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Decentralized Control of Systems Using Switching Methods
Amir G. Aghdam
A thesis submit ted in conformity wit h the requirements for the Degree of Doctor of Philosophy
Graduate Department of Electrical and Computer Engineering Lniversity of Toronto
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To my mother
Abstract
In this thesis, the adaptive control of decentralized systems using switching control
methods is studied.
The motivation for studying this problem arises when the model of the system
being controlled is highly uncertain, i.e., when uncertainty in the plant model is suf-
ficiently large such that a fixed LTI controller or traditional adaptive controller is
ineffective to use. In this case, it is assumed that a dictionary of calculated decen-
tralized controllers has been obtained which has the propem that a t least one of
the controllers always satisfactorily controls the plant; a switching controller is t hen
found mhich sequentially applies a controller from this dictionary of controllers to
control the actual plant. Two approaches are investigated in this thesis. The first
approach requires that a set of plant models be given, and that the real plant rnodel
belongs to this set. The other approach only requires that a family of controllers to
exist with the property that a t least one of them satisfies the control objective for
the problem when applied to the actual system.
In determining a solution to this switching control problem, a study of digital
coatrollers for decentralized systems is made. Such digital controIlers can potentially
improve the overall performance of a decentralized control system, when a LTI contin-
uous controller is ineffective. Conditions under which a digital controller can improve
the performance of a decentralized continuous-time system are discussed, -41~0, it is
shown that using Generalized Sampled-Data Hold Functions (GSHF) instead of a sirn-
ple zero-order hold, in the implementation of the digital controllers, can significantiy
improve the overall performance of the resultant closed-loop system. In particular, it
is shown GSHF can potentially be used to modie the structure of the digraph of the
resultant discrete-time plant, by removing certain interconnections in the equivalent
discrete-time model, to hrrn a hierarchical system model of the plant, and in this
case a decentralized digital switching controller can immediately be irnplemented in
the resultant system.
Acknowledgment s
1 would like to sincerely thank my S u p e ~ s o r , Professor Edward J. Davison, for
providing useful insight in the area of decentralized control theory. 1 have been truly
honoured by his generosity, support and guidance throughout the course of my studies,
and 1 feel priviledged to have been his student.
My gratitude to the members of my Ph.D. thesis cornmittee, professors Daniel E.
Miller, J. D. Lavers, Bruce A. Francis, and Raymond H. S. Kwong for their valuable
advice.
1 am very- grateful for the support of al1 my friends, mentors and administra-
tive staff in ECE: Sarah Cherian, Linda Espeut, Bibiana Chang, Mehran Aliahmad,
Shahriar Mirabbassi, Ali Sheikholeslami, and Kelvin Loveless. Thanks are given to al1
my friends and colleagues in SCG. 1 would like to acknowledge Rashid Kohan, who
was highly supportive, Peyman Gohari, for his help and patience, Steven Postma,
and al1 the network admins for their assistance. To al1 SCG students and professors,
my deepest gratitude.
Many thanks to my wife, my father, and my sisters for their encouragement and
interest.. Without their understanding and support 1 could not have completed this
work.
Finally, and most importantly, 1 would like to thank my rnother whose courage,
faith, and patience !vas inspirational. Mom, my final oral examination was held on
the third anniversary of your passing. I know that o u were there, mom, this is for
VOU.
Contents
1 Introduction 11
1.1 Notation and -4bbreviations . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Decentralized Fixed Modes . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Digital Controllers for Decentralized Systems . . . , . . , . . - . . - . 14
1.4 Decentralized Adaptive Control via Switching . . . . . . . . . . . . . 15
1.4.1 Conventional -4daptive Control . . . . . . . . . . . . . . . . . 16
1.4.2 Switching-4daptiveControl . . . . . . . . . . . . . . . . . . . 18
1.4.3 A New Application for Discretization . . . . . . . . . . . . . . 19
1.5 Outline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Digital Control of Systems with Approximate Decentralized Fixed
Modes, Using Zero-Order Hold 21
2.1 Decentralized Fixed Modes (DFM) . . . . . . . . . . . . . . . . . . . 22
2.2 Sampling and Approximate Decentralized Fised Modes (ADFM) . . . 24
2.3 Theoretical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Decentralized Control of Systems, Using Generalized Sampled-Data
Hold Functions 45
3.1 Formulation of Generalized Sampled-Data Hold Functions (GSHF) . . 45
3.2 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 Pseudo-Decentralized Switching Control 59
B.2 Some Related Material From the Paper [12] "Adaptive Sivitching Con-
trol of LTI MIMO Systems Using a Family of Controllers -4pproach"
by hl. Chang and E.J. Davison . . . . . . . . . . . . . . . . . . . . . 150
C MATLAB Program 154
List of Figures
. . . . . 1.1 The structure of a digital controller as a time-varying system
. . . . 1.2 The configuration of an ideal decentralized switching controller
2.1 -4 decentralized system with two input-output agents . . . . . . . . . .
2.2 The input and output signals in a computer control system . . . . . .
2.3 -A decentralized control system . . . . . . . . . . . . . . . . . . . . .
2.4 -4 quotient system . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 2.5 The general form of the quotient system for system ( S . ' ï ) .
2.6 Digraph of the system in Example 2.1. . . . . . . . . . . . . . . . . .
2.7 Minimum condition numbers of Example 2.2 versus sampling period
for each eigenvalue of the equivalent discrete-time system . (a ) e-T.
(b) eO-lT: (c) e.3T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Closed-loop simulation results for Example 2.2, using optimal decen-
tralized continuous-time LTI controller . (a) Output signal of control
agent #l; (b) input signal of control agent #l; (c) output signal of
. . . . . . . . . control agent #2; (d) input signal of control agent #2
2.9 Closed-loop simulation results for Example 2.2, using optimal decen-
tralized digital controller . (a) Output signal of control agent #l; (b) in-
put signal of control agent #1: (c) output signal of control agent #2;
(d) input signal of control agent #2 . . . . . . . . . . . . . . . . . . .
2.10 Closed-loop simulation results for Example 2.2, using optimal cen-
tralized continuous-time LTI controller. (a) Output signal of control
agent #1; (b) input signal of control agent #l; (c) output signal of
. . . . . . . . control agent #2; (d) input signal of control agent #2. 43
2.11 Minimum performance index versus sampling period for Example 2.2
. . . . . . . . . . . . . . . . . . . . . . . . . . using digital controller. 44
Generalized sarnpled-data hold configuration for a decentralized system. 46
Closed-loop simulation results for Example 3.1, using optimal decen-
tralized digital controller with first-order GSHFs (3.8b) and (3 .8~) .
(a) Output signal of control agent #l; (b) input signal of control
agent #1; (c) output signal of control agent #2; (dj input signal of
control agent #2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimal 2nd order polynomial hold functions obtained for Example 3.1.
(a) The function corresponding to control agent #l; (b) the function
. . . . . . . . . . . . . . . . . . . corresponding to control agent #2.
Closed-loop simulation results for Example 3.1, using optimal decen-
tralized digital controller with second-order GSHFs (3.9b) and (3.9~).
(a) Output signal of control agent #l; (b) input signal of control
agent #l; (c) output signal of control agent #2; (d) input signal of
control agent #S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Minimum condition numbers of Example 3.1 for the GSHFs given by
. . . . . . . . . . . . . (3.9b) and (3.9~). (a) e-T, (b) eo-lT, (c) e-".
Closed-loop simulation for Example 3.1, using optimal continuous-
time state feedback controiler (2.20). (a) Output signal of control
agent #1; (b) input signal of control agent #l; (c) output signal of
. . . . . . . . control agent #2; (d) input signal of control agent #2.
. . . . . . . Block diagram of the decentralized switching controller 1.
Decentralized switching control for Example 4.1. (a) Transient re-
sponse; (b) steady-state response and reference input; (c) upper bound
. . . . . . . . . . . . . . . . . . . . . . signals; (d) switching instants.
Centraiized switching control for Example 4.1. (a) Transient response;
(b) steady-state response and reference input; (c) upper bound signal;
. . . . . . . . . . . . . . . . . . . . . . . . . . (d) switching instants.
The Zinput, 2-output mass-spring system of Example 4.2. . . . . . .
Decentralized switching control for the mass-spring systern of Exam-
ple 4.2. (a) Transient response of control agent #l; (b) steady-state re-
sponse of control agent #l; (c) transient response of control agent #2;
(d) steady-state response of control agent #2; (e) switching instants.
-4 hierarchy of interconnected subsystems which satisfis -kssumption 5.1. 94
Decentralized switching control for Example 5.1. (a) Transient re-
sponse of control agent #l ; (b) steady-state response and reference
input for control agent #l; (c) transient response of control agent #2;
(d) steady-state response and reference input for control agent #2. . .
Decentralized switching controller for Example 5.1. (a) Switching times
for control agent #1; (b) switching times for control agent #2. . . . .
Centraiized switching controller for Example 5.1. (a) Transient re-
sponse of control agent #l; (b) steady-state response and reference
input for control agent #I; (c) transient response of control agent #2;
(d) steady-state response and reference input for control agent #2. . .
Decentralized switching control for Example 5.2 with 912 = O. (a) Tran-
sient response of control agent #l; (b) steady-state response and ref-
erence input for control agent #l; (c) transient response of control
agent #2; (d) steady-state response and reference input for control
agent #2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Decentralized switching control for Example 5.2 with g12 = 0.5. (a) Tran-
sient response of control agent # l ; (b) steady-state respoase and ref-
erence input for control agent #l; (c) transient response cf control
agent #2; (d) steady-state response and reference input for control
agent#2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Decentralized switching control for the mas-spring system of Exam-
ple 5.3. (a) Output response of control agent #1; (b) output response
of control agent #2.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Decentralized switching controller for the mas-spring system of Exam-
pIe 5.3. (a) Switching times for control agent #l; (b) switching times
for control agent #2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
. . . Digraph of a continuous-time LTI system with 3 control agents. 110
Digraph of a continuous-time hierarchical LTI system wit h 3 control
agents.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
. . Sampled-data hold function for control agent #2 in Esample 6.1. 121
Digraph of the mass-spring system of Exarnple 6.2. . . . . . . . . . . 122
Sampled-data hold functions for control agent #2 in Example 6.2. (a)
T = O.5sec; (b) T = 2sec; (c) T = 5sec. . . . . . . . . . . . . . . . . 124
Digraph of the equivalent discrete-time rnass-spring system of Exam-
ple6.2.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Closed-loop simulations for Esample 6.2. (a) Samples of the output
response in control agent #1, for y,,~,, = 1, M . J , ~ = O; (b) samples
of the output response in control agent #2, for y,,,,, = 1, ~ I J ~ ~ J , ~ = 0;
( c ) samples of the output response in control agent #1, for y,.,,, = 0,
~ , , f , ~ = 1; (d) samples of the output response in control agent #2, for
1 / r ~ f , i = O , ~ ~ ~ f , 2 = 1 - . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Closed-loop simulations for Example 6.2. (a) Samples of the control
signal in control agent #1, for Y,.JJ = 1, yr.f,2 = O; (b) samples of the
control signal in control agent #2, for y,,t,, = 1, = O; (c) samples
of the control signal in control agent #1. for y,.~,, = O, y , , ~ , ~ = 1;
( d ) samples of the control signal in control agent #2, for y,,/,l = 0,
y,,f,2=1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Decentralized switching control for the mas-spring system of Esam-
ple 6.3. (a) Samples of the output in control agent #1; (b) samples of
the output in control agent #2. . . . . . . . . . . . . . . . . . . . . . 132
6.10 Decentralized switching control for the mas-spring system of Exam-
ple 6.3. (a) Switching times for control agent #l; (b) switching times
for control agent #2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
List of Tables
2.1 Minimum condition numbers of Esample 2.2 for digital controller and
different sampling periods . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1 .Minimum achievable condition number for Esample 3.1. using polyno-
. . . . . . . . . . . . . . . . . . . . . . . . mials of 3 different degrees 57
3.2 Minimum achievable performance index for Esample 3.1, using poly-
nomials of 3 different degrees . . . . . . . . . . . . . . . . . . . . . . . 58
Chapter 1
Introduction
The decentralized control of uncertain systems using bot h continuous-time and discrete-
time controllers is considered in this thesis by using the so-called "switching control"
technique.
In this chapter, some preliminary information about the notation and abbrevia-
tions used is given in Section 1.1, and then a brief overview of decentralized fixed
modes and the advantage of applying digital controllers to decentralized systems will
be briefly reviewed in Section 1.3. In Section 1.4, a general overview of decentralized
adaptive switching control will be discussed. In Section 1.1.3 a new application of
generalized sampling will be introduced, which can be used in the design of decen-
tralized controllers and in particular, decentralized adaptive controllers. Finally, this
chapter will be concluded by an outline of this thesis.
1.1 Notation and Abbreviations
Sorne of the terms and notation that appear throughout this thesis will now be defined.
IR, R+ and N represent the set of real, positive real, and natural numbers respec-
tively, and 118" and IPxn denote the n-dimensional vector space and rn x n matrix
space respectively.
In the state space representation of a system, capital letters B, C, D, E, and
F are used to denote the system matrices for the control agents al1 together, while
the small letters b j , c j , d j , fj, and ej are used to denote the corresponding system
matrices for each control agent, separately.
sp(.) denotes the spectmm function, and evaluated for a particular matnu, gives
the eigenvalues of that matrix.
The notation diag([DI . . . D,]) is used occasionally to denote a block diagonal r 7
1 O ... D.] E { . } denotes the expected value and is used to defme a performance index mhich
is equal to the average value of a quadratic function for uniform initial conditions
distributed on the unit bal1 [35].
II.Ii denotes the norm of a vector or the induced norm of a rnatr~x. In Chapter 4, i t
is used to represent the 2-nom, while in Chapter 5 it is used to denote the oo-norrn.
It is to be noted that this notation for the norm is consistently used throughout these
two chapters.
PC denotes the set of real vector-valued functions defined on t 2 O whose elements
are piecewise continuous, and PC, denotes the subset of PC functions which are
bounded.
The mod function is defined from W to IV, as foliows:
x mod y := x - floor (3 where floor(r) rounds z to the nearest integer towards -m.
Given a set of plant rnodels, the index of a plant mode1 or controller and cor-
responding parameters are denoted by superscripts, while the number of a control
agent for a plant is denoted by subscripts. For example, K: represents a parameter
1' corresponding to the plant #k and control agent # j .
For continuous-time signals, the independent variable is enclosed in parentheses,
whereas for discrete-time signals, brackets are used to enclose the corresponding in-
dependent variable.
In al1 examples, the international system of units (SI) has been assumed, unless
noted othemise. ,4iso, al1 numerical values are assumed to be represented by a t least
4 significant digits. Therefore, 3.49 will in fact be the same as 3.490 in the thesis.
,411 variables and parameters are denoted by italic fonts, while al1 sets and function
names are represented by Roman fonts. .Alsol the following abbreviations are used
t hroughout the t hesis:
LTI linear t ime-invariant
LTV h e a r t i m e - v a ~ n g
DF hl decentralized fked mode
ADFhI approximate decentralized fixed mode
Q F M quotient fked mode
AQFM approximate quotient fked mode
ZOH zero-order hold
GSHF generalized sampled-date hold func t ion
A 4 IMO multi-input multi-output
BIBO bounded-input bounded-output
MSBF modified strong bounding function
1.2 Decentralized Fixed Modes
Decentralized fixed modes (DFM) were introduced by Wang and Davison in [69],
who showed that they played an essential role in determining if a LTI plant can be
stabilized by applying a decentralized LTI controller. Such DFM can be classified
as either being "structured" or "unstructured" (601. There has been some confusion
in the literature as to what type of fixed modes are also fixed with respect to tirne-
varying output feedback [65]. This question was Iater clarified to a certain extent in
[ïl] and (301. However, to avoid confusion, throughout this thesis, we will consistently
refer to structured DFMs, as the modes that remain "fiued" with respect to any type
of decentralized output feedback, e.g. nonlinear or tirne-varying control. This type
of DFM is sometimes referred to as a "quotient fixed mode (QFM)" [25], which is
associated with strongly connected subsystems of the plant [31]. Thus, systems with
unstable structured DFMs are not stabilizable with respect to any type of nonlinear
or time-varying decentralized controller.
Unstmctured DFMs on the other hand, are modes that can be eliminated by ap-
plying appropriate tirne-varying controllers. Several time-wxying decentralized con-
trollers using vibrational feedback [33], [65], and periodic feedback [l], [30], [68] have
been introduced for this purpose.
1.3 Digital Controllers for Decentralized Systems
It uras shown in [39], that for almost al1 sampling periods, a discretized system has
the same number of stmctured DFMs as the original continuoustime system, and
has no unstructured DFMs; it imrnediately follows that a continuous-time LTI sys-
tem with unstable DFhh which are al1 unstructured, can be stabilized by using a
digital controller with a simple structure, e.g. with a zero-order hold. The equivalent
discrete-time model of the system is first obtained? using a fked sampling period,
and in this case, unstructured DFMs will generically no longer exist in the resultant
discrete-time model, and an appropriate technique can then be used to design a digi-
tal controller to stabilize the system. The overall controller then obtained consists of
the above decentralized LTI controller and a zero-order hold as shown in Figure 1.1.
This configuration is equivalent to a time-varying continuous-time controller. Note
Figure 1.1: The structure of a digital controller as a tiine-varying system.
Y,,, [ kl -
that the hold function corresponding to the Digital to Analog block (D/A) can be
any function defined owr one sampling period.
Cornputer L
' u Ml DIA
with ZOH b A
14 ( f ) - Pl;uit
The idea of using Generalized Sampled-data Hold Functions (GSHF) instead of a
simple zero-order hold (or first-order hold) in control systems n7as first introduced by
Cliammas and Leondes [4]; [5], [6]. Kabamba examined the application of GSHFs in
control systems, and pointed out that by using GSHF, one can obtain much of the
efficiency of state feedback, without the requirement of state estimation [28], [29]. He
also showed that GSHFs can significantly improve the performance of the closed-loop
system.
In this thesis, the application of GSHFs will be extended to decentralized systems.
1.4 Decentralized Adaptive Control via Swit ching
In this t hesis, the adaptive control technique of "switching control" introduced by
Miller and Davison in [41], will be extended to decentralized systems. Switching con-
trollers are nonlinear controllers, which can be used to stabilize and regulate systems
with highly uncertain plant models. This is accomplished by using a dictionary of
controllers, and by switching from one controller to another at appropriate time in-
stants; here a "switching controller," which uses feedback from certain outputs of the
system, is used. This class of adaptive controllers is more effective than conventional
adaptive controllers, when the uncertainty in the plant model is sufficiently large [41];
furt hermore, in applying the switching controller approach, rnany of the classical as-
sumptions required in the adaptive control literature can be relaxed. However, these
controllers reqiiire that the plant be described by a plant model which belongs to a
known finite set P.
In order to implement the above switching control procedure, it is assumed that a
finite set of controllers has been previously designed for each of the respective plant
models contained in P. One then implements an appropriate switching mechanism,
which switches among these controllers, to choose the appropriate controller for the
plant.
Various switching mechanisms have been introduced to stabilize centralized sys-
tems; in this thesis, an extension of switching controllers to decentralized systems is
made.
The block diagram of an ideal decentrdized switching control system is depicted
in Figure 1.2. The system has rn control agents associated with the control inputs
U L , ... , u, and measurable outputs yi, ... , y,. The plant model Pr belongs to a
known set of models P = { P l , ... , PP}. For each plant model, a decentralized LTI
cont roller:
has been designed that stabilizes and regulates the plant model Pk: k = 1, ... ,p . It
is desired to apply a switching mechanism to each control agent, so that after a finite
time, al1 control agents switch to the appropnate controllers that provide stability for
the overall system and tracking of the reference inputs y,.~,~, ... , Yrel,rn in the presence
of the unmeasurable disturbance W . The ideal decentralized switching scheme is one
that acts on each control agent independently of the other agents, Le. the control
input uj is generated and selected by only using the output signal yj and reference
input y,,f corresponding to control agent # j .
