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Studies in Systems, Decision and Control 450
Wenhai Qi
Guangdeng Zong
Analysis
and Design
for Positive
Stochastic Jump
Systems

Studies in Systems, Decision and Control
Volume 450
Series Editor
Janusz Kacprzyk,Systems Research Institute, Polish Academy of Sciences,
Warsaw, Poland

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Wenhai Qi·Guangdeng Zong
AnalysisandDesign
forPositiveStochasticJump
Systems

Wenhai Qi
School of Engineering
Qufu Normal University
Rizhao, Shandong, China
Guangdeng Zong
School of Engineering Qufu Normal University Rizhao, Shandong, China
ISSN 2198-4182 ISSN 2198-4190 (electronic)
Studies in Systems, Decision and Control
ISBN 978-981-19-5489-4 ISBN 978-981-19-5490-0 (eBook)
https://doi.org/10.1007/978-981-19-5490-0
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature
Singapore Pte Ltd. 2023
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This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface
In the operation process of practical systems, many complex factors, such as environ-
mental interference, component failure, and subsystem connection change, always
lead to the jump in system parameters and structures, resulting in deviation of the
measurement process and inaccurate research about system model establishment
and control. Owing to the performance of control systems largely dependent on the
complexity and accuracy of the model, the controller based on the single model has
been unable to achieve the desired control requirements, which has prompted people
to constantly seek new theories to guide the design of control systems. Stochastic
jump systems driven by time and events provide an effective theoretical basis for the
study of such problems and promote the research of related control issues. As a typical
kind of stochastic jump systems, Markov jump systems are described by state space
equations under multiple modes, in which the operating state changes according to
the stochastic Markov switching rule among subsystems. In fact, the jump among
subsystems meets certain statistical laws. In recent years, Markov jump systems have
become one of the research hotspots in the control ﬁeld, and also have found wide
applications in power systems, chemical process, agricultural engineering, aerospace,
biomedicine, and network communication. In these circumstances, more and more
experts have began to study Markov jump systems from different disciplines, thus
promoting the rapid development of the corresponding theory.
As a key factor of Markov jump systems, the transition rate affects the dynamic
characteristics of the system, which is mainly subject to the probability distribution
function of the sojourn time. For Markov jump systems, the sojourn time follows
exponential distribution or geometric distribution. According to the memoryless
property of exponential distribution or geometric distribution, the transition rate
of Markov jump systems does not depend on past modes and has no relationship
with the sojourn time. It is worth noting that, in practice, the sojourn time does
not always obey the exponential distribution or geometric distribution. Compared
with Markov jump systems, the probability distribution function of the sojourn time
in semi-Markov jump systems is relaxed from the special exponential distribution
v

vi Preface
or geometric distribution to the general probability distribution. Then, the transi-
tion rate depends on the past modes and meets the sojourn-time-dependent charac-
teristics, thus describing a wider range of stochastic jump systems. Moreover, the
study of semi-Markov jump systems provides additional insights into some long-
standing and sophisticated problems, such as sliding mode control, adaptive control,
event-triggered control, ﬁnite-time control, and fault detection.
In the past decades, the analysis and synthesis of stochastic jump systems have
been intensively investigated and have attracted increasing attention. Although a
large number of the corresponding works have been developed from various disci-
plines, there still exist many fundamental problems with less well understanding. In
particular, there still lacks a uniﬁed framework to cope with the issues of analysis
and synthesis for positive stochastic jump systems. This motivated us to write the
related work.
The monograph aims to present up-to-date research developments and references
on the analysis and design for all the subsystems of stochastic jump systems belonging
to positive systems. Different from general systems, positive systems are conﬁned
to the positive cone instead of the whole state space and depend on the positivity
of their state signals, output signals, and input signals. Owing to the particularity of
positive systems, many previous approaches for general systems cannot be extended
to positive systems, which makes the analysis and synthesis of positive systems full
of challenges. By using multiple linear co-positive Lyapunov function method and
linear programming technique, a basic theoretical framework is formed towards the
issues of analysis and design for positive stochastic jump systems. The book can be
used by researchers to carry out studies on positive stochastic jump systems and is
suitable for graduate students of control theory and engineering. It may also be a
valuable reference for the control design of switched systems by engineers.
The contents of the book are divided into thirteen chapters which contain several
independent yet related topics, and they are organized as follows. Chapter1intro-
duces some basic background knowledge on positive stochastic jump systems, and
also describes the main work of the book. Chapter
2considers the problems of expo-
nential stability andL1-gain analysis for positive delayed Markov jump systems.
Chapters3–5address the problems of stability, stabilization,L1-gain analysis, and
ﬁnite-time control for positive semi-Markov jump systems. Chapters6–9give theo-
retical developments in detail forL1control,L∞control, robust ﬁnite-time stabi-
lization, and fault detection for positive delayed semi-Markov jump systems. Some
control problems for positive fuzzy semi-Markov jump systems include stochastic
stability,
L1-gain analysis, observer design, and ﬁlter design are discussed in Chap-
ters10–12. Finally, Chap.13concludes some future study directions related to the
contents of the book.
Rizhao, China
Rizhao, China
June 2022
Wenhai Qi
Guangdeng Zong

Acknowledgements
There are numerous individuals without whose constructive comments, useful
suggestions, and wealth of ideas this monograph could not have been completed.
Special thanks go to Prof. Ju H. Park, Yeungnam University; Prof. Shun-Feng Su,
National Taiwan University of Science and Technology; Prof. Jinde Cao, Southeast
University; Prof. Hamid Reza Karimi, Politecnico di Milano; Prof. Xianwen Gao,
Northeastern University; Prof. Yonggui Kao, Harbin Institute of Technology; Prof.
Jun Cheng, Guangxi Normal University and Prof. Xiaoming Chen, Nanjing Univer-
sity of Aeronautics and Astronautics, for their valuable suggestions, constructive
comments, and support. Next, our acknowledgements go to many colleagues who
have offered support and encouragement throughout this research effort. Finally, the
authors would like to express their sincere gratitude to the editors of the book for
their time and kind help.
The monograph was supported in part by the National Natural Science Foundation
of China (62073188 and 61773235), the Postdoctoral Science Foundation of China
(2022T150374), and the Natural Science Foundation of Shandong (ZR2019YQ29
and ZR2021MF083).
vii

Contents
1Introduction..................................................1
1.1Background.............................................1
1.2Markov Jump Systems....................................3
1.3Semi-Markov Jump Systems...............................5
1.4Positive Systems.........................................7
1.5Positive Stochastic Jump Systems..........................8
1.6Organization of the Book..................................10
References....................................................11
Part IPositive Delayed Markov Jump Systems
2Exponential Stability andL1-Gain Analysis.....................21
2.1Introduction.............................................21
2.2Problem Statements and Preliminaries.......................22
2.3Exponential Stability Analysis.............................24
2.4L1-gain Performance Analysis.............................27
2.5Simulation..............................................29
2.6Conclusion..............................................33
References....................................................33
Part IIPositive Semi-Markov Jump Systems
3Stability and Stabilization......................................37
3.1Introduction.............................................37
3.2Problem Statements and Preliminaries.......................38
3.3Mean Stability Analysis...................................40
3.4Controller Design........................................46
3.5Simulation..............................................47
3.6Conclusion..............................................51
References....................................................51
ix

x Contents
4L1-Gain and Control Synthesis.................................55
4.1Introduction.............................................55
4.2Problem Statements and Preliminaries.......................56
4.3Stochastic Stability Analysis...............................57
4.4L1-Gain Performance Analysis............................59
4.5Controller Design........................................60
4.6Simulation..............................................62
4.7Conclusion..............................................65
References....................................................65
5Finite-TimeL1Control........................................67
5.1Introduction.............................................67
5.2Problem Statements and Preliminaries.......................68
5.3Finite-time Boundedness Analysis..........................70
5.4L1Finite-Time Boundedness Analysis......................73
5.5Controller Design........................................74
5.6Simulation..............................................78
5.7Conclusion..............................................83
References....................................................83
Part IIIPositive Delayed Semi-Markov Jump Systems
6L1Control...................................................87
6.1Introduction.............................................87
6.2Problem Statements and Preliminaries.......................88
6.3Stochastic Stability Analysis...............................89
6.4L1-Gain Performance Analysis............................94
6.5Controller Design........................................95
6.6Simulation..............................................98
6.7Conclusion..............................................101
References....................................................102
7L∞Control..................................................103
7.1Introduction.............................................103
7.2Problem Statements and Preliminaries.......................104
7.3Stochastic Stability Analysis...............................107
7.4L∞-Gain Performance Analysis............................110
7.5Controller Design........................................113
7.6Simulation..............................................114
7.7Conclusion..............................................119
References....................................................120
8Robust Finite-Time Stabilization...............................121
8.1Introduction.............................................121
8.2Problem Statements and Preliminaries.......................122
8.3Finite-time Boundedness Analysis..........................124
8.4L1Finite-Time Boundedness Analysis......................128

Contents xi
8.5Controller Design........................................129
8.6Simulation..............................................133
8.7Conclusion..............................................137
References....................................................137
9Fault Detection................................................139
9.1Introduction.............................................139
9.2Problem Statements and Preliminaries.......................140
9.3Stochastic Stability Analysis...............................143
9.4L1-Gain Performance Analysis............................147
9.5Fault Detection Filter Design...............................148
9.6Simulation..............................................151
9.7Conclusion..............................................155
References....................................................156
Part IVPositive Fuzzy Semi-Markov Jump Systems
10Stochastic Stability andL1-Gain Analysis.......................161
10.1Introduction.............................................161
10.2Problem Statements and Preliminaries.......................162
10.3Stochastic Stability Analysis...............................164
10.4L1-Gain Performance Analysis............................168
10.5Simulation..............................................170
10.6Conclusion..............................................173
References....................................................173
11PositiveL1Observer Design...................................177
11.1Introduction.............................................177
11.2Problem Statements and Preliminaries.......................178
11.3Stochastic Stability Analysis...............................179
11.4L1-Gain Performance Analysis............................182
11.5L1Fuzzy Observer Design................................184
11.6Simulation..............................................187
11.7Conclusion..............................................191
References....................................................191
12Filter Design..................................................193
12.1Introduction.............................................193
12.2Problem Statements and Preliminaries.......................194
12.3Stochastic Stability withL1-Gain Performance Analysis.......197
12.4L1Filter Design.........................................199
12.5Simulation..............................................203
12.6Conclusion..............................................209
References....................................................209

xii Contents
Part VSummary
13Conclusion and Future Research Direction......................213
13.1Conclusion..............................................213
13.2Future Research Direction.................................214

Symbols
R Set of real numbers
R
n
Set ofn-column real vectors
R
n
+
Set ofn-dimensional nonnegative real vectors
R
n×m
Set ofn×mreal matrices
x(t) 1
∞
n
i=1
|xi(t)|
x(t) ∞Max1≤i≤n|xi(t)|
x(t) L1

∞
0
E{x(t) 1}dt
x(t) L∞

∞
0
Ex(t) ∞}dt
L1 Space of all vector-valued functions with ﬁniteL1norm
L∞ Space of all vector-valued functions with ﬁniteL∞norm
I Identity matrix
1n All-ones vector inR
n
⊗ Kronecker product
∈ Belong to
A
T
Transpose of matrixA
A
−1
Inverse of matrixA
A>>0A is positive
A≥≥0A is nonnegative
A<<0Aisnegative
A≤≤0A is nonpositive
E{·} Mathematical expectation
xiii

Chapter 1
Introduction
1.1 Background
In numerous practical systems, there exists a special class of signal variables, such as
the number of biological populations, the propagation rate of communication signals,
the commodity price of market economy, and the absolute temperature of physical
processes, etc. It should be pointed out that the values of these variables can never be
negative, and have to meet the requirement of non-negative property. As time goes
on, the trajectories of these signal variables are limited to the ﬁrst quadrant of two-
dimensional space or a cone in the non-negative region of multi-dimensional space,
and the corresponding dynamics are described as positive systems [1]. The charac-
teristic of the above systems is that: for any initial conditions and external inputs,
the state variables and output trajectories of positive systems remain non-negative.
Over the past decades, positive systems have been studied from many different scien-
tiﬁc disciplines, which have widely applied in ecology, market economics, chemical
industry, environmental science, and virus mutation systems, thus tremendously pro-
moting the rapid development of the related theories.
A typical example of positive systems is the network congestion system [2]in
Fig.1.1, expressed as the following dynamic model
˙x(t)=Ax(t)+Bω(t),
y(t)=Cx(t),
x(0)=x
0, (1.1)
wherex(t)=[x
1(t)x2(t)x3(t)]
T
represents the system state,x i(t), (i=1,2,3)
denotes the total data transmitted by network nodes 1,2,3.ω(t)andy(t)are
the external disturbance and the output, respectively.x
0means the total data of
the network node att=0. Coefﬁcient matrixA=[0a
12a13;a210a23;a31a320],
wherea
ij(i,j=1,2,3,iω=j)is the transmission rate between nodeiand nodej;
B=[b
1b2b3]indicates the transmission rate of network nodes affected by external
disturbances;C=[111]shows that the outputy(t)is the total data transmission of
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023
W. Qi and G. Zong,Analysis and Design for Positive Stochastic Jump Systems, Studies
in Systems, Decision and Control 450,https://doi.org/10.1007/978-981-19-5490-0_1
1

2 1 Introduction
Fig. 1.1Communication network model
three network nodes. From (1.1), it can be seen that the total data transmission of
network congestion system should satisfy the special requirements of non-negative
property in line with physical meanings. Such non-negative restrictions about state,
input and output variables make many existing theoretical results for general control
systems no longer applicable to positive systems, such as controllability and reach-
ability, positive realness, design of observer or ﬁlter, etc. Meanwhile, owing to the
non-negativity property of relevant system variables, quite a lot of achievements with
regard to positive systems become more intuitive and novel. Thus, the research on the
analysis and synthesis for positive systems is of important theoretical signiﬁcance
and application value.
On the other hand, most physical systems in the actual operation will be effected
by random parameters, resulting in abrupt changes of structure, which may come
from unexpected complex factors (e.g. sudden changes in the working points or
operating ambient, jumps between subsystems, and component failures). At present,
the aforementioned model can be characterized by Markov jump systems (MJSs).
As one special type of stochastic systems, MJSs consist of a number of subsystems
expressed by differential equations or difference equations and stochastic switching
law. Two kinds of elements are included in MJSs: one is the state of discrete-time
or continuous-time dynamics, and the other is discrete variables taking values in a
ﬁnite set. With the rapid development of economy, science and technology, MJSs
play an important role in the ﬁelds of solar receiver control systems, ﬂight systems,
power systems, economic systems, and network communication systems. The tran-
sition rate, as an important factor, has an inﬂuence on the dynamic characteristics of
MJSs, which is mainly affected by the sojourn time probability density distribution.
Although MJSs can be useful and effective in modeling practical systems to some
extent, they still have some limitations. Because the sojourn time of Markov process
is a random variable following exponential distribution, it may not be suitable in
practice. More generally, the sojourn time may follow some other non-exponential

