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© ShiNung Ching, Yongsoon Eun, Cevat Gokcek, Pierre T. Kabamba,
and Semyon M. Meerkov 2011
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2011
Printed in the United States of America
A catalog record for this publication is available from the British Library.
Library of Congress Cataloging in Publication data
Quasilinear control : performance analysis and design of feedback systems
with nonlinear sensors and actuators / ShiNung Ching ... [et al.].
p. cm.
Includes bibliographical references and index.
ISBN 978-1-107-00056-8 (hardback)
1. Stochastic control theory. 2. Quasilinearization. I. Ching, ShiNung.
QA402.37.Q37 2010
629.8312–dc22 2010039407
ISBN 978-1-107-00056-8 Hardback
Cambridge University Press has no responsibility for the persistence or
accuracy of URLs for external or third-party Internet Web sites referred to in
this publication and does not guarantee that any content on such Web sites is,
or will remain, accurate or appropriate.](https://image.slidesharecdn.com/2031414-250526064453-be191831/85/Quasilinear-Control-Performance-Analysis-And-Design-Of-Feedback-Systems-With-Nonlinear-Sensors-And-Actuators-Shinung-Ching-Et-Al-8-320.jpg)
![1.4 Quasilinear versus Linear Control 5
a single variable – the standard deviation of its argument; thus,
N = N(σu). (1.2)
Note that stochastic linearization is indeed a quasilinear, rather than linear,
operation: the quasilinear gains of αf(·) and f(·)α, where α is a constant, are not the
same, the former being αN(σu) the latter being N(ασu).
In the closed loop environment, σu depends not only on f(u) but also on all
other components of the system (i.e., the plant and the controller parameters) and
on all exogenous signals (i.e., references and disturbances). This leads to transcen-
dental equations that define the quasilinear gains. The study of these equations in
the framework of various control-theoretic problems (e.g., root locus, sensitivity
functions, LQR/LQG, H∞) is the essence of the theory of QLC.
As in the open loop case, a stochastically linearized closed loop system is also
not linear: its output to the sum of two exogenous signals is not equal to the sum of
the outputs to each of these signals, that is, superposition does not hold. However,
since, when all signals and functional blocks are given, the system has a constant gain
N, we refer to a stochastically linearized closed loop system as quasilinear.
1.4 Quasilinear versus Linear Control
Consider the closed-loop LPNI system shown in Figure 1.2(a). If the usual Jacobian
linearization is used, this system is reduced to that shown in Figure 1.2(b), where
all signals are denoted by the same symbols as in Figure 1.2(a) but with a ~. In this
C(s) f (·) P(s)
r
u v y
g(·)
e
ym
−
(a) LPNI system
C(s) P(s)
r
ũ ṽ ỹ
ẽ
ỹm
N J
a = d
dũ f (ũ) ũ *
N J
s = d
dỹ g(ỹ)
ỹ *
−
(b) Jacobian linearization
C(s) P(s)
r
û v̂ ŷ
ê
ŷm
Na = E [ d
dû f (û)]
Ns = E [ d
dŷ g(ŷ)]
−
(c) Stochastic linearization
Figure 1.2. Closed loop LPNI system and its Jacobian and stochastic linearizations.](https://image.slidesharecdn.com/2031414-250526064453-be191831/85/Quasilinear-Control-Performance-Analysis-And-Design-Of-Feedback-Systems-With-Nonlinear-Sensors-And-Actuators-Shinung-Ching-Et-Al-27-320.jpg)
![8 Introduction
As a matter of calculations, consider the LPNI system of Figure 1.2(a) defined
by the following state space equations:
ẋ1
ẋ2
=
−1 −1
1 0
x1
x2
+
1
0
satα(u) +
1
0
w
y = 0 1
x1
x2
, (1.9)
with the feedback
u = Kx, (1.10)
where x = [x1,x2]T is the state of the plant and w is a standard white Gaussian process.
The problem is to select a feedback gain K so that the disturbance is rejected in the
best possible manner, that is, σ2
y is minimized. Based on Jacobian linearization,
this can be accomplished using the LQR approach with a sufficiently small control
penalty, say, ρ = 10−5. Based on stochastic linearization, this can be accomplished
using the method developed in Chapter 5 and referred to as SLQR (where the “S”
stands for “saturating”) with the same ρ. The resulting controllers, of course, are
used in the LPNI system. Simulating this system with the LQR controller and with
the SLQR controller, we evaluated numerically σ2
y for both cases. The results are
shown in Figure 1.4 as a function of the saturation level. From this figure, we conclude
the following:
0.5 1 1.5 2 2.5 3 3.5
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
α
Output
variance
SLQR
LQR
Actual system with SLQR
Actual system with LQR
Figure 1.4. Comparison of LQR, SLQR, and actual system performance.](https://image.slidesharecdn.com/2031414-250526064453-be191831/85/Quasilinear-Control-Performance-Analysis-And-Design-Of-Feedback-Systems-With-Nonlinear-Sensors-And-Actuators-Shinung-Ching-Et-Al-30-320.jpg)
![14 Introduction
The final chapter of the book, Chapter 8, provides the proofs of all formal
statements included in the book.
As it follows from the above overview, this volume transfers most of LC to
QLC. Specifically, the saturating random sensitivity function and the tracking qual-
ity indicators accomplish this for frequency domain techniques, the S-root locus
for time domain techniques, and SLQR/SLQG for state space techniques. In addi-
tion, the LPNI-specific problems, for example, truncation points of the root locus,
instrumentation selection, and the performance recovery, are also formulated and
solved.
1.6 Summary
• The analysis and design of closed loop linear plant/nonlinear instrumentation
(LPNI) systems is the main topic of this volume.
• The goal is to extend the main analysis and design techniques of linear control
(LC) to the LPNI case. Therefore, the resulting methods are referred to as
quasilinear control (QLC).
• The approach of QLC is based on the method of stochastic linearization.
According to this method, an LPNI system is represented by a quasilinear one,
where the static nonlinearities are replaced by the expected values of their
gradients. As a result, stochastic linearization represents the LPNI system
globally (rather than locally, as it is in the case of Jacobian linearization).
• Stochastic linearizations of LPNI systems represent the actual LPNI systems
more faithfully than Jacobian linearization.
• Starting from stochastically linearized versions of LPNI systems, QLC devel-
ops methods for analysis and design that are as rigorous as those of LC (which
starts from Jacobian linearization).
• This volume transfers most LC methods to QLC: The saturated random
sensitivity function and the tracking quality indicators accomplish this for
frequency domain techniques; S-root locus – for time domain techniques; and
SLQR/SLQG – for state space techniques.
• In addition, several LPNI-specific problems, for example, truncation points
of the root locus, instrumentation selection, and the performance recovery,
are formulated and solved.
1.7 Annotated Bibliography
There is a plethora of monographs on design of linear feedback systems. Examples
of undergraduate text are listed below:
[1.1] B.C. Kuo, Automatic Control Systems, Fifth Edition, Prentice Hall,
Englewood Cliffs, NJ, 1987
[1.2] K. Ogata, Modern Control Engineering, Second Edition, Prentice Hall,
Englewood Cliffs, NJ, 1990
[1.3] R.C. Dorf and R.H. Bishop, Modern Control Systems, Eighth Edition,
Addison-Wesley, Menlo Park, CA, 1998](https://image.slidesharecdn.com/2031414-250526064453-be191831/85/Quasilinear-Control-Performance-Analysis-And-Design-Of-Feedback-Systems-With-Nonlinear-Sensors-And-Actuators-Shinung-Ching-Et-Al-36-320.jpg)
![1.7 Annotated Bibliography 15
[1.4] G.C. Godwin, S.F. Graebe, and M.E. Salgado, Control Systems Design,
Prentice Hall, Upper Shaddle River, NJ, 2001
[1.5] G.F. Franklin, J.D. Powel, and A. Emami-Naeini, Feedback Control of
Dynamic Systems, Fourth Edition, Prentice Hall, Englewood Cliffs, NJ, 2002
At the graduate level, the following can be mentioned:
[1.6] I.M. Horowitz, Synthesis of Feedback Systems, Academic Press, London,
1963
[1.7] H. Kwakernaak and R. Sivan, Linear Optimal Control Systems,
Wiley-Interscience, New York, 1972
[1.8] W.M. Wonham, Linear Multivariable Control: A Geometric Approach, Third
Edition, Springer-Verlag, New York, 1985
[1.9] B.D.O. Anderson and J.B. Moore, Optimal Control: Linear Quadratic
Methods, Prentice Hall, Englewood Cliffs, NJ, 1989
[1.10] J.M. Maciejowski, Multivariable Feedback Design, Addison-Wesley,
Reading, MA, 1989
[1.11] K. Zhou, J.C. Doyle, and K. Glover, Robust and Optimal Control, Prentice
Hall, Upper Saddle River, NJ, 1996
The theory of absolute stability has its origins in the following:
[1.12] A.I. Lurie and V.N. Postnikov, “On the theory of stability of control
systems,” Applied Mathematics and Mechanics, Vol. 8, No. 3, pp. 246–248,
1944 (in Russian)
[1.13] M.A. Aizerman, “On one problem related to ‘stability-in-the-large’ of
dynamical systems,” Russian Mathematics Uspekhi, Vol. 4, No. 4,
pp. 187–188, 1949 (in Russian)
Subsequent developments are reported in the following:
[1.14] V.M. Popov, “On absolute stability of nonlinear automatic control systems,”
Avtomatika i Telemekhanika, No. 8, 1961 (in Russian). English translation:
Automation and Remote Control, Vol. 22, No. 8, pp. 961–979, 1961
[1.15] V.A. Yakubovich, “The solution of certain matrix inequalities in automatic
control theory,” Doklady Akademii Nauk, Vol. 143, pp. 1304–1307, 1962 (in
Russian)
[1.16] M.A. Aizerman and F.R. Gantmacher, Absolute Stability of Regulator
Systems. Holden-Day, San Francisco, 1964 (Translated from the Russian
original, Akad. Nauk SSSR, Moscow, 1963)
[1.17] R. Kalman, “Lyapunov functions for the problem of Lurie in automatic
control, Proc. of the National Academy of Sciences of the United States of
America, Vol. 49, pp. 201–205, 1963
[1.18] K.S. Narendra and J. Taylor, Frequency Domain Methods for Absolute
Stability, Academic Press, New York, 1973
The method of harmonic balance has originated in the following:
[1.19] L.S. Goldfarb, “On some nonlinearities in regulator systems,” Avtomatika i
Telemekhanika, No. 5, pp. 149–183, 1947 (in Russian).
[1.20] R. Kochenburger, “A frequency response method for analyzing and
synthesizing contactor servomechanisms,” Trans. AIEE, Vol. 69,
pp. 270–283, 1950
This was followed by several decades of further development and applications. A
summary of early results can be found in the following:](https://image.slidesharecdn.com/2031414-250526064453-be191831/85/Quasilinear-Control-Performance-Analysis-And-Design-Of-Feedback-Systems-With-Nonlinear-Sensors-And-Actuators-Shinung-Ching-Et-Al-37-320.jpg)
![16 Introduction
[1.21] A. Gelb and W.E. Van der Velde, Multiple-Input Describing Function and
Nonlinear System Design, McGraw-Hill, New York, 1968,
while later ones in
[1.22] A.I. Mees, “Describing functions – 10 years later,” IMA Journal of Applied
Mathematics, Vol. 32, No. 1–3, pp. 221–233, 1984
For the justification of this method (based on the idea of “filter hypothesis”) and
evaluation of its accuracy, see the following:
[1.23] M.A. Aizerman, “Physical foundations for application small parameter
methods to problems of automatic control,” Avtomatika i Telemekhanika,
No. 5, pp. 597–603, 1953 (in Russian)
[1.24] E.M. Braverman, S.M. Meerkov, and E.S. Piatnitsky, “A small parameter in
the problem of justifying the harmonic balance method (in the case of the
filter hypothesis),” Avtomatika i Telemekhanika, No. 1, pp. 5–21, 1975
(in Russian). English translation: Automation and Remote Control, Vol. 36,
No. 1, pp. 1–16, 1975
Using the notion of the mapping degree, this method has been justified in the
following:
[1.25] A.R. Bergen and R.L. Frank, “Justification of the describing function
method,” SIAM Journal of Control, Vol. 9, No. 4, pp. 568–589, 1971
[1.26] A.I. Mees and A.R. Bergen, “Describing functions revisited,” IEEE
Transactions on Automatic Control, Vol. AC-20, No. 4, pp. 473–478, 1975
Several monographs that address the issue of stability of LPNI systems with
saturating actuators can be found in the following:
[1.27] T. Hu and Z. Lin, Control Systems with Actuator Saturation, Birkauser,
Boston, MA, 2001
[1.28] A. Saberi, A.A. Stoorvogel, and P. Sannuti, Control of Linear Systems with
Regulation and Input Constraints, Springer-Verlag, New York, 2001
[1.29] V. Kapila and K.M. Grigoriadis, Ed., Actuator Saturation Control, Marcel
Dekker, Inc., New York, 2002
Remarks on the saturating nature of the Chernobyl nuclear accident can be found
in the following:
[1.30] G. Stein, “Respect for unstable,” Hendrik W. Bode Lecture, Proc. ACC,
Tampa, FL, 1989
Reasons for the crash of the YF-22 aircraft are reported in the following:
[1.31] M.A. Dornheim, “Report pinpoints factors leading to YF-22 crash”,
Aviation Week Space Technology., Vol. 137, No. 19, pp. 53–54,
1992
Modern theory of nonlinear control based on the geometric approach has its origin
in the following:
[1.32] R.W. Brockett, “Asymptotic stability and feedback stabilization,” in
Differential Geometric Control Theory, R.W. Brockett, R.S. Millman, and
H.J. Sussmann, Eds., pp. 181–191, 1983](https://image.slidesharecdn.com/2031414-250526064453-be191831/85/Quasilinear-Control-Performance-Analysis-And-Design-Of-Feedback-Systems-With-Nonlinear-Sensors-And-Actuators-Shinung-Ching-Et-Al-38-320.jpg)
![1.7 Annotated Bibliography 17
Further developments are reported in the following:
[1.33] A. Isidori, Nonlinear Control Systems, Third Edition, Springer-Verlag, New
York, 1995
Model predictive control was advanced in the following:
[1.34] J. Richalet, A. Rault, J.L. Testud, and J. Papon, “Model predictive heuristic
control: Applications to industrial processes,” Automatica, Vol. 14, No. 5,
pp. 413–428, 1978
[1.35] C.R. Cutler and B.L. Ramaker, “Dynamic matrix control – A computer
control algorithm,” in AIChE 86th National Meeting, Houston, TX, 1979
Additional results can be found in
[1.36] C.E. Garcia, D.M. Prett, and M. Morari, “Model predictive control: Theory
and practice – a survey,” Automatica, Vol. 25, No. 3, pp. 338–349, 1989
[1.37] E.G. Gilbert and K. Tin Tan, “Linear systems with state and control
constraints: the theory and applications of maximal output admissible sets,”
IEEE Transactions Automatic Control, Vol. 36, pp. 1008–1020, 1995
[1.38] D.Q. Mayne, J.B. Rawlings, C.V. Rao, and P.O.M. Scokaert, “Constrained
model predictive control: Stability and optimality,” Automatica, Vol. 36,
pp. 789–814, 2000
[1.39] E.F. Camacho and C. Bordons, Model Predictive Control, Springer-Verlag,
London, 2004
The term integrator “windup” seems to have appeared in
[1.40] J.C. Lozier, “A steady-state approach to the theory of saturable servo
systems,” IRE Transactions on Automatic Control, pp. 19–39, May 1956
Early work on antiwindup can be found in the following:
[1.41] H.A. Fertic and C.W. Ross, “Direct digital control algorithm with
anti-windup feature,” ISA Transactions, Vol. 6, No. 4, pp. 317–328, 1967
More recent results can be found in the following:
[1.42] M.V. Kothare, P.J. Campo, M. Morari, and C.N. Nett, “A unified framework
for the study of anti-windup designs,” Automatica, Vo. 30, No. 12,
pp. 1869–1883, 1994
[1.43] N. Kapoor, A.R. Teel, and P. Daoutidis, “An anti-windup design for linear
systems with input saturation,” Automatica, Vol. 34, No. 5, pp. 559–574, 1998
[1.44] P. Hippe, Windup in Control: Its Effects and Their Prevention, Springer,
London, 2006
The method of stochastic linearization originated in the following:
[1.45] R.C. Booton, M.V. Mathews, and W.W. Seifert, “Nonlinear
servomechanisms with random inputs,” Dyn. Ana. Control Lab, MIT,
Cambridge, MA, 1953
[1.46] R.C. Booton, “The analysis of nonlinear systems with random inputs,” IRE
Transactions on Circuit Theory, Vol. 1, pp. 32–34, 1954
[1.47] I.E. Kazakov, “Approximate method for the statistical analysis of nonlinear
systems,” Trudy VVIA 394, 1954 (in Russian)
[1.48] I.E. Kazakov, “Approximate probability analysis of operational position of
essentially nonlinear feedback control systems,” Automation and Remote
Control, Vol. 17, pp. 423–450, 1955](https://image.slidesharecdn.com/2031414-250526064453-be191831/85/Quasilinear-Control-Performance-Analysis-And-Design-Of-Feedback-Systems-With-Nonlinear-Sensors-And-Actuators-Shinung-Ching-Et-Al-39-320.jpg)
![18 Introduction
Various extensions can be found in the following:
[1.49] V.S. Pugachev, Theory of Random Functions, Pergamon Press, Elmsford,
NY, 1965 (translation from Russian)
[1.50] I. Elishakoff, “Stoshastic linearization technique: A new interpretation and a
selective review,” The Shock and Vibration Digest, Vol. 32, pp. 179–188, 2000
[1.51] J.B. Roberts and P.D. Spanos, Random Vibrations and Statistical
Linearization, Dover Publications, Inc., Mineola, NY, 2003
[1.52] L. Socha, Linearization Methods for Stochastic Systems, Springer, Berlin
Heidelberg, 2008
Applications to control problems have been described in the following:
[1.53] I.E. Kazakov and B.G. Dostupov, Statistical Dynamics of Nonlinear Control
Systems, Fizmatgiz, Moscow 1962 (In Russian)
[1.54] A.A. Pervozvansky, Stochastic Processes in Nonlinear Control Systems,
Fizmatgiz, Moscow 1962 (in Russian)
[1.55] I.E. Kazakov, “Statistical analysis of systems with multi-dimensional
nonlinearities,” Automation and Remote Control, Vol. 26, pp. 458–464, 1965
and also in reference [1.21]
The stochastic modeling of reference signals in the problem of hard drive control
can be found in the following:
[1.56] A. Silberschatz and P.B. Galvin, Operating Systems Concepts,
Addison-Wesley, 1994
[1.57] T.B. Goh, Z. Li and B.M. Chen, “Design and implementation of a hard disk
servo system using robust abd perfect tracking approach,” IEEE
Transactions on Control Systems Technology, Vol. 9, pp. 221–233, 2001
For the aircraft homing problem, similar conclusions can be deduced from the
following:
[1.58] C.-F. Lin, Modern Navigation, Guidance, and Control Processing, Prentice
Hall, Englewood Cliffs, NJ, 1991
[1.59] E.J. Ohlmeyer, “Root-mean-square miss distance of proportional navigation
missile against sinusoidal target,” Journal of Guidance, Control and
Dynamics, Vol. 19, No. 3, pp. 563–568, 1996
In automotive problems, stochastic reference signals appear in the following:
[1.60] H.S. Bae and J.C. Gerdes, “Command modification using input shaping for
automated highway systems with heavy trucks,” California PATH Research
Report, 1(UCB-ITS-PRR-2004-48), Berkeley, CA, 2004
The usual, Jacobian, linearization is the foundation of all methods for analysis
and design on linear systems, including the indirect Lyapunov method. For more
information see the following:
[1.61] M. Vidyasagar, Nonlinear Systems Analysis, Second Edition, Prentice Hall,
Englewood Cliffs, NJ, 1993
[1.62] H.K. Khalil, Nonlinear Systems, Third Edition, Prentice Hall, Upper Saddle
River, NJ, 2002
A discussion on calculating the 2-norm of a transfer function can be found in
[1.63] K. Zhou and J.C. Doyle, Essentials of Robust Control, Prentice Hall, Upper
Saddle River, NJ, 1999](https://image.slidesharecdn.com/2031414-250526064453-be191831/85/Quasilinear-Control-Performance-Analysis-And-Design-Of-Feedback-Systems-With-Nonlinear-Sensors-And-Actuators-Shinung-Ching-Et-Al-40-320.jpg)
![1.7 Annotated Bibliography 19
For the theory of Fokker-Planck equation turn to
[1.64] L. Arnold, Stochastic Differential Equations, Wiley Interscience, New York,
1973
[1.65] H. Risken, The Fokker-Planck Equation: Theory and Applications,
Springer-Verlag, Berlin, 1989
[1.66] Z. Schuss, Theory and Applications of Stochastic Processes, Springer, New
York, 2009](https://image.slidesharecdn.com/2031414-250526064453-be191831/85/Quasilinear-Control-Performance-Analysis-And-Design-Of-Feedback-Systems-With-Nonlinear-Sensors-And-Actuators-Shinung-Ching-Et-Al-41-320.jpg)
![2.1 Open Loop Systems 21
f (u)
N
u(t)
v(t)
v̂(t)
Figure 2.1. Stochastic linearization of an isolated nonlinearity.
Theorem 2.1. If u(t) is a zero-mean wide sense stationary Gaussian process and
f(u) is an odd, piecewise differentiable function, (2.3) is minimized by
N = E
df(u)
du
u=u(t)
. (2.4)
Since the proof of this theorem is simple and instructive, we provide it here,
rather than in Chapter 8.
Proof. Rewriting (2.3) as
ε(N) = E (f(u) − Nu)2
(2.5)
and differentiating with respect to N, results in the following condition of optimality:
dε
dN
= E[2(f(u) − Nu)u] = 0. (2.6)
It is easy to verify that this is, in fact, the condition of minimality and, therefore, the
minimizer of (2.3) is given by
N =
E[f(u)u]
E
u2
. (2.7)
Taking into account that for zero-mean wss Gaussian u(t) and piecewise
differentiable f(u),
E[f(u)u] = E u2
E
df
du
u=u(t)
, (2.8)
(2.4) follows immediately from the last two expressions.
