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![Preface
Discontinuous control systems are the oldest type of control system and the
most widespread type of nonlinear control system. The theory of discontinuous
control, and the theory of relay feedback systems in particular, is usually con-
sidered a mature subject. However, many problems in discontinuous control
theory still remain open. One problem involves the input-output properties
of these systems, knowledge of which is extremely important to every appli-
cation.
Two types of discontinuous control systems are studied in this book. The
first is the so-called relay feedback system, which normally encompasses re-
lay servomechanisms, various on-off controllers, sigma-delta modulators, relay
feedback tests used for process dynamics identification, and controller tuning.
Relay systems are often considered the main type of nonlinear system, which is
evident by the enormous amount of house temperature control systems (that
are usually implemented as on-off controllers) that exist. The theory of relay
systems is an old subject. The problem of analysis of relay feedback systems
was first considered by L. MacColl in 1945 [71]; the study was motivated by
the development of relay servomechanisms of missile thrusters on the one hand
and vibrational voltage regulators on the other. MacColl’s analysis was based
on an approximate approach close to the describing function method. Later,
exact methods of analysis of relay feedback systems were developed, the most
well-known of which is the Tsypkin locus [94]. The exact approach developed
by Tsypkin, however, did not consider the servo aspect of relay feedback con-
trol. Its purpose was limited to finding periodic motions that may occur in a
relay system in an autonomous mode or under external excitation. The servo
problem in relay feedback control has not received due attention since. In
recent years, a relatively small number of publications on relay feedback sys-
tems theory have appeared, in which only autonomous modes, and not servo
modes, were considered.
The second type of discontinuous control system considered in this volume
is the so-called sliding mode control system, which includes the conventional
first-order sliding mode control system and the second-order sliding mode](https://image.slidesharecdn.com/77293-250317124040-6f4f00b0/85/Discontinuous-Control-Systems-1st-Edition-Igor-Boiko-Auth-11-320.jpg)
![VIII Preface
system. Sliding mode systems are a specific type of discontinuous control
system. There are a number of references on sliding mode control (systems)
devoted to this type of discontinuous control, the most well-known of which
is by V.I. Utkin [97]. This subject has been an active research area during the
past three decades. However, the present volume offers a different treatment
of sliding modes than the traditional approach. The approach presented in
this work allows accounting for the presence of so-called parasitic dynamics in
control loops and uncovers mechanisms of chattering and non-ideal closed-loop
performance in sliding mode control systems.
The purpose of this book is to present a new frequency-domain theory of
discontinuous control systems in which the control systems are viewed and
studied as servo systems. This theory involves a unified frequency-domain ap-
proach to both analysis of possible self-excited periodic motions and analysis
of input-output properties of discontinuous control systems. The servo aspect
of control is very important and was underestimated in the past. Knowledge of
input-output properties is as important as knowledge of autonomous behav-
ior in discontinuous control systems (self-excited oscillations). In fact, these
two aspects complement each other. For example, in on-off house temperature
control system design, it is equally important to know both the frequency of
relay switching and how the average indoor temperature might change in re-
sponse to the outdoor temperature. The latter problem can be solved only if
the servo aspect of the system is considered.
The core approach in this present book is the frequency-domain method
called the locus of a perturbed relay system (LPRS). This method offers an
exact analysis of both oscillatory properties and servo properties of relay
feedback systems. The method is analytical and very convenient for design
applications, which is illustrated by the numerous examples provided. This
approach is exact, which allows for overcoming the drawback of the well-
known describing function method. Further, overcoming the constraint of the
filtering hypothesis of the describing function method allows for extending
the proposed theory to analysis of motion in sliding mode control systems,
where the sliding mode itself is now considered as oscillations of either finite
or infinite frequency. The analysis provided, however, is not merely another
confirmation of available results in sliding mode control theory. The proposed
approach offers a more precise treatment of sliding mode control systems than
does classic sliding mode control theory. Thus, the proposed approach intro-
duces the theory of real sliding mode control versus ideal classic sliding mode
control.
This book is primarily a research monograph, as it is devoted only to
frequency-domain theory of discontinuous control systems, and the theory pre-
sented in the book is novel. However, it also has many features of a textbook,
as the theory presented covers a relatively large classic nonlinear control area.
This theory is illustrated by a number of application examples from different
areas of control engineering and is accessible to students with a background
in linear control. The included MATLAB code can also make understanding](https://image.slidesharecdn.com/77293-250317124040-6f4f00b0/85/Discontinuous-Control-Systems-1st-Edition-Igor-Boiko-Auth-12-320.jpg)
![1
The servo problem in discontinuous control
systems
1.1 Introduction
Discontinuous control systems, and relay feedback systems in particular, are
one of the most important types of nonlinear systems. The term “relay” comes
from electrical applications where on-off control has long been used. To de-
scribe the nonlinear phenomenon typical of electrical relays, the nonlinear
function also received the name “relay” comprising a number of discontinu-
ous nonlinearities. Thus, when applied to any type of control system, the term
“relay” is now associated not with applications but with the kind of nonlin-
earities that are found in the system models. However, what we traditionally
call the “relay system” cannot always be described by the relay system model.
For example, the vibrational voltage regulator is not a relay system in the full
sense, as the charge and discharge time constants are different and the regu-
lator is better described as a variable structure system (which is, however, a
discontinuous control system).
Applications of the discontinuous feedback principle have evolved from vi-
brational voltage regulators and missile thruster servomechanisms of the 1940s
to numerous on-off process parameter closed-loop control systems, sigma-
delta modulators, process identification and automatic tuning of proportional-
integral-derivative (PID) controller techniques, and DC motor, hydraulic and
pneumatic servo systems, to name a few. The enormous number of residential
temperature control systems available throughout the world illustrates how
popular discontinuous control systems are. A number of industrial examples
of relay systems were given in the classic book on relay systems [94]. Further-
more, many existing sliding mode algorithms can be considered and analyzed
via the relay control principle. Perhaps, some aspects of hybrid systems can
also be analyzed via application of discontinuous control system theory. In
fact, discontinuous control systems are probably the most conventional type
of control system in history.
Discontinuous control systems provide many advantages over linear sys-
tems: simplicity of design, cheaper components that were known as early as](https://image.slidesharecdn.com/77293-250317124040-6f4f00b0/85/Discontinuous-Control-Systems-1st-Edition-Igor-Boiko-Auth-18-320.jpg)
![4 1 The servo problem in discontinuous control systems
the 1930s [55], and the ability to adapt the open-loop gain in the relay feed-
back system in response to changing parameters [8, 50, 71]. As a rule, they
also provide a higher open-loop gain and a higher performance [56] than lin-
ear systems. In some applications, smoothing of the Coulomb friction and of
other plant nonlinearities can also be achieved.
Discontinuous control systems theory has been the subject of attention
since the 1940s from the worldwide research community. Traditionally, the
scope of research was composed of the following problems: existence and pa-
rameters of periodic motions, stability of limit cycles, and input-output prob-
lems (set point tracking or external signal propagation through the system),
which includes the disturbance attenuation problem. The theoretical devel-
opment was motivated by the design of missile thruster servomechanisms in
Germany [17] and in the United States [71, 72, 104] as well as vibrational
voltage regulators [51]. Later, a number of publications, some of which be-
came classic works in the area, appeared. Most of these were concerned with
the solution of the periodic problem. A semigrahpical solution [43], a matrix
method [28], a z-transform–based solution [57], frequency-domain methods
[54, 941
] and [9] (which can be considered a generalization of method [94]), a
state-space–based technique [34], and a finite-difference operator method [81]
for the solution of problem of existence and local stability of periodic mo-
tions were proposed in the late 1950s to the mid-1960s. In [79] a systematic
overview of applications of Poincaré maps to the analysis of periodic motions
in relay systems was given. A rigorous solution of the problem of existence
and local stability of limit cycles was presented in [2] and [100] for symmet-
ric and asymmetric limit cycles, respectively. The problem of global stability
of limit cycles was considered in [53]. In [81, 94] and [84], the input-output
problem was analyzed. Relay feedback systems were also studied with the use
of more general approximate methods such as the describing function (DF)
method [71] (where the input-output problem in relay systems was probably
considered for the first time; the authors used the results obtained in [58]),
and [8, 50, 56, 87]. In the recent monographs [103] and [107], extensive cover-
age of the theory of relay feedback test–based identification is given.
Another study of oscillations in the relay and variable structure systems
was motivated by the chattering problem in sliding mode control [97, 105].
This problem is closely related to the original problem, and the results are
applicable to both variable structure systems and relay systems ([4, 16, 26, 46,
47, 98, 99]). A number of techniques of process parameter identification and
PID controller tuning based on the relay feedback test are given in [3, 59–61,
70, 73–75, 83]. Periodic motions in sigma-delta modulators were studied in [41]
and [78].
The method presented in this book is based on the frequency-domain char-
acteristic of the discontinuous control system that is similar in some ways
to the frequency response of a linear system (Nyquist plot) and the Tsypkin
1
The first edition of this book was published in 1955.](https://image.slidesharecdn.com/77293-250317124040-6f4f00b0/85/Discontinuous-Control-Systems-1st-Edition-Igor-Boiko-Auth-19-320.jpg)
![1.2 Fundamentals of frequency-domain analysis of periodic motions 5
locus [94]. Therefore, the presented approach has some resemblance to the DF
method and Tsypkin’s method. Moreover, from the methodological point of
view, the presented theory in some ways replicates the DF method. However,
the presented method offers a few advantages over both methods mentioned
above: better accuracy than that of the DF method (eliminating the necessity
of the filtering hypothesis) and a better solution of the input-output problem
compared with that of the Tsypkin locus, which can only furnish a solution
of the periodic problem.2
All discontinuous control systems have a nonlinear element in the control
loop. Usually the nonlinearity is associated with the controller. In most cases
the discontinuous nonlinearity is an ideal (sign function) or hysteretic relay.
However, it can be a more complex nonlinearity, examples of which are consid-
ered in the chapters devoted to sliding mode control and second-order sliding
mode control.
A mode that may occur in a discontinuous control system is a self-excited
(non-vanishing) oscillation, which is also referred to as a periodic motion. If
the system does not have asymmetric nonlinearities, this periodic motion is
symmetric in the autonomous mode (no external input applied). However, if
an external input (disturbance) is applied to the system that has a periodic
motion, the self-excited oscillations become biased or asymmetric. The key ap-
proach to the analysis of discontinuous control systems is a method of analysis
of self-excited oscillations (symmetric and asymmetric). The following section
gives a general methodology of such an analysis based on frequency-domain
concepts.
