![Parametric Identification for Robust
Control
Piotr Kulczycki
Abstract Many contemporary automatic control applications require parametric
identification, taking into account results of estimation errors unavoidable in practice.
The subject of this chapter constitutes a procedure enabling effective calculation of
optimal, in the sense of minimal expectation of losses, estimator of a parameter in
the case when losses resulting from negative and positive errors have significantly
different influence on system operation, which forces the use of an asymmetric
loss function. The conditional version of the issue is also considered in detail, as
the process is significantly dependent on factors available metrologically during
the system operation. This material has been provided in ready-to-use form, not
requiring additional studies and bibliographical exploration. The presented concept
is directed towards robust control problems. As an appendix, the analysis of solutions
of differential equations with discontinuous right-hand side, subordinated for this
purpose, has been presented, which may be useful for describing the dynamics of
such resultant systems.
1 Introduction
In the early period of the development of control engineering, a fundamental disci-
pline of automatic control, concepts were dominated by the intuition and experience
of designers, without any dedicated mathematical apparatus. A significant change
occurred in the interwar period, thanks to the application of basic elements of oper-
ational calculus: Laplace transformation and the transfer function concept [47]. The
fundamental requirement formulated towards the constructed regulation systems was
stability [4]. The research was facilitated by rich and advantageous mathematical
P. Kulczycki (B)
Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01447 Warsaw, Poland
e-mail: kulczycki@ibspan.waw.pl; kulczycki@agh.edu.pl
Faculty of Physics and Applied Computer Science, AGH University of Science and Technology,
Mickiewicza 30, 30059 Cracow, Poland
© The Editor(s) (if applicable) and The Author(s), under exclusive license
to Springer Nature Switzerland AG 2021
P. Kulczycki et al. (eds.), Automatic Control, Robotics, and Information Processing,
Studies in Systems, Decision and Control 296,
https://doi.org/10.1007/978-3-030-48587-0_1
3](https://image.slidesharecdn.com/89119-250426050546-097cafbd/85/Automatic-Control-Robotics-and-Information-Processing-Piotr-Kulczycki-18-320.jpg)
![4 P. Kulczycki
apparatus provided for that purpose. The main achievement of that period was the
regulator PID, exceptional in its simplicity and convenient in interpretation, and
simultaneously surprisingly effective [2]. To this day it constitutes a fundamental
tool in industrial applications, undergoing permanently various modifications and
modernizations.
However, with the growth in requirements formulated towards automatic control
systems the natural question arose, which among the stable systems should be recog-
nized and applied as the best (optimal)? Using operational calculus there was no
possible way to solve such a defined problem. The conditions of the second World
War and the later period of the Cold War, when the best possible arms quality trans-
ferred into military success and human life, gradually magnified the growth in impor-
tance of optimization and research conducted in this field. In the fifties, operational
calculus was replaced by differential equations, which enabled rapid development in
the optimal control theory [3, 38]. At the end of this decade two fundamental theo-
ries allowing the establishment of control of this kind were published: Pontryagin
Maximum Principle and Bellman Dynamical Programming, at this time engineering
practice unambiguously distinguished two types of optimal control: time-optimal
(minimum-time), relying on reaching the target in the shortest time ([3], Chapter 7),
also with a quadratic performance index, directed primarily towards minimalizing
energy used ([3], Chapter 9).
Although, in many applications of optimal control, sublime and extreme in nature,
it only worked well in the assumed conditions or those not much different from them.
As a consequence this led to a transfer in interest of practitioners in the direction of so-
called robust control, which also worked correctly when real conditions significantly
differed from those assumed [5, 50]. In practice this often means that the values of
the parameter or parameters, used in a given model, can significantly differ from
the real ones occurring in the object. An illustration of this may be for example the
mass of an aircraft, changing due to fuel use, that is impossible to measure directly,
which is more precise. Robust control systems were frequently derived from the
concept of optimal control; the robustness of the former was obtained at the cost of a
small worsening of quality in the sense of the performance index used in the baseline
optimal structure being accepted in advance.
In this chapter the ready-to-use procedure for calculation of the parameter value
is presented, which minimalizes the expected losses resulting from imprecise esti-
mation, different for negative and positive errors. It will provide the solution for
the introduced issue of the robust control problem. Thus, in the next subchapter,
the motivation for the identification task considered here will be presented. The
third subchapter covers the mathematical preliminaries, the methodology of statis-
tical kernel estimators is described here, which will be used in the fourth main
subchapter for the creation of a procedure for calculation of the optimal value of the
studied parameter. In the first two points of this subchapter, the subject of detailed
considerations are basic estimation issues with asymmetric linear and then quadratic
loss function, after which in the third point, the above material will be submitted
in the conditional approach. These issues may be generalized into polynomials of
higher orders and the multidimensional case, where the subject of investigations are](https://image.slidesharecdn.com/89119-250426050546-097cafbd/85/Automatic-Control-Robotics-and-Information-Processing-Piotr-Kulczycki-19-320.jpg)
![Parametric Identification for Robust Control 5
parameters with the correlated impact on estimation errors; these problems will be
briefly outlined in the the fifth subchapter. Finally at the end, the appendix presents
considerations concerning the solutions of differential equations with discontinuous
left-hand side, which maybe helpful for the formal description of a robust control
system, arising as a result of the application of the procedure previously worked out
in this chapter.
2 Motivation and Problem Formulation
As grounds for the concept outlined here, three illustrative examples from the area
of robust control are presented.
Assume the classic dynamic system
ẋ1(t)
ẋ2(t)
=
0 1
0 0
x1(t)
x2(t)
+
0
1
M
u(t), (1)
where the positive parameter M represents a mass submitted to the action of force,
according to the second law of Newtonian dynamics. Here x1, x2 and u denote
the position and velocity of the above mass and the force treated as a control. This
system forms the basis of the majority of research in robotics, leading to considerably
more complex models with specificity appropriate for the problem under research
[35, 37]. Consider the time-optimal control problem, whose basic form relies on
bringing the state of the system to the origin of coordinates by the bounded control,
in a minimal and finite time ([3], Subchapter 7.2). Fundamental importance for
phenomena taking place in the control system has an appropriate identification of
the value of the parameter M, introduced in the formula (1). The control algorithm is
however defined on the basis the value of its estimator M
, different in practice from
the value M occurring in the object.
In the purely hypothetical case M
= M, therefore when the estimator and real
values are equal, the process is regular in nature. The system state reaches the origin
in a minimal and finite time.
In turn, if the estimator is overestimated (i.e. for M
M), then over-regulations
appear in the system. Its state oscillates around the origin and reaches it in a finite
time, however larger than minimal.
Finally, in the case of underestimation (i.e. when M
M), the system state moves
along a so-called sliding trajectory [8, 48] and in the end reaches the origin in a finite
time, again larger than minimal.
Detailed considerations on the topic of the phenomena occurring in the system
under investigation can be found in the paper [30]. On the margin of the above
considerations, in order to illustrate the formal complexity of the task, one can mark
that in the first two cases, the differential equation (1) possesses unique C-solutions
(in the Caratheodory sense) and in the third unique F-solutions (in the Filippov](https://image.slidesharecdn.com/89119-250426050546-097cafbd/85/Automatic-Control-Robotics-and-Information-Processing-Piotr-Kulczycki-20-320.jpg)
![6 P. Kulczycki
sense). For the purpose of unifying all three cases, one can use K-solutions (in the
Krasovski sense), more general than C- and F-solutions. In many issues, the family
of K-solutions turns out to be too numerous, however in the system considered here
it is fortunately unique [13]. Furthermore, in the probabilistic approach, the family of
K-solutions occurring for particular values of the estimated parameter constitutes a
stochasticprocess[14],whichenablesanapplicationofarichmathematicalapparatus
dedicated to this issue. These tasks have been specified and illustrated in the appendix
to this chapter.
Therefore, in each of the above three cases, the system trajectory reaches the
origin in the finite time, however, the phenomena occurring in the system as a result
of under- and overestimation of the parameter M
are completely different and thus
one can expect significantly different times for reaching the target set in the ranges
of negative and positive errors M
− M.
Figure 1 shows the graph of the time to reach the origin by the system (1) state,
with exemplary fixed M = 1. It can be seen that the growth in the value of this index
is approximately directly proportional to the estimation error M
− M, although with
different coefficients for negative and positive errors:
l
M
, M
=
⎧
⎨
⎩
−a
M
− M
when M
− M ≤ 0
b
M
− M
when M
− M ≥ 0
, (2)
while the constants a and b are positive; it is worth underlining they do not necessarily
equal.
In the case of the stronger interaction of the parameter identified on the behavior of
the system, for example where its impact is “multitrack” (multifaceted), the quadratic
form of the performance index may be appropriate. As illustrative example, consider
the dynamic system
Fig. 1 Value of
time-optimal index for
different values of estimator
M
, with M = 1](https://image.slidesharecdn.com/89119-250426050546-097cafbd/85/Automatic-Control-Robotics-and-Information-Processing-Piotr-Kulczycki-21-320.jpg)
![Parametric Identification for Robust Control 7
Fig. 2 Value of quadratic
index for different values of
estimator λ
, with λ = 1
ẋ1(t)
ẋ2(t)
=
λ 1
0 λ
x1(t)
x2(t)
+
0
λ
u(t), (3)
where the constant λ is different from 0. It is worth noting that it occurs both in
the state A and in the control B matrixes. The system (3) will be submitted to the
classic optimal control with the quadratic performance index ([3], Subchapter 9.5)
with unique matrixes and infinite final time. Figure 2 shows the value of the above
index as the function of the estimator λ
value, with exemplary fixed λ = 1. One can
see that the the value of the loss function resulting from the estimation error may be
successfully approximated by a asymmetric quadratic form:
l
λ
, λ
=
⎧
⎨
⎩
a
λ
− λ
2
when λ
− λ ≤ 0
b
λ
− λ
2
when λ
− λ ≥ 0
, (4)
where again the coefficients a and b are positive and not necessarily equal.
Now consider the two-dimensional case. Assume the dynamic system, originated
by the introduction into the system (4) of the drive inertia; therefore, substituting the
control u by the third coordinate of the state vector x3, and so in consequence
⎡
⎣
ẋ1(t)
ẋ2(t)
ẋ3(t)
⎤
⎦ =
⎡
⎣
λ 1 0
0 λ λ
0 0 −τ
⎤
⎦
⎡
⎣
x1(t)
x2(t)
x3(t)
⎤
⎦ +
⎡
⎣
0
0
1
⎤
⎦u(t), (5)
where τ is a positive drive inertia constant. Thus the parameters
λ
τ
originated
from different subsystems (λ from the object itself and τ characterizing the drive)
are subjected to estimation, and therefore can be treated as independent. Once again
the system will be submitted to control with the quadratic performance index ([3],](https://image.slidesharecdn.com/89119-250426050546-097cafbd/85/Automatic-Control-Robotics-and-Information-Processing-Piotr-Kulczycki-22-320.jpg)
![Parametric Identification for Robust Control 9
described by the asymmetric linear (2) or quadratic (4), (6) functions. The polynomial
form seems to be a suitable compromise between the precision and the complexity
and in consequence usefulness of the approach proposed [36].
It is worth furthermore generalizing the classic formulation of the research
problem presented above. Thus, the more complex and refined the models currently
applied become, the less reasoned is a traditional perception of parameters identifica-
tion as the task of calculation of a concrete (existing in reality but unknown) value of
the parameter under investigation. Currently this task covers establishment of value,
which will represent the whole range of multifaceted phenomena, simplified in the
model to one, the only formally existing parameter. In such a situation, evaluation
of the quality of the parameter identification cannot be made in the classical way by
comparing the obtained value to the imagined “real”, although unknown, parameter
value (because no such value exists), but rather through taking into account the impli-
cations of particular values on the functioning of the system under investigation. This
differs from the mathematical apparatus existing in the framework of point estima-
tion from the classic mathematical statistic [34], becoming closer to the currently
expanding data analysis [31–33]. Fortunately, development of modern, advanced
and specialized identification methods enables a rapid expansion of contemporary
computertechnology,supportingfromtheoreticalpointofviewITmethodsdedicated
to it.