1.2 Markov Jump Systems 3
probability distributions. In such a case, the corresponding systems are referred to as
semi-Markov jump systems (S-MJSs). When the random factors in the measurement
process happen, the dynamic (1.1) of network congestion system develops into an
idealized model, which is impossible to accurately describe the dynamics subject to
random changes. Consequently, the network congestion system is more reasonable
to be represented by the corresponding positive S-MJSs, as the following model:
˙x(t)=A(g
t)x(t)+B(g t)ω(t),
y(t)=C(g
t)x(t),
x(0)=x
0, (1.2)
where the stochastic process{g
t,t≥0}represents continuous-time and discrete-
state homogeneous semi-Markov process taking values in a ﬁnite set{1,2, ...,M}
subject to the related transition ratePr{g
t+h=j|g t=i}=π ij(h)h+o(h), (iω=
j);Pr{g
t+h=j|g t=i}=1+π ij(h)h+o(h), (i=j),hmeans the sojourn time.
Furthermore, the air trafﬁc ﬂow systems [3], the gene virus variation systems [4], and
the insect structure population systems [5] can also be described as positive S-MJSs
model in the case of sudden changes.
It is worth noting that positive stochastic jump systems can be recognized as a
class of complex hybrid systems, which mainly include positive MJSs and positive
S-MJSs. For positive stochastic jump systems, it is required that all the subsystems
belong to positive systems and the sojourn time follows the corresponding probability
distribution. Considering the existence of multi-mode switching, the stability of each
subsystem cannot ensure the stability of the whole system. Similarly, the stability of
the whole system no longer stands for the stability of each subsystem. For positive
stochastic jump systems, the coexistence and interaction of continuous and discrete
dynamics, complex transition probability, and positivity bring a new challenge for the
analysis and synthesis of such dynamics. From the above analysis, it can be concluded
that positive stochastic jump systems are not a simple combination of positive systems
and stochastic switching process, and there exist many unique problems. At present,
the research on control theory and application for positive stochastic jump systems
is just getting started. Taking external disturbance and transient performance into
account, the research aiming at stability and robustness for positive stochastic jump
systems will be more complicated.
1.2 Markov Jump Systems
As a special class of stochastic systems, MJSs have become an important research
branch in the control ﬁeld due to their extensive applications in aerospace, indus-
trial process, biomedicine, social economy, and other ﬁelds. In 1961, Krasovskii and
Lidskii proposed the conception of MJSs, composed of subsystems described by
several differential or difference equations and stochastic switching law among

4 1 Introduction
subsystems. In the past few decades, many fruitful theoretical results about sta-
bility and stabilization problem have been reported for MJSs [6–17]. The problems
of transient analysis and almost sure stability have been addressed for a class of the
discrete-time MJSs, in which the expectation of the sojourn time and the activation
number of any mode and the switching number between any two modes have been
presented ﬁrstly [6]. The stability has been analyzed in detail for both the discrete-
time and continuous-time linear MJSs with input quantization, and a mode-dependent
input controller has been obtained in terms of linear part and nonlinear part [7]. By the
help of segmentation technology and linear interpolation, a time-scheduled Lyapunov
functional consisting of an exponential-type looped function has been constructed to
derive the mean square exponential stabilization for sampled-data MJSs [8]. On this
basis, a sufﬁcient condition has been constructed such that the disturbance attenua-
tion performance can be ensured for MJSs subject to mismatched input quantisation
and general transition rate [9]. Considering an appropriate stochastic Lyapunov func-
tional, the issue of exponential stability in mean square sense has been investigated
for stochastic MJSs with mixed time-varying delays and partly unknown transi-
tion rates [11]. By resorting to average dwell time switching approach, sufﬁcient
conditions have been proposed to ensure the exponential mean-square stability for
delayed MJSs with generally incomplete transition rates [12]. The stochastic stability
has been considered for a class of neutral-type Markov jump neural networks with
additive time-varying delays [15]. Based on the Lyapunov function and the Markov
inequality, the problem of globally almost surely attractive sets has been solved for a
class of discrete-time MJSs with stochastic disturbances via impulsive control [17].
Moreover, robust control and ﬁlter have become the hotspots in MJSs, and the
relevant scholars have carried out extensive and in-depth research [18–32]. A state
feedback controller has been devised for a class of MJSs with actuator saturation
[18]. Based on the linear matrix inequalities with equality constraints, the problem
of sliding mode control for nonlinear uncertain MJSs has been studied, and suf-
ﬁcient conditions have been derived such that the sliding motions on the speciﬁed
sliding surfaces are stochastically stable withγ-disturbance attenuation level [19]. An
observer-based mode-dependent sliding mode control scheme has been presented for
a class of stochastic MJSs against sensor fault, actuator fault and input disturbances
[20]. For neutral switching MJSs with generally incomplete transition probabilities,
some sufﬁcient conditions have been derived for exponential mean-square stability in
terms of the Lyapunov-Krasovskii functional procedure [21]. Take into account the
advantages of sliding mode control, the stabilization problem has been considered
for MJSs with time-varying actuator faults and partially known transition proba-
bilities [23]. Furthermore, the robustH
∞ﬁltering problem has been discussed for
mode-dependent time-delay discrete Markov jump singular systems with parameter
uncertainties [24]. Considering the problem of full-order mode-dependent robust ﬁl-
ter for MJSs with norm-bounded parameter uncertainties and time-varying delays,
the Lyapunov-Krasovskii functional under delay-partitioning strategy has been con-
structed to analyze the stochastic stability and performance of the resulting error
system [27]. By the analysis of network-induced delay intervals, the discrete-time
system, the event-triggered scheme, and the network-induced delay have been uniﬁed

1.3 Semi-Markov Jump Systems 5
into discrete-time Markov jump ﬁlter error systems with time-delay [28]. A robust
fault detection ﬁlter has been established such that the discrete-time MJSs with
conic-type non-linearities are stochastically stable and satisfy the given performance
against the external disturbances [30]. Based on the event-triggered communication
scheme, theH
∞ﬁltering problem has been discussed for discrete-time MJSs with
repeated scalar nonlinearities [32].
Owing to some complex factors, such as time delay, packet dropout, and distortion,
system mode is generally asynchronous with controller or ﬁlter mode in the practical
MJSs. In response to this situation, many researchers have made multitudinous out-
standing achievements (see e.g., [33–43]). Hidden Markov model has been adopted
to describe the asynchronous quantized controller and the minimal upper bound of
guaranteed cost control performance has been obtained by exploiting T-S fuzzy tech-
nique [33]. The resilientH
∞ﬁltering problem has been addressed for discrete-time
Markov jump neural networks subject to time-varying delays, unideal measurements,
and multiplicative noises [34]. For discrete-time MJSs with mixed time delays, the
problem of dissipativity-based asynchronous control has been investigated [35]. To
reduce the burden of data transmission, an event-based asynchronousH
∞control
law has been constructed for a class of networked MJSs with missing measurements
[36]. By the designed dynamic event-triggered rule, the issue of the asynchronous
passive controller design has been discussed for singular MJSs with general transition
rates under stochastic cyber-attacks [37]. The asynchronous passive control problem
has been studied for MJSs, in which the asynchronization phenomenon between the
system mode and controller mode has been described by a hidden Markov model
[39]. For a class of MJSs with output quantization, disturbance, and actuator fault,
two cases have been considered, one of which assumes the transition rates being
completely unknown, while the other speciﬁes the uncertain transition rate [42]. The
dissipative asynchronous ﬁltering has been developed for a class of T-S fuzzy MJSs
in continuous-time domain [43].
1.3 Semi-Markov Jump Systems
It is noted that for stochastic jump systems, the transition rate is mainly affected by
the probability distribution function of the sojourn time. Owing to the sojourn time
of Markov process following an exponential distribution, the corresponding MJSs
have some limitations. In order to relax this strict constraint, the concept of semi-
Markov process was put forward in 1954 [44]. Recent years have witnessed many
applications of S-MJSs, such as performance analysis of multi-bus systems, modern
communication technology, fault-tolerant and DNA analysis, etc.
The ﬁrst is continuous-time S-MJSs. The stability problem has been studied for a
class of stochastic S-MJSs with the sojourn time subject to phase distribution, whose
main thought is to transform the phase S-MJSs into MJSs by the use of supplementary
vector and model transformation method [45]. For a class of S-MJSs with the sojourn
time subject to Weibull distribution, the feasibility stochastic stability criterion has

6 1 Introduction
been obtained by deriving the weak inﬁnitesimal operator of mode-dependent Lya-
punov function [46]. Based on a semi-Markov Lyapunov-Krasovskii formulation of
scaled small gain condition combined with projection lemma, aH
∞delay-dependent
memory ﬁlter has been devised for continuous-time S-MJSs with time-varying delays
under an input-output framework [47]. The almost surely exponential stability has
been studied for semi-Markov switched stochastic systems with randomly impul-
sive jumps [48]. So far, many researchers pay close attention to the steady-state
characteristics of dynamical systems in an inﬁnite time interval, while there is no
guarantee about transient performance in a speciﬁc time interval. The system with
steady-state performance may have great overshoot in some operation intervals. In
order to address the transient performance problem, the ﬁnite-time control problem
has been increasingly investigated for S-MJSs [49–54]. The asynchronous event-
triggered sliding mode control has been studied for S-MJSs in a ﬁnite-time time so
as to ensure that the dynamics can reach the sliding mode surface within a given
time bound [50]. The stochastic ﬁnite-time stability has been discussed for uncer-
tain nonlinear semi-Markov jump cyber-physical systems against false data injection
attacks and an event-triggered controller has been established under the framework
of the standard linear matrix inequalities [51]. By applying Lyapunov-Krasovskii
functional and matrix inequality technique, sufﬁcient conditions have been proposed
to guarantee ﬁnite-time boundedness,H
∞ﬁnite-time boundedness, and ﬁnite-time
H
∞state feedback stabilization for S-MJSs [52]. Moreover, numerous associated
works about continuous-time S-MJSs can also be found in [55–71]. A reliable ﬁlter
has been designed to ensure the mixed passivity and performance level of S-MJSs in
the presence of sensor failures [58]. To relax the difﬁculty of ﬁnding upper bounds
in sample-and-hold behaviour of T-S fuzzy S-MJSs, a novel mismatched member-
ship function has been presented in [64]. The problems of stability and stabilization
have been discussed for a class of singular S-MJSs [67]. Considering the multi-agent
systems with semi-Markov jump topology, a reduced-order observer-based control
protocol has been proposed, for two cases that the transition rates are completely
known and partially unknown respectively, which ensures that the containment can
be achieved in the mean square sense [68].
Another is discrete-time S-MJSs. The discrete-time semi-Markov kernel has been
introduced, where the probability density function of sojourn time is dependent on
both current and next system mode. Different types of distributions and/or different
parameters in a same type of distribution of sojourn time, depending on the target
mode towards which the system jumps, can coexist in each mode of semi-Markov
kernel [72–85]. The concept ofσ-error mean square stability has been proposed,
whereσis capable of characterizing the degree of approximation error ofσ-error
mean square stability to mean square stability [72]. On this basis, the stability and
stabilization problems have been further studied for a class of fuzzy S-MJSs with
partial information on jump and sojourn parameters, whose main idea is to construct
a Lyapunov function depending on the current system mode and establish the mean
square stability criterion [73]. Taking advantage of discrete-time semi-Markov kernel
theory and ﬁnite sojourn time constrained condition, the fault-tolerant control prob-
lem has been discussed for S-MJSs [74]. The stability analysis and controller design

1.4 Positive Systems 7
have been addressed for a class of nonhomogeneous hidden S-MJSs with limited
information of sojourn time probability density functions [75]. Under an incomplete
semi-Markov kernel, a dynamic output feedback control scheme has been adopted for
a class of discrete-time S-MJSs [76]. Additionally, a technique has been developed
to estimate the permissible maximum value of singularly perturbed parameter for
discrete-time nonlinear semi-Markov jump singularly perturbed systems by resort-
ing to the T-S fuzzy model approach [81]. In terms of multiple-Lyapunov function,
the stability problem has been developed for discrete-time S-MJSs with bounded
sojourn time [84]. An appropriate sliding mode control law has been established
for uncertain discrete-time S-MJSs, in which an F-404 aircraft engine model has
been shown to demonstrate the applicability of the proposed control strategy [85].
It is noted that most of the relevant achievements for discrete-time S-MJSs focus on
stability analysis or controller design. Nevertheless, combining the key properties
of complex factors and discrete-time semi-Markov kernel, there appears to be very
little literature, which encourages the further research.
1.4 Positive Systems
The research of positive systems can be traced back to the late 1970s,s, in which
stability and stabilization are the most fundamental directions. The concept of pos-
itive systems was introduced in 1979 [86], and a great number of scholars have
taken a keen interest in positive systems, promoting the correlation theories into a
new stage of development and achieving a large number of excellent results [87–
102]. Farina and Rinaldi have systematically expatiated the basic theories related to
stability, positive equilibrium, accessibility, observability, and positive realness of
positive systems [87]. Kaczorek has extended the above results to stability analysis
and control synthesis for positive 2D systems, and further expanded the scope of the
study about positive systems [88]. Moreover, the stabilization problem with posi-
tivity has been investigated for discrete-time linear systems with time delay, and a
delay-independent necessary and sufﬁcient condition has been proposed in terms of
linear matrix inequalities for the existence of desired controllers that guarantee the
closed-loop system to be asymptotically stable and positive [89]. It should be noted
that Lyapunov function becomes one of the most powerful research tools in the sta-
bility analysis of positive systems, which mainly covers quadratic Lyapunov function
and linear Lyapunov function. The stability criterion for discrete positive systems
with time delay has been obtained by resorting to the diagonal quadratic Lyapunov
function [92]. Compared with the quadratic Lyapunov function, the linear Lyapunov
function not only makes full use of positive characteristics, but also is convenient
to reduce the computation load effectively. A mixed time-varying delayed impulsive
positive systems model has been establish for the ﬁrst time and a sufﬁcient criterion
of global exponential stability has been given in terms of copositive Lyapunov-
Krasovskii functional and average impulsive interval method [93]. The stability and
robust stability problems have been discussed for switched positive linear systems

8 1 Introduction
with all unstable subsystems [97] and positive systems with generalised disturbances
[98]. By virtue of the linear copositive Lyapunov function, the uniform stability and
uniformly exponential stability conditions have been presented for positive linear
time-varying systems on time scales [101]. The minimal strongly eventually positive
realization, which relaxes the nonnegativity constraint on the state-space model and
only requires that the state trajectory is nonnegative after a certain number of steps,
has been proposed for externally positive systems [102].
At present, there also have been a great many of achievements in ﬁlter and robust
control for positive systems (see e.g., [103–121]). Based on the quadratic Lyapunov
function approach, the reduced-order positive ﬁltering problem has been addressed
for positive discrete-time systems withH
∞performance [103]. For a class of posi-
tive linear systems with distributed delays, the monotonicity, stability, andL
∞-gain
performance have been analyzed [104]. An analytical method to compute the exact
value ofL
1-induced norm has been ﬁrstly presented, and a novel characterization
has been proposed for stability andL
1-induced performance [105]. The problems
of stability and positive observation have been studied for positive two-dimensional
discrete-time systems with multiple delays in Roesser model [109]. On this basis,
the design problem ofL
1-induced positive observer has been developed for positive
T-S fuzzy systems [110]. On the foundation of linear copositive Lyapunov function,
new approaches about positive observer design have been provided for discrete-time
positive linear systems [114]. For nonlinear discrete-time T-S positive systems, the
stability has been guaranteed by synthesizing a linear co-positive Lyapunov function
and applying the parallel distributed compensation controller [117]. The distributed
model predictive control problem has been investigated for positive systems with
interval and polytopic uncertainties [121]. However, the research on the control the-
ory and application for positive systems is only the beginning, and there still has a
long way to go.
1.5 Positive Stochastic Jump Systems
Along with the in-depth study of control theory about stochastic jump systems, the
research of positive stochastic jump systems has attracted the attention of experts.
Positive systems subject to a stochastic Markov process are investigated [122–129].
The relationship between several kinds of stability for continuous-time positive MJSs
has been analyzed, and necessary and sufﬁcient conditions for stochastic stability
have been given in the form of linear programming [122]. The stochastic stability
and stabilization have been analyzed for positive systems with Markov jump param-
eters in both continuous-time and discrete-time contexts [123]. For a class of positive
MJSs, sufﬁcient conditions have been obtained for the existence of state feedback
controller [126], stochastic stability, andL
∞-gain performance [127]. By the use of
appropriate co-positive type Lyapunov function, theL
1control problem has been
handled for positive MJSs with time-varying delays and partly known transition rates
[128]. A positive full-order ﬁlter has been proposed such that the positive continuous-