1.2 Fundamentals of frequency-domain analysis
of periodic motions in nonlinear systems
The problem of finding possible periodic motions in nonlinear dynamical sys-
tems is a fundamental problem in both mathematics and control theory. It
can be traced back to the works of Poincaré who introduced a widely used
now geometric interpretation of this problem as that of finding a closed orbit
in the state space.
Poincaré’s approach involves finding a certain map in the state space,
which is now called the Poincaré map. Yet Poincaré maps for most nonlinear
systems can hardly be obtained in an analytical form. Analytical expressions
have been obtained only for low-order systems or higher-order systems con-
taining simple types of nonlinear functions. An example of the latter is the
relay feedback system for which the Poincaré map is obtained in an explicit
form. A few formulations of Poincaré maps for relay feedback systems are pre-
sented in [2, 20, 54, 94, 100]. For more complex types of nonlinearities, only an
2
Tsypkin [94] also considers the input-output problem in his book; he proposes
that the solution of the input-output problem should be done via analysis of
asymmetric oscillations.](https://image.slidesharecdn.com/77293-250317124040-6f4f00b0/85/Discontinuous-Control-Systems-1st-Edition-Igor-Boiko-Auth-20-320.jpg)
![6 1 The servo problem in discontinuous control systems
approximate solution can be found as a rule. A number of different approaches
to the problem of finding a periodic solution have been proposed. The most
general approach was given in [64] in which the solution of the periodic prob-
lem was found in the form of a Fourier series. Despite the clear underlying
idea of this method, its application is computationally cumbersome, as the
original problem is transformed into the problem of solving the system of a
high number of nonlinear algebraic equations, which is often impractical. A
modification of this classic approach was given in [96]. However, the compu-
tational efficiency of the proposed approach rapidly decreases with required
accuracy. Among other methods are Volterra series used to solve both the pe-
riodic problem in autonomous systems [32] and forced oscillations [102]. Yet
the Volterra series approach can only handle weak (mild) nonlinearities. A
combination of the perturbation method and the harmonic balance method
was proposed in [29]. But this method also can handle only mild nonlinear-
ities. A singular perturbation method was used for analysis of the periodic
problem in relay systems in [47].
Commonly in engineering practice, only the first-harmonic approximation
is used, which is known as the describing function (DF) method. The main
concepts of this method were developed in the 1940s and 1950s [36, 51, 52,
63, 71, 95]. Comprehensive coverage of this method is given in [8, 50]. The
DF method provides a simple and efficient, but not exact, solution of the pe-
riodic problem. Its applications are limited to systems that can be separated
into nonlinear and linear parts interconnected into a closed loop (Fig. 1.1)
with the nonlinear part being a static nonlinearity (with possible hysteresis)
and the linear part given by linear time-invariant differential equations. More-
over, the linear part must be a low-pass filter to filter out higher harmonics,
so that the input to the nonlinearity could approximately be considered a
sinusoid.
Let us consider the frequency-domain approach to analysis of self-excited
oscillations. Consider the following nonlinear system,
ẋ = Ax + Bu
y = Cx
(1.1)
Fig. 1.1. Block diagram of the system for DF analysis](https://image.slidesharecdn.com/77293-250317124040-6f4f00b0/85/Discontinuous-Control-Systems-1st-Edition-Igor-Boiko-Auth-21-320.jpg)
![1.2 Fundamentals of frequency-domain analysis of periodic motions 7
Fig. 1.2. Poincaré mapping
u = f(σ), (1.2)
σ = −y, (1.3)
where A ∈ Rn×n
, B ∈ Rn×1
, and C ∈ R1×n
are matrices, x ∈ Rn
, y ∈
R1
, σ ∈ R1
, f ∈ R1
, f(σ) is a single-valued symmetric monotone nonlinear
function: f(−σ) = −f(σ) (it will be later extended to a hysteretic function).
Formula (1.1) provides a description of the linear part, formula (1.2) denotes
the nonlinear part of the system, and formula (1.3) gives the condition of
closing the system loop. We shall also use the linear part description in the
form of the transfer function W(s), which can be obtained from formulas (1.1)
as follows: W(s) = C(Is − A)−1
B. Let us also assume that the linear part is
strictly proper, i.e., the relative degree of W(s) is one or higher.
The problem of finding a periodic solution of system (1.1), (1.2), and (1.3)
is usually formulated as the problem of finding a fixed point x∗
of Poincaré
map, which will be a solution to the equation: x = P(x), where P(x) is the
mapping. Figure 1.2 provides two examples of Poincaré mapping. P1 maps
point x1 into point x2, and P2 maps point x3 into itself. Therefore, x3 = x∗
.
In the time domain, the closed orbit that starts from the point x∗
(denote it
x∗
(t) for brevity) will be a periodic vector signal, and y∗
(t) = Cx∗
(t) will be
a periodic scalar signal.
Now consider the open-loop system (1.1), (1.2), which can be obtained by
ignoring equality (1.3). Let σ(t) be a periodic symmetric signal of period T
with zero mean, continuously differentiable with respect to t, so that σ(t +
T/2) = −σ(t), σ ∈ R, t ∈ D ⊂ R, D := t ∈ [0; T]. Consider the following
lemma, which will be instrumental below.
Lemma 1.1. If σ(t) satisfies the conditions of symmetric periodicity and uni-
modality and f(σ) is a symmetric monotone nonlinear function, then x(t) will
be a periodic vector signal, and y(t) will be a periodic scalar signal.
Proof. u(t) is a periodic signal because it is a function of f(σ): u(t + T) =
f(σ(t+T)) = f(σ(t)) = u(t). y(t) as well as x(t) are periodic because they are
responses to the periodic input u(t), which follows from linear system theory.](https://image.slidesharecdn.com/77293-250317124040-6f4f00b0/85/Discontinuous-Control-Systems-1st-Edition-Igor-Boiko-Auth-22-320.jpg)
![8 1 The servo problem in discontinuous control systems
Definition 1.2. Let us call the mapping of space R×D into itself, where σ(t)
satisfies the conditions of symmetric periodicity and unimodality, the periodic
signal mapping (PSM)
y(t) = g(f(σ(t)), (1.4)
which is a chain-rule application of two mappings: g for the linear part and f
for the nonlinear part.
Definition 1.3. Let us call the periodic function σ∗
(t), which provides the
solution to the equation −σ(t) = g(f(σ(t)), t ∈ D, the fixed point of the
periodic signal mapping (the minus sign is attributed to the negative feedback
in the closed-loop system).
The following theorem relates periodic solutions in the state-space and
time domains.
Theorem 1.4. If σ∗
(t) is a fixed point of the periodic signal mapping y(t) =
g(f(σ(t)) in the open-loop nonlinear system (1.1), (1.2) where f(σ) is a single-
valued symmetric monotone nonlinear function, then x∗
(t) = h(f(σ∗
(t)) will
be a fixed point of the Poincaré mapping in the closed-loop system (1.1), (1.2),
(1.3), where h is mapping u(t) → x(t).
Proof. Because σ∗
(t) is a fixed point of the periodic signal mapping (1.4),
σ∗
(t) provides a solution to system (1.1), (1.2), (1.3). This is a periodic so-
lution, as the property of periodicity is “embedded” in the type of function
σ(t). According to Lemma 1.1, x∗
(t) will be a periodic vector function, which
means that in the state space, the trajectory that starts from x∗
is a closed
orbit, i.e., satisfying the condition of being a fixed point of the Poincaré map.
Let us analyze how the PSM can be implemented in practice. Transfer the
analyzed problem including the defined mappings into the spectral domain,
which can be provided via taking the Fourier series of the signals in the system.
Represent the periodic signal σ(t) in the form of the Fourier series as follows,
σ(t) = 2
∞
k=1
|qσk| cos ((2k − 1) ωt + arg qσk) (1.5)
where ω = 2π/T, and all even harmonics are zeros due to σ(t) being symmetric
qσk =
1
T
T/2
−T/2
σ(t) [cos(2k − 1)ωt − j sin(2k − 1)ωt] dt. (1.6)
Therefore, any σ(t) can be characterized by the matrix Qσ as follows:
Qσ = [qσ1 qσ3 qσ5 . . .] subject to the frequency ω being given. We shall call
the matrix Qσ the spectrum of the signal σ(t). The number of elements of Qσ
can be set arbitrarily large, but it makes sense to limit the number of elements](https://image.slidesharecdn.com/77293-250317124040-6f4f00b0/85/Discontinuous-Control-Systems-1st-Edition-Igor-Boiko-Auth-23-320.jpg)
![1.2 Fundamentals of frequency-domain analysis of periodic motions 9
to some reasonable value, so that any further increase of this number will not
provide any noticeable increase in the accuracy of the σ(t) representation. Let
us represent other periodic signals u and y in the system by their spectra: Qu,
Qy. Then the mapping Qσ → Qu can be given by the transformation into the
time domain as per (1.5), nonlinear mapping in the time domain as per (1.3),
and transformation into the spectral domain through the following formulas:
quk =
1
T
T/2
−T/2
u(t) [cos(2k − 1)ωt − j sin(2k − 1)ωt] dt (1.7)
Qu = [qu1 qu3 qu5 . . .] . (1.8)
The mapping Qu → Qy can be given by the following element-by-element
multiplication of the two matrices, which directly follows from linear systems
theory:
Qy = Qu • S (1.9)
where
Qy = [qy1 qy3 qy5 . . .] (1.10)
S = [W(jω) W(j3ω) W(j5ω) . . .] . (1.11)
According to Theorem 1.4, the periodic solution of (1.1), (1.2), (1.3) can be
found as a fixed point of the PSM. We have defined the PSM as the chain
application of two mappings: the first of which is provided by (1.5), (1.2),
and (1.7), and the second is given by (1.9). To find the fixed point, we have
to design a contraction mapping, so that iterates from the initial point will
converge to the solution.