1.4.1 Conventional Adaptive Control
In conventional adaptive control design, it is typically assumed that the actual plant
is fixed, and can be described by a LTI model tvhich is unknown, but that a good
deal of a priari information re the plant is known; this information typically includes
a knourledge of the upper bound on plant's order, the relative degree, the sign of the
high-frequency gain, and minimum phase property. Furthemore, in the conventional
decentralized adaptive control literature, some additional structural restrictions are
also placed on the interconnections of the system. Typically, these restrictions include
some type of Lxed level of interconnection strength assumptions, or various matching
condition assumptions [22], [62], [26], [61], or slowly time-varying interconnection
assumption [59].
Switch 1 (Agent 1)
1 Plant P r I
Figure 1.2: The configuration of an ideal decentralized switching controller.
There have been some developments made to relax some of the classical assump-
tions adopted in conventional adaptive control. For example, some improvements have
been made to remove the required information on the sign of the high-frequency gain
[53], (701, [34], [58], and to weaken the other assumptions [49], [50], [64], [38]. However,
certain assurnptions re the right half plane zeros are required [44]. In the decentral-
ized adaptive control literature, similar assumptions are required for the interacting
su bsystems. However the main assumption generally made in the decentralized case
is on imposing a bound on the magnitude of the subsystems' interconnections [22],
[û2]. There have been some atternpts made to relax some of these assumptions. For
example, in [72] a decentralized adaptive scheme is proposed without requiring any
a priori information on the subsystem7s high-frequency-gain signs. Furthermore, in
[37j and [54] i t is shown that under certain conditions, the interconnection gain as-
sumptions can be relaxed, by estimating the interconnection outputs acting on each
subsystem J however, al1 of the assumptions normally made for the centralized a d a p
tive control case must hold for each subsystem of the decentralized adaptive control
problem.
1.4.2 Switching Adaptive Control
The adaptive centralized control of systems via switching control is a relatively new
line of research which was motivated to weaken the classical a priori information,
and can be traced back to [52], in which a number of questions about the classical
assumptions in conventional adaptive control were raised. There has been a consider-
able amount of interest towards switching control methods and its applications in the
literature recently; e.g. see [70], [43], [42], [46], [47], [45], [48], [55], [3], [SI, [57], [IO],
[Ml, [56] ; [SI, [7], [SI], [63] [20], [2] , [73]. These methods can be very effective mhen
wide-band tracking/disturbance rejection of a physical plant, which can be described
by a family of plants models, is required.
In the adaptive switching control approach using a family of plants, it is typically
assumed that the plant is not necessarily fixed, i.e. the plant may change from one
plant model to anuther; in this case, it is assumed that the plant model belongs to
a known set of models, and so, to implement the adaptive controIler, the first step
required, is to design (using either a model based, or an experimental approach) a
finite set of controllers which provide the required performance for this set of plant
models [21], [46], [57], [al], [41]: [I l ] , 1121. Then, on applying a so called "switch-
ing scheme", each controller is applied to the plant sequentially, and eventually, in
finite time, the switching controller stops switching. This implies that as long as the
plant remains unchanged, the switching controller mil1 remain locked on one of the
appropriate controllers which fulfills the closed-loop performance requirements.
Fu and Barmish [21] considered a compact set of LTI models to represent a plant
and imposed an a priori upper bound on the order of plants in this set. They
showed that Lyapunov stability can be achieved in this case, by applying a finite
set of controllers. Davison and Miller [41] reduced this a priori information, to
the knowledge required re the order of a LTI stabilizing compensator. They then
simplified the compactness assumption required on the set of possible plant models
to just consist of a finite set of plant models. .4s a result of this, one can design a
high-performance LTI controller, e.g. an optimal controller, for each plant model in
the known set.
In (121, a class of multivariable switching control algorithms was introduced which
doeç not require a knowledge of the actual family of plant models. Using this pro-
cedure, the only information which is required to be known, is a set of controllers,
corresponding to the set of plant models, mhich contains a stabilizing controller for
each plant model.
1.4.3 A New Application for Discretization
Sampled-data system applications are widely found in indus t l , mostly in order to
take advantage of the application of cornputers as digital controllers. In a discretized
model, each transfer function in the resulting transfer rnatrix is a function of the sys-
tem parameters, and also the sampling period and hold functions (zero-order hold,
first-order hold, ...). Now, the question arises: using generdized sampled-data hold
functions, to what extent can one modi@ these transfer functions so as to become
equal to zero in the discretized model? This problem is also studied in this thesis and
initiates a new line of research ir, the control of decentralized systerns, with various
applications. Note that the complexity of the controller design problem in decentral-
ized systems greatly depends on the complexity of the structure of the overall system.
For example. to stabilize a hierarchical model (which consists of a set of subsystems
connected by interconnections in certain directions), one can easily design a decen-
tralized stabilizing controller for the system by designing centralized controllers for
each agent independently. Therefore, finding a set of generalized sampled-data hold
functions so that the resulting discretized system has a hierarchical structure, can
greatly simplify the design problem. This is an important observation that can be
applied to decentralized adaptive control systems.
The remainder of this thesis is organized into the following chapters:
In Chapter 2, the application of digital controllers in decentralized systems is dis-
cussed. Initially, an algorithm to find structured DFMs is obtained. It is then shown
that under certain conditions, the application of digital controllers can significantly
improve the performance of a continuous-time decentralized system. In t his chapter,
the hold function considered in the discretization process is a simple zero-order hold.
In Chapter 3, it will be shown how the efficiency of the digital control for decentral-
ized systems can be improved by using a more gelleral form of sampled-data hold
functions instead of a simple zero-order hold- Simulation results obtained for second
order hold functions Nustrate the advantage of using generalized sarnpled-data hold
functions over simple zero-order hold in this case.
In Chapter 4, decentralized switching control using a binary communication link
between different agents is introduced. The proposed switching scheme is a decen-
tralized version of the method introduced by Miller and Davison in (411 and so, it is
assumed that the plant model belongs to a known finite set. Although the switching
controller does not require input or output information flow to exist between different
control agents, the controller is not completely decentralized as it requires a very weak
communication link to exist between the control agents to assure that al1 agents will
switch simultaneously. In Chapter 5, a decentralized switching scheme is proposed
which does not require any information link at al1 to exist between the different con-
trol agents, provided that the systems model has a certain structure. This method is
a decentralized version of the switching mechanism developed by Chang and Davison
in [12]. Therefore, instead of using a set of plant models, it is only assumed tha t a
set of previously designed controllers for the system is given.
In Chapter 6 it is shown how one can use generalized sampted-data hold functions
to convert a continuous-time plant to an equivalent discrete-time system with a certain
desired structure, in particular a hierarchical structure, which can then be directly
used to simplify the design of decentralized control systems for the plant. In this case,
the results can be directly applied to the decentralized adaptive switching method
proposed in Chapter 5 which requires a hierarchical plant model.
Finally, the thesis closes with Chapter 7, the concluding remarks and some sug-
gested future work.
Chapter 2
Digital Control of Systems with
Approximat e Decent ralized Fixed
Modes, Using Zero-Order Hold
In this chapter, the application of digital control with zero-order hold in decentral-
ized systems which have Approximate Decentralized Fixed Modes (rZDFM) will be
discussed. Tmo different types of ADFMs will be defined, and then, using the concept
of -4pproximate Quotient Fked Modes (AQFM), the conditions under which the per-
formance of the closed-loop system can potentially be improved by applying digital
control will be discussed. The efficiency of digital control in decentralized control will
be examined t hrough simulations.
2.1 Decent ralized Fixed Modes (DFM)
As a motivation for introducing the notion of Decentralized Fixed Modes (691, consider
the following LTI system:
(2. l a )
It may be verified that this system is controllable and observable, and thus it is well
known that t here exists a LTI controller which stabilizes the system. Suppose horvever
that the controller is restricted to have the following decentralized configuration:
Then it turns out that it is impossibte to stabilize the system using such a LTI
decentralized controtler, since the system has a so-calted DFh4 at X = L.
Definition 2.1 [69]: Consider a LTI system:
wit h a controller information flow constraint:
Define:
Then X E sp(A) is a decentralized fked mode (DFM) of the system if:
X E sp(.l) fl sp(A + BKC),
1 kp 1 In this case, it can be verified for (2.1) that:
and so X = 1 is indeed a DFM of (2.1).
Consider now the following system:
It can be verified that (2.2) also has a DFM at X = 1 and so this system also cannot be
stabilized using a LTI decentralized controller. However in this example, the systern
can be stabilized by using a linear time-varying decentralized controller, unlike the
case of the system (2.1), and this difference is reflected in the type of DFM's which
(2.1) and (2.2) posses. In the case of (2.1), the DFM X = 1 is said to be a structured
DFM, whereas in the case of (2.2), the DFM is said to be an unstructured DFM.
2.2 Sampling and Approximate Decentralized Fixed
Modes (ADFM)
It is well known that digital control can be used to stabilize an unstabie continuous-
time system which has unstable unstructured DFMs present, e.g. see [39], [l], [30] and
[68] (but not structured DF'vls). In this section, the application of digital control will
be extended to eliminate certain classes of "Approximate Decentralized Fised Modes"
(-\DFM), namely so-called "unstructured -4DFMm, which in turn can thence improve
performance of the closed-loop system: as a result, decentralized digital controllers
can potentially improve the overall performance of the closed-loop system, in cont rast
to the case when the system is controlled by a LTI continuous-time decentralized
çont roller.
-4 brief description of -4DFM as proposed in [67] will now be given. For simplicity,
consider the following decentralized system:
where uj E W' and yj E W1, j = 1,2, denote the inputs and outputs respectively. The
block diagram of this system is depicted in Figure 2.1. This system consists of two
input-output stations corresponding to (u l , yl) and (uZ, y2), and A, a n eigenvalue of
tem:
where cj E RIXn, bj E IRnxL, j E m = {l, ..., m}. The process of sampling and
holding on t his system, will now be investigated. The equivalent discrete-time system,
corresponding to (2.7), is represented by:
where the equations relating the continuous-time system (2.7) to the equivalent
discrete-time system (2.5) depend on the sampling period, holding process, and pro-
cessing delay. Tlie processing delay is the time difierence between sampling the output
signal and computing the control signal and is illustrated in Figure 2.2; its value varies
A u V I l.4 Dl
,, u [O1
Y [41 i - . . - . . . . . time - *
sampling processing p e n d delay
Figure 2.2: The input and output signals in a computer control system.
between zero and the sampling period. In the rest of this thesis, we \\dl consider zero
processing delay, which corresponds to the case when the control law is applied to
the system immediately after sampling the output signal. Assume nonr that a simple
zero-order hold is applied to al1 control agents. The equivalent discrete-time system,
in this case, assuming -4 is invertible, is described by the following matrices:
and a controller with output u[k] can be constructed from ~ [ z ] , i 5 k. If the
continuous-time system (2.3) has unstmctured DFMs, then for generic sampling times
T > O there exist no unstructured DFMs [39] in the equivalent discrete-time system.
There will, hontever, exist ADFMs for these respective modes, and since as the sam-
pling time approaches zero, the behavior of the discrete-time system converges to the
continuous-time system, t his implies t hat for sufficiently small sampling periods, the
discrete-time system mil1 have -4DFMs which tend to infinity as T -, O. This implies
that to avoid large ..2DFMs, one should choose "large" sampling periods: on the other
liand, a large sampling period makes the control signal sluggish. Therefore, the choice
of an "optimal" sampling period involves an optimization problem.
2.3 Theoretical Result s
-4s was previously discussed, decentralized fixed modes (DFM) of a linear dynamic
system are the modes that cannot be shifted by decentralized time-invariant output
feedback. Wang and Davison [69] introduced this notion, and concluded that the
absence of unstable DFMs is a necessary and sufficient condition for the existence of
a decentralized LTI controller to stabilizes a system. Sezer and ~ i l j a k [60] introduced
the concept of structurally decentralized fixed modes to characterize the modes that
are fised with respect to any arbitrary feedback structure. This definition \vas made
using the notion of stmctured matrices and structurally equivalent systems [36].
Structured and unstructured -4DFMs will be defined now. With no loss of general-
ity, consider a system in expanded form
m-agent system:
[66], i.e., consider the following decentralized
where cj E RIXn, bi E PX', i = 2,2, . . . , m. The digraph of this system is defined as
a set of m nodes and directed arcs connecting these nodes. The nodes represent the
control agents of the system and the directed arcs represent the connections between
them. If q ( s 1 - A)-'bj # O, then there exists a directed arc from node j to node i
(i, j = , 2 , m . Consider a subsystern of (2.7) consisting of a subset of the nodes
{1,2, ... , m ) with the property that for each distinct pair of nodes 2 , j in that subset,
t here exists a directed pat h from node i to node j and also a pat h from node j to node
i (a directed path consists of one or more directed arcs). This subsystem is called
a st rongly connected subsystem of (2.7) [3 11 (usually the term "strongly connected"
is referred to the digraph of a system instead of the system itself). -4 system can
always be decornposed uniquely into a number of st rongly connected su bsystems,
which are the largest strongly connected subsysterns in the sense that if an extra
node is added to any of the strongly connected subsystems, that subsystems will no
longer be strongly connected.
Corresponding to the decomposition of a system into strongly connected subsys-
tems, one can define a new decentralized systern wit h a control agent assigned to each
subsystern, which is in fact the quotient system for (2.7) [31]. As a simple example,
consider a decentralized control system with the digraph of Figure 2.3. This sys-
tem consists of three subsystems, and only two of these subsystems (subsystems #l
and #2) are strongly connected. The corresponding quotient systern is depicted in
Figure 2.4.
1 Controller I 1 1 Controller 2 1 ( Controller 31
Figure 2.3: -4 decentralized control system
1 Controller 1 1 1 Controller 2 1 Figure 2.4: -4 quotient system
Definition 2.3 : Given the system (2.71, the DFMs of the so-called quotient system
for (2.7) are called quotient k d modes (QFM) of the system (2.7) [25], which are,
in fact, structured DFMs of (2.7).
-Assume the system (2.7) consists of a set of 1: 1 5 1 5 m strongly connected subsys-
tems {(cl, -4, 61) , (&, -4, &), . . . , (ci, A, bi) ) , where:
The following initial result is defined:
Theorem 2.1 : Consider the system (2.7); then h E sp(A) is a QFM of the sys-
tem (2.7) iff it is a transmission zero of all of the following subsystems:
where & and bi denote any possible partition of C, and bi defined in (2.8) (including
empty sets) respectively (i = 1,2 , . . . ,1) such that the number of rows of C. is equal
to the number of colurnns of bi, i = 1 , 2 , . . . , 1 .
Proof of Theorem 2.1 : The quotient system corresponding to the system (2.7)
lias the general structure of Figure 2.5, where Ci, i = 1,2'. . . , 2 represent the control
agents corresponding to the strongly connected subsysterns and ik, , . . . is are intro-
duced in (2.8). In this case, certain arcs a t each subsystem can be removed without
losing the strong connection among the corresponding agents, and arcs between each
pair of strongly connected subsystems have an appropriate direction such that none
of these pairs are strongly connected. Here each controller Ci has a centralized struc-
Figure 2.5: The general form of the quotient system for system (2.7).
ture. This implies that there can exist feedback from each agent in the subsystem to
any other agent in the same subsystem via the controller Ci- Hence, A E sp(.-l) is a
QFhl of the system (2.7) if and only if it is a DFM of the system with respect to the
block diagonal matrix:
where ATi (i = 1 , 2 , . . . : 1) is a a
(ki - k i - l ) x (Ici - gain matrix (ko = O). NOW,
using the concept of an expanded system (661, one can conclude that A is a QFM of
the system (2.7) iff it is a DFM of the systern:
x = -42 + Bu, y = Cx,
mith the following B and C matrices:
2 (k, - kl-1)
and so, the application of Theorern 2 of [li] immediately leads to the proof of Theo-
rem 2.1 (note that any matriv with identical rows or columns is singular; thus there
Remark 2.1: It is to be noted that Theorem 2.1 provides a new method to identify
structured DFMs, which does not require one to investigate the structure of the non-
zero elements of the system matrices, as was done in [60].
-4s a result of Theorem 2.1, one can use the definition of -4DFM now to define .4p-
prosimate Quotient Fixed Modes (AQFM), and it can be concluded that if X E sp(A)
is an ADFM of magnitude cond(X) for the system (2.9), then it is also an AQFM
of identical magnitude for the system (2.7). A M-4TL-AB program is written and
included in Appendiu C which can be used to obtain the XDFMs and XQFMs of a
LTI system witb no more than 3 control agents, using the proposed method. This
code is used in the next example.
Example 2.1 : Consider a 3-input, 3-output, strictly proper, decentralized LTI sys-
tem with the followïng system matrices:
The digraph of this system is shown in Figure 2.6. This systern consists of tmo
strongly connected subsystems {{l, 2} , {3}} , with sp(. l) = {l. 2' 3}. In order to find
Figure 2.6: Digraph of the system in Example 2.1.
ADFMs in this system, the minimum condition nurnbers of the following matrices are
ob tained for each eigenvalue:
and in this case, using the MATLAB code in Appendk C, the following values are
obtained:
mhich implies that X = 1 and X = 2 are large ADFMs with magnitude 2.382 x I O 4
and 4.836 x lO%espectively.
To find the -4QFMs for this system, ive need to obtain the quotient system corre-
sponding to the strongly connected subsysterns. It can be easily seen from Figure 2.6
that for this esample:
In order to specify AQFMs, the minimum condition numbers of the matrices in (2.10)
are checked together with the following matrices (as described in Theorem 2.1):
Using the bL4TL-4B code in Appendiu C, the following values are obtained:
Therefore, A = 1 and X = 2 can be considered to be large -4DFMs, and the only large
-4QFnlI in this system is X = 1; here, X = 2 is the only unstructured large ADFM in
the system.
Comment 2.1: Throughout this chapter and the next chapter, the terms AQFM
and structured ADFh.1 wiil be used interchangeably. However, there is a special case
for which these two terms are not exactly quivalent. This occurs when a strongly
connected su bsystem includes an interconnection t hat is significantly weak (in terms
of the norm of transfer function representing that interconnection) compared to the
other interconnections, and without that interc~nnection~ the strong connectedness
of the subsystem mil1 no longer exist. In this case, it is possible to have a structured
ADFbI which is not an AQFM In fact, if by removing the weak interconnection, the
corresponding structured ADFh/I becomes a QFM of the new system. that mode is
a structured ADFhl. In the remainder of this chapter and next chapter, it will be
assumed that such a weak interconnection does not exist and thence, AQFM and
structured .4DFM will be considered the sarne.
Now, the effect of sampling on unstructured -4DFMs will be discussed.
Theorem 2.2 : Given the continuous-time decentralized system (2.7), assume that
X E sp(A) is an unstructured ADFM with magnitude cond(A), and let the system (2.7)
have the equivalent discrete-time representation given by (2.6); then there exists a
constant M > O such that if cond(X) > M, this implies that there exists a sampling
period T so that:
tvvhere cond(eAT) denotes the magnitude of the ADFkI for the equivalent discrete-time
system.
Proof of Theorem 2.2 : The proof of this result follou~s by using a continuity
argument. Note that a continuous-time system with an unstructured ADFM, A, can
be rnodeled as arising from perturbing a system which has an unstructured DFM. In
other words, given the system (2.7), represented by:
which has X as an unstructured .4DFM, consider a new system:
where:
and where Ail. Ab,, Ac,. j = 1:. . . . rn are chosen so that X is an unstructured DFM
of (2.12) (here LA, Ab,, and Ac, can always be chosen so that is the case). Consider
the systern (2.12) now; then for any constant M , if 4.4, Abj, Acj are sufficiently
small, it follows that cond(X) of (2.11) is greater than M for almost al1 sampling
periods T. The proof is completed by choosing M > cond(eAT). . Remark 2.2 : Theorem 2.2 implieç that a digitai controller can potentially improve
the performance of a continuous-time closed-loop system, when there exists an un-
structured -4DFIVI in the continuous-time system.
2.4 Numerical Example
Example 2 -2 : Consider a cont rollable, observable, non-minimum phase, unstable
system with the following state space matrices:
The eigenvalues of this system are sp(A) = {-1,0.1, -3}. Applying the minimum
condition number criterion results in:
and so, X = 0.1 c m be considered as being a large ADFM for the system (this is an
unstructured -4DFM since it is not an AQFM). Note that this is an unstable mode
and so to stabilize the system, the decentralized continuous-time LTI controller has
to "shift" it to the left half plane, which irnplies that it will require a large gain.
