![xiv GLOSSARY OF SYMBOLS
D̄ The closure of D (briefly as closed IUD), {s ∈ C : |s| ≤ 1}
Do The closed outer part of the unit disc (briefly as OUD), {s ∈ C : |s| ≥ 1}
D̆o The open outer part of the unit disc (briefly as open OUD), {s ∈ C : |s| 1}
Hn×n
∞ The set of n by n transfer function matrices G(s), with G ∞ ∞
RHn×n
∞ The set of n by n rational transfer function matrices G(s), with G ∞ ∞
Co(r1, · · · , rn) Convex hull of elements r1 · · · , rn
n1, n2 The set of integers from n1 to n2 satisfying n1 ≤ n2, {n1, · · · , n2}
S The index set of source nodes in a communication network, {1, 2, · · · , S}
L The index set of link nodes in a communication network, {1, 2, · · · , L}
AT Transpose of matrix (vector) A
A∗ Conjugate transpose of matrix (vector) A
j Unit of imaginary numbers,
√
−1
1n n × 1 vector [1, · · · , 1]T
e1 n × 1 vector [1, 0, · · · , 0]T
diag{ti, i ∈ 1, n} Diagonal matrix with diagonal entries ti, i ∈ 1, n
λi(A) i-th eigenvalue of matrix A ∈ Cn×n.
σ(A) Spectrum of matrix A ∈ Cn×n, {λ1(A), · · · , λn(A)}
ρ(A) Spectral radius of matrix A ∈ Cn×n, max
i∈1,n
|λi(A)|
σ̄(A) Maximum singular value of matrix A ∈ Cn×m,
max
i∈1,n
{λi(AA∗)}
1
2
rank(A) Rank of matrix A
span(A) The space spanned by all the columns of matrix A
[A](:, i) The i-th column of matrix A
[A](:, i1, i2) The matrix formed by i1-th to i2-th columns of matrix A, where i1 ≤ i2](https://image.slidesharecdn.com/2152340-250603023020-b823ff77/85/Frequencydomain-Analysis-And-Design-Of-Distributed-Control-Systems-Yuping-Tianauth-16-320.jpg)
![INTRODUCTION 7
Digraph and information flow
In a distributed control system, each agent can be considered as a vertex in a digraph, and the
information flow between two agents can be regarded as a directed path between the vertices in
the digraph. Thus, the interconnection topology of a distributed control system can be described
by a digraph. However, differing from the classic signal-flow graph (Chen 1984), in this book
and in many other references on the distributed control system, the direction of an edge in the
digraph does not mean the direction of an information flow. Let us consider the digraph shown
in Figure 1.3 for an instance. Denote by xi ∈ R, i ∈ 1, 5, the state of agent i associated with
vertex i. The existence of edge eij implies that agent i gets the state information xj from agent
j. For example, agent 1 gets information from agent 2.
1.2.2 Graph Operations
We shall construct new graphs from old ones by graph operations.
For two digraphs G1 = (V1, E1) and G2 = (V2, E2), the intersection and union of G1 and
G2 are defined by
G1 ∩ G2 = (V1 ∩ V2, E1 ∩ E2),
G1 ∪ G2 = (V1 ∪ V2, E1 ∪ E2).
For a digraph G = (V, E), the reverse digraph of G is a pair rev(G) = (V, rev(E)), where
rev(E) consists of all edges in E with reversed directions.
If W ⊂ V(G), then G − W = G[VW] is the subgraph of G obtained by deleting the
vertices in W and all edges incident with them. Obviously, G − W is the subgraph of G
induced by VW. Similarly, if E ⊂ E(G), then G − E = (V(G), E(G)E). If W (or E)
contains a single vertex w (or a single edge xy, respectively), the notion is simplified to G − w
(or G − xy, respectively). Similarly, if x and y are non-adjacent vertices of G, then G + xy is
obtained from G by joining x to y.
Undirected graph
An undirected graph (in short, graph) G consists of a vertex set V and a set E of unordered
pairs of vertices. If each edge of the graph G is given a particular orientation, then we get an
oriented graph of G, denoted by G→, which is a digraph. Denote by G← the reverse of G→.
Then, G = G→ ∪ G←.
For an undirected graph G, the in-neighbor set of any vertex is always equal to the out-
neighborsetofthesamevertex.Therefore,intheundirectedcasewesimplyusetheterminations
neighbor, neighbor set and degree.
For an undirected graph, if it contains a globally reachable node, then any other vertex is
also globally reachable. In that case we simply say that the undirected graph is connected.
For an undirected graph, it is said to be a tree if it is connected and acyclic.
Theorem 1.1 G(V, E) is a tree if and only if G is connected and |E| = |V| − 1. Alternatively,
G(V, E) is a tree if and only if G is acyclic and |E| = |V| − 1.
Theorem 1.2 A graph is connected if and only if it contains a spanning tree.](https://image.slidesharecdn.com/2152340-250603023020-b823ff77/85/Frequencydomain-Analysis-And-Design-Of-Distributed-Control-Systems-Yuping-Tianauth-23-320.jpg)
![10 FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS
V1
1
1.5
2
6
3.5
3
3
6
4 4
1
V4
V3 V2 V5
Figure 1.9 A weighted digraph.
1.2.3 Algebraic Graph Theory
Algebraic graph theory studies matrices associated with digraphs.
Weighted digraph and adjacency matrix
A weighted digraph of digraph G(V, E) is a triplet G = (V, E, A), where A = [aij] ∈ Rn×n is
an adjacency matrix satisfying
aij =
0, if (vi, vj) ∈ E,
0, otherwise.
We denote by A(G) the adjacency matrix of a weighted digraph G. Figure 1.9 shows an example
of a weighted digraph. The adjacency matrix of the weighted digraph is
A =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 3 1 4 0
0 0 0 3.5 0
1.5 2 0 6 0
1 0 0 0 6
3 0 0 0 1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
Obviously, a weighted digraph is undirected if and only if aij = aji, that is, A(G) is
symmetric. 1
If digraph G contains no self-loop, then all the diagonal elements of its adjacency matrix
A are zero, i.e., aii = 0, i ∈ 1, n. However, sometimes we may encounter with graphs with
self-loops. In this case, aii may be positive numbers. To avoid confusion in terminology, by
1 If A ∈ Cn×n, under the symmetry, we mean the conjugate symmetry, i.e., A∗ = A, where A∗ is the conjugate
transpose of A.](https://image.slidesharecdn.com/2152340-250603023020-b823ff77/85/Frequencydomain-Analysis-And-Design-Of-Distributed-Control-Systems-Yuping-Tianauth-26-320.jpg)
![12 FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS
Gain, measurement matrix and state-transfer matrix
Let xi(k) ∈ R, i ∈ 1, n, be the state of agent i associated with vertex i, where k represents
the time. By our stipulation of the physical meaning of the edge direction in a digraph, the
existence of eij implies that agent i gets the state information xj from agent j. Then, the
weight aij associated with eij can be regarded as the gain for the information flow. So, for a
system with G(V, E, A) as its interconnection topology digraph, the existence of eij implies
that the agent i gets the amplified state information aijxj(k) from agent j. Let the measurement
yi(k) of agent i be equal to the sum of all the amplified states at the present time received by
agent i, i.e.,
yi(k) = aiixi(k) +
j∈Ni
aijxj(k). (1.9)
Such a measurement will be also referred to as aggregated measurement. Sometimes, each
agent can get only some relative measurement which can be expressed as
yi(k) = aiixi(k) −
j∈Ni
aijxj(k). (1.10)
Denotebyx(k) = [x1(k), · · · , xn(k)]T thestatevectorandy(k) = [y1(k), · · · , yn(k)]T themea-
surement vector. Then, the matrix form of the aggregated measurement (1.9) is
y(k) = Āx(k). (1.11)
Equation (1.11) provides an interpretation of the generalized adjacency matrix Ā from a view-
point of system theory: it can be considered as a measurement matrix.
Suppose the state formation of each agent is updated by the following local law:
x(k + 1) = Ky(k), (1.12)
where K = diag{κi ∈ R, i ∈ 1, n}. Then, we have
x(k + 1) = KĀx(k). (1.13)
So, with the updating law (1.12), KĀ is a state-transfer matrix of a closed-loop system, as
shown in Figure 1.10 .
Weighted bipartite graph
Let G(V, E, A) be a weighted bipartite graph with vertex classes V1 = {v1, · · · , vn1 } and
V2 = {vn1+1, · · · , vn}. Then, by the definition of bipartite graph, the adjacency matrix
x(k) y(k)
x(k + 1)
z–1
K A
Figure 1.10 A closed-loop system.](https://image.slidesharecdn.com/2152340-250603023020-b823ff77/85/Frequencydomain-Analysis-And-Design-Of-Distributed-Control-Systems-Yuping-Tianauth-28-320.jpg)
![14 FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS
Then, G1(V1, E1, ĀI) and G2(V2, E2, ĀII) are weighted equivalent graphs of the bipartite
graph G for V1 and V2, respectively.
Remark. ĀI (or ĀII) is not the un-weighted matrix of Ā1 (or Ā2), i.e.,
(A12)0,1(AT
12)0,1 /
= (A12AT
12)0,1,
(AT
12)0,1(A12)0,1 /
= (AT
12A12)0,1.
Exercise 1.7 Show
[(A12)0,1(AT
12)0,1]0,1 = (A12AT
12)0,1, (1.21)
[(AT
12)0,1(A12)0,1]0,1 = (AT
12A12)0,1. (1.22)
Let x1(k) ∈ Rn1 , x2(k) ∈ Rn−n1 be the state vectors associated with V1 and V2, respectively.
Denote x(k) = [xT
1 (k), xT
2 (k)]T. Then, under the measurement rule (1.11) and the state updated
law (1.12), we have
x1(k + 1)
x2(k + 1)
= K
0 A12
AT
12 0
x1(k)
x2(k)
.
Partition K as K = diag{K1, K2}, where K1 = diag{κi, i ∈ 1, n1}, K2 = diag{κi, i ∈
(n1 + 1), n}. Hence,
x1(k + 1) = K1A12x2(k),
x2(k + 1) = K2AT
12x1(k).
(1.23)
which yields
x1(k + 1) = K1A12x2(k) = K1A12K2AT
12x1(k − 1). (1.24)
If K2 0, we can define a new weight matrix for the bipartite graph G as
A
=
⎡
⎣
0 A12K
1
2
2
K
1
2
2 AT
12
⎤
⎦ .
Then, by Proposition 1.5, A12K2AT
12 is a weighted adjacency matrix of the equivalent graph
for vertex class V1 in G. Equation (1.24) links x(k + 1) to x(k − 1) instead of x(k) because
any vertex in V1 has no (one-level) neighbors in V1. So, K1A12K2AT
12 is also the state-transfer
matrix for the states associated with the vertices in V1.
Laplacian matrix
Now, we can generalize the notions of in- and out-degree to weighted digraphs. In a weighted
digraph G(V, E, A) with V = {v1, · · · , vn}, the weighted out-degree and the weighted](https://image.slidesharecdn.com/2152340-250603023020-b823ff77/85/Frequencydomain-Analysis-And-Design-Of-Distributed-Control-Systems-Yuping-Tianauth-30-320.jpg)
![INTRODUCTION 17
transmission rate xi(t) of source i. The second equation describes the generation of congestion
signal ql(t) at link l, by means of a congestion indication function pl(y), which is assumed
to be monotonically increasing, nonnegative, and not identically zero (a candidate of such a
function can be taken as p(y) = 1 − ey(1 − y), which is positive and strictly increasing for all
y 0).
Suppose that matrix R is of full row rank. Then, one can get the unique equilibrium point
of system (1.29)–(1.30) as x = [x
1, · · · , x
S]T, where x
i = wi/ l∈Li
pl( r∈Sl
x
r ). Define
yi(t) = (xi(t) − x
i )/
kix
i , i ∈ S.
The dynamic property of the system around the equilibrium is determined by the following
linearized model:
ẏi(t) = −κiwix−1
i yi(t) −
κix
i
l∈L
Rlip
l
r∈S
Rlr
κrx
r yr(t) (1.31)
where p
l is the derivative of pl evaluated at y = i∈Sl
x
i , i.e. p
l = p
l( i∈Sl
x
i ). Since pl(y)
is assumed to be monotonically increasing, p
l 0, ∀l ∈ L. By defining
K1 = diag{
p
l, i ∈ L},
K2 = diag{
κrx
r , r ∈ S},
and
ĀS = K2RT
K2
1RK2,
it is easy to verify that the interconnection between yl and yr can be described by a weighted
equivalent graph GS(S, ES, ĀS) for vertex class S in the bipartite graph G(V, E, A), where
V = L ⊕ S and A is given by (1.28). Denote
y(t) = [y1(t), · · · , yN(t)]T
,
K = diag{κiwix−1
i , i ∈ S}.
Then, the matrix form of (1.31) is given by
ẏ(t) = −Ky(t) − ĀSy(t). (1.32)
Model with delays
When propagation delays are considered, the congestion control algorithms can be modeled
by the following delayed differential equations (Johari and Tan 2001)
ẋi(t) = κi(wi − xi(t − Di)
l∈Li
ql(t − d←
li )), i ∈ S, (1.33)
ql(t) = pl(
r∈Sl
xr(t − d→
lr )), (1.34)](https://image.slidesharecdn.com/2152340-250603023020-b823ff77/85/Frequencydomain-Analysis-And-Design-Of-Distributed-Control-Systems-Yuping-Tianauth-33-320.jpg)
![32 FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS
where τij is the communication delay from agent j to agent i which mainly consists of prop-
agation delay and queueing delay, τii is called self-delay which is often introduced to match
the communication delays, aij is the weight for the output channel from agent j to agent i and
aii is the weight for the self-loop of agent i. To incorporate the aggregated measurement (1.9)
and the relative measurement (1.10) in a unified form, throughout this chapter, we assume that
aij in (2.2) can be either positive or negative.
Suppose that agent i is manipulated by the following local controller
˙
x̂i(t) = Ak
i x̂i(t) + Bk
i (ri(t) − zi(t))
ui(t) = Ck
i x̂i(t) + Dk
i (ri(t) − zi(t))
(2.3)
where x̂i(t) ∈ Rmi is the state of the local controller for agent i, Ak
i ∈ Rmi×mi , Bk
i ∈ R, Ck
i ∈ R,
Dk
i ∈ R are parameters of the controller and ri(t) is some reference signal for agent i.
Denote by Gi(s) the transfer function of agent i, i.e.,
Gi(s) = [Ci(sI − Ai)−1
Bi + Di]e−Tis
, (2.4)
and by κi(s) the transfer function of the ith controller, i.e.,
κi(s) = Ck
i (sI − Ak
i )−1
Bk
i + Dk
i . (2.5)
The feedback control system for agent i is shown in Figure 2.1. Let ŷi(s) and ûi(s) be the
Laplace transformation of the output and input, respectively, of the i-agent. Denote
Y(s) = [ŷ1(s), · · ·, ŷn(s)]T
, (2.6)
U(s) = [û1(s), · · ·, ûn(s)]T
, (2.7)
)
(s
Gi
s
ii
ii
e
a
)
(
ˆ s
yi
)
(s
i
)
(
ˆ s
ui
)
(
ˆ s
ri
s
i
i
e
a 1
1
s
i
i
e
a 2
2
s
in
in
e
a
)
(
ˆ1 s
y
)
(
ˆ2 s
y
)
(
ˆ s
yn
Network information
Figure 2.1 Feedback control system of agent i.](https://image.slidesharecdn.com/2152340-250603023020-b823ff77/85/Frequencydomain-Analysis-And-Design-Of-Distributed-Control-Systems-Yuping-Tianauth-47-320.jpg)
![SYMMETRY, STABILITY AND SCALABILITY 35
)
(s
G
)
(s
Ab
)
(
1 s
Y
)
(s
K
)
(
1 s
R
)
(s
Af
)
(s
P
)
(
2 s
Y
)
(s
)
(
2 s
R
Figure 2.4 Bipartite distributed control system.
