong, with a horny
operculum. Lower mandible with the angle long and wide, the dorsal
outline very short, ascending, and straight, the sides convex, toward
the end ascending and flattened, the edges thin and inclinate, the tip
acute, with a sinus behind.
Body full and compact; neck short and thick; head large, ovate. Feet
short, rather stout; tibia bare for two-twelfths of an inch; tarsus very
short, compressed, covered anteriorly with oblique scutella, behind
with angular scales; hind toe wanting; anterior toes connected by
reticulated webs, the inner much shorter than the outer, which is
almost as long as the middle; the scutella numerous. Claws rather
small, moderately arched, compressed, rather acute, that of the
middle toe having its inner edge considerably expanded.
Plumage dense, blended, glossy. Wings of moderate length, narrow,
pointed; primaries pointed, the first longest, the rest rapidly
graduated; secondaries rounded. Tail very short, slightly rounded, of
twelve feathers.
Bill black. Iris dark hazel. Feet pale flesh-coloured; webs dusky;
claws black. Inside of mouth light yellow. The head, upper part of
neck, and all the upper surface, glossy bluish-black. A small spot on
the upper eyelid, another on the lower, several longitudinal streaks
on the scapulars, and a bar along the tips of the secondary quills,
white. The lower parts white; the feathers on the sides under the
wings have the outer webs white, the inner dusky; lower wing-
coverts blackish-grey.
Length to end of tail 7 1/8 inches, to end of claws 7 7/8, to end of
wings 6 7/8, to carpal joint 2 7/8; extent of wings 14 1/4; wing from
1/2
flexure 4 7/8; bill along the ridge 4 /8, along the edge of lower
mandible 1; tarsus 3/4; middle toe 1, its claw 1/4; outer toe 1, claw
1 1/2/ ; inner toe 5/8, its claw 1
1/2
/8. Weight 8 1/2 oz.
8

Adult Female, in winter. Plate CCCXXXIX. Fig. 2.

In winter, the throat and the lower parts of the cheeks are white; the
sides and fore part of the neck white, irregularly barred with blackish-
grey; the upper parts of a duller black than in summer.
There is nothing very remarkable in the anatomy of this bird, beyond
what is observed in the Auks and Guillemots. The ribs extend very
far back, and, having the dorsal and sternal portions much
elongated, are capable of aiding in giving much enlargement to the
body, of which the internal, or thoracic and abdominal cells are very
large. The subcutaneous cells are also largely developed, as in
many other diving and plunging birds.
The roof of the mouth is flat, broad, and covered with numerous
series of short horny papillæ directed backwards. The tongue is
large, fleshy, 10 twelfths of an inch long, emarginate at the base, flat
above, horny on the back. The heart is large, measuring 10 twelfths
in length, 8 1/2 twelfths in breadth. The right lobe of the liver is 1 3/12
inch in length, the left 1 1/12; the gall-bladder is elliptical. The kidneys
are very large.

Fig. 1.

Fig. 2.
 Fig. 3.

The œsophagus, Fig. 1, a b c, is 3 inches 10 twelfths long, its walls

very thin, its inner or mucous coat thrown into longitudinal plates; its
diameter at the middle of the neck 5 eighths, diminishing to 4
twelfths as it enters the thorax. It then enlarges and forms the
proventriculus, c e, which has a diameter of 8 twelfths; the glandules
are cylindrical, very numerous, and arranged in a complete belt, half
an inch in breadth, in the usual manner, as seen in Fig. 2, b c. The
stomach, properly so called, Fig. 1, d g, is oblong, 11 twelfths in
length, 8 twelfths in breadth; its muscular coat moderately thick, and
disposed into two lateral muscles with large tendons; its epithelium,
Fig. 2, c d e, thick, hard, with numerous longitudinal and transverse
rugæ, and of a dark reddish colour. The duodenum, f g h, curves in
the usual manner at the distance of 1 1/4 inch, ascends toward the
upper surface of the right lobe of the liver for 1 inch and 10 twelfths,
then forms 4 loops, and from above the proventriculus, passes
directly backward. The length of the intestine, f g h i, is 16 1/2 inches,
its diameter 2 1/4 twelfths, and nearly uniform as far as the rectum,
which is 1 1/4 inch long, at first 3 twelfths in diameter, enlarged into
an ovate cloaca of great size, Fig. 3. b; the cœca a, a, 41 twelfths
long, cylindrical, 1/2 twelfth in diameter, obtuse.