In summary: in the method presented here, the value of estimator will be sought,
minimizing the expectation value of losses resulting from estimation errors. One
assumes availability of a finite number of metrologically obtained values for the esti-
mated parameter. The knowledge of its distribution is not necessary; its establishment
constitutes an integral part of the procedure. The conditional approach will also be
proposed, designated for the case where the parameter under research is significantly
dependent on the factor, which current value can be successively measured during
realization of the control process, causing an increase in the models precision. The
material is given in the complete form, not requiring additional investigations or
bibliographical research.
3 Mathematical Preliminaries: Kernel Estimators
Kernel estimators are counted among the nonparametric methods. Methods of this
type do not require initial assumptions concerning the type of the existing distribu-
tion. In their basic formula, kernel estimators serve to establish the fundamental func-
tional characteristics of the distribution; its density. First, in Sect. 3.1 the basic one-
dimensional case will be presented, generalized in Sect. 3.2 to the multidimensional
approach.](https://image.slidesharecdn.com/89119-250426050546-097cafbd/85/Automatic-Control-Robotics-and-Information-Processing-Piotr-Kulczycki-24-320.jpg)
![10 P. Kulczycki
3.1 One-Dimensional Case
Let x denote the quantity under investigation. Let the m-elements set of independent
measurements
x1, x2, . . . , xm ∈ R (7)
be given. The kernel estimator of the distribution density of the studied quantity
f
: R → [0, ∞) is defined in its standard form by the following formula
f
(x) =
1
mh
m
i=1
K
x − xi
h
, (8)
where the positive constant h is called the smoothing parameter, while the function
K : R → [0, ∞) measurable, symmetrical with respect to zero and having the weak
global maximum at this point, with unique integral
R K(y)dy = 1, is named as the
kernel.
The interpretation of the above definition is illustrated in Fig. 4. In the case of
the single element xi , the function K transposed by the vector xi and scaled by the
coefficient h, represents estimation of the distribution after obtaining the value xi . For
m independent values x1, x2, . . . , xm, it takes the form of the sum of such individual
estimations. The coefficient 1/mh norms the function obtained to guarantee the
condition
R f
(y)dy = 1. Finally, the formula (8) provides the tool for estimation
of density of the set (7) distribution; it is illustrative and rather less complex for
numerical calculations.
In practice the choice of the kernel form ([20], Section 3.1.3; [49], Subchapters 2.7
and 4.5) has no significant importance, and thanks to this it is possible to primarily
take into account the properties of the estimator obtained, e.g. its continuity, differ-
entiability, or bounding of the support; advantageous from the point of view of the
Fig. 4 One-dimensional
kernel estimator (8)](https://image.slidesharecdn.com/89119-250426050546-097cafbd/85/Automatic-Control-Robotics-and-Information-Processing-Piotr-Kulczycki-25-320.jpg)
![Parametric Identification for Robust Control 11
specific application. In the method presented here, the Cauchy kernel
K(x) =
2
π
x2 + 1
2
(9)
will be applied, because of its convenient integration properties. The constants
W(K) =
∞
−∞ K(y)2
dy and U(K) =
∞
−∞ y2
K(y)dy, which are necessary for
further calculations, equal in this case:
W(K) =
5
4π
(10)
U(K) = 1. (11)
Great importance for the estimation quality is, however, fixing of the smoothing
parameter h value ([20], Section 3.1.5; [49], Chapter 3 and Subchapter 4.7). Fortu-
nately for practitioners, many convenient calculational procedures, based the set (7),
have been investigated. During the initial design phase, as well as in the case of the
set size greater than 1000, one can recommend a simple approximate method with
linear complexity regarding to the size m. In the remaining cases, the effective plug-in
procedure, with quadratic complexity, can be implemented.
The first method relies on the direct use of the formula
h =
8
√
π
3
W(K)
U(K)2
1
m
1/5
σ
, (12)
where σ
denotes the classic standard deviation estimator:
σ
=
1
m − 1
m
i=1
(xi )2
−
1
m(m − 1)
m
i=1
xi
2
. (13)
In turn, the second procedure, plug-in, is based on the application of the above
formula, following several improvements in such obtained value. Thus, in the
case of the third degree procedure, one should sequentially apply the following
dependencies:
d10 =
−945
64
√
πσ
11
, (14)
where σ
is given by the equality (13), and next
gI I I =
−2
K(8)
(0)
d10U
K
m
1/11
(15)](https://image.slidesharecdn.com/89119-250426050546-097cafbd/85/Automatic-Control-Robotics-and-Information-Processing-Piotr-Kulczycki-26-320.jpg)
![14 P. Kulczycki
Fig. 5 Two-dimensional
kernel estimator (26);
elements of set (25) are
marked by crosses, lines
denote the contours of
estimator
found in the monographs [20, 44, 49]. It is also worth recalling the publications [6,
7, 40–42, 45] as well as the new subject book [9], where one can also obtain broad
current literature. The sister problem of the kernel regression can be found in [10].
The implicational possibilities to identification of atypical elements (outliers), clus-
tering, and classification, three basic procedures of data analysis and exploration, are
summarized in the survey article [21] and also in the work [26] for the conditional
approach. In the paper [19] the concept of the sophisticated diagnostic system based
on kernel estimators is presented.
4 Procedure for Parametric Identification
4.1 Linear Case
Assume the asymmetric linear loss function (2), presented below using standard
notation
l
x
, x
=
−a
x
− x
when x
− x ≤ 0
b
x
− x
when x
− x ≥ 0
, (29)
where x represents the investigated parameter, x
its estimator, and the coefficients a
and b are positive (non-necessarily equal to each other). Consider the set](https://image.slidesharecdn.com/89119-250426050546-097cafbd/85/Automatic-Control-Robotics-and-Information-Processing-Piotr-Kulczycki-29-320.jpg)
![16 P. Kulczycki
I (x) =
1
π
arctg(x) +
x
π
x2 + 1
+
1
2
. (35)
Let us introduce the auxiliary notation
Pi (x) = I
x − xi
h
for i = 1, 2, . . . , m. (36)
The criterion (32) takes the form
1
m
m
i=1
Pi
x
−
a
a + b
= 0. (37)
Denoting the left side of the above equation by L
x
, i.e.
L
x
=
1
m
m
i=1
Pi
x
−
a
a + b
, (38)
its derivative is
L
x
= f
x
. (39)
Note that the function I is increasing. This feature, therefore, passes to Pi and in
consequence also to L. Because an analytic form of the derivative is accessible, then
these properties suggest use of the classic Newton algorithm [12] in order to solve the
Eq. (32). Then this solution can be found as a limit of the sequence x
k
!∞
k=0
defined
by the formulas
x
0 =
1
m
m
i=1
xi (40)
x
k+1 = x
k −
L
x
k
L
x
k
for k = 0, 1, . . . . (41)
As a stop criterion the following condition is assumed:
x
k − x
k−1
≤ 0, 01 σ
, (42)
where σ
is defined by the formula (13).
Summarizing, having independent measurements of the investigated parameter
(30), its optimal (in the sense of the expected loss value, in the case when the loss
function is given in the asymmetrical linear form (29)) the estimator can be calculated
by the recurrence algorithm (40)–(41) having the stop condition (42) with (13). The](https://image.slidesharecdn.com/89119-250426050546-097cafbd/85/Automatic-Control-Robotics-and-Information-Processing-Piotr-Kulczycki-31-320.jpg)
![Parametric Identification for Robust Control 17
functions L and L
occurring there are given by the formulas (38)–(39), and in turn
Pi and f
are defined by the dependencies (36) and (8). If one uses the Cauchy kernel
recommended here, then the functions K and I are denoted by the formulas (9) and
(35), respectively. The value of the smoothing parameter h can be calculated directly
from the formula (12) with (13), or by the plug-in method (14)–(24) and (13). When
the Cauchy kernel is used, the values of the functionals U and W existing there are
given by the formulas (10)–(11).
It should be noted that if a = b, the criterion (32) indicates directly the median.
The first results in the range of estimation with the asymmetrical loss function,
concerning the linear case, have been published in the paper [18].
4.2 Quadratic Case
The asymmetric quadratic loss function (3), described using standard notation, takes
the form
l
x
, x
=
#
a
x
− x
2
for x
− x ≤ 0
b
x
− x
2
for x
− x ≥ 0
, (43)
where x, x
and a, b have the same interpretation as in the case of the asymmetric
linear loss function (29), similarly as the set of measurements x1, x2, . . . , xm given
there by the formula (30). Let f
be established, using the material from Subchapter 3,
by the kernel estimator methodology; while the applied kernel K is continuous with
positive values, and additionally such that
∞
−∞
xK(x)dx ∞. (44)
The Cauchy kernel (9) fulfils the above condition.
