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Lecture Notes
in Control and Information Sciences 426
Editors
Professor Dr.-Ing. Manfred Thoma
Institut fuer Regelungstechnik, Universität Hannover, Appelstr. 11, 30167 Hannover,
Germany
E-mail: [email protected]
Professor Dr. Frank Allgöwer
Institute for Systems Theory and Automatic Control, University of Stuttgart,
Pfaffenwaldring 9, 70550 Stuttgart, Germany
E-mail: [email protected]
Professor Dr. Manfred Morari
ETH/ETL I 29, Physikstr. 3, 8092 Zürich, Switzerland
E-mail: [email protected]
Series Advisory Board
P. Fleming
University of Shefﬁeld, UK
P. Kokotovic
University of California, Santa Barbara, CA, USA
A.B. Kurzhanski
Moscow State University, Russia
H. Kwakernaak
University of Twente, Enschede, The Netherlands
A. Rantzer
Lund Institute of Technology, Sweden
J.N. Tsitsiklis
MIT, Cambridge, MA, USA
For further volumes:
http://www.springer.com/series/642

Abdellah Benzaouia
SaturatedSwitchingSystems
ABC

Author
Professor Abdellah Benzaouia
Department of Physics
Faculty of Science Semlalia
University of Cadi Ayyad
B.P. 2390, 40 000 Marrakech
Morocco
E-mail: [email protected]
ISSN 0170-8643 e-ISSN 1610-7411
ISBN 978-1-4471-2899-1 e-ISBN 978-1-4471-2900-4
DOI 10.1007/978-1-4471-2900-4
Springer London Heidelberg New York Dordrecht
Library of Congress Control Number: 2012931319
cSpringer-Verlag London Limited 2012
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Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)

To
Laila, Souﬁane, Rim,
Samy, Mohamed Walid.
To
All my family.

Contents
1 Saturated Linear Systems: Analysis.............................. 1
1.1 Introduction . . ............................................. 1
1.2 Discrete−TimeSystems...................................... 1
1.2.1 Analysis of EquationFA + FBF = HF................... 6
1.2.2 Non−symmetricalSaturations .......................... 9
1.2.3 Asymptotic Stability . . ............................... 12
1.2.4 System Input Augmentation . .......................... 13
1.3 Continuous−Time Systems................................... 16
1.3.1 PreliminaryResults .................................. 18
1.3.2 SystemwithStateMatrixofMetzlerType ............... 24
1.3.3 ControllerDesignbyDirectProcedure .................. 25
1.4 Saturated Singular Systems.................................. 30
1.4.1 ProblemFormulation................................. 31
1.4.2 Conditions of Positive Invariance for Singular Systems..... 33
1.5 Conclusion................................................ 35
2 Saturated Linear Systems: Controller Design..................... 37
2.1 Introduction . . ............................................. 37
2.2 Resolution of the Algebraic EquationXA+XBX=HX ...... 37
2.2.1 PoleAssignmentProblem............................. 38
2.2.2 Resolution of theAlgebraic Equation................... 41
2.2.3 ControllerDesignbyInverseProcedure ................. 51
2.2.4 Reduction of theAlgebraic Equation................... 57
2.3 ControllerDesignUsingLMIs ............................... 59
2.4 StabilizationofLinearSystemsSubjecttoActuatorSaturation..... 62
2.5 Saturated Singular Systems.................................. 66
2.5.1 Solution of EquationXA+XBXE=HXE.................. 66
2.5.2 Controller Design for Singular Systems. . ................ 69
2.6 Conclusion................................................ 75

VIII Contents
3 Introduction to Switched Systems................................ 77
3.1 Introduction . . ............................................. 77
3.2 PhysicalExamples ......................................... 77
3.3 Continuous−Time Systems................................... 78
3.3.1 ProblemFormulation................................. 78
3.3.2 Unstability of the Switched System: Example 1........... 78
3.3.3 Stability of the Switched System: Example 2 . . ........... 79
3.4 Common Lyapunov Quadratic Function . . . ..................... 80
3.4.1 EquivalencewithPolytopicUncertainStructure........... 81
3.5 Stability Analysis of Switched Systems........................ 82
3.5.1 Unstability of Switched Systems: Example 3 . . ........... 83
3.5.2 Stability of Switched Systems: Example 4............... 83
3.5.3 Stability Analysis of Switched Systems . . ................ 84
3.6 Multiple Lyapunov Functions . ............................... 85
3.6.1 TimeBasis.......................................... 85
3.6.2 Example5 ......................................... 85
3.6.3 Example6.......................................... 86
3.6.4 DesignoftheSwitchingScheme ....................... 86
3.7 Discrete−TimeSwitchedSystems ............................. 88
3.7.1 ProblemFormulation ................................ 88
3.7.2 Common Lyapunov Function .......................... 88
3.7.3 Multiple Lyapunov Functions .......................... 88
3.7.4 Stabilization by Multiple Lyapunov Functions . ........... 90
3.7.5 Remarks............................................ 92
3.8 Conclusion................................................ 94
4 Saturated Control Problem of Switching Systems................. 95
4.1 Introduction . . ............................................. 95
4.2 SaturatedSwitchingLinearSystems........................... 95
4.2.1 ProblemStatement................................... 95
4.2.2 LinearSystemsAnalysis.............................. 97
4.2.3 StabilizationProblemwithConstrainedControl...........100
4.3 Saturated Switching Singular Linear Systems...................110
4.3.1 ProblemFormulation.................................110
4.3.2 AnalysisandControllerDesign ........................111
4.3.3 Example............................................114
4.4 Conclusion................................................116
5 Saturated Markovian Switching Systems.........................117
5.1 Introduction . . .............................................117
5.2 Discrete−TimeSystems......................................117
5.2.1 ProblemStatement...................................117
5.2.2 PreliminaryResults ..................................118
5.2.3 DesignoftheController ..............................123
5.2.4 Resolution of EquationG(
α)Ac(α)=H(α)F(α).........126

Contents IX
5.3 Continuous−Time Systems...................................131
5.3.1 ProblemFormulation.................................131
5.3.2 PreliminaryResults ..................................132
5.3.3 ControllerDesign....................................135
5.4 ComplementtotheDiscrete−TimeCase........................141
5.4.1 PreliminaryResult ...................................142
5.4.2 ControllerDesign....................................142
5.5 Conclusion................................................144
6 Stabilization of Saturated Switching Systems......................145
6.1 Introduction . . .............................................145
6.2 ProblemFormulation .......................................145
6.3 PreliminaryResults.........................................146
6.4 Analysis and Synthesis of Stabilizability . . .....................147
6.4.1 Region of Asymptotic Stability........................148
6.4.2 StateFeedbackControl ...............................149
6.4.3 OutputFeedbackControl..............................155
6.5 Conclusion................................................163
7 Stabilization of Saturated Switching Systems with Uncertainties.....165
7.1 Stabilization of Saturated Switching Systems with Polytopic
Uncertainties ..............................................165
7.1.1 ProblemPresentation.................................165
7.1.2 Analysis and Synthesis of Stabilizability . ................167
7.1.3 SynthesisofUnsaturatingControllers...................174
7.2 Stabilization of Saturated Switching Systems with Parametric
Uncertainties ..............................................176
7.2.1 ProblemPresentation.................................176
7.2.2 Analysis and Synthesis of Stabilizability . ................177
7.3 Stabilization of Uncertain State Saturated Discrete−Time Switched
Systems ..................................................185
7.3.1 ProblemPresentation.................................185
7.3.2 PreliminaryResults ..................................186
7.3.3 Stability Analysis ....................................188
7.4 Conclusion................................................193
8 Stability and Stabilization of Positive Switching Linear
Discrete-Time Systems..........................................195
8.1 Introduction . . .............................................195
8.2 Stability Analysis for Positive Switching Linear Discrete−Time
Systems ..................................................196
8.2.1 ProblemFormulation.................................196
8.2.2 PreliminaryResults ..................................197
8.2.3 Stability Analysis with Multiple Lyapunov Functions ......199

X Contents
8.3 Stabilization of Positive Switching Linear Discrete−Time
Systems ..................................................205
8.3.1 ProblemFormulation.................................206
8.3.2 Synthesis of Stabilizing Controllers by State Feedback
Control.............................................208
8.3.3 Synthesis of Stabilizing Controllers by Output Feedback . . . 212
8.4 Conclusion................................................216
9 Stabilization of Discrete 2 D Switching Systems....................217
9.1 Introduction . . .............................................217
9.2 StateFeedbackControl......................................218
9.2.1 ProblemStatement...................................218
9.2.2 PreliminaryResults ..................................220
9.2.3 Stabilizability Analysis and Controller Design . ...........221
9.3 OutputFeedbackControl....................................233
9.3.1 ProblemFormulation.................................233
9.3.2 Preliminaries........................................235
9.3.3 Stabilizability Analysis and Controller Design . ...........235
9.4 Conclusion................................................245
10 Switching Takagi-Sugeno Systems................................247
10.1 Introduction . . .............................................247
10.2 Stabilization of Switching T−S Systems by Multiple Switching
Lyapunov Function . ........................................247
10.2.1 ProblemFormulation.................................247
10.2.2 ConditionsofStabilization ............................250
10.3 Stabilization of Switching T−S Systems by Switched Lyapunov
Function..................................................255
10.3.1 ConditionsofStabilization ............................255
10.4 SaturatedDiscrete−TimeSwitchingT−SFuzzySystems...........262
10.5 SaturatedPositiveSwitchingT−SFuzzyDiscrete−TimeSystems....264
10.6 ApplicationtoaRealPlantModel.............................265
10.7 Conclusion................................................272
10.8 GeneralConclusion.........................................273
References.........................................................275
Index.............................................................285

Acronyms
C. T. S Continuous−time system
D. T. S Discrete−time system
C. T. C Continuous−time case
D. T. C Discrete−time case
T−S Takagi−Sugeno
SOFC Static output feedbcak control
LMI Linear matrix inequality
2D Twodimensional
OAE Output algebraic equation
w.r.t. with respect to
s.t. such that
resp. respectively
Eq. Equation
Eqs. Equations
co{.}Convex hull of{.}
PDC parallel distributed compensation
SISO Single input single output
MIMO Multiple input multiple output

Notations
•Ifx,yare vectors ofR
n
,thenx≤ystands componentwise.
•For a matrixA∈R
n×m
,|A|is the matrix formed by the absolute value of the
components ofA, while
σ(A)denotes its spectrum.
•For a vector
θi∈R
n
,θ
i
l
indicates thel
th
component of the vector.
•intDdenotes the interior of the setD.
•For a square matrixQ>0,(Q≥0)ifQ∈R
n×n
is positive deﬁnite ( positive semi
deﬁnite, respectively).
•Q
j,j=1,...,n, denotes thej
th
row of matrix Q.
•Aρ0 stands for apositivematrixA, that is, a matrix with nonnegative elements:
a
ij≥0.
•
ρ(A)stands for the radius spectrum of matrixA.
•I:={1,...,N}, whileI
2
=I×I.
•A matrix whose its off−diagonal entries are non positive is called Z−matrix.
•Scalar
ηdenotes,η=2
m
.
•Idenotes the identity of appropriate size.
•For a square matrixH∈R
m×m
,˜Hd,αHc∈R
2m×2m
are deﬁned as
˜H
d=
σ
H
+
H
−
H
−
H
+
θ
,
αH
c=
σ
H
1H2
H2H1
θ
with
H
+
(i,j)=h
+
ij
=Sup(h ij,0),H
−
(i,j)=h
−
ij
=Sup(−h ij,0),
fori,j=1, ...,n,whereh
ijdenotes the matrix componentH(i,j)and
H
1(i,j)=
ρ
h
ijif i=j
h
+
ij
if iη =j
H
2(i,j)=
ρ
0if i=j
h −
ij
if iη =j
•For a complex vector
ξ,
ˉξstands for the conjugate vector while
ˉ λstands for the
conjugate of the complex scalar
λ.