The discrete-time equivaient of the above example is now computed using the set of
matrices given by (2.6), and on applying the minimum condition number criterion
for ADFM in the equivalent discrete-time system, the results shown in Table 2.1, and
Figure 2.7 are obtained; these results show the minimum condition number for each
eigenvalue of the discrete-time system as a function of the sampling period. It can
be seen from these results, that for a wide range of sampling intervals, the equivalent
discrete-time system has no large ADFM, e-g., with a sampling interval of T = 2sec,
the condition nurnber of the ADFM e0.lT for the discrete-time system is 250.2 which is
Table 2.1: Minimum condition numbers of Example 2.2 for digital controller and different sampling periods.
sampling period T (sec)
O. 1
approximately 7 times smaller than cond(0.l) = 1778 for the continuous-time system.
-4s a rule of thumb, if the minimum condition numbers associated with the unstable
modes in the continuous-tirne system are significantly greater t han their discrete-time
counterparts, a suitable digital cont-roller can potentially be more efficient than a LTI
one.
It is desired now to find a controller to stabiiize the unstable system (2.13), and
the following performance index, for the continuous-time system, is considered as
a rneasure of the system performance using either continuous-time or discrete-time
static decentralized controllers:
where E denotes the expectation operator [35]. It is to be noted that in the case
of discrete-timc control, this performance index takes intersample ripple effects into
account. The following cases are examined:
pole e-T
2.087 x 10'
i) Optimal continuous-time decentralized LTI controller: Consider applying the
controller:
to (2.13). Then on rninimizing the performance index defined in (2.14), with
pole - 3 ~
1.395 x 10'
pole (?o.~T
3.882 x IO3
Figure 2.7: Minimum condition numbers of Example 2.2 versus sampling period for each eigenvalue of the equivalent discrete-time system. (a) e-T, (b) eO-lT, (c) e-3T.
respect to kl and k2, using Algorithm I (see Appendk .4), the optimal perfor-
mance index of J , = 1.105 x IO5 and the following optimal feedback taw are
obtained:
which produces the following eigenvalues for the closed-loop system:
which irnplies that the closed-loop system has been stabilized. The correspond-
ing performance index for x(0) = [l 1 11' is 1.087 x 105 and Figure 2.8 gives
the corresponding input and output signals for this case. Note that y,(O) = 1,
~ ~ ( 0 ) = -0.995 and yl then peaks to a value of 32.95 before settling down to
zero. The performance of the resultant controller is not satisfactory, and is a
consequence of the large ADFM at 0.1 in the system. In fact, even using a de-
centralized continuous-time LTI dynamic controller will not make a significant
improvement in the performance of the system in presence of the large ADFM
Figure 2.8: Closed-Ioop simulation results for Example 2.2, using optimal decentral- ized continuous-time LTI controller. (a) Output signal of control agent #l; (b) input signal of control agent #l; (c) output signal of control agent #2; (d) input signal of control agent #2.
ii) Decentralized digital controller. Consider applying the digital controller:
to (2.13), with a sampling period of T > O and a zero-order hold. Then on
40
minimizing the perfonnance index (2.14) as \vas done in (i) , using Algorithm II
(see -4ppendix A), with respect to kl, k2 and T, the optimal performance index
of J , = 4896 is obtained which is approximately 20 times smaller than the
result obtained in (i). The corresponding optimal controller is given by:
and the optimal sarnpling penod is TV = 3.085sec. The eigemalues of the
resultant closed-loop system for the equivalent discrete-time system are given
by :
nhich shows that the closed-loop system has been stabilized. The corresponding
performance index for x(0) = [l 1 11' is 2.116 x 104 and Figure 2.9 gives the
resultant input and output signals for this case. The transient response of this
system now has a much more reasonable response compared to Figure 2.8.
For cornpleteness, the case of a cent.ralized controller will now be considered.
i i i ) Centralzzed continuow-time optimal LTI controller: Consider applying the cen-
tralized controller:
to (2.13). Then on minimizing the same performance index given in (i), the
follon4ng optimal feedback law is obtained:
Figure 2.9: Closed-loop simulation results for Example 2.2, using optimal decentral- ized digital controller. (a) Output signal of control agent #l; (b) input signal of control agent #l; (c) output signai of control agent #2; (d) input signal of control agent #S.
which produces the following closed-loop eigenvalues:
The resultant optimal performance index is J, = 8.382. The corresponding
performance index for x(0) = [ l 1 11' is 18.52, and Figure 2.10 gives the
resultant input and output signals obtained for this case.
The above results show that the input and output signals obtained in the proposed
decentralized controller are approximations to the optimal LTI centralized control
system.
It is clear that the performance index obtained for the digital decentralized con-
troller (2.17), which depends on the sampling period T, is superior to the continuous-
Figure 2.10: Closed-loop simulation results for Example 2.2, using optimal centralized continuous-time LTI controller. (a) Output signal of control agent #1; (b) input signal of control agent #l; (c) output signal of control agent #2; (d) input signal of control agent #2.
time decentralized controller (2.15). For completeness, Figures 2.11 (a) and (b) show
how the optimal performance index obtained using (2.15) depends on the sampling
interval.
So far, various output feedback controllers have been considered in this example.
It is ta be noted for this problem, that the minimum achievable performance index for
(2.14) using any type of controller is obtained by the contincous-time state feedback
law :
L
and the corresponding performance index for x(0 ) = [l 1 11' is given by J , = 1.448.
Figure 2.11: Minimum performance index versus sampling period for Example 2.2 using digital controller.
Remark 2.3 : It is to be noted that the optimization algorithm used for digital
controller design in Example 2.2, and similar examples in Chapter 3 (-4lgorithm II,
Appendix A) rninimizes the continuous-time quadratic performance index, and so this
implies that the optimization algorithm takes intersample ripple effects into account.
Chapter 3
Decentralizcd Control of Systems,
Using Generalized Sampled-Dat a
Hold Functions
It was s h o w in Chapter 2 that digital controllers can potentially improve the perfor-
mance of systems with unstructured ADFMs. In that chapter, a very simple form of
digital controllers, using zero-order hold was used. The question now arises: is it pos-
sible to improve the performance of the overall system by using a more generaI form
of hold functions instead of a simple zero-order hold? This problem is investigated in
this chapter.
3.1 Formulation of Generalized Sampled-Data Hold
Functions (GSHF)
The idea of using a GSHF in a closed-loop control system, is to sample the output of
the system, and let the control signal be a linear time-varying weighting of the output
during each time interval [28]; one of the advantages of using GSHFs in this case is
that the resultant output feedback controller acts in a dual way to state feedback,
without the requirement of using state estimation 1281.
Consider a strictly proper decentralized LTI system with rn control agents repre-
sented by:
where x ( t ) E Rn is the state vector, and uj ( t ) E WJ, and y j ( t ) E Rrj, j E 15 =
(1, ..., m) are the control vector and output vector of control agent #j respectively,
and A, b,, and c, are matrices of appropriate dimensions. Generalized Sampled-Data
Hold Functions (GSHF) can be formulated as follows (see Figure 3.1):
mhere üj[k] E HPSl is a control vector of the discrete-time system, for the corresponding
Figure 3.1: GeneraIized sampled-data hold configuration for a decentralized system.
sampled-data system associated with (3.1) and (3.2): given by:
where:
It can be assumed without loss of generality, that rj = s j , j E m. One can now set
üj[k] = gj[k] in equation (3.2a) to achieve closed-loop control. It is to be noted that
this feedback law along nith the system matrices defined in (3.4) correspond to zero
processing delay because the control signal is applied to the system immediately after
sarnpling the output signal. It has been showri in [28] that the resulting closed-loop
continuous-time systern is stable if the eigenvalues of the matriu:
which represents the equivalent closed-loop discrete-time system, are al1 contained
inside the unit circle. We will now show how GSHFs can improve the performance of
the closed-loop system.
Lemma 3.1 : Consider the system (3.1) with sampled-data hold functions (3.2). For
any sampling period T , and any arbitrary n x sj matriw b; whose columns belong
to the controllable subspace of (A, b j ) , there exists a sj x sj GSHF f, so that with
uj (t) = f, ( t)üj [k] , t E [ k T , ( k + l )T) , the corresponding discrete-time system (3.3),
has the property that b4 = b;.
Proof of Lemma 3.1 : The proof follows by using an argument similar to that
presented in [28].
Thus, for each T > O' there exists a sj x 1 function fj(t) that can transform
%, (tm) = O to G, ( tkCL) = bi! where GJ ( t k ) and b*; denote the ith colurnn of b d j ( t k )
and 6; (i E (1' 2, ..., sj)) respective'v, and t k + 1 - tk = T . In other words, one can
always find a sj x sj GSHF f,, (non-unique), so that the matrix bdJ in the resultant
discrete-time system is equal to amy specified n x sj mat ri^ whose columns belong to
the controllability subspace of (A, b j ) - In fact, one can use the following steps to find
a possible solution to the above problem:
i) Assume that the rank of the controllable subspace of (A, 6,) equals nl 5 n.
Find a transformation V to partition the system as:
where the controllable pair (All, 6 , ) represents the controllable subspace of
(A, b,) . The first nl columns of V can be any set of vectors that span the
column space of the controllability matrix [b, Ab, ... 4"-'b,], and the last n - nl
columns can be any set of vectors that results in a nonsingular matrix V.
ii) Find the controllability grammian corresponding to ( A L ,, 61) on [O, T ] as fol-
lows:
iii) Then, the folloring GSHF is a solution (non-unique) to this problem:
where 6;I is obtained from the following equation:
It has been shown in [27] that using the GSHFs of this form, good stability margins
can be achieved by the closed-loop systern, for appropriate choices of T and 6,: j E m.
The following result is now obtained:
Theorem 3.1 : Consider the continuous-time decentralized system (3.1) , and assume
that X E sp(A) is an unstructured ADFM with magnitude cond(A); then for almost
al1 ([4 . . d m ] 4, [ b ... b,]) systems, and for any sampling interval T, there exists
a GSHF such that the condition number representing the ADF-M in the equivalent
discrete-time system is smaller than the same ADFM obtained by a zero-order hold.
Proof of Theorem 3.1 : The proof follows on noting that:
(i) for almost al1 ([d, ... c',]', A, [bl ... b,]) systems, one can choose a matris b; ir.
the controllable subspace of (A, b j ) to minimize the minimum condition number
corresponding to the XDFM in the equivalent discrete-time system (3.3),
(ii) on noting from Lemma 1, that on applying an appropriate GSHF such as (3.5),
bd, can then be made equal to b;, and
(iii) on noting that a zero-order hold lies in the zeros of a non-zero polynomial hold
function (the space spanned by a zero-order hold is limited to the constant
functions while a GSHF can be any function and provides more degrees of
freedom) . Thus for almost al1 ([d, . .. dm]', A, [bl ... bm]) systems, the resultant
GSHF obtained is not a constant.
This implies that the condition number corresponding to any ADFM can be mini-
mized by using an appropriate GSHF, instead of a simple zero-order hold. The special
case for which a zero-order hold provides the minimum value for the condition number
corresponding to a ADFM, occurs when 6; is proportional to ~ ~ e A ( i - r ) b j d ~ (which
is equal to A-' (Ad - I )b , if A is invertible). It is to be noted that the function
which minimizes the corresponding minimum condition numbers is not unit
the function given by (3.5) is just one possible choice. . Remark 3.1 : Note that the GSHF which minimizes the corresponding m
ue and
nimum
condition numbers is not necessarily the optima1 GSHF +th respect to a performance
index that may be defined for the closed-loop system.
Remark 3.2: One can use a simple function, e.g. a c l a s of polynomials of specific
degree, instead of the function defined in equation (3.5), to minimize the condition
number of an ADFM. For example, by using the class of second-order polynomial
functions, one can minimize the condition number over 3 CI=, s: parameters (note
that each second-order function includes three coefficients). Alternately, one could
also apply piece-wise constant functions, instead of polynominal functions to create
a GSHF.
3.2 Numerical Example
Example 3.1 : To illustrate the ad\-antage of using GSHF for decentralized systems,
consider Example 2.2 of the previous chapter, whose corresponding system matrices
are rewritten as fo1lows:
AS discussed in Example 2.2, A = 0.1 can be considered to be a large ADFM for the
system. In fact, X = 0.1 is an unstructured ADFM and so, it can be "shifted" by
means of an appropriate LTI controller. However since this mode is an unstable mode,
any stabilizing controller has to shift this mode into the left half plane, and this will
require a large controller gain for any LTI continuous-tirne decentralized controller.
It was shown in Chapter 2, that on applying a LTI continuous-time decentralized
output feedback controller and on minimizing the performance index:
using Algorithm 1 (see Appendk A), a value of J , = 1.105 x 105 is obtained. In
this case, the corresponding input and output responses for the system when x(0) =
[l 1 11' are given in Figure 2.8, and it is observed that the inputs ui and u ~ , peak to
approximately 110 and 650, respectively. It is desired now to design a decentralized
digital controller with appropriate GSHFs for this system, to improve the performance
of the closed-loop systern. Two different cases will now be investigated:
i) First-order polynomials: Consider applying the sampled-data hold functions
f i ( t ) = ait + bi and f2(t) = a2t + to each control agent. On minimizing the
performance index (3.7) using Algorithm III (see Appendix .4), the following
results are obtained:
Optimal sampling time : T, = 1.152sec7
Optimal fi(t) : fi(t) = -4.8865 + 2.915,
Optimal f2(t) : f2(t) = 1.768t - 1.978,
Optimal performance index : J , = 15.08.
Note that in this case only one parameter has been added to the control design of
each agent compared to digital control with a zero-order hold, but a significant
improvement has been achieved. The corresponding input and output signals
of control agent #1 and control agent #2 for x ( 0 ) = [l 1 11' are depicted in
Figure 3.2 in which (a) and (b) give the output and input signals of control
agent # 1 : and (c) and (d) give the corresponding signals of control agent #2,
respectively. The resultant performance index for x (0 ) = [ l 1 lIt is 50.17,
which is significantly smaller than 1.105 x 105 obtained in the continuous case.
ii) Second-order polynomials: In this case, sampled-data hold functions of the form
f i (t) = al t2 + bit + cl and f2 (t) = a2t2 + &t + cz will be assigned for each agent.
On minimizing the performance index (3.7) as described in the previous case,
the following results are obtained:
Optimal sampling time : T = 1.254sec,
Optimal fi(t) : fl(t) = 5.931t2 - 4.229t - 0.3416,
Optimal f&) : f2(t) = 1.058t2 - 2.565t + 1.400,
Optimal performance index : J, = 14.71.
These optimal hold functions are shown in Figure 3.3. The corresponding per-
formance index for x(0) = [l 1 11' is 6.293 and Figure 3.4 gives the resultant
input and output signals for this case.
Figure 3.2: Closed-loop simulation results for Example 3.1, using optimal decentral- ized digital controller with first-order GSHFs (3.8b) and (3.8~). (a) Output signal of control agent #l; (b) input signal of control agent #l; (c) output signal of control agent #2; (d) input signal of control agent #2.
It can be seen from Figure 3.2 and Figure 3.4, that no "peaking" occurs in the
response of the system, as compared to the continuous-time systern (see Figure 2.8).
In addition, the magnitudes of the input and output signals in Figure 3.2 and Fig-
ure 3.4 are much smaller than those of Figure 2.9 obtained by using a zero-order hold
for the same system.
For interest, the minimum condition number cond(X) for the eigenvalues of (3.6)
using the optimal GSHFs (3.9b) and ( 3 . 9 ~ ) versus sampling time T? are given in
Figure 3.5. It is obsemed that the condition number of the unstable mode has been
reduced by a factor of 100 approximately, using this class of generalized sampled-
data hold functions, instead of a zero-order hold. It can also be seen from this figure
that as the sampling time T approaches 0, the condition number of the unstable
mode (A = 0.1) increases; thus this result confirms the observation often made in
Figure 3.3: Optimal 2nd order polynomial hold functions obtained for Exarnple 3.1. (a) The function corresponding to control agent #l; (b) the function corresponding to control agent #2.
practice, that "fast sampling" is not always a desirable procedure to implement in
digital control design.
In completeness, a continuous-time centralized state feedback controller was de-
signed using a X2 design method with performance index (3.7) (see Example 2.2 and
equation (2.20)). Simulations using the resultant state feedback controller for the
case when x ( 0 ) = [l 1 11' are given in Figure 3.6.
Remark 3.3: Here i t is observed that the input-output response of the resultant
continuous-time centralized system using the optimal state feedback (Figure 3.6), is
comparable to the input-output response of the decentralized digital control system
with optimal first-order polynomial and second-order polynomial GSHFs (Figure 3.2
and Figure 3.4), on ignoring "ripple type of effects". There still however will be a
significant difference of behavior between a centralized continuous-time controller and
-2' I O 5 10 15 20
@l t (sec)
Figure 3.4: Closed-loop simulation results for Example 3.1, using optimal decentral- ized digital controller with second-order GSHFs (3.9b) and (3.9~). (a) Output signal of control agent #l; (b) input signal of coctrol agent #l; (c) output signal of control agent #2; (d) input signal of control agent #2.
a decentralized controller, if one wishes to obtain "high performance" for the system,
i.e. a "rapid response with no 'peaking' occurring", in the system. This is because
the minimum condition number of the unstable ADFM (A = 0.1) will become larger,
as the sampling time decreases (see Figure 3.5). Thus the application of GSHFs
in digital decentralized control is Iimited mainlv to t hose situations where moderate
sampling times are to be applied.
Remark 3.4: Choosing the proper sampled-data hold functions is a trade-off be-
tween the improvement of the overall performance of the closed-loop system achieved
by the functions, and the complexity introduced by the corresponding functions. Ta-
ble 3.1 compares different polynomial orders in ternis of the minimum acliievable
condition number, corresponding to the unstable ADFM X = 0.1 in Example 3.1.
Figure 3.5: Minimum condition numbers of Example 3.1 for the GSHFs given by (3.9b) and (3.9~). (a) e-=, (b) eO-LT, (c) e-3T.
Table 3.2 compares the effect of zero-order, first-order, and second-order polynomi-
als on the minimum achievable performance index defined in (3.7) (these results are
obtained by using -4lgorithm III in Appendix A). Both tables illustrate the effects
of using higher-order sampled-data hold functions. For this example, a first-order
polynomial is a good choice and there is not any significant improvement achieved by
higher-order polynomials. Note that a zereorder polynomial as a GSHF, is equiva-
lent to a zero-order hold. However, the results of Table 3.1 do not match the results
obtained in Chapter 2 (Figure 2.7). The reason is that the matrix Bd defined in (2.6)
mhich is used to obtain the condition numbers, is different from the corresponding
matrix obtained by applying GSHFs (equation (3.4b)), since the latter is in fact a
combination of the former, and the optimal static gain for digital controller.
-0.08 1 O 5 1 O 15 20
(bl (sec)
-1.2 O 5 10 15 20
(4 r (sec)
Figure 3.6: Closed-loop simulation for Example 3.1, using optimal continuous-time state feedback controller (2.20). (a) Output signal of control agent #l; (b) input signal of control agent #l; ( c ) output signal of control agent #2; (d) input signal of control agent #2.
Table 3.1: Minimum achievable condition number for Exarnple 3.1, using polynomials of 3 different degrees.
zero-order
polynomials first-order
polynomiais second-order
polynomiais
optimal sampled-data hold functions
f i = 1066
fi = -2.301 x f = 1.037t - 552.7
f2 = -785.2t + 43.90 f i = -9741t2 + 4028t - 321.4
f2 = 2.015 x 104t2 - 7520t + 489.6
optimal sampling period (sec)
2.805
0.1007
0.3564
minimum value for cond (0.1 )
454.9
2.986
1.375
zero-order
polynomials first-order
optimal sampled-data hold funetions
f 1 = 9.996
optimal sampling period (sec)
polynomiais second-order
Table 3.2: h/linimurn achievable performance index for Example 3.1, using polynomials of 3 different degrees.
minimum value for (3.7)
- -
f2 = -0.7069 fi = -4.886t + 2.915
polynomials
f2 = 1.76St - 1.978 f i = 5.934t2 - 4.231t - 0.3412
3.085
- - f2 = 1.046t2 - 2.550t + 1.396
4896
1.152 15 -08
1.254 14.71
Chapter 4
Pseudo-Decentralized Swit ching
Control
In previous chapters, it was assumed that the model of the plant being studied is
esactly known, which is impossible to achieve in "real world" applications, i-e. the
problem of robustness was not considered. This gives motivation to this chapter,
which studies the decentralized control of plants with uncertain mathematical models.