The form of the definition (43) constitutes directly that the expectation value of
losses, resulting from the accepted value of the estimator x
, is given by the formula
x
= b
x
−∞
x
− x
2
f
(x)dx + a
∞
x
x
− x
2
f
(x)dx. (45)
It is not difficult to show that this function reaches a minimum for the value x
, which
is the solution of the equation](https://image.slidesharecdn.com/89119-250426050546-097cafbd/85/Automatic-Control-Robotics-and-Information-Processing-Piotr-Kulczycki-32-320.jpg)
Automatic Control, Robotics, and Information Processing Piotr Kulczycki
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- 4. Studies in Systems, Decision and Control 296 Piotr Kulczycki Józef Korbicz Janusz Kacprzyk Editors Automatic Control, Robotics, and Information Processing
- 5. Studies in Systems, Decision and Control Volume 296 Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland
- 6. The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. ** Indexing: The books of this series are submitted to ISI, SCOPUS, DBLP, Ulrichs, MathSciNet, Current Mathematical Publications, Mathematical Reviews, Zentralblatt Math: MetaPress and Springerlink. More information about this series at http://www.springer.com/series/13304
- 7. Piotr Kulczycki • Józef Korbicz • Janusz Kacprzyk Editors Automatic Control, Robotics, and Information Processing 123
- 8. Editors Piotr Kulczycki Systems Research Institute Polish Academy of Sciences Warsaw, Poland Faculty of Physics and Applied Computer Science AGH University of Science and Technology Kraków, Poland Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences Warsaw, Poland Józef Korbicz Institute of Control and Computation Engineering University of Zielona Góra Zielona Góra, Poland ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-030-48586-3 ISBN 978-3-030-48587-0 (eBook) https://doi.org/10.1007/978-3-030-48587-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
- 9. To Nestors of Polish automatic control and robotics, our Masters and Mentors, we dedicate this work
- 10. Preface The history of mankind could be viewed as the history of difficult challenges which had to be solved by human beings for, first, simple survival, and then for the necessity of effective action in increasingly complex situations, often in the face of natural disasters, the necessity of existence and operation in adverse environments, and also self-generated processes in the presence of successive development of technology and war art. As intelligent beings, humans have quickly realized that their specific physical, mental, and cognitive limitations are difficult to overcome and, as a consequence, they prevented them from reaching an effective and efficient performance of many activities requiring physical strength or constant, long-term concentration, and/or focus attention or reaction speed. An obvious conclusion has readily been that these basic human abilities, capabilities, and skills should some- how be supported (augmented) by using appropriate technological tools and tech- niques available at the current development level of knowledge and technology. A natural example can be the use of the first draught (working, usually pulling) animals, followed by steam engines, and electric motors, up to the ubiquitous use of computation technology today. In our context, all these activities can be considered as consecutive steps toward an increasingly common automation of all activities, from simple solutions requiring practically only physical strength, through more complex, requiring a high accuracy and speed of action, to the most complex and sophisticated acts, the execution of which needs intelligence, learning skills, planning, effective, and efficient group cooperation or collaboration, to just mention a few. This human quest for developing more effective and efficient tools and techniques for dealing with complex processes and systems has been a natural inspiration for the emer- gence of numerous fields of science and technology, among which control and automation and, recently, robotics, which are the subjects of papers included in this volume, are of a particular relevance. The purpose of this book is to present a wide and comprehensive spectrum of issues and problems related to these fields of science and engineering, both from a theoretical and applied perspective. vii
- 11. To be more specific, the works included involve the development of methods and algorithms to determine the best practices, most often regarding decisions or controls, which would ensure the most effective and efficient attainment of goals assumed, under the satisfaction of some constraints. From an engineering point of view, many papers deal with how to automate a specific process or even a complex system. From a perspective of tools, there are numerous contributions which are concerned with the development of analytic and algorithmic methods and techniques, and then devices and systems which make it possible to attain automation and robotization. For these issues outlined above as examples of various aspects of the wide field of knowledge that is control and automation, and its related robotics, there are also a number of very important, detailed and more specific elements, in particular those connected with the analysis of developed methods and systems. Virtually all of the above aspects are the subjects of consideration in the indi- vidual chapters of this monograph written by outstanding Polish researchers, scholars, and engineers who are well known in the fields of automatic control, automation, robotics, computer science, IT/ITC, as well as in some other fields such as measurement and sensor technology, reliability and damage detection. The first authors of the individual chapters are members of the Committee for Automatic Control and Robotics, Polish Academy of Sciences, selected by the Polish scientific community in recognition of their outstanding publication and application records, as well as their high stature in international science. For many years, Polish control and automation has enjoyed a great recognition in the world's scientific community. Immediately after World War II, despite the destruction of the country, higher education was reinstated by resuming the opera- tion of many renowned universities, and then establishing new ones. Already in the early 1950s, the first research teams, usually within University departments, working on automatic control and automation were formed. These centers quickly obtained valuable scientific results and as a consequence the Polish school in these fields began to be highly valued in the world. At that time, top scientists from around the world decided to set up the IFAC (International Federation of Automatic Control), which is today the largest and most prestigious, opinion-forming organization bringing together top experts, both researchers, scholars and practitioners, in auto- matic control, automation and robotics. Poland was one of the founders of IFAC. Moreover, as both a great distinction for a country so severely damaged by the war, and as a proof of a high recognition for the scientific and technological achieve- ments, Poland was entrusted with the organization of the Fourth IFAC World Congress in 1969. This Congress, held in Warsaw, gathered practically all leading theoreticians and practitioners in the fields from around the world, as well as many young researchers and students. The participation of the Polish authors was partic- ularly significant. To this day, the IFAC World Congresses are held every 3 years and are the most prestigious scientific events in the fields of broadly perceived automatic control and automation, and related areas, at which outstanding innovative theoretical achievements and applications are presented. In addition, representatives of the Polish science organize numerous conferences devoted to various aspects of viii Preface
- 12. automation and robotics, both under the auspices of IFAC and other prestigious international organizations: IEEE (Institute of Electrical and Electronics Engineers), IMECO (International Measurement Confederation), IFIP (International Federation of Information Processing), to just mention a few. The purpose of this volume is to present the current state of knowledge as well as new trends and application potential in the field of broadly understood automatic control automation and robotics in a comprehensible way. The first part, our starting point, is devoted to various basic issues of mathe- matical modeling. Currently, without knowing the model, which properly captures the examined fragment of reality, it is difficult, if not impossible, to perform any analysis of a process or system, and in particular to develop an appropriate control algorithm. Various types of mathematical models, both classic and specialized are presented here. As a more specific example, the modeling of physical flow pro- cesses is discussed. The issues of noise and disturbance and identification of pro- cesses in their presence are widely considered which in modern engineering is a matter of primary importance. The next part of the monograph is devoted to various aspects of control and regulation. Both linear and non-linear systems are presented, along with a detailed analysis of their fundamental properties, especially controllability and stability. Special attention is paid to the design of control systems, using various concepts and methods, both formal and based on computer simulation. One of the most important tools used in solving problems in the area of auto- matic control and automation are optimization methods that make it possible to derive the best, or at least sufficiently good solutions. Some important methods from this area are included in the next part of this book. The use of optimization methods to identify systems with space-time dynamics is shown, the use of discrete optimization methods in industrial automation is presented, and finally, the use of dynamic programming models for multi-stage control tasks under the conditions of uncertain and imprecise (fuzzy) information is discussed. Moreover, some relevant applications of the tools and techniques derived in the area of control, automation and robotics are discussed, exemplified by the use of advanced methods for the modeling and control of walking and humanoid robots. New approaches and methods for the control of robot teams and the use of multi-agent paradigms are also presented. Widely understood intelligent methods, notably, tools and techniques stemming from artificial intelligence, are also discussed. Notably, using the decision support paradigm, new approaches to technical diagnostics, development of fault-tolerant, and damage-tolerant control systems are presented. The application of computa- tional intelligence methods, mainly fuzzy systems, evolutionary computations, and neural networks, to control specific technical objects, in particular vessels, as well as selected industrial processes are also dealt with. A very important part of this volume is devoted to a detailed presentation of specific applications which concern here, to just mention a few, the analysis and control of important issues affecting human comfort and health, such as a reduction of noise or acoustic disturbances, as well as the emission of pollutants into the Preface ix
- 13. atmosphere. Innovative application of control methods in gene-cell networks is also presented. As the editors of this collective work, we hope that various aspects, models, concepts, and methods of modern control, automation and robotics discussed in this volume will be of interest and use for many researchers and practitioners, and also students of various levels, seeking information on both the new trends as well as state-of-the-art concepts in the areas concerned. The material included can also be used for scientific, professional, and teaching purposes. We hope that this volume will be of a particular interest to the younger gener- ation, both students and graduates, opening to them the fascinating world of these disciplines of science and technology which are widely associated with hopes for a better future. Namely, virtually all forecasts, which primarily capture the funda- mental importance of broadly perceived information processing and artificial intelligence for the competitiveness of economies, as well as the ability to solve basic socioeconomic problems, emphasize the key importance of ubiquitous automation and robotization of virtually all systems and activities. The roots of this volume can be traced to the related book “Automatyka, robotyka i przetwarzanie informacji”, eds. P. Kulczycki, J. Korbicz, J. Kacprzyk, issued in Polish by the PWN scientific publisher. Particular topics have been modified and updated in varying degrees. We would like to thank first of all the authors of the chapters for their extremely valuable works, presenting in an insightful but accessible way both the state of the art of modern knowledge as well as the latest trends and tendencies, sometimes too innovative and too early to enter the standard canon of applications. Our gratitude is also due to the peer reviewers whose insightful and constructive remarks and suggestions have helped the authors improve their contributions. And last but not least, we wish to thank Dr. Tom Ditzinger, Dr. Leontina di Cecco, and Mr. Holger Schaepe from Springer Nature for their dedication and help to implement and finish this important publication project on time, while main- taining the highest publication standards. Warsaw/Kraków, Poland Piotr Kulczycki Zielona Góra, Poland Józef Korbicz Warsaw, Poland March 2020 Janusz Kacprzyk x Preface
- 14. Contents Mathematical Modeling Parametric Identification for Robust Control . . . . . . . . . . . . . . . . . . . . . 3 Piotr Kulczycki Flow Process Models for Pipeline Diagnosis . . . . . . . . . . . . . . . . . . . . . . 35 Zdzisław Kowalczuk and Marek Sylwester Tatara Output Observers for Linear Infinite-Dimensional Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Zbigniew Emirsajłow Non-Gaussian Noise Reduction in Measurement Signal Processing . . . . 93 Jerzy Świątek, Krzysztof Brzostowski, and Jarosław Drapała Fractional Order Models of Dynamic Systems . . . . . . . . . . . . . . . . . . . . 115 Andrzej Dzieliński, Grzegorz Sarwas, and Dominik Sierociuk Switched Models of Non-integer Order . . . . . . . . . . . . . . . . . . . . . . . . . 153 Stefan Domek Control Nonlinear Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Piotr Tatjewski and Maciej Ławryńczuk Positive Linear Control Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Tadeusz Kaczorek Controllability and Stability of Semilinear Fractional Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Jerzy Klamka, Artur Babiarz, Adam Czornik, and Michał Niezabitowski Computer Simulation in Analysis and Design of Control Systems . . . . . 291 Ewa Niewiadomska-Szynkiewicz and Krzysztof Malinowski xi
- 15. Optimization Optimal Sensor Selection for Estimation of Distributed Parameter Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 Dariusz Uciński and Maciej Patan Discrete Optimization in the Industrial Computer Science . . . . . . . . . . . 359 Czesław Smutnicki Dynamic Programming with Imprecise and Uncertain Information . . . . 387 Janusz Kacprzyk Robotics Endogenous Configuration Space Approach in Robotics Research . . . . 425 Krzysztof Tchoń Control of a Mobile Robot Formation Using Artificial Potential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 Krzysztof Kozłowski and Wojciech Kowalczyk Biologically Inspired Motion Design Approaches for Humanoids and Walking Machines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 Teresa Zielińska Robotic System Design Methodology Utilising Embodied Agents . . . . . . 523 Cezary Zieliński Computational Intelligence and Decision Support Fault-Tolerant Control: Analytical and Soft Computing Solutions . . . . . 565 Józef Korbicz, Krzysztof Patan, and Marcin Witczak Systems Approach in Complex Problems of Decision-Making and Decision-Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 Jerzy Józefczyk and Maciej Hojda Advanced Ship Control Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 Roman Śmierzchalski and Anna Witkowska On-line Diagnostics of Large-Scale Industrial Processes . . . . . . . . . . . . . 645 Jan Maciej Kościelny Applications of Computational Intelligence Methods for Control and Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671 Jacek Kluska, Tomasz Żabiński, and Tomasz Mączka xii Contents