Assume now that the linear part is an ideal low-pass filter. This assumption
enables one to rewrite (1.9) as
Qy = qu1W(jω) (1.12)
and (1.5), (1.2), (1.7) as just
qu1 =
1
T
T/2
−T/2
u(t) [cos ωt − j sin ωt] dt (1.13)
with σ(t) = aσ cos ωt. The ratio qu1/qσ1 = 0.5aσ gives the describing func-
tion (DF) of the nonlinearity. As a result, considering (1.3) we can write the
harmonic balance equation for the DF approximated system as follows:
N(aσ)W(jΩ) = −1 (1.14)
where N(aσ) is the describing function of the nonlinearity, aσ is the amplitude
of σ(t), and Ω is the frequency of the periodic motion:](https://image.slidesharecdn.com/77293-250317124040-6f4f00b0/85/Discontinuous-Control-Systems-1st-Edition-Igor-Boiko-Auth-24-320.jpg)
![10 1 The servo problem in discontinuous control systems
N(aσ) =
qu1
qσ1
=
2
Taσ
T/2
−T/2
u(t) [cos ωt − j sin ωt] dt.
Equation (1.14) is a well-known formula [8, 50] that is widely used for
analysis of periodic motions. Therefore, the DF analysis can be viewed as a
fixed point of the PSM considered above, subject to the assumption about
the low-pass filtering property of the linear part.
Equation (1.14) provides a basis for analysis of possible periodic motions
in a nonlinear system via application of the DF method. Usually this equation
is presented in the following form:
W(jΩ) = −
1
N(aσ)
(1.15)
which is convenient for graphical interpretation, as the unknown variables are
separated into the left-hand and the right-hand parts of the equation.
1.3 Relay servo systems
Usually, control systems are categorized into two groups depending on the
control problem they are supposed to solve. These two categories are stabi-
lization systems and servo systems. Stabilization systems carry out the task of
stabilization of certain process variables. The main challenge for such systems
is producing a compensating action for the disturbances that inevitably oc-
cur. The servo systems implement the task of tracking certain input signals,
so that the system output follows the input signal as precisely as possible.
Normally, the input to the system is unknown, but some a priori informa-
tion about the input signal such as frequency and amplitude range may be
available. The typical examples of stabilization systems are temperature, pres-
sure, level, flow control systems; and examples of servo systems are actuators,
airplane autopilots, and temperature control systems, which are supposed to
ensure the temperature change in time according to a given profile.
We can see from the above examples that the autonomous mode, when no
external signals are applied to the system, does not normally occur. What is
common for both these types of control systems is the existence of external
signals that affect the functioning of the system. In the first case, this is a
disturbance to which the system is supposed to respond in such a way as
to provide its compensation. In the second case, the system is supposed to
respond to the input to bring the output in alignment with this external
input. In both cases, the problem analyzing the effect of external signals on
the system characteristics is the most important part of system performance
analysis.](https://image.slidesharecdn.com/77293-250317124040-6f4f00b0/85/Discontinuous-Control-Systems-1st-Edition-Igor-Boiko-Auth-25-320.jpg)
Discontinuous Control Systems 1st Edition Igor Boiko (Auth.)
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- 8. Igor Boiko Discontinuous Control Systems Frequency-DomainAnalysis and Design Birkhäuser Boston • Basel • Berlin
- 9. Igor Boiko Honeywell andUniversity of Calgary 2500 University Dr. NW Calgary, Alberta T2N 1N4 Canada i.boiko@ieee.org ISBN: 978-0-8176-4752-0 e-ISBN: 978-0-8176-4753-7 DOI: 10.1007/978-0-8176-4753-7 © 2009 Birkhäuser Boston, a part of Springer Science+Business Media, LLC Printed on acid-free paper. 9 8 7 6 5 4 3 2 1 www.birkhauser.com Library of Congress Control Number: 2008928863 All rights reserved. This work may not be translated or copied in whole or in part without the writ- ten permission of the publisher (Birkhäuser Boston, c/o Springer ScienceCBusiness Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adap- tation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
- 10.
- 11. Preface Discontinuous control systemsare the oldest type of control system and the most widespread type of nonlinear control system. The theory of discontinuous control, and the theory of relay feedback systems in particular, is usually con- sidered a mature subject. However, many problems in discontinuous control theory still remain open. One problem involves the input-output properties of these systems, knowledge of which is extremely important to every appli- cation. Two types of discontinuous control systems are studied in this book. The first is the so-called relay feedback system, which normally encompasses re- lay servomechanisms, various on-off controllers, sigma-delta modulators, relay feedback tests used for process dynamics identification, and controller tuning. Relay systems are often considered the main type of nonlinear system, which is evident by the enormous amount of house temperature control systems (that are usually implemented as on-off controllers) that exist. The theory of relay systems is an old subject. The problem of analysis of relay feedback systems was first considered by L. MacColl in 1945 [71]; the study was motivated by the development of relay servomechanisms of missile thrusters on the one hand and vibrational voltage regulators on the other. MacColl’s analysis was based on an approximate approach close to the describing function method. Later, exact methods of analysis of relay feedback systems were developed, the most well-known of which is the Tsypkin locus [94]. The exact approach developed by Tsypkin, however, did not consider the servo aspect of relay feedback con- trol. Its purpose was limited to finding periodic motions that may occur in a relay system in an autonomous mode or under external excitation. The servo problem in relay feedback control has not received due attention since. In recent years, a relatively small number of publications on relay feedback sys- tems theory have appeared, in which only autonomous modes, and not servo modes, were considered. The second type of discontinuous control system considered in this volume is the so-called sliding mode control system, which includes the conventional first-order sliding mode control system and the second-order sliding mode
- 12. VIII Preface system. Slidingmode systems are a specific type of discontinuous control system. There are a number of references on sliding mode control (systems) devoted to this type of discontinuous control, the most well-known of which is by V.I. Utkin [97]. This subject has been an active research area during the past three decades. However, the present volume offers a different treatment of sliding modes than the traditional approach. The approach presented in this work allows accounting for the presence of so-called parasitic dynamics in control loops and uncovers mechanisms of chattering and non-ideal closed-loop performance in sliding mode control systems. The purpose of this book is to present a new frequency-domain theory of discontinuous control systems in which the control systems are viewed and studied as servo systems. This theory involves a unified frequency-domain ap- proach to both analysis of possible self-excited periodic motions and analysis of input-output properties of discontinuous control systems. The servo aspect of control is very important and was underestimated in the past. Knowledge of input-output properties is as important as knowledge of autonomous behav- ior in discontinuous control systems (self-excited oscillations). In fact, these two aspects complement each other. For example, in on-off house temperature control system design, it is equally important to know both the frequency of relay switching and how the average indoor temperature might change in re- sponse to the outdoor temperature. The latter problem can be solved only if the servo aspect of the system is considered. The core approach in this present book is the frequency-domain method called the locus of a perturbed relay system (LPRS). This method offers an exact analysis of both oscillatory properties and servo properties of relay feedback systems. The method is analytical and very convenient for design applications, which is illustrated by the numerous examples provided. This approach is exact, which allows for overcoming the drawback of the well- known describing function method. Further, overcoming the constraint of the filtering hypothesis of the describing function method allows for extending the proposed theory to analysis of motion in sliding mode control systems, where the sliding mode itself is now considered as oscillations of either finite or infinite frequency. The analysis provided, however, is not merely another confirmation of available results in sliding mode control theory. The proposed approach offers a more precise treatment of sliding mode control systems than does classic sliding mode control theory. Thus, the proposed approach intro- duces the theory of real sliding mode control versus ideal classic sliding mode control. This book is primarily a research monograph, as it is devoted only to frequency-domain theory of discontinuous control systems, and the theory pre- sented in the book is novel. However, it also has many features of a textbook, as the theory presented covers a relatively large classic nonlinear control area. This theory is illustrated by a number of application examples from different areas of control engineering and is accessible to students with a background in linear control. The included MATLAB code can also make understanding
- 13. Preface IX of thepresented theory easy. The book, therefore, can be used by researchers, practitioners, and undergraduate and graduate students. The material is organized into two parts and appendices. Part I is de- voted to the theory of the locus of a perturbed relay system (LPRS) method, and Part II presents applications of the LPRS method. Part I comprises five chapters. Chapter 1 poses the problem and outlines the scope and method of analysis, which is presented in detail in the following chapter. In Chapter 3, the results obtained in Chapter 2 for slow inputs are extended to the analy- sis of the system response to relatively fast input signals. Chapter 4 presents frequency-domain theory of sliding mode control systems. The sliding mode analysis follows the methodology developed in the preceding chapters. Chap- ter 5 is devoted to an emerging area of sliding mode control — second-order sliding mode control systems analysis. Part II gives a number of applications of LPRS theory. These applications include the electro-pneumatic servomechanism (Chapter 6), the relay feed- back test and its application to process dynamics identification and controller tuning (Chapter 7), the sliding mode differentiator and dynamic compen- sator (Chapter 8), and the sliding mode observer (Chapter 9). Some of these applications have been in engineering practice for many years, and other ap- plications are relatively new. In general, the material of each chapter in the second part of the book is independent and only makes general references to topics presented in the first part. The Appendix contains the derivations of the LPRS, the proofs of the theorems, and the MATLAB code used in the text for the LPRS computations. I wish to acknowledge the many colleagues who have contributed to the development of this book. I am especially grateful to Prof. L. Fridman for the many discussions resulting in the refinement of the theory presented, to Dr. A. Pisano for his review of the manuscript, to Prof. V.I. Utkin and Prof. K. Furuta for their encouragement and support of my work on this project, to the late Prof. V. Kainov who years ago introduced me to the world of discontinuous control systems, to Prof. N.V. Faldin who influenced my research in many ways, to Prof. D. Atherton and Prof. Y. Shtessel for their valuable comments and discussions of the theory currently presented in the book, and to Mr. E. Tamayo for providing the opportunity and help in implementation of the presented theory in a loop tuner. I thank Alex and Michael Boiko for their help with the LaTeX typesetting and artwork. I also express gratitude to my family for their patience and support; without them this undertaking would not have been possible. Calgary, Canada Igor Boiko October 2008