1.5 Positive Stochastic Jump Systems 9
time MJSs with partly known transition rates could achieve positivity and stochastic
stability withL
1-gain performance [129]. It is worth mentioning that positive MJSs
still have a vast space to develop although there exist many achievements [130–145].
For example, the stochastic stability problem has been addressed for positive sin-
gular MJSs with mode-dependent derivative-term coefﬁcient, and the condition on
stochastic stability of positive singular MJSs associated with state jumps behavior at
switching instants has been given by means of linear co-positive Lyapunuov function
[134]. For positive systems with piecewise-homogeneous Markov chain, the deﬁni-
tions of positivity and mean stabilization have been introduced, and some sufﬁcient
conditions have been derived to ensure that the considered system is positive and
mean stable, in which the Markov chain follows the conditional probability distribu-
tion and the Bernoulli distribution, respectively [141]. By using the T-S fuzzy model
strategy, an appropriate controller has been obtained to depend on the observation
mode which makes the closed-loop fuzzy hidden MJSs positive and stochastically
ﬁnite-time bounded with a givenL
2performance index [145].
On the other hand, positive systems subject to stochastic semi-Markov process
are investigated in [146–153]. By applying linear Lyapunov-Krasovskii functional
depending on the bound of time-varying delays, the stability analysis and control
synthesis have been addressed for positive S-MJSs with time-varying delays, in
which the stochastic semi-Markov process is related to non-exponential distribution
[146]. A concept of hybrid gain performance has been presented to guarantee that
the designed random event-triggered ﬁlter can attenuate the affect from the distur-
bance and sensor faults [147]. Considering S-MJSs subject to positive constraint,
the problems of mean stability analysis and control synthesis have been studied
under a linear programming framework, and the theoretical ﬁndings have been illus-
trated by the virus mutation treatment model [148]. For delayed positive S-MJSs,
some necessary and sufﬁcient conditions for state-feedback controller satisfyingL
∞
boundedness and positivity have been derived in standard linear programming [149].
The problems of stochastic stability andL
1-gain analysis have been considered for
continuous-time positive fuzzy S-MJSs with time-varying delays, and the application
of Lotka-Volterra population model has been adopted for the validity of the results
[151]. A developed ﬁnite-time feedback controller design method has been proposed
to reduce some constraints of input matrices, which guarantees that the closed-loop
positive S-MJSs achieves positivity, ﬁnite-time boundedness, and has a prescribed
noise attenuation performance index [152]. With a quick glimpse into the literature,
it can be found that positive S-MJSs still have great theoretical research space and
practical applied value, which is worth deepening.
As can be seen in the above illustrations, many control issues of positive systems
and stochastic jump systems have been noticed and developed in the past few years,
some of which, however, have not been successfully solved so far. For example,
sufﬁcient and necessary criterion for positive stochastic jump systems is still an
open problem in time-delay contexts. Also, it is urgent to carry out investigations on
more complicated positive stochastic jump models for practical applications, such
as ﬁnite-time control, nonlinear plant, fault detection, etc.

10 1 Introduction
1.6 Organization of the Book
This book studies the analysis and design for positive stochastic jump systems. Struc-
ture of the book is summarized as follows.
This chapter has introduced the system description and some background knowl-
edge, and also addressed the motivations of the book.
Chapter2investigates the problems of exponential stability andL
1-gain analysis
for positive time-delay Markov jump systems with switching transition rates. By
using a linear co-positive Lyapunov function and average dwell time approach, a
novel exponential stability criterion is obtained for the corresponding system. Based
on this, sufﬁcient conditions for the system withL
1-gain performance are established
in terms of linear programming.
Chapter3deals with the problems of mean stability and control synthesis for pos-
itive semi-Markov jump systems. The stochastic process is subject to semi-Markov
process. By constructing linear stochastic semi-Markov Lyapunov function, neces-
sary and sufﬁcient conditions are proposed for the existence of feedback controller.
Chapter4considers the problems of stochastic stability,L
1-gain and control
synthesis for positive semi-Markov jump systems. For the control input matrix, three
cases are discussed as follows: (i) All the elements ofB
iare greater than or equal
to 0; (ii) The elements ofB
iappear greater than 0 or equal to and less than 0 or equal
to simultaneously; (iii) All the elements ofB
iare less than or equal to 0. An efﬁcient
condition has been given to construct the feedback controller under the semi-Markov
jump signal with linear co-positive Lyapunov function.
In Chap.5, we address the robust ﬁnite-time stabilization for positive semi-Markov
jump systems with delay and external disturbance. By using the gain matrix decom-
position method and linear co-positive Lyapunov function, ﬁnite-time feedback con-
troller is designed to ensure positivity and ﬁnite-time boundedness with a prescribed
L
1-gain performance index for the resulting closed-loop system.
The problem ofL
1control for positive delayed semi-Markov jump systems is
investigated in Chap.6. First, a novel criterion is proposed to achieve positivity
and stochastic stability withL
1-gain performance. Then, three different cases as
B
α≥≥0,{B α}mn≥≥0 and{B α}mn≤≤0 simultaneously, andB α≤≤0are
considered for the developed feedback controller.
Chapter7addresses the issue ofL
∞control for positive delay semi-Markov jump
systems. By introducing an equivalent deterministic linear system with multiple time
delays, sufﬁcient and necessary conditions are developed for positivity,L
∞bound-
edness criteria, and control design in terms of linear programming. The proposed
method is helpful in communication network model.
Chapter8investigates the problem of the robust ﬁnite-time stabilization for pos-
itive semi-Markov jump systems with delay and external disturbance. By the use of
stochastic semi-Markov Lyapunov-Krasovskii functional, ﬁnite-time boundedness
criteria are presented for positive delayed S-MJSs with a prescribedL
1noise attenu-
ation performance index. Furthermore, a feedback controller is designed to depend on

References 11
the gain matrix decomposition method and the bounds of the time-varying transition
rate matrix.
Chapter9focuses on the fault detection problem for positive semi-Markov jump
systems with time-varying delays. By transforming the fault detection problem into
positive ﬁlter design problem, the desiredL
1fault detection ﬁlter is designed in the
strict linear programming framework. The proposed method improves rapidity and
accuracy on fault estimation for positive systems.
Chapter10considers the problems of stochastic stability andL
1-gain analysis
for positive fuzzy semi-Markov jump systems with time delay. By introducing linear
Lyapunov function and T-S fuzzy model approach, stochastic stability criteria with
L
1boundedness are proposed in linear programming.
Chapter11deals with the problem of positiveL
1observer design for positive
fuzzy semi-Markov jump systems. By applying the fuzzy-dependent Lyapunov func-
tion, stochastic stability criteria are given for positive nonlinear S-MJSs with a pre-
scribedL
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is designed such that the resulting closed-loop augmented system is positive and
stochastically stable withL
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The problem of positiveL
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semi-Markov jump systems is studied in Chap.12. First, phase-type semi-Markov
jump systems are transformed into Markov jump systems through the supplemen-
tary variable and the plant transformation technique. Then, the associated nonlinear
Markov jump systems are transformed into the local linear Markov jump systems
with speciﬁc T-S fuzzy rules. A positiveL
1ﬁlter is designed such that the corre-
sponding system is stochastically stable withL
1performance.
Finally, in Chap.13, the perspectives of analysis and synthesis for positive stochas-
tic jump systems are concluded and predicated.
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14 1 Introduction
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Part I
Positive Delayed Markov Jump Systems

Chapter 2
Exponential Stability andL1-Gain
Analysis
In this chapter, the problems of exponential stability andL 1-gain analysis for pos-
itive time-delay Markov jump systems (MJSs) with switching transition rates are
addressed. Another set of useful regime-switching model that is an extension of
ﬁxed transition rate to time-varying transition rate has been given. By resorting to
the multiple linear co-positive Lyapunov function and average dwell time, sufﬁcient
conditions for exponential stability are proposed in standard linear programming.
Based on the obtained results,L
1-gain performance is analyzed. Finally, an exam-
ple illustrates the validity of the main results.
2.1 Introduction
In practical systems, there exists a special class of dynamic systems whose state vari-
ables and output signals remain nonnegative whenever both the initial condition and
the input signal are nonnegative. This unordinary kind of systems, usually denoted
as positive systems [1,2], also known as nonnegative systems, have extensive appli-
cations in communication networks [3] and industrial engineering [4]. This special
category of systems has received ever-increasing research interest in the past decade
and many interesting properties have been unraveled (see e.g., [5–15]).
On the other hand, as a special class of hybrid systems, MJSs have some advan-
tages of describing dynamic systems subject to sudden changes between subsystems
caused by external causes, such as networked control systems, manufacturing sys-
tems, fault-detection systems, and system theory (see e.g., [16–22]). However, many
results about positive MJSs [23–31] have been obtained with completely transition
rates. In fact, different from those uncertain or partly known transition rates, an exten-
sion of transition rate to Markov jump model with time-varying transition rates has
provided another set of useful regime-switching model. To motivate interest for such
systems, one example can be referred to econometric model. In such model, MJSs
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023
W. Qi and G. Zong,Analysis and Design for Positive Stochastic Jump Systems, Studies
in Systems, Decision and Control 450,https://doi.org/10.1007/978-981-19-5490-0_2
21

22 2 Exponential Stability and L1-Gain Analysis
with switching transition rates have advantages over that with ﬁxed transition rates
in terms of ﬂexibility [32]. Recent papers [32,33] in switching transition rates have
been published.
This chapter will study the issues of exponential stability andL
1-gain analysis
for positive MJSs with time-varying delays and switching transition rates. Com-
pared with [34] ﬁrstly, the switching law followed asynchronous switching law with
mode-dependent average dwell time while the switching law in this chapter will
follow the Markov process with time-varying property. Second, the works [23–31]
mainly investigated the positive MJSs with time-invariant transition rates while the
transition rates in this chapter are time-varying. Third, the works [23–28] mainly
investigated the positive MJSs without time delays while the model in this chapter
includes time delays. The problems of exponential stability andL
1-gain analysis for
positive MJSs with time-varying delays and switching transition rates are theoreti-
cally challenging, difﬁcult, and open all the time. First, it is very difﬁcult to handle
positive time-delay MJSs with time-varying transition rates which are different from
positive time-delay MJSs with time-invariant transition rates. Second, one needs to
consider not only stability of dynamic systems governed by the Markov process,
but also the constrained positivity. Third, the dynamical behaviors are affected by
the interaction among the Markov process, the time delay, the switching transition
rates, the disturbance input, and the positivity, which are very complicated. These
above aspects motivate our current research work. The main contributions of this
chapter are given as follows: (i) By resorting to multiple linear co-positive Lyapunov
function and average dwell time, sufﬁcient conditions for exponential stability are
given; (ii) The disturbance attenuation performance (theL
1-gain performance) is
analyzed for positive time-delay Markov jump systems with switching transition
rates; (iii) Additionally, in the presence of the quantity of virus mutation treatment
model satisfying the nonnegative property and abrupt change of genetic mutation,
modeling virus mutation treatment model as positive MJSs shows the validity of the
main algorithms.
2.2 Problem Statements and Preliminaries
Consider the positive time-delay MJSs as follows:
˙x(t)=A(g
t)x(t)+A d(gt)x(t−d(t))+G(g t)w(t),
z(t)=C(g
t)x(t)+D(g t)w(t),
x(t
0+θ)=ϕ(θ),∀θ∈[−d,0], (2.1)
wherex(t)∈R
n
,w(t)∈R
l
, andz(t)∈R
q
are, respectively, the state vector, dis-
turbance input, and estimated output;ϕ(θ)is a vector-valued initial continuous
function;g
tstands for a time-homogeneous Markov process and takes values in
S
1={1,2,...,N}with transition rate as:

2.2 Problem Statements and Preliminaries 23
P{gt+Δt=j|g t=i}=
∀
π
(rt)
ij
Δt+o(Δt),iΔ=j,
1+π
(rt)
ij
Δt+o(Δt),i=j,
(2.2)
whereΔt≥0, lim
Δt→0
(o(Δt)/Δt)=0 andπ
(rt)
ij
≥0, foriΔ=jand
∈
N
j=1,iΔ=j
π
(rt)
ij
=
−π
(rt)
ii
.
The time-varying functiond(t)satisﬁes
0<d(t)≤d,˙d(t)≤h,
wheredandhare known real constant scalars.
In addition, the process{g
t,t≥0}describes the time-varying characters of the
modes of system (2.1). Byr
t=α, we can see that the transition rates are time-
varying. Moreover,r
t∈S2means thatr tis a piecewise constant function of time,
which takes values in a ﬁnite setS
2={1,2,...,M}. For simplicity, forg t=i∈S 1,
A(g
t),Ad(gt),G(g t),C(g t), andD(g t)are, respectively, denoted asA i,Adi,Gi,Ci,
andD
i.
Remark 2.1In [23–31], the transition rates are assumed to be time-invariant. Actu-
ally, the assumption is not true in some applications and the resulting transition rates
may vary throughout the running time of the modeled control system. In such case,
the time-varying transition rates are more suitable to describe positive MJSs. In this
chapter, we deﬁner
t=αthat the transition rates are time-varying.
Remark 2.2System (2.1) with switching transition rates (2.2) obeys both the
stochastic jump process and deterministic switching signal. It should be noted that
whenr
t=1, MJSs (2.1) reduce to a homogeneous MJSs [23–31]. Thus, the homo-
geneous MJSs can be regarded as a special case of MJSs with time-varying transition
rates which are more general and complex than homogeneous MJSs.
Deﬁnition 2.1([1,2]) System (2.1) is said to be positive if, for any initial condition
ϕ(θ)≥≥0, there holdx(t)≥≥0 andz(t)≥≥0,t>0.
Lemma 2.1([1,2])Consider the following positive system:
˙x(t)=Ax(t)+A
dx(t−d(t))+Gw(t),
z(t)=Cx(t)+Dw(t),
x(t
0+θ)=ϕ(θ),∀θ∈[−d,0]. (2.3)
System (2.3) is positive if and only if A is Metzler matrix, A
d≥≥0,G≥≥0,C≥≥0,
and D≥≥0.
Deﬁnition 2.2([15]) System (2.1) is said to be exponentially stable if the following
inequality holds
E{γx(t)γ
1}≤˜αe
−˜β(t−t 0)
γx(t0)γ1,∀t≥t 0, (2.4)