Agent i in S is manipulated by dynamic controller κi(s), i = 1, · · · , n, and agent l in L is
manipulated by dynamic controller γl(s), l ∈ 1, · · · , m. Denote by
R1(s) = [r̂1
1(s), · · · , r̂1
n(s)]T
and
Y1(s) = [ŷ1
1(s), · · · , ŷ1
n(s)]T
the reference input vector and output vector of all the agents in S, respectively; by
R2(s) = [r̂2
1(s), · · · , r̂2
m(s)]T
and
Y2(s) = [ŷ2
1(s), · · · , ŷ2
m(s)]T
the reference input vector and output vector of all the agents in L, respectively. Then, the
bipartite system with forward and backward channels can be shown by Figure 2.4, where
G(s) = diag
Gi(s), i ∈ 1, n
,
K(s) = diag{κi(s), i ∈ 1, n},
P(s) = diag
Pl(s), l ∈ 1, m
,
(s) = diag{γl(s), l ∈ 1, m},
Āb(s) = {ab
ile−τb
lis
, i ∈ 1, n, l ∈ 1, m},
Āf (s) = {a
f
li e−τ
f
li s
, l ∈ 1, m, i ∈ 1, n}.](https://image.slidesharecdn.com/2152340-250603023020-b823ff77/85/Frequencydomain-Analysis-And-Design-Of-Distributed-Control-Systems-Yuping-Tianauth-50-320.jpg)
![I suppose Christmas is observed with great pomp in France. It
is a day which our Puritan forefathers, in their separation from
the Church of England, endeavored to blot out from the days of
religious festivals; and this because it was observed with so
much pomp by the Romish Church. In this, as well as in many
other things, they were as unreasonable as though they had
said they would not eat bread because the Roman Catholics do.
I hope and trust the time is not far distant when Christmas will
be observed by the descendants of the Puritans with all suitable
respect, as the first and highest holiday of Christians; combining
all the feelings and views of New England Thanksgiving with all
the other feelings appropriate to it.
January 31, 1830.
You have seen, perhaps, that the Directors of the Bunker Hill
Monument Association have applied to the Legislature for a
lottery. I am extremely sorry for it. I opposed the measure in all
its stages, and feel mortified that they have done so. They
cannot get it, and I desire that General Lafayette may
understand this; and, if he will write us a few lines during the
coming year, it will help us in getting forward a subscription.
When our citizens shall have had one year of successful
business, they will be ready to give the means to finish the
monument. My feelings are deeply interested in it, believing it
highly valuable as a nucleus for the affections of the people in
after time; and, if my life be spared and my success continue, I
will never cease my efforts until it be completed.
Further details will be given in this volume to show now nobly Mr.
Lawrence persevered in the resolution thus deliberately formed; and,
though he was destined to witness many fruitless efforts, he had the
satisfaction at last of seeing the completion of the monument, and
from its summit of pointing out the details of the battle to the son of
one of the British generals in command[2]
on that eventful day.](https://image.slidesharecdn.com/2152340-250603023020-b823ff77/85/Frequencydomain-Analysis-And-Design-Of-Distributed-Control-Systems-Yuping-Tianauth-70-320.jpg)
Frequencydomain Analysis And Design Of Distributed Control Systems Yuping Tianauth
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- 5. FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS
- 6. FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS Yu-Ping Tian Southeast University, China
- 7. This edition first published 2012 © 2012 John Wiley & Sons Singapore Pte. Ltd. Registered office John Wiley & Sons Singapore Pte. Ltd., 1 Fusionopolis Walk, #07-01 Solaris South Tower, Singapore 138628 For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as expressly permitted by law, without either the prior written permission of the Publisher, or authorization through payment of the appropriate photocopy fee to the Copyright Clearance Center. Requests for permission should be addressed to the Publisher, John Wiley & Sons Singapore Pte. Ltd., 1 Fusionopolis Walk, #07-01 Solaris South Tower, Singapore 138628, tel: 65-66438000, fax: 65-66438008, email: [email protected]. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The Publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloging-in-Publication Data Tian, Yu-Ping. Frequency-domain analysis and design of distributed control systems / Yu-Ping Tian. p. cm. Includes bibliographical references and index. ISBN 978-0-470-82820-5 (hardback) 1. Distributed parameter systems. 2. Time-domain analysis. 3. Frequency curves. I. Title. QA402.37.T53 2012 003’.5—dc23 2012020588 ISBN: 9780470828205 Set in 10/12 pt Times by Thomson Digital, Noida, India
- 8. I dedicate this book to Ningning and Ouya for their love, trust and support.
- 9. Contents Preface xi Glossary of Symbols xiii 1 Introduction 1 1.1 Network-Based Distributed Control System 1 1.2 Graph Theory and Interconnection Topology 4 1.2.1 Basic Definitions 4 1.2.2 Graph Operations 7 1.2.3 Algebraic Graph Theory 10 1.3 Distributed Control Systems 16 1.3.1 End-to-End Congestion Control Systems 16 1.3.2 Consensus-Based Formation Control 22 1.4 Notes and References 25 1.4.1 Graph Theory and Distributed Control Systems 25 1.4.2 Delay in Control and Control by Delay 26 References 26 2 Symmetry, Stability and Scalability 31 2.1 System Model 31 2.1.1 Graph-Based Model of Distributed Control Systems 31 2.1.2 Bipartite Distributed Control Systems 34 2.2 Symmetry in the Frequency Domain 36 2.2.1 Symmetric Systems 36 2.2.2 Symmetry of Bipartite Systems 38 2.3 Stability of Multivariable Systems 39 2.3.1 Poles and Stability 39 2.3.2 Zeros and Pole-Zero Cancelation 41 2.4 Frequency-Domain Criteria of Stability 43 2.4.1 Loop Transformation and Multiplier 44 2.4.2 Multivariable Nyquist Stability Criterion 45 2.4.3 Spectral Radius Theorem and Small-Gain Theorem 50 2.4.4 Positive Realness Theorem 53 2.5 Scalable Stability Criteria 53 2.5.1 Estimation of Spectrum of Complex Matrices 53 2.5.2 Scalable Stability Criteria for Asymmetric Systems 56
- 10. viii CONTENTS 2.5.3 Scalable Stability Criteria for Symmetric Systems 60 2.5.4 Robust Stability in Deformity of Symmetry 61 2.6 Notes and References 64 References 65 3 Scalability in the Frequency Domain 67 3.1 How the Scalability Condition is Related with Frequency Responses 67 3.2 Clockwise Property of Parameterized Curves 71 3.3 Scalability of First-Order Systems 76 3.3.1 Continuous-Time System 76 3.3.2 Discrete-Time System 79 3.4 Scalability of Second-Order Systems 85 3.4.1 System of Type I 85 3.4.2 System of Type II 95 3.5 Frequency-Sweeping Condition 103 3.5.1 Stable Quasi-Polynomials 103 3.5.2 Frequency-Sweeping Test 105 3.6 Notes and References 108 References 109 4 Congestion Control: Model and Algorithms 111 4.1 An Introduction to Congestion Control 111 4.1.1 Congestion Collapse 112 4.1.2 Efficiency and Fairness 114 4.1.3 Optimization-Based Resource Allocation 114 4.2 Distributed Congestion Control Algorithms 116 4.2.1 Penalty Function Approach and Primal Algorithm 116 4.2.2 Dual Approach and Dual Algorithm 117 4.2.3 Primal-Dual Algorithm 118 4.2.4 REM: A Second-Order Dual Algorithm 118 4.3 A General Model of Congestion Control Systems 119 4.3.1 Framework of End-to-End Congestion Control under Diverse Round-Trip Delays 119 4.3.2 General Primal-Dual Algorithm 122 4.3.3 Frequency-Domain Symmetry of Congestion Control Systems 124 4.4 Notes and References 126 References 127 5 Congestion Control: Stability and Scalability 129 5.1 Stability of the Primal Algorithm 129 5.1.1 Johari–Tan Conjecture 129 5.1.2 Scalable Stability Criterion for Discrete-Time Systems 131 5.1.3 Scalable Stability Criterion for Continuous-Time Systems 135 5.2 Stability of REM 138 5.2.1 Scalable Stability Criteria 138
- 11. CONTENTS ix 5.2.2 Dual Algorithm: the First-Order Limit Form of REM 145 5.2.3 Design of Parameters of REM 146 5.3 Stability of the Primal-Dual Algorithm 152 5.3.1 Scalable Stability Criteria 152 5.3.2 Proof of the Stability Criteria 161 5.4 Time-Delayed Feedback Control 163 5.4.1 Time-Delayed State as a Reference 163 5.4.2 TDFC for Stabilization of an Unknown Equilibrium 165 5.4.3 Limitation of TDFC in Stabilization 166 5.5 Stabilization of Congestion Control Systems by Time-Delayed Feedback Control 170 5.5.1 Introduction of TDFC into Distributed Congestion Control Systems 170 5.5.2 Stabilizability under TDFC 171 5.5.3 Design of TDFC with Commensurate Self-Delays 181 5.6 Notes and References 188 5.6.1 Stability of Congestion Control with Propagation Delays 188 5.6.2 Time-Delayed Feedback Control 189 References 190 6 Consensus in Homogeneous Multi-Agent Systems 193 6.1 Introduction to Consensus Problem 193 6.1.1 Integrator Agent System 193 6.1.2 Existence of Consensus Solution 194 6.1.3 Consensus as a Stability Problem 194 6.1.4 Discrete-Time Systems 195 6.1.5 Consentability 195 6.2 Second-Order Agent System 196 6.2.1 Consensus and Stability 196 6.2.2 Consensus and Consentability Condition 199 6.2.3 Periodic Consensus Solutions 203 6.2.4 Simulation Study 204 6.3 High-Order Agent System 206 6.3.1 System Model 206 6.3.2 Consensus Condition 208 6.3.3 Consentability 211 6.4 Notes and References 216 References 217 7 Consensus in Heterogeneous Multi-Agent Systems 219 7.1 Integrator Agent System with Diverse Input and Communication Delays 219 7.1.1 Consensus in Discrete-Time Systems 220 7.1.2 Consensus under Diverse Input Delays 221 7.1.3 Consensus under Diverse Communication Delays and Input Delays 224 7.1.4 Continuous-Time System 229 7.1.5 Simulation Study 230
- 12. x CONTENTS 7.2 Double Integrator System with Diverse Input Delays and Interconnection Uncertainties 233 7.2.1 Leader-Following Consensus Algorithm 233 7.2.2 Consensus Condition under Symmetric Coupling Weights 235 7.2.3 Robust Consensus under Asymmetric Perturbations 238 7.2.4 Simulation Study 240 7.3 High-Order Consensus in High-Order Systems 243 7.3.1 System Model 243 7.3.2 Consensus Condition 245 7.3.3 Existence of High-Order Consensus Solutions 249 7.3.4 Constant Consensus 252 7.3.5 Consensus in Ideal Networks 254 7.4 Integrator-Chain Systems with Diverse Communication Delays 255 7.4.1 Matching Condition for Self-Delay 255 7.4.2 Adaptive Adjustment of Self-Delay 255 7.4.3 Simulation Study 257 7.5 Notes and References 265 References 266 Index 269
- 13. Preface This book is devoted to the study of distributed control systems, a very promising research area given the current rapid development of technologies of micro-sensors, micro-motors, sensor networks and communication networks. The main feature of the book is to adopt a frequency- domain approach to cope with analysis and design problems in distributed control systems. Frequency-domain methods utilize frequency response properties of subsystems (agents) and communication channels to characterize the stability and other performance criteria of the entire system. By comparison with time-domain methods, it is usually assumed to be more convenient to use frequency-domain methods for analyzing the robustness of the system against noise and dynamic perturbations. It will be shown in this book that frequency-domain methods are also powerful in scalability analysis of distributed control system. The book consists of three parts. The first part includes Chapters 1, 2 and 3, and describes common features and mathematical models of distributed control systems; it also introduces basic tools that are useful for further analysis and design of the underlying systems, such as graph theory, frequency-domain stability criteria, scalability analysis based on differential geometric properties of frequency response plots, etc. The second part includes Chapters 4 and 5, and the third part includes Chapters 6 and 7. The second part focuses on the distributed congestion control of communication networks while the third part studies the consensus control of multi-agent systems. The second and third parts can be considered as the application of the general theory introduced in the first part. However, the theory is also developed in the last two parts for two particular types of distributed control systems. Moreover, many interesting and beneficial results are obtained when general theory is applied to concrete systems. It is quite common to choose the congestion control and the consensus control for the study of distributed control systems. There are at least two reasons. Firstly, both types of control schemes emphasize cooperation in a group of agents although the starting points of the cooperation are somehow different. For the congestion control, the cooperation is real- ized through distributed real-time optimization of some common performance indexes, i.e., allocation of limited resources in networks. For the consensus control, the basis of the coop- eration is to manipulate states (or output) of agents to reach some common value. The two cooperation approaches are widely used in many other practical distributed control systems, such as multi-robot control systems, localization and information fusion sensor networks, etc. Secondly, many theoretical problems such as stability, scalability and delay effect encountered in these two kinds of systems can be treated by using a unified frequency-domain method. Although distributed real-time optimization problems in congestion control usually lead to nonlinear models, the local dynamics around an equilibrium of a congestion control system can be described as a linear distributed feedback control system with a bipartite interconnection topology graph, which essentially has the same structure as a multi-agent system controlled by
- 14. xii PREFACE a consensus protocol. Nevertheless, consensus problems are different from congestion control problems in many aspects. For example, a multi-agent system driven by a consensus protocol has a continuum of equilibriums instead of an isolated equilibrium. This is perhaps why the Laplacian matrix of the interconnection topology graph plays a key role in consensus problems. The time-delayed feedback control is introduced in this book as one of the basic design methodsfordistributedcontrolsystems.Westudynotonlytheeffectoftimedelays,inparticular various communication delays, on stability and scalability, but also the potential of time- delayed feedback control for enhancing the stability. This looks somewhat illogical. But in fact it is possible for time delay to have a dual character in feedback control systems, as in many other things in nature. Some notions and results appearing in this book are new. They are mostly related to bipartite systems, symmetric systems, symmetric communication delays, commensurate self- delays, semi-stability test in the frequency domain, and high-order consensus. However, most of the material is based on the research results of the author with his collaborators and students, notably Jiandong Zhu, Hong-Yong Yang, Cheng-Lin Liu and Ya Zhang, to whom the author would like to express his sincere thanks. The author would also like to acknowledge the continuous support of the National Natural Science Foundation of China in the research topics of this book and in other related areas. Parts of the book were taught on short courses listed below. 1. “Time-delayed Feedback Control”, Pre-conference workshop lecture on Chinese Control Conference, Harbin, 2006. 2. “Internet Congestion Control”, Summer school lecture at the School of Mechanics and Engineering Sciences, Peking University, Beijing, 2008. 3. “Coordination Control of Multi-agent Systems”, Summer school lecture at School of Mechanics and Engineering Sciences, Peking University, Beijing, 2010. The author would also like to thank Prof. Zhong-Ping Jiang of New York University, Prof. Lin Huang of Peking University and Prof. Guangrong Chen of City University of Hong Kong for their invitations to present these lectures, which impelled the author to collect related materials scattered in his own and others’ research publications. Yu-Ping Tian Southeast University, Nanjing
- 15. Glossary of Symbols := “defined as” “denoted as” A B Exclude set B from set A, where B ⊂ A A ⊗ B Kronecker product of matrices A and B A ⊕ B Direct sum of set A and B (z)+ x A piece-wise function of x ≥ 0, which takes value z if x 0, or max(z, 0) if x = 0 R, Rn, Rn×n The set of real numbers, n component real vectors, and n by n real matrices R+ The set of nonnegative real numbers C, Cn, Cn×n The set of complex numbers, n component complex vectors, and n by n complex matrices Cn Space of functions which are n-times differentiable C(ω) Curvature of a parametric curve at parameter ω Re(s) The real part of complex number s Im(s) The imaginary part of complex number s C− The open left half of the complex plane (briefly as LHP), {s ∈ C : Re(s) 0} C̄− The closed left half of the complex plane (briefly as closed LHP), {s ∈ C : Re(s) ≤ 0} C+ The closed right half of the complex plane (briefly as RHP), {s ∈ C : Re(s) ≥ 0} C̆+ The open right half of the complex plane (briefly as open RHP), {s ∈ C : Re(s) 0} D The interior of the unit disc (briefly as IUD), {s ∈ C : |s| 1}
- 16. xiv GLOSSARY OF SYMBOLS D̄ The closure of D (briefly as closed IUD), {s ∈ C : |s| ≤ 1} Do The closed outer part of the unit disc (briefly as OUD), {s ∈ C : |s| ≥ 1} D̆o The open outer part of the unit disc (briefly as open OUD), {s ∈ C : |s| 1} Hn×n ∞ The set of n by n transfer function matrices G(s), with G ∞ ∞ RHn×n ∞ The set of n by n rational transfer function matrices G(s), with G ∞ ∞ Co(r1, · · · , rn) Convex hull of elements r1 · · · , rn n1, n2 The set of integers from n1 to n2 satisfying n1 ≤ n2, {n1, · · · , n2} S The index set of source nodes in a communication network, {1, 2, · · · , S} L The index set of link nodes in a communication network, {1, 2, · · · , L} AT Transpose of matrix (vector) A A∗ Conjugate transpose of matrix (vector) A j Unit of imaginary numbers, √ −1 1n n × 1 vector [1, · · · , 1]T e1 n × 1 vector [1, 0, · · · , 0]T diag{ti, i ∈ 1, n} Diagonal matrix with diagonal entries ti, i ∈ 1, n λi(A) i-th eigenvalue of matrix A ∈ Cn×n. σ(A) Spectrum of matrix A ∈ Cn×n, {λ1(A), · · · , λn(A)} ρ(A) Spectral radius of matrix A ∈ Cn×n, max i∈1,n |λi(A)| σ̄(A) Maximum singular value of matrix A ∈ Cn×m, max i∈1,n {λi(AA∗)} 1 2 rank(A) Rank of matrix A span(A) The space spanned by all the columns of matrix A [A](:, i) The i-th column of matrix A [A](:, i1, i2) The matrix formed by i1-th to i2-th columns of matrix A, where i1 ≤ i2
- 17. 1 Introduction Ruling a large country is like cooking a small fish. —Lao Dan (580–500 BC), Tao Te Ching This chapter is concerned with some common characteristics of distributed control systems, such as distributive interconnections, local control rules and scalability. Basic notions and results of algebraic graph theory are introduced as a theoretic foundation for modeling interconnections of subsystems (agents) in distributed control systems. Coordination con- trol systems and end-to-end congestion control systems are also introduced as two typical kinds of distributed control systems. The main topics of this book are also highlighted in the introduction of these two kinds of systems. 1.1 Network-Based Distributed Control System From the “flyball” speed governor of Watt’s steam engine to the control of communication systems including the telephone system, cell phones and the Internet, the control mechanism in industrial, social and many other real-world systems has experienced a development process from centralized control to distributed control. In the past decade, this process has been signifi- cantly speeded up thanks to rapid advances of communication techniques and their application to control systems. Recently, workshops, seminars, short courses and even regular courses on distributed con- trol systems emerge in the control society just like bamboo shoots after spring rain. But the following questions, which are often raised in seminars or lectures by post-graduate students, may also puzzle the reader of this book. What kind of control systems can be called distributed control system? Are there any differences between a distributed control system and a so-called large-scale control system or a decentralized control systems? What’s the advantage of dis- tributed control over centralized control? etc. We will not attempt to answer all the questions; rather, we will try to grasp some common features of distributed control systems. Cooperation among agents Cooperation is perhaps the soul of a distributed control system. Since there is no centralized control unit, the only way through which all the agents in the system can work in harmony Frequency-Domain Analysis and Design of Distributed Control Systems, First Edition. Yu-Ping Tian. © 2012 John Wiley Sons Singapore Pte. Ltd. Published 2012 by John Wiley Sons Singapore Pte. Ltd.