In particular, it is assumed that the plant is described by a continuous LTI model,
which is contained in a specified family P of plant models, and in this case it is
assumed that a family of decentralized controllers has been found to satisfactorily
control the models contained in P. -4 switching control scheme is then proposed
that uses the input-output information of each agent, to switch between the local
candidate controllers in each corresponding agent. -4 binary communication link is
assumed to exist between the local control agents to assure that ail agents switch to
the next candidate controller, simultaneously. Simulation results: using the proposed
decentralized scheme, are given and compared with the centralized scheme given in
Pl]-
4.1 Decentralized Feedback Mode1
Consider the family of strictly proper, controllable and observable, decentralized LTI
systems P = {P'? k E p = (1, . . . , p ) ) , which has m control agents and is described
by :
i . = ~ ~ z + [ b f ...cl [ ] + [ f . . e ] [ w1 ] 7 ,4.w
um Wm
Yre f ,m Ym
(4. lc)
mhere x ( t ) E Wk is the state, q ( t ) , j E m = {l, . . . : m ) is the control input of control
agent #j, y j (t) is the output to be regulated of control agent # j, [ w; ( t ) . . . w&(t ) 1' is
the plant disturbance and ëj(t) is the error at control agent #j, which is the difference
between the specified reference input IJ,.J~ and the output y j - Al1 matrices and
variables here have appropriate dimensions. It is assumed that the plant models Pk,
k E p are known, and that the actual plant mode1 is contained in P = {Pk, k E p}.
It is also assumed that yj and Yrefj, j E m are measurable, and that the system
given by (4.la) and (4.lb) is observable from each agent, i.e. (6, A*) is observable
for V j € ni, Vk E p.
It is assumed that a dynamic decentralized LTI controller represented by the
block diagonal rnatrix K k ( s ) =diag([Kf(s) . . . Kk(s)]) has been designed for each
plant mode1 Pk, k E p, and has the structure:
where z j ( t ) E IRck1 , and where, without Ioss of generality, it is assumed that ly = 1
for al1 k E p and j E fi, by adding unobsenable stable modes if necessary. In
particular, it is assumed that each Kk has been designed so that it stabilizes the
closed-loop system corresponding to Pko and that the controller satisfactorily performs
disturbance rejection and/i>r tracking for Pk, for a specified class of tracking and
disturbance signals.
The previous controller (U), can alternately be written in terms of a static output
feedback controller:
wlien applied to the following augmented controllable and observable system [41]:
Yre f,i
Yref ,m
where:
and:
Thus, at this point, we have an appropriate decentralized controller (4.3) which can
respectively control each plant mode1 PL, k E P, and we wish to find a way to control
the actua1 plant, when it is assumed that the plant can be described by some LTI
mode1 contained in B. -4 decentralized switching technique will now be proposed to
accomplish this goal.
4.2 Main Result
The decent ralized swi tching controller will be obtained by taking the same approach
as used by Miller and Davison in [41] for the centraiized case. The general approach
adopted consists of two phases. We initiaily want t.o find an upper bound for the
error signal, which is associated with each Pk and the corresponding controller gain
k k The upper bound signals will then be used to find an appropriate controller by
switching between the different controllers. Note that since a decentralized switching
scheme is to be applied here, each control agent wïll be examined separately. Hence,
based on the input and output signals of each agent, some auxiliary signals will be
introduced to construct an upper bound signal for each agent.
Decentralized Switching Controiler 1 : Following [41], the decentralized switch-
ing controiler consists of two phases:
Phase 1: Finding a bound o n the initial condition
Here, we want to define for each control agent, parameters in a similar way as was
described in [41] (Lemma 1). Choose any arbitrary positive value for T and define:
and let:
a k j s := the srnailest singular value of Wkj- (4-5)
Since it is assumed that the system is observable from each agent, this implies
a k j 3 # O. Now, define:
and let:
(here, I I . 1 1 denotes the 2-nom of a vector or the corresponding induced n o m of a
matrix). With ü ( t ) = O for t E [O, Tl and [" ] = 0, but y j (O) not necessarily
equal to zero, finci:
and assuming that I lwj(t) 1 1 define:
Assuming that I luj(t) l l 5 &, for t E [O, Tl; and that the plant is described by the LTI
mode1 p k , it then follons from Lemma 1 in [41] (see AppendLv B) that:
Xom? find constants yki > O and At < O as described in Lemma 2 in [41] (see
-4ppendiu B) , so that:
and let:
Define nom the following p x m auxiliarq- signals:
i k j ( t ) = b k j ( t ) + ? k 2 e k j ( t ) rkj(0) = O , t E [O,T], k E p, j E m,
where:
I l
and:
Note that the upper bound signal introduced in [41] is:
With ü( t ) = O for t [O,T] and [I':, ] = O then:
and since
Phase 2 :
( 0 ) 1 5 j this impiies that ëkj( t ) 2 ~ l ~ ~ ( g ( t ) - B k
Searching the gains
Now, control action will be applied. Let the control input be:
Let the p x rn auxiliary signals be given by:
where:
Set t l := T, and for every k E (2, . . . . p + 1) for mhich tk-l # 00, define tk by:
where:
1 : there exists a time f E [T, t] for which
if the set is nonempty (the minimum exists if it is nonempty), and oo otherwise; if
t,+l is defined and finite, then define t,+2 = O and set K;+' = O. j E mo where E is a
small positive constant. and is a matrix consisting- of the rows:
of ck, where s, denotes the number of outputs a t control agent #q, with so = O (in
the special case, when each agent of the plant is only single-input, single-output, the
matrix is just a matrix consisting of the r o m j , m + j , and 2m + j of the matrix
Ck).
4.3 Properties of Decentralized Switching Controller
The following result shows that the switching controller "chooses" a stzbilizing LTI
controller for the plant in finite time.
Theorem 4.1 : Consider the family of LTI plant models (4.l), and the corresponding
family of decentralized LTI controllers (4.3). Suppose that y,.,j, u, E PC,, j E ml
r w1 1 1 4
condition x(0), when the switching controller defined in Phase 1 and Phase 2 is applied
to the plant, the closed-loop systern has the following properties:
(i) The switching controller stops switching in finite time, Le., there exists a time
t' > O, by which no switching occurs Vt > t* .
(ii) The state Z ( t ) defined in (4.4) is bounded, Vt 1 0.
To prove this, we need the following preliminary results.
Lemma 4.1: For any vector x = [il] , there exists j E {1, . . . , m}, so that
Xm
Jm 11xj11 2 11x11.
Proof of Lemma 4.1 : We can write llxll in terms of llxj[l1 as follows:
(note that in the general case, each element xj can be a vector). Choose j now so
that:
which implies that:
Lemma 4.2 : Consider the following differential equations:
+ L ( t ) = ATl ( t ) + ré1 ( t ) , i l (h) = f 1 0 ,
;2 ( t ) = XF2 ( t ) + ré2 ( t ) , F2(t0) = r 2 ,
Assume that 7 > 0, f l 0 , and é l ( t ) 2 ë 2 ( t ) > O , t 2 to. Then, T l ( t ) 1
?2(t) , t L t o -
Proof of Lemma 4.2 : The solutions of the above first order differential equations
are given by:
7eNt-r) ë, dTl FI ( t ) = rloe
Since yël ( T ) 2 y ë 2 ( r ) , tO 5 T < t , and e*(t-T) > O , Vt. r , this implies that:
Also, since iio 2 F20, then T ~ ~ ~ " ~ - ~ o ) > ~ ~ ~ e " ( ' - ~ ) , which implies i l ( t ) 3 F2(t), t 2
t,. . Proof of Theorem 4.1 : Let y,,!,
fi, and J [ [ ] 1 5 6. It the. fbllows fmm Lemma 4.2 th.:
Wm
where rk(t) , defined in (111 (see -4ppendix B), is given by Equation (4.11). Further-
more, on assuming that ü(t) = ~ ' i j ( t ) , since at least for one j E fi, Ekj(t) is greater
e l 1 than IlG(t) - ~ ~ ( ( i t ) - D~ ( i 1 )II, t E (tl, toi] in (4.11), i t follows from
1 %ef,m(t) J Lemma 1.1 and Lemma 4.2 that rkj(t) 3 rt(t), t E (tl, ti+l]. On the other hand, it
can be easily shown that for any j E ni,
Thus, if the appropriate controller ü(t) = KLjj(t) is being applied, then the n o m of
the signai:
is less than fkj, t > tl in (4.12). Since it has been assumed that the system is
observable from each agent, this implies that if any of the applied controllers results in
an unbounded state response, then the outputs of al1 control agents of the system will
become unbounded. The proof now follows from Theorem 1 in [41] (see Appendix B).
œ
Remark 4.1 : Note that despite the fact that the switching scheme is decentralized,
a weak communication link is in fact required to exist between the agents, so that al1 r 1
1 ~re , , j ( t> J of al1 agents have reached their corresponding upper bound signals). The only in-
agents snitch at the same time (this occurs when the signals Ilij,(t) -
formation that we oeed in this case, is the time instant in which each agent hits its
corresponding upper bound. In other words, we need to assign a binary flag to each
control agent. It is to be noted that under some conditions, this binary information
can be communicated via the interconnections between the subsystems.
t h e block diagram of the proposed decentralized switching controller for a system
with two input-output agents is depicted in Figure 4.1. The weak communication link
or binary signals between the switching controllers are shown by the dotted lines.
Decision . - . . . - . . . . Switching Decision
Maker 1 Command
O
O
Switching Command
I I
- . - 1 [ .J -
Agent 1 Agent 2 4 C . YI "2
Figure 4.1: Block diagram of the decentralized switching controller 1.
Remark 4.2 : Sometimes, it is more convenient to deal with continuous signals
instead of discontinuous ones. Thus, in the case that f ' w j is discontinuous, which
makes Il 0, ( t ) - II discontinuous, one can design an appropriate first order
filter such as:
where:
X:=min{Xk: k E p), < A ,
and use ëf&) instead of Il&(t) - 1 II to find the switching times.
Remark 4.3 : One can make similar assumptions as made in [41] (Theorem 2), to
parantee that if the plant is PL, then the final gain is kk.
Remark 4.4: It is to be noted that in using this method of decentralized switching,
we need to compute p x rn awriliary signals (one for each plant and each agent) whereas
in the centralized switching method introduced in [41], only p ausiliary signals are
required (one for each plant). However, the total nurnber of input and output signals
required to find the upper bound signals for each agent in the decentralized case, is
less than the total number of input and output signals required in the centralized
case, and, the required flow of information in the centralized switching is much more
demanding than for the decentralized case.
4.4 Numerical Example
Example 4.1 : Consider the system (4.1) with the following family of plant models:
0.124 0.595 a Model P2 : A2 =
-0.274 -0.107 0.984 0.847
4 = [ -0.144 -0.525 ] , 4 = [ -0.685 -0.207 ] , e: = eg = [ 0-1 0-1 ] , j; = f$ = 0.1,
1 0.953 Model P3 : .A3 =
-0.246 -0.422 0.754 0.872
4 = [ -0.784 -0.608 ] , 6 = [ -0.979 -0.196 ] , ei = es = [ 0.1 0.1 ] , = f: = 0-1,
1 0.597 a Model P5 : Ag =
-0.378 -0.131 0.255 0.730
6 = [ -0.053 -0.064 ] , 4 = [ -0.019 -0.709 ] , e: = ez = [ 0.1 0.1 ] , ff = fi = 0.1.
For each plant model, a decentralized servomechanism controller of the form:
lias been designed to minimize the performance index:
with p = 1, for the case of constant reference inputs Yre,,~ and yre,z, using -41-
gonthm IV in -4ppendiu -4- The corresponding gains obtained for each controller
C$ k E {l, 2 ,3 ,4 ,5 ) , j E {1 ,2) , are given as follows:
0 For plant model PL :
m For plant model p2 :
m For plant model p3 :
m For plant model P4 :
For plant model P' :
which result in the following eigenvalues for the closed-loop system associated with
each plant model:
Setting T = lsec and on applying the switching controller equations in Phase 1, the
following parameters are obtained from (4.5)-(4.10) in order to construct the upper
bound signals for each plant model:
&For plant model P' :
al11 = 2.955 x IO', a12 ' = 5.965 x IO-', a131 = 6.769 x 1od2,
0112 = 4.815 X IO', a i 2 2 = 1.711, a l32 = 4.154 X 10-~,
711 = 3.500, 712 = 4.686' 713 = 3.116, Al = -1.849 x IO-',
0 For plant model p2 :
a 2 1 1 = 1.072 x 103, an1 = 2.346 x IO', azsl = 1.866 x 10-~,
a212 = 8.443 X IO', = 2.819, a232 = 2369 X X O - ~ ,
721 =2.660, 722 =3.701, 723 = 1.194, A 2 = -1.975 x 10-',
0 For plant model P3 :
as11 = 1.674 x 102, a321 = 7.754, = 1.195 x 10-~ ,
a s i 2 = 5.621 x IO', = 2.484, (1332 = 3.558 x 10-~ ,
731 = 1.333, 732 = 1.930, 733 = 5.178 x IO-', A3 = -1.108 x 10-l,
0 For plant model p4 :
a411 = 1.004 x 102, a 4 2 1 = 2.404, a d 3 1 = 1.993 x 10-~ ,
a 4 1 2 = 9.637 x lol, am = 2.221: (1432 = 2.075 x 10-~,
7.11 = 1.103, 7 4 2 = 1.103, 7 4 3 = 3.372 x IO-', X4 = -1.355 x IO-*,
0 For plant model p5 :
a511 = 1.514 x 104, a521 = 1.796 x 102, as31 = 1.321 x 10 -~ ,
(1512 = 3.345 x 102, 0522 = 6.997, -2 = 5.979 x 10-~ ,
751 = 1.479, 752 = 1.502. 753 = 1.246 x IO', Ag = -1.340 x 10-~.
We now have a set of five decentralized two-input, two-output LTI systems. Assume
nonr that 6, = W2 = yrcf,l and ~ ~ ~ 1 . 2 are square waves of magnitude 10 and period
lûO?r, that the unknown plant is P5, and that the order of switching controllers is
C: -t C: + Cf + C: + C:, j = 1,2. Choose ~ = 1 and let x(0) = [0.5 0.51'.
The folloning closed-loop system results of Figure 4.2 are then obtained on applying
the decentralized switching controller 1 to the system. The transient response and
steady-state response of the output are given in Figures 4.2 (a) and (b) respectively
The thick lines indicate the output yl and the thin lines, indicate y*. Figure 4.2
(c) shows the upper bound signals for each agent, with the thick Line and thin line
indicating the signals for control agent #1 and control agent #2 respectively, and the
signal g ( t ) in Figure 4.2 (d) represents the switching instants. It can be seen that
the system switches to controller nurnber 2, 3, 4 and 5 after about 22, 31, 39 and 72
seconds respectivelly.
-51 I O 200 400 600
(ai t (sec)
Figure 4.2: Decentralized switching control for Example 4.1. (a) Transient response; (b) steady-state response and reference input; (c) upper bound signals; (d) switching instants.
For comparison, if we use the centralized approach introduced in [41] with the
above plant models and controllers, and with the same T, E , tracking signals, and
initial values, we obtain the results shown in Figures 4.3 (a), (b), (c), and (d) for
the transient responses, steady-state response, upper bound signal, and switching
instants, respectively. .4gain, the thick lines in Figures 4.3 (a) and (b) represent
the output at control agent #1, while the thin lines represent the output at control
agent #2.
Figure 4.3: Centralized switching control for Example 4.1. (a) Transient response; (b) steady-state response and reference input; (c) upper bound signal; (d) switching instants.
Comparing Figure 4.2 and Figure 4.3, we observe that both techniques provide
stability, tracking and disturbance rejection for the closed-loop system in this example,
but that the centralized technique, results in a better transient response and a faster
switching mechanism. The reason for this is that the decentralized switching scheme
is more conservative in terms of the upper bound signals obtained and the error
signals. On the other hand, the decentralized mitching scheme requires far less data
transmission to exist between the plant's control agents. It is also to be observed
that both controllers result in excessive "peaking" occurring in the output transient
response.
Example 4.2 : Consider a 2-input ,2-output mas-spring system shown in Figure 4.4.
Choosing the state vector x = [gnf Gnf y1 yi 32 y$ and vsuming w = 0, the
Figure 4.4: The 2-input, 2-output mas-spring system of Esample 4.2.
following state space equations are obtained:
Let Ar = 10, ml = rn, = 1, kl = k2 = 1, and fi = f' = 0.1, and define the following
systems matrices:
The follouing set of 3 plant models will now be considered:
O Model P' :
0 Model P2 :
Mode1 P3 :
Note that no fixed LTI controller can simultaneously stabilize this set of plant models
1241. On using the controller equations (4.2), the following decentralized servomech-
anism controllers are designed:
a For plant model PL :
a For plant model PZ :
For plant mode1 p3 :
where:
Assume now that P3 is the correct plant model. Consider the controller switching
order given by Cj + C: + C?, j = 1 , 2 and choose c = 1 and T = ioec. For
zero initial state and with reference inputs yref,i = 1 and Yref,z = -i, the results
given in Figure 4.5 are obtained. Figures 4.5 (a) and (b) give the transient response
and steady-state response of the output of control agent #1, and Figures 4.5 (c)
and (d) give the corresponding output for control agent #2, and Figure 4.5 (e) gives
the switching instants. It is clear from these figures that the system will switch to
the correct controller in approximately 3 seconds. The eigenvalues of the closed-loop
system, when plant model P3 is controlled by the decentralized controller C3, are
given by:
Figure 4.5: Decentralized switching control for the mass-spring system of Exam- ple 4.2. (a) Transient response of control agent #l; (b) steady-state response of con- trol agent #l; ( c ) transient response of control agent #2; (d) steady-state response of control agent #2; (e) switching instants.
Chapter 5
Decentralized Switching Cont rol
for Hierarchical Systems
In this chapter, a decentralized switching scheme will be introduced for systems with
a certain type of structure, and where the switching controller acts on each control
agent independently of each other. in other words, there is no need to communicate
between the different control agents as was done in Chapter 4, using this decentral-
ized switching controller. However, this is accomplished a t the expense of assuming
a specific structural constraint on the direction and strength of interconnections ex-
isting between different subsystems of the plant. The controller does not require any
knowledge re plant models for the system or any bound on disturbance signals in the
systern. The only requirement is that there is a known set of controllers, containing
at least one controller that stabilizes and regulates the actual physical system. Simu-
lation results for the proposed decentralized switching controller and the centralized
switching controller introduced in [12] are giwn and compared.
5.1 Decentralized MIMO Feedback Mode1
Consider the finite dimensional LTI plant mode1 Pk, k E P = { 1, . . . , p ) which has
rn control agents and is described by:
where x ( t ) E Rn* is the state, u j ( t ) E Rrn~ : j E m = ( 1 , . . . , ml is the control input at
control agent # j , and yj(t) E Rmj is the output to be regulated at control agent # j ,
w,(t) E W P j is the disturbance a t control agent #j, and ë,(t) E Rnj is the error at
control agent #j, which is the difference between the specified reference input ym/j
and the output y j - It is assumed that yj and yrejj are the only measurable signals.
-Assume now that a set of candidate decentralized controllers given by:
,\7here zj(t) E g k j , G" ~ l k j , ~5 E P k j xmj, JI E @ k j x m j , K* E IRrn, , J 1 J
L% I Emj x m ~ , and Mk J E Rmj Xmj stabilizes the plant mode1 (5.1) for k = 1 , 2 , . . . , p
respectively. Without l o s of generality, assume that z k j = I for al1 k E P and j E m.
Note that we do not necessarily assume that the plant model (5.1) (or their degree)
is known. In fact, one could assume that the set of decentralized controllers (5.2) for
the plant (5.1) has been obtained by using only experirnental methods, e.g., by using
on-line tuning methods (151.
The set of system modeki dong wïth their corresponding controllers can now be
written in the following forrn:
where (121:
-Cs .- 1, .- ( 1 - ~ C L ~ ) - ~ , 3 J
-
Yrel. 1
Yref .m
w1
W m
Yre j,m
We now wish to constntct a decentralized switching controller for the plant, which
selects an appropriate decentralized controller from (5.2) , so that for an unknown
plant which has a plant model contained in (5.1), the resultant closed-loop system is
stable.