- 16. Applications Consequences and Modeling Challenges Connected with Atmospheric Pollution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 Zbigniew Nahorski and Piotr Holnicki System with Switchings as Models of Regulatory Modules in Genomic Cell Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739 Andrzej Świerniak, Magdalena Ochab, and Krzysztof Puszyński Modelling and Control of Heat Conduction Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767 Wojciech Mitkowski and Krzysztof Oprzędkiewicz Active Suppression of Nonstationary Narrowband Acoustic Disturbances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791 Maciej Niedźwiecki and Michał Meller Methods of Device Noise Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821 Marek Pawełczyk, Stanisław Wrona, and Krzysztof Mazur Contents xiii
- 18. Parametric Identification for Robust Control Piotr Kulczycki Abstract Many contemporary automatic control applications require parametric identification, taking into account results of estimation errors unavoidable in practice. The subject of this chapter constitutes a procedure enabling effective calculation of optimal, in the sense of minimal expectation of losses, estimator of a parameter in the case when losses resulting from negative and positive errors have significantly different influence on system operation, which forces the use of an asymmetric loss function. The conditional version of the issue is also considered in detail, as the process is significantly dependent on factors available metrologically during the system operation. This material has been provided in ready-to-use form, not requiring additional studies and bibliographical exploration. The presented concept is directed towards robust control problems. As an appendix, the analysis of solutions of differential equations with discontinuous right-hand side, subordinated for this purpose, has been presented, which may be useful for describing the dynamics of such resultant systems. 1 Introduction In the early period of the development of control engineering, a fundamental disci- pline of automatic control, concepts were dominated by the intuition and experience of designers, without any dedicated mathematical apparatus. A significant change occurred in the interwar period, thanks to the application of basic elements of oper- ational calculus: Laplace transformation and the transfer function concept [47]. The fundamental requirement formulated towards the constructed regulation systems was stability [4]. The research was facilitated by rich and advantageous mathematical P. Kulczycki (B) Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01447 Warsaw, Poland e-mail: kulczycki@ibspan.waw.pl; kulczycki@agh.edu.pl Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, Mickiewicza 30, 30059 Cracow, Poland © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. Kulczycki et al. (eds.), Automatic Control, Robotics, and Information Processing, Studies in Systems, Decision and Control 296, https://doi.org/10.1007/978-3-030-48587-0_1 3
- 19. 4 P. Kulczycki apparatus provided for that purpose. The main achievement of that period was the regulator PID, exceptional in its simplicity and convenient in interpretation, and simultaneously surprisingly effective [2]. To this day it constitutes a fundamental tool in industrial applications, undergoing permanently various modifications and modernizations. However, with the growth in requirements formulated towards automatic control systems the natural question arose, which among the stable systems should be recog- nized and applied as the best (optimal)? Using operational calculus there was no possible way to solve such a defined problem. The conditions of the second World War and the later period of the Cold War, when the best possible arms quality trans- ferred into military success and human life, gradually magnified the growth in impor- tance of optimization and research conducted in this field. In the fifties, operational calculus was replaced by differential equations, which enabled rapid development in the optimal control theory [3, 38]. At the end of this decade two fundamental theo- ries allowing the establishment of control of this kind were published: Pontryagin Maximum Principle and Bellman Dynamical Programming, at this time engineering practice unambiguously distinguished two types of optimal control: time-optimal (minimum-time), relying on reaching the target in the shortest time ([3], Chapter 7), also with a quadratic performance index, directed primarily towards minimalizing energy used ([3], Chapter 9). Although, in many applications of optimal control, sublime and extreme in nature, it only worked well in the assumed conditions or those not much different from them. As a consequence this led to a transfer in interest of practitioners in the direction of so- called robust control, which also worked correctly when real conditions significantly differed from those assumed [5, 50]. In practice this often means that the values of the parameter or parameters, used in a given model, can significantly differ from the real ones occurring in the object. An illustration of this may be for example the mass of an aircraft, changing due to fuel use, that is impossible to measure directly, which is more precise. Robust control systems were frequently derived from the concept of optimal control; the robustness of the former was obtained at the cost of a small worsening of quality in the sense of the performance index used in the baseline optimal structure being accepted in advance. In this chapter the ready-to-use procedure for calculation of the parameter value is presented, which minimalizes the expected losses resulting from imprecise esti- mation, different for negative and positive errors. It will provide the solution for the introduced issue of the robust control problem. Thus, in the next subchapter, the motivation for the identification task considered here will be presented. The third subchapter covers the mathematical preliminaries, the methodology of statis- tical kernel estimators is described here, which will be used in the fourth main subchapter for the creation of a procedure for calculation of the optimal value of the studied parameter. In the first two points of this subchapter, the subject of detailed considerations are basic estimation issues with asymmetric linear and then quadratic loss function, after which in the third point, the above material will be submitted in the conditional approach. These issues may be generalized into polynomials of higher orders and the multidimensional case, where the subject of investigations are
- 20. Parametric Identification for Robust Control 5 parameters with the correlated impact on estimation errors; these problems will be briefly outlined in the the fifth subchapter. Finally at the end, the appendix presents considerations concerning the solutions of differential equations with discontinuous left-hand side, which maybe helpful for the formal description of a robust control system, arising as a result of the application of the procedure previously worked out in this chapter. 2 Motivation and Problem Formulation As grounds for the concept outlined here, three illustrative examples from the area of robust control are presented. Assume the classic dynamic system ẋ1(t) ẋ2(t) = 0 1 0 0 x1(t) x2(t) + 0 1 M u(t), (1) where the positive parameter M represents a mass submitted to the action of force, according to the second law of Newtonian dynamics. Here x1, x2 and u denote the position and velocity of the above mass and the force treated as a control. This system forms the basis of the majority of research in robotics, leading to considerably more complex models with specificity appropriate for the problem under research [35, 37]. Consider the time-optimal control problem, whose basic form relies on bringing the state of the system to the origin of coordinates by the bounded control, in a minimal and finite time ([3], Subchapter 7.2). Fundamental importance for phenomena taking place in the control system has an appropriate identification of the value of the parameter M, introduced in the formula (1). The control algorithm is however defined on the basis the value of its estimator M , different in practice from the value M occurring in the object. In the purely hypothetical case M = M, therefore when the estimator and real values are equal, the process is regular in nature. The system state reaches the origin in a minimal and finite time. In turn, if the estimator is overestimated (i.e. for M M), then over-regulations appear in the system. Its state oscillates around the origin and reaches it in a finite time, however larger than minimal. Finally, in the case of underestimation (i.e. when M M), the system state moves along a so-called sliding trajectory [8, 48] and in the end reaches the origin in a finite time, again larger than minimal. Detailed considerations on the topic of the phenomena occurring in the system under investigation can be found in the paper [30]. On the margin of the above considerations, in order to illustrate the formal complexity of the task, one can mark that in the first two cases, the differential equation (1) possesses unique C-solutions (in the Caratheodory sense) and in the third unique F-solutions (in the Filippov
- 21. 6 P. Kulczycki sense). For the purpose of unifying all three cases, one can use K-solutions (in the Krasovski sense), more general than C- and F-solutions. In many issues, the family of K-solutions turns out to be too numerous, however in the system considered here it is fortunately unique [13]. Furthermore, in the probabilistic approach, the family of K-solutions occurring for particular values of the estimated parameter constitutes a stochasticprocess[14],whichenablesanapplicationofarichmathematicalapparatus dedicated to this issue. These tasks have been specified and illustrated in the appendix to this chapter. Therefore, in each of the above three cases, the system trajectory reaches the origin in the finite time, however, the phenomena occurring in the system as a result of under- and overestimation of the parameter M are completely different and thus one can expect significantly different times for reaching the target set in the ranges of negative and positive errors M − M. Figure 1 shows the graph of the time to reach the origin by the system (1) state, with exemplary fixed M = 1. It can be seen that the growth in the value of this index is approximately directly proportional to the estimation error M − M, although with different coefficients for negative and positive errors: l M , M = ⎧ ⎨ ⎩ −a M − M when M − M ≤ 0 b M − M when M − M ≥ 0 , (2) while the constants a and b are positive; it is worth underlining they do not necessarily equal. In the case of the stronger interaction of the parameter identified on the behavior of the system, for example where its impact is “multitrack” (multifaceted), the quadratic form of the performance index may be appropriate. As illustrative example, consider the dynamic system Fig. 1 Value of time-optimal index for different values of estimator M , with M = 1
- 22. Parametric Identification for Robust Control 7 Fig. 2 Value of quadratic index for different values of estimator λ , with λ = 1 ẋ1(t) ẋ2(t) = λ 1 0 λ x1(t) x2(t) + 0 λ u(t), (3) where the constant λ is different from 0. It is worth noting that it occurs both in the state A and in the control B matrixes. The system (3) will be submitted to the classic optimal control with the quadratic performance index ([3], Subchapter 9.5) with unique matrixes and infinite final time. Figure 2 shows the value of the above index as the function of the estimator λ value, with exemplary fixed λ = 1. One can see that the the value of the loss function resulting from the estimation error may be successfully approximated by a asymmetric quadratic form: l λ , λ = ⎧ ⎨ ⎩ a λ − λ 2 when λ − λ ≤ 0 b λ − λ 2 when λ − λ ≥ 0 , (4) where again the coefficients a and b are positive and not necessarily equal. Now consider the two-dimensional case. Assume the dynamic system, originated by the introduction into the system (4) of the drive inertia; therefore, substituting the control u by the third coordinate of the state vector x3, and so in consequence ⎡ ⎣ ẋ1(t) ẋ2(t) ẋ3(t) ⎤ ⎦ = ⎡ ⎣ λ 1 0 0 λ λ 0 0 −τ ⎤ ⎦ ⎡ ⎣ x1(t) x2(t) x3(t) ⎤ ⎦ + ⎡ ⎣ 0 0 1 ⎤ ⎦u(t), (5) where τ is a positive drive inertia constant. Thus the parameters λ τ originated from different subsystems (λ from the object itself and τ characterizing the drive) are subjected to estimation, and therefore can be treated as independent. Once again the system will be submitted to control with the quadratic performance index ([3],
- 23. 8 P. Kulczycki Fig. 3 Value of two-dimensional quadratic index for different values of estimators λ τ , with λ τ = 1 1 Subchapter 9.5) with unique matrixes and infinite time. Figure 3 shows the spatial graph of this index as the function of the estimators’ λ τ values, with exemplary fixed λ τ = 1 1 . It can be seen that losses resulting from estimation errors can be approximated with high precision by the two-dimensional asymmetric quadratic function: l λ T , λ T = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ar λ − λ 2 + aru λ − λ T − T + au T − T 2 when λ − λ ≥ 0 i T − T ≥ 0 al λ − λ 2 + alu λ − λ T − T + au T − T 2 when λ − λ ≤ 0 i T − T ≥ 0 al λ − λ 2 + ald λ − λ T − T + ad T − T 2 when λ − λ ≤ 0 i T − T ≤ 0 ar λ − λ 2 + ard λ − λ T − T + ad T − T 2 when λ − λ ≥ 0 i T − T ≤ 0 , (6) where al, ar , au, ad 0, ald, aru ≥ 0 and alu, ard ≤ 0. The mixed coefficients ald, aru, alu, ard represent mutual correlation of estimation errors of both parameters. In the case when they are equal to zero, the problem de facto reduces to two separate tasks with the asymmetric quadratic loss function (4). The above examples illustrate the problem of quality of robust control—orig- inating from the conceptions of optimal structures—with respect to the errors of parameters estimation, in the cases where the losses resulting from this reason can be
- 24. Parametric Identification for Robust Control 9 described by the asymmetric linear (2) or quadratic (4), (6) functions. The polynomial form seems to be a suitable compromise between the precision and the complexity and in consequence usefulness of the approach proposed [36]. It is worth furthermore generalizing the classic formulation of the research problem presented above. Thus, the more complex and refined the models currently applied become, the less reasoned is a traditional perception of parameters identifica- tion as the task of calculation of a concrete (existing in reality but unknown) value of the parameter under investigation. Currently this task covers establishment of value, which will represent the whole range of multifaceted phenomena, simplified in the model to one, the only formally existing parameter. In such a situation, evaluation of the quality of the parameter identification cannot be made in the classical way by comparing the obtained value to the imagined “real”, although unknown, parameter value (because no such value exists), but rather through taking into account the impli- cations of particular values on the functioning of the system under investigation. This differs from the mathematical apparatus existing in the framework of point estima- tion from the classic mathematical statistic [34], becoming closer to the currently expanding data analysis [31–33]. Fortunately, development of modern, advanced and specialized identification methods enables a rapid expansion of contemporary computertechnology,supportingfromtheoreticalpointofviewITmethodsdedicated to it. In summary: in the method presented here, the value of estimator will be sought, minimizing the expectation value of losses resulting from estimation errors. One assumes availability of a finite number of metrologically obtained values for the esti- mated parameter. The knowledge of its distribution is not necessary; its establishment constitutes an integral part of the procedure. The conditional approach will also be proposed, designated for the case where the parameter under research is significantly dependent on the factor, which current value can be successively measured during realization of the control process, causing an increase in the models precision. The material is given in the complete form, not requiring additional investigations or bibliographical research. 3 Mathematical Preliminaries: Kernel Estimators Kernel estimators are counted among the nonparametric methods. Methods of this type do not require initial assumptions concerning the type of the existing distribu- tion. In their basic formula, kernel estimators serve to establish the fundamental func- tional characteristics of the distribution; its density. First, in Sect. 3.1 the basic one- dimensional case will be presented, generalized in Sect. 3.2 to the multidimensional approach.