- 18. 2 FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS is to conduct some kind of cooperation. Ignoring cooperation in biology might be one of a few shortcomings by which one could rebuke the great theory of Darwin (1859). Indeed, cooperation has been observed in many biological colonies, such as in birds, fish, bacilli and so on. Through cooperation they increase the probability of discovering food, get rid of prey and other dangers. Breder (1954) proposed a simple mathematic model to characterize the attracting and excluding actions in schools of fish. To describe the flocks and aggregations of schools more effectively,Reynolds(1987)proposedthreesimplerules:(1)collisionavoidance,(2)agreement on velocity and (3) approaching center, which actually implies approaching any neighbor. He successfully simulated the motion of fish by using these rules. These investigations stimulate researchers to develop artificial systems that make decisions and take actions in distributive but cooperative manners. Nowadays, distributed control mechanisms have been widely used in engineering systems, such as aircraft traffic control (Tomlin, Pappas and Sastry 1998), multi-robot control (Rekleitis, Dudek and Milios 2000), coverage control of sensor networks (Qi, Iyengar and Chakrabarty 2001), formation control of unmanned aerial vehicles (UAVs) (Giulietti, Pollini and Innocenti 2000) etc. Spatially distributed interconnection In a classic feedback control system, the controller gets the output signal of the plant, compares it with some reference signal and makes decisions on how to act. Such a system also serves as one of basic units (subsystems) in a distributed control system. However, to cooperate with other units in the entire system, it should be equipped with a sensor/communication modular besides the classic decision/action modular (controller), as shown in Figure 1.1. Such a subsystem is sometimes called an agent. In a distributed control system, therefore, each subsystem gets not only the information of the output of itself but also the information of some other subsystems via sensor or/and communication networks (Figure 1.2). A distributed control system interconnected with multiple agents is also called a multi-agent system (MAS). For many distributed control systems such as the Internet, power grids, traffic control systems etc., subsystems (agents) are often distributed across multiple computational units in animmensespaceandconnectedthroughlong-distancepacket-basedcommunications.Inthese system packet loss and delay are unavoidable, and hence, computational and communication C S/C modular P r y Figure 1.1 An agent in distributed control system.
- 19. INTRODUCTION 3 1 r P1 C1 Ck Pk Cn Pn Pi Ci ri rn rk Communication network or/and sensor network Figure 1.2 A network-based distributed control system. constraints cannot be ignored. New formalism to ensure stability, performance and robustness is required in the analysis and design of distributed control systems (Murray et al. 2003). Local control rule A distributed control law should be subject to the local control rule, which implies that in most cases there is no kind of centralized supervision or control unit in the system, and each agent makes decisions based only on the information received by its own sensor or from its neighboring agents through communication. The action rules proposed by Reynolds (1987) are typically local control rules. The local control rule is the most important feature of distributed control systems and makes the distributed control distinguished from the decentralized control of large-scale systems, which were extensively studied in the 1970s and 1980s (see, e.g., Sastry (1999) for a survey on large-scale systems). The decentralized control mechanism allows each agent to communicate perfectly with any other agent in the system, ignoring the computational and communication constraints. Moreover, the plant in decentralized control systems is usually a single tightly connected unit and not a rather loosely interconnected group of complete systems, which is the typical case in distributed control systems. Therefore, cooperation among agents in distributed control systems is much more important than in large- scale systems, and hence the coupling between a pair of subsystems can not be dealt with as a disturbance as in some decentralized control designs. A remarkable advantage of the local control rule over the non-local control rules is its higher fault-tolerance capability. This is extremely important for many large-scale engineering systems which must operate continuously even when some individual units fail. Therefore, building a very reliable distributed control system with fault-tolerance ability from unreliable parts is a very promising research direction (Murray et al. 2003), although the topic is beyond the scope of this book. Scalability Scalability is perhaps another very important reason why distributed control systems prefer the local control rule. In this book the scalability of a distributed control system implies that the controller of the system and its maintenance utilize only local information around each agent and rarely depends on the scale of the system. In other words, by scalability we mean that not only the control law but also most important properties of the system rely on the local information. For example, in checking the stability of the entire distributed control system the
- 20. 4 FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS scalability requires that the stability criterion does not need to use information about the global interconnection topology of the system because such global information is usually unavailable to individual agents. A scalable system allows new applications to be designed and deployed without requiring changes to the underlying system. Thescalabilityofheterogeneousdistributedcontrolsystemshasdrawnmuchmoreattention ofresearchersbecausemostpracticalnetworkedsystemssuchastheInternetareheterogeneous. Diverse communication and/or input delays, different channel capacities, non-identical agent dynamics and so on, can make a system heterogeneous. Analysis and design of heterogeneous systems are much more difficult in comparison with homogeneous systems. One reason for that is the analysis and/or design of a large but homogenous systems can usually be treated as a task for a small system through diagonalization of the original system. But in most cases such a simplification method is not applicable to heterogeneous systems. 1.2 Graph Theory and Interconnection Topology 1.2.1 Basic Definitions Graph theory plays a crucial role in describing the interconnection topology of distributed control systems. In this section we only present basic definitions about graph theory. For systematic study of graph theory we refer the reader to, for example, Biggs (1994); Bollobás (1998); Diestel (1997); Godsil and Royle (2001). Digraph A directed graph (in short, digraph) of order n is a pair G = (V, E), where V is a set with n elements called vertices (or nodes) and E is a set of ordered pairs of vertices called edges. In other words, E ⊆ V × V. We denote by V(G) and E(G) the vertex set and edge set of graph G, respectively, and denote by 1, n := {1, · · · , n} a finite set for vertex index. For two vertices vi, vj ∈ V, i.e., i, j ∈ 1, n, i / = j, the ordered pair (vi, vj) represents an edge from vi to vj and is also simply denoted by eij. A digraph G(V, E) is said to be a subgraph of a digraph (V, E) if V ⊂ V and E ⊂ E. In particular, a digraph (V, E) is said to be a spanning subgraph of a digraph (V, E) if it is a subgraph and V = V. The digraph (V, E) is the subgraph of (V, E) induced by V ⊂ V if E contains all edges in E between two vertices in V. Path and connectivity A path in a digraph is an ordered sequence of vertices such that any ordered pair of vertices appearing consecutively in the sequence is an edge of the digraph. A path is simple if no vertices appear more than once in it, except possibly for the initial and final vertices. The length of a path is defined as the number of consecutive edges in the path. For a simple path, the path length is less than the number of vertices contained in the path by unity. A vertex vi in digraph G is said to be reachable from another vertex vj if there is a path in G from vi to vj. A vertex in the digraph is said to be globally reachable if it is reachable from every other vertex in the digraph. A digraph is strongly connected if every vertex is globally reachable. In Figure 1.3, v1, v2, v4, v5 are globally reachable vertices. But the digraph is not strongly connected because v3 is unreachable from the other vertices.
- 21. INTRODUCTION 5 V1 V4 V2 V3 V5 Figure 1.3 A digraph. Cycle and tree A cycle is a simple path that starts and ends at the same vertex. A cycle containing only one vertex is called a self-cycle (or self-loop). The length of a cycle is defined as the number of edges contained in the cycle. A cycle is odd (even) if it’s length is odd (even). If a vertex in a cycle is globally reachable, then any other vertex in the cycle is also globally reachable. In Figure 1.3, the path (v1, v2, v5, v1) is a cycle. The path {v2, v4, v5, v2} and the path {v1, v2, v4, v5, v1} are also cycles. This digraph has no self-cycle. A digraph with self-cycle is shown in Figure 1.4. V1 V4 V2 V3 V5 Figure 1.4 A digraph with a self-cycle.
- 22. 6 FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS V1 V4 V2 V3 V5 Figure 1.5 A directed tree. A digraph is acyclic if it contains no cycles. An acyclic digraph is called a directed tree if it satisfies the following property: there exists a vertex, called the root, such that any other vertex of the digraph can be reached by one and only one path starting at the root. A directed spanning tree of a digraph is a spanning subgraph that is a directed tree. The digraph shown in Figure 1.5 is a directed tree. Obviously, it is a directed spanning tree of both the digraph in Figure 1.3 and the digraph in Figure 1.4. Neighbor and degree If (vi, vj) is an edge of digraph G, then vj is an out-neighbor of vi, and vi is an in-neighbor of vj. The set of out-neighbors (in-neighbors) of vertex vi in digraph G(V, E) is denoted by Nout i (G) (Nin i (G)). The out-degree (in-degree) of vi is the cardinality of Nout i (Nin i ). A digraph is topologically balanced if each vertex has the same in- and out-degrees. Note that neither Nout i nor Nin i contains the vertex i itself even if there is a self-loop at vertex i. Let us consider the example of a digraph shown in Figure 1.3. For vertex 1, its out-neighbor set is Nout 1 = {v2}, and its in-neighbor set is Nin 1 = {v3, v5}. The out-degree and the in-degree of v1 are 1 and 2, respectively. In this book, if the superscript is dropped in formulae or the prefix is omitted in texts, it will be referred to as the out-neighbor, i.e., Ni(G) = {vj ∈ V : (vi, vj) ∈ E}. Sometimes we may use the term multi-level neighbor. If there is an m-length path in digraph G from vertex i to vertex j, then vertex j is said to be an m-level neighbor of vertex i. Of course, a neighbor in conventional sense is a 1-level neighbor. The set of m-level neighbors of agent i in digraph G is denoted by Nm i (G). For example, in the digraph shown in Figure 1.3, N2 1 (G) = {v4, v5}.
- 23. INTRODUCTION 7 Digraph and information flow In a distributed control system, each agent can be considered as a vertex in a digraph, and the information flow between two agents can be regarded as a directed path between the vertices in the digraph. Thus, the interconnection topology of a distributed control system can be described by a digraph. However, differing from the classic signal-flow graph (Chen 1984), in this book and in many other references on the distributed control system, the direction of an edge in the digraph does not mean the direction of an information flow. Let us consider the digraph shown in Figure 1.3 for an instance. Denote by xi ∈ R, i ∈ 1, 5, the state of agent i associated with vertex i. The existence of edge eij implies that agent i gets the state information xj from agent j. For example, agent 1 gets information from agent 2. 1.2.2 Graph Operations We shall construct new graphs from old ones by graph operations. For two digraphs G1 = (V1, E1) and G2 = (V2, E2), the intersection and union of G1 and G2 are defined by G1 ∩ G2 = (V1 ∩ V2, E1 ∩ E2), G1 ∪ G2 = (V1 ∪ V2, E1 ∪ E2). For a digraph G = (V, E), the reverse digraph of G is a pair rev(G) = (V, rev(E)), where rev(E) consists of all edges in E with reversed directions. If W ⊂ V(G), then G − W = G[VW] is the subgraph of G obtained by deleting the vertices in W and all edges incident with them. Obviously, G − W is the subgraph of G induced by VW. Similarly, if E ⊂ E(G), then G − E = (V(G), E(G)E). If W (or E) contains a single vertex w (or a single edge xy, respectively), the notion is simplified to G − w (or G − xy, respectively). Similarly, if x and y are non-adjacent vertices of G, then G + xy is obtained from G by joining x to y. Undirected graph An undirected graph (in short, graph) G consists of a vertex set V and a set E of unordered pairs of vertices. If each edge of the graph G is given a particular orientation, then we get an oriented graph of G, denoted by G→, which is a digraph. Denote by G← the reverse of G→. Then, G = G→ ∪ G←. For an undirected graph G, the in-neighbor set of any vertex is always equal to the out- neighborsetofthesamevertex.Therefore,intheundirectedcasewesimplyusetheterminations neighbor, neighbor set and degree. For an undirected graph, if it contains a globally reachable node, then any other vertex is also globally reachable. In that case we simply say that the undirected graph is connected. For an undirected graph, it is said to be a tree if it is connected and acyclic. Theorem 1.1 G(V, E) is a tree if and only if G is connected and |E| = |V| − 1. Alternatively, G(V, E) is a tree if and only if G is acyclic and |E| = |V| − 1. Theorem 1.2 A graph is connected if and only if it contains a spanning tree.
- 24. 8 FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS 1 2 3 II I III IV Figure 1.6 A bipartite graph. Bipartite graph A graph G is a bipartite graph with vertex classes V1 and V2 if V(G) is a direct sum of V1 and V2, i.e., V = V1 ⊕ V2, which implies that V = V1 ∪ V2 and V1 ∩ V2 = ∅, and every edge joins a vertex of V1 to a vertex of V2. It is also said that G has bipartition (V1, V2). Figure 1.6 shows an example of a bipartite graph. For this graph, the vertex set V is a direct sum of V1 = {v1, v2, v3} and V2 = {vI, vII, vIII, vIV}. Each vertex in V1 has neighbors only in V2, and vice versa. Theorem 1.3 A graph is bipartite if and only if it does not contain an odd cycle. For the bipartite graph G with vertex classes V1 and V2, define E 12 = vl∈V2 {(vi, vj) : vi, vj ∈ Nl}, (1.1) E 21 = vr∈V1 {(vi, vj) : vi, vj ∈ Nr}. (1.2) Note that in both (1.1) and (1.2), vi and vj can be the same vertex. In this case, (vi, vi) is a self-loop. Let G1 = G − V2 + E 12 (1.3) and G2 = G − V1 + E 21. (1.4) G1 is a new graph obtained from G by deleting all the vertices in V2 and adding the new edges in E 12. By definition (1.1), E 12 is the set of new edges, each of which adds a self-loop to one vertex or joins two vertices in V1 that are neighbors of a vertex in V2. Similarly, G2 is a new graph obtained from G by deleting all the vertices in V1 and adding the new edges in E 21.