The trachea, Fig. 1. k, l, is very wide, flattened, its rings unossified,

its length 2 9/12 inches, its breadth 3 twelfths, nearly uniform, but at
the lower part contracted to 2 twelfths. There are 75 rings, with 5
inferior blended rings, which are divided before and behind. The
bronchi, Fig. 1. m, m, are wide and rather elongated, with about 25
half rings. The contractor muscles are extremely thin, the sterno-
tracheal slender; there is a pair of inferior laryngeal attached to the
first bronchial rings.
The above account of the digestive organs of this bird will be seen to
be very different from that given by Sir Everard Home, who has, in all
probability, mistaken the species. “There is still,” says he, “one more
variety in the structure of the digestive organs of birds, that live
principally upon animal food, which has come under my observation;
and with an account of which I shall conclude the present lecture.
This bird is the Alea Alle of Linnæus, the Little Auk. The termination
of the œsophagus is only known by the ending of the cuticular lining,
and the beginning of the gastric glands; for the cardiac cavity is one
continued tube, extending considerably lower down in the cavity of
the abdomen, and gradually enlarging at the lower part; it then turns
up to the right side, about half-way to the origin of the cavity, and is
there connected to a small gizzard, the digastric muscle of which is
strong, and a small portion of the internal surface on each side has a
hard cuticular covering. The gastric glands at the upper part are
placed in four distinct longitudinal rows, becoming more and more
numerous towards the lower part of the cavity, and extend to the
bottom, where it turns up. The extent of the cavity in which the
gastric glands are placed, exceeds any thing met with in the other
birds that live upon fish; and the turn which the cavity takes almost
directly upwards, and the gizzard being at the highest part instead of
the lowest, are peculiarities, as far as I am acquainted, not met with
in any other birds of prey. This mechanism, which will be better
understood by examining the engraving, makes the obstacles to the
food in its passage to the intestines unusually great; and enables the
bird to digest both fishes and sea-worms with crustaceous shells. It
appears to be given for the purpose of economizing the food in two
different ways,—one retaining it longer in the cardiac cavity, the other
supplying that cavity with a greater quantity of gastric liquor than in
other birds. This opinion is further confirmed by the habits of life of
this particular species of bird, which spends a portion of the year in
the frozen regions of Nova Zembla, where the supplies of
nourishment must be both scanty and precarious.”
With respect to this statement and the reasonings founded upon it, it
will be seen from the description and accompanying figures above,
taken directly from nature, and without the least reference to the
dissections or theories of any person, that the œsophagus and
stomach of the Little Auk or Guillemot, Alca Alle of Linnæus, are very
similar to those of other Auks, Guillemots, Divers, and fish-eating
birds in general. The cardiac or proventricular cavity forms no curve;
and the gizzard with which it is connected, is not small, nor has it
merely a small portion of the internal surface on each side covered
with a hard cuticular lining; for the epithelium covers its whole
surface, and is of considerable extent. The gastric glands are not at
all disposed as represented by Sir E. Home, but are aggregated in
the form of a compact belt half an inch broad, Fig. 2. b, c. As to the
ingenious reasoning by which the economy of the Little Auk is so
satisfactorily accounted for, it is enough here to say, that having no
foundation, it is of less than no value. But were there such a
curvature as that in question, there could be no propriety in
supposing that it presented any great obstacle to the passage of the
food, or retained it longer than usual. Nor is the statement as to
scanty and precarious supply of nourishment correct; for the Arctic
Seas, to which this bird resorts in vast numbers, are represented by
navigators as abounding in small crustacea, on which chiefly the
Little Auk feeds, and that to such an extent as to colour the water for
leagues. Besides, if there were such a scarcity of food in Nova
Zembla, why should the birds go there? In short, the whole
statement is incorrect; and the many compilers, from Dr Carus to
the most recent, who have pressed it into their service, may, in their
future editions, with propriety leave it out, and supply its place with
something equally ingenious.
The egg of this species measures one inch and nearly five-eighths in
length, one inch and an eighth in its greatest breadth. It is
remarkably large for the size of the bird, and of a dull uniform pale
greenish-blue.
 LEAST PETREL.

Thalassidroma pelagica, Leach.

PLATE CCCXL. Male and Female.