- 25. 10 P. Kulczycki 3.1 One-Dimensional Case Let x denote the quantity under investigation. Let the m-elements set of independent measurements x1, x2, . . . , xm ∈ R (7) be given. The kernel estimator of the distribution density of the studied quantity f : R → [0, ∞) is defined in its standard form by the following formula f (x) = 1 mh m i=1 K x − xi h , (8) where the positive constant h is called the smoothing parameter, while the function K : R → [0, ∞) measurable, symmetrical with respect to zero and having the weak global maximum at this point, with unique integral R K(y)dy = 1, is named as the kernel. The interpretation of the above definition is illustrated in Fig. 4. In the case of the single element xi , the function K transposed by the vector xi and scaled by the coefficient h, represents estimation of the distribution after obtaining the value xi . For m independent values x1, x2, . . . , xm, it takes the form of the sum of such individual estimations. The coefficient 1/mh norms the function obtained to guarantee the condition R f (y)dy = 1. Finally, the formula (8) provides the tool for estimation of density of the set (7) distribution; it is illustrative and rather less complex for numerical calculations. In practice the choice of the kernel form ([20], Section 3.1.3; [49], Subchapters 2.7 and 4.5) has no significant importance, and thanks to this it is possible to primarily take into account the properties of the estimator obtained, e.g. its continuity, differ- entiability, or bounding of the support; advantageous from the point of view of the Fig. 4 One-dimensional kernel estimator (8)
- 26. Parametric Identification for Robust Control 11 specific application. In the method presented here, the Cauchy kernel K(x) = 2 π x2 + 1 2 (9) will be applied, because of its convenient integration properties. The constants W(K) = ∞ −∞ K(y)2 dy and U(K) = ∞ −∞ y2 K(y)dy, which are necessary for further calculations, equal in this case: W(K) = 5 4π (10) U(K) = 1. (11) Great importance for the estimation quality is, however, fixing of the smoothing parameter h value ([20], Section 3.1.5; [49], Chapter 3 and Subchapter 4.7). Fortu- nately for practitioners, many convenient calculational procedures, based the set (7), have been investigated. During the initial design phase, as well as in the case of the set size greater than 1000, one can recommend a simple approximate method with linear complexity regarding to the size m. In the remaining cases, the effective plug-in procedure, with quadratic complexity, can be implemented. The first method relies on the direct use of the formula h = 8 √ π 3 W(K) U(K)2 1 m 1/5 σ , (12) where σ denotes the classic standard deviation estimator: σ = 1 m − 1 m i=1 (xi )2 − 1 m(m − 1) m i=1 xi 2 . (13) In turn, the second procedure, plug-in, is based on the application of the above formula, following several improvements in such obtained value. Thus, in the case of the third degree procedure, one should sequentially apply the following dependencies: d10 = −945 64 √ πσ 11 , (14) where σ is given by the equality (13), and next gI I I = −2 K(8) (0) d10U K m 1/11 (15)
- 27. 12 P. Kulczycki gI I = −2 K(6) (0) d8, j (gI I I )U K m 1/9 (16) gI = −2 K(4) (0) d6, j (gI I )U K m 1/7 ; (17) finally h = 1 d4(gI ) W(K) U(K)2 1 m 1/5 , (18) while dp(g) = 1 m2gp+1 m i=1 m k=1 K(p) xi − xk g for p = 4, 6, 8. (19) The basic kernel K, used so far, exists in the above procedure only in the last formula (18). In the remaining formulas one can apply any form convenient for multiple differentiation. Commonly the normal kernel K(x) = 1 √ 2π exp − x2 2 (20) is accepted here. Then the factors appearing in the dependencies (15)–(17) and (19) are equal respectively: K(8)(x) = 1 √ 2π x8 − 28x6 + 210x4 − 420x2 − 105 exp − x2 2 , K(8)(0) = − 105 √ 2π (21) K(6)(x) = 1 √ 2π x6 − 15x4 + 45x2 − 15 exp − x2 2 , K(6)(0) = − 15 √ 2π (22) K(4) (x) = 1 √ 2π x4 − 6x2 + 3 exp − x2 2 , K(4) (0) = 3 √ 2π (23) U K = 1. (24)
- 28. Parametric Identification for Robust Control 13 3.2 Multidimensional Case The concept of the kernel estimator will now be extended to the multidimen- sional case. The role of the measurements (7) is taken then by the m-elements set of n-dimensional vectors: x1, x2, . . . , xm ∈ Rn . (25) The kernel estimator of the density of its distribution f : Rn → [0, ∞) can now be defined in the form f (x) = 1 m m i=1 K(x, xi , h), (26) where after indication of coefficients x = ⎡ ⎢ ⎢ ⎢ ⎣ x1 x2 . . . xn ⎤ ⎥ ⎥ ⎥ ⎦ , xi = ⎡ ⎢ ⎢ ⎢ ⎣ xi,1 xi,2 . . . xi,n ⎤ ⎥ ⎥ ⎥ ⎦ for i = 1, 2, . . . , m and h = ⎡ ⎢ ⎢ ⎢ ⎣ h1 h2 . . . hn ⎤ ⎥ ⎥ ⎥ ⎦ , (27) whereas the positive constants h j plays the role of smoothing parameters for partic- ular coordinates j = 1, 2, . . . , n, while with the contemporary applications the multidimensional kernel K is commonly used in the product form K(x, xi , h) = n j=1 1 h j K xj − xi, j h j , (28) therefore, as the product of the one-dimensional kernels K (discussed in details in the previous Sect. 3.1) scaled with the coefficient 1/h j . Because of such decompo- sition with respect to particular coordinates, all the previously formulated comments concerning the fixing of the smoothing parameter value and selection of the kernel form are applicable also in the multidimensional case. In particular, the smoothing parameters h j can be calculated using the approximate or plug-in methods, separately for each coordinate, with the one-dimensional kernel K assumed in the Cauchy form (9). The two-dimensional kernel estimator has been illustrated in Fig. 5. Note also that in the one-dimensional case, after substituting n = 1 and omitting the index j redundant here, the definitions (26)–(28) lead to the basic form (8). In practice, various modifications and generalizations of the presented above stan- dard form of the kernel estimators are possible. One should remember, however, that they increase the complexity of the formulas, and reduce the ease of interpretation and consequently mean that the solution of the problem becomes less convenient for the potential user. Detailed description of the kernel estimators methodology can be
- 29. 14 P. Kulczycki Fig. 5 Two-dimensional kernel estimator (26); elements of set (25) are marked by crosses, lines denote the contours of estimator found in the monographs [20, 44, 49]. It is also worth recalling the publications [6, 7, 40–42, 45] as well as the new subject book [9], where one can also obtain broad current literature. The sister problem of the kernel regression can be found in [10]. The implicational possibilities to identification of atypical elements (outliers), clus- tering, and classification, three basic procedures of data analysis and exploration, are summarized in the survey article [21] and also in the work [26] for the conditional approach. In the paper [19] the concept of the sophisticated diagnostic system based on kernel estimators is presented. 4 Procedure for Parametric Identification 4.1 Linear Case Assume the asymmetric linear loss function (2), presented below using standard notation l x , x = −a x − x when x − x ≤ 0 b x − x when x − x ≥ 0 , (29) where x represents the investigated parameter, x its estimator, and the coefficients a and b are positive (non-necessarily equal to each other). Consider the set
- 30. Parametric Identification for Robust Control 15 x1, x2, . . . , xm ∈ R (30) of values of the parameter under research, obtained metrologically in the framework of independent measurements. If one denotes by f the kernel estimator of its distri- bution density (calculated following Subchapter 3), then the dependence (29) shows directly that the expected value of losses resulting from taking the estimator x can be given as x = b x −∞ x − x f (x)dx − a ∞ x x − x f (x)dx. (31) Let the function f be continuous; this condition can be easily obtained while using the continuous kernel K, e.g. Cauchy (9), for the construction of the kernel estimator. It is then not difficult to show, that the function , defined above, reaches the minimum for the value x fulfilling the equation x −∞ f (x)dx = a a + b . (32) Because 0 a/(a + b) 1, then the above equation has a solution, and if the support of the function f is coherent, it is unique. Such a condition can be obtained using the kernel K with positive values, e.g. Cauchy (9), for the construction of the kernel estimator f . And finally, due to equality a a + b = a b a b + 1 , (33) it can be seen that the separate fixing of the parameters a and b is not necessary, but only the ratio a/b. Assume, therefore, that the function f is a kernel estimator constructed using a continuous kernel with positive values. Also, I denotes the primitive function of the kernel K, i.e. I (x) = x −∞ K(y)dy. (34) In the case of the Cauchy kernel (9), this function can be expressed by the analytical formula
- 31. 16 P. Kulczycki I (x) = 1 π arctg(x) + x π x2 + 1 + 1 2 . (35) Let us introduce the auxiliary notation Pi (x) = I x − xi h for i = 1, 2, . . . , m. (36) The criterion (32) takes the form 1 m m i=1 Pi x − a a + b = 0. (37) Denoting the left side of the above equation by L x , i.e. L x = 1 m m i=1 Pi x − a a + b , (38) its derivative is L x = f x . (39) Note that the function I is increasing. This feature, therefore, passes to Pi and in consequence also to L. Because an analytic form of the derivative is accessible, then these properties suggest use of the classic Newton algorithm [12] in order to solve the Eq. (32). Then this solution can be found as a limit of the sequence x k !∞ k=0 defined by the formulas x 0 = 1 m m i=1 xi (40) x k+1 = x k − L x k L x k for k = 0, 1, . . . . (41) As a stop criterion the following condition is assumed: x k − x k−1 ≤ 0, 01 σ , (42) where σ is defined by the formula (13). Summarizing, having independent measurements of the investigated parameter (30), its optimal (in the sense of the expected loss value, in the case when the loss function is given in the asymmetrical linear form (29)) the estimator can be calculated by the recurrence algorithm (40)–(41) having the stop condition (42) with (13). The
- 32. Parametric Identification for Robust Control 17 functions L and L occurring there are given by the formulas (38)–(39), and in turn Pi and f are defined by the dependencies (36) and (8). If one uses the Cauchy kernel recommended here, then the functions K and I are denoted by the formulas (9) and (35), respectively. The value of the smoothing parameter h can be calculated directly from the formula (12) with (13), or by the plug-in method (14)–(24) and (13). When the Cauchy kernel is used, the values of the functionals U and W existing there are given by the formulas (10)–(11). It should be noted that if a = b, the criterion (32) indicates directly the median. The first results in the range of estimation with the asymmetrical loss function, concerning the linear case, have been published in the paper [18]. 4.2 Quadratic Case The asymmetric quadratic loss function (3), described using standard notation, takes the form l x , x = # a x − x 2 for x − x ≤ 0 b x − x 2 for x − x ≥ 0 , (43) where x, x and a, b have the same interpretation as in the case of the asymmetric linear loss function (29), similarly as the set of measurements x1, x2, . . . , xm given there by the formula (30). Let f be established, using the material from Subchapter 3, by the kernel estimator methodology; while the applied kernel K is continuous with positive values, and additionally such that ∞ −∞ xK(x)dx ∞. (44) The Cauchy kernel (9) fulfils the above condition. The form of the definition (43) constitutes directly that the expectation value of losses, resulting from the accepted value of the estimator x , is given by the formula x = b x −∞ x − x 2 f (x)dx + a ∞ x x − x 2 f (x)dx. (45) It is not difficult to show that this function reaches a minimum for the value x , which is the solution of the equation
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- 34. Camilla thanked him, and ran eagerly to speak to her sister; but she was not in her room; nor was she with her uncle; nor yet with Dr. Orkborne. She returned uneasily to the parlour, and said she would seek her in the park. Edgar followed; but they looked around for her in vain: he then, deeming the danger urgent, left her, to hasten to the spot where he had seen the post-chaise. Camilla ran on alone; and, when she reached the park gate, perceived her sister, Molly Mill, and Bellamy, in the lane. They heard her quick approach, and turned round. The countenance of Bellamy exhibited the darkest disappointment, and that of Eugenia the most excessive confusion. 'Now then, Sir,' she cried, 'delay our separation no longer.' 'Ah, permit me,' said he, in a low voice, 'permit me to hope you will hear my last sad sentence, my final misery, another day!—I will defer my mournful departure for that melancholy joy, which is the last I shall feel in my wretched existence!' He sighed so deeply, that Eugenia, who seemed already in much sorrow, could not utter an abrupt refusal; and, as Camilla now advanced, she turned from him, without attempting to say any thing further. Camilla, in the delight of finding her sister safe, after the horrible apprehensions she had just experienced, could not speak to her for tears. Abashed at once, and amazed, Eugenia faintly asked what so affected her? She gave no explanation, but begged her to turn immediately back. Eugenia consented; and Bellamy, bowing to them both profoundly, with quick steps walked away.