24 2 Exponential Stability and L1-Gain Analysis
for constants˜α>0,˜β>0, whereγx(t 0)γ1=sup−d≤θ≤0 γϕ(θ)γ 1.
Deﬁnition 2.3([15]) For a switching signalr
tand anyt>t 0,letN r(t0,t)be the
switching numbers ofr
tover the interval(t 0,t).ForsomeT a>0 andN 0≥0, if
the inequalityN
r(t0,t)≤N 0+
t−t0
Ta
holds, then the positive constantT ais called an
average dwell time andN
0is called a chatting bound.
For convenience, we chooseN
0=0 in this chapter.
Deﬁnition 2.4([15]) Forγ>0, system (2.1) is said to have a weightedL
1-gain,
if the following conditions hold
(i) System (2.1) is exponentially stable whenw(t)=0;
(ii) The following inequality holds
E
Δ≥
∞
t
0
e
−λ(t−t 0)
||z(t)|| 1dt
→
≤γE
Δ≥
∞
t
0
||w(t)|| 1dt
→
,
whenϕ(θ)=0 andw(t)Δ=0.
In this section, sufﬁcient conditions are proposed such that the positive system
(2.1) with switching transition rates is exponentially stable withL
1-gain perfor-
mance.
2.3 Exponential Stability Analysis
In this part, we will consider the problem of exponential stability analysis for system
(2.1).
Theorem 2.1Letλ>0andμ>1. System (2.1) is exponentially stable, if there
exist a set of vectorsν
α,i,σ1α,i,σ2α,i,σ1α,σ2α,ρα,i∈R
n
+
, such that the following
inequalities hold:
A
T
i
να,i+λνα,i+σ1α,i+σ2α,i+dσ 1α+dσ 2α+ρα,i+
∈
N
j=1
π
α
ij
να,j<<0,
(2.5)
A
T
di
να,i−(1−h)e
−λd
σ1α,i−ρα,i<<0, (2.6)
∈
N
j=1
π
α
ij
σ1α,j≤≤σ 1α,
∈
N
j=1
π
α
ij
σ2α,j≤≤σ 2α, (2.7)
ν
α,i≤≤μν β,j,σ1α,i≤≤μσ 1β,j,σ2α,i≤≤μσ 2β,j,σ1α≤≤μσ 1β,σ2α≤≤μσ 2β,
∀α, β∈S
2,i,j∈S 1,αΔ=β, (2.8)
with the average dwell time satisfying
T
a>T
∗
a
=
lnμ
λ
. (2.9)

2.3 Exponential Stability Analysis 25
ProofThe mode-dependent integral terms in Lyapunov-Krasovskii functional could
provide more freedom and overcome the conservativeness of mode-independent
Lyapunov-Krasovskii functional. For system (2.1), choose the linear co-positive Lya-
punov function candidate as
V(x(t),t,α,i)=x
T
(t)να,i+
≥
t
t−d(t)
e
λ(s−t)
x
T
(s)σ1α,ids+
≥
t
t−d
x
T
(s)e
λ(s−t)
σ2α,ids
+
≥
0
−d
≥
t
t+θ
e
λ(s−t)
x
T
(s)(σ1α+σ2α)dsdθ, (2.10)
whereν
α,i,σ1α,i,σ2α,i,σ1α,σ2α∈R
n
+
and
∈
N
j=1
π
α
ij
σ1α,j≤≤σ 1α,
∈
N
j=1
π
α
ij
σ2α,j≤≤σ 2α.
Along the trajectory of system (2.1), one has
ΓV(x(t),t,α,i)
=x
T
(t)[A
T
i
να,i+λνα,i+σ1α,i+σ2α,i+dσ 1α+dσ 2α+
∈
N
j=1
π
α
ij
να,j]
+x
T
(t−d(t))A
T
di
να,i−(1−˙d(t))e
−λd(t)
x
T
(t−d(t))σ 1α,i
+
∈
N
j=1
π
α
ij
≥
t
t−d(t)
e
λ(s−t)
x
T
(s)σ1α,jds−
≥
t
t−d
e
λ(s−t)
x
T
(s)(σ1α+σ2α)ds
−e
−λd
x
T
(t−d)σ 2α,i+
∈
N
j=1
π
α
ij
≥
t
t−d
e
λ(s−t)
x
T
(s)σ2α,jds−λV(x(t),t,α,i),
whereΓis the weak inﬁnitesimal operator.
Using Leibniz-Newton formula, one can obtain
≥
t
t−d(t)
˙x(s)ds=x(t)−x(t−d(t)), (2.11)
and
≥
t
t−d(t)
˙x(s)ds=
≥
t
t−d(t)
(Aix(s)+A dix(s−d(s)))ds. (2.12)
Then, for any vectorsρ
α,i∈R
n
+
, according to the inequalities (2.11) and (2.12),
we obtain
[x(t)−x(t−d(t))−
≥
t
t−d(t)
(Aix(s)+A dix(s−d(s)))ds]
T
ρα,i=0.(2.13)

26 2 Exponential Stability and L1-Gain Analysis
According to−
≤
t
t−d
e
λ(s−t)
x
T
(s)σ1αds≤−
≤
t
t−d(t)
e
λ(s−t)
x
T
(s)σ1αds,wehave
ΓV(x(t),t,α,i)
≤x
T
(t)[A
T
i
να,i+λνα,i+σ1α,i+σ2α,i+dσ 1α+dσ 2α+ρα,i+
∈
N
j=1
π
α
ij
να,j]
+x
T
(t−d(t))[A
T
di
να,i−(1−h)e
−λd
σ1α,i−ρα,i]
−e
−λd
x
T
(t−d)σ 2α,i−[
≥
t
t−d(t)
(Aix(s)+A dix(s−d(s)))ds]
T
ρα,i
−λV(x(t),t,α,i). (2.14)
From the inequalities (2.5)–(2.7), one obtains
ΓV(x(t),t,α,i)≤−λV(x(t),t,α,i). (2.15)
Multiplying (2.15)bye
−λt
and integrating both sides of (2.15) fromt ktotyield
V(x(t),t,r
t,gt)≤e
−λ(t−t k)
V(x(t k),tk,rtk
,gtk
). (2.16)
Combining conditions (2.8) and (2.16) leads to
V(x(t),t,r
t,gt)≤μe
−λ(t−t k)
V(x(t
−
k
),t
−
k
,r
t
−
k,g
t
−
k). (2.17)
Therefore, it follows from (2.17) and makes the recursion fromt
k−1totk,tk−2to
t
k−1,···, untilt 0, ﬁnally, we can get
V(x(t),t,r
t,gt)≤e
−(λ−
lnμ
Ta
)(t−t 0)
V(x0,t0,r0,g0). (2.18)
Considering the deﬁnition ofV(x(t),t,α,i), and denoting
ε
1=min(α,i,p)∈S 2×S1×S3
{νp,α,i},ε2=max(α,i,p)∈S 2×S1×S3
{νp,α,i},
ε
3=max(α,i,p)∈S 2×S1×S3
{σp,1α,i},ε4=max(α,i,p)∈S 2×S1×S3
{σp,2α,i},
ε
5=max(α,p)∈S 2×S3
{σp,1α},ε6=max(α,p)∈S 2×S3
{σp,2α},S3={1,2,···,n}
yield
V(x(t),t,r
t,gt)≥ε 1γx(t)γ 1,
V(x(t),t,r
t,gt)
≤
2γx(t0)γ1+3
≥
t
t
0−d(t)
e
λ(s−t 0)
γx(s)γ 1ds+ 4
≥
0
−d
≥
t0
t0+θ
e
λ(s−t 0)
γx(s)γ 1ds
+
5
≥
t0
t0−d
e
λ(s−t 0)
γx(s)γ 1ds+ 6
≥
0
−d
≥
t0
t0+θ
e
λ(s−t 0)
γx(s)γ 1ds. (2.19)

2.4L1-gain Performance Analysis 27
Combining (2.18)–(2.19)givesriseto
γx(t)γ
1≤˜αe
−˜β(t−t 0)
γx(t0)γ1, (2.20)
where
˜α=

2+3de
−λd
+
1
2
4d
2
e
−λd
+5de
−λd
+
1
2
6d
2
e
−λd
1
,˜β=(λ−
lnμ
Ta
).
Finally, if the inequalities (2.5)–(2.8) hold, then system (2.1) is exponentially
stable for switching signal with average dwell time (2.9). ∀
Remark 2.3The constraints
∈
N
j=1
π
α
ij
σ1α,j≤≤σ 1αand
∈
N
j=1
π
α
ij
σ2α,j≤≤σ 2α
can be helpful to eliminate the following integral terms
∈
N
j=1
π
α
ij
≥
t
t−d(t)
e
λ(s−t)
x
T
(s)
σ
1α,jds−
≤
t
t−d
e
λ(s−t)
x
T
(s)(σ1α+σ2α)ds,
∈
N
j=1
π
α
ij
≥
t
t−d
e
λ(s−t)
x
T
(s)σ2α,jdsin the
proof of Theorem2.1.
Remark 2.4Generally speaking, for MJSs with time-varying delays, the
Lyapunov function is frequently chosen asV(x(t),t,α,i)=x
T
(t)να,i+
≤
t
t−d(t)
e
λ(s−t)
x
T
(s)σ1αds+
≤
0
−d
≤
t
t+θ
e
λ(s−t)
x
T
(s)σ1αdsdθ, The parameter in integral
term of equality is independent of Markov process, which may lead to some conserva-
tiveness. Here, an appropriate mode-dependent co-positive type Lyapunov function
(2.10) is constructed, and parameter in integral term is dependent of Markov process,
which may reduce some conservativeness.
Remark 2.5In Theorem2.1, due to the existence of switching signal, the Markov
processes are piecewise integrated. Moreover, from conditions (2.17) and (2.18), it
is clear that the Lyapunov function is allowed to increase not only in the same mode,
but also at the switching instants, which may reduce some conservativeness.
2.4L 1-gain Performance Analysis
In this part, we will consider the problem ofL 1-gain performance analysis for system
(2.1).
Theorem 2.2Letλ>0andμ>1. System (2.1) is exponentially stable withL
1-
gain performance, if there exist a set of vectorsν
α,i,σ1α,i,σ2α,i,σ1α,σ2α,ρα,i∈R
n
+
,
such that the inequalities (2.6)–(2.8) and the following inequalities hold

28 2 Exponential Stability and L1-Gain Analysis
A
T
i
να,i+λνα,i+σ1α,i+σ2α,i+dσ 1α+dσ 2α+ρα,i+
∈
N
j=1
π
α
ij
να,j
+C
T
i
1<<0, (2.21)
G
T
i
να,i+D
T
i
1−γ1<<0, (2.22)
with the average dwell time satisfying (2.9).
ProofAccording to (2.21), we can get (2.5), which means that system (2.1)
(w(t)=0) is exponentially stable.
Whenw(t)Δ=0, considering Lyapunov function (2.10)givesriseto
ΓV(x(t),t,α,i)+γz(t)γ 1−γγw(t)γ 1
≤x
T
(t)[A
T
i
να,i+λνα,i+σ1α,i+σ2α,i+dσ1α+dσ2α+ρα,i++C
T
i
1+
∈
N
j=1
π
α
ij
να,j]
+x
T
(t−d(t))[A
T
di
να,i−(1−h)e
−λd
σ1α,i−ρα,i]−e
−λd
x
T
(t−d)σ 2α,i
−[
≥
t
t−d(t)
(Aix(s)+A dix(s−d(s))+G iw(s))ds]
T
ρα,i
+w
T
(t)(G
T
i
να,i+D
T
i
1−γ1)−λV(x(t),t,α,i). (2.23)
Fromt∈[t
k,tk+1), the inequalities (2.6)–(2.8) and (2.21)–(2.22), we have
V(x(t),t,r
t,gt)≤e
−λ(t−t k)
V(x(t k),tk,rtk
,gtk
)−E{
≥
t
t
k
e
−λ(t−s)
Π(s)ds},(2.24)
whereΠ(s)=||z(s)||
1−γ||w(s)|| 1.
Combining (2.8) leads to
E{V(x(t),t,r
t,gt)}≤μe
−λ(t−t k)
E{V{(x(t
−
k
),t
−
k
,r
t
−
k,g
t
−
k)}}
−E{
≥
t
t
k
e
−λ(t−s)
Π(s)ds}
≤μ
k
e
−λ(t−t 0)
V{(x 0,t0,r0,g0)}−μ
k
E{
≥
t1
t0
e
−λ(t−s)
Π(s)ds}
−μ
k−1
E{
≥
t2
t1
e
−λ(t−s)
Π(s)ds}−···−μ
0
E{
≥
t
t
k
e
−λ(t−s)
Π(s)ds}.(2.25)
Under zero initial condition, we get
−E{
≥
t
t
k
e
−λ(t−s)+N σ(s,t)lnμ
Π(s)ds}≥0. (2.26)
Multiplying both sides of (2.26)byN
σ(t0,t)lnμyields

2.5 Simulation 29
E{
≥
t
t
0
e
−λ(t−s)−N σ(t0,s)lnμ
||z(s)|| 1ds}
≤E{γ
≥
t
t
0
e
−λ(t−s)−N σ(t0,s)lnμ
||w(s)|| 1ds}. (2.27)
Noticing thatN
σ(t0,s)≤(s−t 0)/Taand (2.9), we haveN σ(t0,s)lnμ≤
λ(s−t
0), which means that
E{
≥
t
t
0
e
−λ(t−s)−λ(s−t 0)
||z(s)|| 1ds}≤E{γ
≥
t
t
0
e
−λ(t−s)
||w(s)|| 1ds}. (2.28)
Integrating both sides of (2.28) fromt
0to∞leads toE
γ
≤
∞
t
0
e
−λ(t−t 0)
||z(t)|| 1dt

≤
γE
γ
≤
∞
t
0
||w(t)|| 1dt

. ∀
Remark 2.6For given scalarλ>0, we can takeγas the optimized variable to
obtain the minimumL
1-gain value that conditions (2.6)–(2.8) and (2.21)–(2.22)
hold. Then, the optimization problem can be described as follows:
min γ
ν
α,i,σ1α,i,σ2α,i,σ1α,σ2α,ρα,i∈R
n
+
,λ>0,μ>1
s.t.Inequalities(2.6)−(2.8)and(2.21)−(2.22).
(2.29)
Remark 2.7Although the Lyapunov function is complex and the computational
complexity is increased, sufﬁcient conditions in the form of linear programming
have been proposed to reduce the conservativeness in optimization problem (2.29).
The example about mathematical model of virus mutation treatment demonstrates
the validity of the main results.
Remark 2.8The model parametric uncertainties are not considered in this chapter.
As the proposed conditions are based on linear programming, they can be easily
extended to positive time-delay MJSs with model parametric uncertainties.
2.5 Simulation
Consider the mathematical model of virus mutation treatment from [14]:
˙x(t)=(R
i−δI+ζM)x(t)+G iw(t),z(t)=C ix(t)+D iw(t), (2.30)
wherex(t)indicates two different viral genotypes;w(t)andz(t)are the distur-
bance input and the estimated output;i=1,2 means a Markov process;ζ,δ, and
M=[M
mn]represent the mutation rate, the death or decay rate, and the system matri-
ces;M
mn∈{0,1}means the genetic connections between genotypes. The parameter

30 2 Exponential Stability and L1-Gain Analysis
values are given as follows:
R
1=

0.05 0
00.25

,R 2=

0.06 0
00.26

,M=

01
10

,G 1=

0.5
0.1

,G 2=

0.3
0.2

,
C
1=

1.00

,C 2=

1.00

,D 1=

0.1

,D 2=

0.2

,δ=0.68,ζ=0.001.
We assume that there exists the time-varying delays in the mathematical model
of virus mutation treatment. Then, the other parameters are given as follows:
A
d1=

0.20.1
0.10.2

,A d2=

0.20.3
0.10.1

,d(t)=1.2(1−cos(t)),d=2.4,h=1.2.
The transition rate matrices are given as follows:
1
ε
=

−1.21.2
2−2

,
2
ε
=

−0.80.8
0.5−0.5

.
Letλ=0.1019,w(t)=e
−t
(2−sin(t)). Solving optimization problem (2.29)
results in the corresponding parameters as follows:
ν
11=

108.6027
196.1369

,ν 12=

109.3913
197.7968

,ν 21=

106.7682
199.7497

,ν 22=

106.3332
201.2709

,
σ
111=

3.1493
2.7121

,σ 112=

3.3044
2.8499

,σ 121=

3.2993
2.8751

,σ 122=

3.0365
2.7208

,
σ
211=

3.6490
3.1617

,σ 212=

3.8099
3.3031

,σ 221=

3.8104
3.3396

,σ 222=

3.5217
3.1609

,
σ
11=

1.0440
0.8533

,σ 12=

0.9331
0.8878

,σ 21=

1.0450
0.8539

,σ 22=

0.9340
0.8883

,
ρ
11=

43.0631
51.3967

,ρ 12=

44.0012
54.4613

,ρ 21=

43.3435
52.4069

,ρ 22=

42.7013
53.4937

,
μ=1.4902,γ=0.2018.
Then, according to (2.9), we can getT
∗
a
=3.9147 and choose the average dwell
timeT
a=4.
Figures2.1,2.2,2.3and2.4stand for switching signalr
t, jumping modeg t,
state trajectoryx(t)with initial conditionsx(0)=

0.20.1

T
,x(0)=

0.30.4

T
,
x(0)=

0.50.6

T
, and estimated outputz(t)with initial conditionx(0)=

0.30.4

T
.
These ﬁgures demonstrate that our results are effective.