It turns out that (2.4) holds for a more general approximation of f(u). To show
this, let n(t) be the impulse response of a causal linear system and, instead of (2.2),
introduce the approximation
v̂(t) = n(t) ∗ u(t), (2.9)
where ∗ denotes the convolution. The problem is to select n(t) so that the functional
ε(n(t)) = E (f(u(t)) − n(t) ∗ u(t))2
(2.10)
is minimized.](https://image.slidesharecdn.com/2031414-250526064453-be191831/85/Quasilinear-Control-Performance-Analysis-And-Design-Of-Feedback-Systems-With-Nonlinear-Sensors-And-Actuators-Shinung-Ching-Et-Al-43-320.jpg)
![2.2 Closed Loop LPNI Systems 31
FΩr
(s)
FΩd
(s)
C(s) P(s)
f (·)
g(·)
wd
wr
−
e u v y
ym
r
d
Figure 2.4. Closed loop LPNI system.
the actuator and sensor, Fd
(s) and Fr (s) are coloring filters with 3dB bandwidths
d and r, respectively, wr, wd are independent standard Gaussian white noise
processes, and the scalars r, d, y, ym, u, v, and e denote the reference, disturbance,
plant output, sensor (or measured) output, control signal, actuator output, and error
signal, respectively.
Let the quasilinear gains of f(·) and g(·) in isolation be denoted as Na and Ns,
respectively. Assume that the range of Na is Na, the range of Ns is Ns, and the range
of NaNs is Nas. For instance, if f(·) and g(·) are standard saturation functions, then
Na = Ns = Nas = [0,1]. Using these notations, introduce the following:
Assumption 2.1.
(i) P(s) has all poles in the closed left half plane;
(ii) C(s) has all poles in the closed left half plane;
(iii) 1 + γ P(s)C(s) has all zeros in the open left half plane for γ ∈ Nas.
This assumption, as it is shown below, is a sufficient condition for the existence
of variances of various signals in the stochastically linearized versions of LPNI sys-
tems under consideration. Therefore, unless stated otherwise, it is assumed to hold
throughout this volume.
Under Assumption 2.1, we discuss below stochastic linearization of the closed
loop system of Figure 2.4. First, we address the case of nonlinear actuators and
sensors separately and then the case of nonlinearities in both actuators and sensors
simultaneously.
2.2.2 Reference Tracking with Nonlinear Actuator
Quasilinear gain: Consider the closed loop system of Figure 2.5(a), where all func-
tional blocks and signals remain the same as in Figure 2.4. The goal is to obtain its
quasilinear approximation shown in Figure 2.5(b).
The situation here is different from that of Figure 2.2 in two respects. First, the
signal u(t) at the input of the nonlinearity is no longer Gaussian. Second, the signals
u(t) and û(t) are not the same. Therefore, the quasilinear gain (2.12) is no longer
optimal. Nevertheless, proceeding formally, we view the system of Figure 2.5(b) with
Na = E
df(û)
dû
û=û(t)
(2.47)
as the stochastic linearization of Figure 2.5(a).](https://image.slidesharecdn.com/2031414-250526064453-be191831/85/Quasilinear-Control-Performance-Analysis-And-Design-Of-Feedback-Systems-With-Nonlinear-Sensors-And-Actuators-Shinung-Ching-Et-Al-53-320.jpg)
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Title: Fifth Report of the Vestry of the Parish of Chelsea in the County of Middlesex, Appointed
Under the Metropolis Local Management Act, 1855: 1860-1
Author: Charles Lahee
Release date: August 26, 2013 [eBook #43559]
Language: English
Credits: Transcribed from the 1861 edition by David Price. Many thanks to the Royal Borough of
Kensington and Chelsea Library Service for allowing their copy to be used for this
transcription
*** START OF THE PROJECT GUTENBERG EBOOK FIFTH REPORT OF THE VESTRY OF THE PARISH OF
CHELSEA IN THE COUNTY OF MIDDLESEX, APPOINTED UNDER THE METROPOLIS LOCAL
MANAGEMENT ACT, 1855: 1860-1 ***](https://image.slidesharecdn.com/2031414-250526064453-be191831/85/Quasilinear-Control-Performance-Analysis-And-Design-Of-Feedback-Systems-With-Nonlinear-Sensors-And-Actuators-Shinung-Ching-Et-Al-60-320.jpg)
![illuminating power of the gas supplied. In alleged compliance with this requirement the London Gas
light company have appointed a testing station at the house No. 73, Besborough Street, Pimlico.
Mr. Hughes [8] observes with reference to this twenty-seventh section,—
“There was some discussion as to the propriety of making the requirements of this clause
compulsory instead of merely permissive. After the fullest consideration however, it was thought
essential, with the view of avoiding discussions in vestries, and especially with the view of
counteracting the underhand and secret influence which the gas companies exert in many vestries
and district boards, to make the clause compulsory. There are many instances on record, where
powers are given to local authorities, and yet these powers, although highly important to the public
interests, have never hitherto been exercised. Hence an additional reason for positively requiring
them to provide apparatus, c., for testing the gas, and to appoint and pay an inspector for the
purpose. It is true that no time is fixed within which the apparatus is to be provided and the
inspector appointed, but I apprehend it will be competent for any ratepayer to compel, by
mandamus, the performance of this duty by any vestry or district board, within a reasonable time.
“The second section of the clause imposes an obligation on the gas companies—namely, that each
of them shall within six months erect at the prescribed distance from their works, an experimental
meter and other apparatus for testing the illuminating power of the gas.
“Now these two obligations, the one on the local authorities of the metropolis, and the other on the
gas companies, must not be confounded, because they are perfectly distinct, and the one is not to
be a substitute for the other. In the first place each local authority in the metropolis—i.e., each
vestry and district board, about thirty-eight in number, constituted under the Metropolis Local
Management Act, is to erect its own apparatus, and appoint its own inspector; and from future
clauses it appears that the act contemplates a continuous and regular succession of testing by this
inspector, both for purity and illuminating power. The evidence of this inspector however as to any
defects in the gas, will not be conclusive until his report has been confirmed by testing the gas at
the prescribed distance of 1000 yards from the works. Hence the necessity for the obligation on
the company. The next clause will better explain the mode of proceeding by the inspector,
whenever he finds the gas to be below the prescribed standard of illuminating power. With
reference to purity there is nothing about testing for this at the distance of 1000 yards, and
therefore the test for this may be made wherever the inspector pleases.
“With respect to the number of separate places for testing the gas, inasmuch as there are thirteen
companies included within the act, and each must provide a testing house 1000 yards distant from
their works, there must evidently be not less than thirteen of these. [9] But if each local authority
also erects a separate one at some central part of its district, thirty-eight of these will be
necessary. A power is afterwards given for two or more local authorities to combine, and then the
number will probably be somewhat diminished.
“All this necessity for a duplicate set of testing establishments is rendered essential by the absurd
requirement of the act as to testing the gas at 1000 yards distance from the works. Now as the
erection of this apparatus by the companies and the establishment of a permanent testing place
away from their works will be very expensive to the companies, it is just possible they may be very
glad to be relieved from this expense, and may consent to the testing at the establishment of the
local authority being sufficient evidence of the illuminating power as well as the purity of the gas.
This is the more probable as it must be perfectly well known to the companies—at least to their
engineers—that so far as the company is concerned the gas may just as well be tested in the
centre of any district supplied, as at the limited distance of 1000 yards from the works.”
Under the provisions of the fiftieth section of this act, the duty is imposed upon the Metropolitan Board
of Works of raising the costs and charges incident to its passing; and that board have accordingly levied](https://image.slidesharecdn.com/2031414-250526064453-be191831/85/Quasilinear-Control-Performance-Analysis-And-Design-Of-Feedback-Systems-With-Nonlinear-Sensors-And-Actuators-Shinung-Ching-Et-Al-68-320.jpg)
![under their precepts upon the several vestries and district boards, the necessary amounts for meeting
the claims received.
The accounts sent in by the various parties were submitted to the proper officer of the House of
Commons for taxation with the following result:—
Accounts sent in. Amount Certified by Taxing
Officer.
Costs
Allowed.
Board to
Pay.
£ s. d. £ s. d. £ s. d. £ s. d.
Mr. Dangerfield 89 19 6 62 6 0 3 11 8 65 17 8
Mr. Wyatt 3666 0 4 3033 18 10 None. 3033 18 10
Mr. Beal . . . £2685 13 6
Less paid by Mr. Wyatt . . . 1413
0 0
1272 13 6 676 11 7 None. 676 11 7
5022 13 4
House Fees on Taxation to be paid by Dyson Co., (as Parliamentary Agents) 42 0 0
3818 8 1
The proportion of the above with which this parish is debited by the precept of 1861 is £106 16s. 5d.,
but that proportion is based upon the amount of the accounts as sent in; as reduced by taxation it will
only be £81 12s. 8d. But as the vestry has already contributed two hundred pounds to Mr. Beal in
answer to his various appeals, and expended upwards of twenty pounds in making experiments as
suggested by him, they will have a large sum to receive back
In November, 1860, a proposal was made by the vestry of St. James’, Westminster, to recognize the
services of Messrs. Beal and Hughes throughout the enquiry into the supply of gas to the metropolis,
and the passing of the recent measure by raising some fitting testimonial, and a committee of that
vestry having been appointed to carry it out, Messrs. Hall and Perry were deputed to confer with them
as to the best mode of doing so.
The Bill to amend the Metropolis Gas Act, [11] introduced by the government into parliament for
transferring to the metropolitan board of works, so far as regards the metropolis, the powers conferred
by the acts for regulating measures used in the sale of gas (22 23 Vic., cap. 66, and 23 24 Vic., cap.
146) upon the justices of the peace, of appointing inspectors of meters, having received the royal
assent (24 25 Vict., cap. 79), it will be the duty of that board to take the necessary steps for carrying
it into execution. To aid in accomplishing this transfer, the vestry on the tenth July, 1860, expressed
their opinion that the appointment of inspectors of meters in the metropolis would be best placed in the
hands of the metropolitan board of works, and petitioned parliament and memorialised the justices of
the peace upon the subject. See Appendix No. 10 (page 56).
The number of public lamps in the parish on the 25th March, 1861, was as follows—
Lighted by the London Gas Company 696
,, Western ditto at Kensal Town 30
726
WATERING AND SCAVENGERS.
The following tabular statement shows the cost of these two services for the last five years:—](https://image.slidesharecdn.com/2031414-250526064453-be191831/85/Quasilinear-Control-Performance-Analysis-And-Design-Of-Feedback-Systems-With-Nonlinear-Sensors-And-Actuators-Shinung-Ching-Et-Al-69-320.jpg)
![1861, Apr. 9 £1499 12 6 May 1st. 2d. in the £ £220,927 0 0 £1839 3 8
RANELAGH DISTRICT.
1860, July 2 £715 18 5 Oct. 3rd 1d in the £ £184,992 0 0 £770 9 1½
1861, Apr. 9 £600 16 11 May 1st 1d. in the £ £186,421 0 0 £776 3 10
MAIN DRAINAGE.—WHOLE PARISH.
1860, Oct.
23
£2300 0 0 Oct. 31 3d. in the £ £219,477 10 0 £2741 5 3¾
The metropolitan board of works early in February of the present year determined to re-introduce into
parliament the bill for amending the Metropolis Local Management Act, containing the same provisions
as were in the bill of last session, with the exception of the introduction of certain clauses for the re-
apportionment of the Rock Loan; but in consequence of the strong opposition to those clauses, the
board deemed it expedient to withdraw them from the bill, and to embody them in a separate bill, and
the two bills were accordingly introduced into the House of Commons. The first of them, after great
delays, passed through the House of Commons on the nineteenth of July, 1861, and was read a first
time in the House of Lords on the twenty-second of that month, but in consequence of the advanced
period of the session, it was found impossible to procure its passage through the required stages in
order to its becoming law this year. The bill No. 2, having relation to the Rock Loan, being strongly
opposed by certain members of the House of Commons, was withdrawn.
With respect to the abortive bill of 1860, the solicitors of the metropolitan board reported: “We think it
will be nearly correct to estimate the expense to the board, exclusive of printing, c. by the printers of
the board, and expenses of that kind, at about £1650.” It is to be hoped that the failure of 1861 will be
somewhat less costly.
The vestry of Chelsea have at various times responded to the invitation of the metropolitan board for
suggestions during the preparation of the several bills for amending the Metropolis Local Management
Act; for convenience of reference, their labours in this respect will be found in the Appendix No. 19,
(pages 77 to 103).
CHELSEA CHARITIES.
On the twenty-second of May, 1860, upon the motion of Mr. Finch, a committee, consisting of the rector
(the Rev. A. G. W. Blunt), the churchwardens (Dr. Diplock and Mr. Collier), three past churchwardens
(Messrs. Hall, Perry, and Leete), and Messrs. Whitehead, Jones, Miles, Brown, Rabbits, E. O. Symons,
Till, and Finch, was appointed to enquire into the charities of the parish, the present particulars of the
several properties, and the application of the respective annual proceeds thereof, and to report
thereon. At the same time, the vestry clerk reported that, at the request of the rector, he was making
arrangements preliminary to the appointment of new trustees, and laid before the vestry the following
letter from Mr. Druce:
Mr. Lahee,
Dear Sir, Doubtless you are aware that the information contained in last Saturday’s “Chelsea Times”
[20] was provided by me, as far as concerned the Chelsea charities.
The article of this morning [20] would lead me to suppose that it was taken to be the opinion of the
writer of such information that under present circumstances it would be to the advantage of the
parish that the vestry clerk should be a solicitor; nothing can be more foreign to my opinion, and
without compliment, I think the office is now in very good hands. In the wicked old tory times on
Easter Tuesday, a man might blow out the steam of discontent, and tell a few truths profitable to
the parish to hear; now a few highly honourable and sensible parishioners ‘make things pleasant’ in](https://image.slidesharecdn.com/2031414-250526064453-be191831/85/Quasilinear-Control-Performance-Analysis-And-Design-Of-Feedback-Systems-With-Nonlinear-Sensors-And-Actuators-Shinung-Ching-Et-Al-75-320.jpg)
![a comfortable room in a house in the King’s Road. The money of the parish charities is not
properly looked after; for many months I have been trying to get the late rector to put matters to
rights, but without avail. Being therefore left sole trustee for some portion, and knowing more than
most, about the parish school and trust funds, I thought I was the proper person to light the
match. Should the vestry appoint the committee on charity affairs, I shall be happy to attend their
summons to give them any information in my power: and I think they ought to have a committee,
both on this subject and that of the church trustees, and at least chronicle the exact state of facts;
for, seriously speaking, we owe ‘liberal opinions’ no small grudge for shutting up Easter Tuesday. I
do not recollect that in your charity report [21] you mentioned the large sums belonging to the
charity schools, if you did, the vestry ought to have seen that the trustees were dying out. From
my experience of the working of the parish charities, schools, c., I am quite convinced that the
vestry should annually qualify themselves to issue a report on all charity money. I have never
found anything to make me think otherwise than favourably of the honesty of all dealings with the
charity money, but I have found abundant reason to rest satisfied, that without some lay assistance
and watchfulness, all matters get into disorder, especially where there are any accounts to keep.
This letter is at your service, and may be used as you think fit. I am, dear sir, always yours very
faithfully, Wm. Druce. Swan Wharf, Chelsea, 14th April, 1860.
The committee, favoured occasionally with the assistance of Mr. Druce, and strengthened by the
addition to it of Messrs. Butt, Tipper, and Callow, have met upon numerous occasions, and are carefully
pursuing their enquiry.
CREMORNE GARDENS.
It will be recollected that towards the close of the season of 1858, the vestry thought it necessary to
remind Mr. Simpson, the proprietor of Cremorne Gardens, of the arrangement made with them as to the
conduct of his gardens, particularly as to the hour of closing; and that Mr. Simpson’s reply, although
dated the 16th of August, was not received in sufficient time to be considered until the meeting on the
fifth of October, when it was thought to be unsatisfactory. Last year the vestry again moved in the
matter, by reminding him of these promises, and requesting to know before Monday, the tenth of
September, whether it was his intention to comply with their wishes and those of the parish, by closing
the gardens at twelve o’clock at night. No reply having been received from Mr. Simpson at the meeting
of the board on the eleventh of September, it was moved by Mr. Delany that the report of the second of
November, 1857, which the vestry had refrained from making public in consequence of the pledge on
the part of the proprietor, that the gardens should be closed as near to midnight as possible, and that in
other respects he would meet the wishes of the vestry, and the comfort of the inhabitants, should be
printed for the use of the vestry. The chairman, however, having ruled that this motion was irregular, it
was referred to the committee of works and for general purposes to consider the whole question,
particularly with reference to the effect, a change in the proprietorship, from an individual to a
company, might have in increasing the annoyance of the parishioners; and the report of the committee
was presented on the twenty-fifth of September.
That report, which is printed at page 107, recommended the publication of the report of 1857, and the
presentation of a petition to the licensing magistrates that the hour of closing might be made twelve
o’clock at night, as was the case with other public gardens. A memorial from forty-four owners of
property and ratepayers residing near the gardens, urging the vestry to act, was presented at the same
time the report was under consideration.
The vestry then decided, by a majority of 17 to 6, to present the petition, but to withold the publication
of the report of 1857; and the solicitor being in attendance with a form of petition, which was approved,
the seal was affixed to it at once, and it was duly presented.
On the ninth of October, however, a day or two before the licensing day, a communication was received
from the solicitors, which induced the board, on a division, by a majority of twenty-eight to eight, to](https://image.slidesharecdn.com/2031414-250526064453-be191831/85/Quasilinear-Control-Performance-Analysis-And-Design-Of-Feedback-Systems-With-Nonlinear-Sensors-And-Actuators-Shinung-Ching-Et-Al-76-320.jpg)
![pass the following resolution:—
Resolved, that having regard to the communication now made by the solicitors, from which it
appears that the necessary evidence in support of the petition cannot be obtained upon the present
occasion, the solicitors be instructed not to incur any further expense in the matter, on the
understanding that if the gardens should be kept open after twelve at night, and the nuisance,
annoyance, and injury to the parishioners continue, this board, will in sufficient time next year
consider the necessary steps to be taken effectually to oppose the application for the renewal of
the license in 1861.
“THE LOTS.”
“The Lots,” a parcel of land so called lying on the banks of the river [23a] in front of Ashburnham House,
comprises about four acres, and is the meadow ground formerly allotted to Sir Arthur Gorges by the
Lord of the Manor, in lieu of his right of common. [23b] These are, and have been for centuries, the
Chelsea Lammas Lands, and have hitherto been accustomed to be opened on the twelfth of August,
being the first day of the month according to the Old Style. The graziers, butchers, and others, with
their cattle, used formerly to assemble in the lane leading to “The Lots” on the eve of Lammas, and
when the clock had struck twelve they entered upon the meadow.
From the report of a committee, printed at page 108, appointed by the vestry in the year 1834 to
investigate the state of these town meadows, when it was necessary vi et armis to reassert the invaded
privileges of the inhabitants, it appears that “‘The Lots’ are Lammas land, and have been for ages
appurtenant to the manor of Chelsea. The Lord of the Manor possesses the right of letting the land on
lease for the spring and autumn quarters, beginning with March and ending in August; and the
inhabitants at large enjoy the privilege of turning in their cattle from August till February, being the
autumn and winter quarters.” Railways, however, and acts of parliament for smoothing down difficulties
in their way, have sprung up since those pastoral clays; and the Lord of the Manor having sold his
freehold to a railway company, the clerk called the attention of the vestry, on the 8th of May, 1860, to
the following advertisement:—
West London Extension Railway Company.—Notice is hereby given, that a Meeting of the
Householders, Inhabitants, and Land-owners of the Parish of St. Luke, Chelsea, in the County of
Middlesex, and other persons interested in the Lammas Lands called “The Lots,” in the said Parish
of St. Luke, Chelsea, will be held at the house of Mr. John Sparks Alexander situate in Cremorne
Road, in the said parish, and known by the name or sign of the ‘King’s Arms,’ on Wednesday, the
9th day of May next, at Eleven o’clock in the forenoon, for the purpose of appointing a Committee
to treat with the Company for the compensation to be paid by them for the extinction of the
Lammas and other Commonable Rights, in or over certain land called ‘The Lots’ in the said parish of
St. Luke, Chelsea, and which is required by the Company for the purposes of the ‘West London
Extension Railway. Act, 1859.’—Dated the 27th day of April, 1860.—Edward Bellamy, Secretary of
the Company.
A meeting took place consequently on the 9th of May, at which the following resolution was passed:—
Resolved, that Messrs. William Hall (church warden), John Perry, James Miles, George Wevell
Richards, and William Whitehead, being five of the persons entitled to Lammas or other
Commonable Rights over or in the piece of land called ‘The Lots Meadow,’ otherwise ‘Chelsea Lot
Mead,’ situate in the parish of Saint Luke, Chelsea, in the County of Middlesex, and containing by
admeasurement three acres, two roods, and thirty perches, or thereabouts, and marked or referred
to in the map or plan, and in the book of reference of the West London Extension Railway,
deposited with the Clerk of the Peace for the said County of Middlesex, by the number 10 in the
said parish of St. Luke, Chelsea, shall be, and they are accordingly, in pursuance of the provisions
of the ‘Lands Clauses Consolidation Act, 1845,’ appointed to be a Committee having all such powers](https://image.slidesharecdn.com/2031414-250526064453-be191831/85/Quasilinear-Control-Performance-Analysis-And-Design-Of-Feedback-Systems-With-Nonlinear-Sensors-And-Actuators-Shinung-Ching-Et-Al-77-320.jpg)
![as by the ‘Lands Clauses Consolidation Act, 1845’ are conferred upon Committees of like
description, to treat with the West London Extension Railway Company for the compensation to be
paid for the extinction of all Lammas or other Commonable Rights over or in the said piece of land.