5.2 The Main Result
We shall consider the case when the plant/controller is described by (i) a continuous-
time model or (ii) a discrete-time model.
5.2.1 Decentralized Switching Controller for Continuous-Time
Systems
Consider the set of plant models (LI), and partition the corresponding matrices into
the following:
mhere A*. E Rrk' X Q i , bk. E wki Xmij and c*, E Wrnj "'kj '1 '3
( 2 , j E m, k E P), where rkj's
are arbitras- positive real numbers, with the property that r k l + . . . + r h = nk.
Assumpt ion 5.1 : Given c 3 O, assume that:
either for j > i or for j < i, where A:, &:, ej are fked matrices with unity norm.
87
Here if E = 0, the conditions of Assumption 5.1 correspond to a constraint that the
plant (5.1) consists of a hierarchy of interconnected subsystems as illustrated in Fig-
ure 5.1; if E is "small", the assumption corresponds to a plant consisting of a hierarchy
of interconnected subsystems which are weakly interconnected to the remaining sub-
systems of the plant. One can also look a t the magnitude of the interconnection
transfer function a t different frequencies to check the weak connectedness (future re-
search is required to characterize exactly when the conditions of Assumption 5.1 can
be satisfied) .
Define Decentralized Switching Controller II for each agent, as follows:
Decentralized Switchinn Controller II :
mhere j E ra. t E (t,, ts+lj]7 z( tz) 0: s E {l , 2 , . . .}, and i := ((s-1) mod p ) + l .
Set t l j := O, V j E m and for s 2 2 such that t,-u is finite, define the switching time
ts j as:
min t 3 t > ts-lj, and 11 [ z i ( t ) ~ > ~ ( t ) ] II = Fj(s - 1)
tSj := if this minimum exists
I - othenvise
where 11. II denotes the oo-norm, ë,,(t) is the filtered error signal a t control agent # j ,
given by:
and where F is a modijied stmng bounding function (f E MSBF) [12], mhich is defined
to be a strictly increasing function from N to R, such that for al1 positive real constants
(ao, al : a2) the funct ion:
goes to infinity as i + oo. An example of such a MSBF function is ie i2 , i E N
Now, considcr the following assumption:
Assumpt ion 5.2 [12]:
The condition (i) and (ii) of Assumption 5.2 can be easily met by setting z j ( t ) = O
and ëfj(t) = O. j E m; the conditions (iii) and (iv) of Assurnption 5.2 are necessary
conditions for there to exist a solution to the decentralized robust servomechanism
problem [16]. The following main result is obtained:
Theorem 5.1: Consider the set of plailt models (5.1), and assume that the plant
models have the structure defined in Assumption 5.1, and that the actual plant has a
mode1 which belongs to this set. Then there exists c > O sufficiently small such
t hat the following property holds: consider the corresponding decentralized con-
trollers (5.2) for each agent, and apply the switching controller 1 to each agent at
time t = 0; tlien for every Fj E MSBF and X E IR+, for every initial condition
x (O), 7 (O), and ëf (O) satiswng -4ssumption 5.2, and for every bounded piecewise
continuous reference input and disturbance signal, the closed-loop system has the
following properties:
i) there exists a finite time tj,ss > O for each agent, by which no switching occurs
at that agent for t > t,,,,,
ii) al1 the controller states z j ( t ) , the plant states x ( t ) , and the filtereci error signais
ë J j ( t ) are bounded, and
iii) for alrnost al1 controller parameters G:, H:, K:, and L: ( j E m, k E P), for
constant refererice and constant disturbance signals, the error signal a t each
agent asymptoticolly decreases to zero as time increases.
Proof of Theorem 5.1 : The first step in proving this theorem, is to impose the
conditions of .i\ssumption 5.1 onto the closed-loop systern (5.3). Assume that the
actual plant's mode1 is given by Pr, (r E P), and that over an arbitrary time interval,
the controller (G: , H ) , K: , L: ) has been applied to control agent # j. Assume also
that condition (i) in Assurnption 5.1 holds true ~ 4 t h e = O; in this case, we will
have the following structure for the resultant closed-loop system in this time interval.
Define:
In this case, it can be easily seen that the closed-loop system corresponding to control
agent #l is given by:
which implies that the differential equations of the first rrl plant states [XI . . . xrFl]'
and the first l r l controller states [zl . . . zi,,]' are independent from the rest of the
augmented mode1 states. Thus, since al1 conditions in .4ssumption 5.2 hold true,
BIBO stability of the subsystem #1 represented by XI,. . . ,xrr1, 21,. . . , yrl which
corresponds to control agent #1, immediately follows by Theorem 3.1 in [12], and al1
three conditions in Theorem 5.1 are fulfilled for this subsystem. Thus, the switching
mechanism will stop switching in a finite time for control agent #1, and al1 the
states in this subsystem will stay bounded. L.ikewise, if we consider the differential
equations corresponding to x,,,+l, . . . , xrrl zirl +I, . . . , zi,, +lF,, we will similarly see
that they are functions of the first rrl +rr2 plant states and the first Lr1 +lr2 controller
states. Since the states of subsystem #1 are bounded. one can assume that the signal
introduced by subsystem #1 in the differential equations of subsystem #2 acts as
a bounded disturbance. Thus, subsystem #2 also meets the required conditions of
Theorern 3.1 in [12] and it fulfills al1 three conditions in Theorem 5.1. Similarly,
one can conclude that the subsystem associated with control agent #3 and al1 other
subsystems will lock ont0 an appropriate controller in a finite time, resulting in BIBO
stability for the whole system, and asyrnptotic error regulation for al1 of the control
agents.
Assume now that c > O in -4ssumption 5.1. Because of continuity. it can be con-
cluded from [12] (see Appendix B) that if € is "sufficiently small" , then the switching
mechanism can still be successfully carried out using the proposed method. In other
words, if there e-xists a sufficiently weak interconnection from the lower level subsys-
tems to the upper ones in Figure 5.1, then the conditions in Theorem 5.1 will still be
achieved. Systems which have this property are said to be "Weakly interconnected".
Such a system Ml1 be illustrated in Example 5.2.
Remark 5.1: One can use the dual modifiai strong bovnding functions defined in
[12] to compare z j ( t ) and ë f j ( t ) with two different functions separately. This can be
used to potentially improve the transient response of the system-
Remark 5.2 : Note that given a sufficiently small E, -4ssümption 5.1 may not hold
true for a set of state equations, but it may hold true for another set of state equations
which is related to the original set by a similarity transformation. For the case when
i = O, this irnplia that if the transfer rnatrix 1 i 1 (sr - A*)-' [ bf . . b; ] has
L4J a lower-triangular or upper-triangular forrn, then Assumption 5.1 holds true. The
digraph of such systems can be represented by a hierarchy of interconnected subsys-
tems, where the interconnection goes only in one direction as depicted in Figure 3.1.
In other words, there exist no strongly connected pair of subsystems in the whole
system 1311. Since no link exists from the lower level subsystems to the upper ones,
once al1 subsystems above subsystem # j are locked ont0 the proper controller, it is
guaranteed that subsystem # j will also find the proper controller and stop switching
in a finite time.
Remark 5.3: Note that in the proposed decentralized switching scheme, the plant
mode1 need not be stable, strictly proper, minimum phase: controllable and/or ob-
servable. In addition, no a priuri bound on the magnitude of the reference input or
disturbance signal is required.
Figure 5.1: A hierarchy of interconnected subsystems which satisfies Xssumption 5.1.
Remark 5.4 : Note that the proposed decentralized switching scheme acts on each
agent independently, Le., no communication of any type is required to exist between
the control agents.
5.2.2 Decentralized Switching Controller for Discrete-Time
Systems
Using the same approach as given in Section 5.2.1, one can directly obtain the discrete-
time counterpart for the proposed decentralized switching controller. This can be
accomplished by replacing the differential equations (5.1 ), (5.2), (5.3), and (5 .5) .
with the corresponding difference equations, and with the corresponding switching
times defined as:
min k 3 k > k,-v , and [ ( [$[k] c [ k ] ] 11 2 Fj(s - 1)
kS j := if this minimum exists
I - O t herwise
5.3 Numerical Examples
Example 5.1 : Consider a set of LTI systems described by (5.1) with m = 2 and
assume that the actual plant mode1 is represented by the following matrices:
-41~0, assume that a set of controllers CF, k E {l, 2 ,3 ,4 ,5 ) , j E {l, 21, as represented
by (5.2) is given, where G: = O, H: = 1, J: = -1, -44: = O, j E m, k E P, and
where Kt , L: are given by:
C: : L: = 5.676, K: = 1.618; for plant model PL :
Ci : Li = -2.992, Ki = -1.771,
C: : L: = 0.1249, K: = 0.2689; for plant model p2 :
Ci : L; = -0.2751, K: = 0.7505,
C: : L: = 2.808, K: = 1.253; for plant model P3 :
q : L; = -2.100, Ki = -0.9962,
C : L: = 0.8143, 1<: = -0.3355; for plant model p4 :
q : q = 0.6932, 1(: = -0.8341,
C: : L: = -2.530, K: = -0.9202; for plant mode1 p5 :
Cg : L; = 2.743, Kz = 0.9680,
where at least one of these controllers stabilizes the plant (in fact, the decentralized
controller represented by {Cf , Cg} which has been found by using Algorithm III in
Xppend i~ A, accomplishes this). Assume now that the initial plant state x (0 ) is zero,
that the reference input is a 0.02Hz square wave for both agents, and that we have a
constant disturbance of the form w (t) = [1 11'. Set the initial couditions ëlj (0) and
t j ( 0 ) , j E {l, 2) to zero, and let X = 1. On applying the proposed decentralized
switching controller with the following Fi (i), F2(i) E MSBF:
and the controller switching order given by C: -t Cj + C: -t C: + C:, j = 1,2,
we obtain the simulation results given in Figure 5.2. Figures 5.2 (a) and (b) give the
transient and steady-state response of the output a t control agent #L respectively,
while similar results for control agent #2 are given in Figures 5.2 (c) and (d). These
results show for this plant and the given set of controllers (with the particular order
of switching) and hnctions Fl(i) and F2(i) given by (5.7) for each agent, that Ive have
good tracking and disturbance rejection for constant reference inputs and constant
disturbances at both agents. Furthemore, the functions gl(t) and g2(t) in Figures 5.3
(a) and (b) give the switching times corresponding to each agent ( g j ( t ) = k represents
the time interval within which the controller number k is being examined at control
agent #j). It can be seen from these figures that control agent #1 and control
agent #2 lock ont0 the correct controilers (C: , CZ) in less than 6 seconds and less
than 11 seconds, respectively. -41~0, they show that each controller has been applied
to the plant three times.
For cornparison, on applying the centralized switching mechanism introduced in
[12] to the sarne set of decentralized controllers and system parameters given in Ex-
ample 5.1, the results of Figure 5.4 are obtained. The transient response of the
closed-loop system in this case is slightly better than that of the decentralized case
(Figure 5.2) as expected. It is to be noted that there is an undesirable overshoot in
the initial transient of the output response, for both the decentralized and centralized
switching controllers, in t his example.
Figure 5.2: Decentralized switching control for Example 5.1. (a) Transient response of control agent #l; (b) steady-state response and reference input for control agent #l; (c) transient response of control agent #2; (d) steady-state response and reference input for control agent #2.
Example 5.2 : Consider a set of decentralized 2-input, 2-output LTI control systems
and assume the same actual plant as studied in Example 5.1; then this system is
described by:
Yi? ( 4 -0.2357s2 - 0.09293s + 0.0863 g2L(s) := - = Ul (s) s3 + 0.6417s2 - 0.1845s + O.OO669I7
Figure 5.3: Decentralized switching controller for Example 5.1. (a) Switching times for control agent #l; (b) switching times for control agent #2.
Since the transfer function from u2 to y1 equals 0, the plant consists of two subsystems
with a single interconnection represented by gzl(s); in other words, the plant is not
strongly connected, and the conditions of -4ssumption 5.1 hold. Also, assume that
Figure 5.4: Centralized switching controller for Example 5.1. (a) Transient response of control agent #l ; (b) steady-state response and reference input for control agent #l; (c) transient response of control agent #2; (d) steady-state response and reference input for control agent #2.
the following set of controller parameters for the closed-loop system is given:
for plant model :
for plant model PZ :
for plant mode1 P3 :
for plant model P4 :
for plant model f5 :
Here the decentralized controller {Cf, Cz} has been designed using Algorithm III
in Appendix -4. Also, assume that the plant has zero initial states, that the refer-
ence inputs for both agents are square waves of frequency 0.02Hz as in the previous
example, and that there are no disturbances in the s-tem. Set ëjj(0) and t j ( 0 ) ,
j E {1,2} to zero, and let X = 1. On applying the proposed decentralized switching
controller with modified strong bounding functions given in (5.7) and the same con-
troller switching order as the previous example, the results given in Figure 5.5 are
obtained. Here, control agent #1 and control agent #2 lock ont0 Cf and Ci in 5 and
15 steps respectively and good tracking is achieved for constant reference inputs and
constant disturbances at both agents.
Figure 5.5: Decentralized switching control for Example 5.2 with 912 = O. (a) Tran- sient response of control agent #1; (b) steady-state response and reference input for control agent #l; (c) transient response of control agent #2; (d) steady-state response and reference input for control agent #2.
.4ssume now that g&) is equal to a non-zero constant g12 = 0.5 instead of zero,
which introduces a new interconnection in the previous interconnected system. This
implies that the overall system is now strongly connected. The corresponding optimal
decentralized controller obtained for this system using Algorithm III in Appendix -4
is now given by:
Cf : Lf = -2.609, K: = -0.9755; for plant mode1 p5
C; : L; = 2.596, Kz = 0.9842-
Suppose that al1 other controller and plant parameters remain unchanged. In this
case, on applying the previoa decentralized switching controller II to t his system,
the results given in Figure 5.6 are obtained. These figures show that the outputs of
the closed-loop system stiil remain bounded, although the system no longer has a
hierarchicai structure. In fact? this is the case for -1.885 < gl* < 0.7696.
-1.5~ 1 O 20 40 60 80 1 0 0
(cl t (sec)
Figure 5.6: Decentralized switching control for Example 5.2 with g l z = 0.5. (a) Tran- sient response of control agent #l; (b) steady-state response and reference input for control agent #1; (c) transient response of control agent #2; (d) steady-state response and reference input for control agent #2.
Remark 5.5 : It should be noted that the switching controller II will not necessarily
lock ont0 a particular controller which has been designed for the actual plant model.
In other words, if other controllers in the given controller set stabilize the actual plant
model, it is possible that switching will stop when this particular controller is being
examined. This is shown in the next example.
Example 5.3 : Consider the 2-input, 2-output mas-spring system of Figure 4.4. For
ml = 7722 = 1, M = 10, kr = kg = 1, JI = f2 = 0.1 and w = O, this system can be
descri bed by the following LTI model:
This mode1 will be represented by ( [ 9 ] 3 A l [ b l 4 1 ) -
It is desired now to design a decentralized controlIer of the fom:
control agent #l : ol = -El,
Pi = &PI + Bci (-el + S i ~ r ) ,
u1 = ccipi ,
control agent #2 : % = -ë2,
A? = -4&2 + & ( 4 2 + P277-21,
u2 = Cc2P2,
to solve the robust servomechanism problem for t
w and constant y,.~ signals. The following control
iie system, for the case of constant
.er parameters are thence obtained
for t his decentralized controller:
In this case the above decentralized controller can be written in the form of (5.4a)
and (5.4b) as follows:
Assume now that the plant to be controlled has a mode1 which is contained in the
family of models {Pk, k = 1,2 , ..., 5), where Pk, k = 1,2, ..., 5 are given as follows:
This corresponds to assuming that the plant to be controlled may have incorrect
polarity of inputs and outputs, due to incorrect wiring. Note that no fived LTI
controller can simultaneously stabilize t his set of plant models [24].
The following family of controllers \vil1 now be considered to control the plant:
Gf =G1, H: = Hl, J: = Ji,
K: = KI, L: = LI, M; = h11; For p2 : I
For p3 :
G: = Gl, Hf = Hl, J: = JI,
Kt = Ki, L: = Li, Mf = A&; (5.9d)
= G,, H: = -H2? J i = diag([-1 1 l ] ) J z ,
K: = K2, L: = Lz, M i = hf*,
G: = Gl, H; = H l , J; = J I ,
K: = ICI, L: = LI, .I:= Ml; F o r p5 :
-4ssume n o a that the actual plant is described by P5, but that at t = ZOsec, it
clianges to P4 (due to a change in the polarity of the output of control agent #2).
Consider the controiler switching order given by Cj + Cj + Cj + C; + C:,
j = 1,2. Assume zero initial condition on the plant, that the system has refererice
signals yreJv i ( t ) = 1 , gref,a(t) = O , t 2 0 , and that the MSBF functions are given by
(5.7). On applying the proposed decentralized snitching scheme to the system (5.8)
wit h the controllers defined in (5.9), the results given in Figure 5.7 are obtained. These
results show that good tracking of both agents, with reasonable transient responses
are obtained. Figure 5.8 gives the switching times. Note that the final controiler
obtained for control agent #1 is C: and for control agent #2 i t is Ci immediately
before the change of plant a t t = 20sec, while the corresponding final controllers
obtained after the change of plant for control agent #1 is C: and for control agent #2
is Ci. .!%O, note that s defined in Assumption 5.1 is greater than zero for this system,
which implies t hat there exist interconnections in both directions between the two
subsys terns.
Figure 5.7: Decentralized switching control for the mas-spring system of Example 5.3. (a) Output response of control agent #l; (b) output response of control agent #2.
1
,0.8 - - j -
<0.6
0.4
0 2
0
- - - - Plant changes from 9 IO P'
1 1 1 I 1 I 1 1
O 5 1 O 15 20 25 30 35 40 45 50
Figure 5.8: Decentralized switching controller for the mas-spring system of Exam- ple 5.3. (a) Switching times for control agent #l; (b) switching times for control agent #2.
Chapter 6
Application of Generalized
Sampled-Data Hold Functions to
Decentralized Control Structure
Modification
It was shown in Chapter 2 and Chapter 3 that under various circumstances, digital
controllers can potentially improve the overall performance of the closed-loop decen-
tralized system. Decentralized adaptive control n.as then introduced in Chapter 4,
and in Chapter 5 it was sho~vn that for uncertain systems with a certain hierarchical
structure, one can apply a decentralized adaptive switching control scheme to regulate
and stabilize the system? with no communication links required to exist between the
different control agents. In this chapter it will be shown how one can use generalized
sampled-data hold functions to convert a continuous plant to an equivalent discrete-
time system with a desired structure, in particular a hierarchical structure, which can
directly be used to simplify the design of decentralized control systems for the plant.
.4n example application study is included, which applies the decentralized adaptive
switching controller design of Chapter 5 to a plant which has been transformed, using
generalized sampled-data hold functions, to have a hierarchical structure.
6.1 Digraphs and System Structure
Consider the foilowing strictly proper continuous-time decentralized LTI system with
m control agents:
mhere x ( t ) E R" is the state vector, and u,(t) E R'J, and y , ( t ) E Rrj, j E fi' fi :=
(1, ..., n) are the control vector and output vector of control agent # j respectively,
and -4' b,, and cj are matrices of appropriate dimensions. The transfer matrix relating
the inputs and outputs of this system can be written as:
where:
As discussed in Chapter 2, g , ( s ) represents the directed arc from node j to node i in
the digraph of this system. For exarnple, the digraph of a systern with 3 input-output
agents is depicted in Figure 6.1.