- 14. Contents Preface . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII Part I The locus of a perturbed relay system theory 1 The servo problem in discontinuous control systems . . . . . . . 3 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Fundamentals of frequency-domain analysis of periodic motions in nonlinear systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Relay servo systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Symmetric oscillations in relay servo systems: DF analysis . . . . 12 1.5 Asymmetric oscillations in relay servo systems: DF analysis . . . 14 1.6 Slow signal propagation through a relay servo system . . . . . . . . 16 1.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 The locus of a perturbed relay system (LPRS) theory . . . . . 19 2.1 Introduction to the LPRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Computing the LPRS for a non-integrating plant . . . . . . . . . . . . 21 2.2.1 Matrix state-space description approach . . . . . . . . . . . . . . 21 2.2.2 Partial fraction expansion technique . . . . . . . . . . . . . . . . . 23 2.2.3 Transfer function description approach . . . . . . . . . . . . . . . 24 2.2.4 Orbital stability of relay systems . . . . . . . . . . . . . . . . . . . . 26 2.3 Computing the LPRS for an integrating plant . . . . . . . . . . . . . . . 26 2.3.1 Matrix state-space description approach . . . . . . . . . . . . . . 26 2.3.2 Transfer function description approach . . . . . . . . . . . . . . . 29 2.3.3 Orbital stability of relay systems . . . . . . . . . . . . . . . . . . . . 30 2.4 Computing the LPRS for a plant with a time delay . . . . . . . . . . 31 2.4.1 Matrix state-space description approach . . . . . . . . . . . . . . 31 2.4.2 Orbital asymptotic stability . . . . . . . . . . . . . . . . . . . . . . . . 32 2.5 LPRS of first-order dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.6 LPRS of second-order dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
- 15. XII Contents 2.7 LPRSof first-order plus dead-time dynamics . . . . . . . . . . . . . . . . 38 2.8 Some properties of the LPRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.9 LPRS of nonlinear plants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.9.1 Additivity property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.9.2 The LPRS extended definition and open-loop LPRS computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.10 Application of periodic signal mapping to computing the LPRS of some special nonlinear plants . . . . . . . . . . . . . . . . . . 48 2.11 Comparison of the LPRS with other methods of analysis of relay systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.12 An example of analysis of oscillations and transfer properties. . 53 2.13 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3 Input-output analysis of relay servo systems . . . . . . . . . . . . . . . 57 3.1 Slow and fast signal propagation through a relay servo system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Methodology of input-output analysis . . . . . . . . . . . . . . . . . . . . . . 63 3.3 Example of forced motions analysis with the use of the LPRS . 63 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4 Analysis of sliding modes in the frequency domain . . . . . . . . . 67 4.1 Introduction to sliding mode control . . . . . . . . . . . . . . . . . . . . . . . 67 4.2 Representation of a sliding mode system via the equivalent relay system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.3 Analysis of motions in the equivalent relay system . . . . . . . . . . . 73 4.4 The chattering phenomenon and its LPRS analysis . . . . . . . . . . 77 4.5 Reduced-order and non–reduced-order models of averaged motions in a sliding mode system and input-output analysis . . 85 4.6 On fractal dynamics in sliding-mode control . . . . . . . . . . . . . . . . 88 4.7 Examples of chattering and disturbance attenuation analysis . . 95 4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5 Performance analysis of second-order SM control algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.2 Sub-optimal algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.3 Describing function analysis of chattering . . . . . . . . . . . . . . . . . . . 105 5.4 Exact frequency-domain analysis of chattering . . . . . . . . . . . . . . 106 5.5 Describing function analysis of external signal propagation . . . . 108 5.6 Exact frequency-domain analysis of external signal propagation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.7 Example of the analysis of sub-optimal algorithm performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
- 16. Contents XIII Part IIApplications of the locus of a perturbed relay system 6 Relay pneumatic servomechanism design . . . . . . . . . . . . . . . . . . 125 6.1 Relay pneumatic servomechanism dynamics and characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.2 LPRS analysis of uncompensated relay electro-pneumatic servomechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.3 Compensator design in the relay electro-pneumatic servomechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6.4 Examples of compensator design in the relay electro-pneumatic servomechanism . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.5 Compensator design in the relay electro-pneumatic servomechanism with the use of the LPRS of a nonlinear plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7 Relay feedback test identification and autotuning . . . . . . . . . . 139 7.1 The relay feedback test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.2 The LPRS and asymmetric relay feedback test . . . . . . . . . . . . . . 140 7.3 Methodology of identification of the first-order plus dead-time process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.4 Analysis of potential sources of inaccuracy . . . . . . . . . . . . . . . . . . 143 7.5 Performance analysis of the identification algorithm . . . . . . . . . . 145 7.6 Tuning algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 8 Performance analysis of the sliding mode–based analog differentiator and dynamical compensator . . . . . . . . . . . . . . . . . 153 8.1 Transfer function “inversion” via sliding mode. . . . . . . . . . . . . . . 153 8.2 Analysis of SM differentiator dynamics . . . . . . . . . . . . . . . . . . . . . 154 8.3 Temperature sensor dynamics compensation via SM application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.4 Analysis of the sliding mode compensator . . . . . . . . . . . . . . . . . . 160 8.5 An example of compensator design . . . . . . . . . . . . . . . . . . . . . . . . 162 8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 9 Analysis of sliding mode observers . . . . . . . . . . . . . . . . . . . . . . . . . 167 9.1 The SM observer as a relay servo system . . . . . . . . . . . . . . . . . . . 167 9.2 SM observer performance analysis and characteristics . . . . . . . . 170 9.3 Example of SM observer performance analysis . . . . . . . . . . . . . . . 172 9.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
- 17. XIV Contents 10 Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 10.1 The LPRS derivation for a non-integrating linear part . . . . . . . . 177 10.2 Orbital stability of a system with a non-integrating linear part 181 10.3 The LPRS derivation for an integrating linear part . . . . . . . . . . 183 10.4 Orbital stability of a system with an integrating linear part . . . 191 10.5 The LPRS derivation for a linear part with time delay . . . . . . . 194 10.6 MATLAB code for LPRS computing . . . . . . . . . . . . . . . . . . . . . . . 198 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
- 18. 1 The servo problemin discontinuous control systems 1.1 Introduction Discontinuous control systems, and relay feedback systems in particular, are one of the most important types of nonlinear systems. The term “relay” comes from electrical applications where on-off control has long been used. To de- scribe the nonlinear phenomenon typical of electrical relays, the nonlinear function also received the name “relay” comprising a number of discontinu- ous nonlinearities. Thus, when applied to any type of control system, the term “relay” is now associated not with applications but with the kind of nonlin- earities that are found in the system models. However, what we traditionally call the “relay system” cannot always be described by the relay system model. For example, the vibrational voltage regulator is not a relay system in the full sense, as the charge and discharge time constants are different and the regu- lator is better described as a variable structure system (which is, however, a discontinuous control system). Applications of the discontinuous feedback principle have evolved from vi- brational voltage regulators and missile thruster servomechanisms of the 1940s to numerous on-off process parameter closed-loop control systems, sigma- delta modulators, process identification and automatic tuning of proportional- integral-derivative (PID) controller techniques, and DC motor, hydraulic and pneumatic servo systems, to name a few. The enormous number of residential temperature control systems available throughout the world illustrates how popular discontinuous control systems are. A number of industrial examples of relay systems were given in the classic book on relay systems [94]. Further- more, many existing sliding mode algorithms can be considered and analyzed via the relay control principle. Perhaps, some aspects of hybrid systems can also be analyzed via application of discontinuous control system theory. In fact, discontinuous control systems are probably the most conventional type of control system in history. Discontinuous control systems provide many advantages over linear sys- tems: simplicity of design, cheaper components that were known as early as
- 19. 4 1 Theservo problem in discontinuous control systems the 1930s [55], and the ability to adapt the open-loop gain in the relay feed- back system in response to changing parameters [8, 50, 71]. As a rule, they also provide a higher open-loop gain and a higher performance [56] than lin- ear systems. In some applications, smoothing of the Coulomb friction and of other plant nonlinearities can also be achieved. Discontinuous control systems theory has been the subject of attention since the 1940s from the worldwide research community. Traditionally, the scope of research was composed of the following problems: existence and pa- rameters of periodic motions, stability of limit cycles, and input-output prob- lems (set point tracking or external signal propagation through the system), which includes the disturbance attenuation problem. The theoretical devel- opment was motivated by the design of missile thruster servomechanisms in Germany [17] and in the United States [71, 72, 104] as well as vibrational voltage regulators [51]. Later, a number of publications, some of which be- came classic works in the area, appeared. Most of these were concerned with the solution of the periodic problem. A semigrahpical solution [43], a matrix method [28], a z-transform–based solution [57], frequency-domain methods [54, 941 ] and [9] (which can be considered a generalization of method [94]), a state-space–based technique [34], and a finite-difference operator method [81] for the solution of problem of existence and local stability of periodic mo- tions were proposed in the late 1950s to the mid-1960s. In [79] a systematic overview of applications of Poincaré maps to the analysis of periodic motions in relay systems was given. A rigorous solution of the problem of existence and local stability of limit cycles was presented in [2] and [100] for symmet- ric and asymmetric limit cycles, respectively. The problem of global stability of limit cycles was considered in [53]. In [81, 94] and [84], the input-output problem was analyzed. Relay feedback systems were also studied with the use of more general approximate methods such as the describing function (DF) method [71] (where the input-output problem in relay systems was probably considered for the first time; the authors used the results obtained in [58]), and [8, 50, 56, 87]. In the recent monographs [103] and [107], extensive cover- age of the theory of relay feedback test–based identification is given. Another study of oscillations in the relay and variable structure systems was motivated by the chattering problem in sliding mode control [97, 105]. This problem is closely related to the original problem, and the results are applicable to both variable structure systems and relay systems ([4, 16, 26, 46, 47, 98, 99]). A number of techniques of process parameter identification and PID controller tuning based on the relay feedback test are given in [3, 59–61, 70, 73–75, 83]. Periodic motions in sigma-delta modulators were studied in [41] and [78]. The method presented in this book is based on the frequency-domain char- acteristic of the discontinuous control system that is similar in some ways to the frequency response of a linear system (Nyquist plot) and the Tsypkin 1 The first edition of this book was published in 1955.