The important question of the boundaries of the parish has occupied considerable attention during the
past year, and has been the subject of a reference to a sub-committee of the committee of works and
for general purposes, who have still the subject under consideration.
Various memorials and petitions have been presented during the year, to which, as they sufficiently
elucidate the subjects of them, and are printed in the appendix (Nos. 23 to 27) no further allusion need
be made here.
The “Public Indicator” erected in Sloane Square by permission of the board in May, 1860, was removed
in the fallowing March, in consequence of the failure of the proprietors to light it at night, and to provide
the promised clock. It is retained in the custody of the vestry, under a clause in the agreement, as a
lien for the expenses of its removal and making good the paving.
Plans, books of reference, and parliamentary notices of the “London, Buckinghamshire, and West
Midland Railway” (whose intended terminus was in the Pavilion grounds), and of “The London Tramway
and Dispatch Company,” were deposited with the Vestry, but both of their bills were unsuccessful. Mr.
Train also made an application to the vestry for permission to lay down his tramways in Sloane Street
and the King’s Road, but its consideration has been adjourned sine die.
On the twenty-ninth of January, 1861, Mr. Finch laid before the Board a copy of a “Bill to Transfer the
Seats in Parliament forfeited by the Borough of St. Albans to the proposed Borough of Chelsea and
Kensington,” which Viscount Enfield, M.P. for Middlesex had given notice of his intention to introduce in
the House of Commons. Immediately the Government [25] brought in their “Bill for the Appropriation of
the Seats Vacated by the Disfranchisement of the Boroughs of Sudbury and Saint Albans,” which
contained the following clause: “The parishes of Chelsea and Kensington in the county of Middlesex
shall, for the purposes of this act, together form a borough, to be called the borough of Chelsea, and
such borough shall, from and after the said first day of November, 1861, return one member to serve in
parliament.”
An influential deputation from both parishes, accompanied by the two county members, had an
interview with Lord Palmerston on the fifth of March, on the subject of the bill, and laid before his
lordship various statistical information (see appendix No. 29). But on a division in the House of
Commons on the tenth of June, the clause was rejected by a majority of two hundred and seventy-five
to one hundred and seventy-two. Mr. Tite thus reported the circumstance to the board:—
42, Lowndes Square,
Tuesday Morning, 1 o’clock.
My dear Mr. Lahee,—I never was so taken by surprise as by the division I have just left. The
Government Bill was affirmed by two large majorities, but when we came to the Chelsea question,
to my astonishment we were beaten by one hundred and three. The house was excessively
impatient, and would hardly listen to anything; but it appeared to me Sir James Graham, whom
they would hear, said all that could be said on the subject. I was also astonished at some of those
I saw voting against us. I will send you the division list as soon as I get it. The impression in the
House just now was that the bill would be withdrawn, so in happier times we may have another
chance.—Yours very truly,
William Tite.
P.S.—Rather sharp work for M.P.’s yesterday. I was on a committee from 12 to 4, in the House from
5 to just now; to-day we begin again at 12, I suppose until 2 o’clock Wednesday morning.](https://image.slidesharecdn.com/2031414-250526064453-be191831/85/Quasilinear-Control-Performance-Analysis-And-Design-Of-Feedback-Systems-With-Nonlinear-Sensors-And-Actuators-Shinung-Ching-Et-Al-78-320.jpg)
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- 5. QUASILINEAR CONTROL Performance Analysis and Design of Feedback Systems with Nonlinear Sensors and Actuators This is a textbook on quasilinear control (QLC). QLC is a set of methods for performance analysis and design of linear plant/nonlinear instrumentation (LPNI) systems. The approach of QLC is based on the method of stochastic linearization, which reduces the nonlinearities of actuators and sensors to quasilinear gains. Unlike the usual – Jacobian linearization – stochastic linearization is global. Using this approximation, QLC extends most of the linear control theory techniques to LPNI systems. In addition, QLC includes new problems, specific for the LPNI scenario. Examples include instrumented LQR/LQG, in which the controller is designed simultaneously with the actuator and sensor, and partial and complete performance recovery, in which the degradation of linear performance is either contained by selecting the right instrumentation or completely eliminated by the controller boosting. ShiNung Ching is a Postdoctoral Fellow at the Neurosciences Statistics Research Laboratory at MIT, since completing his Ph.D. in electrical engineering at the University of Michigan. His research involves a systems theoretic approach to anesthesia and neuroscience, looking to use mathematical techniques and engineering approaches – such as dynamical systems, modeling, signal processing, and control theory – to offer new insights into the mechanisms of the brain. Yongsoon Eun is a Senior Research Scientist at Xerox Innovation Group in Webster, New York. Since 2003, he has worked on a number of subsystem technologies in the xerographic markingprocessandimageregistrationtechnologyfortheinkjetmarkingprocess. Hisinterests are control systems with nonlinear sensors and actuators, cyclic systems, and the impact of multitasking individuals on organizational productivity. Cevat Gokcek was an Assistant Professor of Mechanical Engineering at Michigan State University. His research in the Controls and Mechatronics Laboratory focused on automo- tive, aerospace, and wireless applications, with current projects in plasma ignition systems and resonance-seeking control systems to improve combustion and fuel efficiency. Pierre T. Kabamba is a Professor of Aerospace Engineering at the University of Michigan. His research interests are in the area of linear and nonlinear dynamic systems, robust control, guidance and navigation, and intelligent control. His recent research activities are aimed at the development of a quasilinear control theory that is applicable to linear plants with nonlinear sensors or actuators. He has also done work in the design, scheduling, and operation of multi- spacecraft interferometric imaging systems, in analysis and optimization of random search algorithms, andinsimultaneouspathplanningandcommunicationschedulingforUAVsunder the constraint of radar avoidance. He has more than 170 publications in refereed journals and conferences and numerous book chapters. Semyon M. Meerkov is a Professor of Electrical Engineering at the University of Michigan. He received his Ph.D. from the Institute of Control Sciences in Moscow, where he remained until 1977. He then moved to the Department of Electrical and Computer Engineering at the Illinois Institute of Technology and to Michigan in 1984. He has held visiting positions at UCLA (1978–1979); Stanford University (1991); Technion, Israel (1997–1998 and 2008); and Tsinghua, China (2008). He was the editor-in-chief of Mathematical Problems in Engineering, department editor for Manufacturing Systems of IIE Transactions, and associate editor of several other journals. His research interests are in systems and control with applications to production systems, communication networks, and the theory of rational behavior. He is a Life Fellow of IEEE. He is the author of numerous research publications and books, including Production Systems Engineering (with Jingshang Li, 2009).
- 7. Quasilinear Control Performance Analysis and Design of Feedback Systems with Nonlinear Sensors and Actuators ShiNung Ching Massachusetts Institute of Technology Yongsoon Eun Xerox Research Center Webster Cevat Gokcek Michigan State University Pierre T. Kabamba University of Michigan Semyon M. Meerkov University of Michigan
- 8. CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo, Mexico City Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9781107000568 © ShiNung Ching, Yongsoon Eun, Cevat Gokcek, Pierre T. Kabamba, and Semyon M. Meerkov 2011 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2011 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication data Quasilinear control : performance analysis and design of feedback systems with nonlinear sensors and actuators / ShiNung Ching ... [et al.]. p. cm. Includes bibliographical references and index. ISBN 978-1-107-00056-8 (hardback) 1. Stochastic control theory. 2. Quasilinearization. I. Ching, ShiNung. QA402.37.Q37 2010 629.8312–dc22 2010039407 ISBN 978-1-107-00056-8 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.
- 9. To my parents, with love, SHINUNG CHING To my wife Haengju, my son David, and my mother Ahn Young, with love and gratitude, YONGSOON EUN To my family, with love and gratitude, PIERRE T. KABAMBA To my dear wife Terry and to our children, Meera, Meir, Leah, and Rachel, with deepest love and admiration, SEMYON M. MEERKOV
- 11. Brief Contents Preface page xiii 1 Introduction 1 2 Stochastic Linearization of LPNI Systems 20 3 Analysis of Reference Tracking in LPNI Systems 66 4 Analysis of Disturbance Rejection in LPNI Systems 114 5 Design of Reference Tracking Controllers for LPNI Systems 134 6 Design of Disturbance Rejection Controllers for LPNI Systems 167 7 Performance Recovery in LPNI Systems 204 8 Proofs 225 Epilogue 275 Abbreviations and Notations 277 Index 281 vii
- 13. Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . page xiii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Linear Plant/Nonlinear Instrumentation Systems and Quasilinear Control 1 1.2 QLC Problems 3 1.3 QLC Approach: Stochastic Linearization 4 1.4 Quasilinear versus Linear Control 5 1.5 Overview of Main QLC Results 9 1.6 Summary 14 1.7 Annotated Bibliography 14 2 Stochastic Linearization of LPNI Systems . . . . . . . . . . . . . . . . . . . . . . 20 2.1 Stochastic Linearization of Open Loop Systems 20 2.1.1 Stochastic Linearization of Isolated Nonlinearities 20 2.1.2 Stochastic Linearization of Direct Paths of LPNI Systems 29 2.2 Stochastic Linearization of Closed Loop LPNI Systems 30 2.2.1 Notations and Assumptions 30 2.2.2 Reference Tracking with Nonlinear Actuator 31 2.2.3 Disturbance Rejection with Nonlinear Actuator 36 2.2.4 Reference Tracking and Disturbance Rejection with Nonlinear Sensor 37 2.2.5 Closed Loop LPNI Systems with Nonlinear Actuators and Sensors 40 2.2.6 Multiple Solutions of Quasilinear Gain Equations 46 2.2.7 Stochastic Linearization of State Space Equations 50 2.3 Accuracy of Stochastic Linearization in Closed Loop LPNI Systems 53 2.3.1 Fokker-Planck Equation Approach 53 2.3.2 Filter Hypothesis Approach 55 ix
- 14. x Contents 2.4 Summary 57 2.5 Problems 57 2.6 Annotated Bibliography 64 3 Analysis of Reference Tracking in LPNI Systems . . . . . . . . . . . . . . . . 66 3.1 Trackable Domains and System Types for LPNI Systems 67 3.1.1 Scenario 67 3.1.2 Trackable Domains and Steady State Errors 67 3.1.3 System Types 74 3.1.4 Application: Servomechanism Design 75 3.2 Quality Indicators for Random Reference Tracking in Linear Systems 79 3.2.1 Scenario 79 3.2.2 Random Reference Model 79 3.2.3 Random Sensitivity Function 81 3.2.4 Tracking Quality Indicators 86 3.2.5 Application: Linear Hard Disk Servo Design 88 3.3 Quality Indicators for Random Reference Tracking in LPNI Systems 90 3.3.1 Scenario 90 3.3.2 Saturating Random Sensitivity Function 92 3.3.3 Tracking Quality Indicators 98 3.3.4 Application: LPNI Hard Disk Servo Design 101 3.4 Summary 105 3.5 Problems 106 3.6 Annotated Bibliography 112 4 Analysis of Disturbance Rejection in LPNI Systems . . . . . . . . . . . . . 114 4.1 Basic Relationships 114 4.1.1 SISO Systems 115 4.1.2 MIMO Systems 116 4.2 Fundamental Limitations on Disturbance Rejection 124 4.3 LPNI Systems with Rate-Saturated Actuators 125 4.3.1 Modeling Rate-Saturated Actuators 126 4.3.2 Bandwidth of Rate-Saturated Actuators 127 4.3.3 Disturbance Rejection in LPNI Systems with Rate-Saturated Actuators 128 4.4 Summary 130 4.5 Problems 132 4.6 Annotated Bibliography 133 5 Design of Reference Tracking Controllers for LPNI Systems . . . . . . 134 5.1 Admissible Pole Locations for Random Reference Tracking 134 5.1.1 Scenario 134
- 15. Contents xi 5.1.2 Admissible Domains for Random Reference Tracking by Prototype Second Order System 137 5.1.3 Higher Order Systems 141 5.1.4 Application: Hard Disk Servo Design 141 5.2 Saturated Root Locus 143 5.2.1 Scenario 143 5.2.2 Definitions 144 5.2.3 S-Root Locus When Ke(K) Is Unique 145 5.2.4 S-Root Locus When Ke(K) Is Nonunique: Motivating Example 149 5.2.5 S-Root Locus When Ke(K) Is Nonunique: General Analysis 153 5.2.6 Approach to Controller Design to Avoid Nonunique Ke(K) 154 5.2.7 S-Root Locus and Amplitude Truncation 155 5.2.8 Calibration of the S-Root Locus 157 5.2.9 Application: LPNI Hard Disk Servo Design 159 5.3 Summary 161 5.4 Problems 162 5.5 Annotated Bibliography 165 6 Design of Disturbance Rejection Controllers for LPNI Systems. . . . 167 6.1 Saturated LQR/LQG 167 6.1.1 Scenario 167 6.1.2 Problem Formulation 168 6.1.3 SLQR Theory 169 6.1.4 SLQG Theory 174 6.1.5 Application: Ship Roll Damping Problem 178 6.1.6 Generalizations 181 6.2 Instrumented LQR/LQG 182 6.2.1 Scenario 182 6.2.2 ILQR Theory 184 6.2.3 ILQG Theory 188 6.2.4 Generalizations 193 6.2.5 Application: Ship Roll Damping Problem 195 6.3 Summary 197 6.4 Problems 198 6.5 Annotated Bibliography 202 7 Performance Recovery in LPNI Systems . . . . . . . . . . . . . . . . . . . . . . . 204 7.1 Partial Performance Recovery 204 7.1.1 Scenario 204 7.1.2 Problem Formulation 205 7.1.3 Main Result 206
- 16. xii Contents 7.1.4 Examples 207 7.2 Complete Performance Recovery 209 7.2.1 Scenario 209 7.2.2 Problem Formulation 211 7.2.3 a-Boosting 212 7.2.4 s-Boosting 214 7.2.5 Simultaneous a- and s-Boosting 214 7.2.6 Stability Verification in the Problem of Boosting 215 7.2.7 Accuracy of Stochastic Linearization in the Problem of Boosting 215 7.2.8 Application: MagLev 217 7.3 Summary 218 7.4 Problems 219 7.5 Annotated Bibliography 224 8 Proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 8.1 Proofs for Chapter 2 225 8.2 Proofs for Chapter 3 226 8.3 Proofs for Chapter 4 234 8.4 Proofs for Chapter 5 236 8.5 Proofs for Chapter 6 241 8.6 Proofs for Chapter 7 269 8.7 Annotated Bibliography 272 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Abbreviations and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
- 17. Preface Purpose: This volume is devoted to the study of feedback control of so-called linear plant/nonlinear instrumentation (LPNI) systems. Such systems appear naturally in situations where the plant can be viewed as linear but the instrumentation, that is, actuators and sensors, can not. For instance, when a feedback system operates effectively and maintains the plant close to a desired operating point, the plant may be linearized, but the instrumentation may not, because to counteract large perturbations or to track large reference signals, the actuator may saturate and the nonlinearities in sensors, for example, quantization and dead zones, may be activated. The problems of stability and oscillations in LPNI systems have been studied for a long time. Indeed, the theory of absolute stability and the harmonic balance method are among the best known topics of control theory. More recent literature has also addressed LPNI scenarios, largely from the point of view of stability and anti- windup. However, the problems of performance analysis and design, for example, reference tracking and disturbance rejection, have not been investigated in sufficient detail. This volume is intended to contribute to this end by providing methods for designing linear controllers that ensure the desired performance of closed loop LPNI systems. The methods developed in this work are similar to the usual linear system techniques, for example, root locus, LQR, and LQG, modified appropriately to account for instrumentation nonlinearities. Therefore, we refer to these methods as quasilinear and to the resulting area of control as quasilinear control. Intent and prerequisites: This volume is intended as a textbook for a graduate course on quasilinear control or as a supplementary textbook for standard graduate courses on linear and nonlinear control. In addition, it can be used for self-study by practic- ing engineers involved in the analysis and design of control systems with nonlinear instrumentation. The prerequisites include material on linear and nonlinear systems and control. Some familiarity with elementary probability theory and random processes may also be useful. xiii
- 18. xiv Preface C(s) P(s) f(·) g(·) d r u y ym − Figure 0.1. Linear plant/nonlinear instrumentation control system Problems addressed: Consider the single-input single-output (SISO) system shown in Figure 0.1, where P(s) and C(s) are the transfer functions of the plant and the controller; f(·), g(·) are static odd nonlinearities characterizing the actuator and the sensor; and r, d, u, y, and ym are the reference, disturbance, control, plant output, and sensor output, respectively. In the framework of this system and its multiple- input multiple-output (MIMO) generalizations, this volume considers the following problems: P1. Performance analysis: Given P(s), C(s), f(·), and g(·), quantify the quality of reference tracking and disturbance rejection. P2. Narrow sense design: Given P(s), f(·), and g(·), design a controller C(s) so that the quality of reference tracking and disturbance rejection meets specifications. P3. Wide sense design: Given P(s), design a controller C(s) and select instru- mentation f(·) and g(·) so that the quality of reference tracking and disturbance rejection meets specifications. P4. Partial performance recovery: Let C(s) be a controller, which is designed under the assumption that the actuator and the sensor are linear and which meets reference tracking and disturbance rejection specifications. Given C(s), select f(·) and g(·) so that the performance degradation is guaranteed to be less than a given bound. P5. Complete performance recovery: Given f(·) and g(·), modify, if possible, C(s) so that performance degradation does not take place. This volume provides conditions under which solutions of these problems exist and derives equations and algorithms that can be used to calculate these solutions. Nonlinearities considered: We consider actuators and sensors characterized by piecewise continuous odd scalar functions. For example, we address: • saturating actuators, f(u) = satα(u) := α, u +α, u, −α ≤ u ≤ α, −α, u −α, (0.1) where α is the actuator authority;
- 19. Preface xv • quantized sensors, g(y) = qn(y) := ++y/, y ≥ 0, −−y/, y 0, (0.2) where is the quantization interval and u denotes the largest integer less than or equal to y; • sensors with a deadzone, g(y) = dz(y) := y − , y +, 0, − ≤ u ≤ +, y + , y −, (0.3) where 2 is the deadzone width. The methods developed here are modular in the sense that they can be modified to account for any odd instrumentation nonlinearity just by replacing the general function representing the nonlinearity by a specific one corresponding to the actuator or sensor in question. Main difficulty: LPNI systems are described by relatively complex nonlinear differ- ential equations. Unfortunately, these equations cannot be treated by the methods of modern nonlinear control theory since the latter assumes that the control signal enters the state space equations in a linear manner and, thus, saturation and other nonlinearities are excluded. Therefore, a different approach to treat LPNI control systems is necessary. Approach: The approach of this volume is based on the method of stochastic lin- earization, which is applicable to dynamical systems with random exogenous signals. Thus, we assume throughout this volume that both references and disturbances are random. However, several results on tracking deterministic references (e.g., step, ramp) are also included. According to stochastic linearization, the static nonlinearities are replaced by equivalent or quasilinear gains Na and Ns (see Figure 0.2, where û, ŷ, and ŷm replace u, y, and ym). Unlike the usual Jacobian linearization, the resulting approximation is global, that is, it approximates the original system not only for small but for large signals as well. The price to pay is that the gains Na and Ns depend not only on the nonlinearities f(·) and g(·), but also on all other elements of Figure 0.1, including the transfer functions and the exogenous signals, since, as it turns out, Na and Ns are functions of the standard deviations, σû and σŷ, of û and ŷ, respectively, that is, Na = Na(σû) and Ns = Ns(σŷ). Therefore, we refer to the system of Figure 0.2 as a quasilinear control system. Systems of this type are the main topic of study in this volume. Thus, instead of assuming that a linear system represents the reality, as in linear control, we assume that a quasilinear system represents the reality and carry out
- 20. xvi Preface C(s) P(s) Na Ns d r û ŷ ŷm − Figure 0.2. Quasilinear control system control-theoretic developments of problems P1–P5, which parallel those of linear control theory, leading to what we call quasilinear control (QLC) theory. The question of accuracy of stochastic linearization, that is, the precision with which the system of Figure 0.2 approximates that of Figure 0.1, is clearly of impor- tance. Unfortunately, no general results in this area are available. However, various numerical and analytical studies indicate that if the plant, P(s), is low-pass filtering, the approximation is well within 10% in terms of the variances of y and ŷ and u and û. More details on stochastic linearization and its accuracy are included in Chapter 2. It should be noted that stochastic linearization is somewhat similar to the method of harmonic balance, with Na(σû) and Ns(σŷ) playing the roles of describing functions. Book organization: The book consists of eight chapters. Chapter 1 places LPNI systems and quasilinear control in the general field of control theory. Chapter 2 describes the method of stochastic linearization as it applies to LPNI systems and derives equations for quasilinear gains in the problems of reference tracking and disturbance rejection. Chapters 3 and 4 are devoted to analysis of quasilinear con- trol systems from the point of view of reference tracking and disturbance rejection, respectively (problem P1). Chapters 5 and 6 also address tracking and disturbance rejection problems, but from the point of view of design; both wide and narrow sense design problems are considered (problems P2 and P3). Chapter 7 addresses the issues of performance recovery (problems P4 and P5). Finally, Chapter 8 includes the proofs of all formal statements included in the book. Each chapter begins with a short motivation and overview and concludes with a summary and annotated bibliography. Chapters 2–7 also include homework problems. Acknowledgments: The authors thankfully acknowledge the stimulating environ- ment at the University of Michigan, which was conducive to the research that led to this book. Financial support was provided for more than fifteen years by the National Science Foundation; gratitude to the Division of Civil, Mechanical and Manufacturing Innovations is in order. Thanks are due to the University of Michigan graduate students who took a course based on this book and provided valuable comments: these include M.S. Holzel, C.T. Orlowski, H.-R. Ossareh, H.W. Park, H.A. Poonawala, and E.D. Summer. Special thanks are due to Hamid-Reza Ossareh, who carefully read every chapter of the manuscript and made numerous valuable suggestions. Also, the
- 21. Preface xvii authors are grateful to University of Michigan graduate student Chris Takahashi, who participated in developing the LMI approach to LPNI systems. The authors are also grateful to Peter Gordon, Senior Editor at Cambridge University Press, for his support during the last year of this project. Needless to say, however, all errors, which are undoubtedly present in the book, are due to the authors alone. The list of corrections is maintained at http://www.eecs.umich.edu/∼smm/monographs/QLC/. Last, but not least, we are indebted to our families for their love and support, which made this book a reality.