Introduction and Preview
Introduction
Switched systems are a class of hybrid systems encountered in many practical sit−
uations which involve switching between several subsystems depending on various
factors [43, 46, 54, 75, 118, 142]. Generally, a switching system consists of a fam−
ily of continuous−time subsystems and a rule that supervises the switching between
them. For example, many process in thechemical and pharmaceutical industries
operate following batches, composed of different operations that are carried out in
sequence. This changes discontinuously the dynamics of the operation [144]. In
manufacturing, hybrid switched systems are found in steel rolling mills [112], used
for producing thin metal sheets, following several steps based on pressing the metal
strip with rolling cylinders: the dynamics are known to change at each pass due to
the variation in thickness [50]. Many other examples can be found in the automotive
industry, in aircraft and air trafﬁc control, and many other ﬁelds. Some stabilizability
problems for switching/switched systems are studied in [77, 120, 159, 160].
Two main problems are widely studied in the literature according to the classiﬁ−
cation given in [45]: The ﬁrst one, which is the one solved in this book, looks for
testable conditions that guarantee the asymptotic stability of a switched system un−
der arbitrary switching rules, called in this book switching system, while the second
is to determine a switching sequence that renders the switched system asymptoti−
cally stable (see [118]) and the reference therein).
A main problem which is always inherent to all dynamical systems is the
presence of actuator saturations. Even for linear systems, this problem has been
an active area of research for many years. Besides approaches using anti−windup
techniques [131], model predictive controls [56] and asymmetricl
1[136], two main
approaches have been developed in the literature: The ﬁrst is the so−called positive
invariance approach which is based on the design of controllers which work inside
a region of linear behavior where saturations do not occur (see [10, 15, 17, 44],
and the references therein). This approach has already being applied to a class of
hybrid systems involving jumping parameters [51]. It has also been used to design
controllers for switching systems with constrained control under complete modelling

XVI Introduction and Preview
taking into account reset functions at each switch and different system’s dimension.
The second approach, however allows saturations to take effect while guarantee−
ing asymptotic stability (see [26, 103, 104, 150] and the references therein). The
main challenge in these two approaches is to obtain large domains of initial states
which ensures asymptotic stability for the system despite the presence of saturations
[17, 18, 94, 103].
The objective of this book is to present theavailable results in the literature for
switching systems subject to actuator saturations. These results follow generally two
ways: the ﬁrst concerns the synthesis of non saturating controllers (controllers work−
ing inside a large region of linear behavior where the saturations do not occur), while
the second extends the results obtained for unsaturated switching systems leading to
saturating controllers (controllers tolerating saturations to take effect)[65, 66, 85].
The ﬁrst approach was successfully used to study a general class of switching sys−
tems, with reset functions and different subsystem orders [27]. Necessary and suf−
ﬁcient condition of positive invariance is obtained. A design method based on the
resolution of equationXA+XBX=HXis proposed. A new topic, using the same
approach, presented for the ﬁrst time in this book, concerns the study of saturated
singular switching systems. The controller design is developed upon the resolution
of the algebraic equationXA+XBXE=HEalso presented in this book. The second
method was ﬁrstly used in [22] with the use of a multiple Lyapunov function. How−
ever, only the intersection of all the corresponding level sets of the local functions
was considered as a region of asymptotic stability of the switching system. This
drawback is improved in [25, 35] by considering, for the ﬁrst time, a large set of
asymptotic stability composed by the union of all the level sets. In this context, dif−
ferent sufﬁcient conditions of asymptotic stability were obtained for switching sys−
tems subject to actuator saturations. Furthermore, these conditions were presented
in the form of LMIs for the state feedback control case.
The static output feedback control (SOFC) for dynamical systems plays a very
important role in control theory and applications. The purpose is to design con−
trollers such that the resulting closed−loop system is asymptotically stable without
using any reconstruction method of the unavailable states. This type of control was
already used for switching systems in [66] and extended to saturated switching sys−
tems in [25] and positive switching systems in [29]. In this book, a particular atten−
tion is given to the output feedback case which has an additive complexity due to the
output equation. It is also shown, upon examples, that the LMIs obtained for com−
puting controllers working inside a large region of linear behavior are not always
more conservative. The obtained results are then extended to uncertain switching
systems subject to actuator saturations as developed in [30, 31]. The uncertainty
types considered in these two works are the polytopic one and the structured one.
This second type of uncertainty was also studied, without saturation, in [98].
A frequent and inherent constraint in dynamical systems is the nonnegativity
of the controls and/or the states. Systems with nonnegative states are important in
practice because many physical and chemical processes involve quantities that have
intrinsically a constant and nonnegative sign: temperatures, level of liquids, con−
centration of substances, etc, are of course positive or nonnegative. This problem

Introduction and Preview XVII
is of relevance in many practical applications of switched systems, as the states
frequently cannot take negative values for safety or production regulations. For ex−
ample, control of the force applied in rolling mills must takeinto account that the
width of the metal strip must always be positive. Unfortunately, in the presence of
switches it is not simple to check that a given control technique will respect this non−
negativity, which prompts the development of speciﬁc control design techniques for
switched systems that take into account this request.
In the literature, systems where the states are nonnegative whenever the initial
conditions are nonnegative are referred to as being positive [84]. The design of con−
trollers for these positive systems has been extensively studied even with constraints,
see for example [1, 2, 99, 100] and references therein. To our knowledge, few works
have directly considered positive switching linear systems. For example, the sta−
bility of switching positive systems composed of two subsystems has been studied
by [96, 146]. In these works, the authors studied the difﬁculty of constructing a
common Lyapunov function for a positive switching continuous−time system. The
problem of switching positive systems was investigated in [29, 34] by using multi−
ple Lyapunov functions but without saturation. The obtained results are recalled in
this book and extended to saturated positive switching systems in discrete−time.
This book also studies the stabilization problem by state and output feedback con−
trol for linear two dimensional (2D) switching systems. In the last two decades, the
2D system theory has received a considerable attention by many researchers. The
2D linear models were introduced in the seventies [87, 88] and have found many
applications, in digital data ﬁltering, image processing, modeling of partial differ−
ential equations [127], among others. On the other hand, the stabilization problem
is not fully investigated and still not completely solved. It has been shown that the
stability of 2D systems can be reduced to checking the stability of 2D characteristic
polynomial [4, 141, 157]. Further, in the literature, various types of easily check−
able, but only sufﬁcient conditions, for asymptotic stability and stabilization for 2D
linear systems have been proposed [90, 91, 101, 116, 164]. However, in the best of
our knowledge, no works on 2D switching systems were developed before [33, 36].
The last problem studied in this book concerns discrete−time switching nonlinear
systems. Each subsystem is written as an equivalent T−S fuzzy model. Since the
introduction of T−S fuzzy models by Takagi and Sugeno [147] in 1985, fuzzy model
control has been extensively studied because T−S fuzzy models provide an effective
representation of complex non linear systems [28, 38, 39, 132, 133, 135, 154]. In
the literature, few works were interested to switching T−S fuzzy systems. One can
consult [148, 166] where a detailed study of the problem of stability and controller
switching for switching fuzzy systems is presented. In all the previous works, only
a common Lyapuniv function was used for all the T−S fuzzy subsystems. Even for
switching linear systems, it was proven that the use of such a common Lyapunov
function leads to conservative results [54, 146]. A different approach using sampled−
data fuzzy controller for nonlinear systems based on switching T−S fuzzy model
is proposed in [115]. While robust control problem for uncertain switching fuzzy
systems is studied in [121, 153].

XVIII Introduction and Preview
In this book, sufﬁcient conditions of stabilizability, by state feedback control, are
obtained by using two types of multiple Lyapunov functions. These conditions are
then worked out to be presented in a LMI form.
Preview of Chapters
This book is composed of ten chapters: The ﬁrst and second chapters present the
background of the approach called positive invariance which was developed in the
eighteen and nineteen for linear systems to deal with saturations on the control of
the system. This approach is known by designing controllers working in a linear
behavior and does not allow saturation of the control. The use of this technique is
extended to the class of switching systems in ChapterIVand ChapterV.Themain
idea of the technique of saturation is also recalled in this chapter. This technique
presents conditions of building stabilizing controllers, for linear systems, allowing
saturation to take effect on the control. These results are then extended to switching
systems in ChapterVIand ChapterVII.
ChapterIIIintroduces the class of switching systems by showing their particular
comportment and the difﬁculties of studying the stability of this class. Some exam−
ples are presented to illustrate the different evolutions depending on the choice of
the switching sequence which can stabilize a switching system composed of two
unstable subsystems and destabilize a switching system composed of two stable
subsystems. The main tools used to study the stability of switching systems are then
recalled. A particular sight is given to multiple Lyapunov functions.
ChapterIVpresents the ﬁrst extension of the approach called positive invariance
to the class of switching systems. The switching system here is taken in a general
form with reset functions. In this chapter, necessary and sufﬁcient condition of pos−
itive invariance is obtained. A design method based on the resolution of equation
XA+XBX=HXis proposed. A main result obtained for the ﬁrst time with this ap−
proach is also presented. This result consists in proving that the union of all the sets
of local positive invariance constitutes a set of positive invariance of the switching
system. The developed controllers with this technique work only in regions of linear
behavior.
ChapterVpresents a different type of switching systems: systems where the se−
quence of switching is random and follows a Markovian law. This class is called
Systems with Markovian Jumping Parameters. For this class of systems, the ap−
proach of positive invariance is again applied. Necessary and sufﬁcient condition of
positive invariance of a common set to all the subsystems is proposed. The cases of
discrete−time and continuous−time systems are studied.
In ChapterVI, the technique of saturation is applied to switching systems al−
lowing the design of stabilizing controllers tolerating saturations to take effect on
the control. The obtained results are presented in LMI form. The objective of using
this technique is to obtain larger domains of initial values. However, this objective is
not theoretically guaranteed at all. As particular results, one ﬁnd those obtained with
positive invariance approach. A comparison between the two techniques is presented

Introduction and Preview XIX
for some examples showing the advantage and the drawback of each technique. The
problem of stabilizing switching systems by using output feedback is also studied.
ChapterVIIextends the results obtained in ChapterVIto uncertain switching
systems. Two types of uncertainties are considered: polytopic uncertainties and
parametric uncertainties. Stabilizing controllers are developed while tolerating satu−
rations despite the presence of uncertainties. Illustrative examples are presented for
each type. While ChapterVIIdeals with saturated discrete−time uncertain switch−
ing systems with arbitrary switching sequences, is also concerned with saturated
switched discrete−time uncertain systems where the switching sequence is consid−
ered as the control.
ChapterVIIIdeals with the stability and stabilizability analysis of positive
switching systems with multiple Lyapunov functions. A necessary and sufﬁcient
condition for the positive switching system to admit a multiple Lyapunov function is
proposed. A linear programming is presented to select positive diagonal matrices to
construct this Lyapunov function. The obtained results are used to study the problem
of stabilization of positive switching systems. Stabilizing controllers are designed
by using state feedback and output feedback control. All the results are presented
in LMIs form making easy their application. Illustrative examples are also studied.
The extension to saturated positive switching systems is also developed.
ChapterIXstudies the stabilization problem by sate feedback and output feed−
back control respectively for linear two dimensional (2D) switching systems. New
conditions of stabilizability are obtained in this chapter by using common and mul−
tiple Lyapunov functions for both state feedback and output feedback controls. The
case of saturated switching 2D systems is also studied. The unsaturating controllers
ensuring asymptotic stability for this class of systems is then presented.
In ChapterX, sufﬁcient conditions of stabilizability are obtained for switch−
ing nonlinear systems described under Takagi−Sugeno form. A switched Lyapunov
function and state feedback control are used. The proposed Lyapunov function is
multiple for both the T−S fuzzy subsystems and the switching modes. This choice
allows a general approach which may be less conservative. The obtained conditions
are then worked out to be presented in LMI form.