- 20. 1.2 Fundamentals offrequency-domain analysis of periodic motions 5 locus [94]. Therefore, the presented approach has some resemblance to the DF method and Tsypkin’s method. Moreover, from the methodological point of view, the presented theory in some ways replicates the DF method. However, the presented method offers a few advantages over both methods mentioned above: better accuracy than that of the DF method (eliminating the necessity of the filtering hypothesis) and a better solution of the input-output problem compared with that of the Tsypkin locus, which can only furnish a solution of the periodic problem.2 All discontinuous control systems have a nonlinear element in the control loop. Usually the nonlinearity is associated with the controller. In most cases the discontinuous nonlinearity is an ideal (sign function) or hysteretic relay. However, it can be a more complex nonlinearity, examples of which are consid- ered in the chapters devoted to sliding mode control and second-order sliding mode control. A mode that may occur in a discontinuous control system is a self-excited (non-vanishing) oscillation, which is also referred to as a periodic motion. If the system does not have asymmetric nonlinearities, this periodic motion is symmetric in the autonomous mode (no external input applied). However, if an external input (disturbance) is applied to the system that has a periodic motion, the self-excited oscillations become biased or asymmetric. The key ap- proach to the analysis of discontinuous control systems is a method of analysis of self-excited oscillations (symmetric and asymmetric). The following section gives a general methodology of such an analysis based on frequency-domain concepts. 1.2 Fundamentals of frequency-domain analysis of periodic motions in nonlinear systems The problem of finding possible periodic motions in nonlinear dynamical sys- tems is a fundamental problem in both mathematics and control theory. It can be traced back to the works of Poincaré who introduced a widely used now geometric interpretation of this problem as that of finding a closed orbit in the state space. Poincaré’s approach involves finding a certain map in the state space, which is now called the Poincaré map. Yet Poincaré maps for most nonlinear systems can hardly be obtained in an analytical form. Analytical expressions have been obtained only for low-order systems or higher-order systems con- taining simple types of nonlinear functions. An example of the latter is the relay feedback system for which the Poincaré map is obtained in an explicit form. A few formulations of Poincaré maps for relay feedback systems are pre- sented in [2, 20, 54, 94, 100]. For more complex types of nonlinearities, only an 2 Tsypkin [94] also considers the input-output problem in his book; he proposes that the solution of the input-output problem should be done via analysis of asymmetric oscillations.
- 21. 6 1 Theservo problem in discontinuous control systems approximate solution can be found as a rule. A number of different approaches to the problem of finding a periodic solution have been proposed. The most general approach was given in [64] in which the solution of the periodic prob- lem was found in the form of a Fourier series. Despite the clear underlying idea of this method, its application is computationally cumbersome, as the original problem is transformed into the problem of solving the system of a high number of nonlinear algebraic equations, which is often impractical. A modification of this classic approach was given in [96]. However, the compu- tational efficiency of the proposed approach rapidly decreases with required accuracy. Among other methods are Volterra series used to solve both the pe- riodic problem in autonomous systems [32] and forced oscillations [102]. Yet the Volterra series approach can only handle weak (mild) nonlinearities. A combination of the perturbation method and the harmonic balance method was proposed in [29]. But this method also can handle only mild nonlinear- ities. A singular perturbation method was used for analysis of the periodic problem in relay systems in [47]. Commonly in engineering practice, only the first-harmonic approximation is used, which is known as the describing function (DF) method. The main concepts of this method were developed in the 1940s and 1950s [36, 51, 52, 63, 71, 95]. Comprehensive coverage of this method is given in [8, 50]. The DF method provides a simple and efficient, but not exact, solution of the pe- riodic problem. Its applications are limited to systems that can be separated into nonlinear and linear parts interconnected into a closed loop (Fig. 1.1) with the nonlinear part being a static nonlinearity (with possible hysteresis) and the linear part given by linear time-invariant differential equations. More- over, the linear part must be a low-pass filter to filter out higher harmonics, so that the input to the nonlinearity could approximately be considered a sinusoid. Let us consider the frequency-domain approach to analysis of self-excited oscillations. Consider the following nonlinear system, ẋ = Ax + Bu y = Cx (1.1) Fig. 1.1. Block diagram of the system for DF analysis
- 22. 1.2 Fundamentals offrequency-domain analysis of periodic motions 7 Fig. 1.2. Poincaré mapping u = f(σ), (1.2) σ = −y, (1.3) where A ∈ Rn×n , B ∈ Rn×1 , and C ∈ R1×n are matrices, x ∈ Rn , y ∈ R1 , σ ∈ R1 , f ∈ R1 , f(σ) is a single-valued symmetric monotone nonlinear function: f(−σ) = −f(σ) (it will be later extended to a hysteretic function). Formula (1.1) provides a description of the linear part, formula (1.2) denotes the nonlinear part of the system, and formula (1.3) gives the condition of closing the system loop. We shall also use the linear part description in the form of the transfer function W(s), which can be obtained from formulas (1.1) as follows: W(s) = C(Is − A)−1 B. Let us also assume that the linear part is strictly proper, i.e., the relative degree of W(s) is one or higher. The problem of finding a periodic solution of system (1.1), (1.2), and (1.3) is usually formulated as the problem of finding a fixed point x∗ of Poincaré map, which will be a solution to the equation: x = P(x), where P(x) is the mapping. Figure 1.2 provides two examples of Poincaré mapping. P1 maps point x1 into point x2, and P2 maps point x3 into itself. Therefore, x3 = x∗ . In the time domain, the closed orbit that starts from the point x∗ (denote it x∗ (t) for brevity) will be a periodic vector signal, and y∗ (t) = Cx∗ (t) will be a periodic scalar signal. Now consider the open-loop system (1.1), (1.2), which can be obtained by ignoring equality (1.3). Let σ(t) be a periodic symmetric signal of period T with zero mean, continuously differentiable with respect to t, so that σ(t + T/2) = −σ(t), σ ∈ R, t ∈ D ⊂ R, D := t ∈ [0; T]. Consider the following lemma, which will be instrumental below. Lemma 1.1. If σ(t) satisfies the conditions of symmetric periodicity and uni- modality and f(σ) is a symmetric monotone nonlinear function, then x(t) will be a periodic vector signal, and y(t) will be a periodic scalar signal. Proof. u(t) is a periodic signal because it is a function of f(σ): u(t + T) = f(σ(t+T)) = f(σ(t)) = u(t). y(t) as well as x(t) are periodic because they are responses to the periodic input u(t), which follows from linear system theory.
- 23. 8 1 Theservo problem in discontinuous control systems Definition 1.2. Let us call the mapping of space R×D into itself, where σ(t) satisfies the conditions of symmetric periodicity and unimodality, the periodic signal mapping (PSM) y(t) = g(f(σ(t)), (1.4) which is a chain-rule application of two mappings: g for the linear part and f for the nonlinear part. Definition 1.3. Let us call the periodic function σ∗ (t), which provides the solution to the equation −σ(t) = g(f(σ(t)), t ∈ D, the fixed point of the periodic signal mapping (the minus sign is attributed to the negative feedback in the closed-loop system). The following theorem relates periodic solutions in the state-space and time domains. Theorem 1.4. If σ∗ (t) is a fixed point of the periodic signal mapping y(t) = g(f(σ(t)) in the open-loop nonlinear system (1.1), (1.2) where f(σ) is a single- valued symmetric monotone nonlinear function, then x∗ (t) = h(f(σ∗ (t)) will be a fixed point of the Poincaré mapping in the closed-loop system (1.1), (1.2), (1.3), where h is mapping u(t) → x(t). Proof. Because σ∗ (t) is a fixed point of the periodic signal mapping (1.4), σ∗ (t) provides a solution to system (1.1), (1.2), (1.3). This is a periodic so- lution, as the property of periodicity is “embedded” in the type of function σ(t). According to Lemma 1.1, x∗ (t) will be a periodic vector function, which means that in the state space, the trajectory that starts from x∗ is a closed orbit, i.e., satisfying the condition of being a fixed point of the Poincaré map. Let us analyze how the PSM can be implemented in practice. Transfer the analyzed problem including the defined mappings into the spectral domain, which can be provided via taking the Fourier series of the signals in the system. Represent the periodic signal σ(t) in the form of the Fourier series as follows, σ(t) = 2 ∞ k=1 |qσk| cos ((2k − 1) ωt + arg qσk) (1.5) where ω = 2π/T, and all even harmonics are zeros due to σ(t) being symmetric qσk = 1 T T/2 −T/2 σ(t) [cos(2k − 1)ωt − j sin(2k − 1)ωt] dt. (1.6) Therefore, any σ(t) can be characterized by the matrix Qσ as follows: Qσ = [qσ1 qσ3 qσ5 . . .] subject to the frequency ω being given. We shall call the matrix Qσ the spectrum of the signal σ(t). The number of elements of Qσ can be set arbitrarily large, but it makes sense to limit the number of elements
- 24. 1.2 Fundamentals offrequency-domain analysis of periodic motions 9 to some reasonable value, so that any further increase of this number will not provide any noticeable increase in the accuracy of the σ(t) representation. Let us represent other periodic signals u and y in the system by their spectra: Qu, Qy. Then the mapping Qσ → Qu can be given by the transformation into the time domain as per (1.5), nonlinear mapping in the time domain as per (1.3), and transformation into the spectral domain through the following formulas: quk = 1 T T/2 −T/2 u(t) [cos(2k − 1)ωt − j sin(2k − 1)ωt] dt (1.7) Qu = [qu1 qu3 qu5 . . .] . (1.8) The mapping Qu → Qy can be given by the following element-by-element multiplication of the two matrices, which directly follows from linear systems theory: Qy = Qu • S (1.9) where Qy = [qy1 qy3 qy5 . . .] (1.10) S = [W(jω) W(j3ω) W(j5ω) . . .] . (1.11) According to Theorem 1.4, the periodic solution of (1.1), (1.2), (1.3) can be found as a fixed point of the PSM. We have defined the PSM as the chain application of two mappings: the first of which is provided by (1.5), (1.2), and (1.7), and the second is given by (1.9). To find the fixed point, we have to design a contraction mapping, so that iterates from the initial point will converge to the solution. Assume now that the linear part is an ideal low-pass filter. This assumption enables one to rewrite (1.9) as Qy = qu1W(jω) (1.12) and (1.5), (1.2), (1.7) as just qu1 = 1 T T/2 −T/2 u(t) [cos ωt − j sin ωt] dt (1.13) with σ(t) = aσ cos ωt. The ratio qu1/qσ1 = 0.5aσ gives the describing func- tion (DF) of the nonlinearity. As a result, considering (1.3) we can write the harmonic balance equation for the DF approximated system as follows: N(aσ)W(jΩ) = −1 (1.14) where N(aσ) is the describing function of the nonlinearity, aσ is the amplitude of σ(t), and Ω is the frequency of the periodic motion:
- 25. 10 1 Theservo problem in discontinuous control systems N(aσ) = qu1 qσ1 = 2 Taσ T/2 −T/2 u(t) [cos ωt − j sin ωt] dt. Equation (1.14) is a well-known formula [8, 50] that is widely used for analysis of periodic motions. Therefore, the DF analysis can be viewed as a fixed point of the PSM considered above, subject to the assumption about the low-pass filtering property of the linear part. Equation (1.14) provides a basis for analysis of possible periodic motions in a nonlinear system via application of the DF method. Usually this equation is presented in the following form: W(jΩ) = − 1 N(aσ) (1.15) which is convenient for graphical interpretation, as the unknown variables are separated into the left-hand and the right-hand parts of the equation. 1.3 Relay servo systems Usually, control systems are categorized into two groups depending on the control problem they are supposed to solve. These two categories are stabi- lization systems and servo systems. Stabilization systems carry out the task of stabilization of certain process variables. The main challenge for such systems is producing a compensating action for the disturbances that inevitably oc- cur. The servo systems implement the task of tracking certain input signals, so that the system output follows the input signal as precisely as possible. Normally, the input to the system is unknown, but some a priori informa- tion about the input signal such as frequency and amplitude range may be available. The typical examples of stabilization systems are temperature, pres- sure, level, flow control systems; and examples of servo systems are actuators, airplane autopilots, and temperature control systems, which are supposed to ensure the temperature change in time according to a given profile. We can see from the above examples that the autonomous mode, when no external signals are applied to the system, does not normally occur. What is common for both these types of control systems is the existence of external signals that affect the functioning of the system. In the first case, this is a disturbance to which the system is supposed to respond in such a way as to provide its compensation. In the second case, the system is supposed to respond to the input to bring the output in alignment with this external input. In both cases, the problem analyzing the effect of external signals on the system characteristics is the most important part of system performance analysis.