- 23. 1 Introduction Motivation: This chapter is intended to introduce the class of systems addressed in this volume – the so-called Linear Plant/Nonlinear Instrumentation (LPNI) systems – and to characterize the control methodology developed in this book – Quasilinear Control (QLC). Overview: After introducing the notions of LPNI systems and QLC and listing the problems addressed, the main technique of this book – the method of stochastic lin- earization – is briefly described and compared with the usual, Jacobian, linearization. In the framework of this comparison, it is shown that the former provides a more accurate description of LPNI systems than the latter, and the controllers designed using the QLC result, generically, yield better performance than those designed using linear control (LC). Finally, the content of the book is outlined. 1.1 Linear Plant/Nonlinear Instrumentation Systems and Quasilinear Control Every control system contains nonlinear instrumentation – actuators and sensors. Indeed, the actuators are ubiquitously saturating; the sensors are often quantized; deadzone, friction, hysteresis, and so on are also encountered in actuator and sensor behavior. Typically, the plants in control systems are nonlinear as well. However, if a con- trol system operates effectively, that is, maintains its operation in a desired regime, the plant may be linearized and viewed as locally linear. The instrumentation, how- ever, can not: to reject large disturbances, to respond to initial conditions sufficiently far away from the operating point, or to track large changes in reference signals – all may activate essential nonlinearities in actuators and sensors, resulting in fundamen- tally nonlinear behavior. These arguments lead to a class of systems that we refer to as Linear Plant/Nonlinear Instrumentation (LPNI). The controllers in feedback systems are often designed to be linear. The main design techniques are based on root locus, sensitivity functions, LQR/LQG, H∞, and so on, all leading to linear feedback. Although for LPNI systems both linear and 1
- 24. 2 Introduction nonlinearcontrollersmaybeconsidered, totransfertheabove-mentionedtechniques to the LPNI case, we are interested in designing linear controllers. This leads to closed loop LPNI systems. This volume is devoted to methods for analysis and design of closed loop LPNI systems. As it turns out, these methods are quite similar to those in the linear case. For example, root locus can be extended to LPNI systems, and so can LQR/LQG, H∞, and so on. In each of them, the analysis and synthesis equations remain prac- tically the same as in the linear case but coupled with additional transcendental equations, which account for the nonlinearities. That is why we refer to the resulting methods as Quasilinear Control (QLC) Theory. Since the main analysis and design techniques of QLC are not too different from the well-known linear control theoretic methods, QLC can be viewed as a simple addition to the standard toolbox of control engineering practitioners and students alike. Although the term “LPNI systems” may be new, such systems have been consid- ered in the literature for more than 50 years. Indeed, the theory of absolute stability was developed precisely to address the issue of global asymptotic stability of linear plants with linear controllers and sector-bounded actuators. For the same class of systems, the method of harmonic balance/describing functions was developed to provide a tool for limit cycle analysis. In addition, the problem of stability of systems with saturating actuators has been addressed in numerous publications. However, the issues of performance, that is, disturbance rejection and reference tracking, have been addressed to a much lesser extent. These are precisely the issues considered in this volume and, therefore, we use the subtitle Performance Analysis and Design of Feedback Systems with Nonlinear Actuators and Sensors. In view of the above, one may ask a question: If all feedback systems include nonlinear instrumentation, how have controllers been designed in the past, lead- ing to a plethora of successful applications in every branch of modern technology? The answer can be given as follows: In practice, most control systems are, indeed, designed ignoring the actuator and sensor nonlinearities. Then, the resulting closed loop systems are evaluated by computer simulations, which include nonlinear instru- mentation, and the controller gains are readjusted so that the nonlinearities are not activated. Typically, this leads to performance degradation. If the performance degradation is not acceptable, sensors and actuators with larger linear domains are employed, and the process is repeated anew. This approach works well in most cases, but not in all: the Chernobyl nuclear accident and the crash of a YF- 22 airplane are examples of its failures. Even when this approach does work, it requires a lengthy and expensive process of simulation and design/redesign. In addition, designing controllers so that the nonlinearities are not activated (e.g., actuator saturation is avoided) leads, as is shown in this book, to performance losses. Thus, developing methods in which the instrumentation nonlinearities are taken into account from the very beginning of the design process, is of signif- icant practical importance. The authors of this volume have been developing such methods for more than 15 years, and the results are summarized in this volume.
- 25. 1.2 QLC Problems 3 As a conclusion for this section, it should be pointed out that modern Nonlin- ear Control Theory is not applicable to LPNI systems because it assumes that the control signals enter the system equations in a linear manner, thereby excluding saturation and other nonlinearities in actuators. Model Predictive Control may also be undesirable, because it is computationally extensive and, therefore, complex in implementation. 1.2 QLC Problems Consider the closed loop LPNI system shown in Figure 1.1. Here the transfer func- tions P(s) and C(s) represent the plant and controller, respectively, and the nonlinear functions f(·) and g(·) describe, respectively, the actuator and sensor. The signals r, d, e, u, v, y, and ym are the reference, disturbance, error, controller output, actuator output, plant output, and measured output, respectively. These notations are used throughout this book. In the framework of the system of Figure 1.1, this volume considers the following problems (rigorous formulations are given in subsequent chapters): P1. Performance analysis: Given P(s), C(s), f(·), and g(·), quantify the perfor- mance of the closed loop LPNI system from the point of view of reference tracking and disturbance rejection. P2. Narrow sense design: Given P(s), f(·), and g(·), design, if possible, a controller so that the closed loop LPNI system satisfies the required performance specifications. P3. Wide sense design: Given P(s), design a controller C(s) and select the instru- mentation f(·) and g(·) so that the closed loop LPNI system satisfies the required performance specifications. P4. Partial performance recovery: Assume that a controller, Cl(s), is designed so that the closed loop system meets the performance specifications if the actuator and sensor were linear. Select f(·) and g(·) so that the performance degradation of the closed loop LPNI system with Cl(s) does not exceed a given bound, as compared with the linear case. P5. Complete performance recovery: As in the previous problem, let Cl(s) be a controller that satisfies the performance specifications of the closed loop system with linear instrumentation. For given f(·) and g(·), redesign Cl(s) so that the closed loop LPNI exhibits, if possible, no performance degradation. C(s) P(s) f (·) g(·) d r – + u y ym v e Figure 1.1. Closed loop LPNI system.
- 26. 4 Introduction The first two of the above problems are standard in control theory, but are considered here for the LPNI case. The last three problems are specific to LPNI systems and have not been considered in linear control (LC). Note that the last problem is reminiscent of anti-windup control, whereby Cl(s) is augmented by a mechanism that prevents the so-called windup of integral controllers in systems with saturating actuators. 1.3 QLC Approach: Stochastic Linearization The approach of QLC is based on a quasilinearization technique referred to as stochastic linearization. This method was developed more than 50 years ago and since then has been applied in numerous engineering fields. Applications to feed- back control have also been reported. However, comprehensive development of a control theory based on this approach has not previously been carried out. This is done in this volume. Stochastic linearization requires exogenous signals (i.e., references and distur- bances) to be random. While this is often the case for disturbances, the references are assumed in LC to be deterministic – steps, ramps, or parabolic signals. Are these the only references encountered in practice? The answer is definitely in the negative: in many applications, the reference signals can be more readily modeled as random than as steps, ramps, and so on. For example, in the hard disk drive control problem, the read/write head in both track-seeking and track-following operations is affected by reference signals that are well modeled by Gaussian colored processes. Similarly, the aircraft homing problem can be viewed as a problem with random references. Many other examples of this nature can be given. Thus, along with disturbances, QLC assumes that the reference signals are random processes and, using stochastic linearization, provides methods for designing controllers for both reference tracking and disturbance rejection problems. The standard, deterministic, reference signals are also used, for example, to develop the notion of LPNI system types and to define and analyze the notion of the so-called trackable domain. The essence of stochastic linearization can be characterized as follows: Assume that the actuator is described by an odd piecewise differentiable function f(u(t)), where u(t) is the output of the controller, which is assumed to be a zero-mean wide sense stationary (wss) Gaussian process. Consider the problem: approximate f(u(t)) by Nu(t), where N is a constant, so that the mean-square error is minimized. It turns out (see Chapter 2) that such an N is given by N = E df(u) du u=u(t) , (1.1) where E denotes the expectation. This is referred to as the stochastically linearized gain or quasilinear gain of f(u). Since the only free parameter of u(t) is its standard deviation, σu, it follows from (1.1) that the stochastically linearized gain depends on
- 27. 1.4 Quasilinear versus Linear Control 5 a single variable – the standard deviation of its argument; thus, N = N(σu). (1.2) Note that stochastic linearization is indeed a quasilinear, rather than linear, operation: the quasilinear gains of αf(·) and f(·)α, where α is a constant, are not the same, the former being αN(σu) the latter being N(ασu). In the closed loop environment, σu depends not only on f(u) but also on all other components of the system (i.e., the plant and the controller parameters) and on all exogenous signals (i.e., references and disturbances). This leads to transcen- dental equations that define the quasilinear gains. The study of these equations in the framework of various control-theoretic problems (e.g., root locus, sensitivity functions, LQR/LQG, H∞) is the essence of the theory of QLC. As in the open loop case, a stochastically linearized closed loop system is also not linear: its output to the sum of two exogenous signals is not equal to the sum of the outputs to each of these signals, that is, superposition does not hold. However, since, when all signals and functional blocks are given, the system has a constant gain N, we refer to a stochastically linearized closed loop system as quasilinear. 1.4 Quasilinear versus Linear Control Consider the closed-loop LPNI system shown in Figure 1.2(a). If the usual Jacobian linearization is used, this system is reduced to that shown in Figure 1.2(b), where all signals are denoted by the same symbols as in Figure 1.2(a) but with a ~. In this C(s) f (·) P(s) r u v y g(·) e ym − (a) LPNI system C(s) P(s) r ũ ṽ ỹ ẽ ỹm N J a = d dũ f (ũ) ũ * N J s = d dỹ g(ỹ) ỹ * − (b) Jacobian linearization C(s) P(s) r û v̂ ŷ ê ŷm Na = E [ d dû f (û)] Ns = E [ d dŷ g(ŷ)] − (c) Stochastic linearization Figure 1.2. Closed loop LPNI system and its Jacobian and stochastic linearizations.
- 28. 6 Introduction system, the actuator and sensor are represented by constant gains evaluated as the derivatives of f(·) and g(·) at the operating point: NJ a = df(ũ) dũ ũ=ũ∗ , (1.3) NJ s = dg(ỹ) dỹ ỹ=ỹ∗ . (1.4) Clearly, this system describes the original LPNI system of Figure 1.2(a) only locally, around the fixed operating point. If stochastic linearization is used, the system of Figure 1.2(a) is reduced to the quasilinear one shown in Figure 1.2(c), where all signals are again denoted by the same symbols as in Figure 1.2(a) but with aˆ; these notations are used throughout this book. As it is indicated above and discussed in detail in Chapter 2, here the actuator and sensor are represented by their quasilinear gains: Na(σû) = E df(û) dû |û=û(t) , (1.5) Ns(σŷ) = E dg(ŷ) dŷ |ŷ=ŷ(t) . (1.6) Since Na(σû) and Ns(σŷ) depend not only on f(·) and g(·) but also on all elements of the system in Figure 1.2(c), the quasilinearization describes the closed loop LPNI system globally, with “weights” defined by the statistics of û(t) and ŷ(t). The LC approach assumes the reduction of the original LPNI system to that of Figure 1.2(b) and then rigorously develops methods for closed loop system analysis and design. In contrast, the QLC approach assumes that the reduction of the original LPNI system to that of Figure 1.2(c) takes place and then, similar to LC, develops rigorous methods for quasilinear closed loop systems analysis and design. In both cases, of course, the analysis and design results are supposed to be used for the actual LPNI system of Figure 1.2(a). Which approach is better, LC or QLC? This may be viewed as a matter of belief or a matter of calculations. As a matter of belief, we think that QLC, being global, provides a more faithful description of LPNI systems than LC. To illustrate this, consider the disturbance rejection problem for the LPNI system of Figure 1.2(a) with P(s) = 1 s2 + s + 1 , C(s) = 1, f(u) = satα(u), g(y) = y, r(t) = 0 (1.7) and with a standard white Gaussian process as the disturbance at the input of the plant. In (1.7), satα(u) is the saturation function given by satα(u) = α, u +α, u, −α ≤ u ≤ α, −α, u −α. (1.8)
- 29. 1.4 Quasilinear versus Linear Control 7 0 0.5 1 1.5 2 0.2 0.25 0.3 0.35 0.4 0.45 0.5 α Output variance Stochastic linearization Jacobian linearization Actual system Figure 1.3. Comparison of stochastic linearization, Jacobian linearization, and actual system performance. For this LPNI system, we construct its Jacobian and stochastic linearizations and calculate the variances, σ2 ỹ and σ2 ŷ , of the outputs ỹ(t) and ŷ(t) as functions of α. (Note that σ2 ỹ is calculated using the usual Lyapunov equation approach and σ2 ŷ is calculated using the stochastic linearization approach developed in Chapter 2.) In addition, we simulate the actual LPNI system of Figure 1.2(a) and numerically eval- uate σ2 y . All three curves are shown in Figure 1.3. From this figure, we observe the following: • The Jacobian linearization of satα(u) is independent of α, thus, the predicted variance is constant. • When α is large (i.e., the input is not saturated), Jacobian linearization is accurate. However, it is highly inaccurate for small values of α. • Stochastic linearization accounts for the nonlinearity and, thus, predicts an output variance that depends on α. • Stochastic linearization accurately matches the actual performance for all values of α. We believe that a similar situation takes place for any closed loop LPNI sys- tem: Stochastic linearization, when applicable, describes the actual LPNI system more faithfully than Jacobian linearization. (As shown in Chapter 2, stochastic linearization is applicable when the plant is low-pass filtering.)
- 30. 8 Introduction As a matter of calculations, consider the LPNI system of Figure 1.2(a) defined by the following state space equations: ẋ1 ẋ2 = −1 −1 1 0 x1 x2 + 1 0 satα(u) + 1 0 w y = 0 1 x1 x2 , (1.9) with the feedback u = Kx, (1.10) where x = [x1,x2]T is the state of the plant and w is a standard white Gaussian process. The problem is to select a feedback gain K so that the disturbance is rejected in the best possible manner, that is, σ2 y is minimized. Based on Jacobian linearization, this can be accomplished using the LQR approach with a sufficiently small control penalty, say, ρ = 10−5. Based on stochastic linearization, this can be accomplished using the method developed in Chapter 5 and referred to as SLQR (where the “S” stands for “saturating”) with the same ρ. The resulting controllers, of course, are used in the LPNI system. Simulating this system with the LQR controller and with the SLQR controller, we evaluated numerically σ2 y for both cases. The results are shown in Figure 1.4 as a function of the saturation level. From this figure, we conclude the following: 0.5 1 1.5 2 2.5 3 3.5 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 α Output variance SLQR LQR Actual system with SLQR Actual system with LQR Figure 1.4. Comparison of LQR, SLQR, and actual system performance.
- 31. 1.5 Overview of Main QLC Results 9 • Since ρ is small and the plant is minimum phase, LQR provides a high gain solution that renders the output variance close to zero. Due to the underlying Jacobian linearization, this solution is constant for all α. • Due to the input saturation, the performance of the actual system with the LQR controller is significantly worse than the LQR design, even for relatively large values of α. • The SLQR solution explicitly accounts for α and, thus, yields a nonzero output variance. • The performance of the actual system with an SLQR controller closely matches the intended design. • The actual SLQR performance exceeds the actual LQR performance for all values of α. As shown, using LQR in this situation is deceiving since the actual system can never approach the intended performance. In contrast, the SLQR solution is highly representative of the actual system behavior (and, indeed, exceeds the actual LQR performance). In fact, it is possible to prove that QLC-based controllers (e.g., controllers designed using SLQR) generically ensure better performance of LPNI systems than LC-based controllers (e.g., based on LQR). These comparisons, we believe, justify the development and utilization of QLC. 1.5 Overview of Main QLC Results This section outlines the main QLC results included in this volume. Chapter 2 describes the method of stochastic linearization in the framework of LPNI systems. After deriving the expression for quasilinear gain (1.1) and illustrat- ing it for typical nonlinearities of actuators and sensors, it concentrates on closed loop LPNI systems (Figure 1.2(a)) and their stochastic linearizations (Figure 1.2(c)). Since the quasilinear gain of an actuator, Na, depends on the standard deviation of the signal at its input, σû and, in turn, σû depends on Na, the quasilinear gain of the actuator is defined by a transcendental equation. The same holds for the quasilinear gain of the sensor. Chapter 2 derives these transcendental equations for various sce- narios of reference tracking and disturbance rejection. For instance, in the problem of reference tracking with a nonlinear actuator and linear sensor, the quasilinear gain of the actuator is defined by the equation Na = F Fr (s)C(s) 1 + P(s)NaC (s) 2 , (1.11) where F (σ) = ∞ −∞ d dx f (x) 1 √ 2πσ exp − x2 2σ2 dx. (1.12) Here, Fr (s) is the reference coloring filter, f(x) is the nonlinear function that describes the actuator, and || · ||2 is the 2-norm of a transfer function. Chapter 2
- 32. 10 Introduction provides a sufficient condition under which this and similar equations for other per- formance problems have solutions and formulates a bisection algorithm to find them with any desired accuracy. Based on these solutions, the performance of closed loop LPNI systems in problems of reference tracking and disturbance rejection is inves- tigated. Finally, Chapter 2 addresses the issue of accuracy of stochastic linearization and shows (using the Fokker-Planck equation and the filter hypothesis) that the error between the standard deviation of the plant output and its quasilinearization (i.e., σy and σŷ) is well within 10%, if the plant is low-pass filtering. The equations derived in Chapter 2 are used throughout the book for various problems of performance analysis and design. Chapter 3 is devoted to analysis of reference tracking in closed loop LPNI sys- tems. Here, the notion of system type is extended to feedback control with saturating actuators, and it is shown that the type of the system is defined by the plant poles at the origin (rather than the loop transfer function poles at the origin, as it is in the linear case). The controller poles, however, also play a role, but a minor one compared with those of the plant. In addition, Chapter 3 introduces the notion of trackable domains, that is, the ranges of step, ramp, and parabolic signals that can be tracked by LPNI systems with saturating actuators. In particular, it shows that the trackable domain (TD) for step inputs, r(t) = r01(t), where r0 is a constant and 1(t) is the unit step function, is given by TD = {r0 : |r0| 1 C0 + P0 α}, (1.13) where C0 and P0 are d.c. gains of the controller and plant, respectively, and α is the level of actuator saturation. Thus, TD is finite, unless the plant has a pole at the origin. While the above results address the issue of tracking deterministic signals, Chapter 3 investigates also the problem of random reference tracking. First, lin- ear systems are addressed. As a motivation, it is shown that the standard deviation of the error signal, σe, is a poor predictor of tracking quality since for the same σe track loss can be qualitatively different. Based on this observation, the so-called tracking quality indicators, similar to gain and phase margins in linear systems, are introduced. The main instrument here is the so-called random sensitivity function (RS). In the case of linear systems, this function is defined by RS() = ||F(s)S(s)||2, (1.14) where, as before, F(s) is the reference signal coloring filter with 3dB bandwidth and S(s) is the usual sensitivity function. The bandwidth of RS(), its d.c. gain, and the resonance peak define the tracking quality indicators, which are used as specifications for tracking controller design. Finally, Chapter 3 transfers the above ideas to tracking random references in LPNI systems. This development is based on the so-called saturating random
- 33. 1.5 Overview of Main QLC Results 11 sensitivity function, SRS(,σr), defined as SRS(,σr) = RS() σr , (1.15) whereRS()istherandomsensitivityfunctionofthestochasticallylinearizedversion of the LPNI system and σr is the standard deviation of the reference signal. Using SRS(,σr), an additional tracking quality indicator is introduced, which accounts for the trackable domain and indicates when and to what extent amplitude truncation takes place. In conclusion, Chapter 3 presents a diagnostic flowchart that utilizes all tracking quality indicators to predict the tracking capabilities of LPNI systems with saturating actuators. These results, which transfer the frequency (ω) domain methods of LC to the frequency () domain methods of QLC, can be used for designing tracking controllers by shaping SRS(,σr). The theoretical developments of Chapter 3 are illustrated using the problem of hard disk drive control. Chapter 4 is devoted to analysis of the disturbance rejection problem in closed loop LPNI systems. Here, the results of Chapter 2 are extended to the multiple- input-multiple-output (MIMO) case. In addition, using an extension of the LMI approach, Chapter 4 investigates fundamental limitations on achievable disturbance rejection due to actuator saturation and shows that these limitations are similar to those imposed by non-minimum-phase zeros in linear systems. The final section of this chapter shows how the analysis of LPNI systems with rate saturation and with hysteresis can be reduced to the amplitude saturation case. Chapter 5 addresses the issue of designing tracking controllers for LPNI systems in the time domain. The approach here is based on the so-called S-root locus, which is the extension of the classical root locus to systems with saturating actuators. This is carried out as follows: Consider the LPNI system of Figure 1.5(a) and its stochastic linearization of Figure 1.5(b). The saturated root locus of the system of Figure 1.5(a) is the path traced by the poles of the quasilinear system of Figure 1.5(b) when K changes from 0 to ∞. If N were independent of K, the S-root locus would coincide with the usual root locus. However, since N(K) may tend to 0 as K → ∞, the behavior of the S-root locus is defined by limK→∞KN(K). If this limit is infinite, the S-root locus coincides with the usual root locus. If this limit is finite, the S-root locus terminates prematurely, prior to reaching the open-loop zeros. These points FΩ(s) C(s) P(s) sat(u) wr − y K u e r (a) LPNI system with saturating actuator and gain K FΩ(s) C(s) P(s) N(K) wr − ŷ K û ê r (b) Stochastically linearized version with the equivalent gain KN(K) Figure 1.5. Systems for S-root locus design.