In August 1830, being becalmed on the banks of Newfoundland, I

obtained several individuals of this species from a flock composed
chiefly of Thalassidroma Leachii, and Th. Wilsoni. Their smaller size,
and the more rapid motions of their wings, rendered them quite
conspicuous, and suggested the idea of their being a new species,
although a closer inspection shewed them to belong to the present.
In their general manners, while feeding, floating on the water, or
rambling round the boat in which I went in pursuit of them, they did
not differ materially from the other species. Their flight, however, was
more hurried and irregular, and none of them uttered any note or cry,
even when wounded and captured. I have been assured that this
bird breeds on the sandy beaches of Sable Island on the coast of
Nova Scotia; but not having had an opportunity of visiting it, or any
other breeding place, I here present you with Mr Hewitson’s
observations on this subject.
“In an excursion,” says this amiable and enterprising naturalist,
“through the Shetland Islands during the present summer, in search
of rarities for this work (the British Oology), I had the very great
satisfaction of seeing and taking many of these most interesting
birds alive; they breed in great numbers on several of the islands,
principally upon Foula, the north of Hunst, and upon Papa, and
Oxna, two small islands in the Bay of Scalloway; the last of these I
visited on the 31st of May in hopes of procuring their eggs (it being
the season in which most of the sea-birds begin to lay); but in this I
was disappointed; the fishermen who knew them well by the name of
Swallows, assured me that my search would be quite useless, that
they had not yet “come up from sea,” and so it proved. Sixteen days
after this (June 16th and three following days) I was at Foula, but
was alike unsuccessful, the birds had arrived at their breeding
places, but had not yet begun laying their eggs; numbers of them
were sitting in their holes, and were easily caught; one man brought
me about a dozen tied up in an old stocking, two of which I kept alive
in my room for nearly three days, and derived very great pleasure
from their company; during the day they were mostly inactive, and
after pacing about the floor for a short time, poking their head into
every hole, they hid themselves between the feet of the table and the
wall; I could not prevail upon them to eat any thing, though I tried to
tempt them with fish and oil; their manner of walking is very light and
pleasing, and differing from that of every other bird which I have
seen; they carry their body so far forward and so nearly horizontal,
as to give them the appearance of being out of equilibrium. In the
evening, toward sun-set, they left their hiding places, and for hours
afterwards, never ceased in their endeavours to regain their liberty;
flying round and round the room, or fluttering against the windows;
when flying, their length of wing, and white above the tail, gives them
a good deal the appearance of our House-Martin. I went to bed and
watched them in their noiseless flight long ere I fell asleep, but in the
morning they had disappeared; one had fortunately made its escape
through a broken pane in the window which a towel should have
occupied, the other had fallen into a basin, full of the yolks of eggs
which I had been blowing, and was drowned. I regretted much the
fate of a being so interesting, by its very remarkable, wandering,
solitary, and harmless life. Before leaving Shetland I again visited the
island of Oxna, and though so late as the 30th of June, they were
only just beginning to lay their eggs. In Foula they breed in the holes
in the cliff, at a great height above the sea; but here under stones
which form the beach, at a depth of three or four feet, or more,
according to that of the stones; as they go down to the earth,
beneath them, on which to lay their eggs. In walking over the
surface, I could hear them, very distinctly, singing in a sort of
warbling chatter, a good deal like swallows when fluttering above our
chimneys, but harsher; and in this way, by listening attentively, was
guided to their retreat, and, after throwing out stones as large as I
could lift on all sides of me, seldom failed in capturing two or three
seated on their nests, either under the lowest stone or between two
of them. The nests, though of much the same materials as the
ground on which they were placed, seem to have been made with
care; they were of small bits of stalks of plants, and pieces of hard
dry earth. Like the rest of the genus, the Stormy Petrel lays
invariably one egg only. During the day-time they remain within their
holes; and though the fishermen are constantly passing over their
heads (the beach under which they breed being appropriated for the
drying of fish), they are then seldom heard, but toward night become
extremely querulous; and when most other birds are gone to rest,
issue forth in great numbers, spreading themselves far over the
surface of the sea. The fishermen then meet them very numerously;
and though they have not previously seen one, are sure to be
surrounded by them upon throwing pieces of fish overboard.”
The egg measures one inch and an eighth in length, six and a half
eighths in breadth, is nearly equally rounded at both ends, rather
thick-shelled, and pure white, but generally with numerous minute
dots of dull red at the larger end, sometimes forming a circular band.