2.5 Simulation 31
0510152025303540
00.511.522.53
Time(s)
Switching signal
Fig. 2.1Switching signal
0510152025303540
00.511.522.53
Time(s)
Jumping mode
Fig. 2.2Jumping mode

32 2 Exponential Stability and L1-Gain Analysis
00.10.20.30.40.50.60.70.8
00.10.20.30.40.50.60.7
Time(s)
State trajectory
Fig. 2.3State trajectory
0510152025303540
00.050.10.150.20.250.30.350.40.45
Time(s)
z(t)
Fig. 2.4Estimated output

References 33
Remark 2.9If Lyapunov function is chosen asV(x(t),t,α,i)=x
T
(t)να,i+
≤
t
t−d(t)
e
λ(s−t)
x
T
(s)σ1αds+
≤
0
−d
≤
t
t+θ
e
λ(s−t)
x
T
(s)σ1αdsdθ, we can getγ=1.3652, which
means that sufﬁcient conditions in this chapter can effectively suppress the distur-
bance and reduce some conservativeness. Although the computational complexity
has increased, the smaller parameterγcan be obtained.
2.6 Conclusion
This chapter has concerned with exponential stability andL 1-gain analysis for pos-
itive time-delay MJSs with switching transition rates subject to average dwell time
approach. By using a linear co-positive Lyapunov function, sufﬁcient conditions for
exponential stability of the system withL
1-gain performance are proposed. As the
proposed conditions are based on linear programming, they can be easily extended
to other problems for the underlying systems, such as 2D systems, MJSs with model
parametric uncertainties, fuzzy control, fault detection ﬁlter design, etc.
References
1. Farina, L., Rinaldi, S.: Positive Linear Systems: Theory and Applications. Wiley, New York
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Part II
Positive Semi-Markov Jump Systems

Chapter 3
Stability and Stabilization
In this chapter, the problems of mean stability analysis and control synthesis are
studied for stochastic jump systems subject to positive constraint. Such a switching
is governed by a semi-Markov process subject to a special non-exponential distri-
bution. Considering a linear Lyapunov-Krasovskii function, necessary and sufﬁcient
conditions are proposed to realize mean stability for the open-loop system. Based
on this, the solvability conditions for the desired stabilizing controller can be deter-
mined under a linear programming framework. Finally, the theoretical ﬁndings are
illustrated by the virus mutation treatment model.
3.1 Introduction
In the previous chapter, the positive MJSs have been investigated. However, there
exists one obvious limitation for MJSs, that is, the Markov process demands that
the sojourn time is exponentially distributed [1,2]. This strict restriction means
that the transition rate matrix is constant and many results obtained for MJSs are
intrinsically conservative. Owing to the relaxed conditions on the probability distri-
bution of exponential distribution, the sojourn time in semi-Markov jump systems
(S-MJSs) obeys a more general non-exponential distribution, which leads to the time-
varying characteristic of the transition rate matrix and brings both challenges and
chances to analysis and synthesis of dynamics. It should be pointed out that S-MJSs
cover the traditional MJSs as a special case, which can better model the dynamical
systems subject to inevitable stochastic changes, such as bunch-train cavity interac-
tion, credit risk assessment, and other aspects. This special category of systems has
received extensive research interest (see e.g., [3–20]). Considering the case in which
all the subsystems of S-MJSs belong to positive systems [21–28], many previous
approaches for general systems cannot be extended to positive systems due to the
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023
W. Qi and G. Zong,Analysis and Design for Positive Stochastic Jump Systems, Studies
in Systems, Decision and Control 450,https://doi.org/10.1007/978-981-19-5490-0_3
37

38 3 Stability and Stabilization
particularity of such systems, which makes analysis and synthesis of positive S-MJSs
full of challenges.
In this chapter, the problems of mean stability and control synthesis problems for
a class of positive S-MJSs are considered. Compared with recent works [1,2,24–28]
ﬁrstly, the sojourn time obeying the exponential distribution limits application scope
of MJSs that are unable to describe more complex system models such as more
complex network environment in the presence of packet loss and channel delay.
Second, the switching law in [22,23] is subject to average dwell time switching
while the switching law in this chapter obeys stochastic semi-Markov jump. And
also, as a basic problem in control theory, an interesting question naturally arises:
Can necessary and sufﬁcient conditions for stability criteria and control synthesis
of positive S-MJSs be given? This research is full of theoretical challenges, since it
needs to consider not only stability and controller design for dynamic systems, but
also constrained positivity. In summary, there are still many open problems associated
with stability and stabilization of positive S-MJSs, some of which will be discussed
towards in the end.
These above aspects stimulate our research interest to address the mean stability
and control synthesis problems for a class of positive S-MJSs. The main contributions
can be concluded as follows: (i) Due to the effect of complex working environment,
one unrealistic assumption, i.e. the sojourn time in stochastic jump systems follows
the exponential distribution, is removed in this chapter by applying S-MJSs model.
(ii) Necessary and sufﬁcient conditions for mean stability rely on the analysis of
linear Lyapunov-Krasovskii function. (iii) Some necessary and sufﬁcient conditions
for the existence of controller design are given in an appropriate linear programming
formulation. (iv) Additionally, representing a virus mutation treatment model as
positive S-MJSs illustrates the effectiveness of the controller design algorithm.
3.2 Problem Statements and Preliminaries
Consider the positive S-MJSs as
˙z(t)=A(o
t)z(t),z(t 0)=z 0,ot0
=o0,t0=0, (3.1)
wherex(t)∈R
n
is the state.{o t,ι}t≥0:= {o l,ιl}l∈N≥1
represents continuous-time
and discrete-state homogeneous semi-Markov process with right continuous trajec-
tories in a ﬁnite set℘={1,2,...,ℵ}, where{o
l}l∈N≥1
is system mode index at thelth
transition in℘, and{d
l}l∈N≥1
is the sojourn time of modeo l−1between the(l−1)th
transition andlth transition [4,5,10]. The transition rate is given by

3.2 Problem Statements and Preliminaries 39
⎧
⎪
⎪
⎨
⎪
⎪
⎩
Pr{o
l+1=ν, ιl+1≤ι+¯Δ|o l=μ, ιl+1>ι}
=σ
μν(ι)¯Δ+o(¯Δ), μσ=ν,
Pr{o
l+1=μ, ιl+1>ι+¯Δ|o l=μ, ιl+1>ι}
=1+σ
μμ(ι)¯Δ+o(¯Δ), μ=ν,
whereιmeans the sojourn time thatι≥0,¯Δ>0 and lim
¯Δ→0(o(¯Δ)/¯Δ)=0,
σ
μν(ι) >0 represents transition rate fromμtoνforμσ=ν, andσ μμ(ι)=
−
σ
ℵ
ν=1,νσ=μ
σμν(ι) <0. Wheno t=μ,∀μ∈℘,A(o t)is denoted asA μ.
Remark 3.1It is well known that the sojourn time of semi-Markov process follows
the non-exponential distribution, which means that the transition rate of semi-Markov
process depends on the sojourn timeι. However, the sojourn time of Markov process
is related to the exponential distribution, which means that the transition rate of
Markov process is independent of the sojourn timeι. This shows that the semi-
Markov process stands for a class of more general stochastic process than Markov
process and covers Markov process as a special case.
Remark 3.2The problems of stability analysis and control synthesis for S-MJSs
have been studied in recent years [3–20]. However, there is little attention to control
for S-MJSs with positive constraint. As is well known, absolute temperature, mate-
rial density, and displacement in physics, population position in biology, reactant
concentration in chemistry, number of vehicles in transportation system, demand,
supply, and price in economics can be described by positive systems. When com-
bining S-MJSs and positive constraint together, it needs to reanalysis the stability.
For this reason, the stability and control for positive S-MJSs are investigated in this
chapter.
Deﬁnition 3.1([29]) Consider
˙z(t)=Az(t),z(t
0)=z 0,t0=0. (3.2)
For anyz
0≥≥0, there holdsz(t)≥≥0 for allt≥0, then system (3.2)issaidtobe
positive.
Lemma 3.1([29])For system (3.2), the followings are equivalent:
(i) System (3.2) is asymptotically stable (i.e.,Ais a Hurwitz matrix);
(ii) There existsη>>0such thatA
T
η<<0.
Lemma 3.2([21])For system (3.2), the followings are equivalent:
(i) System (3.2) is asymptotically stable (i.e.,Ais a Hurwitz matrix);
(ii) There existsη>>0such thatAη<<0.

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XIII
Mehiläismetsästys
Aurinko oli tuskin noussut taivaanrannalle, kun kenraali, jonka
hevonen oli satuloitu, astui ulos ruokomajasta, jota hän piti
makuuhuoneenaan, ja valmistautui lähtemään. Juuri kun hän nousi
ratsunsa selkään, kohotti herttainen käsi teltan oviverhoa ja doña
Luz tuli näkyviin.
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lähden. Se tuottaa ehkä minulle onnea", lisäsi hän huokaisten.
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otsansa, jolle eno painoi suudelman.
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kelpaavan, ennenkuin nousette ratsaille. Ettehän kieltäydy siitä, eikö
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lapset, jotka aina saavat vanhat ukot hyvälle tuulelle.

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kauan odottaa aamiaista, jonka minulle niin ystävällisesti tarjoat, sillä
minulla on vähän kiire."
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"No olkoon menneeksi pari minuuttia", myönsi toinen mennen
hänen perässään.
Nuori tyttö taputti iloisesti käsiään.
Muutamassa hetkessä aamiainen oli valmis, ja kenraali istuutui
sisarentyttärensä kera pöytään.
Koko ajan enoaan palvellessaan ja huolehtiessaan siitä, ettei tältä
puuttunut mitään, nuori tyttö katseli häntä niin hämillään ja
hartaasti, että vanha sotilas vihdoin sen huomasi.
"Kas niin", sanoi hän keskeyttäen syöntinsä ja katsoen
sisarentytärtään. "Sinulla on minulta jotakin pyydettävää, Lucita.
Tiedäthän hyvin, että tapanani ei ole kieltää sinulta mitään."
"Se on totta, eno. Mutta luulen, että minun tällä kertaa on
vaikeampi voittaa teitä puolelleni."
"Ah, eikö mitä!" huudahti kenraali iloisena. "Onko sinulla siis joku
perin tärkeä asia?"
"Päinvastoin, eno. Ja kuitenkin tunnustan, että pelkään teidän sen
minulta kieltävän."

"Anna tulla vain, lapseni", vastasi vanha sotilas, "puhu
pelkäämättä.
Sanottuasi, mistä on kysymys, minä kyllä vastaan sinulle."
"No niin, eno", aloitti nuori tyttö punastuen, mutta käyden asiaan
suoraan käsiksi. "Tunnustan teille, että leirielämä ei tunnu erikoisen
miellyttävältä."
"Se käsitys minullakin on siitä, lapseni, mutta mitä tahtoisit minua
tekemään?"
"Kaikki."
"Mitä tarkoitat?"
"Kah, eno, jos olisitte täällä luonani, niin en olisi millänikään.
Saisinhan silloin viettää aikani teidän seurassanne."
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poistuessani joka aamu en voi olla täällä."
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"Niinkö luulet?"
"Siitä olen aivan varma."
"En vielä käsitä, miten se kävisi päinsä. Sillä tavalla tosin, että
jäisin luoksesi, mutta se on mahdotonta."

"Oh! On toinenkin keino, jolla asian voi aivan hyvin järjestää."
"Ah, vielä mitä!"
"Kyllä, eno. Ja varsin yksinkertainen keino onkin. Haluatteko
kuulla?"
"Kas, kas, ja mikä se keino on, sydänkäpyseni?"
"Etkö toru minua, eno?"
"Pikku hupsu! Minäkö sinua toruisin?"
"Se on totta, te olette niin hyvä!"
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"Niin, eno. Keino on se…"
"Keino on se?"
"Että otatte minut mukaanne retkillenne joka aamu."
"Oh, oh!" lausui kenraali, ja hänen kulmakarvansa menivät
ryppyihin.
"Mitä minulta pyydätkään, rakas lapsi!"
"Mutta sehän on varsin selvä asia, eno, niin ainakin minusta
tuntuu."
Kenraali ei vastannut, vaan vaipui ajatuksiin. Nuori tyttö seurasi
tuskastuneena merkkejä, joita mietteet jättivät hänen kasvoilleen.
Hetken kuluttua kenraali kohotti päätään.

"Itse asiassa", mutisi hän, "se olisikin ehkä parasta." Ja luoden
nuoreen tyttöön läpitunkevan katseen hän jatkoi: "Sinusta olisi siis
hauskaa tulla mukaan retkilleni?"
"Kyllä, eno", vastasi tyttö.
"No niin, valmistaudu siis, rakas lapsi. Tästä lähtien saat seurata
minua retkilläni."
Nuori tyttö nousi pöydästä yhdellä hyppäyksellä, syleili kiihkeästi
enoaan ja antoi määräyksen satuloida hevosensa.
Neljännestuntia myöhemmin doña Luz ja hänen enonsa Lörpön
johtamina ja parin lanceron seuraamina poistuivat leiristä ja
tunkeutuivat metsään.
"Minne suunnalle haluatte tänään kulkea, kenraali?" kysyi opas.
"Opasta minut niiden erämiesten majalle, joista eilen minulle
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Opas kumarsi merkiksi, että hän tahtoi noudattaa käskyä. Pieni
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pitkin, jolla hevoset melkein joka askeleella kompastuivat liaaneihin
tai lankesivat maata pitkin luikertelevia puunjuuria vasten.
Doña Luz oli onnellinen. Kenties hän näillä retkillä tapaisi
Uskollisen
Sydämen.
Lörppö, joka ratsasti muutamia askeleita edellä muita, huudahti
äkkiä.