- 34. 12 Introduction –30 –25 –20 –15 –10 –5 0 –30 –20 –10 0 10 20 30 jω σ RL SRL Figure 1.6. Saturated root locus. are referred to as termination points, and Chapter 5 shows that they can be evaluated using the positive solution, β∗, of the following equation: β − F (s)C (s) 1 + α √ 2/π β P(s)C (s) 2 = 0, (1.16) where, as before, F(s), P(s), and C(s) are the reference coloring filter, the plant and the controller, respectively. An example of the S-root locus (SRL) and the classical root locus (RL) is shown in Figure 1.6, where the termination points are indicated by white squares, the shaded area is the admissible domain, defined by the tracking quality indicators of Chapter 3, and the rest of the notations are the same as in the classical root locus. In addition, Chapter 5 introduces the notion of truncation points, which indi- cate the segments of the S-root locus corresponding to poles leading to amplitude truncation. These points are shown in Figure 1.6 by black squares; all poles beyond these locations result in loss of tracking due to truncations. To “push” the truncation points in the admissible domain, the level of saturation must be necessarily increased. These results provide an approach to tracking controller design for LPNI systems in the time domain. Chapter 6 develops methods for designing disturbance rejection controllers for LPNIsystems. First, itextendstheLQR/LQGmethodologiestosystemswithsaturat- ing actuators, resulting in SLQR/SLQG. It is shown that the SLQR/SLQG synthesis engine includes the same equations as in LQR/LQG (i.e., the Lyapunov and Riccati equations) coupled with additional transcendental equations that account for the quasilinear gain and the Lagrange multiplier associated with the optimization prob- lem. These coupled equations can be solved using a bisection algorithm. Among various properties of the SLQR/SLQG solution, it is shown that optimal disturbance rejection indeed requires the activation of saturation, which contradicts the intuitive opinion that it should be avoided. So the question “to saturate or not to saturate” is
- 35. 1.5 Overview of Main QLC Results 13 answered in the affirmative. Another technique developed in Chapter 6 is referred to as ILQR/ILQG, where the “I” stands for “Instrumented.” The problem here is to design simultaneously the controller and the instrumentation (i.e., actuator and sensor) so that a performance index is optimized. The performance index is given by J = σ2 ŷ + ρσ2 û + W(α,β), (1.17) where ρ 0 is the control penalty and W models the “cost” of the instrumentation as a function of the parameters α of the actuator and β of the sensor. Using the Lagrange multipliers approach, Chapter 6 provides a solution of this optimization problem, which again results in Lyapunov and Riccati equations coupled with transcendental relationships. The developments of Chapter 6 are illustrated by the problem of ship roll stabilization under sea wave disturbance modeled as a colored noise. Chapter 7 is devoted to performance recovery in LPNI systems. The problems here are as follows: Let the controller, Cl(s), be designed to satisfy performance specifications under the assumption that the actuator and sensor are linear. How should the parameters of the real, that is, nonlinear, actuator and sensor be selected so that the performance of the resulting LPNI system with the same Cl(s) will not degrade below a given bound? This problem is referred to as partial performance recovery. The complete performance recovery problem is to redesign Cl(s) so that the LPNI system exhibits the same performance as the linear one. The solution of the partial performance recovery problem is provided in terms of the Nyquist plot of the loop gain of the linear system. Based on this solution, the following rule of thumb is obtained: To ensure performance degradation of no more than 10%, the actuator saturation should be at least twice larger than the standard deviation of the controller output in the linear system, that is, α 2σul . (1.18) The problem of complete performance recovery is addressed using the idea of boosting Cl(s) gains to account for the drop in equivalent gains due to actuator and sensor nonlinearities. The so-called a- and s-boosting are considered, referring to boosting gains due to actuator and sensor nonlinearities, respectively. In particular, it is shown that a-boosting is possible if and only if the equation xF x P(s)C(s) 1 + P(s)C (s) 2 = 1 (1.19) with F defined in (1.12) has a positive solution. Based on this equation, the following rule of thumb is derived: Complete performance recovery in LPNI systems with saturating actuators is possible if α 1.25σul , (1.20) where all notations are the same as in (1.18). Thus, if the level of actuator saturation satisfies (1.20), the linear controller can be boosted so that no performance degra- dation takes place. A method for finding the boosting gain is also provided. The validation of the boosting approach is illustrated using a magnetic levitation system.
- 36. 14 Introduction The final chapter of the book, Chapter 8, provides the proofs of all formal statements included in the book. As it follows from the above overview, this volume transfers most of LC to QLC. Specifically, the saturating random sensitivity function and the tracking qual- ity indicators accomplish this for frequency domain techniques, the S-root locus for time domain techniques, and SLQR/SLQG for state space techniques. In addi- tion, the LPNI-specific problems, for example, truncation points of the root locus, instrumentation selection, and the performance recovery, are also formulated and solved. 1.6 Summary • The analysis and design of closed loop linear plant/nonlinear instrumentation (LPNI) systems is the main topic of this volume. • The goal is to extend the main analysis and design techniques of linear control (LC) to the LPNI case. Therefore, the resulting methods are referred to as quasilinear control (QLC). • The approach of QLC is based on the method of stochastic linearization. According to this method, an LPNI system is represented by a quasilinear one, where the static nonlinearities are replaced by the expected values of their gradients. As a result, stochastic linearization represents the LPNI system globally (rather than locally, as it is in the case of Jacobian linearization). • Stochastic linearizations of LPNI systems represent the actual LPNI systems more faithfully than Jacobian linearization. • Starting from stochastically linearized versions of LPNI systems, QLC devel- ops methods for analysis and design that are as rigorous as those of LC (which starts from Jacobian linearization). • This volume transfers most LC methods to QLC: The saturated random sensitivity function and the tracking quality indicators accomplish this for frequency domain techniques; S-root locus – for time domain techniques; and SLQR/SLQG – for state space techniques. • In addition, several LPNI-specific problems, for example, truncation points of the root locus, instrumentation selection, and the performance recovery, are formulated and solved. 1.7 Annotated Bibliography There is a plethora of monographs on design of linear feedback systems. Examples of undergraduate text are listed below: [1.1] B.C. Kuo, Automatic Control Systems, Fifth Edition, Prentice Hall, Englewood Cliffs, NJ, 1987 [1.2] K. Ogata, Modern Control Engineering, Second Edition, Prentice Hall, Englewood Cliffs, NJ, 1990 [1.3] R.C. Dorf and R.H. Bishop, Modern Control Systems, Eighth Edition, Addison-Wesley, Menlo Park, CA, 1998
- 37. 1.7 Annotated Bibliography 15 [1.4] G.C. Godwin, S.F. Graebe, and M.E. Salgado, Control Systems Design, Prentice Hall, Upper Shaddle River, NJ, 2001 [1.5] G.F. Franklin, J.D. Powel, and A. Emami-Naeini, Feedback Control of Dynamic Systems, Fourth Edition, Prentice Hall, Englewood Cliffs, NJ, 2002 At the graduate level, the following can be mentioned: [1.6] I.M. Horowitz, Synthesis of Feedback Systems, Academic Press, London, 1963 [1.7] H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, Wiley-Interscience, New York, 1972 [1.8] W.M. Wonham, Linear Multivariable Control: A Geometric Approach, Third Edition, Springer-Verlag, New York, 1985 [1.9] B.D.O. Anderson and J.B. Moore, Optimal Control: Linear Quadratic Methods, Prentice Hall, Englewood Cliffs, NJ, 1989 [1.10] J.M. Maciejowski, Multivariable Feedback Design, Addison-Wesley, Reading, MA, 1989 [1.11] K. Zhou, J.C. Doyle, and K. Glover, Robust and Optimal Control, Prentice Hall, Upper Saddle River, NJ, 1996 The theory of absolute stability has its origins in the following: [1.12] A.I. Lurie and V.N. Postnikov, “On the theory of stability of control systems,” Applied Mathematics and Mechanics, Vol. 8, No. 3, pp. 246–248, 1944 (in Russian) [1.13] M.A. Aizerman, “On one problem related to ‘stability-in-the-large’ of dynamical systems,” Russian Mathematics Uspekhi, Vol. 4, No. 4, pp. 187–188, 1949 (in Russian) Subsequent developments are reported in the following: [1.14] V.M. Popov, “On absolute stability of nonlinear automatic control systems,” Avtomatika i Telemekhanika, No. 8, 1961 (in Russian). English translation: Automation and Remote Control, Vol. 22, No. 8, pp. 961–979, 1961 [1.15] V.A. Yakubovich, “The solution of certain matrix inequalities in automatic control theory,” Doklady Akademii Nauk, Vol. 143, pp. 1304–1307, 1962 (in Russian) [1.16] M.A. Aizerman and F.R. Gantmacher, Absolute Stability of Regulator Systems. Holden-Day, San Francisco, 1964 (Translated from the Russian original, Akad. Nauk SSSR, Moscow, 1963) [1.17] R. Kalman, “Lyapunov functions for the problem of Lurie in automatic control, Proc. of the National Academy of Sciences of the United States of America, Vol. 49, pp. 201–205, 1963 [1.18] K.S. Narendra and J. Taylor, Frequency Domain Methods for Absolute Stability, Academic Press, New York, 1973 The method of harmonic balance has originated in the following: [1.19] L.S. Goldfarb, “On some nonlinearities in regulator systems,” Avtomatika i Telemekhanika, No. 5, pp. 149–183, 1947 (in Russian). [1.20] R. Kochenburger, “A frequency response method for analyzing and synthesizing contactor servomechanisms,” Trans. AIEE, Vol. 69, pp. 270–283, 1950 This was followed by several decades of further development and applications. A summary of early results can be found in the following:
- 38. 16 Introduction [1.21] A. Gelb and W.E. Van der Velde, Multiple-Input Describing Function and Nonlinear System Design, McGraw-Hill, New York, 1968, while later ones in [1.22] A.I. Mees, “Describing functions – 10 years later,” IMA Journal of Applied Mathematics, Vol. 32, No. 1–3, pp. 221–233, 1984 For the justification of this method (based on the idea of “filter hypothesis”) and evaluation of its accuracy, see the following: [1.23] M.A. Aizerman, “Physical foundations for application small parameter methods to problems of automatic control,” Avtomatika i Telemekhanika, No. 5, pp. 597–603, 1953 (in Russian) [1.24] E.M. Braverman, S.M. Meerkov, and E.S. Piatnitsky, “A small parameter in the problem of justifying the harmonic balance method (in the case of the filter hypothesis),” Avtomatika i Telemekhanika, No. 1, pp. 5–21, 1975 (in Russian). English translation: Automation and Remote Control, Vol. 36, No. 1, pp. 1–16, 1975 Using the notion of the mapping degree, this method has been justified in the following: [1.25] A.R. Bergen and R.L. Frank, “Justification of the describing function method,” SIAM Journal of Control, Vol. 9, No. 4, pp. 568–589, 1971 [1.26] A.I. Mees and A.R. Bergen, “Describing functions revisited,” IEEE Transactions on Automatic Control, Vol. AC-20, No. 4, pp. 473–478, 1975 Several monographs that address the issue of stability of LPNI systems with saturating actuators can be found in the following: [1.27] T. Hu and Z. Lin, Control Systems with Actuator Saturation, Birkauser, Boston, MA, 2001 [1.28] A. Saberi, A.A. Stoorvogel, and P. Sannuti, Control of Linear Systems with Regulation and Input Constraints, Springer-Verlag, New York, 2001 [1.29] V. Kapila and K.M. Grigoriadis, Ed., Actuator Saturation Control, Marcel Dekker, Inc., New York, 2002 Remarks on the saturating nature of the Chernobyl nuclear accident can be found in the following: [1.30] G. Stein, “Respect for unstable,” Hendrik W. Bode Lecture, Proc. ACC, Tampa, FL, 1989 Reasons for the crash of the YF-22 aircraft are reported in the following: [1.31] M.A. Dornheim, “Report pinpoints factors leading to YF-22 crash”, Aviation Week Space Technology., Vol. 137, No. 19, pp. 53–54, 1992 Modern theory of nonlinear control based on the geometric approach has its origin in the following: [1.32] R.W. Brockett, “Asymptotic stability and feedback stabilization,” in Differential Geometric Control Theory, R.W. Brockett, R.S. Millman, and H.J. Sussmann, Eds., pp. 181–191, 1983
- 39. 1.7 Annotated Bibliography 17 Further developments are reported in the following: [1.33] A. Isidori, Nonlinear Control Systems, Third Edition, Springer-Verlag, New York, 1995 Model predictive control was advanced in the following: [1.34] J. Richalet, A. Rault, J.L. Testud, and J. Papon, “Model predictive heuristic control: Applications to industrial processes,” Automatica, Vol. 14, No. 5, pp. 413–428, 1978 [1.35] C.R. Cutler and B.L. Ramaker, “Dynamic matrix control – A computer control algorithm,” in AIChE 86th National Meeting, Houston, TX, 1979 Additional results can be found in [1.36] C.E. Garcia, D.M. Prett, and M. Morari, “Model predictive control: Theory and practice – a survey,” Automatica, Vol. 25, No. 3, pp. 338–349, 1989 [1.37] E.G. Gilbert and K. Tin Tan, “Linear systems with state and control constraints: the theory and applications of maximal output admissible sets,” IEEE Transactions Automatic Control, Vol. 36, pp. 1008–1020, 1995 [1.38] D.Q. Mayne, J.B. Rawlings, C.V. Rao, and P.O.M. Scokaert, “Constrained model predictive control: Stability and optimality,” Automatica, Vol. 36, pp. 789–814, 2000 [1.39] E.F. Camacho and C. Bordons, Model Predictive Control, Springer-Verlag, London, 2004 The term integrator “windup” seems to have appeared in [1.40] J.C. Lozier, “A steady-state approach to the theory of saturable servo systems,” IRE Transactions on Automatic Control, pp. 19–39, May 1956 Early work on antiwindup can be found in the following: [1.41] H.A. Fertic and C.W. Ross, “Direct digital control algorithm with anti-windup feature,” ISA Transactions, Vol. 6, No. 4, pp. 317–328, 1967 More recent results can be found in the following: [1.42] M.V. Kothare, P.J. Campo, M. Morari, and C.N. Nett, “A unified framework for the study of anti-windup designs,” Automatica, Vo. 30, No. 12, pp. 1869–1883, 1994 [1.43] N. Kapoor, A.R. Teel, and P. Daoutidis, “An anti-windup design for linear systems with input saturation,” Automatica, Vol. 34, No. 5, pp. 559–574, 1998 [1.44] P. Hippe, Windup in Control: Its Effects and Their Prevention, Springer, London, 2006 The method of stochastic linearization originated in the following: [1.45] R.C. Booton, M.V. Mathews, and W.W. Seifert, “Nonlinear servomechanisms with random inputs,” Dyn. Ana. Control Lab, MIT, Cambridge, MA, 1953 [1.46] R.C. Booton, “The analysis of nonlinear systems with random inputs,” IRE Transactions on Circuit Theory, Vol. 1, pp. 32–34, 1954 [1.47] I.E. Kazakov, “Approximate method for the statistical analysis of nonlinear systems,” Trudy VVIA 394, 1954 (in Russian) [1.48] I.E. Kazakov, “Approximate probability analysis of operational position of essentially nonlinear feedback control systems,” Automation and Remote Control, Vol. 17, pp. 423–450, 1955
- 40. 18 Introduction Various extensions can be found in the following: [1.49] V.S. Pugachev, Theory of Random Functions, Pergamon Press, Elmsford, NY, 1965 (translation from Russian) [1.50] I. Elishakoff, “Stoshastic linearization technique: A new interpretation and a selective review,” The Shock and Vibration Digest, Vol. 32, pp. 179–188, 2000 [1.51] J.B. Roberts and P.D. Spanos, Random Vibrations and Statistical Linearization, Dover Publications, Inc., Mineola, NY, 2003 [1.52] L. Socha, Linearization Methods for Stochastic Systems, Springer, Berlin Heidelberg, 2008 Applications to control problems have been described in the following: [1.53] I.E. Kazakov and B.G. Dostupov, Statistical Dynamics of Nonlinear Control Systems, Fizmatgiz, Moscow 1962 (In Russian) [1.54] A.A. Pervozvansky, Stochastic Processes in Nonlinear Control Systems, Fizmatgiz, Moscow 1962 (in Russian) [1.55] I.E. Kazakov, “Statistical analysis of systems with multi-dimensional nonlinearities,” Automation and Remote Control, Vol. 26, pp. 458–464, 1965 and also in reference [1.21] The stochastic modeling of reference signals in the problem of hard drive control can be found in the following: [1.56] A. Silberschatz and P.B. Galvin, Operating Systems Concepts, Addison-Wesley, 1994 [1.57] T.B. Goh, Z. Li and B.M. Chen, “Design and implementation of a hard disk servo system using robust abd perfect tracking approach,” IEEE Transactions on Control Systems Technology, Vol. 9, pp. 221–233, 2001 For the aircraft homing problem, similar conclusions can be deduced from the following: [1.58] C.-F. Lin, Modern Navigation, Guidance, and Control Processing, Prentice Hall, Englewood Cliffs, NJ, 1991 [1.59] E.J. Ohlmeyer, “Root-mean-square miss distance of proportional navigation missile against sinusoidal target,” Journal of Guidance, Control and Dynamics, Vol. 19, No. 3, pp. 563–568, 1996 In automotive problems, stochastic reference signals appear in the following: [1.60] H.S. Bae and J.C. Gerdes, “Command modification using input shaping for automated highway systems with heavy trucks,” California PATH Research Report, 1(UCB-ITS-PRR-2004-48), Berkeley, CA, 2004 The usual, Jacobian, linearization is the foundation of all methods for analysis and design on linear systems, including the indirect Lyapunov method. For more information see the following: [1.61] M. Vidyasagar, Nonlinear Systems Analysis, Second Edition, Prentice Hall, Englewood Cliffs, NJ, 1993 [1.62] H.K. Khalil, Nonlinear Systems, Third Edition, Prentice Hall, Upper Saddle River, NJ, 2002 A discussion on calculating the 2-norm of a transfer function can be found in [1.63] K. Zhou and J.C. Doyle, Essentials of Robust Control, Prentice Hall, Upper Saddle River, NJ, 1999
- 41. 1.7 Annotated Bibliography 19 For the theory of Fokker-Planck equation turn to [1.64] L. Arnold, Stochastic Differential Equations, Wiley Interscience, New York, 1973 [1.65] H. Risken, The Fokker-Planck Equation: Theory and Applications, Springer-Verlag, Berlin, 1989 [1.66] Z. Schuss, Theory and Applications of Stochastic Processes, Springer, New York, 2009
- 42. 2 Stochastic Linearization of LPNI Systems Motivation: This chapter is intended to present the main mathematical tool of this book – the method of stochastic linearization – in terms appropriate for the sub- sequent analysis and design of closed loop LPNI systems. Those familiar with this method are still advised to read this chapter since it derives equations used throughout this volume. Overview: First, we present analytical expressions for the stochastically linearized (or quasilinear) gains of open loop systems. Then we derive transcendental equations that define the quasilinear gains of various types of closed loop LPNI systems. Finally, we discuss the accuracy of stochastic linearization in predicting the standard deviations of various signals in closed loop LPNI systems. 2.1 Stochastic Linearization of Open Loop Systems 2.1.1 Stochastic Linearization of Isolated Nonlinearities Quasilinear gain: Consider Figure 2.1, where f(u) is an odd piece wise differentiable function, u(t) is a zero-mean wide sense stationary (wss) Gaussian process, v(t) = f(u(t)), (2.1) N is a constant, and v̂(t) = Nu(t). (2.2) The problem is to approximate f(u) by Nu(t) so that ε(N) = E v(t) − v̂(t) 2 (2.3) is minimized, where E denotes the expectation. The solution of this problem is given by the following theorem: 20
- 43. 2.1 Open Loop Systems 21 f (u) N u(t) v(t) v̂(t) Figure 2.1. Stochastic linearization of an isolated nonlinearity. Theorem 2.1. If u(t) is a zero-mean wide sense stationary Gaussian process and f(u) is an odd, piecewise differentiable function, (2.3) is minimized by N = E df(u) du u=u(t) . (2.4) Since the proof of this theorem is simple and instructive, we provide it here, rather than in Chapter 8. Proof. Rewriting (2.3) as ε(N) = E (f(u) − Nu)2 (2.5) and differentiating with respect to N, results in the following condition of optimality: dε dN = E[2(f(u) − Nu)u] = 0. (2.6) It is easy to verify that this is, in fact, the condition of minimality and, therefore, the minimizer of (2.3) is given by N = E[f(u)u] E u2 . (2.7) Taking into account that for zero-mean wss Gaussian u(t) and piecewise differentiable f(u), E[f(u)u] = E u2 E df du u=u(t) , (2.8) (2.4) follows immediately from the last two expressions. It turns out that (2.4) holds for a more general approximation of f(u). To show this, let n(t) be the impulse response of a causal linear system and, instead of (2.2), introduce the approximation v̂(t) = n(t) ∗ u(t), (2.9) where ∗ denotes the convolution. The problem is to select n(t) so that the functional ε(n(t)) = E (f(u(t)) − n(t) ∗ u(t))2 (2.10) is minimized.