List of Figures
1.1 Blockschemaofsaturatedsystems............................. 1
1.2 Validityregionsforthelinearandthenonlinearmodel ............ 3
1.3 Non−symmetricaldomainasapositivelyinvariantset ............. 15
2.1 Trajectories obtained with unsaturating controller with the
polyhedral setL(F)insolidline .............................. 62
2.2 Trajectories obtained with saturating controller with the polyhedral
setL(F)in dotted line andL(H)insolidline .................. 65
2.3 The evolution of the three states of the singular system in closed
loop emanating fromx
o=[−1−65]
t
......................... 74
2.4 The evolution of the two controls of the singular system in closed
loop obtained with the initial statex
o=[−1−65]
t
.............. 74
3.1 The evolution of a trajectory of the switching system starting at
x
0=[1;1]
T
together with the switching ruleα................... 79
3.2 The evolution of a trajectory of the switching system starting at
x
0=[1;1]
T
and the switching ruleα........................... 80
3.3 Trajectory of the switching system and the corresponding
α(t)..... 84
3.4 The evolution of the switched system and the multiple
Lyapunov−like function . . . .................................... 86
3.5 The ellipsoid sets for the switching discrete−time linear system
computed with LMIs (3.33) and
ρ=20 ........................ 94
3.6 The ellipsoid sets for the switching discrete−time linear system
computed with LMIs (3.36) and
ρ=20 ........................ 94
4.1 The regionO
aas a set of asymptotic stability for the switching
systemHin the state space(x
1,x2)...........................110
5.1 Set of positive invariance and stochastic stabilityK
cin the state
space(x
1,x2), a jumping Markovian process in( α,t)and the
admissiblecontrol...........................................141

XXII List of Figures
5.2 A jumping Markovian process, the admissible control and the set
of positive invariance and stochastic stabilityF
ugiven as (5.60)
withatrajectory.............................................144
6.1 The ellipsoids sets of invariance and contractivity for the switching
discrete−timelinearsystemcomputedwithLMIs(6.16)−(6.17)......154
6.2 The ellipsoids sets of invariance and contractivity for the switching
discrete−timelinearsystemcomputedwithLMIs(6.27)−(6.28) .....154
6.3 The level set as the union of two ellipsoid sets of invariance and
contractivity for the switching discrete−time linear system obtained
by Corollary 6.1 together with three trajectories and a switching
sequence...................................................161
6.4 The level set as the union of two ellipsoid sets of invariance and
contractivity for the switching discrete−time linear system obtained
byCorollary6.4.............................................162
6.5 The level set as the union of two ellipsoid sets of invariance and
contractivity for the switching discrete−time linear system obtained
byCorollary6.2.............................................163
7.1 Switching signals
α(k)anduncertaintiesevolution ...............172
7.2 Motion of the system with controllers obtained with Theorem 6.5
andCorollary7.1 ...........................................172
7.3 Motion of the system with controllers obtained with Theorem7.2 . . . . 173
7.4 Inclusion of the ellipsoids inside the polyhedral sets obtained with
Theorem7.3................................................173
7.5 Level sets and the corresponding
polyhedral sets of saturations obtained with
Theorem7.4................................................174
7.6 Level sets and the corresponding polyhedral
sets of linear behavior obtained with
Theorem7.5................................................175
7.7 Inclusion of the ellipsoids inside the polyhedral sets using Theorem
7.7........................................................183
7.8 Theswitchingsequences .....................................183
7.9 Theuncertaintiesevolution ...................................183
7.10 Inclusion of the ellipsoids inside the polyhedral sets using Theorem
7.2........................................................184
7.11 Inclusion of the ellipsoids inside the polyhedral sets using Theorem
7.7........................................................184
7.12 Inclusion of the ellipsoids inside the polyhedral sets using Theorem
7.6........................................................184
7.13 Uncertaintiesevolution.......................................192
7.14 Switching signals
α(k).......................................192

List of Figures XXIII
7.15 Inclusion of the ellipsoids inside the
polyhedral set of constraint together with two
trajectories .................................................192
8.1 Cases where conditions of Theorem 8.4 with LMIs (3.25) withX
i
diagonal matrices (*),conditions of Corollary 8.1 given by (8.29)
(+)andconditionsofTheorem8.3(o)aresatisﬁed ...............204
8.2 Two Trajectories of the system in the
positive orthant and the arbitrary sequence of
switching ..................................................206
8.3 Four trajectories in the positive orthant of the state space of the
closed−loopswitchingsystem(8.45)............................211
8.4 An arbitrary switching sequence corresponding to one of the
trajectoriesinFigure8.3......................................211
8.5 Four trajectories in the positive orthant of the state space of the
closed−loopswitchingsystem(8.56)............................215
8.6 An arbitrary switching sequence corresponding to one of the
trajectoriesin8.5............................................215
9.1 How to compute the states at coordinates(k+1,l)and(k,l+1)
situated on the same line with respect to the previous ones at
coordinates(k,l)fork,l=0,...,3 situated on the same previous line 219
9.2 Transmissionline ...........................................224
9.3 The trajectory of the statesx
h
1
(k,l),x
h
2
(k,l),x
v
(k,l)and the
corresponding sequence of switching obtained with Theorem 9.5. . . . 230
9.4 The states evolution ofx
h
1
(k,l),x
h
2
(k,l),x
v
(k,l)and the
corresponding sequence of switching obtained with Corollary 9.4 . . . 239
9.5 The states evolution ofx
h
1
(k,l),x
h
2
(k,l),x
v
(k,l)and the
corresponding sequence of switching with Corollary 9.6 ...........243
10.1 Feasibility of LMIs (10.37) depending on scalarsa,b..............262
10.2 Processcomposedoftwolinkedtanks ..........................266
10.3 This ﬁgure plots the evolution of the statesx
1andx 2in liter . ......269
10.4 Evolution of the two pump ﬂows in liter/mn .....................270
10.5 Evolutionofsequenceofswitching ............................270
10.6 Evolution of the statesx
1andx 2in liter.........................272
10.7 Evolution of the two pump ﬂows in liter/mn .....................272

Chapter 1
Saturated Linear Systems: Analysis
1.1 Introduction
This chapter recalls the main results developed on saturated linear systems to be
used for saturated switching systems. Namely, two approaches are recalled:
•The so−called positive invariance approach based on the idea of working inside
linear behavior regions without tolerating the control to be saturated;
•The technique of saturation tolerating the control to be saturated while guaran−
teeing the asymptotic stability of the system.
This chapter is devoted to the study of saturated linear systems as presented in
Fig. 1.1:
Fig. 1.1Block schema of saturated systems
1.2 Discrete-Time Systems
In this section, we deal with linear discrete−time systems described by
x
k+1=Axk+Buk (1.1)
withk∈N,xis the state vector inR
n
anduis the constrained control satisfying
u
k∈Ω⊂R
m
,m≤n. (1.2)
A. Benzaouia: Saturated Switching Systems, LNCIS 426, pp. 1–35.
springerlink.com cαSpringer−Verlag London Limited 2012

2 1 Saturated Linear Systems: Analysis
MatricesAandBare constant and satisfy the assumption :
(A,B)is stabilizable. (1.3)
As it generally occurs in practical situations, the set of admissible controls
Ωis an
asymmetric polyhedral set deﬁned as
Ω=
η
u∈R
m
:−q2≤u≤q 1,q1,q2∈R
m
+
ξ
. (1.4)
In order to study the problem control design under inequality constraints, we follow
the approach adopted by [97]. Let us ﬁrst consider the unconstrained case where the
regulator problem for system (1.1) consists in the design of a feedback law as
u
k=Fxk,withF∈R
mxn
. (1.5)
Applying the control law as deﬁned above, system (1.1) becomes
x
k+1=(A+BF)x k=Aoxk (1.6)
Letx(k;x
o)be the motion of system (1.6), at timek, starting atx o.
Generally speaking, the matrixFis chosen in order to speed up the closed−loop
system dynamics with (1.6) asymptotically stable, that is,
and
ρ(A+BF)< ρ(A)andρ(A+BF)<1
rank(F)= m.
In the constrained case, the feedback law is deﬁned as
u
k=sat(Fx k)=
⎧
⎨
⎩
q
1 forFx k>q1
FxkforFx k∈Ω
−q2forFx k<−q 2,
This feedback law implies two possible models for the system in the closed loop :
(i)the linear model
x
k+1=(A+BF)x k=Aoxk, forFx k∈Ω, (1.7)
(ii)the non−linear model
x
k+1=Axk+Bsat(Fx k), forFx kη∈Ω. (1.8)
Both representations are obtained in two different regions of the state space as
indicated in Fig. 1.2.

1.2 Discrete−Time Systems 3
αx1
σ
0
x
2
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θθ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
D(F,q 1,q2)
Fig. 1.2Validity regions for the linear and the nonlinear model
The approach we deal with inthis chapter consists inproceeding in such a way
that the model (1.7) remains valid every time. This is only possible if the state is
constrained to evolve in a speciﬁed region deﬁned by :F
−1
Ω=D(F,q 1,q2).
whereF
−1
Ωstands for the inverse image of theΩwithout the requirement of the
invertibility ofF. From (1.4), (1.7) and (1.7), the set of admissible states is deﬁned
as
D(F,q
1,q2)={x∈R
n
/−q 2≤Fx≤q 1;q1,q2∈R
m
+
}. (1.9)
Note that this domain is unbounded in the general case whenm<n.
Clearly, ifx
k∈D(F,q 1,q2)we may getx k+1η∈D(F,q 1,q2).
Further, if a Lyapunov functionV(x)is known for the system (1.7), then there
always exists a scalarc∈IntRsuch that the set
D
L={x∈R
n
/V(x)≤c}, (1.10)
is a subset ofD(F,q
1,q2). Hence, for everyx k∈DL⊂D(F,q 1,q2),wehaveu k∈
Ω; consequently, the model (1.7) remains valid. Further, sinceV(x)is a Lyapunov
function for the system (1.7), then for everyx∈D
o⊂DL,whereD odenotes the
set of admissible initial states, we obtainx(k,x
o)∈D L,∀k∈Nandx(k,x o)→
0askgoes to∞.
It may be noted, from (1.4), that the setD(F,q
1,q2)is generally of a
polyhedral asymmetric nature. Thus, the largest domain of admissible initial
values of system (1.1) is obtained if
D
o=D L=D(F,q 1,q2). (1.11)

4 1 Saturated Linear Systems: Analysis
The use of a quadratic Lyapunov function only allows one to obtain an ellipsoidal
stability domain [97]. The idea of constructing the largest polyhedral stability
domainD
L⊂D(F,q 1,q2)was put forward by [62] by using simplicial cones.
Its formulation in the symmetrical case, was given by [47] who gives necessary and
sufﬁcient conditions for the setD(F,q
1,q2)withq 1=q2>0 to be positively
invariant w.r.t. the system.
Consider the following deﬁnitions:
Deﬁnition 1.1.A subsetSofR
n
is said to be positively invariant with respect to
(w.r.t.) the system (1.1)-(1.2), if for every initial state x
o∈S, every admissible
sequence
U
k={u o,u1, ...,u k−1;ui∈Ω},
the motion x(x
o,Uk,k)∈S, for every k∈N.
Deﬁnition 1.2.A subsetSofR
n
is said to be
•contractive w.r.t. the system (1.1)-(1.2), if for every x
k∈∂(τkS), there exists
τk+1>0, satisfyingτk+1<τksuch that xk+1∈∂(τk+1S), for every admissible
control u
kand k∈N.If τk<1(resp.τk>1),wesaythatthesetSis in-
contractive (resp. out-contractive) w.r.t. the system.
•attractive for a subsetTofR
n
w.r.t. the system (1.1)-(1.2) if, for every xo∈
T\S, there exists k
o∈Nsuch that x(x o,Uk,k)∈S, for every k≥k o,
and every admissible sequenceU
k.
•globally attractive w.r.t. the system (1.1)-(1.2) ifT=R
n
.
Note that the contractiveness property deﬁned here is a one−step contractivity.
In the approach proposed by Gutman and Hagander [97], the necessity of the
positive invariance property of domainD
Lw.r.t. the system (1.7) (i.e.,A oDL⊂
D
L), when we are interested in achieving (1.11), requires one to ﬁnd conditions
under which the setD(F,q
1,q2)is positively invariant w.r.t. the system (1.7), the
same system. This will be the main purpose of this chapter.
Hence, we will present in the subsequent paragraphs of this section, the nec−
essary and sufﬁcient conditions allowing the design of a regulator for linear
discrete−time systems with symmetric and asymmetric constrained control.
Let us now deﬁne the null spaceKer(F)ofFas follows :
Ker(F)=
η
x∈R
n
/Fx=0,F∈R
mxn
ξ
. (1.12)
Consider the following state transformation
z
k=Fx k, F∈R
mxn
, (1.13)
then, from (1.7), one can obtain
z
k+1=F(A+BF)x k. (1.14)