- 25. INTRODUCTION 9 1 2 3 (a) 1 2 3 (b) Figure 1.7 Graph operation and equivalent graph for V1. E 21 is the set of new edges, each of which adds a self-loop to one vertex or joins two vertices in V2 that are neighbors of a vertex in V1. G1 (G2) is said to be the equivalent graph for V1 (V2) deduced from G. Obviously, neither G1 nor G2 is a subgraph of G. But the interconnection topology between any two vertices in one vertex class remains unchanged for G and the equivalent graph G1 (or G2). The graph operation defined by (1.3) for the bipartite graph shown in Figure 1.6 is illustrated by Figure 1.7 (a). Since only one edge is defined between any pair of vertices in graph, the equivalent graph for V1 is given by Figure 1.7 (b). Similarly, the graph operation defined by (1.4) for the same bipartite graph is illustrated by Figure 1.8 (a). The equivalent graph for V2 is given by Figure 1.8 (b). II I III IV (a) (b) II I III IV Figure 1.8 Graph operation and equivalent graph for V2.
- 26. 10 FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS V1 1 1.5 2 6 3.5 3 3 6 4 4 1 V4 V3 V2 V5 Figure 1.9 A weighted digraph. 1.2.3 Algebraic Graph Theory Algebraic graph theory studies matrices associated with digraphs. Weighted digraph and adjacency matrix A weighted digraph of digraph G(V, E) is a triplet G = (V, E, A), where A = [aij] ∈ Rn×n is an adjacency matrix satisfying aij = 0, if (vi, vj) ∈ E, 0, otherwise. We denote by A(G) the adjacency matrix of a weighted digraph G. Figure 1.9 shows an example of a weighted digraph. The adjacency matrix of the weighted digraph is A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 3 1 4 0 0 0 0 3.5 0 1.5 2 0 6 0 1 0 0 0 6 3 0 0 0 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . Obviously, a weighted digraph is undirected if and only if aij = aji, that is, A(G) is symmetric. 1 If digraph G contains no self-loop, then all the diagonal elements of its adjacency matrix A are zero, i.e., aii = 0, i ∈ 1, n. However, sometimes we may encounter with graphs with self-loops. In this case, aii may be positive numbers. To avoid confusion in terminology, by 1 If A ∈ Cn×n, under the symmetry, we mean the conjugate symmetry, i.e., A∗ = A, where A∗ is the conjugate transpose of A.
- 27. INTRODUCTION 11 adjacency matrix we still mean the matrix A with zero entries in the diagonal. Actually, A(G) is associated with the digraph that is obtained by cutting off all the self-loops in G. We denote by Ā(G) the matrix that characterizes the existence of all edges including self-loops in G, i.e., Ā = diag{aii, i ∈ 1, n} + A, (1.5) and refer to Ā as the generalized adjacency matrix of G. For a weighted digraph G(V, E, Ā) of order n, we use Ā0,1 = {dij}n×n to denote its un-weighted adjacency matrix, where dij = 1, if aij 0, 0, if aij = 0. Let G(V, E, Ā) be a weighted digraph of G(V, E) of order n. Given a matrix Ā = {a ij}n×n with a ij ∈ R+, if (Ā )0,1 = Ā0,1, (1.6) then, by the definition of adjacency matrix, G(V, E, Ā) is also a weighted digraph of G(V, E). Naturally, a digraph G = (V, E) can be considered as a weighted digraph with {0, 1}- weights, i.e., aij = 1, if (vi, vj) ∈ E, 0, otherwise. The weighted out-degree matrix of digraph G, denoted by Dout(G), is the diagonal matrix with the weighted out-degree of each node along its diagonal, i.e., Dout (G) = diag{dout (vi), i ∈ 1, n}. (1.7) The weighted in-degree matrix of digraph G, denoted by Din(G), is the diagonal matrix with the weighted in-degree of each node along its diagonal, i.e., Din (G) = diag{din (vi), i ∈ 1, n}. (1.8) While the adjacency matrix characterizes the location of edges among vertices in a digraph, the following result shows that powers of the adjacency matrix characterize the relationship between directed paths and vertices in the digraph. Lemma 1.4 Let G(V, E, Ā) be a weighted digraph of order n possibly with self-loops. Ā0,1 is its un-weighted adjacency matrix. Then, for all i, j, k ∈ 1, n, (1) the (i, j) entry of Āk 0,1 equals the number of directed paths of length k (including paths with self-loops) from vertex i to vertex j; and (2) the (i, j) entry of Āk is positive if and only if there exists a directed path of length k (including paths with self-loops) from vertex i to vertex j.
- 28. 12 FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS Gain, measurement matrix and state-transfer matrix Let xi(k) ∈ R, i ∈ 1, n, be the state of agent i associated with vertex i, where k represents the time. By our stipulation of the physical meaning of the edge direction in a digraph, the existence of eij implies that agent i gets the state information xj from agent j. Then, the weight aij associated with eij can be regarded as the gain for the information flow. So, for a system with G(V, E, A) as its interconnection topology digraph, the existence of eij implies that the agent i gets the amplified state information aijxj(k) from agent j. Let the measurement yi(k) of agent i be equal to the sum of all the amplified states at the present time received by agent i, i.e., yi(k) = aiixi(k) + j∈Ni aijxj(k). (1.9) Such a measurement will be also referred to as aggregated measurement. Sometimes, each agent can get only some relative measurement which can be expressed as yi(k) = aiixi(k) − j∈Ni aijxj(k). (1.10) Denotebyx(k) = [x1(k), · · · , xn(k)]T thestatevectorandy(k) = [y1(k), · · · , yn(k)]T themea- surement vector. Then, the matrix form of the aggregated measurement (1.9) is y(k) = Āx(k). (1.11) Equation (1.11) provides an interpretation of the generalized adjacency matrix Ā from a view- point of system theory: it can be considered as a measurement matrix. Suppose the state formation of each agent is updated by the following local law: x(k + 1) = Ky(k), (1.12) where K = diag{κi ∈ R, i ∈ 1, n}. Then, we have x(k + 1) = KĀx(k). (1.13) So, with the updating law (1.12), KĀ is a state-transfer matrix of a closed-loop system, as shown in Figure 1.10 . Weighted bipartite graph Let G(V, E, A) be a weighted bipartite graph with vertex classes V1 = {v1, · · · , vn1 } and V2 = {vn1+1, · · · , vn}. Then, by the definition of bipartite graph, the adjacency matrix x(k) y(k) x(k + 1) z–1 K A Figure 1.10 A closed-loop system.
- 29. INTRODUCTION 13 A(G) can be partitioned as A = 0 A12 AT 12 0 , (1.14) where A12 ∈ Rn1×(n−n1). Hence, A2 = A12AT 12 0 0 AT 12A12 . (1.15) By Lemma 1.4 we know that any vertex in V1 has two-level neighbors in V1, and any vertex in V2 has two-level neighbors in V2. Furthermore, the existence of paths of length 2 (including self-loops) in G between any pair of vertices in V1 (or V2) is characterized by the sign of entries of A12AT 12 (or AT 12A12). So, by the definition of equivalent graph, A12AT 12 (or AT 12A12) can be defined as a weighted adjacency matrix associated with the equivalent graph of the bipartite graph G for V1 (or V2). Summarizing the discussion we have the following proposition. Proposition 1.5 Let G1(V1, E1) and G2(V2, E2) be the equivalent graphs of the bipartite graph G(A) for V1 and V2, respectively, where A is given by (1.14). Then, Ā1 = A12AT 12, (1.16) Ā2 = AT 12A12, (1.17) are weighted adjacency matrices associated with G1 and G2, respectively. By the definition of un-weighted adjacency matrix, A0,1 = 0 (A12)0,1 (AT 12)0,1 0 . (1.18) So, we have A2 0,1 = (A12)0,1(AT 12)0,1 0 0 (AT 12)0,1(A12)0,1 . So, by Lemma 1.4, the number of paths of length 2 (including self-loops) in G from vertex i to vertex j (i, j ∈ 1, n1) in V1 equals the value of the (i, j) entry of (A12)0,1(AT 12)0,1; and the number of paths of length 2 (including self-loops) in G from vertex i to vertex j (i, j ∈ n − n1, n) in V2 equals the value of the (i, j) entry of (AT 12)0,1(A12)0,1). However, the signs of entries of (A12)0,1(AT 12)0,1 (or (AT 12)0,1(A12)0,1) also characterize the existence of paths of length 2 (including self-loops) in G between any pair of vertices in V1 (or V2) (see Exercise 1.7). So, by the definition of equivalent graph, the following proposition is also true. Proposition 1.6 Let G1(V1, E1) and G2(V2, E2) be the equivalent graphs of the bipartite graph G(A) for V1 and V2, respectively, where A is given by (1.14). Denote ĀI = (A12)0,1(AT 12)0,1, (1.19) ĀII = (AT 12)0,1(A12)0,1. (1.20)
- 30. 14 FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS Then, G1(V1, E1, ĀI) and G2(V2, E2, ĀII) are weighted equivalent graphs of the bipartite graph G for V1 and V2, respectively. Remark. ĀI (or ĀII) is not the un-weighted matrix of Ā1 (or Ā2), i.e., (A12)0,1(AT 12)0,1 / = (A12AT 12)0,1, (AT 12)0,1(A12)0,1 / = (AT 12A12)0,1. Exercise 1.7 Show [(A12)0,1(AT 12)0,1]0,1 = (A12AT 12)0,1, (1.21) [(AT 12)0,1(A12)0,1]0,1 = (AT 12A12)0,1. (1.22) Let x1(k) ∈ Rn1 , x2(k) ∈ Rn−n1 be the state vectors associated with V1 and V2, respectively. Denote x(k) = [xT 1 (k), xT 2 (k)]T. Then, under the measurement rule (1.11) and the state updated law (1.12), we have x1(k + 1) x2(k + 1) = K 0 A12 AT 12 0 x1(k) x2(k) . Partition K as K = diag{K1, K2}, where K1 = diag{κi, i ∈ 1, n1}, K2 = diag{κi, i ∈ (n1 + 1), n}. Hence, x1(k + 1) = K1A12x2(k), x2(k + 1) = K2AT 12x1(k). (1.23) which yields x1(k + 1) = K1A12x2(k) = K1A12K2AT 12x1(k − 1). (1.24) If K2 0, we can define a new weight matrix for the bipartite graph G as A = ⎡ ⎣ 0 A12K 1 2 2 K 1 2 2 AT 12 ⎤ ⎦ . Then, by Proposition 1.5, A12K2AT 12 is a weighted adjacency matrix of the equivalent graph for vertex class V1 in G. Equation (1.24) links x(k + 1) to x(k − 1) instead of x(k) because any vertex in V1 has no (one-level) neighbors in V1. So, K1A12K2AT 12 is also the state-transfer matrix for the states associated with the vertices in V1. Laplacian matrix Now, we can generalize the notions of in- and out-degree to weighted digraphs. In a weighted digraph G(V, E, A) with V = {v1, · · · , vn}, the weighted out-degree and the weighted
- 31. INTRODUCTION 15 in-degree of vertex vi are defined by, respectively, dout (vi) = n j=1 aij, (1.25) and din (vi) = n j=1 aji. (1.26) The weighted digraph G is weight-balanced if dout(vi) = din(vi) for all vi ∈ V. Definition 1.8 The Laplacian matrix of the weighted digraph G(V, E, A) is defined as L(G) = Dout (G) − A(G). (1.27) If we let aii = n j=1 aij, then, the relative measurement (1.10) can be rewritten as yi(k) = j∈Ni lijxj(k), or in the matrix form y(k) = Lx(k). So, the Laplacian matrix can be interpreted as a kind of relative-measurement matrix. The following theorems give some useful properties of Laplacian matrices. Theorem 1.9 Let G be a weighted digraph of order n. The following statements hold: (1) L(G)1n = 0n, that is, 0 is an eigenvalue of L(G) with eigenvector 1n; (2) for any eigenvalue λi, i = 1 · · · , n, of L(G), either λi = 0 or Reλi 0 (thus, if G is undirected, then L(G) is positively semi-definite); (3) digraph G contains a globally reachable vertex (or say, undirected graph G is con- nected) if and only if rank(L(G))=n − 1. Theorem 1.10 Digraph G is weight-balanced if and only if one of the following statements holds: (1) 1T nL(G) = 0T n; (2) L(G) + L(G)T is positively semi-definite.
- 32. 16 FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS 1.3 Distributed Control Systems In this section we introduce two typical kinds of distributed control systems, which will be studied more carefully in the remaining chapters of this book. 1.3.1 End-to-End Congestion Control Systems One effective way for agents in a distributed control system to cooperate with each other is to optimize some performance index which concerns all of the agents in the system, such as allocation of limited resources in a communication network (Kelly 1997; Kelly, Maulloo and Tan 1998), coverage of as much as possible an area by a sensing network (Cortés et al. 2004), etc. Congestion control of the Internet is a typical system designed by using a distributed real-time optimization approach, which will be presented in Chapter 4. Here we just introduce some basic models with some necessary notions of end-to-end congestion control systems. Basic model Let us consider an end-to-end congestion control system for a network with a set of L link nodes shared by a set of S source nodes. Denote by L = {1, · · · , L} the set of all link nodes and by S = {1, · · · , S} the set of source nodes. Link node l ∈ L is used by a set of sources denoted by Sl ⊂ S, and source node i ∈ S sends packets to a set of links denoted by Li ⊂ L. The sets Li define an L × S routing matrix R = {Rli}, where Rli = 1, if l ∈ Li 0, otherwise. From a viewpoint of graph theory, each node in S has neighbors only in L and each node in L has neighbors only in S. So, the topology of such a system can be described by a bipartite graph G(V, E), where V = L ⊕ S. A weighted adjacency matrix of the bipartite graph can be given by A = 0 K1RK2 K2RTK1 0 , (1.28) where K1 ∈ RL×L, K2 ∈ RS×S are two positively definite diagonal matrices. Source i ∈ S transmits packets to all the links used by source i at a rate xi(t); in other words, all the links in Li receive packets at rate xi(t). The primal algorithm for the congestion-avoidance rate control is given by (Kelly, Maulloo and Tan 1998) ẋi(t) = κi(wi − xi(t) l∈Li ql(t)), i ∈ S (1.29) ql(t) = pl( r∈Sl xr(t)), (1.30) where the control gain ki is a positive constant, and wi is some desired value of the rate of marked packets received back at sources i. The first equation describes the time evolution of the
- 33. INTRODUCTION 17 transmission rate xi(t) of source i. The second equation describes the generation of congestion signal ql(t) at link l, by means of a congestion indication function pl(y), which is assumed to be monotonically increasing, nonnegative, and not identically zero (a candidate of such a function can be taken as p(y) = 1 − ey(1 − y), which is positive and strictly increasing for all y 0). Suppose that matrix R is of full row rank. Then, one can get the unique equilibrium point of system (1.29)–(1.30) as x = [x 1, · · · , x S]T, where x i = wi/ l∈Li pl( r∈Sl x r ). Define yi(t) = (xi(t) − x i )/ kix i , i ∈ S. The dynamic property of the system around the equilibrium is determined by the following linearized model: ẏi(t) = −κiwix−1 i yi(t) − κix i l∈L Rlip l r∈S Rlr κrx r yr(t) (1.31) where p l is the derivative of pl evaluated at y = i∈Sl x i , i.e. p l = p l( i∈Sl x i ). Since pl(y) is assumed to be monotonically increasing, p l 0, ∀l ∈ L. By defining K1 = diag{ p l, i ∈ L}, K2 = diag{ κrx r , r ∈ S}, and ĀS = K2RT K2 1RK2, it is easy to verify that the interconnection between yl and yr can be described by a weighted equivalent graph GS(S, ES, ĀS) for vertex class S in the bipartite graph G(V, E, A), where V = L ⊕ S and A is given by (1.28). Denote y(t) = [y1(t), · · · , yN(t)]T , K = diag{κiwix−1 i , i ∈ S}. Then, the matrix form of (1.31) is given by ẏ(t) = −Ky(t) − ĀSy(t). (1.32) Model with delays When propagation delays are considered, the congestion control algorithms can be modeled by the following delayed differential equations (Johari and Tan 2001) ẋi(t) = κi(wi − xi(t − Di) l∈Li ql(t − d← li )), i ∈ S, (1.33) ql(t) = pl( r∈Sl xr(t − d→ lr )), (1.34)
- 34. 18 FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS Source 2 Source 1 Link 1 Link 2 Destination 2 Destination 1 Link 3 Figure 1.11 Communication network. where d→ li denotes the forward delay of delivering packets from source i to link l, d← li denotes the backward delay of sending the feedback signal from link l to source i, and Di is the round- trip delay associated with source i. In the current Internet, it is assumed that propagation delay is much more significant than queueing delay. Therefore, ignoring the queueing delay, the following relationship holds: d→ li + d← li = Di, ∀l ∈ Li ⊆ L. The linearized model around the equilibrium is given by ẏi(t) = −κiwix−1 i yi(t − Di) − κix i l∈L Rlip l r∈S Rlr κrx r yr(t − d→ lr − d← li ). (1.35) Delay effects on stability Let us use the congestion control system to check the influence of delays. Consider a simple network shown in Figure 1.11. There are two sources transmitting packages and three links through which packages are sent to destinations. Source 1 uses link 1 and link 2, source 2 uses link 2 and link 3. The interconnection topology of sources and links can be represented by a bipartite graph shown in Figure 1.12. According to (1.33) the congestion control algorithm is given by ẋi(t) = κi(wi − xi(t − Di) l∈Li ql(t − d← li )), i = 1, 2, Source 1 Link 1 Link 2 Link 3 Source 2 Figure 1.12 Interconnection graph.