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- 27. themselves the revengingjustice of God, from verse 10. to the end of the Chapter. In the description of the false teachers he makes, 1. A comparison of the like betwixt the false teachers and those that were false prophets among the people of Israel, partly that he might joyne together this discourse with the former, where he spake of true Prophets; and partly that he might meet with the scandall which might arise by such seducers; and partly that he might stirre up the faithfull so much the more, to beware of their seducing. In the first respect, the last words of the former Chapter, and the first of this Chapter, make such a discreet axiom as this; although there were many Prophets among the People of Israel, that were holy men of God, and were moved by the Spirit of God in those things that they spake; yet among the same people also there were many false Prophets. This discretion is shewed in these words: But there were. In the second respect these words make a copulate axiom thus: Like as there were false Prophets among the People of Israel, so also shall there be false teachers among you. In the third respect there is intimated and implicitely contained such a connex axiom as this: If there shall be false teachers among you, like as there were false Prophets among the people of Israel, then must you be alike careful to beware of these, as the Israelites were warned heretofore to beware of them. After that description which is made by way of comparison, followes an absolute description; wherein the false teachers are described by their effects. Which effects have either an immediate reference unto the false teachers themselves, or unto others by them. Those that have reference unto the false teachers themselves, are 1. The bringing in of heresies, which are set forth by their adjunct, that they are damnable, whereof he gives a reason also, taken from the nature of heresies, which is said to be such, that it brings in a denying of the Lord. 2. The second effect is, that they bring upon themselves swift destruction. In respect of others the first effect is, that they shall have many followers, who are pointed out by another effect accompanying this seducing, namely, a speaking evill of the truth, or of the Gospell, which shall come to passe by their meanes. The other effect is the abuse of those whom they do seduce for gain; which abuse is set forth by the helping cause, namely by fained words.
- 28. The Doctrines arisingherehence. Doctrine 1. There are alwayes false prophets and false teachers in the Church. This is gathered from these words: there were, there shall bee. Reason. This comes to passe in a threefold respect, 1. in respect of God. 2. In respect of the Devill. 3. In respect of men. First, in respect of God, and that 1. that they which are approved in the Church may be made manifest, 1 Corinthians 11.19. Deuteronomy 13.3. 2. That the obstinate wicked may perish in their ungodlinesse and cruelty, 2 Thessalonians 2.10,11,12. 1 Kings 22.20,21,22. Secondly, in respect of the Devill, because he is alwayes the father of lyes, and a murtherer, and that envious one, which soweth tares among the good seed. Thirdly, In respect of men, because the wisdome of the flesh is enmity against God and his truth. The use hereof according to the Apostles intention is twofold: Vse 1. To admonish us, not to be offended, nor to be too much troubled in minde, when we see heresies to arise in the Church, but to say as the Apostle did, There were, there shall be. 2. To instruct us, not to believe every spirit, but to try whether they are of God, 1 Iohn 4.1. Doctrine 2. Every heresie hath damnation joyned with it. This is gathered from these words: damnable heresies. Some are more damnable then others, yet all do in their nature tend to damnation. For all are contrary unto the way of truth, whereof mention is made, verse 2. which way alone leads unto life. But there are some heresies which can by no meanes consist with salvation or eternall life. Hereupon such opinions are by a kind of appropriation called heresies in the Church, that is, opinions altogether damnable. For there is a difference to be observed betwixt things rashly spoken, bare errors in faith, and heresies properly so called. In the heresies
- 29. themselves also wemust distinguish betwixt the materiality, and the formality of it. For to make a formall heresie, for which a man is properly called an heretick, there is required besides an errour in those things that pertaine unto the fundamentals of religion, a reluctation against the cleare light, and obstinacy in that reluctation. Vse. This may serve to condemne those that under the name of peace and charity, make so little reckoning of the bringing in of heresies into the Church. They would have all things received in and tollerated, which is all one, as if they would for peace and charity admit of damnation it selfe. For heresies have damnation joyned with it. 2. To admonish us, 1. to take diligent heed that we do not by any means further such heresies. 2. Not rashly or for a light cause to pronounce any man an heretick. 3. As carefully to beware of heresies, as we should of damnation. Question. In the reason which is here given of this property of heresies, that they are damnable, a question is moved, What is meant by denying the Lord? denying the Lord that bought them. For many gather from this place, that all and every particular man were redeemed by Christ, because it is here said that he bought even those wicked hereticks: but it may be answered divers wayes. Answer 1. That these false teachers were members of the visible Church, and therefore, 1. they accounted themselves such as were redeemed by Christ, and made shew of that profession. 2. Men were to account them such, as long as they continued in the Church; and in this respect they may be said to be such as were redeemed by Christ; like as the Apostles are wont to call whole Churches justified, sanctified, and elect. There is not the same reason therefore of all and every particular man, as there was of these. Answer 2. He doth not here speak of Christ properly, as he is Redeemer, nor of the redemption of his blood, but of that right, whereby masters chalenge their servants, as having bought them; which may be thus shewed: 1. Here is no mention of Christ as being
- 30. Lord properly, butMaster. For in the Greek it is δεσπώτης, which is properly, a Master in respect of a servant, not κώριος, Lord. And these two names are clearly distinguished, Iude verse 4. where the same thing is spoken of the same false teachers: there they are said to deny their Dominum herum, Lord and master; although he may be called their Herus, Master, he is not properly called their Dominus, Lord. Then againe, the Lord is not said to have redeemed all men, but only to have bought them: now although the word ἀγορώζω, to buy, be sometimes used in the same sense that the other is, yet it is not alwayes so. Answer 3. The comparison here is plainly made betwixt the false teachers and the false Prophets, ♦betwixt the visible Church and the people of Israel; for the Lord is said to have bought those men in the same sense, that he is said to have redeemed and delivered all the people of Israel; that is, as he delivered them out of Egypt, and challenged them to himselfe in a peculiar manner, that by a solemne covenant they should be his servants; so also all those that give their names unto the Church, he may in this respect be said to have bought, as a Master, that they should serve him for ever. To deny the Lord is, not to acknowledge God or Christ for the Lord, Matthew 10.32,33. to confesse and to deny are made contrary one to another, Luke 12.8,9. The reason is, because it hath once professed the name of the Lord, and afterwards departs from that profession, it is all one as if he did plainly say, that he would no longer be a servant of that Lord. For this denying is twofold, either verball or reall. A verball denying was in Peter, a reall denying is in all those that after they have made profession of the faith, return unto a profane life, 1 Timothy 5.8. 2 Timothy 3.5. The denying in this place is properly meant of a reall denying. For the description of these men doth rather expresse their profane courses, then their open renouncing of Christ. ♦ “bewixt” replaced with “betwixt”
- 31. Doctrine 3. Itis a damnable impiety to deny Christ either in word or deed. Reason 1. ♦Because by this means Christ is greatly dishonoured; for men, our lusts, and Sathan himselfe are preferred before him, Mark 8.38. where the indignity of this sinne is shewed, that an adulterous and sinfull generation is preferred before Christ. Christ also by this means is accused of iniquity: for no servant usually goeth away from his master, if he confesseth him to be a just master; therefore this going away is as it were a professing that he doth accuse that master from whom he goes of some iniquity. Hereupon was that contesting of the Lord against his rebellious people, that he was a just and right master, neither did he give them any cause to depart from him, Ieremiah 2.5. Micah 6.3. Deuteronomy 32.4. ♦ “Becawse” replaced with “Because” 2. Because Christ threatned a most heavie judgement against this sin, when he saith that he will deny those men; that he wil be ashamed of them before God and his Angels, Mark 8.38. Vse 1. This may serve to condemne, not only the manifest Apostates, but also all those, that with the profession of godlinesse joyne a profane life, or turn aside from that profession. The sinne of these men is more grievous, then of Turks and Infidels, that never gave their names unto Christ; for the neerer any man commeth unto God, the more doth he dishonour his name, if he doth fall back againe. He that hath in some sort forsaken Sathan, and given himselfe up unto God, and afterward forsakes God again, and gives himselfe up unto Sathan, seems to have known both masters, and advisedly to pronounce that he is the better, whose servant he would rather be, that is, Sathan. 2. To admonish us, 1. Carefully to beware, not only of the damnable sin it selfe, but also of all appearance, and every degree of it, and every occasion that might leade us thereunto. 2. If in any
- 32. respect we haveturned towards such like abomination, to recall our selves betimes, and seriously to bewaile our offence, like as Peter did after his denying of Christ, Matthew 26.75. Doctrine 4. It is the nature of such sinnes to bring destruction upon men. This is gathered from these words; bring upon themselves destruction. Reason. Because sinne in respect of the guilt that is joyned with it, hath a kind of power whereby it drawes all evill; like as faith and prayers obtain every good gift from God. Hence is it, that our sins are said to cry for vengeance; and also to draw punishment unto it as it were with strong cords, Isaiah 5.18. For while men draw sin with such-like cords, they do also draw the punishment of sinne, which is joyned unto it. Vse 1. This may serve to admonish us, in this respect to beware of sin. 2. To exhort us, by faith, repentance, and prayers, to turn away that destruction from us, which our sins would bring upon us. Doctrine 5. The destruction which hangs over the heads of grievous offenders will come swiftly. This is gathered from these words; swift destruction. Doctrine 6. Men are wonderfully prone to imbrace errors. This is gathered from these words; And many shall follow. Reason 1. By reason of that darknesse, which hath taken hold of mens mindes; thereupon they may be easily drawne under any shew or glimps of light. 2. By reason of that disposition which they have, alien from the truth; thereupon they easily imbrace those things that are contrary to
- 33. this truth. 3. Byreason of that disposition which they have, agreeing with the nature of errors. For sinners have in themselves the seeds, of all sins, errors and heresies; so that the nature of corrupt man doth as easily receive errors, as the tinder fire. 4. By reason of the just judgement of God, whereby for the punishment of other sins they are given over to the power of errors, 2 Thessalonians 2.10. Vse 1. This may serve to instruct us, to shew the corruption and wretchednesse of our nature, how hardly are we brought to imbrace the heavenly and saving truth, when we are carried headlong unto damnable errors. 2. To admonish us, that it should not be an offence unto us, if we see many given unto errors; because this was foretold us by the Apostle, and it hath a manifest reason; so that it is not to be wondred at, that many are given unto errors, but that rather, that any one doth believe the truth. 3. To exhort us, 1. Unto humiliation, in respect of this pronenesse unto error, 2. To care and circumspection that we be not overswayed by it. And an Argument whereby we may and should be stirred up unto this care, we have in the text, whiles they which follow heresies and hereticks, are said to follow their destruction. For not only the authors of heresies run unto destruction, but also their followers, as in the place of the Thessalonians before cited. In those which perish, so that it is a signe of destruction, to be seduced by hereticks. Doctrine 7. The fruit of errors and heresies is speaking evill of the way of truth. This is gathered from these words, By reason of whom the way of truth shall be evill spoken of. By the way of truth is meant the Gospell, or the will of God revealed in the Scripture, which shewes the true way of salvation. It is said to be evill spoken of, when the