1.2 Discrete−Time Systems 5
If a matrixH∈R
mxm
exists such that,
FA
o=HF, (1.15)
or equivalently,
FA+FBF=HF,
then−order dynamical system (1.7) can be transformed to an m−order dynamical
system given by
z
k+1=Hzk,zk∈R
m
, (1.16)
and domain (1.9) becomes
D(I,q
1,q2)=

z∈R
m
/−q 2≤z≤q 1,q1,q2∈R
m
+
⊂
. (1.17)
The invariance positivity of (1.17) implies necessarily the stability ofH.Further,
comparing (1.9) with (1.17) leads to
x
k∈D(F,q 1,q2)iffz k∈D(I,q 1,q2)∀k∈N. (1.18)
It is obvious, that in this case, the domain (1.9) is positively invariant (resp. posi−
tively invariant and contractive) w.r.t. the system (1.7) if and only if domain (1.17)
is positively invariant (resp. positively invariant and contractive) w.r.t. the system
(1.16).
In this approach, matricesA,BandFare given while matrixHis obtained as
a solution to Equation (1.15). This approach is known as the direct approach and
is based on the state transformation (1.13), Equation (1.15) and the transformed
dynamical system (1.16) with domain (1.17) and property (1.18).
The ﬁrst result presented in this section, concerns a necessary and sufﬁcient con−
dition for domainD(I,q
1,q2)to be positively invariant w.r.t. the system (1.16).
Theorem 1.1.The subsetD(I,q
1,q2)ofR
n
deﬁned by (1.17) is positively invariant
(resp. positively invariant and contractive) w.r.t. the system (1.16) if and only if
˜H
dq≤q(resp.˜H dq<q), (1.19)
with
˜H
d=
∈
H
+
H
−
H
−
H
+
≥
, (1.20)
q=
∈
q
1
q2
≥
, (1.21)
and
H
+
ij
=sup(0,H ij),H
−
ij
=sup(0,−H ij),fori,j=1...n. (1.22)

6 1 Saturated Linear Systems: Analysis
Proof:
(Sufﬁciency):Letz
k∈D(I,q 1,q2),thatis,
−q
2≤zk≤q1. (1.23)
DecomposeHas follows,H=H
+
−H
−
withH
+
andH
−
both non−negative
matrices. By pre−multiplying inequality (1.23) byH
+
(resp. by−H
−
, one
obtains
−H
+
q2≤H
+
zk≤H
+
q1(resp.−H
−
q1≤−H
−
zk≤H
−
q2).(1.24)
Summing up both inequalities yields
−H
+
q2−H
−
q1≤Hzk≤H
+
q1+H
−
q2. (1.25)
Taking account of (1.19), one can deduce thatz
k+1∈D(I,q 1,q2),forevery
k∈N. Note that the contractivity property is guaranteed with˜H
dq<q.
(Necessity:)Assume that the setD(I,q
1,q2)is positively invariant w.r.t. the
system (1.16) and condition (1.19) is violated. Thus, without loss of generality,
assume there exists an integerk∈Nand a subscripti∈[1, ...,n]such that,
H
+
i
q1+H
−
i
q2>(q 1)i, (1.26)
whereH
+
i
is thei
th
row vector of the matrixH
+
, the same deﬁnition applies for
H
−
i
.Thei
th
component ofq 1is denoted as(q 1)i. Consider the following state
χkgiven by
(
χk)j=
⎧
⎨
⎩
(q
1)jifHij>0,
0ifH
ij=0, j=1...n.
−(q
2)jifHij<0,
Note that
χk∈D(I,q 1,q2).The stateχk+1=Hχkhas thei
th
component
(
χk+1)i=H iχk (1.27)
=H
+
i
q1+H
−
i
q2
which, by virtue of (1.26), leads to( χk+1)i>(q1)i. The consequence is that
χk+1η∈D(I,q 1,q2)forχk∈D(I,q 1,q2)which contradicts our initial
assumption. α
1.2.1 Analysis of Equation FA + FBF = HF
This section deals with the analysis of Equation (1.15) by relating the existence of
the matrixHtoKer(F),matrixAand matrixF. This equation represents the

1.2 Discrete−Time Systems 7
pivot of this approach like the Riccati equation in the linear optimal control. The
following results can be found in [10, 13].
Lemma 1.1.The setKer(F)is positively invariant with respect to the motion of
the system (1.7) if and only if there exists a matrix H∈R
mxm
satisfying (1.15).
Proof:
(Sufﬁciency):Assume that there exists a matrixHsatisfying (1.15) and letx∈
Ker(F),thatisFx=0. Then,FA
ox=HFx=0,i.e.,A ox∈Ker(F).
(Necessity:)Letw∈Ker(F)and assume thatKer(F)is positively invariant
w.r.t. (1.7), then it follows thatFA
ow=0. Besides, without loss of generality,
letF,FA
oandwbe partitioned as :
F=[F
1F2],withF 1∈R
mxm
,F2∈R
mx(n−m)
,
rankF
1=m,and
FA
o=[M 1M2],withM 1∈R
mxm
andM 2∈R
mx(n−m)
,
and
w=[w
t
1
w
t
2
]
T
,withw 1∈R
m
andw 2∈R
n−m
.
Hence,Fw=0 can be written as follows :
F
1w1+F2w2=0,
which impliesw
1=−F
−1
1
F2w2. Let us underline that for everyw 2∈R
n−m
,
we have ∈
−F
−1
1
F2w2
w2
≥
∈Ker(F).
By substituting the latter equality intoFA
ow=0, it follows that for every
w
2∈R
n−m
we haveM 1F
−1
1
F2w2=M2w2, which can also be written as
HF
2=M 2, where the matrixHis taken to be
H=M
1F
−1
1
∈R
mxm
(1.28)
or equivalently, satisfying the linear equationHF
1=M 1, then we get ﬁnally
HF=[M
1M2]=FA o. ≤
Lemma 1.2.If the setD(F,q
1,q2)is positively invariant w.r.t. the system (1.7),
then, the setKer(F)is also positively invariant w.r.t. the system.
Proof:Letw∈Ker(F), then it is clear thatw∈D(F,q
1,q2).SetFA ow=r.
From the development in the proof of Lemma 1.1, it can be easily seen that
FA
ow=rcan be written, with the same notations as above, asGw 2=r, with
G=M
2−M1F
−1
1
F2∈R
mx(n−m)
.
Recall that for everyw
2∈R
n−m
, the vectorw=[w
t
1
w
t
2
]
T
∈Ker(F),
withw
1=−F
−1
1
F2w2.

8 1 Saturated Linear Systems: Analysis
Assume there exists, at least, one nonzero component ofG, for instance
gil.Forw2chosen such that,
(w2)l=α
(q1)i
gil
,withα>1,and(w2)j=0,forjη=l=1, ... ,n−m.
Further,Gw2=ryields
rs=α(q1)i
gsl
gil
fors=1, ...,m.
In particular we have for thei
th
component,ri=α(q1)iwhich yields, asα>1,
(FAow)i=(Gw2)i=ri>(q1)i,orinotherwords
FAowη∈D(F,q1,q2),
forw∈D(F,q1,q2). Hence, the vectorrwill belong to the domainD(F,q1,q2)
for everyw2∈R
n−m
only ifG=0, which yieldsr=FAow=0. α
Lemma 1.3.If a stable matrix H∈R
mxm
satisfying (1.15) exists, then the spec-
trum of the matrix A contain a set of n−m stable eigenvalues, closed under
complex conjugacy, corresponding to n−m common eigenvectors to matrices A
and Ao, belonging toKer(F)and parallel to∂D(F,q1,q2).
Proof:LetHbeamatrixinR
mxm
satisfying (1.15), and consider an eigenvectorv
ofAoassociated to the eigenvalueλ(real or complex). Then Equation (1.15)
allows us to writeFAov=HFv.SinceAov=λv, it follows thatH(Fv)=
λ(Fv). HenceFvis an eigenvector ofHcorresponding to the same eigenvalue
λ. The matrixH∈R
mxm
could admit onlymeigenvalues from the spectrum
of the matrixAo.Letσ(Ao)be the spectrum ofAogiven byσ(Ao)=Λ=
Λ1∪Λ2, withσ(H)=Λ1⊂C
m
andΛ2⊂C
n−m
.
Letαbe an eigenvector ofAoassociated toσ∈Λ2,then
FAoα=HFα=σFα,
which leads toFα=0sinceσis not assumed to belong toσ(H).Further,
Fα=0 implies thatAoα=Aα=σα, or in other words, thatσis an eigenvalue
ofAwhich means thatΛ2⊂σ(A).
Further, condition (1.7) ensures that the spectrum ofAois stablei.e.,
|λi(Ao)|<1, fori=1, ...n. Consequently,Λ2containsn−mstable
eigenvalues corresponding ton−mcommon eigenvectors to matricesAand
Aoand belonging toKer(F).Further,Fχ=r, withr=q1orr=−q2,
represents the equations ofmboundary hyper plans of the setD(F,q1,q2)and
are obviously parallel toKer(F).
Then, thesen−mvectors are parallel to∂D(F,q1,q2). Finally, since the
matrixAois real, its spectrumΛis obviously closed under complex conjugacy.
The required matrixHis also real, then its spectrumΛ1is also closed under

1.2 Discrete−Time Systems 9
complex conjugacy. All this implies that the setΛ2is necessarily closed
under complex conjugacy. α
Equation (1.15) was also studied by [139]. The following lemma is the direct
application of Porter’s result. We ﬁrst write Equation (1.15) in the following equiv−
alent form:
FA=H
oF,withH o=H−FB. (1.29)
Lemma 1.4.[139] There exists a solution H∈R
mxm
of Equation (1.29), where
F∈R
mxn
and rank(F)=m, with m≤n, if and only if :
rank(S)=m,withS=
σ
FA
F
θ
. (1.30)
Remark 1.1.
(i)When condition (1.30) is satisﬁed, matrix H can be computed from (1.28).
Lemma 1.3 allows us to relate the resolution of Equation (1.15) to an eigenvalue
assignment problem. According to this Lemma, it is clear that, if A does not
possess n−m stable eigenvalues which are closed under complex conjugacy, the
required matrix H, solution to Equation (1.15) or (1.29), will not exist for every
given matrix F satisfying (1.7). However, the necessary and sufﬁcient condition
given by Lemma 1.4 has to be satisﬁed, after Lemma 1.3, for a given matrix F
satisfying (1.7).
(ii)If A is non-singular, i.e. rank(A)=n, it follows that N=FA and F have the
same rank which yields rank(S)≥m. Consequently, a solution H to Equation
(1.29) may not always exist.
1.2.2 Non-symmetrical Saturations
In this section, we extend some results given previously to the asymmetric case. A
necessary and sufﬁcient condition for domain (1.9) to be positively invariant and
contractive w.r.t. the system (1.7) is given by the result below [10, 13].
Theorem 1.2.The subsetD(F,q
1,q2)ofR
n
given by (1.9), with rank(F)=m,
is positively invariant (resp. positively invariant and contractive) with respect to
the motions of system (1.7) if and only if the conditions below hold :
(i)There exists a matrix H∈R
mxm
satisfying (1.15), i.e.,
FA+FBF=HF, (1.31)
(ii)for q
1and q2∈R
m
+
, we have
(I−˜H
d)q≥0(resp.(I−˜H d)q>0andq∈IntR
2m
+
), (1.32)