- 35. Camilla asked a thousand questions; but Eugenia seemed unable to answer them. In a few minutes they were joined by Edgar, who, walking hastily up to them, took Camilla apart. He told her he firmly believed a villainous scheme to have been laid: he had found the chaise still in waiting, and asked the postillion to whom he belonged. The man said he was paid for what he did; and refused giving any account of himself. Bellamy then appeared: he seemed confounded at his sight; but neither of them spoke; and he left him and his chaise, and his postillion, to console one another. He doubted not, he said, but the design had been to carry Eugenia off, and he had probably only pretended to take leave, that the chaise might advance, and the postillion aid the elopement: though finding help at hand, he had been forced to give up his scheme. Camilla even with rapture blest his fortunate presence; but was confounded with perplexity at the conduct of Eugenia. Edgar, who feared her heart was entangled by an object who sought only her wealth, proposed dismissing Molly Mill, that he might tell her himself the opinion he had conceived of Bellamy. Camilla overtook her sister, who had walked on without listening to or regarding them; and, sending away Molly, told her Edgar wished immediately to converse with her, upon something of the utmost importance. 'You know my high esteem of him,' she answered; 'but my mind is now occupied upon a business of which he has no information, and I entreat that you will neither of you interrupt me.' Camilla, utterly at a loss what to conjecture, joined Mandlebert alone, and told him her ill success. He thought every thing was to be feared from the present state of the affair, and proposed revealing at once all he knew of it to Mr. Tyrold: but Camilla desired him to take no step till she had again expostulated with her sister, who might else be seriously hurt or offended. He complied, and said he would
- 36. continue in the house, park, or environs, incessantly upon the watch, till some decisive measure were adopted. Joining Eugenia then again, she asked if she meant seriously to encourage the addresses of Bellamy. 'By no means,' she quietly answered. 'My dear Eugenia, I cannot at all understand you; but it seems clear to me that the arrival of Edgar has saved you from some dreadful violence.' 'You hurt me, Camilla, by this prejudice. From whom should I dread violence? from a man who—but too fatally for his peace—values me more than his life?' 'If I could be sure of his sincerity,' said Camilla, 'I should be the last to think ill of him: but reflect a little, at least, upon the risk that you have run; my dear Eugenia! there was a post-chaise in waiting, not twenty yards from where I stopt you!' 'Ah, you little know Bellamy! that chaise was only to convey him away; to convey him, Camilla, to an eternal banishment!' 'But why, then, had he prevailed with you to quit the park?' 'You will call me vain if I tell you.' 'No; I shall only think you kind and confidential.' 'Do me then the justice,' said Eugenia, blushing, 'to believe me as much surprised as yourself at his most unmerited passion: but he told me, that if I only cast my eyes upon the vehicle which was to part him from me for ever, it would not only make it less abhorrent to him, but probably prevent the loss of his senses.' 'My dear Eugenia,' said Camilla, half smiling, 'this is a violent passion, indeed, for so short an acquaintance!'
- 37. 'I knew you would say that,' answered she, disconcerted; 'and it was just what I observed to him myself: but he satisfied me that the reason of his feelings being so impetuous was, that this was the first and only time he had ever been in love.—So handsome as he is!— what a choice for him to make!' Camilla, tenderly embracing her, declared, 'the choice was all that did him honour in the affair.' 'He never,' said she, a little comforted, 'makes me any compliments; I should else disregard, if not disdain him: but indeed he seems, notwithstanding his own extraordinary manly beauty, to be wholly superior to external considerations.' Camilla now forbore expressing farther doubt, from the fear of painful misapprehension; but earnestly entreated her to suffer Edgar to be entrusted and consulted: she decidedly, however, refused her consent. 'I require no advice,' cried she, 'for I am devoted to my uncle's will: to speak then of this affair would be the most cruel indelicacy, in publishing a conquest which, since it is rejected, I ought silently, though gratefully, to bury in my own heart.' She then related the history of all that had passed to Camilla; but solemnly declared she would never, to any other human being, but him who should hereafter be entitled to her whole heart, betray the secret of the unhappy Bellamy.
- 38. CHAPTER II An Adviser The wish of Camilla was to lay this whole affair before her father; but she checked it, from an apprehension she might seem displaying her duty and confidence at the expence of those of her sister; whose motives for concealment were intentionally the most pure, however, practically, they might be erroneous; and whom she both pitied and revered for her proposed submission to her uncle, in opposition to her palpable reluctance. She saw not, however, any obstacle to consulting with Edgar, since he was already apprised of the business, and since his services might be essentially useful to her sister: while, with respect to herself, there seemed, at this time, more of dignity in meeting than shunning his friendly intercourse, since his regard for her seemed to have lost all its peculiarity. He has precisely, cried she, the same sentiments for my sisters as for me,—he is equally kind, disinterested, and indifferent to us all! anxious alike for Eugenia with Mr. Bellamy, and for me with the detestable Major! Be it so!—we can no where obtain a better friend; and I should blush, indeed, if I could not treat as a brother one who can treat me as a sister. Tranquil, though not gay, she returned to converse with him; but when she had related what had passed, he confessed that his uneasiness upon the subject was increased. The heart of Eugenia appeared to him positively entangled; and he besought Camilla not to lose a moment in acquainting Mr. Tyrold with her situation. She pleaded against giving this pain to her sister with energetic affection: her arguments failed to convince, but her eloquence powerfully touched him; and he contented himself with only
- 39. entreating that she would again try to aid him with an opportunity of conversing with Eugenia. This she could not refuse; nor could he then resist the opportunity to inquire why Mrs. Arlbery had left her and Lavinia at the play. She thanked him for remembering his character of her monitor, and acknowledged the fault to be her own, with a candour so unaffected, that, captivated by the soft seriousness of her manner, he flattered himself his fear of the Major was a chimæra, and hoped that, as soon as Sir Hugh was able to again join his family, no impediment would remain to his begging the united blessings of the two brothers to his views. When Camilla told her sister the request of Edgar, she immediately suspected the attachment of Bellamy had been betrayed to him; and Camilla, incapable of any duplicity, related precisely how the matter had passed. Eugenia, always just, no sooner heard than she forgave it, and accompanied her sister immediately down stairs. 'I must rest all my hope of pardon,' cried Edgar, 'for the part I am taking, to your conviction of its motive; a filial love and gratitude to Mr. Tyrold, a fraternal affection and interest for all his family.' 'My own sisterly feelings,' she answered, 'make me both comprehend and thank your kind solicitude: but, believe me, it is now founded in error. I am shocked to find you informed of this unhappy transaction; and I charge and beseech that no interference may wound its ill-fated object, by suffering him to surmise your knowledge of his humiliating situation.' 'I would not for the world give you pain,' answered Edgar: 'but permit me to be faithful to the brotherly character in which I consider myself to stand with you ... all.' A blush had overspread his face at the word Brotherly; while at that of all, which recovered him, a still deeper stained the cheeks of Camilla: but neither of them looked at the other; and Eugenia was too self-absorbed to observe either.
- 40. 'Your utter inexperience in life,' he continued, 'makes me, though but just giving up leading-strings myself, an adept in the comparison. Suffer me then, as such, to represent to you my fears, that your innocence and goodness may expose you to imposition. You must not judge all characters by the ingenuousness of your own; nor conclude, however rationally and worthily a mind such as yours might—may—and will inspire a disinterested regard, that there is no danger of any other, and that mercenary views are out of the question, because mercenary principles are not declared.' 'I will not say your inference is severe,' replied Eugenia, 'because you know not the person of whom you speak: but permit me to make this irrefragable vindication of his freedom from all sordid motives; he has never once named the word fortune, neither to make any inquiries into mine, nor any professions concerning his own. Had he any inducement to duplicity, he might have asserted to me what he pleased, since I have no means of detection.' 'Your situation,' said Edgar, 'is pretty generally known; and for his ... pardon me if I hint it may be possible that silence is no virtue. However, since I am unacquainted, you say, with his character, will you give me leave to make myself better informed?' 'There needs no investigation; to me it is perfectly known.' 'Forgive me if I ask how!' 'By his letters and by his conversation.' A smile which stole upon the features of Edgar obliged him to turn his head another way; but presently recovering, 'My dear Miss Eugenia,' he cried, 'will it not be most consonant to your high principles, and scrupulous delicacy, to lay the whole of what has passed before Mr. Tyrold?' 'Undoubtedly, if my part were not strait forward. Had I the least hesitation, my father should be my immediate and decisive umpire.
- 41. But ... I am not at liberty even for deliberation!—I am not ... I know ... at my own disposal!'— She blushed and looked down, confused; but presently, with firmness, added, 'It is not, indeed, fit that I should be; my uncle completely merits to be in all things my director. To know his wishes, therefore, is not only to know, but to be satisfied with my doom. Such being my situation, you cannot misunderstand my defence of this unhappy young man. It is but simple justice to rescue an amiable person from calumny.' 'Let us allow all this,' said Edgar; 'still I see no reason why Mr. Tyrold....' 'Mr. Mandlebert,' interrupted she, 'you must do what you judge right. I can desire no one to abstain from pursuing the dictates of their own sense of honour. I leave you, therefore, unshackled: but there is no consideration which, in my opinion, can justify a female in spreading, even to her nearest connexions, an unrequited partiality. If, therefore, I am forced to inflict this undue mortification, upon a person to whom I hold myself so much obliged, an uneasiness will remain upon my mind, destructive of my forgetfulness of an event which I would fain banish from my memory.' She then refused to be any longer detained. 'How I love the perfect innocence, and how I reverence the respectable singularity of that charming character!' cried Edgar; 'yet how vain are all arguments against such a combination of fearless credulity, and enthusiastic reasoning? What can we determine?' 'I am happy to retort upon you that question,' replied Camilla; 'for I am every way afraid to act myself, lest I should hurt this dear sister, or do wrong by my yet dearer father.' 'What a responsibility you cast upon me! I will not, however, shrink from it, for the path seems far plainer to me since I have had this conversation. Eugenia is at present safe; I see, now, distinctly, her
- 42. heart is yet untouched. The readiness with which she met the subject, the openness with which she avows her esteem, the unembarrassed, though modest simplicity with which she speaks of his passion and his distress, all shew that her pity results from generosity, not from love. Had it been otherwise, with all her steadiness, all her philosophy, some agitation and anxiety would have betrayed her secret soul. The internal workings of hopes and fears, the sensitive alarms of repressed consciousness....' A deep glow, which heated his face, forced him here to break off; and, abruptly leaving his sentence unfinished, he hastily began another. 'We must not, nevertheless, regard this as security for the future, though it is safety for the present; nor trust her unsuspicious generosity of mind to the dangerous assault of artful distress. I speak without reserve of this man; for though I know him not, as she remonstrated, I cannot, from the whole circumstances of his clandestine conduct, doubt his being an adventurer.... You say nothing? tell me, I beg, your opinion.' Camilla had not heard one word of this last speech. Struck with his discrimination between the actual and the possible state of Eugenia's mind, and with the effect the definition had produced upon himself, her attention was irresistibly seized by a new train of ideas, till finding he waited for an answer, she mechanically repeated his last word 'opinion?' He saw her absence of mind, and suspected his own too palpable disturbance had occasioned it: but in what degree, or from what sensations, he could not conjecture. They were both some time silent; and then, recollecting herself, she said it was earnestly her wish to avoid disobliging her sister, by a communication, which, made by any one but herself, must put her into a disgraceful point of view. Edgar, after a pause, said, they must yield, then, to her present fervour, and hope her sounder judgment, when less played upon, would see clearer. It appeared to him, indeed, that she was so free,
- 43. at this moment, from any dangerous impression, that it might, perhaps, be even safer to submit quietly to her request, than to urge the generous romance of her temper to new workings. He undertook, meantime, to keep a constant watch upon the motions of Bellamy, to make sedulous inquiries into his character and situation in life, and to find out for what ostensible purpose he was in Hampshire: entreating leave to communicate constantly to Camilla what he might gather, and to consult with her, from time to time, upon what measures should be pursued: yet ultimately confessing, that if Eugenia did not steadily persist in refusing any further rejections, he should hold himself bound in conscience to communicate the whole to Mr. Tyrold. Camilla was pleased, and even thankful for the extreme friendliness and kind moderation of this arrangement; yet she left him mournfully, in a confirmed belief his regard for the whole family was equal. Eugenia, much gratified, promised she would henceforth take no step with which Edgar should not first be acquainted.