"Mitä nyt?" tiedusti kenraali. "Onko jotakin erikoista, Lörppö?"
"Mehiläisiä, teidän ylhäisyytenne."
"Mitä? Mehiläisiäkö? Onko täälläkin mehiläisiä?"
"Joku aika sitten niitä on tullut."
"Mitä? Onko niitä tänne siirtynyt?"
"On. Tiedättehän, että valkoihoiset ovat tuoneet Amerikkaan
mehiläisiä."
"Asiassa on perää. Mutta miten ihmeessä niitä tavataan täällä?"
"Ei mikään ole yksinkertaisempaa. Mehiläiset ovat valkoihoisten
etuvartioita: sitä mukaa kuin valkoihoiset tunkeutuvat Amerikan
sisäosiin, etenevät mehiläiset heidän edellään viitoittaen heille tietä
ja osoittaen uutismaan paikkoja. Niiden ilmestyminen asumattomaan
seutuun ennustaa aina uutisviljelijän tai squatterijoukon tuloa."
"Sepä omituista", mutisi kenraali; "oletko varma siitä, mitä minulle
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"Olen, aivan varma. Kaikille preirien asujille on tunnettua, että
esimerkiksi intiaanit luottavat mehiläisiin, jotka erehtymättömän
varmasti ilmaisevat squatterien tulon."
"Merkillistä", lausui kenraali ihmetellen.
"Hunaja on varmaankin hyvää", sanoi doña Luz.
"Mainiota, señorita, ja jos niin haluatte, on varsin helppoa sitä
meille hankkia."

"Tehdään niin", päätti kenraali.
Opas, joka vähän aikaisemmin oli pensaikkoon jättänyt syöttiä
mehiläisille, joiden hänen tarkka katseensa oli havainnut parvittain
lentelevän tiheiköissä, viittasi seuralaisilleen merkiksi, että he
pysähtyisivät.
Mehiläiset olivat tosiaan asettuneet syötille ja tutkiskelivat sitä
parhaillaan innokkaasti. Hankittuaan tarpeeksi varastoonsa tätä
ainetta ne kohosivat hyvin korkealle ilmaan ja suuntasivat sitten
lentonsa suoraa viivaa pitkin niin nopeasti kuin kiväärinluoti.
Opas tarkkasi huolellisesti suuntaa, jota ne alkoivat noudattaa, ja
viitattuaan kenraalille hän lähti seuraamaan niiden jälkiä koko
partiojoukon saattamana, joka raivasi itselleen tietä toisiinsa
kietoutuneitten puunjuurien, sortuneiden puiden, pensaikkojen ja
viidakkojen lävitse, silmät lakkaamatta suunnattuina taivasta kohti.
Täten he eivät kadottaneet näkyvistään hunajaa keränneitä
mehiläisiä, ja tunnin ajan mitä tukalimmissa oloissa ajettuaan niitä
takaa he huomasivat niiden saapuvan pesälleen, joka oli rakennettu
kuivuneen ebenpuun onteloon. Hetken aikaa suristuaan ne menivät
pesään aukosta, joka oli korkeintaan kahdenkymmenen jalan
korkeudella maasta.
Kehoitettuaan seuralaisiansa pysymään kunnioitettavan välimatkan
päässä, jotta he olisivat suojassa puun kaatuessa ja turvattuina sen
asujainten kostolta, opas otti esille kirveensä ja alkoi reippaasti
hakata poikki ebenpuuta sen tyvestä.
Mehiläiset eivät näyttäneet lainkaan säikähtävän kirveen iskuja.
Yhä edelleen niitä meni pesään ja tuli sieltä ulos, ja ne jatkoivat

kaikessa rauhassa uutteraa työtään. Eipä edes raju ryskekään, joka
oli merkkinä rungon murtumiseen, siirtänyt niiden huomiota pois
tavallisista askareista.
Vihdoin puu kaatui hirvittävästi ryskyen ja halkesi pitkin
pituuttaan, niin että siinä asuneen yhteiskunnan keräämät rikkaudet
tulivat näkyviin.
Opas otti heti heinätukon, jonka oli varannut valmiiksi, ja sytytti
sen tuleen puolustautuakseen mehiläisiltä.
Mutta ne eivät hyökänneet kenenkään kimppuun. Ne eivät edes
yrittäneet kostaa. Nuo hyönteisraukat olivat hämmästyneitä, ne
juoksivat ja lentelivät joka suunnalle hävitetyn valtakuntansa
ympärillä ajattelematta muuta kuin kiirehtiä ottamaan selvää
onnettomuudesta.
Silloin opas ja lancerot ryhtyivät toimeen puukoin ja kauhoin
saadakseen vahakakut esille ja sulloakseen ne nahkaleileihinsä.
Useat olivat tummanruskeita ja vanhoja, toiset taas kauniinvaaleita,
niin että kennoissa oleva hunaja oli melkein läpikuultavaa.
Sillä aikaa, kun kiireesti koottiin paraimpia vahakakkuja, saapui
joka ilmansuunnalta täyttä vauhtia lukemattomin parvin mehiläisiä,
jotka tunkeutuivat rikottujen vahakakkujen kammioihin, keräten
niistä itselleen varaston, samalla kun pesän entiset omistajat
synkkinä ja tylsinä katsoivat hunajansa ryöstöä koettamatta pelastaa
siitä edes pientä osaakaan.
Mahdotonta on kuvailla niiden mehiläisten ällistystä, jotka
onnettomuuden aikana olivat poissa tapahtumapaikalta ja jotka
vähitellen kuormineen saapuivat pesälle. Ne lentelivät kehässä

ilmassa sillä paikalla, missä puu oli ollut, hämmästyneinä siitä, että
huomasivat paikan tyhjäksi. Vihdoin ne näyttivät käsittävän niitä
kohdanneen tuhon ja kokoontuivat ryhmittäin erään läheisen puun
kuivuneelle oksalle, josta ne tuntuivat ihmetellen tarkastelevan
pesän maassa olevia jäännöksiä ja valittelevan valtakuntansa
hävitystä.
Doña Luz tunsi väkisinkin liikutusta hyönteisparkojen surun
johdosta.
"Menkäämme", sanoi hän, "minua kaduttaa, että halusin hunajaa,
herkutteluni tuottaa liian paljon onnettomuutta."
"Lähtekäämme", lausui kenraali hymyillen, "jättäkäämme niille nuo
muutamat vahakakut."
"Eikö mitä!" lausui opas olkapäitään kohauttaen, "silloin
syöpäläiset ne pian korjaisivat."
"Mitä? Syöpäläisetkö? Mitä syöpäläisiä tarkoitatte?" tiedusteli
kenraali.
"No, pesukarhut, pussirotat ja ennen kaikkea karhut."
"Karhutko?" kysyi doña Luz.
"Niin, señorita!" jatkoi opas, "ne ovat maailman taitavimpia
syöpäläisiä etsiessään mehiläisten asumia puita ja käyttäessään niitä
hyväkseen."
"Ne siis pitävät hunajasta?" kysyi nuori tyttö huvitettuna.

"Se merkitsee, että ne ovat siihen silmittömästi ihastuneita,
señorita", sanoi opas näyttäen käyvän hilpeämmäksi. "Kuvitelkaa,
että ne ovat sille niin persoja, että jyrsivät puuta viikkomääriä,
kunnes saavat siihen aukon voidakseen työntää sinne kämmenensä.
Sitten ne ahmivat hunajaa ja mehiläisiä, vaivautumatta tekemään
valintaa."
"Lähtekäämme taas jatkamaan matkaamme", toisti kenraali, "ja
pyrkikäämme erämiesten luokse."
"Oh! Saavumme kyllä pian sinne, teidän ylhäisyytenne", vastasi
opas, "tuolla jonkun matkan päässä meistä on Canadianjoki, ja
erämiehet ovat asettuneet sen sivuhaarojen varrelle."
Pieni joukko lähti taas marssimaan.
Mehiläisajo oli nuoren tytön tietämättä jättänyt häneen
alakuloisuuden tunteen, jota hän ei voinut karkoittaa. Nuo pienet
eläinparat, jotka olivat niin suloisia ja uutteria ja jotka nyt oikun
vuoksi olivat joutuneet vainottaviksi ja tuhottaviksi, saattoivat hänet
murheelliseksi ja panivat hänet väkisinkin miettimään.
Hänen enonsa huomasi hänen mielentilansa.
"Rakas lapsi", sanoi hän tytölle, "mikä sinua oikein vaivaa? Et ole
enää iloinen kuten lähtiessämme. Mistä johtuu mielentilasi äkillinen
muutos?"
"Hyvä Jumala, älä ole huolissasi sen johdosta, eno. Olen
samanlainen kuin muutkin nuoret tytöt, siis hiukan hupsu ja
haaveellinen. Tämä metsästys, josta arvelin itselleni koituvan niin

paljon huvia, on tietämättäni jättänyt minuun surumielisyyden
tunteen, josta en voi vapautua."
"Onnellinen lapsi", mutisi kenraali, "jonka näin vähäpätöinen
seikka voi vielä tehdä murheelliseksi. Suokoon Jumala,
sydänkäpyseni, että vielä kauan pysyisit samanlaisena ja etteivät
suuremmat ja todelliset murheet koskaan sinua kohtaisi!"
"Eno hyvä, enkö aina olisi onnellinen saadessani olla luonanne!"
"Kas niin, lapseni! Kuka tietää, salliiko Jumala minun kauan sinusta
huolehtia?"
"Älkää puhuko sellaisia, eno. Toivon, että saamme vielä monet
vuodet viettää yhdessä."
Kenraali vain huokasi vastaukseksi.
"Eno", jatkoi tyttö hetken kuluttua, "eikö teistäkin meitä
ympäröivän suurenmoisen ja ylevän luonnon näkemisessä ole jotakin
tenhoavaa, joka jalostuttaa aatteet, kohottaa mielen ja tekee
ihmisen paremmaksi? Ne, jotka asuvat näissä äärettömissä
erämaissa, ovat varmaankin onnellisia!"
Kenraali katsoi häneen hämmästyneenä.
"Mistä johtuvat nuo ajatukset, rakas lapsi?" kysyi hän.
"En tiedä, eno", vastasi tyttö arasti, "olen vain tietämätön lapsi,
jonka vielä niin lyhyt elämä tähän asti on kulunut rauhallisesti ja
seikkailuitta teidän luonanne, en muuta! On hetkiä, jolloin minusta
tuntuu, että on onnellista asua näissä suunnattomissa eräseuduissa."

Hämmästyneenä ja sisimmässään ihastuneena sisarentyttärensä
viattomasta tunteenpurkauksesta kenraali aikoi juuri vastata hänelle,
kun opas äkkiä lähestyi heitä, viittasi merkiksi, että hän kehoitti heitä
vaitioloon, ja lausui hiljaa kuiskaten:
"Ihminen!…"

XIV
Musta Hirvi
Jokainen pysähtyi.
Erämaassa sana ihminen merkitsee melkein aina vihollista.
Ihminen on preiriellä lähimmäiselleen peloittavampi kuin julmin
villipeto.
Ihminen merkitsee kilpailijaa, pakollista liittolaista, joka
väkevämmän oikeuteen perustuen tulee jakamaan saaliin
ensimmäisen valtaajan kanssa ja usein, ellei aina, yrittää korjata
viimemainitun työläiden ponnistusten tulokset.
Valkoihoiset, intiaanit tai sekarotuiset tervehtivätkin toisiaan
tavatessaan preiriellä aina varoen, korvat hörössä ja sormi
rihlakiväärin liipasimella.
Kun oli kuulunut sana ihminen, olivat kenraali ja lancerot kiireesti
valmistautuneet äkillistä hyökkäystä vastaanottamaan ladaten
aseensa ja piiloutuen parhaansa mukaan pensaitten suojaan.

Viidenkymmenen askeleen päässä seisoi mies, joka pyssynperä
maassa ja molemmin käsin pidellen pitkän rihlapyssyn piippua
tarkkaili heitä tutkien.
Hän oli suurikokoinen mies, jolla oli tarmokkaat piirteet, avoin ja
päättäväinen katse.
Hänen pitkä, huolellisesti kammattu tukkansa oli palmikoitu ja
sidottu saukonnahkarihmoilla ja erivärisillä nauhoilla.
Siro, nahkainen metsästyspusero ulottui polviin asti;
erikoiskutoiset säärystimet, joissa oli nauhat ja tupsut ja paljon
helistimiä, suojasivat hänen jalkojaan. Jalkineina oli pari mainioita
mokkasineja, joita koristivat tekohelmet.
Heleänpunainen loimi riippui hänen olkapäillään, ja vyötäisillä oli
punainen vyö, johon oli pistetty kaksi pistoolia, veitsi ja
intiaanipiippu.
Hänen kiväärinsä oli kauniiksi maalattu sinoberivärillä ja koristettu
pienillä kuparinauloilla.
Jonkun askeleen päässä hänestä hänen hevosensa söi
terhometsässä.
Omistajansa tavoin oli sekin mitä eriskummaisimmin koristettu. Se
oli sinoberilla tehty täplikkääksi ja juovikkaaksi, suitset ja häntävyö
oli koristettu tekohelmillä ja nauharuusuilla, pää, harja ja häntä
runsaasti somistettu kotkansulilla, jotka heiluivat tuulessa.
Huomatessaan tämän miehen kenraali ei voinut pidättää
hämmästyksen huutoa.