10 1 Saturated Linear Systems: Analysis
where
˜H
d=
σ
H
+
H
−
H
−
H
+
θ
,q=
σ
q
1
q2
θ
. (1.33)
Proof: (Sufﬁciency):Assume that there exists a matrixHsatisfying conditions
(1.31)−(1.32). Applying the transformationz
k=Fxkto the closed−loop system (1.7),
leads to the setD(I,q
1,q2)and the reduced order systemz k+1=Hzk.UsingThe−
orem 1.1, condition (1.32) implies that the setD(I,q
1,q2)is positively invariant
(resp. positively invariant and contractive) with respect to the reduced order system
z
k+1=Hzk. This means that the controlu k=Fxkis always admissible which im−
plies that the setD(F,q
1,q2)given by (1.9) is positively invariant (resp. positively
invariant and contractive) with respect to the system (1.7).
(Necessity):Assume that the setD(F,q
1,q2)given by (1.9), withrank(F)=m,
is positively invariant (resp. positively invariant and contractive) with respect to the
system (1.7). According to Lemma 1.2, the setKer(F)is also positively invari−
ant (resp. positively invariant and contractive) with respect to the system (1.7). By
virtue of Lemma 1.1, there exists a matrixHsolution of Equation (1.31). Apply−
ing again the transformationz
k=Fxkto the closed−loop system (1.7), leads to the
setD(I,q
1,q2)and the reduced order systemz k+1=Hzk.ByusingTheorem1.1,
condition (1.32) is obtained. α
Remark 1.2.
(i)Note that conditions (1.32) can be rewritten as follows
˜H
dq≤q(resp.˜H dq<q), (1.34)
with˜H
d∈R
2mx2m
and q∈R
2m
+
.
Furthermore, it is worth noting that the asymmetric case requires more com-
putation time.
(ii)Condition (1.32) of Theorem 1.2 containsconditions of the symmetrical case.
To show this, let q
1=q2=ρ,then˜H dq≤q is equivalent to
H
+
ρ+H
−
ρ≤ρandH
−
ρ+H
+
ρ≤ρ
which leads to(H
+
+H
−
)ρ=|H|ρ≤ρ.
Two particular interesting cases of Theorem 1.2 can now be stated by means of the
following result.
Corollary 1.1.
(i)DomainD(F,q
1,q2)is positively invariant (resp. positively invariant and
contractive) w.r.t. the system (1.7) if and only if domainD(F,q
2,q1)is positively
invariant (resp. positively invariant and contractive) w.r.t. the system.

1.2 Discrete−Time Systems 11
(ii)If domainD(F,q 1,q2)is positively invariant (resp. positively invariant and
contractive ) w.r.t. the system (1.7), then the set
D(F,q
1,q2)∪D(F,q 2,q1)
is positively invariant (resp. positively invariant and contractive) w.r.t. the
system.
Proof:Obvious. α
When we are interested in a control vectorwith non−negative components, condition
(1.32) allows this possibility only in the case of positive invariance property of
domainD(F,q
1,q2). If the control vector is non−negative (resp. non−positive),i.e.,
0≤u≤
ρ(resp.− ρ≤u≤0), we need the following result :
Corollary 1.2.The subsetD(F,
ρ,0),(resp.D(F,0, ρ)), ofR
n
given by (1.9),
with rank(F)=m, is positively invariant w.r.t. the motion of the system (1.7) if and
only if there exists a non-negative matrix H∈R
mxm
(i)solution to the equation :
FA+FBF=HF,
(ii)and satisfying
H
ρ≤ρ.
Proof:Follows readily by replacingq
1=ρ,q2=0 in condition (1.32).α
Theorem 1.3.The domain
D(F,
ρ,∞)={x∈R
n
/Fx≤ ρ,ρ∈IntR
m
+
},
with rank(F)=m, is positively invariant (resp. positively invariant and contrac-
tive) w.r.t. the motion of system (1.7) if and only if there exists a non-negative matrix
H∈R
mxm
(i)solution to the equation
FA+FBF=HF,
(ii)and satisfying
H
ρ≤ρ(resp.H ρ<ρ).
Proof:Follows readily by following the same reasoning used to obtain condition
(1.32). The same conditions hold for the setD(F,∞,
ρ). The same result is obtained
in [47]. α
Example 1.1.Consider the linear discrete-time system described by the following
matrices :
A=
σ
−0.5−1.5
1.83.2
θ
, B=
σ
−1
2
θ
.

12 1 Saturated Linear Systems: Analysis
The matrix A is unstable, withλ1(A)=2.2and λ2(A)=0.5, then it contains
n−m=1stable real eigenvalue. The set of admissible controls
Ωis given by
Ω={u∈R
m
:−δ≤u≤10, δ≥0}.
In this case, we choose the following spectrum
Λand matrix F:
Λ={0.5,0.2},
F=[−1−1.5] with rank(F)=rank(S)=1.
Hence,
A
o=A+BF=
σ
0.50
−0.20.2
θ
.
It is obvious that, equation (1.15) yields H=0.2. Then, for
δ=1, we have
(I−˜H
d)q=[80 .8]
t
>0,
and for
δ=0, we have
(I−˜H
d)q=[00 .8]
t
≥0,
that is, condition (1.32) is satisﬁed in both cases of
δ=1andδ=0.
Consequently, the setD(F,q
1,q2)is positively invariant with respect to the
motion of the system (1.7).
1.2.3 Asymptotic Stability
In this section, we link both Lemma 1.4 and Theorem 1.2 to the asymptotic stability
of a linear discrete−time system with constrained control (1.1)−(1.5).
It is now obvious that the positive invariance property of domainD(F,q
1,q2)
w.r.t. the system (1.7) guarantees that every motion emanating from this domain can
not leave it. This property ensures that the linear model (1.7) remains valid despite
the existence of input saturations (1.4). Besides, the contractiveness property of the
same domain w.r.t. the system (1.7) implies the asymptotic stability of system
(1.1)−(1.5) for everyx
o∈D(F,q 1,q2). In addition, the positive invariance
property guarantees also the asymptotic stability of system (1.1)−(1.5).
Theorem 1.4.If there exists a matrix F∈R
mxn
such that:
(i)rank(F)=m,
(ii)
ρ(A+BF)<min(1, ρ(A)),
(iii)rank
σ
FA
F
θ
=m,
(vi)the solutionH∈R
mxm
to equationFA+FBF=HF
satisﬁes˜H
dq≤q,

1.2 Discrete−Time Systems 13
then, the system (1.1)-(1.5) with (1.7) is asymptotically stable and the set
D(F,q
1,q2)is positively invariant w.r.t. the closed-loop system.
Proof:Obvious. α
Let us underline that in Example (1.1), the system is also asymptotically stable.
1.2.4 System Input Augmentation
In this section, we present a technique allowing to avoid the conservatism of Lemma
1.3 by augmenting the system entries such that the required numbern−msta−
ble eigenvalues of matrixAbecomes equal to zero. The price of this operation
will be the augmentation of the computation time and the restriction of domain
D(F,q
1,q2), which is unbounded, to a bounded set.
Let us rewrite Equation (1.1) as follows:
x
k+1=Axk+[B0]
σ
u
k
vk
θ
(1.35)
withv
ka ﬁctitious input of the system.The set of admissible control Ωgiven by
(1.2) remains unchanged when the vectorv
ksatisﬁes
v
k∈Ωf=
η
v∈R
n−m
/−r 2≤v≤r 1;r1,r2∈R
n−m
+
ξ
, (1.36)
where vectorsr
1andr 2stand for ﬁctitious saturations. Let
g
1=
σ
q
1
r1
θ
;g
2=
σ
q
2
r2
θ
;g=
σ
g
1
g2
θ
.
The purpose remains unchanged and the feedback law becomes
w
k=
σ
u
k
vk
θ
=
σ
F
E
θ
x
k=Gx k,withG∈R
nxn
. (1.37)
In this case, domain (1.9) takes the form
D(G,g
1,g2)=D(F,q 1,q2)∩D(E,r 1,r2). (1.38)
This technique obviously implies a limitation of domainD(F,q
1,q2).The
existence of a matrixH∈R
nxn
solution to Equation (1.15) is now guaranteed if
the matrixGis invertible which implies that matrixEand vectorsr
1andr 2are
chosen such that:
(i)rank(G)=n
(ii)D
o⊂D(G,g 1,g2).
(1.39)

14 1 Saturated Linear Systems: Analysis
Recall that the setD ois the set of initial values and note that the domain
D(G,g
1,g2)is a bounded polyhedral set. In this case, the necessary and suf−
ﬁcient condition of Theorem 1.2 is expressed inR
2nx2n
instead ofR
2mx2m
.
Example 1.2.Here, we consider the same example as the one given in [97]
A=
σ
11
01
θ
,B=
σ
0.5
1
θ
,
with a different set of admissible controls
Ω={u∈R/− δ≤u≤10},
instead of|u|≤1as used in [97].
In this case, the matrix A does not contain n−m=1stable eigenvalue. Thus, we
proceed according to the previous technique. Let us take any matrix F of full rank
such that A
o=A+BF is stable, in particular the one computed in [97], i.e.,
F=[−1 −1.5].
This leads to
A
o=
σ
0.50.25
−1−0.5
θ
.
We now choose a matrix E such that
rank
σ
F
E
θ
=2,
for instance
E=[−1 −0.5].
Moreover the ﬁctitious constraints r
1,r2are chosen as r1=δand r2=10; then,
one gets g=[10
δδ10]
T
. Further, the matrix H is given by
H=
σ
0−1
00
θ
,
which yields˜H
dg=[10 0 δ10]
T
≤g, for everyδ≥0. If the domainD ois
known, many choices of E, r
1and r2are possible. Thus the setD(G,g 1,g2)is
positively invariant w.r.t. the closed-loop system for every x
o∈D(G,g 1,g2)(cf.
Fig. 1.3).
In Figure 1.3, we show domainD(G,g
1,g2)forδ=1as a positively invariant
and asymptotic stability set w.r.t. the system (1.7) with two motions : one of them is
starting inside the setD(G,g
1,g2)belonging toD(G,g 1,g2)whereas the second,
starting outside, leaving the setD(G,g
1,g2).

1.2 Discrete−Time Systems 15
−2 0 2 4 6 8 10 12 14 16 18
−30
−20
−10
0
10
x1
x2
Fig. 1.3Non−symmetrical domain as a positively invariant set
1.2.4.1 Algorithm
In order to recapitulate the steps required to satisfy our purpose, the following
algorithm is presented taking into accountthe assumption (1.3) of stabilizability of
the pair(A,B).
Algorithm1 Computation of matricesHandFfrom the dataAandB
Step 1 : If the matrixAdoes not possess a set ofn−mnon null and
stable eigenvalues, closed under complex conjugacy form<n, proceed
according to paragraph 1.2.4.
Step 2 : Take a set
Λ={λ1λ2...λn}(subject to complex conjugacy) con−
tainingn−mnon null and stable eigenvalues ofAclosed under complex
conjugacy, such as
ρ(Λ)<min(1, ρ(A)).
Step 3 : ComputeFsuch as
σ(A+BF)= Λ. Recall that, according to as−
sumption (1.3), the matrixFalways exists.
Step 4 : Ifrank(F)=rank(S)=m,gotostep5,elsegotostep1andchange
the set
Λ.
Step 5 : ComputeHby means of Equation (1.28).
Step 6 : If˜H
dq≤q, stop, else go to step 1.
In this section, necessary and sufﬁcient conditions, given in [10, 13], for an asym−
metric bounded polyhedral set to be positively invariant and contractive w.r.t. the
system (1.7) are successfully used. The application of these conditions to the reg−
ulator problem of linear discrete−time system with asymmetric constrained control
enables us to specify conditions under which, system (1.7) admits a largest posi−
tively invariant and contractive asymmetric polyhedral subset ofD(F,q
1,q2).