- 35. INTRODUCTION 19 where L1 = {1, 2}, L2 = {2, 3}, q1(t) = p1(x1(t − d→ 11 )), q2(t) = p2(x1(t − d→ 21 ) + x2(t − d→ 22 )), q3(t) = p3(x2(t − d→ 32 ))), p1(y) = p2(y) = p3(y) = 1 − (1 − y)ey . The parameters in the simulation are selected as follows: κ1 = w1 = 0.05, κ2 = w2 = 0.07, D1 = 22(s), D2 = 25(s), d→ 21 + d← 22 = d→ 22 + d← 21 = 1 2 (D1 + D2) = 23.5(s). Under this set of parameters, the system is asymptotically stable. Suppose that the transmission process of the network is influenced by some external factors so that D2 changes from 25(s) to 50(s) when t ≥ 1000(s), and suppose that the increments of both d← 22 and d→ 22 are equal to 12.5(s). In this case, the equilibrium point becomes unstable and periodic oscillation takes place. The simulation of this process is shown in Figure 1.13. The above example shows that the size of delay constant plays a significant role in the sta- bility analysis. The problem of the stability of time-delayed feedback control systems has been extensively studied in recent years (see, e.g., Gu, Kharitonov and Chen (2003) and Niculescu (2001)). In this book we will pay more attention to one specific aspect of the problem: under what condition can one get a scalable stability criterion for a time-delayed distributed control 0 2000 4000 6000 8000 10000 0 0.1 0.2 0.3 x1(t) 0 2000 4000 6000 8000 10000 0 0.1 0.2 0.3 0.4 t/second x2(t) Figure 1.13 Periodic oscillations due to communication delays.
- 36. 20 FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS system? Here, under the scalability we mean the criterion checks only dynamics and parame- ters locally around each individual agent. The stability and scalability of distributed congestion control systems will be studied in Chapter 5. Stabilization by time-delayed feedback control On the one hand, propagation delay may destroy the stability of the congestion system as we have shown by numerical experiments. On the other hand, an appropriately introduced time-delayed feedback controller may serve as a stabilizer. Let us introduce the following time-delayed feedback control (TDFC) ud(t) = h τ (x(t) − x(t − τ)) into the equation of x2(t). Then, the closed-loop system is modified as ẋ1 = k1(w1 − x1(t − D1)(p(x1(t − D1)) + p(x1(t − D1) + x2(t − d← 21 − d→ 22 )))) ẋ2 = k2(w2 − x2(t − D2)(p(x2(t − D2)) + p(x2(t − D2) + x1(t − d← 22 − d→ 21 )))) − h τ (x2(t) − x2(t − τ)). Now, let us conduct numerical experiments again for the foregoing oscillating system. In the simulation, the gain and the delay time are selected as h = 0.8 and τ = 4(s), respectively. For the system, the TDFC is introduced when t ≥ 5000(s). Figure 1.14 shows that the closed-loop system becomes stable again at a steady transmission rate: x = (0.2271, 0.2901). Figure 1.15 shows that the control force returns to zero when the equilibrium is stabilized by the TDFC. 0 2000 4000 6000 8000 10000 0 0.1 0.2 0.3 x1(t) 0 2000 4000 6000 8000 10000 0 0.1 0.2 0.3 0.4 t/second x2(t) Figure 1.14 Stabilization of oscillations by TDFC.
- 37. INTRODUCTION 21 0 2000 4000 6000 8000 10000 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 x 10–3 t/second control force:u(t) Figure 1.15 Control signal of TDFC. It is natural to guess that the delay constant of the TDFC can not be arbitrarily large. In the simulation, the algorithm becomes unstable again when τ is greater than a certain value. The simulation shown in Figure 1.16 demonstrates that oscillations of the sending rates of the sources come about when τ ≥ 84(s). So, an interesting problem is why the TDFC works and 0 2000 4000 6000 8000 10000 0 0.1 0.2 0.3 x1(t) 0 2000 4000 6000 8000 10000 0 0.1 0.2 0.3 0.4 t/second x2(t) Figure 1.16 Oscillations due to large delay in TDFC.
- 38. 22 FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS how to select the gains and delay constants of a TDFC. Related problems are also discussed in Chapter 5. 1.3.2 Consensus-Based Formation Control Trying to reach some agreement (or consensus) among agents is another effective way to conductcooperationinadistributedcontrolsystem.Herewebrieflyreviewtheconsensus-based formation control approach. Formation control is one of the most important and fundamental issues in the coordination control of multi-agent systems, which requires that each agent moves according to the prescribed trajectory, and all the agents keep a particular spatial formation pattern at the same time (Balch and Arkin 1998; Hu 2001). Consider the multi-agent system modeled as a group of Newtonian particles ṗi = qi, q̇i = ui, i ∈ 1, n, (1.36) where pi ∈ R2 and qi ∈ R2 denote the position and velocity of agent i, respectively; ui ∈ R2 denotes its control input. Here we just consider the formation control in the plane. To describe the desired geometric pattern of the multi-agent system, let us introduce vector ci ∈ R2 to specify the position of agent i in the pattern. Then, the desired formation can be described by a set of vectors, F = {ci ∈ R2, i ∈ 1, n}. Figure 1.17 shows a pentagon formation with vertex-position vectors ci, i ∈ 1, 5. Note that for any c0 ∈ R2, F = {(ci + c0) ∈ R2, i ∈ 1, n} describes the same formation as F. The objective of formation control includes that the agents asymptotically converge to the prescribed geometric pattern F and each agent’s velocity asymptotically approaches to a desired constant v0 ∈ R2. Suppose the desired velocity v0 is known by only one or a few agents which are called leaders among the other agents. Let us denote by L ⊂ 1, n the index set of leaders. p2 c1 0 c5 c4 c3 c2 p1 Figure 1.17 A pentagon formation.
- 39. INTRODUCTION 23 For the system (1.36), a formation control algorithm can be given by ui(t) = ui1 + ui2, if i ∈ L ui2, otherwise (1.37) where ui1 = −β(qi(t) − v(t)) (1.38) is the velocity tracking part for leader agents, and ui2 = −γ1 j∈Ni aij(qi(t) − qj(t)) − γ0 j∈Ni aij((pi(t) − ci) − (pj(t) − cj)) (1.39) is the coordination control part. In the control law, γi 0, i = 0, 1, and β 0 are control parameters,aij 0 isaweightusedbyagent ifortheinformationobtainedfromitsneighboring agent j, and Ni is the set of neighbors of agent i. Obviously, the interconnection between the agents in such a system can be described by a digraph G = (V, E, A): each agent can be considered as a vertex in V; the information flow between two agents can be considered as a directed edge in E, i.e., if agent i uses position and velocity information of agent j, then there is an edge from vertex i to vertex j; and the weight aij can be regarded as the entry of the adjacency matrix A. The formation control algorithm (1.37) is indeed a local control law because each agent uses only the information of itself and its neighbors. For such a distributed control system, of course, the first problem we are interested in is stabilizability and stability, i.e., under what conditions do there exist available control param- eters such that the closed-loop system is stable? and how can control parameters be selected in order to ensure the stability? The second problem we are interested in is: can the desired for- mation indeed be achieved when the system is asymptotically stabilized? In the first problem, the scalability of the stabilizability and/or the stability criteria is the profile to which we will pay more attention due to the possible huge scale of distributed control systems. While the problem of stabilizability and stability is of concern in all feedback control systems, the second problem raised here is a more distinct feature of distributed control systems. Indeed, even if the system is stable, does it imply the conclusion ui1 → 0, ∀i ∈ L and ui2 → 0, ∀i ∈ 1, n? Even if ui1 → 0, ∀i ∈ L and ui2 → 0, ∀i ∈ 1, n, does it imply that all the agents track the desired velocity v0 and that the relative displacement of each pair of agents satisfies the formation requirement F? By taking a first look at the questions, we can see that the condition of ui2 → 0, ∀i ∈ 1, n, (1.40) and the implication ui2 → 0, ∀i ∈ 1, n =⇒ qi(t) = qj(t), and pi(t) − pj(t) = ci − cj, ∀i, j ∈ 1, n, i / = j (1.41) are critical for getting the answer to all the questions.
- 40. 24 FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS To consider the problem formulated by (1.40) and (1.41) one can let ci = 0, ∀i ∈ 0, n. In this case, the formation problem reduces to the so-called rendezvous problem (i.e., position agreement). Actually, the basic algorithm for the rendezvous problem is ui(t) = −γ1 j∈Ni aij(qi(t) − qj(t)) − γ0 j∈Ni aij(pi(t) − pj(t)), (1.42) This algorithm is also called the consensus (or agreement) algorithm. If the multi-agent system (1.36) is stable under this control, then the states (velocities and positions) of all the agents in the system approach to the same value. From the above discussion, we see that the stability of the consensus algorithm is at the center of the problem. The problem, in particular, includes the analysis and design of the available values of control parameters γi, β and even adjacent weights aij in the sense of stability. We should note the difference between the stability of the consensus problem and the stability of conventional control systems. In most conventional (linear) feedback control systems, the origin is the only equilibrium and the stability implies that all the eigenvalues of the system matrix of the closed-loop system have positive real parts. But, it is easy to see that the system matrix of the closed-loop system of (1.36) with (1.42) is singular, and thus the system has a continuum of equilibrium points. And therefore, the stability of the consensus algorithm does not imply that all the eigenvalues of the closed-loop have positive realparts.Thestabilityofconsensuscontrolsystemsandotherrelatedproblemsarediscussedin Chapter 6. If the agents are distributed in a huge range of space and connected with the help of a communication network, delay is inevitable in information transmission. At current time t agent i can get only the delayed information pij(t − τij) and qij(t − τij) from its neighboring agent j, where τij 0 is the communication delay of information flow from agent j to agent i. In this case, if the value of the delay constant is available for agent i, then the consensus algorithm (1.42) can be modified as follows: ui(t) = −γ1 j∈Ni aij(qi(t − τij) − qj(t − τij)) −γ0 j∈Ni aij(pi(t − τij) − pj(t − τij)). (1.43) Note that when diverse communication delays are involved in the communication network, the stability and scalability analysis of such consensus algorithms will become much more complicated and challenging. Related problems are discussed in Chapter 7. In the algorithm (1.43) the communication delay τij is assumed to be available for agent i and is used as a “self-delay”. In practice, however, only approximate estimations of commu- nication delays are available. If the self-delay in the (1.43) is replaced by the estimation of the communication delay, then we have to use the following consensus algorithm: ui(t) = −γ1 j∈Ni aij(qi(t − τ ij) − qj(t − τij)) −γ0 j∈Ni aij(pi(t − τ ij) − pj(t − τij)). (1.44)
- 41. INTRODUCTION 25 In this case, a new question is raised naturally: does there exists any consensus solution of the closed-loop system (1.36) under feedback control (1.44) when τ ij / = τij? This question will be also answered in Chapter 7 for this and more generalized systems. In closing, we would like to note that although the consensus control and the congestion control are designed from quite different principles of cooperation, they share a lot of simi- larities in system structure, and hence the stability and scalability of the two kinds of systems can be dealt with by using some common tools which are introduced in this chapter and the following two chapters as well. 1.4 Notes and References 1.4.1 Graph Theory and Distributed Control Systems Graph theory is a fundamental tool in distributed control and distributed computation. The basic definitions of graph theory introduced in this book are quite standard; see, for example, Biggs (1994), Bollobás (1998), Diestel (1997), Godsil and Royle (2001). For powers of the adjacency matrix the reader is referred to literature on algebraic graph theory (Biggs 1994; Bullo, Cortés and Martı́nez 2009; Godsil and Royle 2001). Laplacian matrices have numerous remarkable properties and some of them play a key role in the analysis of consensus problems. The reader is referred to Mohar (1991) and Merris (1994) for elegant surveys of Laplacian matrices. Theorem 1.9 and Theorem 1.10, characterizing the properties of the Laplacian matrix, contain some recent results. A proof of statement (2) of Theorem 1.9 was given in Olfati-Saber and Murray (2004). Statement (3) of Theorem 1.9 was proved by Lin, Francis and Maggiore (2005). An equivalent version of statement (3) was proved in Ren and Beard (2005). Statement (1) of Theorem 1.10 was proved by Olfati-Saber and Murray (2004), and statement (2) of Theorem 1.10 was proved by Moreau (2005). Two typical kinds of distributed control systems are introduced in this chapter. The first is the end-to-end congestion control and the second is the consensus-based formation control. The first congestion avoidance control algorithm for the Internet was proposed by Jacobson (1988). Motivated by the theory of economics, Kelly (1997) and Kelly, Maulloo and Tan (1998) introduced the concept of price into the congestion control, and formulated the rate control in communication networks as a problem of optimization of allocation re- sources (utilization of channel capacities) in networks. For a mathematic treatment of con- gestion control problems the reader is referred to, for example, Low and Lapsley (1999); Low, Paganini and Doyle (2002); Srikant (2004), but this is the first book to use bipartite graphs to describe the architecture of congestion control systems. It will be shown in the remaining chapters of this book that we benefit from such a treatment in characterizing the symmetry of congestion control systems and relating them to multi-agent systems discussed in consensus problems. Consensus problems are closely related to many other problems in coordination control, such as formation control (Balch and Arkin 1998; Fax and Murray 2004; Hu 2001; Wang and Hadaegh 1994) and flocking (Olfati-Saber and Murray 2006; Toner and Tu 1998). For a survey on the relationship between consensus problems and coordination control strategies the reader is referred to Olfati-Saber, Fax and Murray (2007).
- 42. 26 FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS 1.4.2 Delay in Control and Control by Delay The fact that delay in feedback may cause oscillation and instability was pointed out as early as the 1930s (see, e.g., Wiener (1948)). The delay effects on the stability of feedback control systems have been extensively studied with tools from both classic Lyapunov stability theory and contemporary robust control theory (Gu, Kharitonov and Chen 2003; Niculescu 2001). Johari and Tan (2001) studied the stability of the primal algorithm of the end-to-end con- gestion control system with propagation delays. They gave a stability criterion for systems with identical round-trip delay and also proposed a conjecture for systems with diverse round-trip delays. The conjecture was proved in its original version and generalized to a less conservative form by Tian and Yang (2004) via a frequency-domain method. Tian (2005) further showed that such a frequency-domain method was powerful in studying the stability of second-order congestion control systems with diverse propagation delays. The study on the consensus problem with communication delays started as early as the 1980s in the context of distributed computation (Bertsekas and Tsitsiklis 2007; Tsitsiklis, Bertsekas and Athans 1986). In some other communities such as physics, this problem has also been extensively studied as synchronization of coupled oscillators (see, e.g., Yeung and Strogatz 1999). Recently, these kinds of consensus protocols have been studied for fixed or even switched graphs by using different analysis methods, such as the contraction analysis method (Wang and Slotine 2006), the passivity-based method (Chopra and Spong 2006), the method based on delayed and hierarchical graphs (Cao, Morse and Anderson 2006), among others. Tian and Liu (2008, 2009) noticed the symmetry of multi-agent systems with diverse input delays, and used the frequency-domain method developed initially for congestion control to study the stability of consensus control systems. The idea of using a time-delayed feedback controller (TDFC) to stabilize unstable periodic orbits of chaotic systems was first proposed by Pyragas (1992). Kokame et al. (2001) illustrated the TDFC as a kind of differential control and applied it to stabilize uncertain steady states. Liu and Tian (2008) noticed the fact that the equilibrium point of the Internet is usually not available for each source node or link node and thus proposed a stabilization scheme for the congestion control system of the Internet by introducing the TDFC into the primal algorithm and proved that the system with any round-trip delay can be locally stabilized by the modified algorithm. Many other applications of the TDFC method have also been reported in the literature, such as stabilization of coherent models of lasers (Bleich and Socolar 1996; Naumenko et al. 1998) and magneto-elastic systems (Hilkihara, Touno and Kawagoshi 1997), control of cardiac conduction model (Brandt, Shih and Chen 1997), control of stick-slip friction oscillations (Elmer 1998), traffic models (Konishi, Kokame and Hirata 1999) and PWM-controlled buck convertors (Battle, Fossas and Olivar 1999), just to name a few. The reader is referred to Tian, Zhu and Chen (2005) for a recent survey of time-delayed feedback control. References Balch T and Arkin RC (1998). Behavior-based formation control for multirobot terms. IEEE Transanc- tions on Robotics and Automation, 14, 926–939. Battle C, Fossas E and Olivar G (1999). Stabilization of periodic orbits of the buck convertor by time- delayed feedback. International Journal of Circuits Theory and Application, 27(6), 617–631.