- 44. 22 Stochastic Linearization of LPNI Systems Theorem 2.2. Under the assumptions of Theorem 2.1, ε(n(t)) is minimized by n(t) = E df(u) du u=u(t) δ(t), (2.11) where δ(t) is the δ-function. Proof. See Section 8.1. Thus, in this formulation as well, the minimizer of the mean square error is a static system with gain N = E df(u) du u=u(t) . (2.12) The gain N is referred to as the stochastic linearization or the quasilinear gain of f(u). Unlike the local, Jacobian, linearization of f(u), that is, NJ = df(u) du u=u∗ , (2.13) where u∗ is an operating point, N of (2.12) is global in the sense that it characterizes f(u) at every point with the weight defined by the statistics of u(t). This is the main utility of stochastic linearization from the point of view of the problems considered in this volume. Since the expectation in (2.12) is with respect to a Gaussian probability den- sity function (pdf) defined by a single parameter – the standard deviation, σu, the quasilinear gain N is, in fact, a function of σu, that is, N = N (σu). (2.14) With this interpretation, the quasilinear gain N can be understood as an analogue of the describing function F(A) of f(u), where the role of the amplitude, A, of the harmonic input u(t) = Asinωt isplayedbyσu. Itisnosurprise, therefore, thattheaccuracyofstochasticlinearization is similar to that of the harmonic balance method. As follows from (2.12), N is a linear functional of f(u). This implies that if N1 and N2 are quasilinear gains of f1(u) and f2(u), respectively, then N1 + N2 is the quasilinear gain of f1(u) + f2(u). Note, however, that the quasilinear gain of γ f(·), where γ is a constant, is not equal to the quasilinear gain of f(·)γ : if, in a serial connection, γ precedes f(·) the quasilinear gain of f(·)γ is N(γ σu); if f(·) precedes γ the quasilinear gain of γ f(·) is γ N(σu). In general, of course, γ N(σu) = N(γ σu). (2.15) This is why N is referred to as the quasilinear, rather than the linear, gain of f(·).
- 45. 2.1 Open Loop Systems 23 Examples: The stochastic linearization for typical nonlinearities of actuators and sensors is carried out below. The illustrations are provided in Table 2.1. Saturation nonlinearity: Consider the saturation function defined by f(u) = satα(u) := +α, u +α, u, −α ≤ u ≤ +α, −α, u −α, (2.16) where α 0. Since f (u) = 1, −α u +α, 0, u +α or u −α, (2.17) and u is zero-mean Gaussian, it follows that N = +∞ −∞ d du satα (u) 1 √ 2πσu exp − u2 2σ2 u du Table 2.1 Common nonlinearities and their stochastic linearizations Nonlinearity Quasilinear Gain Saturation v u a a 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 su Na Relay v u a a = 1 a = 1 0 0.5 1 1.5 2 0 10 20 30 40 50 60 70 80 Na (Continued) su
- 46. 24 Stochastic Linearization of LPNI Systems Table 2.1 (continued) Nonlinearity Quasilinear Gain Deadzone v u a = 1 a = 1, = 1 a = 1 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 su Na Saturation with deadzone v u a a = 1 0 2 4 6 8 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Na Friction v u a 0 1 2 3 4 5 0 1 2 3 4 5 6 7 8 9 10 (Continued) su Na su
- 47. 2.1 Open Loop Systems 25 Table 2.1 (continued) Nonlinearity Quasilinear Gain Piecewise-linear v u m1 m2 a m1 = 1, m2 = 2, a = 1 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 su Na Quantization v u = 0.25 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Saturation with quantization v u a = 0.25, a = 1 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 su Na su Na
- 48. 26 Stochastic Linearization of LPNI Systems = +α −α 1 √ 2πσu exp − u2 2σ2 u du = erf α √ 2σu , (2.18) where erf(x) is the error function defined by erf(x) = 1 √ π +x −x exp(−t2 )dt. (2.19) Note that, as it follows from (2.18), N = P{|u| ≤ α}, that is, N is the proba- bility that no saturation takes place. Clearly, N(σu) is a decreasing function of σu. Moreover, when σu is small, N ≈ 1, and when σu is large, N ≈ √ 2/π(α/σu). When α = 1, the function sat1(u) is referred to as the standard saturation and denoted as sat(u). Relay nonlinearity: For the relay function, f(u) = relα(u) := +α, u 0, 0, u = 0, −α, u 0, (2.20) the derivative of f(u) is f(u) = 2αδ(u). (2.21) Hence, taking the expectation in (2.12), it follows that N = 2 π α σu . (2.22) Clearly, the quasilinear gain is infinite at σu = 0 and decreases to zero hyperbolically as σu → ∞. Deadzone nonlinearity: Consider the deadzone nonlinearity, f(u) = dz(u) := u − , u +, 0, − ≤ u ≤ +, u + , u −. (2.23) Writing dz(u) as dz(u) = u − sat(u), (2.24) we obtain N = 1 − erf √ 2σu . (2.25)
- 49. 2.1 Open Loop Systems 27 Note that N = P{|u| ≥ }. Obviously, for σu , N ≈ 0, while N → 1 as σu → ∞. Saturation with deadzone nonlinearity: Consider the saturated deadzone nonlin- earity, f(u) = satα(dz(u)) := +α, u +α + , u − , + ≤ u ≤ +α + , 0, − u +, u + , −α − ≤ u ≤ −, −α, u −α − . (2.26) Since f (u) = 1, |u| α + , 0, otherwise, (2.27) it follows that N = − −α− 1 √ 2πσu exp − u2 2σ2 u du + +α+ + 1 √ 2πσu exp − u2 2σ2 u du = erf α + √ 2σu − erf √ 2σu . (2.28) A characteristic feature of this nonlinearity is that N is a nonmonotonic function of σu: increasing for small σu and decreasing for large ones. Friction nonlinearity: Consider the friction nonlinearity, f(u) = friα(u) := u + α, u 0, 0, u = 0, u − α, u 0. (2.29) Since friα(u) = u + relα(u), (2.30) it follows that N = 1 + 2 π α σu . (2.31) Again, N = ∞ for σu = 0 and decreases to 1 hyperbolically as σu → ∞. Piecewise-linear function: For the piecewise-linear function, f(u) = pwlα(u) := m2u + (m1 − m2)α, u +α, m1u, −α ≤ u ≤ +α, m2u + (m2 − m1)α, u −α, (2.32)
- 50. 28 Stochastic Linearization of LPNI Systems the derivative of f(u) is f (u) = m1, |u| α, m2, |u| α. (2.33) Thus, N = m2 + (m1 − m2)erf α √ 2σu . (2.34) Note that N = m2 + (m1 − m2)P{|u| ≤ α}. Quantization nonlinearity: The quantization nonlinearity is defined as f(u) = qn(u) := ++u/, u ≥ 0, −−u/, u 0, (2.35) where is the quantizer step size and u denotes the largest integer less than or equal to u. Clearly, f (u) = +∞ k=−∞ k=0 δ(u − k). (2.36) Hence, N = 2 2πσ2 u ∞ k=1 exp − 2 2σ2 u k2 . (2.37) For σu , N is akin to the deadzone and approaches 1 as σu → ∞. Saturation with quantization nonlinearity: The saturated quantization is defined as f(u) = satα(qn(u)) := +α, u ≥ +α, ++u/, 0 ≤ u α, −−u/, −α ≤ u 0, −α, u −α, (2.38) where it is assumed that α = m for a positive integer m. The derivative of f(u) is f (u) = +m k=−m k=0 δ(u − k). (2.39) Thus, N = 2 2πσ2 u m k=1 exp − 2 2σ2 u k2 . (2.40) Here, again, N is nonmonotonic in σu.
- 51. 2.1 Open Loop Systems 29 C(s) f(·) P(s) wr y − u FΩr (s) v r e (a) Open loop LPNI system C(s) Na P(s) wr ŷ − u v̂ r e (b) Open loop quasilinear system FΩr (s) Figure 2.2. Open loop LPNI system and its quasilinearization. 2.1.2 Stochastic Linearization of Direct Paths of LPNI Systems Quasilinear gain: Consider the open loop LPNI system shown in Figure 2.2(a), where Fr (s), P(s), and C(s) are transfer functions with all poles in open left half plane (OLHP)representingthecoloringfilterwiththe3dBbandwidthr, theplant, andthe controller, respectively, f(u) is the actuator nonlinearity, and wr and r are standard white noise and the reference signal. Since, in such a system, the steady state input to the nonlinearity is still a zero- mean wss Gaussian process, its stochastically linearized gain remains the same as in Subsection 2.1.1, and the corresponding quasilinear system is shown in Figure 2.2(b). Since the standard deviation σu is given by the 2-norm of the transfer function from wr to u, that is, σu = Fr (s)C(s) 2 = 1 2π +∞ −∞ |Fr (jω)|2|C(jω)|2 dω, the stochastically linearized gain in Figure 2.2(b) is Na (σu) = F Fr (s)C(s) 2 , (2.41) where F (σ) = +∞ −∞ d dx f (x) 1 √ 2πσ exp − x2 2σ2 dx. (2.42) Computational issues: To evaluate the standard deviation of the signal at the input of the actuator in Figure 2.2(b), one has to evaluate the 2-norm of a transfer function. A computationally convenient way to carry this out is as follows: Let {A,B,C} be a minimal realization of the strictly proper transfer function G(s) with all poles in the open left half plane. Consider the Lyapunov equation AR + RAT + BBT = 0. (2.43)
- 52. 30 Stochastic Linearization of LPNI Systems 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 a Na Figure 2.3. Quasilinear gain as a function of α in Example 2.1. Thenitiswellknownthatitssolution, R, ispositivedefinitesymmetricand, moreover, G(s) 2 = tr CRCT 1/2 . (2.44) Clearly, this provides a constructive way for calculating G(s) 2. In the MATLAB computational environment, the function norm can be used for this purpose. Example 2.1. Consider the system of Figure 2.2(a), with P(s) = 10 s(s + 10) , C(s) = 5, Fr (s) = √ 3 s3 + 2s2 + 2s + 1 , f(u) = satα(u). (2.45) The closed loop version of this system is a servomechanism with the reference signal, r, defined by a 3rd order Butterworth filter with bandwidth r = 1 and d.c. gain selected so that σr = 1. For this system, taking into account (2.18), from (2.41), Na = erf α √ 2 5 √ 3 s3+2s2+2s+1 2 . (2.46) The behavior of Na as a function of α is illustrated in Figure 2.3. From this figure we conclude the following: • For α ∈ (0,2), Na(α) is practically linear with slope 0.3. • For α 7, Na(α) is practically 1, that is, the effect of saturation may be ignored. 2.2 Stochastic Linearization of Closed Loop LPNI Systems 2.2.1 Notations and Assumptions The block diagram of the LPNI feedback systems studied in this volume is shown in Figure 2.4. As in Figure 2.2, P(s) and C(s) are the plant and controller, f(·) and g(·) are
- 53. 2.2 Closed Loop LPNI Systems 31 FΩr (s) FΩd (s) C(s) P(s) f (·) g(·) wd wr − e u v y ym r d Figure 2.4. Closed loop LPNI system. the actuator and sensor, Fd (s) and Fr (s) are coloring filters with 3dB bandwidths d and r, respectively, wr, wd are independent standard Gaussian white noise processes, and the scalars r, d, y, ym, u, v, and e denote the reference, disturbance, plant output, sensor (or measured) output, control signal, actuator output, and error signal, respectively. Let the quasilinear gains of f(·) and g(·) in isolation be denoted as Na and Ns, respectively. Assume that the range of Na is Na, the range of Ns is Ns, and the range of NaNs is Nas. For instance, if f(·) and g(·) are standard saturation functions, then Na = Ns = Nas = [0,1]. Using these notations, introduce the following: Assumption 2.1. (i) P(s) has all poles in the closed left half plane; (ii) C(s) has all poles in the closed left half plane; (iii) 1 + γ P(s)C(s) has all zeros in the open left half plane for γ ∈ Nas. This assumption, as it is shown below, is a sufficient condition for the existence of variances of various signals in the stochastically linearized versions of LPNI sys- tems under consideration. Therefore, unless stated otherwise, it is assumed to hold throughout this volume. Under Assumption 2.1, we discuss below stochastic linearization of the closed loop system of Figure 2.4. First, we address the case of nonlinear actuators and sensors separately and then the case of nonlinearities in both actuators and sensors simultaneously. 2.2.2 Reference Tracking with Nonlinear Actuator Quasilinear gain: Consider the closed loop system of Figure 2.5(a), where all func- tional blocks and signals remain the same as in Figure 2.4. The goal is to obtain its quasilinear approximation shown in Figure 2.5(b). The situation here is different from that of Figure 2.2 in two respects. First, the signal u(t) at the input of the nonlinearity is no longer Gaussian. Second, the signals u(t) and û(t) are not the same. Therefore, the quasilinear gain (2.12) is no longer optimal. Nevertheless, proceeding formally, we view the system of Figure 2.5(b) with Na = E df(û) dû û=û(t) (2.47) as the stochastic linearization of Figure 2.5(a).
- 54. 32 Stochastic Linearization of LPNI Systems C(s) f (·) P(s) wr y − u FΩr (s) v 0 e r (a) Closed loop LPNI system C(s) Na P(s) wr ŷ − û v̂ 0 ê r (b) Closed loop quasilinear system FΩr (s) Figure 2.5. Reference tracking closed loop LPNI system with nonlinear actuator and its quasilinearization. Although the accuracy of this approximation is discussed in Section 2.3, we note here that the first of the above obstacles to optimality is alleviated by the fact that, if the plant is low-pass filtering, the signal u(t) is close to Gaussian, even if v(t) is not. The second obstacle, however, is unavoidable. Nevertheless, as shown in Section 2.3 and numerous previous studies, stochastic linearization of closed loop systems results in accuracy well within 10%, as far as the difference between the standard deviations of the outputs, σy and σŷ, is concerned. Since the standard deviation of û is σû = Fr (s)C(s) 1 + P(s)NaC (s) 2 , it follows from (2.12) that the quasilinear gain of Figure 2.5(b) is defined by Na = F Fr (s)C(s) 1 + P(s)NaC (s) 2 , (2.48) where F (σ) = ∞ −∞ d dx f (x) 1 √ 2πσ exp − x2 2σ2 dx. (2.49) Thus, Na is a root of the equation x − F Fr (s)C(s) 1 + xP(s)C (s) 2 = 0, (2.50) which is referred to as the reference tracking quasilinear gain equation. This equation is used in Chapters 3 and 5 for analysis and design of LPNI systems from the point of view of reference tracking.
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- 59. The Project Gutenberg eBook of Fifth Report of the Vestry of the Parish of Chelsea in the County of Middlesex, Appointed Under the Metropolis Local Management Act, 1855: 1860-1
- 60. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: Fifth Report of the Vestry of the Parish of Chelsea in the County of Middlesex, Appointed Under the Metropolis Local Management Act, 1855: 1860-1 Author: Charles Lahee Release date: August 26, 2013 [eBook #43559] Language: English Credits: Transcribed from the 1861 edition by David Price. Many thanks to the Royal Borough of Kensington and Chelsea Library Service for allowing their copy to be used for this transcription *** START OF THE PROJECT GUTENBERG EBOOK FIFTH REPORT OF THE VESTRY OF THE PARISH OF CHELSEA IN THE COUNTY OF MIDDLESEX, APPOINTED UNDER THE METROPOLIS LOCAL MANAGEMENT ACT, 1855: 1860-1 ***
- 61. Transcribed from the 1861 edition by David Price, email ccx074@pglaf.org. Many thanks to the Royal Borough of Kensington and Chelsea Library Service for allowing their copy to be used for this transcription. FIFTH REPORT OF THE Vestry of the Parish of Chelsea, IN THE COUNTY OF MIDDLESEX, APPOINTED UNDER THE METROPOLIS LOCAL MANAGEMENT ACT, 1855. 1860–1. Ordered to be printed September 24th, 1861. LONDON: C. F. BELL, MACHINE PRINTERS, (By Appointment to the Vestry of Chelsea), 133, KING’S ROAD. 1861.
- 62. TABLE OF CONTENTS. Page Fifth Report of the Vestry of the Parish of Chelsea. 2–26 NO. APPENDIX. 1. List of Vestrymen, Auditors, and Officers, with Plans and Descriptions of the Wards 28–36 2. Return of Members’ Attendances . . . facing page 36 3. List of Parish Officers elected at Easter, 1860, and of the Overseers previously nominated for selection 37 4. List of Committees and Sub-Committees 38–41 5. Names and Places of Abode of the Clergy and other Parochial Officers 42–46 6. Salaries, c.—Return asked for by the Committee of Works and for General Purposes 47 7. General Works and Revenue 48–49 8. Surveyor’s Return of General Works, and of Works remain in progress 50 9. Chelsea Bridge Road 51–55 10.Lighting—Petition and Memorial 56 11.Dust, Ashes, c.—Regulations for their Removal 57–59 12.Return of Proceedings taken for the Removal of Nuisances, and for the Sanitary Improvement of the Parish 60 13.List of Slaughter Houses in the Parish Licensed October, 1860 61 14. Sewers Works and Revenue 62 15.Surveyor’s Return of Sewerage Works 63 16.The late Hans Town Commission 64 17. The late Chelsea Improvement Commission 64 18.Vestry Hall 65–76 19.Metropolis Local Management Act—Suggestions for its Amendment 77–103 20.Chelsea Charities 104– 106 21.Cremorne Gardens 107 22.“The Lots” 108– 109 23.Chelsea Hospital Grounds 110– 111 24. Thames Embankment—Petition in favour of the “London Coal and Wine Dues Continuance 112
- 63. Bill” 25.Horse Ride in Kensington Gardens—Petition to the Queen 112 26.Local Magistracy—Correspondence with the Lord Lieutenant of the County of Middlesex 113 27. Parochial Assessments Bill—Petition against 114 28.Houses and Population in 1851 and 1861 compared; with some other Statistics 115 29.Enfranchisement of the Parish of Chelsea 116, 117 30.Water Companies’ Accounts 118, 119 31.An Account in Abstract of Receipt and Expenditure 120– 123 32.Establishment Charges 124 33.Reports of the Auditors, 1860 and 1861 125, 126 34. Interest Account 127 35.A Statement of all Arrears of Rates, c. 128 36.A Statement of all Moneys excepting Rates, c. 129 37. A Statement of all Mortgages, c. 130 38.A Statement of all other Debts and Liabilities 131, 132 39.Balance Sheet 133 40.A Statement of the Amount of all Contracts, c. 134, 136 41.Bye Laws 137– 140
- 64. FIFTH REPORT OF THE Vestry of the Parish of Chelsea, For the Year ended March 25th, 1861. MEMBERS OF THE BOARD, AND ESTABLISHMENT ARRANGEMENTS. The result of the ward elections on the 29th of May, 1860, was to effect the following changes in the constitution of the board:— Retiring Members. Mr. BOWERS Mr. LEETE Mr. THIRST ,, BURTON ,, OSBORN ,, TILL ,, BUTT ,, PITT ,, WAIN ,, CARTER ,, PORTER ,, WATKINS ,, COMPTON ,, RABBITS WHITEHEAD ,, CURRY ,, E. RICHARDS ,, R. WOOD, and ,, GABLE ,, G. W. RICHARDS ,, YAPP. ,, JACKSON Members Re-Elected. Mr. BUTT Mr. LEETE Mr. THIRST ,, COMPTON ,, OSBORN ,, TILL, and ,, GABLE ,, RABBITS WAIN. New Members. Mr. BADCOCK Mr. GURNEY Mr. PARKER ,, BLAZDELL ,, HULSE ,, ROOPE ,, DELANY ,, LAWRENCE ,, WALKER, and ,, DOUBELL ,, OXFORD ,, L. E. WOOD. ,, FOY Descriptions, with plans, of the several wards, with the names and addresses of the members of the vestry representing them, of the auditors of the accounts, and of the officers, as required by the Act to be given in each report, will be found in the Appendix No. 1 (pages 28 to 36). At Easter, 1860, the usual election of churchwardens, sidesmen, church trustees, and auditors of the church trustees’ accounts took place; their names and addresses, together with those of the persons previously nominated as fit to serve the office of overseers of the poor, and submitted for the choice of the magistrates in special sessions, will be found in the Appendix No. 3, (page 37).
- 65. The vestry on Easter Tuesday also elected John Moore Segar, son of John Moore Segar deceased, for admission into the parochial charity school, pursuant to the will of the late John Chamberlayne, Esquire. The return usually ordered by the board previous to the annual election, showing the attendances of each member, is reprinted in the Appendix No. 2 (and faces page 36); but as the period embraced in that return does not entirely correspond with the parochial year, the tabular arrangement commenced in the last report is continued:— From March 25th to March 25th. Vestry Meetings. Committee Meetings. Evening Committees. Average Attendances At Vestries. 1856–7 50 52 nil 23 1857–8 53 59 31 23 1858–9 45 85 37 27 1859–60 45 92 32 25 1860–1 44 109 74 28 A list of the members who have served upon the various committees, and sub-committees, during the year, is given in the Appendix No. 4 (pages 38 to 41); and the list annually issued by the vestry clerk, containing the names and places of abode of the parochial clergy and lay officers, immediately follows it. Appendix No. 5 (pages 42 to 46). In November, 1860, in consequence of the numerous additional matters connected with the new building, into which the vestry had now removed, requiring the hall-keeper’s attention, the clerk reported that the messenger’s duties were not satisfactorily performed, and the subject being referred to one of the standing committees for consideration, a youth was appointed upon their recommendation at ten shillings per week to commence with. During this year the vestry have contemplated some alterations in the salaries of the officers, and upon the application of Mr. J. E. Salway, one of the office clerks, for increased remuneration, in December, 1860, referred the whole question of their duties and salaries to the committee of works and for general purposes for consideration. That committee for the purpose of comparison, and in order to have some data before them upon which to form a satisfactory opinion, requested Mr. Tite, M.P., to move the House of Commons for a return from each vestry and district board in the metropolis, of its population; number of houses; extent of its area; aggregate length of its streets; and the value of property, as assessed to live county rate, and the poor rate. The committee wished this return also to show the number of meetings held during the year, whether of the full board or of committees, or sub- committees; as well as the names and salaries of the officers, with particulars of any other benefits enjoyed by them: and that this information might be tabularly arranged in the order of the amount of population of each place. See Appendix No. 6 (page 47). Mr. Tite however replied that Sir John Shelley had moved for such a return, referring to one similar to that printed at page 69 in the second report, but as the return moved for by Sir John Shelley differed from that suggested by the committee, in several important particulars, the vestry requested Mr. Tite to obtain the introduction into it of the additional information. The act for regulating the supply of gas to the metropolis rendered it necessary that the vestry should consider the appointment of a Gas Examiner; and having themselves determined to put in operation in this parish the act for preventing the adulteration of articles of food and drink by appointing an Analyst; a committee was appointed in March, 1861, to confer with Dr. Barclay as to his acceptance of the latter office, and the terms upon which he would undertake its duties, having regard to his diminished duties as medical officer of health. This Committee was amalgamated with one appointed subsequently, and specially, to consider the duties and salary of the medical officer of health, to whom was referred also, the question of the appointment of the gas examiner.