- 44. CHAPTER III Various Confabulations Mr. Tyrold saw, at first, the renewed visits of Edgar at Cleves with extreme satisfaction; but while all his hopes were alive from an intercourse almost perpetual, he perceived, with surprise and perplexity, that his daughter became more and more pensive after every interview: and as Edgar, this evening, quitted the house, he observed tears start into her eyes as she went up stairs to her own room. Alarmed and disappointed, he thought it now high time to investigate the state of the affair, and to encourage or prevent future meetings, as it appeared to him to be propitious or hopeless. Penetrated with the goodness, while lamenting the indifference of Edgar, Camilla had just reached her room; when, as she turned round to shut her door, Mr. Tyrold appeared before her. Hastily, with the back of her hand, brushing off the tears from her eyes, she said, 'May I go to my uncle, Sir?... can my uncle admit me?' 'He can always admit you,' he answered; 'but, just now, you must forget him a moment, and consign yourself to your father.' He then entered, shut the door, and making her sit down by him, said, 'What is this sorrow that assails my Camilla? Why is the light heart of my dear and happy child thus dejected?' Speech and truth were always one with Camilla; who, as she could not in this instance declare what were her feelings, remained mute and confounded.
- 45. 'Hesitate not, my dear girl,' cried he kindly, 'to unbosom your griefs or your apprehensions, where they will be received with all the tenderness due to such a confidence, and held sacred from every human inspection; unless you permit me yourself to entrust your best and wisest friend.' Camilla now trembled, but could not even attempt to speak. He saw her disorder, and presently added, 'I will forbear to probe your feelings, when you have satisfied me in one doubt;—Is the sadness I have of late remarked in you the effect of secret personal disturbance, or of disappointed expectation?' Camilla could neither answer nor look up: she was convinced, by this question, that the subject of her melancholy was understood, and felt wholly overcome by the deeply distressing confusion, with which wounded pride and unaffected virgin modesty impress a youthful female, in the idea of being suspected of a misplaced, or an unrequited partiality. Her silence, a suffocating sigh, and her earnest endeavour to hide her face, easily explained to Mr. Tyrold all that passed within; and respecting rather than wishing to conquer a shame flowing from fearful delicacy, 'I would spare you,' he said, 'all investigation whatever, could I be certain you are not called into any action; but, in that case, I know not that I can justify to myself so implicit a confidence, in youth and inexperience so untried in difficulties, so unused to evil or embarrassment as yours. Tell me then, my dear Camilla, do you sigh under the weight of any disingenuous conduct? or do you suffer from some suspence which you have no means of terminating?' 'My dearest father, no!' cried she, sinking upon his breast. 'I have no suspence!' She gasped for breath.
- 46. 'And how has it been removed, my child?' said Mr. Tyrold, in a mournful tone; 'has any deception, any ungenerous art....' 'O no, no!... he is incapable ... he is superior ... he....' She stopt abruptly; shocked at the avowal these few words at once inferred of her partiality, of its hopelessness, and of its object. She walked, confused, to a corner of the room, and, leaning against the wainscot, enveloped her face in her handkerchief, with the most painful sensations of shame. Mr. Tyrold remained in deep meditation. Her regard for Edgar he had already considered as undoubted, and her undisguised acknowledgment excited his tenderest sympathy: but to find she thought it without return, and without hope, penetrated him with grief. Not only his own fond view of the attractions of his daughter, but all he had observed, even from his childhood, in Edgar, had induced him to believe she was irresistibly formed to captivate him; and what had lately passed had seemed a confirmation of all he had expected. Camilla, nevertheless, exculpated him from all blame; and, while touched by her artlessness, and honouring her truth, he felt, at least, some consolation to find that Edgar, whom he loved as a son, was untainted by deceit, unaccused of any evil. He concluded that some unfortunate secret entanglement, or some mystery not yet to be developed, directed compulsatorily his conduct, and checked the dictates of his taste and inclination. Gently, at length, approaching her, 'My dearest child,' he said, 'I will ask you nothing further; all that is absolutely essential for me to know, I have gathered. You will never, I am certain, forget the noble mother whom you are bound to revere in imitating, nor the affectionate father whom your ingenuousness renders the most indulgent of your friends. Dry up your tears then, my Camilla, and command your best strength to conceal for ever their source, and, most especially ... from its cause.' He then embraced, and left her.
- 47. 'Yes, my dearest father,' cried she, as she shut the door, 'most perfect and most lenient of human beings! yes, I will obey your dictates; I will hide till I can conquer this weak emotion, and no one shall ever know, and Edgar least of all, that a daughter of yours has a feeling she ought to disguise!' Elevated by the kindness of a father so adored, to deserve his good opinion now included every wish. The least severity would have chilled her confidence, the least reproof would have discouraged all effort to self-conquest; but, while his softness had soothed, his approbation had invigorated her; and her feelings received additional energy from the conscious generosity with which she had represented Edgar as blameless. Blameless, however, in her own breast, she could not deem him: his looks, his voice, his manner, ... words that occasionally dropt from him, and meanings yet more expressive which his eyes or his attentions had taken in charge, all, from time to time, had told a flattering tale, which, though timidity and anxious earnestness had obscured from her perfect comprehension, her hopes and her sympathy had prevented from wholly escaping her. Yet what, internally, she could not defend she forgave; and, acquitting him of all intentional deceit, concluded that what he had felt for her, he had thought too slight and immaterial to deserve repressing on his own part, or notice on her's. To continue with him her present sisterly conduct was all she had to study, not doubting but that what as yet was effort, would in time become natural. Strengthened thus in fortitude, she descended cheerfully to supper, where Mr. Tyrold, though he saw with pain that her spirits were constrained, felt the fondest satisfaction in the virtue of her exertion. Her night passed in the consolation of self-applause. My dear father, thought she, will see I strive to merit his lenity, and that soothing consideration with the honourable friendship of Edgar, will be sufficient for the happiness of my future life, in the single and tranquil state in which it will be spent.
- 48. Thus comforted, she again met the eye of Mr. Tyrold the next day at breakfast; in the midst of which repast Edgar entered the parlour. The tea she was drinking was then rather gulped than sipped; yet she maintained an air of unconcern, and returned his salutation with apparent composure. Edgar, while addressing to Mr. Tyrold his inquiries concerning Sir Hugh, saw, from the window, his servant, whom he had out- galloped, thrown with violence from his horse. He rushed out of the parlour; and the first person to rise, with involuntary intent to follow him, was Camilla. But, as she reached the hall-door, she saw that the man was safe, and perceived that her father was the only person who had left the room besides herself. Ashamed, she returned, and found the female party collected at the windows. Hoping to retrieve the error of her eagerness, she seated herself at the table, and affected to finish her breakfast. Eugenia told her they had discovered the cause of the accident, which had been owing to a sharp stone that had penetrated into the horse's hoof, and which Edgar was now endeavouring to extract. A general scream, just then, from the window party, and a cry from Eugenia of 'O Edgar!' carried her again to the hall-door with the swiftness of lightning, calling out, 'Where?... What?... Good Heaven!...' Molly Mill, accidentally there before her, said, as she approached, that the horse had kicked Mr. Mandlebert upon the shoulder. Every thing but tenderness and terror was now forgotten by Camilla; she darted forward with unrestrained velocity, and would have given, in a moment, the most transporting amazement to Edgar, and to herself the deepest shame, but that Mr. Tyrold, who alone had his face that way, stopt, and led her back to the house, saying, 'There is no mischief; a bee stung the poor animal at the instant the stone was extracted, and the surprise and pain made it kick; but,
- 49. fortunately, without any bad effect. I wish to know how your uncle is; I should be glad you would go and sit with him till I can come.' With these words he left her; and, though abashed and overset, she found no sensation so powerful as joy for the safety of Edgar. Still, however, too little at ease for conversing with her uncle, she went straight to her own chamber, and flew involuntarily to a window, whence the first object that met her eyes was her father, who was anxiously looking up. She retreated, utterly confounded, and threw herself upon a chair at the other end of the room. Shame now was her only sensation. The indiscretion of her first surprise, she knew, he must forgive, though she blushed at its recollection; but a solicitude so pertinacious, an indulgence so repeated of feelings he had enjoined her to combat ... how could she hope for his pardon? or how obtain her own, to have forfeited an approbation so precious? She could not go to her uncle; she would have remained where she was till summoned to dinner, if the house-maid, after finishing all her other work, had not a third time returned to inquire if she might clean her room. She then determined to repair to the library, where she was certain only to encounter Eugenia, who would not torment, or Dr. Orkborne, who would not perceive her: but at the bottom of the stairs she was stopt by Miss Margland, who, with a malicious smile, asked if she was going to hold the bason? 'What bason?' cried she, surprised. 'The bason for the surgeon.' 'What surgeon?' repeated she, alarmed. 'Mr. Burton, who is come to bleed Mr. Mandlebert.'
- 50. She asked nothing more. She felt extremely faint, but made her way into the park, to avoid further conference. Here, in the most painful suspence, dying for information, yet shirking whoever could give it her, she remained, till she saw the departure of the surgeon. She then went round by a back way to the apartment of Eugenia, who informed her that the contusion, though not dangerous, was violent, and that Mr. Tyrold had insisted upon immediate bleeding. The surgeon had assured them this precaution would prevent any ill consequence; but Sir Hugh, hearing from the servants what had happened, had desired that Edgar would not return home till the next day. The joy of Camilla, that nothing was more serious, banished all that was disagreeable from her thoughts, till she was called back to reflections less consoling, by meeting Mr. Tyrold, as she was returning to her own room; who, with a gravity unusual, desired to speak with her, and preceded her into the chamber. Trembling, and filled with shame, she followed, shut the door, and remained at it without daring to look up. 'My dear Camilla,' cried he with earnestness, 'let me not hope in vain for that exertion you have promised me, and to which I know you to be fully equal. Risk not, my dear girl, to others, those outward marks of sensibility which, to common or unfeeling observers, seem but the effect of an unbecoming remissness in the self-command which should dignify every female who would do herself honour. I had hoped, in this house at least, you would not have been misunderstood; but I have this moment been undeceived: Miss Margland has just expressed a species of compassion for what she presumes to be the present state of your mind, that has given me the severest pain.' He stopt, for Camilla looked thunderstruck. Approaching her, then, with a look of concern, and a voice of tenderness, he kindly took her hand, and added: 'I do not tell you
- 51. this in displeasure, but to put you upon your guard. You will hear from Eugenia that we shall not dine alone; and from what I have dropt you will gather how little you can hope to escape scrutiny. Exert yourself to obviate all humiliating surmises, and you will amply be repaid by the balm of self-approbation.' He then kissed her, and quitted the room. She now remained in utter despair: the least idea of disgrace totally broke her spirit, and she sat upon the same spot on which Mr. Tyrold had left her, till the ringing of the second dinner bell. She then gloomily resolved to plead an head-ache, and not to appear. When a footman tapt at her door, to acquaint her every body was seated at the table, she sent down this excuse: forming to herself the further determination, that the same should suffice for the evening, and for the next morning, that she might avoid the sight of Edgar, in presence either of her father or Miss Margland. Eugenia, with kind alarm, came to know what was the matter, and informed her, that Sir Hugh had been so much concerned at the accident of Edgar, that he had insisted upon seeing him, and, after heartily shaking hands, had promised to think no more of past mistakes and disappointments, as they had now been cleared up to the county, and desired him to take up his abode at Cleves for a week. Camilla heard this with mixt pleasure and pain. She rejoiced that Edgar should be upon his former terms with her beloved uncle; but how preserve the caution demanded from her for so long a period, in the constant sight of her now watchful father, and the malicious Miss Margland? She had added to her own difficulties by this present absconding, and, with severe self-blame, resolved to descend to tea. But, while settling how to act, after her sister had left her, she was struck with
- 52. hearing the name of Mandlebert pronounced by Mary, the house- maid, who was talking with Molly Mill upon the landing place. Why it had been spoken she knew not; but Molly answered: 'Dearee me, never mind; I'll help you to do his room, if Nanny don't come in time. My little mistress would rather do it herself, than he should want for anything.' 'Why, it's natural enough,' said Mary, 'for young ladies to like young gentlemen; and there's none other comes a nigh 'em, which I often thinks dull enough for our young misses. And, to be certain, Mr. Mandlebert would be as pretty a match for one of 'em as a body could desire.' 'And his man,' said Molly, 'is as pretty a gentleman sort of person, to my mind, as his master. I'm sure I'm as glad as my young lady when they comes to the house.' 'O, as to Miss Eugeny,' said Mary, 'I believe, in my conscience, she likes our crack-headed old Doctor as well as e'er a young gentleman in Christendom; for there she'll sit with him, hour by hour, poring over such a heap of stuff as never was seed, reading, first one, then t'other, God knows what; for I believe never nobody heard the like of it before; and all the time never give the old Doctor a cross word.—' 'She never given nobody a cross word,' interrupted Molly; 'if I was Mr. Mandlebert, I'd sooner have her than any of 'em, for all she's such a nidging little thing.' 'For certain,' said Mary, 'she's very good, and a deal of good she does, to all as asks her; but Miss Camilla for my money. She's all alive and merry, and makes poor master young again to look at her. I wish Mr. Mandlebert would have her, for I have overheard Miss Margland telling Miss Lynmere she was desperate fond of him, and did all she could to get him.' Camilla felt flushed with the deepest resentment, and could scarcely command herself to forbear charging Miss Margland with this persecuting cruelty.