16 1 Saturated Linear Systems: Analysis
1.3 Continuous-Time Systems
This section is devoted to the study of linear continuous−time systems described by
Equation (1.40)
˙x=Ax+Bu, (1.40)
wherexis the state vector inR
n
,anduis the constrained control, satisfying
u∈
Ω⊂R
m
. (1.41)
MatricesAandBare constant, of appropriate size and satisfy the following
assumption.
(A,B)is stabilizable. (1.42)
Ωis the set of admissible controls deﬁned as
Ω={u∈R
m
/−q 2≤u≤q 1;q1,q2∈R
m
+
−{0}}. (1.43)
This is an asymmetric polyhedral set as is generally the case in practical situations.
Let us ﬁrst consider the unconstrained case where the regulator problem for sys−
tem (1.40) consists in realizing a feedback law as
u=Fx,F∈R
mxn
withrank(F)=m. (1.44)
In such a case, system (1.40) becomes
˙x=(A+BF)x=A
ox, (1.45)
whereFis generally chosen in such a way that an increase of system dynam−
ics is obtained with the asymptotic stability of the closed−loop system (1.45), or
equivalently
Re(
λi(A+BF))<0;i=1, ...,n. (1.46)
In the constrained case, we follow the approach proposed in Section 1.2. Recall
that this approach consists in giving conditions allowing the choice of a stabilizing
controller (1.44) such that the model (1.45) remains valid every time. This is only
possible if the state is constrained to evolve in a speciﬁed region deﬁned by
D(F,q
1,q2)={x∈R
n
/−q 2≤Fx≤q 1;q1,q2∈R
m
+
−{0}}.(1.47)
Note that this domain is unbounded whenm<n. In addition, if
x(t)∈D(F,q
1,q2),
one would get
x(t+
τ)η∈D(F,q 1,q2),∀τ≥0.

1.3 Continuous−Time Systems 17
Recall thatx(t+τ)=e
Aoτ
x(t), with
e
Aot
=
∞
∑
k=o
1
k!
t
k
A
k
o.
This matrix makes the problem very difﬁcult to deal with compared to the case of the
discrete−time systems (cf.1.2). However, an attempt will be given in the following
sections.
The present problem has been studied by many authors. For instance, [48, 49]
give a solution to the regulator problem for linear continuous−time systems with
constrained control. Their approach consists in constructing a positively invariant
domain w.r.t. the system (1.45) included in the set (1.47) and it allows to construct
an admissible control law with the asymptotic stability for the closed−loop.
Bitsoris [48] presents necessary and sufﬁcient conditions for domain (1.47) to be
positively invariant w.r.t. the system (1.45) derived by means of an indirect proof.
While in [149, 55] the domains used are given by
D
[a,b]{K}=(K+−b)∩(K−−a),
withK=K+=−K−is a proper cone, and
K+−a={x∈R
n
,/x=y+a,y∈K+}.
They give necessary and sufﬁcient conditions for such domains to be positively
invariant w.r.t. a large class of continuous−time linear systems given by
˙x=Aox+c(t),withe
Aot
K+⊂K+,
andc(t)deﬁned in a compact set. Their results can be extended to systems described
by (1.45), whose state belongs to the set (1.47), with the restrictive assumption that
e
Aot
K+⊂K+.
Benzaouia and Hmamed [14, 15] present necessary and sufﬁcient conditions for
domain (1.47) to be positively invariant w.r.t. the system (1.45) based on a new and
direct proof using asymmetric Lyapunov function given for the ﬁrst time in [11, 12].
In particular, domainD(I,q1,q2)deﬁned by
D(I,q1,q2)={z∈R
m
/−q2≤z≤q1}, (1.48)
which is described by the function
ϑ(z)=Max
i
max
ρ
z
+
i
q
i
1
,
z
−
i
q
i
2

,whenq1>0andq2>0 (1.49)
i.e.,
D(I,q1,q2)={z∈R
m
/ϑ(z)≤1}.

18 1 Saturated Linear Systems: Analysis
It follows from above that the main purpose of this chapter is to present nec−
essary and sufﬁcient conditions under which the polyhedral asymmetric domain
D(F,q1,q2)is positively invariant w.r.t. the system (1.45).
1.3.1 Preliminary Results
In this section, we present some useful deﬁnitions and results. Consider a continuous−
time nonlinear system
˙z(t)=f(z(t)),z∈R
m
,f(0)=0. (1.50)
Consider a functionϕ:R
m
→R+withϕ(0)=0 and assume thatϕis directionally
differentiable at each point in each direction and deﬁne˙ϕ(z)by
˙ϕ(z(t)) =
d
+
dt
[ϕ(z(t))] =
∂ϕ
∂z
T
f(z(t)),
=lim
ε→0
+
ϕ(z+εf(z))−ϕ(z)
ε
,
(1.51)
Deﬁnition 1.3.[151]˙ϕ(z)is the directional derivative ofϕat z in the direction
f(z), with f(0)=0and˙z(t)=f(z(t)).
Theorem 1.5.[151]
(i)If a functionϕis positive deﬁnite on a setR⊆R
m
,ϕis a Lyapunov function
of system (1.50) onRif and only if˙ϕ(z)is negative semi-deﬁnite along any
motion of the system (1.50) starting at z(to), for every z(t)∈Rand∀t>to, i.e.,
˙ϕ(z)<0,∀z(t)∈Rand∀t>to.
(ii)If a functionϕis a Lyapunov function of system (1.50) on a subsetRofR
m
then the set
D(ϕ,c)={z∈R/ϕ(z)≤c,c>0}, (1.52)
is a stability domain of the system.
Consider the following linear continuous−time stationary system,
˙z(t)=Hz(t),z∈R⊆R
m
and 0∈IntR. (1.53)
Lemma 1.5.DomainD(ϕ,1)given by (1.52) withϕ(z)a positive deﬁnite function
satisfying
ϕ(αz)=α
σ
ϕ(z);α>0,σ>0 (1.54)
is positively invariant w.r.t. the system (1.53) if and only ifϕ(z)is a Lyapunov func-
tion of the system.

1.3 Continuous−Time Systems 19
Proof:
(Sufﬁciency):Letϕ(z)be a Lyapunov function of the system, then domain
D(ϕ,1)is a stability domain , which is obviously positively invariant w.r.t. the
system.
(Necessity:)Let domainD(ϕ,1)be positively invariant w.r.t. the system, that is,
ϕ(z(t))≤1=⇒ϕ(z(t+τ))≤1,∀τ>0.
In this case, we have to show that domainD(ϕ,c)is also positively invari−
ant w.r.t. the system for every positive scalarc.Letz(t)∈D(ϕ,c),thatis,
ϕ(z(t))<c. Since functionϕ(z(t))satisﬁes (1.54), we should havec
−1
ϕ(z(t)) =
ϕ(c
−χ
z(t))≤1, withχ=σ
−1
. It follows thatc
−χ
z(t)∈D(ϕ,1), this implies
thatc
−χ
z(t+τ)∈D(ϕ,1). Using relation (1.54) a second time, we can write
equivalentlyϕ(z(t+τ))≤c, for every positive scalarc. In particular, consider
the case whenc=ϕ(z(t)), this allow us to state that
ϕ(z(t+τ))≤ϕ(z(t)),∀τ≥0,
and
lim
τ→0
+
ϕ(z(t+τ))−ϕ(z(t))
τ
≤0.
Consequently, the positive deﬁnite functionϕ(z(t))is a Lyapunov function of
the system.
(Necessity bis:)Let domainD(ϕ,1)be positively invariant w.r.t. the system
˙z(t)=f(z(t)), or explicitly
ϕ(z(t))≤1 impliesϕ(z(t+τ))≤1,∀τ>0.
Besides, without loss of generality, take a vectorzinD(ϕ,1)such asϕ(z(t)) =
c
−σ
, withc>0. It follows thatc
σ
ϕ(z(t)) =ϕ(cz(t)) =1, which means that
for everyτ≥0,ϕ(cz(t+τ))is inD(ϕ,1)since the latter set is assumed to be
positively invariant. Further, using (1.54) one can deduce easily that
ϕ(z(t+τ))≤ϕ(z(t)),
for everyτ≥0, or explicitly thatϕ(·)is a Lyapunov function. α
This lemma enables us to give a necessary and sufﬁcient condition for domain
D(I,q1,q2)deﬁned by (1.48) to be a positively invariant set w.r.t. the system (1.53)
by using the associated asymmetric function.
Theorem 1.6.[15] DomainD(I,q1,q2)⊆Rgiven by (1.48) with q1,q2>0is
positively invariant w.r.t. the system (1.53) if and only if
˜Hcq≤0, (1.55)

20 1 Saturated Linear Systems: Analysis
with
˜Hc=
σ
H1H2
H2H1
θ
; q=
σ
q1
q2
θ
H1=
ρ
hiifori=j
h
+
ij
foriη=j
;H2=
ρ
0fori=j
h
−
ij
foriη=j
.
(1.56)
Proof:
(Sufﬁciency):Consider the setD(ϕ,1)deﬁned by (1.52) with functionϕ(z)
given by (1.49). This function is continuous, positive deﬁnite [11, 12] and sat−
isﬁes relation (1.54) withσ=1. According to Lemma 1.5, domainD(ϕ,1)is
positively invariant w.r.t. the system (1.53) if and only ifϕ(z)is a Lyapunov
function for the system. It is now obvious thatD(ϕ,1)=D(I,q1,q2).Let
us compute the directional derivative of the functionϕ(·)atzin the direction
f(z)=Hz. For this, recall thatϕis a continuous positive deﬁnite function. By
virtue of Deﬁnition 1.3
˙ϕ(z)=lim
ε→0
+
ϕ(z+εHz)−ϕ(z)
ε
. (1.57)
DenoteC=I+εHand compute the rate of increaseΔϕ(z)=ϕ(Cz)−ϕ(z).For
this, we use the result of [11, 12].
Δϕ(z)≤Max
i
max

(C
+
q1)i+(C
−
q2)i
q
i
1
−1
Λ
ϕ(z);

(C
−
q1)i+(C
+
q2)i
q
i
2
−1
Λ
ϕ(z)
δ
(1.58)
It is worth noting that in the vicinity of zero,εcan be chosen sufﬁciently small,
for instance, such as the term 1+εhii≥0orinotherwords
1
ε
>Max
i
|hii|.
The matricesC
+
andC
−
whenε→0
+
are given by
C
+
=(I+εH)
+
=
ρ
εh
+
ij
foriη=j
1+εhiifori=j
=I+εH1,
C
−
=(I+εH)
−
=
ρ
εh
−
ij
foriη=j
0for i=j
=εH2.