"Mihin intiaaniheimoon tuo mies kuuluu?" kysyi hän oppaalta.
"Ei mihinkään", vastasi viimemainittu.
"Mitä? Eikö mihinkään?"
"Ei! Hän on valkoihoinen metsästäjä."
"Ja tuolla tavalla pukeutunut!"
Opas kohautti olkapäitään.
"Olemme preiriellä", sanoi hän.
"Se on totta", mutisi kenraali.
Mies, jota olemme kuvailleet, kyllästyi varmaankin edessään
olevan pienen joukon varovaiseen käyttäytymiseen ja halusi päästä
selville siitä, mitä hänen puolestaan oli tehtävä. Siksi hän
päättäväisesti alkoi puhua.
"Ohoo, mitä lemmon väkeä te olette", lausui hän englanninkielellä,
"ja mitä te tulette täältä etsimään?"
"Caramba!" vastasi kenraali heittäen kiväärin luotaan ja käskien
seuralaistensa tehdä samoin. "Me olemme matkalaisia, jotka olemme
lopen uuvuksissa pitkän matkan jälkeen. Aurinko paahtaa kuumasti,
pyydämme siis teiltä lupaa saada levätä hetken aikaa majassanne."
Koska nämä sanat oli lausuttu espanjaksi, vastasi erämies samalla
kielellä.
"Tulkaa pelotta! Musta Hirvi on hyvä paholainen, kun häntä ei
aiota suututtaa. Saatte käyttää hyväksenne sitä vähää, mitä minulla

on, ja tehköön se teille hyvää."
Kuullessaan nimen Musta Hirvi opas ei voinut pidättää säikähdystä
ilmaisevaa liikettä. Hän aikoi sanoa jotakin, mutta siihen ei hänellä
ollut aikaa, sillä metsästäjä oli heittänyt kiväärin olalleen, yhdellä
hyppäyksellä lentänyt satulaan ja lähestynyt meksikkolaisia jo aivan
likelle.
"Majani on muutaman askeleen päässä täältä", sanoi hän
kenraalille. "Jos señorita haluaa maistaa hyvin paistettua
biisoninlihaa, niin olen valmis sen kohteliaisuuden hänelle
osoittamaan."
"Ei mikään ole sen ystävällisempää. Kiitämme tarjouksesta", lausui
kenraali.
"Minä puolestani kaipaan etupäässä lepoa", huomautti nuori tyttö.
"Kaikki tulee aikanaan", sanoi erämies opettavasti, "sallitteko
minun muutamaksi hetkeksi astua oppaanne tilalle?"
"Kuten tahdotte", myöntyi kenraali, "menkää, me seuraamme
teitä."
"Siis matkaan", huudahti erämies asettuen joukon etunenään.
Samassa hänen silmänsä sattumalta osuivat oppaaseen, ja hänen
tuuheat kulmakarvansa rypistyivät. "Hm!" mumisi hän hampaittensa
välistä, "mitä tämä merkinnee? Saammehan nähdä", lisäsi hän.
Kiinnittämättä sen enempää huomiota tähän mieheen ja
osoittamatta tuntevansa hänet hän antoi lähtökäskyn.

Kun joukkue oli jonkun aikaa vaiteliaana marssinut jokseenkin
leveän joen rantaa pitkin, teki erämies jyrkän mutkan ja kääntyi
sivulle päin tunkeutuen takaisin metsään muiden seuratessa häntä.
"Pyydän anteeksi", sanoi hän, "että vaivaan teitä kulkemaan tätä
kiertotietä, mutta täällä on majavalammikko, ja pelkään säikäyttää
sen asujaimia."
"Oi!" huudahti nuori tyttö, "olisin niin iloinen saadessani nähdä
noiden uutterien eläinten puuhia!"
Erämies pysähtyi.
"Se käy varsin helposti päinsä, señorita", sanoi hän, "jos vain
tahdotte seurata minua seuralaistenne jäädessä tänne meitä
odottamaan."
"Kyllä! Kyllä!" vastasi doña Luz kiitollisena, mutta malttaen
mielensä hän äkkiä lisäsi: "Anteeksi, mutta enoni"…
Kenraali katsahti erämieheen.
"Mene, lapseni, odotamme sinua täällä", sanoi hän.
"Kiitos, eno", huudahti nuori tyttö iloisena hypäten alas satulasta.
"Vastaan hänestä", lausui erämies vilpittömästi, "älkää lainkaan
pelätkö!"
"En pelkää mitään uskoessani hänet teidän huostaanne, ystäväni",
vastasi kenraali.
"Kiitos!" ja viitaten merkiksi doña Luzille Musta Hirvi katosi hänen
kanssaan pensaitten ja puiden sekaan.

Kun he olivat päässeet jonkun matkan päähän, pysähtyi erämies.
Kuunneltuaan tarkasti ja silmäiltyään joka puolelle ympärilleen hän
kääntyi nuoren tytön puoleen ja laskien kevyesti kätensä hänen
käsivarrelleen sanoi:
"Kuulkaa!"
Doña Luz pysähtyi levottomana ja vavisten.
Erämies huomasi hänen mielentilansa.
"Älkää pelätkö", jatkoi hän, "minä olen kunniallinen mies, ja te
olette yhtä varmassa turvassa, vaikka olettekin yksin täällä
erämaassa kanssani, kuin jos olisitte Meksikon tuomiokirkossa
pääalttarin juurella."
Nuori tyttö katsahti salavihkaa erämieheen. Omituisesta puvusta
huolimatta oli hänen kasvoillaan sellainen vilpittömyyden leima, ja
hänen silmänsä olivat niin lempeät ja kirkkaat, kun hän katsahti
tyttöön, että tämä tunsi jälleen rauhoittuvansa.
"Puhukaa", sanoi hän.
"Kuulutte, sillä nyt tunnen teidät, siihen ulkomaalaisjoukkueeseen,
joka jo muutaman päivän on tutkinut preirietä joka suunnalla, eikö
niin?"
"Kyllä."
"Joukossanne on kummallinen mies, jolla on vihreät silmälasit ja
vaalea tekotukka ja joka huvittelee, en tiedä mistä syystä keräämällä
kasveja ja kiviä sensijaan, että koettaisi kunnon metsästäjän lailla
pyydystää majavia tai kaataa kuusipeuroja."

"Tunnen miehen, josta puhutte. Hän kuuluu tosiaankin
joukkoomme, ja on sangen etevä lääkäri."
"Sen tiedän, hän on sen itse minulle sanonut, hän kulkee usein
tätä kautta, olemme hyviä ystäviä. Pulverilla, jonka hän pakoitti
minut nauttimaan, hän täydellisesti poisti minusta kuumeen, joka jo
kahden kuukauden aikana oli minua kiusannut ja josta en päässyt
vapaaksi."
"Sepä hauskaa. Olen iloinen lääkkeen vaikutuksesta."
"Haluaisin tehdä jotakin hyväksenne, osoittaakseni kiitollisuuttani
tästä palveluksesta."
"Kiitos, ystäväni, mutta en todellakaan tiedä, miten te voisitte olla
minulle hyödyksi, paitsi että näytätte minulle majavat."
Erämies painoi päänsä alas.
"Ehkä toisella tavalla", sanoi hän, "ja kenties pikemmin kuin
luulettekaan. Kuulkaa minua tarkkaavaisesti, señorita. Olen vain
köyhä mies, mutta täällä preiriellä tiedämme asioita, joita Jumala
meille ilmaisee, koska elämme kasvoista kasvoihin hänen kanssaan.
Haluan antaa teille neuvon: mies, joka palvelee teitä oppaana, on
paatunut konna, sellaiseksi hänet tunnetaan kaikkialla lännen
preirieillä. Erehdyn suuresti, ellei hän saata teitä jonkun salakavalan
väijytyksen alaiseksi. Täällä on runsaasti pahanilkisiä lurjuksia, joiden
kanssa hän voi tehdä sopimuksen teidän turmiostanne tai ainakin
ryöstämisestänne."
"Oletteko varma siitä, mitä sanoitte?" huudahti nuori tyttö
kauhistuneena näistä sanoista, jotka, merkillistä kyllä, pitivät yhtä

sen kanssa, mitä Uskollinen Sydän oli sanonut.
"Olen siitä yhtä varma kuin joku henkilö yleensä voi väittää
asiasta, josta hänellä ei ole todistuksia. Toisin sanoen: Lörpön
entisyyden perusteella täytyy hänen puoleltaan odottaa kaikkea.
Uskokaa minua, ellei hän vielä ole teitä pettänyt tekee hän sen
ennen pitkää."
"Hyvä Jumala. Varoitan kyllä enoani!"
"Sitä karttakaa tekemästä, sillä silloin olisi kaikki menetetty!
Roistoja, joiden kanssa hän on liittoutunut tai kohta liittoutuu, ellei
hän vielä ole sitä tehnyt, on paljon, lukemattomia, ja he tuntevat
täydellisesti preirien."
"Mitä on siis tehtävä?" kysyi nuori tyttö tuskastuneena.
"Ei mitään. Pitää odottaa ja huolellisesti tarkata kaikkia oppaanne
puuhia hänen sitä huomaamattaan."
"Mutta…"
"Ymmärrätte hyvin", keskeytti hänet erämies, "että kun kehoitan
teitä pitämään häntä silmällä, se ei tarkoita sitä, että hetken tullessa
ja tarvitessanne apuani jättäisin teidät pulaan."
"Sen uskon."
"No niin, menetelkää seuraavasti. Heti kun olette varma siitä, että
oppaanne pettää teitä, te lähetätte luokseni vanhan hupsun
lääkärinne. Häneenhän voitte luottaa, vai mitä?"
"Täydellisesti."

"Hyvä. Lähetätte siis hänet luokseni, kuten sanoin, ja käskette
hänen lausua minulle vain: Musta Hirvi. Musta Hirvi olen minä."
"Tiedän sen, sanoitte jo sen meille."
"Mainiota, hän siis sanoo minulle: 'Musta hirvi, hetki on tullut'. Ei
muuta. Muistatteko nyt nuo sanat?"
"Tarkalleen. En vain oikein ymmärrä, miten se saattaisi meitä
hyödyttää."
Erämies hymyili salaperäisen näköisenä.
"Hm!" sanoi hän hetken kuluttua, "nuo muutamat sanat
toimittavat kahden tunnin kuluessa avuksenne viitisenkymmentä
preirien rohkeinta miestä. Miehiä, jotka päällikkönsä viittauksesta
antavat tappaa itsensä pelastaakseen teidät niiden käsistä, jotka
mahdollisesti ovat teidät riistäneet, kuten aavistan tapahtuvan."
Hetkisen oltiin vaiti, doña Luz näytti vaipuneen ajatuksiin.
Erämies hymyili.
"Älkää hämmästykö sitä innokasta harrastusta, jota teitä kohtaan
osoitan", sanoi hän; "muuan mies, jonka täydessä vallassa olen, on
vannottanut minua huolehtimaan teistä hänen ollessaan pakosta
muualla."
"Mitä tarkoitatte?" kysäisi tyttö uteliaana, "kuka mies se on?"
"Hän on metsästäjä, joka hallitsee kaikkia valkoisia metsästäjiä
preiriellä. Tietäen, että Lörppö toimii oppaananne, hän arvelee, että
tämä mestitsi aikoo johtaa teidät väijytyksen uhriksi."

"Mutta mikä on sen miehen nimi?" huudahti tyttö tuskaisella
äänellä.
"Uskollinen Sydän. Luotatteko minuun nyt?"
"Kiitos, ystäväni, kiitos", vastasi nuori tyttö innostuneena, "en
unohda suositustanne, ja vaaran uhatessa, jos sellainen
onnettomuudekseni tulee, aion muistuttaa teille lupauksestanne."
"Ja siinä teette oikein, señorita, sillä silloin se on ainoa pelastuksen
tie, mikä teillä on jälellä. Lähtekäämme! Olette ymmärtänyt minut,
kaikki on hyvin, pitäkää salaisuutenanne keskustelumme. Ennen
kaikkea älkää näyttäkö, että olette puhellut tästä asiasta kanssani.
Tuo kirottu mestitsi on tarkkavainuinen kuin majava. Jos hän vain
epäileekin jotakin, niin hän luikertelee sormienne lomitse kuin kyy,
joka hän onkin."
"Olkaa huoletta, minä olen mykkä."
"Jatkakaamme nyt matkaamme majavalammelle. Uskollinen Sydän
huolehtii teistä."
"Hän on jo kerran pelastanut henkemme, preirien tulipalossa",
huudahti tyttö innostuneesti.
"Ah, ah!" mutisi erämies katsahtaen häneen omituisesti, "kaikki on
siis parhain päin"; sitten hän lisäsi kovaa: "Olkaa huoletta, señorita,
jos kohta kohdalta seuraatte teille antamiani neuvoja, ei teille
tapahdu preiriellä mitään pahaa, olkoot sitten millaisia hyvänsä ne
petolliset juonet, joiden uhriksi joudutte."
"Oi!" huudahti tyttö haltioissaan, "vaaran hetkellä en epäröi
rientäessäni turviinne, sen vannon!"

"Asia on siis sovittu", lausui Musta Hirvi hymyillen; "nyt
lähtekäämme katsomaan majavia."
He jatkoivat marssiaan, ja muutaman minuutin kuluttua he
saapuivat metsän reunaan.
Silloin erämies pysähtyi viitaten nuorelle tytölle merkiksi, että hän
pysyisi liikkumattomana, ja kääntyen häneen päin kuiskasi:
"Katsokaa."

XV
Majavat
Nuori tyttö työnsi syrjään pajujen oksat ja kurottaen päänsä
eteenpäin katseli.
Majavat eivät olleet uutteran yhteiskuntansa patoamistyöllä
katkaisseet ainoastaan itse pääjoen juoksua, vaan vieläpä kaikki
siihen virtaavat joet, niin että ympäröivä maa oli muuttunut laajaksi
rämeeksi.
Yksi ainoa majava työskenteli tällä hetkellä pääsulun luona, mutta
pian ilmestyi viisi muuta tuoden muassaan puunpalasia, savea ja
kiviä. Sitten ne kaikki yhdessä menivät eräälle kohtaa sulkua, joka,
sen huomasi nuori tyttökin, kaipasi korjausta. Ne laskivat
kantamuksensa sortuneeseen paikkaan ja sukelsivat veteen, mutta
ilmestyivät melkein heti senjälkeen takaisin pinnalle.
Kukin niistä toi mukanaan määrätyn kantamuksen savea, jota ne
käyttivät muurilaastina liittääkseen yhteen puunpalaset ja kivet ja
valmistaakseen niistä tukevan kohdan patoon. Kerta toisensa jälkeen

ne poistuivat palatakseen jälleen tuomaan savea ja puunpalasia.
Muuraustyötä jatkui, kunnes repeämä oli kokonaan korjattu.
Kun työ oli valmis, alkoivat nuo viisaat eläimet hauskutella
ajamalla toisiaan takaa lammessa. Ne sukelsivat veteen tai
uiskentelivat sen pinnalla lyöden vettä hännällään, niin että se
räiskyi.
Doña Luz katseli yhä ihastuneempana tätä erikoista näytelmää.
Hän olisi mielellään jäänyt vaikka koko päiväksi ihailemaan näitä
merkillisiä eläimiä.
Sillä aikaa kun ensinmainitut täten pitivät hauskaa, saapui paikalle
pari muuta majavayhteiskunnan jäsentä. Jonkun aikaa ne totisina
katselivat toveriensa kisailua ollen sennäköisiä, kuin olisivat
mielellään liittyneet leikkiin, mutta sitten ne kiipesivät ylös
mäenrinnettä, joka oli melko lähellä nuoren tytön ja erämiehen
tähystyspaikkaa, istuutuivat takajaloilleen, nojasivat etukäpälillään
nuorta mäntyä vastaan ja alkoivat jyrsiä sen kuorta. Väliin ne siitä
eroittivat pienen palasen pitäen sitä käpäliensä välissä ja pysyen yhä
istumassa. Ne nakertelivat irvistellen ja väännellen kasvojaan aivan
kuin apina, joka jyrsii pähkinää.
Majavilla oli ilmeisesti tarkoituksena kaataa puu, ja sen
saavuttamiseksi ne työskentelivät kuumeisesti. Nuori mänty oli
läpimitaltaan lähes kahdeksantoista tuumaa siltä kohdalta, mistä ne
sitä jyrsivät; se oli suora kuin tikku ja jokseenkin korkea. Ne olisivat
epäilemättä voineet lyhyessä ajassa jyrsiä sen kokonaan poikki,
mutta kenraali, joka oli tullut levottomaksi sisarentyttärensä
viipyessä niin kauan poissa, oli päättänyt lähteä etsimään häntä, ja
majavat, jotka säikähtivät hevosten melua, sukelsivat veteen ja
katosivat äkkiä.