- 34. dignity thereof isviolated, most properly by reproachfull words. It is said to be evill spoken of by them, namely, by the authors of the heresies and their followers, in a two-fold respect, 1. When they themselves speak reproachfully of it, as the Scribes and Pharisees did of Christ. 2. While they give others occasion to speak reproachfully of it, as the Jewes did unto the Gentiles, Romans 2.24. both respects seeme to be meant here by these, for the Greek word διὰ signifies per propter, by, and by reason of. Vse 1. This may serve to instruct us, to judge aright of the nature of sinne, that never staies till it hath brought men to blasphemy against God, and to speak evill of the way of God. 2. To admonish us, 1. For this cause to withstand the beginnings of sin, lest at length we be brought also to blasphemy against the holy Ghost, which Christ saith is a sin that shall never be forgiven. 2. To beware also of those men, that durst openly speak evill of the truth; amongst the number of whom they are to be put, that dare mock and scoffe at the name of predestination and the predestinate. Doctrine 8. They which propagate errors under faire pretence, do often times seek their own gaine. This is gathered from these words; through covetousnesse with fained words: Covetousnesse is commonly the companion of heresies, oftentimes the mother, very often the Nurse of them, as we may see in the Papists. Vse. This may serve to admonish us, 1. Not to believe mens fained words. 2. To beware of covetousnesse, which is the root, as of all evils, so also of heresies. Doctrine 9. Miserable is the condition of those men that are seduced by hereticks; for they are sold like beasts and cattell. This is gathered from these words, they shall make merchandize of you.
- 35. H Verse 3, atthe end. Whose judgement now of a long time lingreth not, and their damnation slumbreth not. The Analysis. itherto we have seen the description of false teachers, and their sins; now followes their reproofe, or the punishment for their sins, that shall certainly befall them. This punishment is set forth, 1. Metonymically, by the cause, in that it is called judgement. 2. By the effect, in that it is called damnation. 3. By the adjunct of swiftnesse and speed, which is shewed by the negation of those things that are opposed unto swiftnesse. Now in generall, lingring or slownesse is opposed thereunto; in speciall, slumbring, which is put as a species of idlensse or slacknesse. This swiftnesse of the punishment that shall befall them, is set forth by the adjunct of time, that it did not now first of all begin to make hast, but now of a long time. The Doctrines arising herehence. Doctrine 1. The punishment of sinne followes after sinne. This is gathered from the connexion of these words with the words foregoing: there was the description of sin; and here presently followes the threatning of punishment. Vse. This may serve to admonish us, never to think of sin, but to adjoyne also the consideration of the punishment, which is joyned unto sin. Doctrine 2. This punishment proceeds from the just judgement of God; for that it is here called judgement. For God as judge of all the earth, pronounceth sentence upon men for their sins, and the execution of that sentence is called punishment. Now he pronounceth sentence according to right and a most holy law, and therefore the punishment which God inflicteth is in a proper sense called judgement.
- 36. Vse 1. Thismay serve to refute, 1. prophane men, which think that those evils that men suffer, both publick and private, come by chance. 2. Those that are wont to complaine of the evils which they suffer, as if God did deale unjustly with them, Genesis 18.25. Deuteronomy 32.4. 2. To admonish us, continually to walke in the feare of the Lord, because we live in his sight, that exerciseth judgement. Doctrine 3. This judgement of God brings damnation upon sinners, that is, a privation of all good, which pertaines unto the happinesse or well-being of man, and the sense of all misery; which makes that very being bitter and unpleasant. Vse. This may serve to admonish us, with feare to flye away from sin, and to seek the face of God. Doctrine 4. This damnation comes swiftly upon sinners. Reason 1. Because it comes long before their expectation of it. 2. Because it is not deferred beyond the time which is most fit, and appointed by God, and in that respect it is said not to linger. 3. Because no impediment can hinder it, but it is deferred through Gods mercy and long-suffering; in that respect it is said not to slumber, because slumbring and sleepe bindes the mind, and hinders the actions. 4. Because now of a long time before, way is made and prepared for this damnation. 5. Because the damnation it selfe, although the wicked oftentimes are not sensible of it, yet it takes hold of them, and spiritually worketh in their minds and consciences. Vse 1. This may serve to condemne those, that give themselves wholly unto sin, either because they think that punishment commeth
- 37. N slowly, or elsebecause they hope to escape unpunished, Ecclesiastes 8.10,11,12. 2. To admonish us, to make haste to repent of our sinnes, not to linger nor slumber. Doctrine 5. In all those things which pertaine unto judgement against sinners, like as in all other things, God doth nothing rashly, but foreseeth, premeditateth, and prepareth all things. This is gathered from the adjunct of time, Now of a long time. So Iude verse 4. Verse 4. For if God spared not the Angels that sinned, but cast them downe to hell, and delivered them into chaines of darknesse to be reserved unto judgement. The Analysis. ow the Apostle proves that which he had before laid downe concerning the punishment that should befall false teachers. The conclusion of this arguing is, 1. In generall propounded of all the wicked, verse 9. For there is the consequent part of that connex axiom, the antecedent whereof begins verse 4. and is shewed by the connexion If, as, If God spared not the Angels, c. Then afterwards in speciall, of these false teachers themselves and their followers, verse 10. In the generall conclusion, verse 9. he doth not barely set downe the condemnation of the wicked, but also by a copulate axiom joynes together with it, the deliverance of the godly from that temptation, which ariseth unto them by means of the wicked. And the reason is, because the Apostles purpose was, so to reprove the wicked, that he might in the mean time comfort the godly, and strengthen them against all the temptations, whereunto they were obnoxious by means of the wicked. In the antecedent part of this arguing he sets downe three arguments, which are as so many examples judgements of God against the wicked; they may make three syllogismes to prove one and the same conclusion. The
- 38. first example isof the Angels that fell, verse 4. The second is of the old world, verse 5. The third is of Sodome and Gomorrha. The syllogismes are knit together, after this manner: If God spared not the Angels, nor the old world, nor Sodome and Gomorrha; then he will not spare these false teachers, nor the like wicked men; But the former is true, by the testimony of Scripture, Therefore, c. These examples in respect of the generall conclusion, which is set downe, verse 9. are as species unto their genus. But in respect of the speciall conclusion wᶜʰ is set down, verse 10. they are as comparata, partly alike, partly greater. That wherein they are alike, consists therein, as God condemned those sinners, so also will he condemn these. Their inequality therein; if God spared not his most excellent creatures, the Angels, nor the old world, nor those cities; then much lesse will he spare these false teachers. The Doctrines arising herehence. Doctrine 1. The judgements of God which he hath executed from the beginning of the world were ordained to be examples for us. This is evidently gathered from all this arguing of the Apostle, and it is plainly taught us, verse 6. So 1 Corinthians 10.6. they are called types, that is, ensamples; which appears also by verse 11. which sense of the words Beza and others have missed. Reason. The reason of this truth is Gods unchangeable nature and constant justice, whereby he is alwayes like himself, not only inwardly and in himselfe, but also outwardly towards us in the dispensation of his punishments and rewards, if we look to the substance of God. Hence is it, that by this argument the godly are both confirmed in their hope, Isaiah 59.1. 9.9,10. and assured of the destruction of all their enemies, and of all those that give themselves over unto impiety. Vse 1. This may serve to instruct us, so to read the sacred History, not as we read profane histories and fables, either for delight, or to be informed only in the knowledge of the times, or in
- 39. things that havebin done; but alwayes to observe Gods warnings and apply them unto our own use. 2. To admonish us, not to deceive our selves, as if we had a priviledge to sin above other men, or as if God ♦would not deale so severely with us, if we should continue in our sins, as he did heretofore with those whom he destroyed: *Other mens harmes should teach us to beware of the like sins. In the first example we have, first, The object of Gods judgement, The Angels that sinned. Secondly, the manner how he dealt with them, If he spared them not. Thirdly, the degree of this judgement, present, and to come: present, that they were adjudged unto damnation; and to come, that they were to be reserved unto judgement. The sin of the Angels is not particularly set downe in the Scriptures, because it doth not so much pertaine to our edification and salvation, to know their sin, as it doth to know our own sins. But in generall we are taught, that they kept not their first estate. Iude verse 6. that is, they left that estate wherein God had placed them, and afterwards exercised envie, lying, murther towards men. We are taught also that the number of those that fell was great; thence it is that the Apostle here speaks in the plurall number. They are said to have been cast down to hell, 1. By reason of the change of their estate, because from that wonderfull high condition, which they had received by creation, they were cast downe to the lowest of all. 2. By reason of the change of their place, because from the place of blisse, where they were round about the throne of God with the other Angels, they were thrust downe into a lower place fit for sin and misery: But that this place is in the lowest parts of the earth, as the Papists would have it, it cannot be shewed out of the Scriptures, but rather the contrary: for they are said to be in the aire, and to rule there, and to go about the earth seeking to devoure men. This at least is plaine out of the Scripture, and it should suffice such as are not over-curious: 1. That they suffer a great change of estate. 2. That they are excluded from their first habitation. 3. That they are in that place, where they receive pœnam damni sensus, the punishment of losse, and the punishment also of sense. They are said to be delivered into darknesse, partly in respect of sin, partly in respect of misery: for
- 40. both are signifiedby darknesse in the Scriptures. They are said to be delivered into chains, by a metaphor taken from malefactors that are condemned, who are kept in prison bound with chains. Now these chains are, 1. A hardning in their sins. 2. A despaire of all deliverance. 3. A fearfull looking for that misery whereof mention is made, Hebrews 10.27. 4. The providence of God which alwayes watcheth over them, to keep and punish them. They are said to be reserved unto judgement, because they are bound so fast with these evils, that they can never escape; and yet these are but the beginnings of the evils, which they must farther undergo. ♦ “wold” replaced with “would” Doctrine 2. There is no dignity that can exempt a sinner from the judgement of God. This is gathered from the nature of the Angels. Reason. Because dignity doth not lessen the sin, but aggravate it. For he that hath received much, owes much, and of him much is required. Use. This may serve for admonition, that men should not trust to such staffes of reed. Doctrine 3. That severity is due unto sin, that sinners should not be spared. This is gathered from these words, He spared not. Reason 1. Because sinners alwayes abuse the goodnesse of God. 2. They spare not the glory of God, but themselves. Vse 1. This may serve to instruct us, to ascribe it unto the grace and mercy of God, that he hath hitherto spared us.