In the control of decentralized systems, the structure of the interconnections gener-
ally has a significant role to play, and in some decentralized control design procedures,
i t is often assurned that there exists a bound on the interconnections between different
nodes. For example, in decentraiized adaptive control methods, such an assumption
Figure 6.1: Digraph of a continuous-time LTI system with 3 control agents.
is often made, or alternatively, certain structural constraints on the interconnections
are often assumed in order to assure the stability of the overall system [22], [62], [26],
and [61]. In the special case of systems which have a hierarchical structure, one can
directly apply centraiized control methods to each subsystem (represented by each
node in the digraph of the system), with no assumptions required to be made on the
bound of the interconnections, since signals coming from a higher level subsystems
to a lower level subsystems d l always be bounded, once the higher level subsys-
tems are stabilized. As an example of this, assume that the transfer functions g,,(s),
gI3(s), and fi3(s) in Figure 6.1 are al1 equal to zero, and that there are no external
disturbances applied to the system; then the corresponding system will have the hier-
archical structure shown in Figure 6.2, and assuming that the system has no unstable
Figure 6.2: Digraph of a continuous-time hierarchical LTI system with 3 control agents.
decentralized fked modes, one can then design a centralized stabilizing controller for
each subsystem separately. In this case, subsystem #1 which has the highest level in
the hierarchicai structure, wilI be internally stable and since there is no input signal
coming into this subsystem other than its control input which is bounded, this implies
that al1 output signals coming out of this subsystem will be bounded. The output
signals of subsystem #1 includes the interconnection signal going to subsystem #2
and subsystem #3 through gzl ( s ) and 931 ( s ) respectivefy. Now subsystem #2, which
has the second highest level will also be internally stable. and since al1 input signals
coming to this subsystem, including the control input and interconnection signal from
subsystem #1 are bounded, this implies that al1 output signals of this subsystem will
also be bounded. Thus, al1 interconnection signals coming from the higher level sub-
systems to subsystem #3 will aow be bounded and since this subsystem will also
be internally stable, using a similar argument, one can conclude that the input and
output signals of subsystem #3 are also bounded. Thence, it can be concluded that
the complete system will be internally stable. This implies that the decentralized
controller design problem for a hierarchical system can be carried out in a centralized
way for each subsystem.
Yote t hat the transfer matrix representing the hierarchical system of Figure 6.2
has the following lower-triangular forrn:
In the general case, the transfer ma t rk of a hierarchical system always has a lower-
triangular form or can be transforrned to such a form by renumbering the subsystems
appropriately (exchanging the number of subsystem #i with subsystem # j is equiv-
alent to exchanging the ith row with the j th row, and the ith column with the j t h
column in the transfer matrk) . I t is also to be noted that any LTI system with a
lower-triangular transfer matrix always represents such a hierarchical system. As an
example, consider a LTI system with the following input-output representation:
Define the new set of input and output vectors as:
The equation relating the new input and output vectors can then be written as:
-4s discussed earlier, in many decentralized control design procedures, i t is often
assumed that some type of condition on the interconnections esisting between the
subsystems hold. Note t hat even a weak interconnection existing between two stable
subsystems can destabilize the whole system. As a simple example, consider the
following second order 2-input, 2-output system:
Let a be equal to zero; in this case, the transfer matrix of the resultant system is:
Because of its hierarchical structure, this model can easily be stabilized by using the
following decentralized controller:
which results in the following closed-loop system:
The resulting closed-loop system can be interpreted as consisting of two stable sub-
systems connected by a unidirectional interconnection. Nom let us introduce a weak
interconnection in the opposite direction of the existing one, by assigning a nonzero
value to a; in this case, it can be verified that the resultant closed-loop system is
unstable for a 2 0.01.
Our goai now is to determine if and hom one can modify the structure of a system
by using generalized sampling. It is obsewed that if certain interconnections of the
system can be eliminated in the equivalent discrete-time model, then many decentral-
ized control problems can be easily solved by applying centralized design methods to
the individual subsystems of the resultant discrete-time system. In the next section,
we will show how this can be accomplished.
6.2 Main Result
Consider the continuous-time LTI system represented by (6.1). As discussed in Chap
ter 3, the equivalent discrete-time model obtained by applying generalized sampled-
data hold functions f j ( t ) , j E m: with a sampling period T, to each control agent is
described by:
where:
It is desired now to determine if one can choose a set of sampled-data hold functions
f j ( t ) , j E m, and a sampling period T, so that certain elements of the transfer
rnatris corresponding to the equivalent discrete-time model, become equal to zero.
The following result is obtained.
Theorem 6.1 : Consider the systern (6 .1 ) . There exists a sainpled-data hold function
f,(t), for each agent p E fi, and a sarnpling penod T, so that the equivalent discrete-
time model has a hierarchical structure, if and only if there exist distinct integers
il, ..., i, E m, a positive scalar h > 0, and a nonzero vector xi,, j = 2, ..., m in the
(A, b i j ) -
Proof of Theorem 6.1 : Since xi, belongs to the controllability subspace of (A, $), from Lemma 3.1 we know that for the sampling period T = h, there exists a sampled-
data hold function fi, (which may be obtained from equation (3.5)), so that in the
equivalent discrete-time model described by (6.3) and (6.4), b 4 = xi j - On the other
hand, since it is known that xiJ belongs to the null-space of
1 e-I J this irnplies that:
where Ad = eAT and cdik = îJ a s defined in (6.4). This implies that by rearranging
the elements of the input and output vectors, the transfer function corresponding to
the equivalent discrete-time mode1 has the following form:
which corres~onds to a hierarchical structure. On the other hand, if there esists no
arrangement for the control agents, so that the null-space of
and the controllability subspace of (A, bi,) have a common vector for h > O, this im-
plies that the upper-diagonal terms in the transfer function matriv of the equivalent
discrete-time model (6.3) cannot be set equal to zero for any input-output arrange-
ment. This completes the proof of the theorem. .
Xote that Theorem 6.1 gives necessary and sufficient conditions for a system to
have a hierarchical discrete-time structure. In the nest step, sufficient conditions are
given, based on the concept of controllability and observability, which are very easy
to check.
Theorem 6.2 : Consider the system (6.1). There exists a sampied-data hold function
f,(t), for each agent p E m and a sampling period T > O, so that the equivalent
discrete-time model (6.3), (6.4) has a hierarchical structure, if there e-xist distinct
integers 11, l2 E fi, so that the following two conditions both hold: r
a) the pair is not observable.
b) the pair (-4, b j ) is controllable for every j E ni, j # 1, .
Proof of Theorem 6.2 : Assume that conditions (a) and (b) of Theorem 6.2 both
hold. Let il = l , , i, = 12 , choose i2, . . . , im -1 so that {il,i2, ..., im) = {l, 2, ..., m), and
choose an arbitras- T > O. Now since the pair ( [ i1 ] , A) , is unobservable, this
c i m - 1
implies that the pair ( [ 7 1 , &) corresponciing to the quivalem discrete-tirne
Gm- 1
model will also be unobservable for any arbitrary sampling period T [13]. This in . .
turn means that the pair ( [ ? ] , .Ad ) , j = 2, ..., rn is also unobservable, which
Cij - ,
implies that the columns of the rnatrix [ ] (11 - A d ) are linearly dependent
C i j - 1
(Theorem 9-13 in [32]). Therefore, there exîsts a nonzero vector xi, so that:
On the other hand, since the pair (A, biJ ) , j = 2, ..., m is controllable, it foIlows from
Lemma 3.1 that for any sampling period T, there esists a sampled-data hold function
f ,] for each agent j (j = 2: ..., rn) such that b4 given by (6.4b) has the property that
bd,, = xily where xi, is defined in (6.6). Such generalized sampled-data hold function
may be obtained by using equation (3.5)) (non-unique) as follows:
f i , ( t ) = b;, e(T-t)A w<:,z~,,
where:
This results in:
Thence, using any arbitrary nonzero sampled-data hold function (such as a simple
zero-order hold) for f i , , along with the set of sampled-data hold functions given in
(6.7), the corresponding transfer matriv for the equivalent discrete-time mode1 will
have the lower-trîangular form given in (6.5). This completes the proof. . Theorem 6.1 and Theorem 6.2 provide the conditions under which the digraph of
a systern can be modified to a hierarchical structure by using sampling. In the next
step, the conditions under which the resulting discretized mode1 does not have any
decentralized fixed modes will be discussed.
Theorem 6.3: Given (6.1): assume that the pair (A, b,) is controllable for al1 j E m.
Then eAT E sp(Ad) is not a decentrdized fixed mode of the equivalent discrete-time
mode1 (6.3) , with respect to the block diagonal gain matrix K = diag(Kl, ... , Km) , Kj E RSj X r ~ , if the following three conditions hold:
i) for every Al,, XI, E sp(A), the relation Re (All) = Re (Al2) implies that
Irn ( X i , - Xi, ) # F, Z1,12E {1, ..-, n}, k = f l , f 2 ,...
ii) ~ ~ ë " f , ( t ) d t # O, A E sp(-A)
iii) the pair ([Il , A ) is observable.
Proof of Theorem 6.3 :
ehT E sp(Ad) is a decentralized kxed mode of (6.3) if and only if any one of the
following conditions hold 11 71 :
1) rank i 2) rank ([
3) rank [-
that {il, ..., i,} = m
rank [- for some i, E ni, j = 1, ... m such that {il ,..., 2,) = m
Ad - eXTI bd
7' ci, - d
.-
O ] < n
for some
m+') .an. ([ Ad - eATr bd, , hi,2 - - - b4,,,-L
G.7, O O ... O for some ij E m, j = 1, ... m such that {il, ..., i,} = rfi
-Assume now that conditions (i)7 (ii), and (iii) are al1 satisfied. Conditions (i) and (ii)
imply that the resulting discretized system will not lose controlIability and obsenr-
ability (corresponding to any controllable or observable pairs in the continuous-time
model) [40]. Thus, the pair (ild, b d j ) is controllable for d l j E mo and the pair ([Il , &) is observable. This ixnplies that the rank conditions (1) to (1 + 1)
for the existence of a decentralized fked mode do not hold. W
Discussion: The results of Theorem 6.1 and Theorem 6.2 are interesting in the
sense that they introduce a completely new application for sampling in control. The
results can be very useful in the design of decentralized controllers, when the struc-
ture of the original systern does not permit one to directly apply centralized controller
design methods to decentralized systems. In this case, if the conditions given in Theo-
rem 6.1 and Theorem 6.2 are met, one can find a set of GSHFs to modify the structure
of the system in the equivalent discrete-time model (6 .3) , (6.4): so that centralized
digital control methods can now be directly applied to each interconnected subsys-
tem. In particular, the results obtained may be applied to the decentralized adaptive
control problem discussed in 5. In this case, one applies sampling to obtain a hierar-
chical discrete-t ime model, and t hence directly applies digital adaptive controIlex-s to
each of the subsystems.
Remark 6.1 : Note that conditions (i) and (ii) in Theorem 6.3 ensure non-pathological
sampling for generalized sampled-data hold functions. In the case of a zero-order hold,
condition (i) is suficient to guarantee non-pathological sampling [13]. In addition, if
the eigenvahes of the continuous-time system are al1 real, condition (i) in Theorem 6.3
will be met.
Remark 6.2: It can be easily seen that asymptotic stability of the equivalent
discret e- time model will imply asymptotic stability of the corresponding cont inuous-
time model [28]. Therefore, if one designs a digital controller to stabilize the plant's
discretized model, it will result in stability for the original continuous-time system.
6.3 Numerical Examples
Example 6.1 : Consider the following 2-input, 2-output continuous-time LTI model:
The inputs and outputs of this system are related through a 2 x 2 transfer matrix as
The eigenvalues of this system are given by:
One can easily check the coiitrollability and observability matrices corresponding to
each subsystem to verify that the pairs (A? b l ) and (A, b2) are controllable, but that the
pairs (cl, A) and (c2, A) are not observable (which implies that there exist pole-zero
cancellations in the transfer functions of the rnatrix representation (6.9)). This implies
t hat the results of Theorem 6.2 can be used to obtain a sampled-data hold function
so that the transfer rnatrk of the resulting discretized system has a triangular forrn.
Both lowver-diagonal and upper-diagonal forms can be achieved, which rneans that any
of the subsystems can be assigned at a higher level in the digraph of the discretized
model. As an example, let us assume ive wish to obtain a lower-triangular transfer
matrix for the equivalent discrete-time rnodel, which implies that subsystem #1 will
have a higher level in the digraph of the discretized mode1 (this means that we choose
II = 1 and l2 = 2 in Theorem 6.2). In this case, it can be verified that for every
sampling period T, the vector [O 1 O 11' belongs to the null-space of cl (21 - Ad)-'.
üsing equation (6.7) with T = lsec, and x2 = bd, = [O 1 O l]', the sampled-
data hold function shown in Figure 6.3 is thence obtained, and one can apply this
r (sec)
Figure 6.3: Sampled-data hold function for control agent # S in Esample 6.1.
sampled-data hold function to control agent #2, together with any nonzero sampled-
data hold function for control agent #1, to obtain a lower-triangular transfer matrix
for the resulting discretized system. For instance, on using a simple zero-order hold
for control agent #1, and the generalized sampled-data hold function of Figure 6.3
for control agent #2, the following equivalent discrete-time system representation is
obtained:
mhich corresponds to a hierarchical digraph structure. It can be easily verified that
conditions (i), (ii), and (iii) in Theorem 6.3 hold true, and thus the discretized model
does not have any DFMs. This implies that one can design a digital controller for
each agent, independently from the other control agent. In other words, the problem
of designing a decentralized controller for the system (6.8) becomes very simple, Le. it
reduces to designing a digital controller for each subsystem in the resulting discretized
model, using only centralized design met hods.
Example 6.2 : Consider the 2-input, 2-output mas-spring system of Figure 4.4 and
assume that the measured outputs are y w := yl and y,2 := y2 + (instead of y1
and y*). The system is described in this case by the following system matrices:
The digraph of this systern is depicted in Figure 6.4. The transfer functions in this
Figure 6.4: Digraph of the mas-spring system of Example 6.2.
figure are given by:
Here, (cl, A) and (c2, A) are unobservable? while (A, bl) and (A, b2) are controllable.
Thus, the conditions of Theorem 6.2 hold for any arrangement of control agents. In
other words, one can use sampling with specific sampled-data hold functions to con-
struct two different hierarchical discrete-time modets, with either one of subsystems
a t a higher level. For instance, let us assume that il = 1 and i2 = 2 (which means that
11 = 1 and Z2 = 2 in Theorem 6.2), and let us use equation (6.7) to obtain a sampled-
data hold function for control agent #2, so that the resultant discretized model will
ùe hierarchical with subsystem #l a t the higher level. In this case, it can be shon-n
tha t for al1 values of the sampling period T , the vector [l O 1 O 1 O]' is a basis for
the null-space of cl ( z I - Ad) - l (in fact, the null-space is one dimensional). Consider
xz = bd2 = [0.1 O 0.1 O 0.1 O]'. On applying equation (6.7), the sampled-data
hold functions of Figures 6.5 (a), (b), and (c) are obtained for T = O.5sec, T = 2sec,
and T = 5sec respectively. Let us choose T = 5sec. Applying the sampled-data hold
function of Figure 6.5 (c) to control agent #2, and a simple zero-order hold to control
agent #1, the discrete-time hierarchical model of Figure 6.6 is obtained. The transfer
functions in this digraph, are given by:
Figure 6.5: Sampled-data hold functions for control agent #2 in Example 6.2. (a) T = 0.5sec; (b) T = 2sec; (c) T = 5sec.
In addition, it can be easily verified that the conditions of Theorem 6.3 for the given
model, GSHF, and sampling period hold true, which implies that the equivalent
discrete-tirne model has no DFMs. This implies that one can design a decentral-
ized controller for this sÿstem by appiying a digital controller for each subsystem
independently, using centralized methods. For instance: consider the controllers:
vli [k + 11 = ~ i l [ k ] + 5(3/mi[k] - yrel,i),
~ l z [ k + 11 = 0.6876v12[k] + yml[k]:
ui [k] = -0.35911/,l [k] - 0.005629~~~[k] - 0-04669u12[k],
(6.1 la)
(6.11b)
(6. I Ic)
Figure 6.6: Digraph of the equivalent discrete-time mas-spring system of Exam- ple 6.2.
and:
which have been designed for subsystem #1 and subsystem #2 respectively, indepen-
dently of each other, using centralized methods to solve the robust semomechanism
problem. Here Yres,i and yrej,2 in (6.11) and (6.12) denote constant reference sig-
n a l ~ for control agent #1 and control agent #2 respectively. The eigenvalues of the
resultant closed-loop discrete-time system corresponding to subsystem #1 and sub-
system #2 are given by:
and:
sp, = {-0.8004,0.5303 f 0.2824i]
respectively. The eigenvalues of the overall closed-loop discrete-time system are thus
given by:
-4ssume now that z[0] = [l 1 1 1 1 11' and uI1[0] = vi2[O] = V ~ ~ [ O ] = u ~ ~ [ O ] = 0-
Figures 6.7 (a) and (b) give the outputs of the resultant closed-loop system, for a
unit step reference input in control agent #1 and a zero reference input in control
agent #2, using the decentralized controller (6.11), (6.12). Figures 6.7 (c) and (d)
give the output signals for a zero reference input in control agent #1 and a unit
step reference input in control agent #2. The input signais, corresponding to control
agent #1 and control agent #2, are given in Figures 6.8 (a)? (b), (c) and (d) for
both set of reference inputs, respectiveiy. It is to be noted that these figures give the
samples of the output and input signals. The corresponding continuous-time signals
will have intersample ripple effects caused by the sampled-data hold functions in each
agent. For instance, the magnitude of the intersample x-ipples in the input signal
corresponding to Figure 6.8 (b) in the steady state, is given by the magnitude of the
samples times the magnitude of the GSHF of control agent #2, which is approximately
50 x 17 = 850. The magnitude of the the corresponding intersamples in the output
signal in Figure 6.7 (b) is approximately 150. These results show that "good tracking"
behaviour occurs for both control agents.
It is to be noted that if an incorrect polarity of inputs and outputs occurs in the
plant to be controlled, due to for example incorrect wiring, the equivalent discrete-
time mode1 obtained preserves its hierarchical structure, In other wvords, the digraph
Figure 6.7: Closed-loop simulations for
(a (-4
Example 6.2. (a) Samples of the output response in control agent #1, for Y,.JJ = 1, y , , j ~ = O; (b) samples of the output response in control agent #2, for y r e ~ , l = 1, yre!,2 = 0; ( c ) samples of the output response in control agent #1, for y,.,,, = O , yrej,f = 1; ( d ) sarnples of the output response in control agent #2, for yref,l = O, Y , . J , ~ = 1 .
of the systems:
Figure 6.8: Closed-loop simulations for Example 6.2. (a) Samples of the control signal in control agent #1, for YreI,l = 1? yref72 = O ; ( b ) samples of the control signal in control agent #2, for y,.~,, = 1, y,,j,z = O; (c) samples of the control signal in control agent #1, for y,,f71 = O, y,,j,~ = 1; (d) samples of the control signal in control agent #2, for yre / , i = 0, Yre/,2 = 1.
al1 have the same hierarchical structure of Figure 6.6 (the corresponding transfer
functions may have different signs however). The reason for this is that if the term
cj (zI - Ad)-'bdi is equal to zero, then the terms cj(zI - Ad)-'(-bd, ) and (-c,) ( d -
-&)-lbdi are both equal to zero also. Thus this implies that one can obtain a solution
to the decentralized adaptive switching control problem introduced in Chapter 5 for
this system by applying centralized saitching control methods to each control agent,
and in this case it is guaranteed that each subsystem will be "stabilized in a finite
time". This observation obtained is important: for esample when the parameters of
the mass-spring system are such that the corresponding interconnections are not weak
enough to be ignored, this implies that the continuous-time decentralized switching
control method proposed in Chapter 5 may not be applicable, but that its discrete-
time counterpart, which employs the proposed sampled-data hold functions described
in this chapter can still be very effective.
Example 6.3 : Consider the 2-input 2-output mas-spring system of Example 6.2,
and a family of plant models P = {PL, P2, P3, p, P5} given by (6.13) (P5 describes
the actual plant). Assume now that each plant mode1 in is discretized by applying
the sampled-data hold function of Figure 6.5(c) i;o control agent #2, and a simple
zero-order hold to control agent #1, with the sampling period T = 5sec. Consider
the following discrete- time controller structure:
The following set of controllers is designed for the set of discretized plant models:
For piant model P' :
For plant model p3 :
0 For plant model P' :
G: = Gl, Hf = Hi, J: = Ji,
KI =K1,Lt = L 1 , d C = M i ;
G: = G2, H i = -H2, J: = - 5 2 ,
K i = K2, Li = -L2; AI; = hi2,
This results in the following eigenvalues for the closed-loop system associated with
the plant mode1 Pa, and the controllers Ci, Ci, i E {1,2}:
Consider the controller switching order given by Cj -t Cf + C: + Cf + CS, j = 1 ,2 , and assume that x[0] = [O O O O O O]', q [ O ] = [O O]' and z2[0] = 0.