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- 46. 2 Symmetry, Stability and Scalability People today see not the moon the ancients saw, and yet this moon shone once upon the ancients. —Li Bai (701–762), ‘Inquiring of the moon, with wine in hand’ This chapter introduces general models of distributed control systems based on graph theory andlinearsystemtheory.Thesymmetrypropertyofdistributedcontrolsystemsinthefrequency domain is defined and investigated. Classic results of the frequency-domain test of the stability of multivariable feedback control systems are reviewed with the emphasis on their extension to the semi-stability and the scalability for distributed control systems. 2.1 System Model 2.1.1 Graph-Based Model of Distributed Control Systems Consider a distributed control system with n subsystems, each of which is also called an agent in this book. Let the model of the dynamics of the ith agent be given by the following state-space model ẋi(t) = Aixi(t) + Biu(t − Ti) yi(t) = Cixi(t) + Diu(t − Ti) (2.1) where xi(t) ∈ Rni , ui(t) ∈ R and yi(t) ∈ R denote the state, input and output, respectively, of the ith agent, Ti is the input delay. Note that in distributed control systems input delays are usually caused by the processing time for the packets arriving at each agent and/or the communication time between actuators and controllers. The agents are interconnected according a fixed topology which can be described by a digraph G = (V, E, Ā) of order n. Denote by Ni the set of neighbors of agent i. By (1.9), the aggregated measurement of agent i is based on the information from its neighbors and itself: zi(t) = aiiyi(t − τii) + j∈Ni aijyj(t − τij), (2.2) Frequency-Domain Analysis and Design of Distributed Control Systems, First Edition. Yu-Ping Tian. © 2012 John Wiley Sons Singapore Pte. Ltd. Published 2012 by John Wiley Sons Singapore Pte. Ltd.
- 47. 32 FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS where τij is the communication delay from agent j to agent i which mainly consists of prop- agation delay and queueing delay, τii is called self-delay which is often introduced to match the communication delays, aij is the weight for the output channel from agent j to agent i and aii is the weight for the self-loop of agent i. To incorporate the aggregated measurement (1.9) and the relative measurement (1.10) in a unified form, throughout this chapter, we assume that aij in (2.2) can be either positive or negative. Suppose that agent i is manipulated by the following local controller ˙ x̂i(t) = Ak i x̂i(t) + Bk i (ri(t) − zi(t)) ui(t) = Ck i x̂i(t) + Dk i (ri(t) − zi(t)) (2.3) where x̂i(t) ∈ Rmi is the state of the local controller for agent i, Ak i ∈ Rmi×mi , Bk i ∈ R, Ck i ∈ R, Dk i ∈ R are parameters of the controller and ri(t) is some reference signal for agent i. Denote by Gi(s) the transfer function of agent i, i.e., Gi(s) = [Ci(sI − Ai)−1 Bi + Di]e−Tis , (2.4) and by κi(s) the transfer function of the ith controller, i.e., κi(s) = Ck i (sI − Ak i )−1 Bk i + Dk i . (2.5) The feedback control system for agent i is shown in Figure 2.1. Let ŷi(s) and ûi(s) be the Laplace transformation of the output and input, respectively, of the i-agent. Denote Y(s) = [ŷ1(s), · · ·, ŷn(s)]T , (2.6) U(s) = [û1(s), · · ·, ûn(s)]T , (2.7) ) (s Gi s ii ii e a ) ( ˆ s yi ) (s i ) ( ˆ s ui ) ( ˆ s ri s i i e a 1 1 s i i e a 2 2 s in in e a ) ( ˆ1 s y ) ( ˆ2 s y ) ( ˆ s yn Network information Figure 2.1 Feedback control system of agent i.
- 48. SYMMETRY, STABILITY AND SCALABILITY 33 G(s) = diag Gi(s), i ∈ i, n , (2.8) K(s) = diag{κi(s), i ∈ i, n}, (2.9) Ā(s) = {aije−τijs }, (2.10) (s) = diag{aiie−τiis , i ∈ 1, n}, (2.11) and A(s) = Ā(s) − (s). (2.12) A(s) and Ā(s) can be considered as the weighted adjacency matrix and generalized weighted adjacency matrix, respectively, of topology digraph G. As a matter of fact, in our model the matrix Ā(s) can be further extended to the following form Ā(s) = {αij(s)e−τijs }, (2.13) where the transfer function αij(s) ∈ RH∞ describes the possible dynamics of the communi- cation channel from agent j to agent i. With notations introduced above, the closed-loop system composed of (2.1), (2.2) and (2.3) can be sketched in Figure 2.2. The transfer function matrix from R(s) to Y(s) is given by W(s) = (I + G(s)K(s)Ā(s))−1 G(s)K(s). (2.14) The system shown in Figure 2.2 is an interconnection of two parts. The first part G(s)K(s) is a diagonal matrix, which represents the agent dynamics. The second part A(s) + (s) can be considered as a generalized complex adjacency matrix of the topology graph of the network; it reflects both the topology and the channel dynamics of the network used by the multi-agent system. Definition2.1 ThedynamicsoftheagentsinthedistributedcontrolsystemshowninFigure2.1 are said to be homogeneous if κi(s)Gi(s) = κj(s)Gj(s) for all i, j ∈ 1, n; the network through which the agents are interconnected is said to be homogeneous if τij = τ0 and αij(s) = α0(s), ∀i, j ∈ 1, n, where τ0 ∈ R+ and α0(s) ∈ RH∞. The distributed control system is said to be homogeneous if all the agents and the network are all homogeneous. Otherwise, the agents (network) are said to be heterogeneous if they are not homogeneous. The system is said to be heterogeneous if it contains heterogeneous agents or heterogeneous network. G(s) (s) A(s) Y(s) K(s) U(s) R(s) Figure 2.2 Diagram of distributed control system.
- 49. 34 FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS n S 2 S 1 S 1 L 2 L m L Bi-directional Network Figure 2.3 Bipartite graph for distributed control systems. 2.1.2 Bipartite Distributed Control Systems There are many distributed control systems in which information exchanges are conducted through forward and backward channels. The interconnection of agents in such systems can be described by the weighted bipartite digraph shown in Figure 2.3. We call these systems bipartite systems. Denote by (S, L) and A the vertex bipartition and the weighted adjacency matrix of the graph, respectively, where S = {S1, · · · , Sn}, (2.15) L = {L1, · · · , Lm}, (2.16) A = 0 A12 AT 12 0 . (2.17) Let the agents associated with vertex class S be described by transfer functions Gi(s), i ∈ 1, n; and the agents associated with L be described by transfer functions Pj(s), j ∈ 1, m. Agent i in S receives output information from its neighbors in L, and agent j in L receives output information from its neighbors in S, i.e., zi(t) = l∈Ni ab ilyl(t − τb li), i ∈ S, (2.18) zl(t) = i∈Nl a f li yi(t − τ f li ), l ∈ L, (2.19) where τb li, ab il are the communication delay and the weight of the backward channel from agent l in L to agent i in S, respectively; τ f li , a f li are the communication delay and the weight of the forward channel from agent i in S to agent l in L. Considering (2.17), the following relationship should hold ab il = a f li , ∀i ∈ S, ∀l ∈ L. (2.20)
- 50. SYMMETRY, STABILITY AND SCALABILITY 35 ) (s G ) (s Ab ) ( 1 s Y ) (s K ) ( 1 s R ) (s Af ) (s P ) ( 2 s Y ) (s ) ( 2 s R Figure 2.4 Bipartite distributed control system. Agent i in S is manipulated by dynamic controller κi(s), i = 1, · · · , n, and agent l in L is manipulated by dynamic controller γl(s), l ∈ 1, · · · , m. Denote by R1(s) = [r̂1 1(s), · · · , r̂1 n(s)]T and Y1(s) = [ŷ1 1(s), · · · , ŷ1 n(s)]T the reference input vector and output vector of all the agents in S, respectively; by R2(s) = [r̂2 1(s), · · · , r̂2 m(s)]T and Y2(s) = [ŷ2 1(s), · · · , ŷ2 m(s)]T the reference input vector and output vector of all the agents in L, respectively. Then, the bipartite system with forward and backward channels can be shown by Figure 2.4, where G(s) = diag Gi(s), i ∈ 1, n , K(s) = diag{κi(s), i ∈ 1, n}, P(s) = diag Pl(s), l ∈ 1, m , (s) = diag{γl(s), l ∈ 1, m}, Āb(s) = {ab ile−τb lis , i ∈ 1, n, l ∈ 1, m}, Āf (s) = {a f li e−τ f li s , l ∈ 1, m, i ∈ 1, n}.
- 51. 36 FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS The transfer function matrix from R1(s) to Y1(s) is given by W1(s) = (I + G(s)K(s)Ab(s)P(s)(s)Af (s))−1 G(s)K(s), (2.21) and the transfer function matrix from R2(s) to Y2(s) is given by W2(s) = (I + P(s)(s)Af (s)G(s)K(s)Ab(s))−1 P(s)(s). (2.22) Denote Ā1(s) = Ab(s)P(s)(s)Af (s) and Ā2(s) = Af (s)G(s)K(s)Ab(s). Then, the open-loop transfer function of W1(s) or W2(s) has the same structure as that of W(s) given by (2.14), i.e., it consists of a diagonal matrix corresponding to the agents’ dynamics and a square matrix corresponding to the topology graph. This suggests that it is possible to deal with the stability problem of two kinds of distributed control systems in a unified framework. 2.2 Symmetry in the Frequency Domain 2.2.1 Symmetric Systems In practice, many multi-agent systems are based on networks with undirected topology graph. In this case, if there is no delay and all the channels have constant dynamics, i.e., τij = 0 αij(s) = 1, ∀i, j ∈ 1, n, then the system shown in Figure 2.2 is said to be symmetric because the adjacency matrix of the topology graph is symmetric. Can such a conception of symmetry be extended to the case when there are non-zero delays or non-constant dynamics in network channels? In this book, we define the symmetry of networked-based distributed control systems as follows. Definition 2.2 Let the open-loop transfer function of a distributed control system be given by Ḡ(s)Ā(s), where Ḡ(s) is a diagonal matrix and Ā(s) is a square matrix. The system is said to be symmetric if Ā(s) can be decomposed as Ā = (s)Â(s), (2.23) where (s) is a diagonal matrix and Â(s) satisfies the following condition: Â(s) = ÂT (−s). (2.24) A starting point for this definition is that a complex generalized adjacency matrix of an undirected graph should be a Hermitian matrix as we defined in Chapter 1. Indeed, letting s = jω, then from (2.24) we have Â(jω) = Â∗(jω). So, if we absorb the diagonal matrix (s) in the agents’ dynamics which also form a diagonal matrix, then Â(jω) can be regarded as a generalized adjacency matrix of the undirected topology graph in the frequency domain. Example 2.3 Symmetry in the frequency domain. Consider the simplest undirected graph of two nodes, shown by Figure 2.5. Suppose the communication delays from node 1 to node 2 and from node 2 to node 1 are τ21 0 and τ12 0,
- 52. SYMMETRY, STABILITY AND SCALABILITY 37 s e 12 s e 21 Figure 2.5 Symmetry in the frequency domain. respectively. It is natural to set e−τ12s and e−τ21s as the weights in the Laplace domain. But, the matrix A(jω) = 0 e−τ12s e−τ21s 0 does not satisfy conjugate symmetry even in the case τ12 = τ21 = τ. In the frequency domain, A(jω) is not a Hermitian matrix, either. Now, let us scale the matrix A(s) by multiplying 1 0 0 e−Ts 1 0 0 eTs , where T = τ12 + τ21, (2.25) which is called the round-trip time between node 1 and node 2. Then, we find that A(s) = 1 0 0 e−Ts 0 e−τ12s eτ12s 0 (s)Â(s), where = 1 0 0 e−Ts is a diagonal matrix, Â(s) = 0 e−τ12s eτ12s 0 satisfies Â(s) = ÂT(−s), and hence, Â(jω) is a Hermitian matrix. So, by Definition 2.2, the system based on the graph shown in Figure 2.5 is symmetric in the frequency domain even if τ12 / = τ21. Absorbing (s) into the diagonal transfer function matrix of the agents implies that the agent associated with node 2 obtains an additional time delay T. So, this example shows that sometimes the frequency-domain symmetry of a communication topology can be achieved by introducing a lag in the time domain for agents! Now, let us consider the general distributed control system composed of (2.1), (2.2) and (2.3) or equivalently the system shown in Figure 2.1. Assume the interconnection topology graph of the system is undirected. Then, to match the symmetry requirement of Definition 2.2, there should exist Ti ∈ R, i ∈ 1, n, such that τij + τji = Ti + Tj, ∀i, j ∈ 1, n. (2.26) It is said that the system has symmetric communication delays if the requirement (2.26) is satisfied. Note that the symmetric communication delay does not imply τij = τji.
- 53. 38 FREQUENCY-DOMAIN ANALYSIS AND DESIGN OF DISTRIBUTED CONTROL SYSTEMS 2.2.2 Symmetry of Bipartite Systems Before we study the symmetry property of bipartite systems with forward and backward chan- nels, let us introduce the notion of semi-homogeneousness as follows. Definition 2.4 Consider a bipartite system which is based on the bipartite graph G(S ⊕ L). It is said to be semi-homogeneous by vertex L if for each agent i in S, the following conditions are satisfied: (1) all the neighbors of agent i have homogeneous dynamics, i.e., Pl(s)l(s) = γlαi(s), ∀l ∈ Ni (2.27) where γl ∈ R is a constant and αi(s) is a scalar transfer function that is independent of l; (2) the round-trip delay from agent i to its any neighbor in L is a constant, i.e., τ f li + τb li = Ti, ∀l ∈ Ni. (2.28) Similarly, one can also define the semi-homogeneousness of the system by vertex class S. Remark. Obviously, if all the agents in vertex class L have homogeneous dynamics, (2.27) holds. The next theorem shows that the technique used in Example 2.3 can be applied to general semi-homogeneous bipartite systems. Theorem 2.5 If the system based on the bipartite graph G(S ⊕ L) is semi-homogeneous by L or by S, then it is symmetric in the frequency domain. Proof. We just consider the case that the system is semi-homogeneous by L. The proof for the other case is the same. Denote 0 = diag{γ1, · · · , γm} and (s) = diag{α1(s), · · · , αn(s)}. Then, under assump- tion (2.27) we have Ab(s)P(s)(s) = (s)Ab(s)0. Denote T(s) = diag{e−T1s, · · · , e−Tns}. Then, by using (2.20) and (2.28), one can see that Ab(s) = T(s)T(−s)Ab(s) = T(s)AT f (−s). So, under the assumption of the semi-homogeneousness of the system by L, we get W1(s) = (I + G(s)K(s)Ab(s)P(s)(s)Af (s))−1 G(s)K(s) = (I + G(s)K(s) (s)Ab(s)0Af (s))−1 G(s)K(s) = (I + G(s)K(s) (s)T(s)AT f (−s)0Af (s))−1 G(s)K(s).