- 41. 2. To exhortus, not to contemne this long-suffering of God, but thereby to be led unto repentance, Romans 2.4. Doctrine 4. By sinne a sinner is cast downe from the place and condition which he had before. This is gathered from these words; cast down to hell. Reason. Because sin is an aversion from God. Now as the communion which we have with God, makes us to ascend into a high condition, even into Heaven it selfe, so that our conversation is in heaven, as it is Philippians 3.20. So also aversion from him makes us to descend, even untill we come unto hell it selfe. Vse. This may serve to admonish us, not to trust the devill, nor our own imaginations, as if we could ever attain any good by sinning. For so our first parents were deceived, when they thought to make themselves like unto God by sin, they were made like unto the devill. Doctrine 5. The darknesse of misery followes the darknesse of sinne. This gathered from these words; delivered them into chaines of darknesse. Reason. Because the light is the same, whereby we are directed, and whereby we are perfected; therefore the privation of perfection and happinesse followes the privation of direction. Use. This may serve to admonish us, alwayes to walk in the light, as children of the light, if we desire to avoid utter darknesse. Doctrine 6. Sinners are held in sinne and in their misery, as it were with bonds. This is gathered from that: Chaines. Reason. Because the sins themselves are the snares of the devill, 2 Timothy 2.last. Lamentations 1.14. There are other reasons
- 42. T given of thisthing in the Analysis. Vse. This may serve to admonish us, betimes to pray for deliverance from our sins, which then we shall obtaine, when we do flye unto Christ, who gives deliverance unto the Captives; Luke 4.18. Doctrine 7. The extreamest punishment of sinners is deferred unto the last judgement of God. Use. This may serve for instruction, that no man should please himselfe therein, that he suffers afflictions in this world, as if therefore he should escape free in the world to come, which is the miserable comfort of some men. Verse 5. And spared not the old world, but saved Noah, the eighth person, a Preacher of righteousnesse, bringing in the flood upon the world of the ungodly. The Analysis. he second and third example of Gods justice towards sinners are not barely propounded, as the first was, but are illustrated by different examples, or by examples of Gods mercy, which he shewed towards the godly, while he destroyed the wicked. The reason is, that both parts of the conclusion, which is set down, verse 9. might be fitly inferred; where the deliverance of the godly is joyned together with the destruction of the wicked. In the former of these examples is set downe, 1. The object of Gods justice, materially, the old world, which is more formally explained by the adjunct of impiety, whence it is called the world of the ungodly. 2. This justice is also illustrated by the instrument thereof, namely, the bringing in of the flood upon the world of the ungodly. The object of Gods mercy is made to be Noah with his family: who are described, 1. By their small number, wherein his family was contained, which is said to be eight. This Peter shewes, 1 Epistle chapter 3. verse 20. 2. By the effect, that he was a Preacher of righteousnesse. For the office or duty is not here commended, but the act or exercise of this duty. For the duty was
- 43. common to himwith many more, but the exercise of it was proper unto himselfe. The Doctrines arising here-hence. Doctrine 1. The multitude of those that sinne doth not hinder Gods justice from executing his judgements upon them for their sinnes. This is gathered from this; He spared not the whole world. For as it was with dignity in the Angels, so it is also with the multitude now in the world. Reason 1. Because the multitude doth not lessen, but rather increase the guilt of sinne, and more fill up the measure thereof, as it is in the Proverbe, Qui peccat exemplo, bis peccat, he that sins with example, sins twice: So he that sins with many examples, sins often times, if the thing be rightly considered. As if a man should see another fall in a slippery way, and not take the more heed unto himselfe therein, but fall into the same place and after the same manner, he is much more taxed by men, then if he had fallen alone, or first of all: so is it also in these. 2. Because the multitude which seemeth unto us a great thing, as if it could withstand Gods anger, is before God like unto water, that fals from a bucket. Vse. This may serve to admonish us, not to follow a multitude to do evill. And so much the more necessary is this admonition, because we are all too prone to go in the way of the multitude; which fault also we seeme to commit with some reason. 1. Because that we presume that many eyes see more then one, whence it is that the voice of the people is often taken for the voice of God. 2. Because the labour of examination seemes too difficult, and almost impossible.
- 44. 3. Because thatwhich all men do, usually goes unpunished among men, and we conceive the like thing of God. 4. Because he that doth not follow the multitude, shall gaine a multitude of enemies, and such as hate him. But the contrary reason, which is of greater value then all those, is that of Christ, Matthew 7.13. Broad is the way that leadeth to destruction, and many there be which go in thereat. To the first reason or objection we answer; that one eye of a man that seeth, perceives more, then a thousand blind eyes. The world or the multitude of the world is blinded; therefore one Christian which hath the eye of his mind inlightned by the Spirit, and followes the light of Gods word, can see more then a great multitude of others. To the second we answer, that the duty of examination lies upon all and every one. And they had need to do it, not only as they are Christians, and as Christ commands them, to prove all things, and hold fast that which is good, but also as they are men: for it is not a humane, but a brutish thing to follow the multitude of those that go before, without judgement. To the third we answer, that the wayes of God are not as the wayes of men: Men winck at many, either out of ignorance, or impotency, or for feare, and the like imperfections which are not incident to God. To the fourth we answer, If God be with us, who can be against us? Doctrine 2. The creatures which are most necessary and profitable unto us, by sin are made our enemies and hurtfull unto us. This is gathered from the flood of waters, that destroyed the old world. Now nothing is more necessary and profitable unto us then water. Reason. Because the creatures are the servants and hoast of God, and therefore they depend upon his will, and pleasure, and command both for war and peace, Hosea 2.18. Vse 1. This may serve to admonish us, so much the more to beware of sin.
- 45. 2. To instructus, if we would have any true comfort from the creatures, to lay the foundation thereof in our reconciliation with God. Doctrine 3. God in the midst of judgement towards the ungodly, is mercifull towards the godly. This is gathered therehence, that when he destroyed the world, he saved Noah. Reason. Because God doth not execute judgement according to the fiercenesse of his anger or fury, as men oftentimes use to do, but according to the counsell of his most perfect wisdome. Use 1. This may serve to instruct us, not to ascribe those things unto fortune or chance, which happen unto us or others in this kind. 2. To comfort us, because in a world of all dangers and judgements, we may flye unto the mercy of God. Doctrine 4. There are wont to be but a few godly men in the world. This is gathered therehence, that Noah is called the eighth of the godly, or rather he may be called the seventh, because Cham was cursed. Reason 1. In respect of God, because he gives this grace but unto a few, that his goodnesse and mercy towards them may appeare the more abundant. 2. In respect of the second causes, and of the nature of the thing it selfe, because that which is most difficult, and of the greatest care and perfection, is not wont to be found but in few. The gifts of nature are common unto all; of art and industry to fewer, but the gifts of grace to fewest of all. There are more good Taylors, then good Physitians, and more good Physitians then good Kings. So that that hath the greatest perfection, which is given but to a few.
- 46. Vse 1. Thismay serve to admonish us, to be so much the more diligent to make our election and calling sure. 2. To exhort us, to give God thanks, that he hath made us partakers of that happinesse, which he grants but to a few. Doctrine 5. All the godly, that will be made partakers of Gods mercy, must be preachers of righteousnesse, as Noah was. Now righteousnesse is preached not only in words, but also in deeds. For so it is wont to be said, The thing it selfe speaketh, not the tongue only, Philippians 2.15,16. Reason. Because no man can escape the revenging justice of God, unlesse he labour to follow and advance the righteousnesse that is prescribed by him. Vse 1. This may serve to condemne and reprove those, that are rather Preachers of unrighteousnesse, then of righteousnesse. 2. To exhort us, more and more to stir up our selves to have a care of this duty. Verse 6. And turning the Cities of Sodome and Gomorrha into ashes, condemned them with an overthrow, making them an ensample unto those that after should live ungodly. Verse 7. And delivered just Lot, vexed with the filthy conversation of the wicked: Verse 8. (For that righteous man dwelling among them, in seeing and hearing, vexed his righteous soule from day to day, with their unlawfull deeds.) The Analysis.
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