Procellaria pelagica, Linn. Syst. Nat. vol. i. p. 212.—Lath. Ind. Ornith. vol.
ii. p. 826.
Stormy Petrel, Nuttall, Manual, vol. ii. p. 327.
Adult Male. Plate CCCXL. Fig. 1.
Bill shorter than the head, slender, compressed towards the end,
straight, with the tips curved. Upper mandible with the nostrils
forming a tube at the base, beyond which, for a short space, the
dorsal line is nearly straight, then suddenly decurved, the sides
declinate, the edges sharp, the tip compressed and acute. Lower
mandible with the angle rather long, narrow, and pointed, the dorsal
line beyond it very slightly concave and decurved, the sides erect,
the edges sharp, the tip slightly decurved.
Head of moderate size, roundish, anteriorly narrowed. Neck short.
Body rather slender. Feet of moderate length, very slender; tibia bare
at its lower part; tarsus very slender, reticulate; hind toe extremely
minute, being reduced, as it were, to a slightly decurved claw;
anterior toes rather long and extremely slender, obscurely scutellate
above, connected by striated webs with concave margins. Claws
slender, arched, compressed, acute.
Plumage very soft, blended, the feathers distinct only on the wings,
which are very long and narrow; primary quills tapering, but rounded,
the second longest, the first three and a half twelfths, the third a
twelfth and a half shorter; secondaries short, the outer incurved,
obliquely rounded. Tail rather long, broad, slightly rounded, of twelve
broad rounded feathers.
Bill and feet black. Iris dark brown. The general colour of the upper
parts is greyish-black, with a tinge of brown, and moderately
glossed; the lower parts of a sooty brown; the secondary coverts
margined externally with dull greyish-white; the feathers of the rump
and the upper tail-coverts white, with the shafts black, the tail-coverts
broadly tipped with black.
Length to end of tail 5 3/4 inches, to end of claws 5 1/4, to end of
wings 6 1/4; extent of wings 13 1/2; wing from flexure 5 1/8; tail 2 1/8;
bill above (4 1/2/8, along the edge of lower mandible 5/8; tarsus 7/8;
middle toe and claw 7/8; outer toe nearly equal; inner toe and claw
5 1/2/8. Weight 4 1/2 drachms; the individual poor.
Adult Female. Plate CCCXL. Fig. 2.
The Female resembles the male.

Fig. 1.

A male bird, from Nova Scotia, examined. The upper mandible

internally has a longitudinal median ridge; the palate is convex, with
two lateral ridges. The tongue is 5 1/2 twelfths long, emarginate and
serrulate at the base, very much flattened, tapering to a horny point.
The heart, Fig. 1, a, is of a very elongated narrow conical form, 2
twelfths in length, 4 twelfths in breadth at the base. The lobes of the
liver, b, c, are equal, 6 1/2 twelfths long. The œsophagus, d, e, is 1
inch 10 twelfths long, of a uniform diameter of 2 1/2 twelfths; behind
the liver, it enters as it were a large sac, f, g, h, 9 twelfths of an inch
long, which gradually expands to a diameter of 6 twelfths, forming a
broad rounded fundus g, then curves forwards on the right side, and
at h terminates in a small gizzard, about 3 twelfths long, and nearly
of the same breadth, from the left side of which comes off the
intestine. The latter passes forward, curving to the right, behind and
in contact with the posterior surfaces of the liver, then forms the
duodenal fold, h, j, k, in the usual manner. The intestine, on arriving
at the right lobe of the liver, at k, receives the biliary duct, curves
backward beneath the kidneys, and forms several convolutions,
which terminate above the proventriculus. It then becomes much
narrower, and passes directly backward, in a straight course to the
rectum, which is only 4 twelfths of an inch long. The cœca are
oblong, 1 1/4 twelfth in length, and 1/2 twelfth in diameter. The
intestine is 8 1/2 inches long, its diameter diminishing gradually from
2 twelfths to 3/4 of a twelfth.

Fig. 2.

In Fig 2. are represented:—the lower part of the œsophagus, d, e, f;

the proventricular sac, f, g, h; the very small gizzard, h; the duodenal
fold of the intestine, i, j, k. Here the parts are viewed from the left
side.