- 54. SYMMETRY, STABILITY AND SCALABILITY 39 So, we get the open-loop transfer function of the system as L(s) = G(s)K(s) (s)T(s)AT f (−s)0Af (s) Ĝ(s)Â(s), whereĜ(s) = G(s)K(s) (s)T(s)isadiagonalmatrix,Â(s) = AT f (−s)0Af (s)satisfiesÂ(s) = ÂT(−s). Then, by Definition 2.2, the system is symmetric in the frequency domain. 2.3 Stability of Multivariable Systems 2.3.1 Poles and Stability Let the state-space representation of the closed-loop system (2.14) be ẋ(t) = A0x + nd i=1 Aix(t − τi) + Br(t) (2.29) y(t) = Cx(t) + Dr(t) (2.30) where x(t) ∈ Rn is the state-vector of the system, {A0, B, C, D} is the minimal state-space realization of the system with all delays being zero, τi, i ∈ 1, nd, are all possible time-delays in the closed-loop system, Ai, i ∈ 1, nd, are the matrices corresponding to time-delayed states. Then, the poles of the system are the roots of the characteristic equation φ(s) := det(sI − A0 − nd i=1 e−sτi ) = 0, (2.31) where is φ(s) is the characteristic (or pole) quasi-polynomial of the system. We also call φ(s) characteristic (or pole) quasi-polynomial corresponding to the transfer function W(s) given by (2.14). By the theory of the time-delayed system (Gu, Kharitonov and Chen 2003), the stability of system (2.29) is fully determined by the characteristic quasi-polynomial φ(s). Namely, the system is stable if and only if φ(s) has no pole in the closed right half of the complex plane. In this book we call the open left half of the complex plane simply the LHP and denote it as C−, and call the closed right half of the complex plane simply as RHP and denote it as C+. For simplicity of statement we also call the closed left half of the complex plane simply the closed LHP and denote it as C̄−, and call the open right half of the complex plane simply the open RHP and denote it as C̆+. Definition 2.6 The time-delayed system (2.29) is said to be stable if φ(s) / = 0, ∀s ∈ C+ . (2.32) By this definition there is no difference between “stable” and “asymptotically stable”, which are both used in this book. The following notion of semi-stability is also used in this book.
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- 56. recovered its tone: his health was restored, and he was once more enabled to give his full powers to the growing interests of his firm. For the few succeeding years, he was engaged in the usual routine of mercantile affairs, and has left but few memorials or letters, except those relating to his business. In the winter of 1820, he made a visit to New York, which he describes in his diary under date of February 15, 1846: Yesterday was one of the most lovely winter days. To-day the snow drives into all the cracks and corners, it being a boisterous easterly snow-storm, which recalls to my mind a similar one, which I shall never forget, in February, 1820. I went to New York during that month, for the New England Bank, with about one hundred thousand dollars in foreign gold, the value of which by law at the mint was soon to be reduced from eighty-seven to eighty-five cents per pennyweight, or about that. I also had orders to buy bills with it, at the best rate I could. Accordingly I invested it, and had to analyze the standing of many who offered bills, as drawers or endorsers. Some of the bills were protested for non-acceptance, and were returned at once, and damages claimed. This was new law in New York, and resisted; but the merchants were convinced by suits, and paid the twenty per cent. damages. The law of damage was altered soon after. On my return, I took a packet for Providence, and came at the rate of ten knots an hour for the first seven hours of the night. I was alarmed by a crash, which seemed to me to be breaking in the side of the ship, within a few inches of my head. I ran upon deck, and it was a scene to be remembered. Beside the crew, on board were the officers of a wrecked vessel from Portsmouth, N. H., and some other old ship-masters, all at work, and giving directions to a coaster, which had run foul of us, and had lost its way. By favor and labor, we were saved from being wrecked; but were obliged to land at some fifteen miles from
- 57. Providence, and get there as we could through the snow. I arrived there almost dead with headache and sickness. Madam Dexter and her daughter left the day before, and reached home in perfect safety before the storm. Such are the scenes of human life! Here am I enjoying my own fireside, while all who were then active with me in the scenes thus recalled are called to their account, excepting Philip Hone, M. Van Schaick, N. Goddard, Chancellor Kent, and his son-in-law, Isaac Hone.
- 58. CHAPTER X. MARRIAGE.—ELECTED TO LEGISLATURE.— ENGAGES IN MANUFACTURES.—REFLECTIONS. In April, 1821, Mr. Lawrence was married to Mrs. Nancy Ellis, widow of the late Judge Ellis, of Claremont, N. H., and daughter of Robert Means, Esq., of Amherst, in the same State. His children, who had been placed with his parents and sisters at Groton, were brought home; and he was now permitted again to unite his family under his own roof, and to enjoy once more those domestic comforts so congenial to his taste, and which each revolving year seemed to increase until the close of his life. Mr. Lawrence was elected a representative from Boston to the Legislature for the session of 1821 and 22; and this was the only occasion on which he ever served in a public legislative body. Although deeply engaged in his own commercial pursuits, he was constantly at his post in the House of Representatives; and attended faithfully to the duties of his office, although with much sacrifice to his own personal interests. Very little is found among his memoranda relating to this new experience. As a member of a committee of the Legislature having in charge the subject of the erection of wooden buildings in Boston, he seems to have had a correspondence with the late Hon. John Lowell, who took strong ground before the committee against the multiplication of buildings of this material, and backed his arguments with some very characteristic statements and observations. On one of these letters Mr. Lawrence made a memorandum, dated March, 1845, as follows: The Boston Rebel was a true man, such as we need more of in these latter days. The open-mouthed lovers of the dear people
- 59. are self-seekers in most instances. Beware of such. The following extract is taken from a letter, dated January 4th, 1822, addressed by Mr. Lawrence to Hon. Frederic Wolcott, of Connecticut, respecting a son who was about to be placed in his counting-room, and who, in after years, became his partner in business: H. will have much leisure in the evening, which, if he choose, may be profitably devoted to study; and we hope he will lay out such a course for himself, as to leave no portion of his time unappropriated. It is on account of so much leisure, that so many fine youths are ruined in this town. The habit of industry once well fixed, the danger is over. Will it not be well for him to furnish you, at stated periods, an exact account of his expenditures? The habit of keeping such an account will be serviceable, and, if he is prudent, the satisfaction will be great, ten years hence, in looking back and observing the process by which his character has been formed. If he does as well as he is capable, we have no doubt of your experiencing the reward of your care over him. For the several following years, Mr. Lawrence was deeply engaged in business; and the firm of which he was the senior partner became interested in domestic manufactures, which, with the aid of other capitalists, afterwards grew into so much importance, until now it has become one of the great interests of the country. Apart from all selfish motives, he early became one of the strongest advocates for the protection of American industry, believing that the first duty of a government is to advance the interests of its own citizens, when it can be accomplished with justice to others; and in opposition to the system of free trade, which, however plausible in theory, he considered prejudicial to the true interests of our own people. He was conscientious in these opinions; and, in their support, corresponded largely with some of the leading statesmen at Washington, as well as with prominent opponents at the South, who
- 60. combatted his opinions while they respected the motives by which he was actuated. He tested his sincerity, by embarking a large proportion of his property in these enterprises; and, to the last, entertained the belief that the climate, the soil, and the habits of the people, rendered domestic manufactures one of the permanent and abiding interests of New England. During seasons of high political excitement and sectional strife, he wrote to various friends at the South, urging them to discard all local prejudices, and to enter with the North into manly competition in all those branches of domestic industry which would tend, not only to enrich, but also to improve the moral and intellectual character of their people. He watched, with increasing interest, the progress of Lowell and other manufacturing districts, and was ever ready to lend a helping hand to any scheme which tended to advance their welfare. Churches, hospitals, libraries, in these growing communities, had in him a warm and earnest advocate; and it was always with honest pride that he pointed out to the intelligent foreigner the moral condition of the operative here, when compared with that of the same class in other countries. On the 1st of January, in each year, Mr. Lawrence was in the habit of noting down, in a small memorandum-book, an accurate account of all his property, in order that he might have a clear view of his own affairs, and also as a guide to his executors in the settlement of his estate, in case of his death. This annual statement commences in 1814, and, with the exception of 1819, when he was in great affliction on account of the death of his wife, is continued every year until that of his own death, in 1852. In this little volume the following memorandum occurs, dated January 1, 1826: I have been extensively engaged in business during the last two years, and have added much to my worldly possessions; but have come to the same conclusions in regard to them that I did in 1818. I feel distressed in mind that the resolutions then made have not been more effectual in keeping me from this overengagedness in business. I now find myself so engrossed
- 61. with its cares, as to occupy my thoughts, waking or sleeping, to a degree entirely disproportioned to its importance. The quiet and comfort of home are broken in upon by the anxiety arising from the losses and mischances of a business so extensive as ours; and, above all, that communion which ought ever to be kept free between man and his Maker is interrupted by the incessant calls of the multifarious pursuits of our establishment. After noting down several rules for curtailing his affairs, he continues: Property acquired at such sacrifices as I have been obliged to make the past year costs more than it's worth; and the anxiety in protecting it is the extreme of folly. 1st of January, 1827.—The principles of business laid down a year ago have been very nearly practised upon. Our responsibilities and anxieties have greatly diminished, as also have the accustomed profits of business; but there is sufficient remaining for the reward of our labor to impose on us increased responsibilities and duties, as agents who must at last render an account. God grant that mine be found correct!
- 62. CHAPTER XI. REFLECTIONS.—BUNKER HILL MONUMENT.— LETTERS. 1st of January, 1828.—After an account of his affairs, he remarks: The amount of property is great for a young man under forty- two years of age, who came to this town when he was twenty- one years old with no other possessions than a common country education, a sincere love for his own family, and habits of industry, economy, and sobriety. Under God, it is these same self-denying habits, and a desire I always had to please, so far as I could without sinful compliance, that I can now look back upon and see as the true ground of my success. I have many things to reproach myself with; but among them is not idling away my time, or spending money for such things as are improper. My property imposes upon me many duties, which can only be known to my Maker. May a sense of these duties be constantly impressed upon my mind; and, by a constant discharge of them, God grant me the happiness at last of hearing the joyful sound, 'Well done, good and faithful servant, enter thou into the joy of thy Lord!' Amen. Amen. Previous to this date, but few private letters written by Mr. Lawrence were preserved. From that time, however, many volumes have been collected, a greater part of them addressed to his children. Out of a very large correspondence with them and with friends, such selections will be made as are thought most interesting, and most worthy to be preserved by his family and their descendants. The nature of this correspondence is such, involving many personal matters of transient interest that often scraps of letters only can be
- 63. given; and, although it will be the aim of the editor to give an outline of the life of the author of these letters, it will be his object to allow him to speak for himself, and to reveal his own sentiments and character, rather than to follow out, from year to year, the details of his personal history. This correspondence commences with a series of letters extending through several years, and addressed to his eldest son, who was, during that time, at school in France and Spain. Boston, November 11, 1828. I trust that you will have had favoring gales and a pleasant passage, and will be safely landed at Havre within twenty days after sailing. You will see things so different from what you have been accustomed to, that you may think the French are far before or behind us in the arts of life, and formation of society. But you must remember that what is best for one people may be the worst for another; and that it is true wisdom to study the character of the people among whom you are, before adopting their manners, habits, or feelings, and carrying them to another people. I wish to see you, as long as you live, a well-bred, upright Yankee. Brother Jonathan should never forget his self- respect, nor should he be impertinent in claiming more for his country or himself than is due; but on no account should he speak ungraciously of his country or its friends abroad, whatever may be said by others. Lafayette in France is not what he is here; and, whatever may be said of him there, he is an ardent friend of the United States; and I will venture to say, if you introduce yourself to him as a grandson of one of his old Yankee officers, he will treat you with the kindness of a father. You must visit La Grange, and G. will go with you. He will not recollect your grandfather, or any of us. But tell him that your father and three uncles were introduced to him here in the State House; that they are much engaged in forwarding the Bunker Hill Monument; and, if ever he return to this country, it will be the pride of your father to lead him to the top of it.
- 64. Among Mr. Lawrence's papers, this is the first allusion to the Bunker Hill Monument, in the erection of which he afterwards took so prominent a part, and to which he most liberally contributed both time and money. From early associations, perhaps from the accounts received from his father, who was present during the battle, his mind became strongly interested in the project of erecting a monument, and particularly in that of reserving the whole battle-ground for the use of the public forever. He had been chosen one of the Building Committee of the Board of Directors in October, 1825, in company with Dr. John C. Warren, General H. A. S. Dearborn, George Blake, and William Sullivan. From this time until the completion of the monument, the object occupied a prominent place in his thoughts; and allusion to his efforts in its behalf during the succeeding years will, from time to time, be introduced. On December 13, 1828, he thus alludes to the death of an invalid daughter six years of age: She was taken with lung fever on the 4th, and died, after much suffering and distress, on the 8th. Nothing seemed to relieve her at all; and I was thankful when the dear child ceased to suffer, and was taken to the bosom of her Saviour, where sickness and suffering will no more reach her, and the imperfections of her earthly tenement will be corrected, and her mind and spirit will be allowed to expand and grow to their full stature in Christ. In his hands I most joyfully leave her, hoping that I may rejoin her with the other children whom it has pleased God to give me. (TO HIS SON.) December 29. My thoughts are often led to contemplate the condition of my children in every variety of situation, more especially in sickness, since the death of dear M. Although I do not allow myself to indulge in melancholy or fearful forebodings, I cannot but feel
- 65. the deepest solicitude that their minds and principles should be so strengthened and stayed upon their God and Saviour as to give them all needed support in a time of such trial and suffering. You are so situated as perhaps not to recall so frequently to your mind as may be necessary the principles in which you have been educated. But let me, in the absence of these objects, remind you that God is ever present, and sees the inmost thoughts; and, while he allows every one to act freely, he gives to such as earnestly and honestly desire to do right all needed strength and encouragement to do it. Therefore, my dear son, do not cheat yourself by doing what you suspect may be wrong. You are as much accountable to your Maker for an enlightened exercise of your conscience, as you would be to me to use due diligence in taking care of a bag of money which I might send by you to Mr. W. If you were to throw it upon deck, or into the bottom of the coach, you would certainly be culpable; but, if you packed it carefully in your trunk, and placed the trunk in the usual situation, it would be using common care. So in the exercise of your conscience: if you refuse to examine whether an action is right or wrong, you voluntarily defraud yourself of the guide provided by the Almighty. If you do wrong, you have no better excuse than he who had done so willingly and wilfully. It is the sincere desire that will be accepted. To his second son, then at school in Andover, he writes: I received your note yesterday, and was prepared to hear your cash fell short, as a dollar-bill was found in your chamber on the morning you left home. You now see the benefit of keeping accounts, as you would not have been sure about this loss without having added up your account. Get the habit firmly fixed of putting down every cent you receive and every cent you expend. In this way you will acquire some knowledge of the relative value of things, and a habit of judging and of care which will be of use to you during all your life. Among the
- 66. numerous people who have failed in business within my knowledge, a prominent cause has been a want of system in their affairs, by which to know when their expenses and losses exceeded their profits. This habit is as necessary for professional men as for a merchant; because, in their business, there are numerous ways to make little savings, if they find their income too small, which they would not adopt without looking at the detail of all their expenses. It is the habit of consideration I wish you to acquire; and the habit of being accurate will have an influence upon your whole character in life. (TO HIS SON IN FRANCE.) April 28, 1829. I beseech you to consider well the advantages you enjoy, and to avail yourself of your opportunities to give your manners a little more ease and polish; for, you may depend upon it, manners are highly important in your intercourse with the world. Good principles, good temper, and good manners, will carry a man through the world much better than he can get along with the absence of either. The most important is good principles. Without these, the best manners, although, for a time, very acceptable, cannot sustain a person in trying situations. If you live to attain the age of thirty, the interim will appear but a span; and yet at that time you will be in the full force of manhood. To look forward to that period, it seems very long; and it is long enough to make great improvement. Do not omit the opportunity to acquire a character and habits that will continue to improve during the remainder of life. At its close, the reflection that you have thus done will be a support and stay worth more than any sacrifice you may ever feel called on to make in acquiring these habits.
- 67. (TO THE SAME.) June 7, 1829. I was forcibly reminded, on entering our tomb last evening, of the inroads which death has made in our family since 1811, at the period when I purchased it. How soon any of us who survive may mingle our dust with theirs, is only known to Omniscience; but, at longest, it can be in his view but a moment, a mere point of time. How important, then, to us who can use this mere point for our everlasting good, that we should do it, and not squander it as a thing without value! Think upon this, my son; and do not merely admit the thought into your mind and drive it out by vain imaginations, but give it an abiding and practical use. To set a just value upon time, and to make a just use of it, deprives no one of any rational pleasure: on the contrary, it encourages temperance in the enjoyment of all the good things which a good Providence has placed within our reach, and thankfulness for all opportunities of bestowing happiness on our fellow-beings. Thus you have an opportunity of making me and your other friends happy, by diligence in your studies, temperance, truth, integrity, and purity of life and conversation. I may not write to you again for a number of weeks, as I shall commence a journey to Canada in a few days. You will get an account of the journey from some of the party.