- 66. In consequence of the heavy amount of the solicitors’ (Messrs. Lee and Pemberton) bill for the year 1859–60, which included the greater part of the law charges in the actions brought against the vestry by the London Gas Light Company and Mr. Hornsby, the contractor, those gentlemen suggested in the following letter to the vestry clerk that they should be taxed:— 44, Lincoln’s Inn Fields, W.C., London, 18th May, 1860. Dear Sir, We have given the subject of our Mr. Pemberton’s conversation with you respecting our bill of costs, mature consideration, and feeling the delicate position in which the vestry are placed as trustees accountable to their constituents, we would suggest that for the future all our bills be referred to Mr. Richard Dax, of the Temple, barrister at law, for taxation, and to certify the amount properly payable to us. This course is adopted by the Great Western and North Western railway companies, the Oxford, Worcester and Wolverhampton railway company, the Shrewsbury railway company, the Metropolitan railway company now in course of formation, all of whom refer their solicitors’ bills to Mr. Dax for the satisfaction of the shareholders, and we believe it to be the practice with nearly all railway and other large companies; should the vestry resolve to do so, we shall feel great pleasure in submitting our bills to Mr. Dax, and being bound by his certificate, as it is extremely distasteful to us that any doubt should exist in the minds of the vestry as to the propriety or amount of any item charged by us, which however there almost necessarily must be, from the technical manner in which solicitors are obliged to make out their bills. We ought to add, that we have no sort of personal acquaintance with Mr. Dax, who is the son of the late Master of the Exchequer, and a well known author and authority on the subject of solicitors’ costs. Should the vestry adopt our suggestion, we must ask to be allowed to make out the bills now before them, afresh, as they are not framed for taxation, and that in future, our bills should be delivered half- yearly and paid within a short time after they have been certified, and we remain, dear Sir, your’s faithfully, Lee Pemberton. The vestry, having accepted the proposal, returned the bills to the solicitors as desired, and requested Mr. Finch, a solicitor and a member of the vestry, to attend the taxation. This, after a strong protest from the solicitors, he did; and the accounts amounting to £914. 3s. 5d., as originally delivered, were settled by Mr. Dax at £998. 8s. 7d., including the costs of taxation. The law costs for the present year have been submitted to the same process, and have been certified by Mr. Dax to be correct at the amount at which they were sent in,—viz., £177. 1s. 6d. Mr. Miles having been elected to preside at eighty-three meetings, out of the one hundred and fifty- three held during the year, his services as chairman were again acknowledged by a vote of thanks on the twenty-first of May. The thanks of the vestry have been also given during the year to Mr. Perry on the eighth of May, for his exertions as delegate at the metropolitan and county association for the equalization of the poor rates; and to Mr. Foy on the fourth of December, for the tasteful way in which the decorations of the vestry hall, on the occasion of the inauguration dinner were carried out by him. GENERAL WORKS. Appendix No. 7 (pages 48–49), represents the cost, during the past year under the several sub-heads of paving, lighting, watering, cleansing, c., and the provision made for meeting the same, by orders upon the board of guardians. The surveyor’s return upon this subject forms Appendix No. 8 (page 50). PAVING.
- 67. The cost of the item of paving during the past year, exceeding the estimate by four hundred and fifty pounds, has been unusually heavy,—viz., £3042 2s. 10d., after allowing for the sum chargeable to the several public companies and others for works executed for them. This is about twelve hundred and fifty pounds more than the cost of the same item of expenditure in the year 1856–7, after making the same allowances; upwards of four hundred pounds more than in 1857–8, eight hundred pounds more than in 1858–9, and six hundred pounds more than last year. It has been occasioned mainly by the purchase of the following materials, and by works executed in the places hereinafter named:— Broken Granite for Roads £1558 7 10 Flints for ditto 238 3 1 Gravel for ditto 137 0 6 Lombard and Duke Street . . . Works executed by Contractors for Masons’ Work 177 15 9 Hans Street . . . ditto 29 9 0 Queen’s Road East . . . ditto 30 1 6 Lower Sloane Street . . . ditto 13 9 5 Green’s Row . . . ditto 170 10 1 Ann’s Place, Milman’s Row . . . ditto 27 13 9 George Street . . . ditto 260 18 5 Moore Street . . . ditto 19 1 3 King Street . . . ditto 10 10 10 Milman’s Row . . . ditto 10 0 0 King’s Road—various parts . . . ditto 49 1 2 Halsey Street . . . ditto 13 15 0 Walton Street . . . ditto 17 7 9 Sloane Street . . . ditto 17 10 6 Queen’s Road West . . . ditto 24 8 7 Caversham Street . . . ditto 57 4 1 Robert Street . . . ditto 34 3 3 Stone sent to the Depôt . . . ditto 36 5 9 In the last report it was stated that the question at issue between the Chief Commissioner of Her Majesty’s Works, c., and the vestry, as to the taking charge and maintenance of the Chelsea Bridge Road, was in an unsatisfactory state, and the position of that question was stated in the appendix to the fourth report (No. 9), at pages 45 to 52, up to the receipt of Mr. Austin’s letter of the 27th June, 1860. The proceedings since that date will be found in the Appendix to this report, No. 9 (pages 51 to 55). LIGHTINGS The Metropolis Gas Act received the royal assent on the 28th August, 1860: it contains fifty-seven clauses, many of them most important for the protection of public and private consumers. By the twenty-seventh section of the act (23 24 Vic., cap. 125) the vestry are required to provide apparatus for testing the illuminating power and purity of the gas, and to appoint a competent person as examiner; and by the same section the gas companies are required to erect, at a distance not less than one thousand yards from their works, experimental meters with the necessary apparatus for testing the
- 68. illuminating power of the gas supplied. In alleged compliance with this requirement the London Gas light company have appointed a testing station at the house No. 73, Besborough Street, Pimlico. Mr. Hughes [8] observes with reference to this twenty-seventh section,— “There was some discussion as to the propriety of making the requirements of this clause compulsory instead of merely permissive. After the fullest consideration however, it was thought essential, with the view of avoiding discussions in vestries, and especially with the view of counteracting the underhand and secret influence which the gas companies exert in many vestries and district boards, to make the clause compulsory. There are many instances on record, where powers are given to local authorities, and yet these powers, although highly important to the public interests, have never hitherto been exercised. Hence an additional reason for positively requiring them to provide apparatus, c., for testing the gas, and to appoint and pay an inspector for the purpose. It is true that no time is fixed within which the apparatus is to be provided and the inspector appointed, but I apprehend it will be competent for any ratepayer to compel, by mandamus, the performance of this duty by any vestry or district board, within a reasonable time. “The second section of the clause imposes an obligation on the gas companies—namely, that each of them shall within six months erect at the prescribed distance from their works, an experimental meter and other apparatus for testing the illuminating power of the gas. “Now these two obligations, the one on the local authorities of the metropolis, and the other on the gas companies, must not be confounded, because they are perfectly distinct, and the one is not to be a substitute for the other. In the first place each local authority in the metropolis—i.e., each vestry and district board, about thirty-eight in number, constituted under the Metropolis Local Management Act, is to erect its own apparatus, and appoint its own inspector; and from future clauses it appears that the act contemplates a continuous and regular succession of testing by this inspector, both for purity and illuminating power. The evidence of this inspector however as to any defects in the gas, will not be conclusive until his report has been confirmed by testing the gas at the prescribed distance of 1000 yards from the works. Hence the necessity for the obligation on the company. The next clause will better explain the mode of proceeding by the inspector, whenever he finds the gas to be below the prescribed standard of illuminating power. With reference to purity there is nothing about testing for this at the distance of 1000 yards, and therefore the test for this may be made wherever the inspector pleases. “With respect to the number of separate places for testing the gas, inasmuch as there are thirteen companies included within the act, and each must provide a testing house 1000 yards distant from their works, there must evidently be not less than thirteen of these. [9] But if each local authority also erects a separate one at some central part of its district, thirty-eight of these will be necessary. A power is afterwards given for two or more local authorities to combine, and then the number will probably be somewhat diminished. “All this necessity for a duplicate set of testing establishments is rendered essential by the absurd requirement of the act as to testing the gas at 1000 yards distance from the works. Now as the erection of this apparatus by the companies and the establishment of a permanent testing place away from their works will be very expensive to the companies, it is just possible they may be very glad to be relieved from this expense, and may consent to the testing at the establishment of the local authority being sufficient evidence of the illuminating power as well as the purity of the gas. This is the more probable as it must be perfectly well known to the companies—at least to their engineers—that so far as the company is concerned the gas may just as well be tested in the centre of any district supplied, as at the limited distance of 1000 yards from the works.” Under the provisions of the fiftieth section of this act, the duty is imposed upon the Metropolitan Board of Works of raising the costs and charges incident to its passing; and that board have accordingly levied
- 69. under their precepts upon the several vestries and district boards, the necessary amounts for meeting the claims received. The accounts sent in by the various parties were submitted to the proper officer of the House of Commons for taxation with the following result:— Accounts sent in. Amount Certified by Taxing Officer. Costs Allowed. Board to Pay. £ s. d. £ s. d. £ s. d. £ s. d. Mr. Dangerfield 89 19 6 62 6 0 3 11 8 65 17 8 Mr. Wyatt 3666 0 4 3033 18 10 None. 3033 18 10 Mr. Beal . . . £2685 13 6 Less paid by Mr. Wyatt . . . 1413 0 0 1272 13 6 676 11 7 None. 676 11 7 5022 13 4 House Fees on Taxation to be paid by Dyson Co., (as Parliamentary Agents) 42 0 0 3818 8 1 The proportion of the above with which this parish is debited by the precept of 1861 is £106 16s. 5d., but that proportion is based upon the amount of the accounts as sent in; as reduced by taxation it will only be £81 12s. 8d. But as the vestry has already contributed two hundred pounds to Mr. Beal in answer to his various appeals, and expended upwards of twenty pounds in making experiments as suggested by him, they will have a large sum to receive back In November, 1860, a proposal was made by the vestry of St. James’, Westminster, to recognize the services of Messrs. Beal and Hughes throughout the enquiry into the supply of gas to the metropolis, and the passing of the recent measure by raising some fitting testimonial, and a committee of that vestry having been appointed to carry it out, Messrs. Hall and Perry were deputed to confer with them as to the best mode of doing so. The Bill to amend the Metropolis Gas Act, [11] introduced by the government into parliament for transferring to the metropolitan board of works, so far as regards the metropolis, the powers conferred by the acts for regulating measures used in the sale of gas (22 23 Vic., cap. 66, and 23 24 Vic., cap. 146) upon the justices of the peace, of appointing inspectors of meters, having received the royal assent (24 25 Vict., cap. 79), it will be the duty of that board to take the necessary steps for carrying it into execution. To aid in accomplishing this transfer, the vestry on the tenth July, 1860, expressed their opinion that the appointment of inspectors of meters in the metropolis would be best placed in the hands of the metropolitan board of works, and petitioned parliament and memorialised the justices of the peace upon the subject. See Appendix No. 10 (page 56). The number of public lamps in the parish on the 25th March, 1861, was as follows— Lighted by the London Gas Company 696 ,, Western ditto at Kensal Town 30 726 WATERING AND SCAVENGERS. The following tabular statement shows the cost of these two services for the last five years:—
- 70. From March to March. Watering. Cleansing. Total. 1856–7 £1109 5 10 £1977 19 4 £3087 5 2 1857–8 709 13 8 1563 9 3 2273 2 11 1858–9 941 4 0 114315 0 2084 19 0 1859–60 1192 1 2 121818 5 2410 19 7 1860–1 1126 0 7 114512 6 2271 13 1 The contractors were fined £40 10s. for neglect during the year. IMPROVING. The principal charges under this head are for works over the Ranelagh sewer at Sloane Square in rebuilding the south wall; and in fencing the enclosure opposite Royal Avenue Terrace. REMOVING NUISANCES. The removal of the dust, ashes and refuse from the houses of the inhabitants, and the public courts and alleys during this year has cost £311 2s. An alteration in the form of the contract as to the mode of executing the works by which the parish is divided into daily districts, was made during this year, which has very greatly lessened the public inconvenience, and consequently the number of complaints. See Appendix No. 11 (page 57). SANITARY MEASURES. The return of the proceedings taken by the vestry under the metropolis local management act and the nuisances removal act, as required by the former statute to be published, will be found in the Appendix No. 12 (page 60). Forty-seven slaughter houses in the parish were licensed by the magistrates in October, 1860, not one having been opposed by the vestry. A list of them is given in the Appendix No. 13 (page 61). The tenant of No. 3, Queen’s Road East, having been detected in slaughtering upon his premises which were not licensed for that purpose, the attention of the board was called to the subject, but as it appeared that he had been acting under the impression that the premises were licensed, in place of other proceedings, a letter of warning was written to him, and the offence was discontinued. The notices of the vestry having been neglected by Mr. Peter Augustus Halahan, owner of Nos. 1 to 10, Wickham Place, application was made under the nuisances removal act to the magistrate and his orders obtained for the execution of the necessary works; these orders being also neglected and nothing being done at the expiration of the time allowed, the clerk was instructed to apply for the enforcement of the penalties provided by the twenty-seventh section. The defendant was fined thirty-six shillings and costs, and in default of payment was committed to prison. A manuscript copy of all the reports made to the vestry by the medical officer of health during the year is sent to the metropolitan board of works with a copy of this report. GENERAL RATES. The orders issued by the vestry, and the rates made by the overseers (the board of guardians) thereunder during the year, have been as follows:—
- 71. Date and Amount of Order issued by the Vestry. Date and Amount of Rate issued by the Guardians. Rateable Value of Property charged. Gross Amount of Rate. 1860, Mar. 13, £4800 Mar. 28, 6d. in £ £213,443 10 0 £5336 1 9 ,, Oct. 23, 5600 Oct. 31, 7d. in £ 219,477 10 0 6401 8 6½ SEWERAGE WORKS. Again no general sewers rate for local purposes having been made during the year, there is a deficit upon this account, of one hundred and fifty-five pounds nineteen shillings and one penny. In the Appendix No. 14 (page 62), and No. 15 (page 63), will be found statements giving particulars of a similar kind to those given under the head of General Works. The orders issued by the vestry for special sewers rates during the year, and the rates made by the board of guardians in consequence, have been as follows:— Date and Amount of Order issued by the Vestry. Date and Amount of Rate made by the Guardians. Rateable Value of Property charged. Gross Amount of Rate. Sept. 25 £24 11 10 Nov. 21 4d. in £ £1772 0 0 £29 10 8 ,, 4 4 6 ,, 9d. in £ 120 0 0 4 10 0 ,, 4 19 2 ,, 7d. in £ 168 10 0 4 18 3½ ,, 4 16 6 ,, 3½d. in £ 412 0 0 6 0 2 ,, 1 13 9 ,, 2d. in £ 325 0 0 2 14 2 APPLICATIONS. During the period between the 26th March, 1860, and the 25th March, 1861, the vestry have adjudicated upon forty-six applications for directions upon the subject of house drainage, twenty-three of which had reference to the drainage of twenty-five houses already built, and twenty-three to forty- three intended houses and the new barracks at Pimlico; they have also brought under the cognisance of the Commissioners of Police, the dangerous condition of forty-five buildings, and have decided upon seventeen miscellaneous applications. Of the applications so made, fifty-nine have been granted, and four negatived. Eight buildings have been commenced without previous notice to the vestry, and their builders having been summoned before the board for neglect, the explanations given by them, or their apologies, have in most cases been considered satisfactory. Fourteen proposals have been made to build before the general frontage line, and one hundred and three obstructions and offences upon the public highway have been dealt with by the Board. The late HANS TOWN COMMISSION, and the late CHELSEA IMPROVEMENT COMMISSION. Nos. 16 and 17 in the Appendix (page 64), explain the alteration which these debts have undergone during the year. No special rate has been necessary upon the Hans Town district, during the past twelvemonth, but the order and rate made for the reduction of the debt upon the remainder of the parish has been as follows:—
- 72. Date and Amount of Order issued by Vestry. Date and Amount of Rate made by Guardians. Rateable Value of Property charged. Gross Amount of Rate. 1860, Mar. 13 £1150 Mar. 28 2d. in £ £163,097 0 0 £1359 2 10 ,, Oct. 23 920 Oct. 31 1½d. in £ 167,515 10 0 1046 19 5¼ VESTRY HALL. The first meeting of the vestry in the new building took place on the ninth of October, 1860, and shortly afterwards it was determined to celebrate the public opening of the large hall with an inauguration dinner. A committee being formed for the purpose the necessary arrangements were made, and on the thirtieth of November one hundred and twenty of the most influential parishioners, presided over by Viscount Chelsea, with the county members as their guests, commemorated the completion of a building, which it may be permitted the vestry to hope, will be of service to the ratepayers and the parish. The report and balance sheet of the dinner committee is printed in the Appendix No. 18 (pages 65–66). After the dinner the first public use of the hall was given gratuitously on three days to the Chelsea, Brompton, and Belgrave Dispensary, and by means of an oratorio, “The Creation,” on one day; a lecture by the Rev. J. B. Owen, of St. Jude’s Church, on another; and a concert on a third day, that useful local charity realised nearly one hundred pounds. Having obtained a license for public music and dancing, the applications for the hire of the hall were soon found to be so numerous that a committee was appointed to regulate the letting, and they drew up the scale of charges which was approved by the vestry, and will be found at page 67. A debtor and creditor account in connexion with the letting of the hall will be also found at pages 68–69. One of the most important applications referred to the letting committee was from the proposed Literary and Scientific Institution, that the suite of rooms on the east side of the building might be set apart for their use. The letting committee made a special report upon the subject, to the vestry on the eighteenth December, and a deputation from the council of the institution, consisting of the Rev. F. Blunt, Messrs. McCullagh, Lawrance and Mead, with Mr. Bull, the honorary secretary, had an interview with the vestry. The report which is printed at page 70, read in connection with the following resolution which was passed by the vestry, will explain the nature of the engagement subsisting between the two bodies:—“Resolved, that the suite of rooms upon the ground floor (east side) including the lighting and warming of the same, and the use of the Hall for forty nights during the year, including the lighting, warming, and cleaning of the same, be let to the council of the Institution for £50 per annum, payable half-yearly, from Christmas next. The tenancy to be subject to six months’ notice of determination by either party at any time.” A debtor and creditor account for one winter quarter will be found at page 74. Upon the application of the vestry, Sir R. Mayne, the chief commissioner of police, appointed a standing for ten cabs at the King’s Road end of Robert Street, exactly opposite the hall; but unfortunately the post-office authorities were unable to comply with their request that a pillar letter box might be placed near to the hall. The vestry hall buildings are insured in the Union Assurance Office for six thousand pounds; and the fixtures, fittings, and furniture, in the Manchester Fire Assurance Office for seven hundred pounds. Ten thousand pounds, the sum originally borrowed, not having been sufficient to complete the building together with the fitting and furnishing, an application was made to the metropolitan board of works for their sanction to a further loan of two thousand pounds; and as it was necessary to accompany that application with a statement of the entire estimated cost, and to fortify it with a memorial from the vestry, those documents, as they contain much matter of interest, are printed at pages 75 and 72–73. COUNTER’S CREEK DISTRICT.