1.3 Continuous−Time Systems 21
By substituting matricesC
−
andC
+
into (1.58), we obtain :
Δϕ(z)≤Max
i
max
ρσ
ε
(H1q1+H2q2)i
q
i
1
θ
ϕ(z);
σ
ε
(H2q1+H1q2)i
q
i
2
θ
ϕ(z)

then,
˙ϕ(z)≤Max
i
max
ρσ
(H1q1+H2q2)i
q
i
1
θ
ϕ(z);
σ
(H2q1+H1q2)i
q
i
2
θ
ϕ(z)

From condition (1.55), we conclude that
˙ϕ(z)≤0,∀z∈R⊆R
m
.
That is,ϕ(z)is a Lyapunov function of the system on the setR. Consequently,
domainD(I,q1,q2)is positively invariant w.r.t. the system (1.53).
(Necessity:)Assume that there exists a subscriptkin the set[1, ...,m]such that
Max
i
{(˜Hcq)i}=(˜Hcq)k>0.
Moreover, without loss of generality, assume thatk≤mand deﬁne the vectorχ
as follows :
χj=
⎧
⎨
⎩
(q1)jforj=k,
(q1)jforhkj≥0,
−(q2)jforhkj<0,
forj=1...m,orinotherwordshkjχj≥0foreveryjin the set[1, ...,m]−{k}
andχk=(q1)k.
It is obvious that for everyk∈[1, ...,m],wehave
(˜Hcq)k=(H1q1+H2q2)k,
=hkk(q1)k+
m
∑
j=1,jη=k

h
+
kj
(q1)j+h
−
kj
(q2)j
δ
,
=(Hχ)k,
which implies that(Hχ)k>0 and consequently that(Cχ)k>0.
Note also that
(Cχ)k=χk+ε(Hχ)k=(q1)k+ε(Hχ)k=(Cχ)
+
k
>0,
from which we deduce that
(Cχ)
+
k
(q1)
k
>1 or equivalently thatϕ(Cχ)>1.Moreover,
as it was assumed thatϕ(χ)=1, it follows thatΔϕ(χ)≥0 which leads to
˙ϕ(χ)=lim
ε→0
+
ϕ(χ+εHχ)−ϕ(χ)
ε
≥0,
or similarly, thatϕis not a Lyapunov function and this contradicts surely the
result of Lemma 1.5. α

22 1 Saturated Linear Systems: Analysis
It is possible to give the same result by using directly the development of the
matrixe
Ht
.
Theorem 1.7.The domainD(I,q1,q2)given by (1.48), with q≥0, is positively
invariant w.r.t. the system (1.53) if and only if (1.55) and (1.56) are satisﬁed.
Proof:
(Sufﬁciency):Consider the solution to (1.53) given byz(t+ε)=e
Hε
z(t)which
can be developed as follows:
z(t+ε)=z(t)+εHz(t)+ε
2
∑
∞
∑
k=2
1
k!
ε
k−2
H
k
z(t)
ϑ
.
Take the matrixΓas follows :
Γ=I+εH+ε
2
Γo,
with
Γo=
∞
∑
k=2
1
k!
ε
k−2
H
k
.
In this case, the system (1.53) can be equivalently described as
z(t+ε)=Γz(t), (1.59)
for everyε>0 as small as possible.
Letz(t)be in the setD(I,q1,q2), withq≥0,i.e.,
−q2≤z(t)≤q1.
The same arguments used in the proof of Theorem 1.2 for the discrete−time sys−
tems lead to
z(t+ε)∈D(I,q1,q2),
withq≥0, if
˜Γq≤q,
or equivalently
ρ
Γ
+
q1+Γ
−
q2≤q1,
Γ
−
q1+Γ
+
q2≤q2.
(1.60)
Further, computeΓ
+
andΓ
−
forε>0 as small as possible, as follows:
Γ
+
=I+εH1+ε
2
Γ
+
o,
Γ
−
=εH2+ε
2
Γ
−
o,

1.3 Continuous−Time Systems 23
and by substituting the latter into (1.60), one can deduce, forε>0,
ρ
H
1q1+H2q2+ε(Γ
+
o
q1+Γ
−
o
q2)≤0,
H
2q1+H1q2+ε(Γ
−
o
q1+Γ
+
o
q2)≤0.
(1.61)
Taking into account that˜H
cq≤0andεis as small as possible, one can always
satisfy (1.61).
(Necessity:)Let the domainD(I,q
1,q2)withq≥0 be positively invariant w.r.t.
the system (1.53) and condition˜H
cq≤0 is violated. For that, leti∈[1,m]
exists in such a way that equation
h
ii(q1)i+
m
∑
j=1,jη =i

h
+
ij
(q1)j+h
−
ij
(q2)j
δ
>0, (1.62)
is satisﬁed.
Besides, consider the vector
ξ(t)∈D(I,q 1,q2)deﬁned as
ξ(t)=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
(q
1)jifj=i,
(q
1)jifhij>0,jη =i,
−(q
2)jifhij<0,jη =i,
0if h
ij=0,jη =i.
j=1,...m, (1.63)
Note that we can also compute
ξ(t+ε)by using the solution to (1.53), given by
(1.59).
It follows that
ξ(t+ε)−ξ(t)=εHξ(t)+ε
2
Γoξ(t).
This implies, for a small
ε>0, that we can write,
ξi(t+ε)−ξi(t)=ε(Hξ(t))i+o(ε
2
).
Moreover, taking into account Equations (1.62) and(1.63), one can conclude that,
ξi(t+ε)−ξi(t)>0,
or in other words that
ξi(t+ε)>(q 1)i,i.e.,ξ(t+ε)η∈D(I,q 1,q2)forξ(t)∈D(I,q 1,q2)
which means that the motion of the system leaves the domainD(I,q
1,q2).This
contradicts our previous assumption. Consequently,˜H
cq≤0. α
The symmetrical case is directly obtained by the corollary below.
Corollary 1.3.The domain
D(I,
ρ)=
η
z∈R
m
/−ρ≤z≤ ρ,ρ∈R
m
+
ξ
,

24 1 Saturated Linear Systems: Analysis
is positively invariant w.r.t. the system (1.53) if and only if
ˆHρ≤0 withˆH=H1+H2=
⎧
⎨
⎩
hii,ifi=j,
|hij|otherwise.
(1.64)
Proof:Follows readily from Theorems 1.6 and 1.7. Note that in the case when
ρ>0,
ϑ(z)=Max
i
|zi|
ρi
,
is a Lyapunov function of the system. α
Based on the arguments above, we are ableto extend the results of Theorem 1.7 as
follows.
Corollary 1.4.
(i) The domainD(I,q1,q2)with q≥0is positively invariant w.r.t. the system
(1.53) if and only ifD(I,q2,q1)is positively invariant w.r.t. the system.
(ii) If the domainD(I,q1,q2)with q≥0is positively invariant w.r.t. the system
(1.53) then
D(I,q1,q2)∪D(I,q2,q1),
is positively invariant w.r.t. the system.
Proof:Follows the same arguments as in the previous section. α
1.3.2 System with State Matrix of Metzler Type
Consider the continuous−time system
˙
ζ(t)=Lζ(t),ζ∈R
m
, (1.65)
whereLis a Metzler matrix,i.e.,Lij≥0,foriη=j.
Corollary 1.5.The DomainD(I,ρ,0)withρ≥0is positively invariant w.r.t. the
system (1.65) if and only if
Lρ≤0. (1.66)
Proof:Consider the same development given in Theorem 1.7, that is,
z(t+ε)=Mz(t), (1.67)
with,
M=I+εL+ε
2
Mo
Mo=
∞
∑
k=2
1
k!
ε
k−2
L
k
.

1.3 Continuous−Time Systems 25
Note that,
M
ii=1+ εLii+ε
2
(Mo)ii
Mij=εLij+ε
2
(Mo)ij,for iη =j.
For
εas small as possible, one can always obtainMa non−negative matrix. The
use of Theorem 1.2 enables one to state that a necessary and sufﬁcient condition
for domainD(I,
ρ,0)withρ≥0 to be positively invariant w.r.t. the system
(1.67) is given for
εas small as possible andε>0by,M ρ≤ρ, or equivalently,
L
ρ≤0. α.
Corollary 1.6.The DomainD(I,
ρ,∞)withρ≥0is positively invariant w.r.t. the
system (1.65) if and only if condition (1.66) holds.
Proof:Follows readily by the use of the same arguments as in the proof of Corollary
1.5 and Theorem 1.2. α
In addition, the following lemma states the stability of the matrixH, as follows.
Lemma 1.6.If H satisﬁes˜H
cq<0, with q>0, then H is Hurwitz.
Proof:LetHbe a matrix satisfying˜H
cq<0. Then from (1.56), one can deduce
easily thatˆHw<0, withw=q
1+q2>0andˆHgiven by (1.64) which
means thatHis Hurwitz. α
1.3.3 Controller Design by Direct Procedure
1.3.3.1 Analysis of EquationFA + FBF = HF
In this section, we apply the results of Theorem 1.7 to the problem of the constrained
regulator described in Section 1.3.
Consider the system (1.40)−(1.43) with the feedback law (1.44) and (1.46). The
closed−loop system is then given by (1.45). Let us make the transformation
z=Fx,F∈R
mxn
, (1.68)
with the matrixFgiven by (1.44) and (1.46). It follows that ˙z=F(A+BF)x.If
there exists a matrixH∈R
mxm
such that
FA+FBF=HF (1.69)
then, the coordinate transformation (1.68) allows one to obtain the dynamical sys−
tem (1.53) from (1.45). The controller design for the system (1.45) withx∈
D(F,q
1,q2)deﬁned by (1.47), becomes possible by the use of (1.53) and
Theorem 1.7, or Theorem 1.6, withz∈R=D(I,q
1,q2).
Before going further, let us present some useful lemmas.

26 1 Saturated Linear Systems: Analysis
Lemma 1.7.The setKerF is positively invariant w.r.t. the system (1.45) if and only
if there exists a matrix H∈R
mxm
such that FAo=HF.
Proof:
(Sufﬁciency):Letx(t)∈Ker(F),thatisFx(t)=0. For everyτ∈R,
x(t+τ)=e
Aoτ
x(t). It follows thatFx(t+τ)=Fe
Aoτ
x(t). EquationFAo=HF
implies obviously thatFA
k
o
=H
k
F,foreveryk∈N. Thus,Fx(t+τ)=e
Hτ
Fx(t).
Consequently,x(t+τ)∈Ker(F),foreveryτ∈R.
(Necessity:)Let the kernelKer(F)be positively invariant w.r.t. the system
(1.45). Then, stating that forw(t),Fw(t)=0 implies necessarilyFw(t+τ)=0,
∀τ∈R. It is clear that in this situation the ﬁrst derivative is surely zero, that is,
d
dt
(Fw(t)) =0,
which obviously leads to
F˙w(t)=FAow(t)=0.
The same decomposition method used in the proof of Lemma 1.1 leads to the
existence of a matrixHsolution toFAo=HF. α
Lemma 1.8.If domainD(F,q1,q2)is positively invariant w.r.t. the system (1.45),
thenKer(F)is also positively invariant w.r.t. the system.
Proof:The proof is similar to that of Lemma 1.2 of Section 1.3, except that we
change in the development, the matrixAobye
Aoτ
. α
Lemma 1.9.If there exists a stable matrix H∈R
mxm
satisfying (1.69), the spec-
trum of Aomust contain a set of n−m stable eigenvalues, closed under complex
conjugacy, corresponding to n−m common eigenvectors to both matrices A and Ao
belonging toKer(F)and parallel to∂D(F,q1,q2).
Proof:It follows the same arguments as in the proof of Lemma 1.3.α
1.3.3.2 Application to the Regulator Problem with Saturated Control
We are now able to give the main result of this section which is a necessary and
sufﬁcient condition for domainD(F,q1,q2)to be positively invariant w.r.t. the
motion of system (1.45).
This result was presented by [48, 15, 59] for the caseq>0.
Theorem 1.8.DomainD(F,q1,q2)is positively invariant w.r.t. the system (1.45),
if and only if there exists a matrix H∈R
mxm
, solution to
FA+FBF=HF, (1.70)

1.3 Continuous−Time Systems 27
and satisfying
˜H
cq≤0, (1.71)
where˜H
cand q are deﬁned by (1.56).
Proof:The proof given here is taken from [15] forq>0.
(Sufﬁciency):Consider the coordinates transformation (1.68). Condition (1.70)
implies that (1.45) can be transformed to (1.53) and domainD(F,q
1,q2)with
q>0todomainR=D(I,q
1,q2).
Theorem 1.6, means that condition (1.71) ensures the positive invariance prop−
erty ofD(I,q
1,q2)w.r.t. the system (1.53).
As a consequence, it follows that domainD(F,q
1,q2)is positively invariant
w.r.t. the system (1.45).
(Necessity:)Let domainD(F,q
1,q2)withq>0 be positively invariant w.r.t.
the system (1.45). By virtue of Lemma 1.8,Ker(F)is also positively invariant
w.r.t. (1.45). According to Lemma 1.7, there exists a matrixH∈R
mxm
satisfying
Equation (1.69). The same coordinates transformation (1.68) allows one to obtain
(1.53) from (1.45) and domainD(I,q
1,q2)fromD(F,q 1,q2), underlying that
the former is also positively invariant w.r.t. the system (1.53). Taking account of
Theorem 1.6, condition (1.71) holds. α
The symmetrical case is obtained directly by the result below.
Corollary 1.7.If q
1=q2=ρ, domainD(F, ρ)is positively invariant w.r.t. the
system (1.45) if and only if there exists a matrix H∈R
mxm
, such that :
(i)FA+FBF=HF,
(ii)ˆH
ρ≤0,
whereˆH is given as in Corollary 1.3.
Proof:Follows readily from Theorem 1.8. α
Recall that the result of this theorem is based on the existence of a matrixH∈R
mxm
satisfying (1.71). A necessary and sufﬁcient condition of the existence of a matrix
His given by [139] that is,
rank
σ
FA
F
θ
=m. (1.72)
Comment 1.1.
- The use of Theorem 1.7 leads to the same result as that of Theorem 1.8 with q≥0.
In this case, the design approach becomes more interesting as it was the case for
discrete-time systems (cf. Section 1.2).
- Note that the conditions presented by [59], are slightly different when compared
to the present conditions. More precisely, conditions (1.71) are to be replaced by
(i)There exist a positive scalar s
oand a matrix P∈R
mxm
such that
FA+FBF=(−s
oI+P)F