- 68. CHAPTER XII. JOURNEY TO CANADA.—LETTERS.—DIARY.— CHARITIES. Mr. Lawrence, with a large party, left Boston on the 13th of June, and passed through Vermont, across the Green Mountains, to Montreal and Quebec. Compared with these days of railroad facilities, the journey was slow. It was performed very leisurely in hired private vehicles, and seems to have been much enjoyed. He gives a glowing account of the beauty of the country through which he passed, as well as his impressions of the condition of the population. From Quebec the party proceeded to Niagara Falls, and returned through the State of New York to Boston, greatly improved in health and spirits. This, with one other visit to Canada several years before, was the only occasion on which Mr. Lawrence ever left the territory of the United States; for, though sometimes tempted, in after years, to visit the Old World, his occupations and long- continued feeble health prevented his doing so. (TO HIS SON.) July 27. If, in an endeavor to do right, we fall short, we shall still be in the way of duty; and that is first to be looked at. We must keep in mind that we are to render an account of the use of those talents which are committed to us; and we are to be judged by unerring Wisdom, which can distinguish all the motives of action, as well as weigh the actions. As our stewardship has been faithful or otherwise, will be the sentence pronounced
- 69. upon us. Give this your best thoughts, for it is a consideration of vast importance. August 27. Bring home no foreign fancies which are inapplicable to our state of society. It is very common for our young men to come home and appear quite ridiculous in attempting to introduce their foreign fashions. It should be always kept in mind that the state of society is widely different here from that in Europe; and our comfort and character require it should long remain so. Those who strive to introduce many of the European habits and fashions, by displacing our own, do a serious injury to the republic, and deserve censure. An idle person, with good powers of mind, becomes torpid and inactive after a few years of indulgence, and is incapable of making any high effort; highly important it is, then, to avoid this enemy of mental and moral improvement. I have no wish that you pursue trade. I would rather see you on a farm, or studying any profession. October 16. It should always be your aim so to conduct yourself that those whom you value most in the world would approve your conduct, if all your actions were laid bare to their inspection; and thus you will be pretty sure that He who sees the motive of all our actions will accept the good designed, though it fall short in its accomplishment. You are young, and are placed in a situation of great peril, and are perhaps sometimes tempted to do things which you would not do if you knew yourself under the eye of your guardian. The blandishments of a beautiful city may lead you to forget that you are always surrounded, supported, and seen, by that best Guardian. December 27.
- 70. I suppose Christmas is observed with great pomp in France. It is a day which our Puritan forefathers, in their separation from the Church of England, endeavored to blot out from the days of religious festivals; and this because it was observed with so much pomp by the Romish Church. In this, as well as in many other things, they were as unreasonable as though they had said they would not eat bread because the Roman Catholics do. I hope and trust the time is not far distant when Christmas will be observed by the descendants of the Puritans with all suitable respect, as the first and highest holiday of Christians; combining all the feelings and views of New England Thanksgiving with all the other feelings appropriate to it. January 31, 1830. You have seen, perhaps, that the Directors of the Bunker Hill Monument Association have applied to the Legislature for a lottery. I am extremely sorry for it. I opposed the measure in all its stages, and feel mortified that they have done so. They cannot get it, and I desire that General Lafayette may understand this; and, if he will write us a few lines during the coming year, it will help us in getting forward a subscription. When our citizens shall have had one year of successful business, they will be ready to give the means to finish the monument. My feelings are deeply interested in it, believing it highly valuable as a nucleus for the affections of the people in after time; and, if my life be spared and my success continue, I will never cease my efforts until it be completed. Further details will be given in this volume to show now nobly Mr. Lawrence persevered in the resolution thus deliberately formed; and, though he was destined to witness many fruitless efforts, he had the satisfaction at last of seeing the completion of the monument, and from its summit of pointing out the details of the battle to the son of one of the British generals in command[2] on that eventful day.
- 71. On the same page with the estimate of his property for the year 1830, he writes: With a view to know the amount of my expenditures for objects other than the support of my family, I have, for the year 1829, kept a particular account of such other expenses as come under the denomination of charities, and appropriations for the benefit of others not of my own household, for many of whom I feel under the same obligation as for my own family. This memorandum was commenced on the 1st of January, 1829, and is continued until December 30, 1852, the last day of his life. It contains a complete statement of his charities during that whole period, including not only what he contributed in money, but also all other donations, in the shape of clothing materials, books, provisions, c. His custom was to note down at cost the value of the donation, after it had been despatched; whether in the shape of a book, a turkey, or one of his immense bundles of varieties to some poor country minister's family, as large, as he says in addressing one, as a small haycock. Two rooms in his house, and sometimes three, were used principally for the reception of useful articles for distribution. There, when stormy weather or ill health prevented him from taking his usual drive, he was in the habit of passing hours in selecting and packing up articles which he considered suitable to the wants of those whom he wished to aid. On such days, his coachman's services were put in requisition to pack and tie up the small haycocks; and many an illness was the result of over-exertion and fatigue in supplying the wants of his poorer brethren. These packages were selected according to the wants of the recipients, and a memorandum made of the contents. In one case, he notifies Professor ——, of —— College, that he has sent by railroad a barrel and a bundle of books, with broadcloth and pantaloon stuffs, with odds and ends for poor students when they go out to keep school in the winter. Another, for the president of a college at the West, one piece of silk and worsted, for three dresses; one piece of plaid, for M. and mamma; a lot of pretty books; a piece of lignum-vitæ from
- 72. the Navy Yard, as a text for the support of the navy; and various items for the children: value, twenty-five dollars. To a professor in a college in a remote region he sends a package containing dressing-gown, vest, hat, slippers, jack-knife, scissors, pins, neck-handkerchiefs, pantaloons, cloth for coat, 'History of Groton,' lot of pamphlets, c. Most of the packages forwarded contained substantial articles for domestic use, and were often accompanied by a note containing from five to fifty dollars in money. The distribution of books was another mode of usefulness to which Mr. Lawrence attached much importance. In his daily drives, his carriage was well stored with useful volumes, which he scattered among persons of all classes and ages as he had opportunity. These books were generally of a religious character, while others of a miscellaneous nature were purchased in large numbers, and sent to institutions, or individuals in remote parts of the country. He purchased largely the very useful as well as tasteful volumes of the American Tract Society and the Sunday-School Union. An agent of the latter society writes: I had almost felt intimately acquainted with him, as nearly every pleasant day he visited the depository to fill the front seat of his coach with books for distribution. Old and young, rich and poor, shared equally in these distributions; and he rarely allowed an occasion to pass unimproved when he thought an influence could be exerted by the gift of an appropriate volume. While waiting one day in his carriage with a friend, in one of the principal thoroughfares of the city, he beckoned to a genteelly- dressed young man who was passing, and handed him a book. Upon being asked whether the young man was an acquaintance, he replied:
- 73. No, he is not; but you remember where it is written, 'Cast thy bread upon the waters, for thou shalt find it after many days.' A barrel of books is no uncommon item found in his record of articles almost daily forwarded to one and another of his distant beneficiaries.
- 74. CHAPTER XIII. CORRESPONDENCE WITH MR. WEBSTER.— LETTERS. (TO HIS SON.) February 5, 1830. Be sure and visit La Grange before you return; say to General Lafayette that the Bunker Hill Monument will certainly be finished, and that the foolish project of a lottery has been abandoned. If, in the course of Providence, I should be taken away, I hope my children will feel it a duty to continue the efforts that are made in this work, which I have had so much at heart, and have labored so much for. To his son, then at school at Versailles, he writes on Feb. 26, 1830: After hearing from you again, I can judge better what to advise respecting your going into Spain. At all events, let no hope of going, or seeing, or doing anything else, prevent your using the present time for improving yourself in whatever you find to do. My greatest fear is, that you may form a wrong judgment of what constitutes your true respectability, happiness, and usefulness. To a youth just entering on the scenes of life, the roses on the wayside appear without thorns; but, in the eagerness to snatch them, many find, to their sorrow, that all which appears so fair is not in possession what it was in prospect, and that beneath the rose there is a thorn that sometimes wounds like a serpent's bite. Let not appearances deceive you; for, when once you have strayed, the second temptation is more likely to be fallen into than the first.
- 75. March 6, 1830. We are all in New England deeply interested by Mr. Webster's late grand speech in the Senate, vindicating New England men and New England measures from reproach heaped upon them by the South; it was his most powerful effort, and you will see the American papers are full of it. You should read the whole debate between him and Mr. Hayne of South Carolina; you will find much to instruct and interest you, and much of what you ought to know. Mr. Webster never stood so high in this country as, at this moment; and I doubt if there be any man, either in Europe or America, his superior. The doctrines upon the Constitution in this speech should be read as a text-book by all our youth. After reading the great speech of Mr. Webster, Mr. Lawrence addressed to that gentleman a letter, expressing his admiration of the manner in which New England had been vindicated, and also his own personal feelings of gratitude for the proud stand thus taken. Mr. Webster replied as follows: Washington, March 8, 1830. Dear Sir: I thank you very sincerely for your very kind and friendly letter. The sacrifices made in being here, and the mortifications sometimes experienced, are amply compensated by the consciousness that my friends at home feel that I have done some little service to our New England. I pray you to remember me with very true regard to Mrs. Lawrence, and believe me Very faithfully and gratefully yours, Daniel Webster. To Amos Lawrence, Esq.
- 76. EXTRACTS OF LETTERS TO HIS SON. April 13, 1830. You may feel very sure that any study which keeps your mind engaged will be likely to strengthen it; and that, if you leave your mind inactive, it will run to waste. Your arm is strengthened by wielding a broadsword, or even a foil. Your legs by various gymnastic exercises, and the organs of sight and hearing by careful and systematic use, are greatly improved; even the finger is trained, by the absence of sight, to perform almost the service of the eye. All this shows how natural it is for all the powers to grow stronger by use. You needed not these examples to convince you; but my desire to have you estimate your advantages properly induces me to write upon them very often. Every American youth owes his country his best talents and services, and should devote them to the country's welfare. In doing that, you will promote not only your own welfare, but your highest enjoyment. The duty of an American citizen, at this period of the world, is that of a responsible agent; and he should endeavor to transmit to the next age the institutions of our country uninjured and improved. We hope, in your next letter, to hear something more of General Lafayette. The old gentleman is most warm in his affection for Americans. May he live long to encourage and bless by his example the good of all countries! In contemplating a life like his, who can say that compensation even here is not fully made for all the anguish and suffering he has formerly endured? Long life does not consist in many years; but in the period being filled with good services to our fellow-beings. He whose life ends at thirty may have done much, while he who has reached the age of one hundred may have done little. With the Almighty, a thousand years are a moment; and he will therefore give no credit to any talents not used to his glory; which use is the
- 77. same thing as promoting, by all means in our power, the welfare and happiness of the beings among whom we are placed. May 7, 1830. I have been pretty steady at my business, without working hard, or having anxious feelings about it. It is well to have an agreeable pursuit to employ the mind and body. I think that I can work for the next six years with as good a relish as ever I did; but I make labor a pleasure. I have just passed into my forty-fifth year, you know. At my age, I hope you will feel as vigorous and youthful as I now do. A temperate use of the good things of life, and a freedom from anxious cares, tend, as much as anything, to keep off old age. June 17, 1830. To-day completes fifty-five years since the glorious battle of Bunker Hill, and five years since the nation's guest assisted at the laying of the corner-stone of the monument which is to commemorate to all future times the events which followed that battle. If it should please God to remove me before this structure is completed, I hope to remember it in my will, and that my sons will live to see it finished. But what I deem of more consequence is to retain for posterity the battle-field, now in the possession of the Bunker Hill Monument Association. The Association is in debt, and a part of the land may pass out of its possession; but I hope, if it do, there will be spirit enough among individuals to purchase it and restore it again; for I would rather the whole work should not be resumed for twenty years, than resume it by parting with the land. I name this to you now, that you may have a distinct intimation of my wishes to keep the land open for our children's children to the end of time. July 17, 1830.
- 78. Temptation, if successfully resisted, strengthens the character; but it should always be avoided. 'Lead us not into temptation' are words of deep meaning, and should always carry with them corresponding desires of obedience. At a large meeting of merchants and others held ten days ago, it was resolved to make an effort to prevent the licensing of such numbers of soda-shops, retailers of spirits and the like, which have, in my opinion, done more than anything else to debase and ruin the youth of our city. It is a gross perversion of our privileges to waste and destroy ourselves in this way. God has given us a good land and many blessings. We misuse them, and make them minister to our vices. We shall be called to a strict account. Every good citizen owes it to his God and his country to stop, as far as he can, this moral desolation. Let me see you, on your return, an advocate of good order and good morals. * * * Our old neighbor the sea-serpent was more than usually accommodating the day after we left Portsmouth. He exhibited himself to a great number of people who were at Hampton Beach last Saturday. They had a full view of his snakeship from the shore. He was so civil as to raise his head about four feet, and look into a boat, where were three men, who thought it the wisest way to retreat to their cabin. His length is supposed to be about one hundred feet, his head the size of a ten-gallon cask, and his body, in the largest part, about the size of a barrel. I have never had any more doubt respecting the existence of this animal, since he was seen here eleven years ago, than I have had of the existence of Bonaparte. The evidence was as strong to my mind of the one as of the other. I had never seen either; but I was as well satisfied of the existence of both, as I should have been had I seen both. And yet the idea of the sea- serpent's existence has been scouted and ridiculed. September 25.
- 79. The events of the late French Revolution have reached us up to the 17th August. The consideration of them is animating, and speaks in almost more than human language. We are poor, frail, and mortal beings; but there is something elevating in the thought of a whole people acting as with the mind and the aim of one man, a part which allies man to a higher order of beings. I confess it makes me feel a sort of veneration for them; and trust that no extravagance will occur to mar the glory and the dignity of this enterprise. Our beloved old hero, too, acting as the guiding and presiding genius of this wonderful event! May God prosper them, and make it to the French people what it is capable of being, if they make a right use of it! I hope that you have been careful to see and learn everything, and that you will preserve the information you obtain in such a form as to recall the events to your mind a long time hence. We are all very well and very busy, and in fine spirits, here in the old town of Boston. Those who fell behind last year have some of them placed themselves in the rear rank, and are again on duty. Others are laid up, unfit for duty; and the places of all are supplied with fresh troops. We now present as happy and as busy a community as you would desire to see.
- 80. CHAPTER XIV. TESTIMONIAL TO MR. WEBSTER.— DANGEROUS ILLNESS.—LETTERS. During the autumn of 1830, in order to testify in a more marked manner his appreciation of Mr. Webster's distinguished services in the Senate of the United States, Mr. Lawrence presented to that gentleman a service of silver plate, accompanied by the following note: Boston, October 23, 1830. Hon. Daniel Webster. Dear Sir: Permit me to request your acceptance of the accompanying small service of plate, as a testimony of my gratitude for your services to the country in your late efforts in the Senate; especially for your vindication of the character of Massachusetts and of New England. From your friend and fellow-citizen, Amos Lawrence. P. S.—If by any emblem or inscription on any piece of this service, referring to the circumstances of which this is a memorial, the whole will be made more acceptable, I shall be glad to have you designate what it shall be, and permit me the opportunity of adding it. To which Mr. Webster replied, on the same evening, as follows: Summer-street, October 23, 1830.
- 81. My dear Sir: I cannot well express my sense of your kindness, manifested in the present of plate, which I have received this evening. I know that, from you, this token of respect is sincere; and I shall ever value it, and be happy in leaving it to my children, as a most gratifying evidence of your friendship. The only thing that can add to its value is your permission that it may be made to bear an inscription expressive of the donation. I am, dear sir, with unfeigned esteem, Your friend and obedient servant, Daniel Webster. Amos Lawrence, Esq. (TO HIS SON.) Boston, January 16, 1831. Our local affairs are very delightful in this state and city. We have no violent political animosities; and the prosperity of the people is very great. In our city, in particular, the people have not had greater prosperity for twenty years. There is a general industry and talent in our population, that is calculated to produce striking results upon their character. In your reflections upon your course, you may settle it as a principle, that no man can attain any valuable influence or character among us, who does not labor with whatever talents he has to increase the sum of human improvement and happiness. An idler, who feels that he has no responsibilities, but is contriving to get rid of time without being useful to any one, whatever be his fortune, can find no comfort in staying here. We have not enough such to make up a society. We are literally all working-men; and the attempt to get up a 'Working-men's party' is a libel upon the whole population, as it implies that there are among us large numbers who are not working-men. He is a working-man whose mind is employed, whether in making researches into the meaning of hieroglyphics or in demonstrating any invention in
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