-Alsol assume a unit step reference input for both control agents. On applying the
discrete-time version of the switching controller of Chapter 5 with A = 0.5 and the
modified strong bounding functions given by (5.7), the results given in Figure 6.9
and Figure 6.10 are obtained. Figures 6.9 (a) and (b) give the samples of output in
control agent #1 and control agent #2 respectively. Figures 6.10 (a) and (b) give the
corresponding switching times for control agent #1 and control agent #2 respectively.
Tliese figures show that control agent #1 switches to controller Ct in 16 seconds and
remains unchanged aftenvards. Control agent #2 on the other hand, switches to
Cz and Cz in 18 and 22 seconds respectively, and remains unchanged aftenvards.
(one can see that C: and Ci are in fact equal to the correct controllers Cf and Cz,
respectively) .
This example illustrates the application of GSHFs in rnodifymg the structure of
the system, so that the adaptive switching controller introduced in Chapter 5 may be
applied.
Figure 6.9: Decentralized switching control for the mass-spring system of Esample 6.3. (a) Samples of the output in control agent #1; (b) samples of the output in control agent #2.
Figure 6.10: Decentralized switching control for the mas-spring system of Exam- ple 6.3. (a) Switching times for control agent #l; (b) switching times for control agent #2.
Chapter 7
Conclusion
This chapter is devoted to concluding remarks and contributions of this work, and
suggestions re future research which can be carried out as extensions of this work.
7.1 Concluding Remarks
In this thesis, a class of decentralized LTV controllers has been proposed for contin-
uous LTI plants, using digital control (Chapter 2 and Chapter 3). The controllers
obtained have the property that when the plant has a certain type of approximate
decentralized h e d mode (-4DFM), namely so-called "unstructured ADFM", they can
outperform their continuous LTI counterparts. To accomplish this, an algorithm to
find structured and unstructured DFMs is initially presented; then, using this algo-
rithm, unstructured ADFMs can be determined. The performance of the closed-loop
system depends on the type of hold function used in the digital to analogue process,
the sampling interval, and the control law itself.
In Chapter 2, the proposed decentralized digital controller obtained uses a simple
zero-order hold. Simulation results obtained in this case, using an optimal sampling
interval and optimal stat ic output feedback, show significant improvements compared
to the corresponding continuous-time optimal decentralized controller.
In Chapter 3, it is shown that using generalized sampled-data hold functions
(GSHF), instead of a simple zero-order hotd, in the proposed decentralized digital
controller, can introduce more degree of freedoms in the digital controller design,
mhich results in an improvement in the overall performance of the system. In this
case, simulation results obtained for the same numerical example as considered in
Chapter 2, using an optimal sampling intemal and optimal second order hold functions
for each agent, show a significant improvement in terms of the performance index
compared to the results of Chapter 2.
In Chapters 4 and 5, the case when the plant model is highly uncertain is con-
sidered, and two different switching control schemes are proposed to stabilize and
regulate the decentralized control system. In these control schemes, it is assurned
that the plant can be described by a family of plant models, and that a set of high-
performance controllers has been obtained, so that the actual plant model can be
stabilized and regulated by a t least one controller contained in this set.
The first switching control approach is described in Chapter 4 and is a decentral-
ized version of the centralized controller introduced in [41], which assumes that the
set of plant models is given, and that upper bounds on the disturbance and refer-
ence input magnitudes are avaiiable at each agent. It is also assumed that the plant
is controllable, and is observable from each agent. Smitching in the system occurs
when the norm of the error signals of al1 agents become greater than or equal to the
corresponding upper bound signal of each respective agent's switching controller. ,411
agents switch at the same time. This means that a weak communication link is re-
quired to exist between the control agents to indicate when a subsystem's error signai
hits the corresponding upper bound. This required information flow however is far
less than the information 0ow required in the centralized case, especially when the
number of control agents is greater than 2. The simulation results obtained show that
good tracking is achieved by applying this method. However, the transient response
which results has large peaking compared to the centralized method, which is due to
the conservative switching strategy of the proposed decentralized met hod.
The second switching control approach is described in Chapter 5 and is a de-
centralized version of the switching controller introduced in [12], and has al1 of the
properties of this controller. I t is to be noted that no a priori information about
the actual plant models is required in this method, and that no communication be-
tween different control agents is required. However, the plant's model is assumed to
have a specific structure, which implies t hat the interconnections between different
subsystems of the plant, are either hierarchical or weakly interconnected. In this
case, simulation results obtained in the examples studied, show that the overall per-
formance of the decentralized switching controller is comparable to the centralized
switching controller case.
In Chapter 6, it is shown that the methods of approach as used in Chapter 3 can,
in certain circumstances, be used to transform the structure of a continuous plant
rnodel to take on a hierarchical structure, which then implies that the decentral-
ized switching control approach of Chapter 5 can directly be appiied. In particular,
Chapter 6 introduces a new application for generalized sampled-data system control,
which has the property that i t can be used to simplify the structure of the resulting
discretized model. Conditions under which the resultant discretized model can have
a hierarchical digraph are discussed, and a method to synthesis the corresponding
sampled-data hold functions for each control agent is presented. The motivation for
simplifying the plant structure to become hierarchical, is that controller design, par-
ticularly for decentralized control, can then be greatly simplified. For example, if the
plant model is hierarchical, then in decentralized control problems, one can design
a decentralized controlier for the system by applying centralized methods directly
to each control agent; in particular, the adaptive switching controller of Chapter 5
can be directly applied. Simulation results given in Chapter 6 show how such a
discretization procedure can result in a simple hierarchical digraph, and thence sim-
plify the decentralized controller design problem in obtaining a solution to the robust
servomechanism problem, and the adaptive switching controller of Chapter 5.
7.2 Suggested Future Work
An open area for research exists in the application of digital controllers in decen-
tralized systems. For example in the research carried out, it was assumed that the
sampling period for al1 control agents is identical. One may investigate the possibil-
ity of using different sampling i n t e d s for the control agents to improve the overall
performance of the closed-loop system. Also, the effect of applying varying sampling
periods in digital control was studied through an example; one can formulate the
general problem of applying non-constant sampling intervals in digital control to de-
termine to what extent it can be effective in decentralized digital control. Robustness
of decentralized digital control is also an important subject for future work.
Previous research in applying snitching control to uncertain systems has focused
mainly on centralized systems. This thesis is one of the first studies on decentralized
switching control, and t herefore it introduces an open research area. For instance,
the two decentralized switching methods proposed in this thesis, both have some
limitations in the sense that either a weak communication link between the control
agents is required, or a specific system structure is assumed to exist. I t would be
interesting to investigate if these conditions can be relaved by applying some other
approaches or by imposing milder restrictions. This could be achieved by irnposing
some constraints on the controllers in the design process, or by applying some type
of synchronous switching for al1 control agents, using appropriate time intervals to
switch between different controllers. One mas also investigate procedures to limit the
magnitude of the transient response obtained which can be very important in potential
industrial implementation of those controllers. It may also be of interest to apply the
proposed switching control procedure, using different generalized sarnpled-data hold
functions in conjunction with the different controllers. In this case, a dictionary of
GSHFs and a dictionary of controllers would be obtained. Finally, it mould be of
interest to generalize the results of Chapter 6 to enlarge the class of outputs which
can be considered in the decentralized servomechanism problem.
Appendix A
Opt imizat ion Algorit hms
This appendix proposes optimization algorithms which have been used throughout
this thesis, to design either continuous-time or discrete-time decentralized controllers
to minimize a continuous-time quadratic performance index- It is to be noted that
the results obtained may not be globally optimal.
A. 1 Decentralized Continuous-Time Static Con-
troller
Consider the following controllable and observable LTI system:
(A. la)
(-4.1 b)
where B = [bi . . . bm] and C = , and assume that it is desired to design a
decentralized controller: L c - J
where:
K =
to stabilize the system such that the following performance index is minimized:
J = E {l=iY'(t)Y(t) + pu'(t)u(t))dt} 9 (-4.3)
where E denotes the expectation operator [35]; in this case the performance index J
(-4.3) is given by J =trace(P), where P > O is the solution to the Lyapunov function:
A'P + P.4 = -(CCf + pKCC'K'),
assuming that the matriu -4 + BKC is Humitz. This implies that the problem of
minimizing the performance index J can be reduced to the problem of minimizing
the following function:
where xl(0), xz(O), ... x.(O) are given by:
with the constraint imposed that the closed-loop system must be stable. This leads
to the following algorithm to find the optimal decentralized controller (A.2).
Algorithm 1:
Assume that there exists a decentralized stabilizing controller (-4.2) with K = K, so
that the system:
is stable, and define the performance index for the system:
for the case when x(0) = ~ ~ ( 0 ) given by (A.4). Let:
and consider the following parameter optimization problem:
Parameter Optimization Problem
min given by (A.2)
J
subject to the constraint that (A.5) is stable, using as the starting point K = K. -4 solution to this optimization problem can be achieved by using the parameter
optimization method of [23].
A. 2 Decentralized Discrete-Time Static Controller
Consider the system (A.l) and assume that the discrete-time equivalent of this system,
obtained by using a zero-order hold with sampling i n t e m l T is controllable and
observable and is described by:
A. 2.1 Zero-Order Hold
Assume that (-4.6) corresponds to a sarnpled system with a zero-order hold, and it is
desired to design a decentralized controller of the fom:
where:
(A. 7)
to minimize the performance index (-4.3) corresponding to the continuous-time sys-
tem.
Algorithm II:
-Assume that there exists a decentralized stabilizing controller (-4.7) with K = K, so
that the system:
is stable. Let N be a large positive integer, and define the performance index for the
system:
for the case when x ( 0 ) = ~ ~ ( 0 ) given by (-4.1). Let:
and consider the following parameter optimization problem:
Parameter Optimization Problem
min K given by (X.7)
Jd
subject to the constraint that (-4.11) is stable, using as the starting point I< = K. In
this case. for sufficiently large N , the optimal value of the performance index Jd will
approach the minimum of the performance index (A.3).
A. 2.2 Generalized Sampled-Data Hold Function
-Assume that (A.6) corresponds to a sampled system with generalized sampled-data
hold functions as folloms:
Also, assume that fj(t), j E m, are polynomials of the following form:
(A. 10)
mhere:
It is desired to find the matri; a, which minimizes the performance indes (A.3)
corresponding to the continuous-time system.
Algorit hm III:
Assume that there exists a set of generalized sampled-data hold functions (.4.10) with
a = 5 , so that the equivalent discrete-time system:
(-4.1 la)
(-A. 11 b)
is stable. Let N be a large positive integer, and define the performance index for the
systern:
for the case when z(0) = ~ ~ ( 0 ) given by (-4.4). One can use (-1.9) to replace the first
term in the right hand side of (A.12) nrith:
where F = dia&$, ..., Fm), and:
Let:
and consider the following parameter optimization problem:
Parameter O~timization Problem
min a gben by (A.10)
Jd
subject to the constraint that (-4.11) is stable, using as the starting point a = 6. In
this case, for sufficiently large N , the optimal value of the performance index Jd will
approach the minimum of the performance indes (-4.3).
A.3 Decentralized Servomechanism Problem
Consider the following controHable and observable LTI system:
xit) = -4x(t) + Bu(t) + Ew(t) ,
y ( t ) = C x ( t ) + Fw(t ) ,
where:
B = [ b l ... b , ] , C =
Here, u( t ) and w ( t ) denote the control input and disturbance signal respectively.
Assume that the reference input for the control agent j is denoted by yret , j ( t ) Assume
that there esists a solution to the robust decentralized servomechanism problem [16].
It is desired to design a LTI decentralized controller of the form:
to minimize the performance index [19]:
for constant disturbance signals w ( t ) and constant reference inputs yrelj(t), j =
1 , 2 ,..., m.
Algorit hm N:
To formulate this optirnization problem, take the derivative of (-4.13) to obtain [19]:
Now, define the augmented model with the state vector + := [xt(t) ef ( t ) ] ' , the control
input ~ ( t ) := ù(t) , and the output Y ( t ) := ë(t). This augmented mode1 can be
represented by the followving matrices:
One can now apply Algorithm 1 to the augmented model for the system defined in
(A.15) to minimize the performance index (-4.14)- Note that the initial conditions
z,+~ (O), ..., zntm(0) in this case, correspond to the constant reference inputs for the
control agents 1, ..., m.
Appendix B
Supporting Theorems and Lemmas
This appendix contains some of the theorems and iemmas in [41] and [12] which have
been referred to throughout this thesis.
B.1 Some Related Material Rom the Paper [41]
"Adaptive Control of a Farnily of Plants" by
D.E. Miller and E. J. Davison
Consider the stnctly proper, controllable and observable, LTI system P r , k E p,
p = (1,. . . , p } : described by:
(B-la)
(B.lb)
(B. lc)
where x ( t ) E IW"k is the state, u( t ) E Rm is the control input, y ( t ) E IlQf is the output
to be regulated, w ( t ) E WV is the disturbance, and e( t ) E W' is the error, mhich is the
difference between the specified reference input yref and the output y. It is assumed
that the parameters of each Pk are known exactly. The actual plant is assumed to be
contained in the family of plants P = {Pk, k E p}, and it is assumed that the signals
y and y,,f are only rneasurable, and that u can be excited.
For each k E p, design a high-performance LTI controller K' (by any means at
your disposal) for Pk of the form:
(B. 2a)
(B2b)
where z ( t ) E @; without loss of generality we can assume that Z k = l , Vk E PT by
adding unobsemable stable modes if necessary. In particular, assume that each Kk
has been designed so that it stabilizes the closed-loop system corresponding to pkT
i-e. the closed-loop system matrix has al1 of its eigenvalues in the open left half of
the complex plane; more importantly, it perfoms satisfactory disturbance regulation
and/or tracking for Pk, depending on the control objective.
Finding a controller (B.2) which provides stabilization of the closed-loop system
for a plant described by Pk is equivdent to finding a matrku K' so that the static
output feedback controller:
stabilizes the following augmented controllable and observable system:
where:
and:
Lemma 1 [41] : Suppose that u(.) = O in (B.1). Then for every T > O, there exist
constants a k l and a&-*, so that for every initial condition x ( 0 ) and every disturbance
w E PC,, we have:
Lemma 2 [41] : There exist constants y k l , ~ 2 , 7 k 3 , and Xk < 0, so that for every
fi; y,,!, w E PC and initial condition 5(0) , the solution of (B.3) satisfies:
for t 2 O.
We define Controller 1 as follows. Fiu T > 0, 8 > 0, and E > O, choose a k l , a k l ,
k E p, so that the inequality given in Lemma 2 holds. Then define:
and Li < A.
The controller is effected in two phases.
Phase 1: Finding a bound on the initial condition.
With ü(t) = O for t E [O, Tl and r(0) = 0, find:
and define the following p auiliary signals:
Define:
if Ilw(t) 11 $ 6 for t E [O, Tl and the plant is PL, then it follows from Lemma 1 that
I!5(o)ll pk-
Phase 2: Searching the gains.
Now we apply the control action. Let the control input be:
let the p auxiliary signals be:
mith rk (T+) = rk(T) + ycie*kTW, and let the filtering signal be:
(notice that Dkl = f i k 2 for al1 kl, k2 E p, so we define := D I ) . So rk(t) generates
an upper bound on the norm of the state vector for t >_ T in the event that the plant
is Pk, nhile i filters ij - D~,, / . We define the switching times recursively as follows:
set t t :=Tl and for every k E (2, . . . , p+ 1) for which tt-1 # oo, define tk by:
min{t 2 tk-l : there exists a time f E [Tl t] for which
- k I -k-1- i ( f ) = JlC - Ilrk-l(i) + llF b + E )
i f the set is nonempty (the minimum exists if it is nonempty), and oc othenvise; if
t,+l is defined and finite, then define t,+2 = m and set I?P+' = O. Observe that it
could be that tk = tk-1 for some k; if this occurs then it means that by the time that
we are ready to try the gain K ~ , Ive have already mled it out.
Theorem 1 [41] : Suppose that y,,,,w E PC,, and that Ilw(t)lJ 5 6 for t 2 0. For
every initial condition x(0 ) , when Controller 1 is applied to the plant, the closed-loop
system has the following properties:
(i) the gain eventually remains constant at an element of (kk : k E p), and
(ii) the state Z(t) is bounded.
B.2 Some Related Material F'rom the Paper [12]
'<Adaptive Switching Control of LTI MIMO
Systems Using a Family of Controllers Ap-
proach" by M. Chang and E.J. Davison
Let each element:
P* := (A" B*, c*, D', E', F'), k E {l, 2, ..., p } , p E N
belonging to the finite set of possible plants to be controlled:
be of the finite dimensional form:
where x ( t ) E R n k is the state: u E iP is the control input, y E W' is the plant output
to be regulated, w E WQ is the disturbance, and ë E lR7 is the difference betwveen the
specified reference input p,.~ and the plant output y. The set of candidate controllers
which will be considered is assumed to have the general form given by:
where 7 E IF", Gk E R9kxgk, HL E R9kxr, J k E R9kxr: Kk E IRmx9r9 Lk E IRmxr, and
i V k E W m x r . In the discussions that follow, we do not necessarily assume that n.k, A*,
B', Cr, Dk, Ek, or Fk are knowvn, and we do not necessarily restrict sp(Ak) c @-,
k E {1,2, ..., p } .
Construction of Controllers FCk: Given plant mode1 Pk, assume now that it is desired
to find controllers ECk to solve the robust servomechanism problem (RSP) [18] for Pk
for the class of constant reference input signals y,,/ E W' and constant disturbance
signals w E RQ, i-e. so that the following requirements are met:
(a) The resultant closed-loop system is asymptotically stable.
(b) -4symptotic tracking and regulation occurs, Le. for al1 constant reference input
signals yr,j E K and for al1 constant disturbance signals w E W,
(c) The controller is robust, i.e. for al1 perturbations of the plant described by
P b h i c h do not dest abilize the resu!tant closed-loop system, property (b) still
holds.
Stÿitchzng Controller 1: With f E MSBF, X E Et+, and filtered error signal ef ( t ) given
by :
define Switching Controller 1 as follows:
where s E { 1 , 2 , 3 ,... }, i := ( ( s - 1 ) mod p) + 1,
M t ) , ~ ( t ) , ~ ( t ) , IW), ~ ( t ) , ~ ( t ) ) := (G', H=, S, K', L*, M'). t E (t,, t,,,],
(here (Ga , Hi , J i , K i , Li: M t ) corresponds to Controller Ki given by (8.4)) t := 0,
and where, for each s >_ 2 such that t,-L # m, switching time t , is defined by:
( min t 3
i) t > t,-L, and if this minimum exists t , :=
ii) Il[ri(t) e>(t)lll = f (s - 1)
I - O t henvise
iii) (Ck, AL, Bk) is stabilizable and detectable, and rank ([c: O:]) =%+.
for k E {l, 2, ..., p } .
Theorem 3.1 [12] : Consider a plant P E with Switching Controller 1 applied
a t time t = O; then for every f E MBSF and A E Et+, for every bounded piecewise
cont inuous reference and dist urbance signal, and for every initial condition .Z (O) : =
[xf(0) qJ(0) e>(O)]' for which Assumption 1 holds, the closed-hop system has the
properties that:
i>
ii)
iii )
there exist a finite time t,, 2 O and constant matrices (Gss, Hu, Jss7 K m 7 Lss, M N )
such that ( G ( t ) , H ( t ) , J ( t ) , K ( t ) , L ( t ) , M ( t ) ) = (GsS , H M 7 Jss , K y LsL", iIP) for
ail t 2 tsS;
the controller states q E C,, the plant states x E L, and the filtered error
signal e l E C,; and
if the reference and disturbance inputs are constant signals, then for almost al1
controller parameters (Gk , Hk, K k , Lk) , asymptotic error regulation occurs, i.e.
e ( t ) + O as t + oo.
Appendix C
MATLAB Program
This appendix contains a MATLAB program which is used in Chapter 2 and Chap-
ter 3 to obtain the approximate decentralized fixed modes (ADFM) and approxirnate
structured decentralized fised modes or approxirnate quotient h e d modes (-4SFM or
AQFM) of a continuous-time LTI system with 2 or 3 control agents.