Kenraali moitti lievästi sisarentytärtään, koska tämä oli viipynyt
niin kauan, mutta ihastuneena näkemästään nuori tyttö ei
kiinnittänyt siihen paljonkaan huomiota, vaan aikoi palata vielä toiste
paikalle näkymättömänä katselijana tarkastamaan majavien
temmellystä.
Pieni joukko suuntasi erämiehen johdolla kulkunsa majaa kohti,
missä hän oli tarjonnut suojaa keskitaivaalle nousseen auringon
paahtavilta säteiltä.
Doña Luz, jonka uteliaisuutta oli mitä suurimmassa määrässä
kiihoittanut äskeinen mielenkiintoinen näytelmä, korvasi vahingon,
jonka hänen enonsa häiritsevä keskeytys oli aiheuttanut, kysellen
Mustalta Hirveltä pienimpiäkin yksityisseikkoja majavien tavoista ja
siitä, miten niitä metsästetään.
Erämies, niinkuin yleensä kaikki ihmiset, jotka tavallisesti elävät
yksinäisyydessä, halusi mielellään tilaisuuden tarjoutuessa ottaa
korvauksen harvapuheisuudesta, jota hän suurimman osan ajastaan
pakostakin noudatti, eikä siis antanutkaan itseään moneen kertaan
pyytää.
"Kuulkaas, señorita", sanoi hän, "punanahat sanovat, että majava
on ihminen, joka ei puhu, ja he ovat oikeassa. Se on viisas,
ymmärtäväinen, rohkea, uuttera ja säästäväinen. Kun talvi lähenee,
ryhtyy koko perhe varaamaan ruokaa. Nuoret ja vanhat, kaikki
tekevät työtä. Usein niiden täytyy tehdä pitkiä matkoja löytääkseen
puunkuorta, josta ne pitävät enemmän kuin muusta. Ne kaatavat
väliin sangen suuria puita ja karsivat niistä oksat, joiden kuori on
niiden parasta herkkua. Ne leikkaavat ne noin kolmen jalan pituisiin
palasiin, kuljettavat ne vesireitille ja uittavat majalleen asti, missä
panevat ne varastoon. Niiden asunnot ovat siistit ja mukavat. Ne

heittävät huolellisesti aina ruoka-ajan jälkeen virtaan sulun
alapuolelle ne puunpalaset, joista ovat kalvaneet kuoren. Ne eivät
koskaan salli vieraan majavan tulla rakentamaan yhdyskuntaa
viereensä, ja usein ne taistelevat mitä rajuimmin turvatakseen
alueensa loukkaamattomuuden."
"Kaikki tuohan on varsin merkillistä", lausui nuori tyttö.
"On, mutta siinä ei ole vielä kaikki", jatkoi erämies. "Keväällä,
jolloin on karvojen luomisaika, uros jättää naaraan kotiin ja lähtee
kuin suuri herra ainakin huviretkelle poistuen usein varsin loitolle. Se
pitää hauskaa läpikuultavan kirkkaissa lampivesissä, joita se tapaa
matkallaan, nousee rannoille jyrsiäkseen nuorten poppeli- ja
pajupuiden hentoja runkoja. Mutta kun kesä lähenee, luopuu se
vanhanpojan elämästä, muistaa velvollisuutensa perheen päänä ja
palaa puolisonsa ja poikastensa luokse, jotka se vie etsimään
varastoja talveksi."
"Täytyy tunnustaa", huomautti kenraali, "että tämä eläin on
luomakunnan merkillisimpiä."
"Niin", puuttui doña Luz puheeseen, "en voi ymmärtää, missä
tarkoituksessa niitä metsästetään kuin vahingollista petoeläintä."
"Mitä tarkoitatte, señorita?" vastasi erämies filosofisesti. "Kaikki
eläimethän ovat ihmistä varten luodut, varsinkin tämä, jonka turkki
on varsin arvokas."
"Tosiaankin", myönsi kenraali. "Mutta miten toimitatte tuon
metsästyksen? Eivät kaikki majavat ole yhtä rohkeita kuin nämä.
Useimmat lienevät niin varovaisia, että ne kätkevät pesänsä
äärimmäisen huolellisesti", lisäsi hän.

"Totta kyllä", vastasi Musta Hirvi, "mutta kokemus on varustanut
tottuneen metsämiehen niin terävällä katseella, että hän usein
sangen vähäisenkin merkin avulla keksii majavan jäljet ja pesän,
vaikka viimemainittua peittäisikin tiheä, nuori näreikkö ja se olisi
pajujen varjossa. Harvoin tapahtuu, ettei erämies osaisi tarkalleen
määrätä pesän asujainten lukuakin. Keksittyään pesän hän asettaa
paikoilleen pyydyksensä. Se pannaan lähelle joen rantaa kaksi tai
kolme tuumaa vedenpinnan alapuolelle ja kiinnitetään ketjuilla lujasti
saveen tai hiekkaan vajotettuun paaluun. Pieni puunpalanen, joka
kuoritaan, ja johon sivellään rohtoa, — niin nimitetään syöttiainetta,
jota yleensä käytetään, — asetetaan noin kolme tai neljä tuumaa
vedenpinnan yläpuolelle, ja toinen pää kiinnitetään vedenpinnan alla
olevaan pyydykseen. Majavaa, jolla on erikoisen hieno vainu, kiehtoa
puoleensa syötin tuoksu. Mutta heti kun se lähestyy ja pistää
kuononsa lähelle syöttiä, alkaakseen sitä kalvaa, tarttuvat sen
käpälät pyydykseen. Kauhistuneena se sukeltaa veden alle, sen
jalassa riippuva pyydys tekee kuitenkin turhiksi kaikki sen
ponnistukset päästä pakoon. Se taistelee jonkun aikaa, sitten se
vihdoin voimien loppuessa painuu pohjaan ja hukkuu. Sillä tapaa
tavallisesti metsästetään majavia, señorita, mutta kallioperäisillä
paikoilla, missä paalua ei voi vajottaa maahan pyydyksen pitimeksi,
on meidän usein pakko kauan etsiskellä, vieläpä uidakin pitkiä
matkoja löytääksemme pyydystetyn majavan. Tapahtuu myöskin,
että kun samasta perhekunnasta on pyydystetty useampia jäseniä,
muut tulevat varovaisiksi. Vaikka mitä metkuja silloin käyttäisimme,
on mahdotonta saada niitä tarttumaan syöttiin. Ne lähestyvät
varovaisesti pyydyksiä, päästävät kepin avulla pontimen vireestä,
vieläpä usein vääntävät pyydykset ylösalasin, laahaavat ne sulkunsa
alapuolelle ja hautaavat ne maahan."
"Miten silloin menetellään?" kysyi nuori tyttö.

"Silloinko?" toisti Musta Hirvi. "Sellaisissa tapauksissa meillä on
vain yksi keino käytettävänämme, nimittäin heittää pyydykset
selkäämme, tunnustaa majavien voittaneen meidät ja matkata
etäämmäksi etsimään toisia, tottumattomampia otuksia. Mutta
tässähän on majani."
Matkalaiset saapuivat samassa kurjannäköiselle majalle. Se oli
tehty punotuista oksista ja kykeni töintuskin suojaamaan
auringonsäteiltä. Hoidon puutteessa se joka suhteessa muistutti
preirieiden muiden erämiesten majoja, erämiehet kun vähimmin
kiinnittävät huomiota elämän mukavuuksiin.
Vaikka maja oli sellainen kuin se oli, kutsui Musta Hirvi kohteliaasti
vieraita sisään.
Toinen erämies istui kyyristyneenä majan edessä kypsentämässä
bisoninpaistia, jonka Musta Hirvi oli edeltäkäsin kertonut tulevan
ateriaksi.
Mies, jonka puku täysin muistutti Mustan Hirven pukua, oli lähes
neljänkymmenen vuoden ikäinen, mutta lukemattomat rasitukset ja
kurjuus, jotka kuuluivat partiomiehen kovaan ammattiin, olivat
uurtaneet hänen kasvoihinsa verkon ristiin rastiin kulkevia ryppyjä,
jotka tekivät hänet paljon vanhemman näköiseksi kuin hän
todellisuudessa oli.
Maailmassa ei tosiaankaan ole vaarallisempaa, rasittavampaa ja
vähemmän tuottavaa ammattia kuin erämiehen. Nuo ihmisparat
saavat usein havaita joko intiaanien tai muiden metsästäjien
ryöstäneen heidän vaivalla hankkimansa saaliin, heiltä nyljetään
päänahka tai heidät teurastetaan, eikä kukaan koskaan ota selvää,
mihin he ovat joutuneet.

"Istuutukaa, señorita, ja te myöskin hyvät herrat", sanoi Musta
Hirvi kohteliaasti. "Niin vähäpätöinen kuin kotini onkin, on se
kuitenkin siksi suuri, että siinä on tilaa teille kaikille."
Matkailijat vastaanottivat kiitollisina pyynnön. He laskeutuivat
maahan, ja pian he olivat pitkällään kuivista lehdistä valmistetuilla
vuoteilla, joita peittivät karhun-, hirven- ja bisoninnahat.
Aterian, oikean metsästäjien aterian, aikana juotiin palan paineeksi
muutamia kupillisia mainiota mezcal-viiniä, jota kenraali aina kuljetti
mukanaan retkillään ja jota erämiehet arvostelivat hyväksi, kuten se
ansaitsikin.
Sillä aikaa kun doña Luz, opas ja lancerot nauttivat päivällisunta
antaakseen auringonsäteiden synnyttämän lämmön vähän haihtua,
pyysi kenraali Mustaa Hirveä kanssaan kävelemään ja poistui hänen
seurassaan majasta.
Heti kun he olivat päässeet jonkun matkan päähän, istuutui
kenraali erään ebenpuun juurelle pyytäen seuralaistansa tekemään
samoin, minkä tämä tekikin.
Hetken aikaa vaiti oltuaan kenraali alkoi puhua:
"Ystäväni", sanoi hän, "sallikaa minun ensin kiittää teitä suuresta
vieraanvaraisuudestanne. Täytettyäni tämän velvollisuuteni haluaisin
tehdä teille eräitä kysymyksiä."
"Caballero!" vastasi erämies vältellen. "Olette kai kuullut, mitä
punanahat sanovat. 'Vedä joka sanan välissä haiku piipustasi
harkitaksesi visusti sanojasi.'"

"Puhutte minulle kuin varovainen mies ainakin, mutta olkaa
levollinen. Minulla ei ole vähäisintäkään aikomusta kysellä teiltä
sellaista, mikä kuuluu ammattisalaisuuksiinne, tai mitään muutakaan,
joka koskisi henkilökohtaisesti teitä itseänne."
"Jos osaan vastata kysymyksiinne, caballero, niin olkaa varma
siitä, etten emmi tehdä mieliksenne."
"Kiitos, ystäväni, en ole aikonut pyytääkään teiltä enempää.
Kuinka kauan olette asunut preiriellä?"
"Jo kymmenen vuotta, ja Jumala suokoon, että saisin jäädä tänne
vielä pitkäksi aikaa."
"Partioelämä siis miellyttää teitä?"
"Enemmän kuin kuvata osaan. Täytyy alkaa, niinkuin minä, jo
lapsena, kestää kaikki koettelemukset, kärsiä kaikki vaivat ja olla
osallisena kaikenlaisissa seikkailuissa voidakseen ymmärtää tämän
elämän hurmaavaa luontoa, sen tarjoamia taivaallisia nautintoja ja
merkillistä mielihyväntunnetta, johon se meidät saattaa! Oi,
caballero, Euroopan kaunein ja suurin kaupunki on sangen pieni,
sangen likainen ja sangen viheliäinen erämaahan verrattuna. Liian
ahdas, säännöllinen ja jäykkä elämä, jota te vietätte, on varsin
kurjaa meikäläisen elämään verrattuna! Vain täällä ihminen tuntee
ilman vaivattomasti tunkeutuvan keuhkoihinsa, saa tuntea elävänsä
ja ajattelevansa. Sivistys vetää hänet alas melkein järjettömän
luontokappaleen tasalle jättämättä hänelle muuta vaistoa kuin mitä
välttämättömästi on tarpeen saastaisten harrastusten
perilleajamiseksi. Mutta sensijaan preiriellä, keskellä erämaata,
kasvoista kasvoihin Jumalan kanssa, hänen aatoksensa laajenevat,

hänen mielensä avartuu, ja hänestä tulee todellisuudessa se, joksi
korkein olento hänet on aikonutkin, nimittäin luomakunnan herra."
Lausuessaan nämä sanat erämies oli tavallaan muuttunut, hänen
kasvoilleen oli tullut innostuksen ilme, silmät salamoivat, ja ryhti
suoristausi yleväksi.
Kenraali huokasi syvään, ja salaa vieri kyynel hänen
harmaantuneille viiksilleen.
"Se on totta", sanoi hän surumielisenä, "tämä elämä tarjoaa outoa
riemua sille, joka sitä haluaa, riemua, joka punoo siteitä sellaisia,
ettei niitä voi katkoa. Kun te tulitte preirieille, niin mistä te
saavuitte?"
"Tulin Quebecista. Olen kanadalainen."
"Vai niin."
Syntyi hetken vaitiolo, ja sitten kenraali jatkoi:
"Eikö toverienne joukossa ole meksikolaisia?"
"On, useita."
"Haluaisin saada tietoja heistä."
"Vain yksi mies voi niitä teille antaa, mutta se mies ei, ikävä kyllä,
ole tällä hetkellä täällä."
"Ja mikä on hänen nimensä?"
"Uskollinen Sydän."

"Uskollinen Sydän", toisti kenraali vilkkaasti. "Muistelen tuntevani
sen miehen."
"Todellako?"
"Oh! Hyvä Jumala, mikä kova onni!"
"Kenties teidän on helpompi kuin luulettekaan hänet tavata, jos
tosiaankin haluatte hänet nähdä."
"Haluaisin äärettömän mielelläni!"
"Olkaa siis levollinen. Kohta saatte hänet nähdä."
"Mitenkä niin?"
"No, aivan yksinkertaisesta syystä! Uskollinen Sydän virittää ansoja
aivan lähellä minua. Minä huolehdin niistä tällä hetkellä, mutta ei
kestä kauan, ennenkuin hän palaa."
"Oi, jospa Jumala soisi, että puheenne pitäisi paikkansa!" lausui
kenraali liikutettuna.
"Heti kun hän saapuu, ilmoitan siitä teille, ellette jo siihen
mennessä ole lähtenyt leiristänne."
"Tiedättekö sitten, minne joukkoni on leiriytynyt?"
"Täällä erämaissa me tiedämme kaikki", vastasi erämies hymyillen.
"Luotan lupaukseenne."
"Teillä on sanani, kenraali."

"Kiitos."
Samassa doña Luz tuli ulos majasta, viitaten Mustalle Hirvelle
merkiksi siitä, että tämä olisi vaiti. Kenraali kiirehti häntä vastaan.
Matkalaiset nousivat ratsuilleen ja kiitettyään erämiehiä heidän
sydämellisestä vieraanvaraisuudestaan lähtivät takaisin leirille.

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Analysis And Design For Positive Stochastic Jump Systems Wenhai Qi

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Analysis And Design For Positive Stochastic Jump Systems Wenhai Qi

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