- 53. Nanny, the under house-maid, now joining them, said she had been detained to finish altering a curtain for Miss Margland. 'And the cross old Frump,' she added, 'is in a worse spite than ever, and she kept abusing that sweet Mr. Mandlebert to Miss Lynmere all the while, till she went down to dinner, and she said she was sure it was all Miss Camilla's doings his staying here again, for she could come over master for any thing: and she said she supposed it was to have another catch at the young 'Squire's heart, but she hoped he would not be such a fool.' 'I'm sure I wish he would,' cried Molly Mill, 'if it was only to spite her, she's such a nasty old viper. And Miss Camilla's always so good- natured, and so affable, she'd make him a very agreeable wife, I dare say.' 'And she's mortal fond of him, that's true,' said Mary, 'for when they was both here, I always see her a running to the window, to see who was a coming into the park, when he was rode out; and when he was in the house, she never so much as went to peep, if there come six horses, one after t'other. And she was always a saying, Mary, who's in the parlour? Mary, who's below? while he was here; but before he come, duce a bite did she ask about nobody.' 'I like when I meets her,' said Molly Mill, 'to tell her Mr. Mandlebert's here, Miss; or Mr. Mandlebert's there, Miss;—Dearee me, one may almost see one self in her eyes, it makes them shine so.' Camilla could endure no more; she arose, and walked about the room; and the maids, who had concluded her at dinner, hearing her step, hurried away, to finish their gossiping in the room of Mandlebert. Camilla now felt wholly sunk; the persecutions of Miss Margland seemed nothing to this blow: they were cruel, she could therefore repine at them; they were unprovoked, she could therefore repel them: but to find her secret feelings, thus generally spread, and familiarity commented upon, from her own unguarded conduct,
- 54. exhausted, at once, patience, fortitude, and hope, and left her no wish but to quit Cleves while Edgar should remain there. Certain, however, that her father would not permit her to return to Etherington alone, a visit to Mrs. Arlbery was the sole refuge she could suggest; and she determined to solicit his permission to accept immediately the invitation of that lady.
- 55. CHAPTER IV A Dodging Camilla waited in the apartment of Mr. Tyrold till he came up stairs, and then begged his leave to spend a few days at the Grove; hinting, when he hesitated, though with a confusion that was hardly short of torture, at what had passed amongst the servants. He heard her with the tenderest pity, and the kindest praise of her sincerity; and, deeply as he was shocked to find her thus generally betrayed, he was too compassionate to point out, at so suffering a moment, the indiscretions from which such observations must have originated. Yet he saw consequences the most unpleasant in this rumour of her attachment; and though he still privately hoped that the behaviour of Mandlebert was the effect of some transient embarrassment, he wished her removed from all intercourse with him that was not sought by himself, while the incertitude of his intentions militated against her struggles for indifference. The result, therefore, of a short deliberation was to accede to her request. Camilla then wrote her proposition to Mrs. Arlbery, which Mr. Tyrold sent immediately by a stable-boy of the baronet's. The answer was most obliging; Mrs. Arlbery said she would herself fetch her the next morning, and keep her till one of them should be tired. The relief which this, at first, brought to Camilla, in the week's exertions it would spare, was soon succeeded by the most acute uneasiness for the critical situation of Eugenia, and the undoubted disapprobation of Edgar. To quit her sister at a period when she might serve her; ... to forsake Cleves at the moment Edgar was
- 56. restored to it, seemed selfish even to herself, and to him must appear unpardonable. 'Alas!' she cried, 'how for ever I repent my hasty actions! Why have I not better struggled against my unfortunate feelings?' She now almost hated her whole scheme, regretted its success, wished herself suffering every uneasiness Miss Margland could inflict, and all the shame of being watched and pitied by every servant in the house, in preference to deserting Eugenia, and making Mandlebert deem her unworthy. But self-upbraiding was all that followed her contrition: Mrs. Arlbery was to fetch her by appointment; and it was now too late to trifle with the conceding goodness of her father. She did not dare excuse herself from appearing at breakfast the next morning, lest Mr. Tyrold should think her utterly incorrigible to his exhortations. Edgar earnestly inquired after her health as she entered the room; she slightly answered she was better; and began eating, with an apparent eagerness of appetite: while he, who had expected some kind words upon his own accident, surprised and disappointed, could swallow nothing. Mr. Tyrold, seeing and pitying what passed in her mind, gave her a commission, that enabled her, soon, to leave the room without affectation; and, happy to escape, she determined to go down stairs no more till Mrs. Arlbery arrived. She wished to have conversed first upon the affairs of Eugenia with Edgar: but to name to him whither she was herself going, when she could not possibly name why; to give to him a surprise that must recoil upon herself in disapprobation, was more than she could endure. She had invested him with full powers to counsel and to censure her; he would naturally use them to dissuade her from a visit so ill-timed; and what could she urge in opposition to his arguments that would not seem trifling or wilful?
- 57. The present moment was all that occupied, the present evil all that ever alarmed the breast of Camilla: to avoid him, therefore, now, was the whole of her desire, unmolested with one anxiety how she might better meet him hereafter. She watched at her window till she saw the groom of Mrs. Arlbery gallop into the Park. She hastened then to take leave of Sir Hugh, whom Mr. Tyrold had prepared for her departure; but, at the door of his apartment, she encountered Edgar. 'You are going out?' cried he, perceiving an alteration in her dress. 'I am ... just going to ... to speak to my uncle,' cried she, stammering and entering the room at the same moment. Sir Hugh kindly wished her much amusement, and hoped she would make him long amends when he was better. She took leave; but again, on the landing-place, met Edgar, who, anxious and perlexed, watched to speak to her before she descended the stairs. Eagerly advancing, 'Do you walk?' he cried; 'may I ask? or ... am I indiscreet?' She answered she had something to say to Eugenia, but should be back in an instant. She then flew to the chamber of her sister, and conjured her to consult Edgar in whatever should occur during her absence. Eugenia solemnly consented. Jacob presently tapped at the door, to announce that Mrs. Arlbery was waiting below in her carriage. How to pass or escape Edgar became now her greatest difficulty; she could suggest nothing to palliate to him the step she was taking, yet could still less bear to leave him to wild conjecture and certain blame: and she was standing irresolute and thoughtful, when Mr. Tyrold came to summon her. After mildly representing the indecorum of detaining any one she was to receive by appointment, he took her apart, and putting a packet into her hand, 'I would not,' he said, 'agitate your spirits this
- 58. morning, by entering upon any topic that might disturb you: I have therefore put upon paper what I most desire you to consider. You will find it a little sermon upon the difficulties and the conduct of the female heart. Read it alone, and with attention. And now, my dearest girl, go quietly into the parlour, and let one brief and cheerful good-morrow serve for every body alike.' He then returned to his brother. She made Eugenia accompany her down stairs, to avoid any solitary attack from Edgar; he suffered them to pass; but followed to the parlour, where she hastily bid adieu to Miss Margland and Indiana; but was stopt from running off by the former, who said, 'I wish I had known you intended going out, for I designed asking Sir Hugh for the chariot for myself this morning, to make a very particular visit.' Camilla, in a hesitating voice, said she should not use her uncle's chariot. 'You walk then?' 'No, ... ma'am ... but—there is—there is a carriage—I believe, now at the door.' 'O dear, whose?' cried Indiana; 'do, pray, tell me where you are going?' while Edgar, still more curious than either, held out his hand to conduct her, that he might obtain better information. 'I am very glad your head-ache is so well,' said Miss Margland; 'but, pray—is Mr. Mandlebert to be your chaperon?' They both blushed, though both affected not to hear her: but, before they could quit the room, Indiana, who had run to a bow- window, exclaimed, 'Dear! if there is not Mrs. Arlbery in a beautiful high phaeton!' Edgar, astonished, was now as involuntarily drawing back, as Camilla, involuntarily, was hurrying on: but Miss Margland, insisting upon an answer, desired to know if she should return to dinner?
- 59. She stammered out, No. Miss Margland pursued her to ask at what time the chariot was to fetch her; and forced from her a confession that she should be away for some days. She was now permitted to proceed. Edgar, impressed with the deepest displeasure, leading her in silence across the hall: but, stopping an instant at the door, 'This excursion,' he gravely said, 'will rescue you from no little intended importunity: I had purposed tormenting you, from time to time, for your opinion and directions with respect to Miss Eugenia.' And then, bowing coldly to Mrs. Arlbery, who eagerly called out to welcome her, he placed her in the phaeton, which instantly drove off. He looked after them for some time, almost incredulous of her departure: but, as his amazement subsided into certainty, the most indignant disappointment succeeded. That she could leave Cleves at the very moment he was reinstated in its society, seemed conviction to him of her indifference; and that she could leave it in the present state of the affairs of Eugenia, made him conclude her so great a slave to the love of pleasure, that every duty and all propriety were to be sacrificed to its pursuit. 'I will think of her,' cried he, 'no more! She concealed from me her plan, lest I should torment her with admonitions: the glaring homage of the Major is better adapted to her taste,—She flies from my sincerity to receive his adulation,—I have been deceived in her disposition,—I will think of her no more!'
- 60. CHAPTER V A Sermon The kind reception of Mrs. Arlbery, and all the animation of her discourse, were thrown away upon Camilla. An absent smile, and a few faint acknowledgments of her goodness were all she could return: Eugenia abandoned when she might have been served, Edgar contemning when he might have been approving ... these were the images of her mind, which resisted entrance to all other. Tired of fruitless attempts to amuse her, Mrs. Arlbery, upon their arrival at the Grove, conducted her to an apartment prepared for her, and made use of no persuasion that she would leave it before dinner. Camilla then, too unhappy to fear any injunction, and resigned to whatever she might receive, read the discourse of Mr. Tyrold. For Miss Camilla Tyrold. It is not my intention to enumerate, my dear Camilla, the many blessings of your situation; your heart is just and affectionate, and will not forget them: I mean but to place before you your immediate duties, satisfied that the review will ensure their performance. Unused to, because undeserving control, your days, to this period, have been as gay as your spirits. It is now first that your tranquillity is ruffled; it is now, therefore, that your fortitude has its first debt to pay for its hitherto happy exemption. Those who weigh the calamities of life only by the positive, the substantial, or the irremediable mischiefs which they produce,
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