% AFM
% This program finds the minimum condition numbers representing
% Approximate Decentralized Fixed Modes (ADFM) and Approximate
% Structured Decentralized Fixed Modes (ASFM or AQFM) of a system
% with 2 or 3 control agents.
% The system matrices must be given in the form of A, bl, cl, b2, c2
% (and b3, c3 if necessary) . % The output of the progran is s (eigenvalues of A), ADFM (the
% magnitude of the minimum condition numbers that represent
% approximate decentralized fixed modes for each eigenvalue), and
% ASFM (the magnitude of the minimum condition numbers that represent
% approximate s t ~ c t u e d fixed modes for each eigenvalue).
n=size ( A , 1) ;
m-input ( ' How many input-output agents (2 or 3) ? ' ) ;
s=eig(A) ;
i f m > 3 1 m < 2 ,
disp('This program is only for systems uith 2 or 3 control agents')
pause
end
if m==2,
for i=l:n,
ml(i)=cond([A-s(i) *eye(n) b1;cl zeros(size(cl*bl) ) l ) ;
m2 Ci) =cond( [A-s (i) *eye (n) b2; c2 zeros (size (c2*b2) ) 1 ) ;
m3 (i) =cond([A-s (i) *eye (n) [bl b2] ; [cl ; c2] zeros (size( [cl ; c2]*. . .
Cbl b2l))l);
ml1 (i)=cond( [A-s (i) *eye(n) bl; c2 zeros (size (c2*bl) )] ) ;
ml2(i)=cond( [A-s(i) *eye(n) b2;cl zeros(size(cl*bS) )]) ;
ADFM(i)=min([ml(i) ,m2(i) ,m3(i)]) ;
[ni2 ,dl21 =ss2tf (A, bl, c2 Dzeros(size(c2*b1) , 1) ;
Ln21 ,d21] =ss2tf (A ,b2,ci,zeros(size(ci*b2)) , 1) ;
ASFM(i)=ADFM(i) ;
if norm(nl2) '= O & norm(n21) -= 0,
ASFM(i>=min([ml(i) ,m2(i) ,m3(i) ,mll(i),ml2(i)l) ;
end
end
else,
for i=l:n,
ml(i)=cond([A-s(i) *eye(n) b1;cl zeros(size(cl*bl) )]) ;
m2 (i) =cond( [A-s (i) *eye (n) b2; c2 zeros (size (c2*b2) ) 1 ) ;
rn3(i)=cond( [A-s (i) *eye(n) b3;c3 zeros (size(c3*b3) 11 ) ;
m4(i)=cond( [A-s (i) *eye(n) [bl b21; [cl ; CS] zerodsize ( [cl; c2] * . . . [bl b21))J);
m5 (i) =cond( [A-s (i) *eye (n) [bl b3] ; [cl ; c3] zeros (size ( [cl ; c3] * . . .
Cb1 b31))I);
m6 (i)=cond( [A-s (i) *eye (n) Cb2 b31; k 2 ; c3] zeros (size ( Cc2; c3]* . . .
Cb2 b33))I);
m?(i>=cond([A-s(i)*eye(n) b;c dl) ;
ml1 (i)=cond([A-s (i) *eye (n) b2 ;cl zeros (size(cl*bZ) 11 ) ;
ml2(i)=cond( [A-s(i) *eye(n) bl ;c2 zeros (size(c2*bl) 11 ) ;
ml3 (1) =cond ( [A-s (i) *eye (n) [b2 b31; [cl ; c3] zeros (size ( [cl ; c3] * . . . b 2 b31))J);
ml4 (i) =cond( [A-s (i) *eye (n) [bl b33; cc2 ; c3] zeros (size ( Cc2 ; c31* . . .
Cbi b31))I);
m2l(i)=cond([A-s(i)*eye(n) b3;cl zeros(size(cl*b3) 11 ) ;
rn22(i)=cond([A-s(i)*eye(n) bl;c3 zeros(size(c3*bl))]);
m23 (i) =cond( [A-s Ci) *eye (n) [b3 b21; [cl ; c2] zeros (size ( [ci ; c21* . . . [b3 b2l))l);
m24(i)=cond([A-s (il *eye(n) [bl b21; Cc3; c21 zeros(size ( Cc3; c2] * . . . Cb1 b23))I);
m3l (i)=cond([A-s (i) *eye (n) b3; c2 zeros (size(c2*b3) )] ) ;
1x132 Ci) =cond ( [A-s (il *eye (n) b2 ; c3 zeros (size (c3*b2) ) 1 ) ;
m33 (i)=cond( [A-s (i) *eye (n) [b3 bl] ; Cc2; ci] zeros (size ( Cc2; cl] *. . . Cb3 bll))]);
m34 (i) =cond( [A-s (i) *eye (n) [b2 bll ; Cc3 ; cl] zeros (size ( Cc3 ; cl) * . . . [b2 b1J))I);
ADFM(i)=min([ml(i) ,m2(i) ,rn3(i) ,m4(i) ,mS(i) ,m6(i) ,m7(1)] 1 ;
In12 ,dl23 =ss2tf (A,bl, c2 ,zeros (size(c2*bl) a 1) ;
Ln21 ,d21]=ss2tf (A,b2,cl,zeros(size(cl*b2) , 1) ;
Ln13 ,dl31 =ss2tf (A, bL, c3 ,zeros (size (c3*bl) 1) ;
En31 ,d31]=ss2tf (A, b3,cl ,zeros(size(cl*b3)), 1) ;
Cn23, d23] =ss2tf (A, b2, c3 ,zeros (size (c3*b2) , 1) ;
Cn32, d321 =ss2tf (A,b3, c2 ,zeros b i z e (c2*b3) 1 1) ;
ASFM(i)=ADFM(i) ;
end
end
s=s '
ADFM
ASFM
end
Bibliography
[l] B. O. D. Anderson and J. B. Moore. Time-varyiog feedback laws for decentralized
control. IEEE Transactions on Automatic Control, AG26(5):1133-1138, May
1981.
[2] C. J. Bet t and M. D. Lemmon. Bounded amplitude performance of switched Ipv
systems with applications to hybrid systems. Autonzatica, 35(3):491-503, Mach
1999.
[3] M. S. Branicky. Analyzing continuous switching systems: Theory and exam-
ples. In Proceedings of the 1994 Arnerican Control Conference, pages 31 10-31 14.
IEEE, 1994.
[4] A. B. Chammas and C. T. Leondes. On the design of linear time invariant
systems by periodic output feedback, part i and ii. International Journal of
Control, 27:885-903, 1978.
[5] -4. B. Chammas and C. T. Leondes. Optimal control of stochastic linear systems
by discrete output feedback. IEEE Transactions on Automatic Control, A G
23(5):92 1-926, 1978.
[6] A. B. Chammas and C. T. Leondes. On the finite time control of linear systems by
piecewise constant output feedback. International Journal of Control, 30(2):227-
234, 1979.
[il M. Chang and E. J. Davison. Intelligent control: Some preliminary results. In
B. .4. Francis and A. R. Tannenbaum, editor, Feedback Control, Nonlinear Sys-
tems, and Complexz~ty, volume 202 of Lecture Notes in Control and Information
Sciences, pages 67-87. Springer-Verlag, London, 1995.
[8] M. Chang and E. J. Davison. New directions in industrial control: Intelligent
control - an experimental study applied to a rnultivariable hydraulic system. In 1.
Lasiecka and B. Morton, editors, Control Problems in Industry: Proceedings from
the SIAM Symposium on Control Problems, San Diego, California, July 1994,
volume 21 of Progress in Systems and Control Theory, pages 65-95. Birkhauser
Press, Boston: 1995-
[9] M. Chang and E. J. Davison. Control of unknown systems using switching
controllers: an experimental study. In Proceedings of the 1994 American Control
Conference, pages 2984-2989. IEEE, 1994.
(101 M. Chang and E. J . Davison. Control of unknown systerns using switching
controllers: The self-tuning robust servomechanism problem. In Proceedings of
the 33'rd IEEE Conference on Decision and Controk pages 2833-2838. IEEE,
1994.
[ I l ] M. Chang and E. J . Davison. Switching control of a family of plants. In Proceed-
ings of the 1995 Amencan Control Conference, pages 1015-1020. IEEE, 1995.
(121 M. Chang and E. J. Davison. Xdaptive switching control of LTI MIMO systems
using a family of controllers approach. Automatica, 33(3):453-465, March 1999.
[13] T- Chen and B. Francis. Optimal Sampied-Data Contml Systems. Springer-
Verlag, 1995.
[14] K. Ciliz and K. S. Narendra. Multiple mode1 based adaptive control of robotic
manipulators. In Proceedings of the 33'rd IEEE Conference on Decision and
Control, pages 1305-1310. IEEE, 1994.
[15] E. J. Davison. Decentralized robust control of unknown systems using tuning
regulators. IEEE Transacttons on Autornatic Control, .4G23(2):276-289, April
f 978.
[16] E. J. Davison. The robust decentralized control of a general servornechanism
problem. IEEE Transactions on Automatic Control, -4C-21(1):14-24: February
1976.
[17] E. J. Davison and T. N. Chang. Decentralized stabilization and pole assign-
ment for general proper systems. IEEE Transactions on Automatic Control,
AC-35(6):652-664, June 1990.
[18] E. J. Davison and A. Goldenberg. Robust control of a general servomechanism
problem: The servo compensator. Automatica, 11 (5):46 1-471, 1989.
[19] E. J. Davison and B. M. Scheninger. Perfect control of the robust servomech-
anism problem. IEEE Transactions on Automatic Control, -4C-32(8):689-702,
-4ugust 1987.
[20] -4. V. S. Efstratios Skafidas, Robin J. Evans and 1. R. Petersen. Stability results
for switched controller systems. Automatica, 35(4):553-564, -4pril 1999.
[21] M. Fu and B. R. Barmish. Adaptive stabilization of linear systems via switch-
ing control. IEEE Transactions on Automatic Control, ,%C-31(12) : 1097-1 103,
December 1986.
[22] D. S. Gave1 and D. D. ~ i l j a k . Decentralized adaptive control: Structural condi-
tions for stability. IEEE Transactions on Automatic Control, -4G34(4):413-426,
-4pril 1989.
[23] W. Gesing and E. J. Davison. A penalty function algorit hm for solving general
constrained parameter optimization problems. Automatica, l S ( 2 ) : 173-183, 1979.
[24] B. K. Ghosh and C. 1. Byrnes. Simultaneous stabilization and simultaneous
pole-placement by nonswitching dynamic compensation. IEEE Transactions on
Automatic Control, AC-28(6):735-741, June 1983.
[25] 2. Gong and M. Aldeen. Stabilization of decentralized control systems. Mathe-
matical Systerns, Estimation, and Control, 7(1):1-16, 1997.
[26] P. -4. Ioannou. Decentralized adaptive control of interconnected systems. IEEE
Transactions on Autornatic Control, .%C-31(4):291-298, April 1986.
[27] P. T. Kabamba. Multi-channel output gain margin improvement using gener-
alized sampled-data hold functions. IEEE Transactions on Automatic Control,
39(3):657-661, March 1994.
[28] P. T. Kabamba. Control of linear systems using generalized sampled-data hold
functions. IEEE Transactions on Autornatic Control, AC-32(9):772-783, Septem-
ber 1987.
1291 P. T. Kabamba and C. Yang. Simultaneous controller design for linear time-
invariant systems. IEEE lYansactions on Automatic Control, 36 ( l ) : 106-1 11,
January 1991.
[30] P. P. Khargonekar and -4. B. 0zgüler. Decentralized control and periodic feed-
back. IEEE Transactions on Automatic Control? 39(4):877-882. -4pril 1994.
[3 11 H. Kobayashi and T. Yoshikawa. Graph-t heoretic approach to controllability and
localizability of decentralized control systems. IEEE Transactions on Automatic
Control? -AG27(5):lO96-llû8, May 1982.
[32j B. C. Kuo. Digital Control Systems. HOLT, RINEHART -4ND WINSTON,
1980.
[33] S. Lee, M. Meerkov, and T. Runolfsson. Vibrational feedback control: Zero
placement capabilities. IEEE Transactions on Automatic Contml, -4G32(7) :604-
611, July 1987.
[34] T. H. Lee and K. S. Narendra. Removing the high-frequency gain assumption
in discrete adaptive control. In Proceedings of the ,%?d'th IEEE Conference on
Decision and Control, pages 1198-1202. IEEE, 1985.
[35] W. S. Levine and M. Athans. On the determination of the optimal constant
output feedback gains for linear multivariable systems. IEEE Transactions on
Automatic Control, -4G15:44-48, February 1970.
[36] C.-T. Lin. Structural controllability. IEEE Transactions on Automatic Control,
19:201-208, June 1974-
1371 M. Makoudi and L. Radouane. A robust decentralized mode1 reference adaptive
control for non-minimum-phase interconnected systems. Autonaatica, 35(8):1499-
1508, August 1999.
[38] B. Martensson. The order of any stabilizing regulator is sufficient information
for adaptive stabilization. Systems and Contro l Letters, 6(2) :87-91, July 1985.
[39] Ü. 0zgüner and E. J. Davison. Sampling and decentralized fked modes. In
Proceedings of the 1985 American Control Con ference, pages 257-262. IEEE,
1985.
[40] R. Middleton and J. Freudenberg. Non-pathological sarnpling for generalized
sampled-data hold functions. Automatica, 31 (2):3 15-319, 1995.
[41] D. E. Miller and E. J. Davison. Maptive control of a family of plants. In D.
Hinrichsen and B. Miirtensson, editors, Control of Uncertain Systems: Proceed-
ings of an International workshop, Bremen, West Germany, - June 198. vol-
ume 6 of Progress in Systems and Control Theory. pages 197-219, Boston. 1990.
Birkhausser Press.
[42] D. E. Miller and E. J. Davison. An adaptive controller which can stabilize any
stabilizable and detectable LTI system. In C. 1. Byrnes, C.F. Martin, and R. E.
Saeks, editors, Analysis and Control of Nonlinear Systems, pages 51-58. Elsevier
Science Publishers B. V., Amsterdam, 1988.
[43] D. E. Miller and E. J. Davison. A new self-tuning controller to solve the ser-
vomechanism problem. In Proceedings of the 26'th IEEE Conference o n Decision
and Control, pages 843-849. IEEE, 1987.
[44] D. E. Miller and E. J. Davison. On necessary assumption in continuous time
mode1 reference adaptive control. In Proceedings of the 28'th IEEE Conference
on Decision and Control, pages 1573-1578. IEEE, 1989.
[15] D. E. Miller and E. J. Davison. -4n adaptive tracking problem. International
Journal of Adaptive Control and Signal Processing, 6 ( 1 ) : 4 5 4 3 , January 1992.
[46] D. E. Miller and E. J. Davison. An adaptive controller which provides lyapunov
stability. IEEE Transactions on Automatic Control, AC-34(6):599-609, June
1989.
[47] D. E. Miller and E. J. Davison. The self-tuning robust servomechanism problem.
IEEE Transactions on Automatic Control, 34(5):511-523, May 1989.
[48] A. S. Morse. Supervisory control of farnilies of linear set-point controllers. In
Proceedzngs of the 32'nd IEEE Conference on Deczsion and Control, pages 1055-
1060. IEEE, 1993.
[dg] A. S. Morse. A three-dimensional universal controller for the adaptive stabi-
lization of any strictly proper minimum-phase system with relative degree not
exceeding two. IEEE Transactions on Automatic Control, AG3O(l2) : 1 188-1 191,
December 1985.
[SOI A. S. Morse. A 4(n+l)-dimensional mode1 reference adaptive stabilizer for any
relative degree one or two, minimum phase system of dimension n or less. Auto-
matica, 23(1):123-125, Januax-y 1987.
[51] .A. S. Morse. Supervisory control of families of linear set-point controllers - part
1: Esact matching. IEEE Transactions on Automatic Control, AC-41(10):1413-
1431, October 1996.
(521 A. S. Morse. Recent problems in parameter adaptive control. Proc. CNRS Col-
loquium on Development and Utilzzation of Mathematical Models in Automatic
Control, Belle-Isle, France, pages 733-740, September 1982.
[53] D. R. Mudgett and A. S. Morse. Adaptive stabiiization of linear systems with
unknown high frequency gains. IEEE ~ n s a c t i o n s on Automatic Control, AG
3O(6) :549-554, June 1985.
[54] S. Mukhopadhyay and K. S. Narendra. Decentralized adaptive control using
partial information. In Proceedings of the 1995 Amerïcan Control Conference,
pages 34-38. IEEE, 1999.
[55] K. S. Narendra and J. Balakrishnan. Improving transient response of adaptive
control systems using multiple models and switching. In Proceedings of the 32'nd
IEEE Conference on Decision and Control, pages 1067-1072. IEEE, 1993.
[56] K . S . Narendra and J. Balakrishnan. Intelligent control using h e d and adaptive
models. In Proceedings of the 33'rd IEEE Conference o n Decision and Control,
pages 1680-1685. IEEE, 1994.
[57] K. S. Narendra and J. Balakrishnan. Improving transient response of adaptive
control systems using multiple models and switching. I E E E Transactions on
Automatic Control, 39(9):1861-1866, September 1994.
[58] R. D. Nussbaum. Some remarks on a conjecture in parameter adaptive control.
Systems and Control Letters, 3(5):243-246, Novernber 1983.
[59] H. Seraji. Decentralized adaptive control of manipulators: Theory, simulation,
and experimentation. I E E E Transactions on Robotics and Automation, 5(2):183-
201, 1989.
[60] M. E. Sezer and D. D. ~ i l j a k . Structurally fked modes. Systems and Control
Letters, 1(1):60-64, July 1981.
[ G l ] L. Slii and S. K. Singh. On lyapunov function based decentralized adaptive con-
trollers. In Proceedings of the 29'th IEEE Conference o n Decision and Contro.
pages 299-304. IEEE, 1990.
[62] L. Shi and S. K. Singh. Decentralized adaptive controller design for large-scale
systems with higher order interconnections. IEEE Transactions on Automatic
Control, .4G37(8) : 1106-1 118, August 1992.
M. W. Spong and L. Praly. Energy based control of underactuated mechanical
systems using switching and saturation. In -4. S. Morse, editor, Preprints of
the Blick Island Workshop on Control using Logic-Based Sun'tching, pages 86-95,
1995.
G. Tao and P. A. loannou. Mode1 reference adaptive control for plants with un-
known relative degree. In Proceedings of the 1989 American Control Conference,
pages 2297-2302. IEEE, 1989.
L. Trave, -4. Tarras, and -4. Titli. An application of vibrational control to cancel
unstable decentralized fixed modes. IEEE Transactions on Automatic Control,
AC-30(3) :283-286, March 1985.
A. F. Vaz and E. J. Davison. The decentralized control expanding system prob-
lem. Systems and Control Letters, 10:301-308, 1988.
-4. F. Vaz and E. J. Davison. On the quantitative characterization of approxi-
mate decentralized fixed modes using transmission zeros. Mathematics of Control
Szgnals, and Systems, 2987-302, 1989.
S. H. Wang. Stabilization of decentraiized control systems via time-varying
controllers. IEEE Transactions on Automatic Control, AG27(3):741-744, June
1982.
S. H. Wang and E. J. Davison. On the stabilization of decentralized control
systems. IEEE Transactions on Automatic Control, AC-18:473-478, 1973.
J. C. MWems and C. 1. Bymes. Global adaptive stabilization in the absence of
information on the sign of the high frequency gain. In -4. Bensoussan and J. L.
Lions, editors, Analysis and Optirnzzation of Systems: Proceedings of the Sixth
International Conference on Analysis and Optimàzation of Systems, volume 62
of Lecture Notes in Control und Information Sciences, pages 49-57, Berlin, 1984.
Springer-I7erlag.
[71] J. L. Willems. Time-varying feedback for the stabilization of fked modes in
decentralized control systems. Automatica, 25 ( 1 ) : lZ"i'l31, 1989.
[72] Y. Xudong. Decentralized adaptive regulation with unknown high-frequency-gain
signs. IEEE Transactions on Automatic Control, 44(l l):2072-20?6, November
1999.
[73] Y. Xudong. Logic-based switching adaptive stabilization of feedfonvard nonlinear
systems. IEEE Transactions on Automatic Corztrol, 44(11):2174-2 178, November
1999.