- 73. Since the last report the sum assessed by the metropolitan board of works upon this portion of the parish has been, for the year 1861, four hundred and seventy-nine pounds, eleven shillings and eleven pence; but the vestry have abstained from issuing their order to the board of guardians for its collection. They have, however, in consequence of the receipt of the following letter, upon the recommendation of the Finance Committee, paid the amount for 1858:— Metropolitan Board of Works, 1, Greek Street, Soho, W., 17th October, 1860, Dear Sir,—I am directed by the metropolitan board of works to acquaint you that they have directed their attention to the subject of the amounts outstanding on their precepts issued to the vestries and district boards of the metropolis, and they desire me to apprize you for the information of the vestry of Chelsea, that having been unsuccessful in their attempt to carry a measure through Parliament in the past session for conferring upon the board the necessary powers to deal with the question of the redistribution of the Counter’s Creek charges, they have had under consideration the subject of the payment of the amount due from the vestry on the precept for the ordinary expenses of the board for the year 1858, being the sum of £447 4s. 2d. payment of which was, as you will recollect, allowed to stand over until the re-apportionments of the debts of the Commissioners of Sewers had been discussed in Parliament. The board now feel that the time has arrived when it is necessary that they should adopt measures for obtaining payments of the amounts outstanding, and after an attentive consideration of the facts of the case, and having regard to the arrangement with the vestry in reference to the parish contributing to the cost of the Counter’s Creek diversion sewer, the conclusion they have arrived at as to the most convenient course to be adopted is, that the board should proceed to appoint persons to make a rate on the parish in default, for the amount due on the precept, and in the event of the vestry continuing their objection to the payment, that some ratepayer should then raise the question of the liability of the parish by an appeal against the rate, on which a special case might be reserved for the opinion of the Court of Queen’s Bench. I am directed to add that the Board have deferred further proceedings in the matter for a fortnight from the date of their last meeting, the 12th instant, in order to afford the vestry an opportunity of paying the amount in question should they think proper so to do; and meanwhile I am to request that you will be good enough to lay this letter before the vestry, and to favor me with a reply with the least possible delay. I am, dear sir, yours faithfully, John Pollard, Clerk of the Board. The amount assessed for 1859, four hundred and fifty-six pounds, nine shillings and one penny, was paid by order of the board on the sixth December, 1859, before any portion of it was received from the board of guardians. January, 1861, the metropolitan board were very pressing for payment of the amount for 1860 (five hundred and eighteen pounds, nineteen shillings, and one penny), threatening immediate proceedings if it were not made before the 25th of that month; the vestry, however, appealed through Mr Tite against this undue pressure, contrasting with it, the treatment received by the vestry of Kensington, who were in arrear many, if not all, of their instalments, and eventually the metropolitan board consented to receive the amount of the precept for their ordinary expenses of 1860 less this sum which was included in it. Thus it is that the sums assessed for the first four years have been collected in the district, and paid to the metropolitan board; but for the two last, 1860 and 1861, they have not been paid to the metropolitan board, nor have any steps been taken to collect the money in the district. METROPOLITAN BOARD OF WORKS. The estimate upon which the sums were originally proposed to be raised by the metropolitan board of works for the services of the year ending the thirty-first of December, 1860, gives the following particulars as applicable to this parish:—
- 74. £ s. d. Sewerage and Drainage 116817 1 Metropolis Improvements 411 5 2 General Purposes, viz.:— Salaries, Wages, c. 271 16 8 Printing, Stationery, c. 45 14 6 Rent, Taxes, c. 23 9 0 Repayment of Loans, c., special:— Principal 283 12 5 Interest 272 17 5 Ditto ditto, general:— Clergy Mutual, £140,000 255 15 2 Bank of England, £400,000 351 14 10 Clergy Mutual, £42,000 31 16 1 Contingencies 40 0 11 £3156 19 3 This estimate, however, having been printed and sent to the various vestries and district boards, underwent considerable discussion and some modifications, the principal of which latter, were, that instead of including the whole cost of the new buildings in Spring Gardens in the precepts for 1860, it was determined to spread it over three years; an item of £20,000 for deodorization was reduced to £10,000; the amount for special charges was reduced by £5,000; and the sum of £2,000 was substituted for £5,000 for minor improvements. These alterations, with a credit of £364 10s. 10d. (the remaining portion of the £3053 16s. 3d. adverted to in former reports), had the effect of diminishing the amount required from this parish to £2239 10s. 9d., and it was thus apportioned by the precept dated 11th May, 1860:— £ s. d. The Whole Parish 1004 13 3 The Counters Creek District 518 19 1 The Ranelagh District 715 18 5 £2239 10 9 In September, the third precept for the Main Drainage rate was received, amounting to £2928 2s.; and in the following February the ordinary precept for 1861, £2580 1s. 4d.; therefore within the period comprised in this report this parish has been required to provide for metropolitan purposes the large sum of £7,747 14s. 1d. The orders made by the vestry in pursuance of these precepts, and the rates made by the board of guardians thereunder, are shewn in the following table:— Date and amount of Order of Vestry. Date and amount of Rate made by Guardians. Rateable value of property charged. Gross amount of Rate. WHOLE PARISH. 1860, July 2 £1004 13 3 Oct. 3rd ¼d. in the £ £219,477 10 0 £228 8 9⅜
- 75. 1861, Apr. 9 £1499 12 6 May 1st. 2d. in the £ £220,927 0 0 £1839 3 8 RANELAGH DISTRICT. 1860, July 2 £715 18 5 Oct. 3rd 1d in the £ £184,992 0 0 £770 9 1½ 1861, Apr. 9 £600 16 11 May 1st 1d. in the £ £186,421 0 0 £776 3 10 MAIN DRAINAGE.—WHOLE PARISH. 1860, Oct. 23 £2300 0 0 Oct. 31 3d. in the £ £219,477 10 0 £2741 5 3¾ The metropolitan board of works early in February of the present year determined to re-introduce into parliament the bill for amending the Metropolis Local Management Act, containing the same provisions as were in the bill of last session, with the exception of the introduction of certain clauses for the re- apportionment of the Rock Loan; but in consequence of the strong opposition to those clauses, the board deemed it expedient to withdraw them from the bill, and to embody them in a separate bill, and the two bills were accordingly introduced into the House of Commons. The first of them, after great delays, passed through the House of Commons on the nineteenth of July, 1861, and was read a first time in the House of Lords on the twenty-second of that month, but in consequence of the advanced period of the session, it was found impossible to procure its passage through the required stages in order to its becoming law this year. The bill No. 2, having relation to the Rock Loan, being strongly opposed by certain members of the House of Commons, was withdrawn. With respect to the abortive bill of 1860, the solicitors of the metropolitan board reported: “We think it will be nearly correct to estimate the expense to the board, exclusive of printing, c. by the printers of the board, and expenses of that kind, at about £1650.” It is to be hoped that the failure of 1861 will be somewhat less costly. The vestry of Chelsea have at various times responded to the invitation of the metropolitan board for suggestions during the preparation of the several bills for amending the Metropolis Local Management Act; for convenience of reference, their labours in this respect will be found in the Appendix No. 19, (pages 77 to 103). CHELSEA CHARITIES. On the twenty-second of May, 1860, upon the motion of Mr. Finch, a committee, consisting of the rector (the Rev. A. G. W. Blunt), the churchwardens (Dr. Diplock and Mr. Collier), three past churchwardens (Messrs. Hall, Perry, and Leete), and Messrs. Whitehead, Jones, Miles, Brown, Rabbits, E. O. Symons, Till, and Finch, was appointed to enquire into the charities of the parish, the present particulars of the several properties, and the application of the respective annual proceeds thereof, and to report thereon. At the same time, the vestry clerk reported that, at the request of the rector, he was making arrangements preliminary to the appointment of new trustees, and laid before the vestry the following letter from Mr. Druce: Mr. Lahee, Dear Sir, Doubtless you are aware that the information contained in last Saturday’s “Chelsea Times” [20] was provided by me, as far as concerned the Chelsea charities. The article of this morning [20] would lead me to suppose that it was taken to be the opinion of the writer of such information that under present circumstances it would be to the advantage of the parish that the vestry clerk should be a solicitor; nothing can be more foreign to my opinion, and without compliment, I think the office is now in very good hands. In the wicked old tory times on Easter Tuesday, a man might blow out the steam of discontent, and tell a few truths profitable to the parish to hear; now a few highly honourable and sensible parishioners ‘make things pleasant’ in
- 76. a comfortable room in a house in the King’s Road. The money of the parish charities is not properly looked after; for many months I have been trying to get the late rector to put matters to rights, but without avail. Being therefore left sole trustee for some portion, and knowing more than most, about the parish school and trust funds, I thought I was the proper person to light the match. Should the vestry appoint the committee on charity affairs, I shall be happy to attend their summons to give them any information in my power: and I think they ought to have a committee, both on this subject and that of the church trustees, and at least chronicle the exact state of facts; for, seriously speaking, we owe ‘liberal opinions’ no small grudge for shutting up Easter Tuesday. I do not recollect that in your charity report [21] you mentioned the large sums belonging to the charity schools, if you did, the vestry ought to have seen that the trustees were dying out. From my experience of the working of the parish charities, schools, c., I am quite convinced that the vestry should annually qualify themselves to issue a report on all charity money. I have never found anything to make me think otherwise than favourably of the honesty of all dealings with the charity money, but I have found abundant reason to rest satisfied, that without some lay assistance and watchfulness, all matters get into disorder, especially where there are any accounts to keep. This letter is at your service, and may be used as you think fit. I am, dear sir, always yours very faithfully, Wm. Druce. Swan Wharf, Chelsea, 14th April, 1860. The committee, favoured occasionally with the assistance of Mr. Druce, and strengthened by the addition to it of Messrs. Butt, Tipper, and Callow, have met upon numerous occasions, and are carefully pursuing their enquiry. CREMORNE GARDENS. It will be recollected that towards the close of the season of 1858, the vestry thought it necessary to remind Mr. Simpson, the proprietor of Cremorne Gardens, of the arrangement made with them as to the conduct of his gardens, particularly as to the hour of closing; and that Mr. Simpson’s reply, although dated the 16th of August, was not received in sufficient time to be considered until the meeting on the fifth of October, when it was thought to be unsatisfactory. Last year the vestry again moved in the matter, by reminding him of these promises, and requesting to know before Monday, the tenth of September, whether it was his intention to comply with their wishes and those of the parish, by closing the gardens at twelve o’clock at night. No reply having been received from Mr. Simpson at the meeting of the board on the eleventh of September, it was moved by Mr. Delany that the report of the second of November, 1857, which the vestry had refrained from making public in consequence of the pledge on the part of the proprietor, that the gardens should be closed as near to midnight as possible, and that in other respects he would meet the wishes of the vestry, and the comfort of the inhabitants, should be printed for the use of the vestry. The chairman, however, having ruled that this motion was irregular, it was referred to the committee of works and for general purposes to consider the whole question, particularly with reference to the effect, a change in the proprietorship, from an individual to a company, might have in increasing the annoyance of the parishioners; and the report of the committee was presented on the twenty-fifth of September. That report, which is printed at page 107, recommended the publication of the report of 1857, and the presentation of a petition to the licensing magistrates that the hour of closing might be made twelve o’clock at night, as was the case with other public gardens. A memorial from forty-four owners of property and ratepayers residing near the gardens, urging the vestry to act, was presented at the same time the report was under consideration. The vestry then decided, by a majority of 17 to 6, to present the petition, but to withold the publication of the report of 1857; and the solicitor being in attendance with a form of petition, which was approved, the seal was affixed to it at once, and it was duly presented. On the ninth of October, however, a day or two before the licensing day, a communication was received from the solicitors, which induced the board, on a division, by a majority of twenty-eight to eight, to
- 77. pass the following resolution:— Resolved, that having regard to the communication now made by the solicitors, from which it appears that the necessary evidence in support of the petition cannot be obtained upon the present occasion, the solicitors be instructed not to incur any further expense in the matter, on the understanding that if the gardens should be kept open after twelve at night, and the nuisance, annoyance, and injury to the parishioners continue, this board, will in sufficient time next year consider the necessary steps to be taken effectually to oppose the application for the renewal of the license in 1861. “THE LOTS.” “The Lots,” a parcel of land so called lying on the banks of the river [23a] in front of Ashburnham House, comprises about four acres, and is the meadow ground formerly allotted to Sir Arthur Gorges by the Lord of the Manor, in lieu of his right of common. [23b] These are, and have been for centuries, the Chelsea Lammas Lands, and have hitherto been accustomed to be opened on the twelfth of August, being the first day of the month according to the Old Style. The graziers, butchers, and others, with their cattle, used formerly to assemble in the lane leading to “The Lots” on the eve of Lammas, and when the clock had struck twelve they entered upon the meadow. From the report of a committee, printed at page 108, appointed by the vestry in the year 1834 to investigate the state of these town meadows, when it was necessary vi et armis to reassert the invaded privileges of the inhabitants, it appears that “‘The Lots’ are Lammas land, and have been for ages appurtenant to the manor of Chelsea. The Lord of the Manor possesses the right of letting the land on lease for the spring and autumn quarters, beginning with March and ending in August; and the inhabitants at large enjoy the privilege of turning in their cattle from August till February, being the autumn and winter quarters.” Railways, however, and acts of parliament for smoothing down difficulties in their way, have sprung up since those pastoral clays; and the Lord of the Manor having sold his freehold to a railway company, the clerk called the attention of the vestry, on the 8th of May, 1860, to the following advertisement:— West London Extension Railway Company.—Notice is hereby given, that a Meeting of the Householders, Inhabitants, and Land-owners of the Parish of St. Luke, Chelsea, in the County of Middlesex, and other persons interested in the Lammas Lands called “The Lots,” in the said Parish of St. Luke, Chelsea, will be held at the house of Mr. John Sparks Alexander situate in Cremorne Road, in the said parish, and known by the name or sign of the ‘King’s Arms,’ on Wednesday, the 9th day of May next, at Eleven o’clock in the forenoon, for the purpose of appointing a Committee to treat with the Company for the compensation to be paid by them for the extinction of the Lammas and other Commonable Rights, in or over certain land called ‘The Lots’ in the said parish of St. Luke, Chelsea, and which is required by the Company for the purposes of the ‘West London Extension Railway. Act, 1859.’—Dated the 27th day of April, 1860.—Edward Bellamy, Secretary of the Company. A meeting took place consequently on the 9th of May, at which the following resolution was passed:— Resolved, that Messrs. William Hall (church warden), John Perry, James Miles, George Wevell Richards, and William Whitehead, being five of the persons entitled to Lammas or other Commonable Rights over or in the piece of land called ‘The Lots Meadow,’ otherwise ‘Chelsea Lot Mead,’ situate in the parish of Saint Luke, Chelsea, in the County of Middlesex, and containing by admeasurement three acres, two roods, and thirty perches, or thereabouts, and marked or referred to in the map or plan, and in the book of reference of the West London Extension Railway, deposited with the Clerk of the Peace for the said County of Middlesex, by the number 10 in the said parish of St. Luke, Chelsea, shall be, and they are accordingly, in pursuance of the provisions of the ‘Lands Clauses Consolidation Act, 1845,’ appointed to be a Committee having all such powers
- 78. as by the ‘Lands Clauses Consolidation Act, 1845’ are conferred upon Committees of like description, to treat with the West London Extension Railway Company for the compensation to be paid for the extinction of all Lammas or other Commonable Rights over or in the said piece of land. The important question of the boundaries of the parish has occupied considerable attention during the past year, and has been the subject of a reference to a sub-committee of the committee of works and for general purposes, who have still the subject under consideration. Various memorials and petitions have been presented during the year, to which, as they sufficiently elucidate the subjects of them, and are printed in the appendix (Nos. 23 to 27) no further allusion need be made here. The “Public Indicator” erected in Sloane Square by permission of the board in May, 1860, was removed in the fallowing March, in consequence of the failure of the proprietors to light it at night, and to provide the promised clock. It is retained in the custody of the vestry, under a clause in the agreement, as a lien for the expenses of its removal and making good the paving. Plans, books of reference, and parliamentary notices of the “London, Buckinghamshire, and West Midland Railway” (whose intended terminus was in the Pavilion grounds), and of “The London Tramway and Dispatch Company,” were deposited with the Vestry, but both of their bills were unsuccessful. Mr. Train also made an application to the vestry for permission to lay down his tramways in Sloane Street and the King’s Road, but its consideration has been adjourned sine die. On the twenty-ninth of January, 1861, Mr. Finch laid before the Board a copy of a “Bill to Transfer the Seats in Parliament forfeited by the Borough of St. Albans to the proposed Borough of Chelsea and Kensington,” which Viscount Enfield, M.P. for Middlesex had given notice of his intention to introduce in the House of Commons. Immediately the Government [25] brought in their “Bill for the Appropriation of the Seats Vacated by the Disfranchisement of the Boroughs of Sudbury and Saint Albans,” which contained the following clause: “The parishes of Chelsea and Kensington in the county of Middlesex shall, for the purposes of this act, together form a borough, to be called the borough of Chelsea, and such borough shall, from and after the said first day of November, 1861, return one member to serve in parliament.” An influential deputation from both parishes, accompanied by the two county members, had an interview with Lord Palmerston on the fifth of March, on the subject of the bill, and laid before his lordship various statistical information (see appendix No. 29). But on a division in the House of Commons on the tenth of June, the clause was rejected by a majority of two hundred and seventy-five to one hundred and seventy-two. Mr. Tite thus reported the circumstance to the board:— 42, Lowndes Square, Tuesday Morning, 1 o’clock. My dear Mr. Lahee,—I never was so taken by surprise as by the division I have just left. The Government Bill was affirmed by two large majorities, but when we came to the Chelsea question, to my astonishment we were beaten by one hundred and three. The house was excessively impatient, and would hardly listen to anything; but it appeared to me Sir James Graham, whom they would hear, said all that could be said on the subject. I was also astonished at some of those I saw voting against us. I will send you the division list as soon as I get it. The impression in the House just now was that the bill would be withdrawn, so in happier times we may have another chance.—Yours very truly, William Tite. P.S.—Rather sharp work for M.P.’s yesterday. I was on a committee from 12 to 4, in the House from 5 to just now; to-day we begin again at 12, I suppose until 2 o’clock Wednesday morning.
- 79. By order of the Vestry, CHARLES LAHEE, Vestry Clerk.
- 80. APPENDIX No. 1. LIST OF VESTRYMEN, AUDITORS, AND OFFICERS, WITH PLANS AND DESCRIPTIONS OF THE WARDS. No. 1, or Stanley Ward. All such parts of the said Parish of Chelsea as lie on the south side of the boundary line dividing the said parish from the parish of St. Mary Abbott’s, Kensington, in the Fulham-road, and on the west side of a line drawn from the point of the said boundary line in the Fulham-road, opposite the middle of Upper Church-street, in a southerly direction, along the middle of Upper Church-street, and along the middle of Church-street, to the south end thereof, and thence in the same direction to the southern boundary of the said Parish of Chelsea, in the river Thames. VESTRYMEN. Breun, John Cowan, 6, Lower Sloane-street. Delany, James, 3, Albion Place, Fulham-road. Foy, William, 46, Paultons-square. Gable, Isaac Cosson, 2, Belle Vue. Garner, Thomas Betts, sen., Mason’s-place, Fulham-road. Hall, William, 22, Paultons-square. Perry, John, 7, Danvers-street. Tipper, William, 7, Odell’s-place, Fulham-road. Wood, William, “Adam and Eve,” Duke-street. AUDITOR. D’Oyle, Henry, 190, Sloane-street. PLAN OF No. 1, OR STANLEY WARD.
- 81. No. 2, or Church Ward. All such parts of the said Parish of Chelsea as are bounded as follows:—that is to say: Bounded on the north-west side by the boundary line dividing the said Parish of Chelsea from the said parish of St. Mary Abbott’s, Kensington, in the Fulham-road, from the point thereof opposite the middle of Upper Church- street to the point thereof opposite the middle of Marlborough-road; bounded towards the north-east, and partly towards the south-east, by a line commencing at the last-mentioned point, and drawn south- east along the middle of Marlborough-road to the point thereof opposite the middle of Whitehead’s- grove, then turning south-west along the middle of Whitehead’s-grove to the centre of the south-east end of College-street, thence turning south-east across College-place, to and along the middle of Markham-street to the middle of King’s-road, and turning north-east along King’s-road, to the point thereof opposite the middle of Smith-street, thence turning south-east along the middle of Smith-street to the south eastern end thereof, and thence along the middle of the Royal Hospital Creek to the river Thames, and in the same direction to the southern boundary of the said Parish of Chelsea, at a point in the river Thames; bounded on the south by the southern boundary of the said Parish in the river Thames, from the point thereof lastly above mentioned to the boundary line of Ward No. 1; and bounded towards the west by the eastern boundary of Ward No. 1 hereinbefore described. VESTRYMEN, Alexander, John Sparks, “King’s Arms,” Cremorne-road. Blazdell, Alexander, 25, Manor-street.
- 82. Callow, John, 41, Queen’s-road West. Carter, Charles, “Red House,” College-street. Dancocks, Samuel Sharman, Fulham-road. Doubell, William, 144, King’s-road. Finch, William Newton, 181, King’s-road. Goss, William John, “Duke’s Head,” Queen-street. Hulse, Robert, 40, Radnor-street. Hunt, Henry, New King’s-road. Lawrence, William, 141, King’s-road. Miles, James, 180, King’s-road. Osborn, William, 8, Queen-street. Oxford, Henry, 40, Riley-street. Parker, Edward, 21, Paultons-square. Robson, Joseph, 24, Smith’s-terrace. Symons, Thomas, Alpha House, Fulham-road. Todd, John, Stanley House, Milner-street. AUDITOR. Mead, George Edward, 2, Durham-place. PLAN OF No. 2, OR CHURCH WARD. No. 3, or Hans Town Ward.
- 83. All parts of the said Parish of Chelsea, not included in Wards No. 1 and No. 2, hereinbefore mentioned, or in Ward No. 4, hereinafter mentioned and described. VESTRYMEN. Badcock, John, 19, Smith-street. Birch, Abel Francis Faulkner, 14, Sloane-street. Butt, John, 1, Bayley’s-place. Chelsea, Viscount, 28, Lowndes-street. Collier, Caleb, 209, Sloane-street. Compton, James, 1, Smith-street. Fisher, John, 60, Cadogan-place. Gurney, George Edward, “The Earl of Cadogan,” 1, Marlborough-road. Handover, William, Kensal New Town. Hopwood, Owen Thomas, 195, Sloane-street. Jones, Benjamin William, 81, Cadogan-place. Rope, Robert Northern, 49, Sloane-street. Shelton, Richard, 22, Halsey-street. Symons, Elias Octavius, 3, Exeter-street. Till, John, “The Australian,” Milner-street. Tite, William, M.P., 42, Lowndes Square. Thirst, Edward, 11, Halsey-terrace. Todd, George, Stanley House, Milner-street. Walker, Thomas Humble, 6, Wellington-square. Williams, William, Kensal New Town. Wood, Lancelot Edward, 28a, College-street. AUDITOR. Rhind, William, 189, Sloane-street. PLAN OF No. 3, OR HANS TOWN WARD.
- 84. No. 4, or Royal Hospital Ward. All such parts of the said Parish of Chelsea as are bounded as follows;—that is to say: Bounded towards the north-west by a line commencing at the intersection of the central line of College-street with the central line of Whitehead’s-grove, and drawn in a north-east direction along the middle of Whitehead’s- grove to and along the middle of Cadogan-street, to and along the middle of Halsey-terrace, to and along the middle of Cadogan-terrace, crossing Sloane-street, to and along the middle of Ellis-street, to and along the party-wall between the public-house known as “The Woodman” public-house, to the south, and No. 12, D’Oyley-street, to the north, and thence in the same direction to the eastern boundary of the said Parish; bounded towards the east by the eastern boundary of the said Parish, from the point wherein the north-western boundary line of the said Ward No. 4, hereinbefore described, meets the same, to the south point of the said eastern boundary in the river Thames; bounded towards the south by the south boundary of the said Parish in the river Thames, from the point lastly hereinbefore mentioned to the boundary of Ward No. 2; bounded towards the south-west by the north- east boundary of Ward No. 2 from the point lastly hereinbefore mentioned, to the said point of intersection of the central line of College-street with the central line of Whitehead’s-grove. VESTRYMEN. Dunkley, Thomas, 18, Lower Sloane-street. Fuge, George Frederick, 13, Sloane-square. Leete, John Hurstwaite, 143, Sloane-street. Livingston, Alexander, 8, King’s-road. Mowels, Samuel Alfred, 142, Sloane-street. Rabbits, William, 20, Sloane-square.
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