28 1 Saturated Linear Systems: Analysis
(ii)and

−s
oI+
∈
P
+
P
−
P
−
P
+

q≤0.
These conditions are obviously equivalent to (1.71) by taking s
0>Max
i
{p
+
ii
}
and p
−
ii
=0and setting H=−s oI+P which implies that
˜H
c=−s oI+˜P,
with
(H
1)ij=

−s
o+p
+
ii
,fori=j
p
+
ij
fori =j,
and(H
2)ij=

0fori=j,
p −
ij
,fori =j.
- Conditions (1.70) and (1.71) guarantee that domainD(F,q
1,q2)deﬁned by
(1.47) is positively invariant w.r.t. the system (1.40)-(1.45), despite the existence
of asymmetric constraints on the control. The choice of F∈R
mxn
satisfying (1.46),
(1.72) and (1.70)-(1.71) allows the system (1.40)-(1.46) to be asymptotically stable,
even when˜H
cq=0. Further, Lemma 1.9 gives a necessary condition on A for the
existence of H solution to Equation (1.69), i.e., the matrix A must possess n−m
stable eigenvalues. When it is not the case, we proceed by augmenting the control
vector dimension without losing the assumption (1.42) by writing the system (1.40)
into the equivalent representation.
˙x=Ax+[B0]
∈
u
v
≥
.
This technique is presented in Section 1.3.
The result below can be easily deduced from Lemmas 1.6 and 1.9.
Theorem 1.9.If there exists a matrix H∈R
mxm
satisfying
(i)FA+FBF=HF,
(ii)˜H
cq≤0,[−2mm]
then system (1.40)-(1.46) is asymptotically stable for every x
o∈D(F,q 1,q2).
Corollary 1.8.
(i)The domainD(F,q
1,q2)with q≥0is positively invariant w.r.t. the system
(1.45) if and only ifD(F,q
2,q1)is positively invariant w.r.t. the system.
(ii)If the domainD(F,q
1,q2)with q≥0is positively invariant w.r.t. the system
(1.45) then
D(F,q
1,q2)∪D(F,q 2,q1),
is positively invariant w.r.t. the system.

1.3 Continuous−Time Systems 29
Proof:Follows readily from Corollary 1.4. α
Corollary 1.9.The domainD(F,
ρ,0)withρ≥0is positively invariant w.r.t. the
system (1.45) if and only if there exists a matrix H∈R
mxm
with hij≥0for iη =j,
such that :
(i)FA+FBF=HF,
(ii)H
ρ≤0.
(1.73)
Proof:Follows readily from Corollary 1.5 . α
Corollary 1.10.The domainD(F,
ρ,∞)withρ≥0is positively invariant w.r.t.
the system (1.45) if and only if conditions (1.73) hold.
Proof:Obtained by following the same reasoning as Corollary 1.6. This result is
also given by [48]. α
Example 1.3.Consider the following double integrator system with matrices A and
B given by
A=
σ
01
00
θ
andB=
σ
0
1
θ
,
and let−5≤u≤10. According to (1.44) and (1.46), we have to ﬁnd a gain matrix
F∈R
mxn
such that A+BF is stable. A solution to this is given by
F=[−2−3].
In this case we have
A
o=A+BF=
σ
01
−2−3
θ
.
Note that matrix A is unstable. The eigenvalue
λ=0has a degree of multiplicity
equal to2.
Further, this matrix does not possess n−m stable eigenvalues. For this, we pro-
ceed as indicated in Section 1.2. That is,
˙x=Ax+[B0]
σ
u
v
θ
,withv∈R
n−m
−r2≤v≤r 1,
where r
1and r2are some ﬁctitious constraints on the control law v. The whole
control law is given by
σ
u
v
θ
=
σ
F
E
θ
x, where the matrix E is chosen in such a way
that the matrix G=
σ
F
E
θ
is invertible.

30 1 Saturated Linear Systems: Analysis
Take for instance E=[−10], then H is given by
H=GA
oG
−1
=
∈
−2.3333−1.3333
0.3333−0.6667
≥
.
The choice of the ﬁctitious constraints is given such that domainD(G,g
1,g2)
becomes the largest possible domain with conditions˜H
cg≤0standing with
g=
∈
g
1
g2
≥
,g
1=
∈
q
1
r1
≥
andg
2=
∈
q
2
r2
≥
.
We ﬁnd out that r
1=5and r2=11which means that they satisfy condition (1.71).
We also obtain H
1=
∈
−2.3333 0
0.3333−0.6667
≥
and H 2=
∈
01.3333
00
≥
.
Thus,
˜H
cg=[−8.67 0−5−0.61]
t
≤0.
Consequently,D(G,g
1,g2)is a positively invariant set w.r.t. the system in closed-
loop.
In this section, the regulator problem of linear continuous−time systems with
asymmetric constrained control is studied. Necessary and sufﬁcient conditions for
domainD(F,q
1,q2), which generates admissible control by feedback law, to be
a positively invariant set w.r.t. the system (1.45), are given. These conditions guar−
antee that system (1.40)−(1.46) is asymptotically stable for every motion emanating
from domainD(F,q
1,q2).
1.4 Saturated Singular Systems
This section studies the class of saturated singular systems by using the concept of
positive invariance.
Singular systems have been of great interest in the control literature since they
can model many systems in electrical circuits networks, robotic and economics [67].
Some problems of observers and synthesis of stabilizing controllers for linear singu−
lar systems can be cited in this category [67, 69] and the references therein. Among
the subjects of continuous interest, the pole assignment problem takes a remarkable
place due to the large amount of works on this area [63, 67, 68, 79, 86, 108, 137, 167]
and the references therein. Two main algebraic approaches are usually followed, the
ﬁrst consists in transforming the singular system into a slow and fast subsystem.
The pole assignment is then formed indirectly [67]. The second approach deals with
the solution of a generalized Sylvester equation as used in [68, 167] and the ref−
erences therein. A different way was, however, followed by [86] and [108], where
the pole assignment is deeply studied, giving the necessary and sufﬁcient conditions
of existence of a state feedback allowing an arbitrary ﬁnite set of self−conjugate
eigenvalues to be placed. The expression of this controller is also given.

1.4 Saturated Singular Systems 31
In this section, new, necessary and sufﬁcient conditions of positive invariance are
presented for continuous−time singular system with constraints on the control. These
conditions are obtained directly and withoutuse of any transformation of the initial
system, as used in [89, 140, 161], by using state feedback control. These results can
be found in [42].
1.4.1 Problem Formulation
In this section, we give the problem formulation related to singular linear systems
studied in this section.
Consider the following singular linear system described by
E
δx(t)=Ax(t)+Bu(t) (1.74)
x(0)=x
0
wherex∈R
n
is the state,u∈R
m
is the control. MatricesA,B,Eare real of appro−
priate size withEa square matrix such thatRank(E)=r≤n.
δx(t)denotes ˙x(t)
for continuous−time singular systems while
δx(t)denotesx(t+1)for discrete−time
singular systems. The control is assumed here to be constrained as follows:
u∈
Ω={u∈R
m
|−umin≤u≤u max;umax,umin≥0};g=
σ
u
max
umin
θ
.(1.75)
Deﬁnition 1.4.[67]
•The pair(E,A)is said to be regular if det(sE−A)is not identically zero.
•The pair(E,A)is said to be impulse free if deg(det(sE−A)) =rank(E).
In order to present some useful results, matricesEandBare decomposed as follows:
E=[R0][S
0S∞]
T
(1.76)
B=U[Z
T
B
0]
T
,U=[U 0U1] (1.77)
where matrix[S
0S∞]is orthogonal,R∈R
n×r
is of full column rank and matrixUis
orthogonal withrankZ
B=rankB. These orthogonal matrices always exist and can
be obtained by using the singular value decomposition.
Lemma 1.10.[108]
The pencil(E,A)is regular and has r ﬁnite eigenvalues if and only if
rank[E+AS
∞S
T
∞
]=n. (1.78)
Note that the pencil(E,A)hasrﬁnite eigenvalues ifdeg(det(sE−A)) =rankE=r,
according to Deﬁnition 1.4, the pencil is said impulse free.

32 1 Saturated Linear Systems: Analysis
We assume that,
H1) The pencil(E,A)is regular and impulse free,i.e., rank[E+AS
∞S
T
∞
]=n.
H2) The singular system (1.74) is controllable,i.e., rank[B,(
λE−A)] =n,∀ λ∈C.
Consider a state feedback given byu(t)=Fx(t), the closed−loop singular system is
then obtained as
E
δx(t)=(A+BF)x(t)=A cx(t). (1.79)
The problem of pole assignment can be stated as follows: given real matrices
E,A,BwhererankE=r≤nand a set ofrﬁnite self−conjugate complex numbers
L={
λ1...λr},ﬁndamatrixF∈R
m×n
such that
det(A+BF−
λE)=0,∀ λ,|λ|<∞,∈L (1.80)
det(A+BF−
λE)η =0,∀ λ,|λ|<∞,/∈L. (1.81)
In this case, one should have
(A+BF)V
r=EVrΛ (1.82)
where,
Λ=diag{ λ1...λr}andV r∈C
n×r
a given matrix. Equations (1.80)−(1.81)
mean that only therelements of the setLare the generalized eigenvalues of the
pencil(A+BF,E). One can notice that only therﬁnite eigenvalues are placed while
the inﬁnite eigenvalues remain unchanged. Besides, since the open−loop system is
impulse free and the closed−loop system must be kept impulse free by state feedback
control, these eigenvalues have no impulsive impact in closed−loop. According to
[108], this problem has a solution if and only if the singular system is controllable
and condition (1.78) holds. This justiﬁes the assumptions H1) and H2).
We recall hereafter an interesting result of pole assignment.
Theorem 1.10.[108]
For matrix B of full rank, given
Λand a matrix Vr∈C
n×r
such that[V rS∞]is non-
singular, then there exists F satisfying (1.82) and such that the pencil(A+BF,E)is
regular if and only if
U
T
1
(AVr−EVrΛ)=0 (1.83)
U
T
1
(E+AS ∞S
T
∞
)has full rank. (1.84)
Then matrix F is given by
F=
Γ
Z
−1
B
U
T
0
(EVrΛ−AVr)Z
−1
B
W
ζ
[V rS∞]
−1
(1.85)
where W is any matrix such that
rank[E+AS
∞S
T
∞
+U0WS∞]=n. (1.86)

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This sudden spate of falsehood had come upon him, as it were, from
the outside.
"If the truth will not help me," he muttered, "why, I can lie with any
man. Else wherefore was I born a Dane? But, by my faith, my
mistress must have done some rare tall lying on her own account,
and now I am reaping that which she hath sown."
As he kneeled thus the Princess bent over him with a quizzical
expression on her face.
"You are sure that you speak the truth now? Your wound is not
again causing you to dote?"
"Nay," said the Sparhawk; "indeed, 'tis almost healed."
"Where was the wound?" queried the Princess anxiously.
"There were two," answered Von Lynar diplomatically; "one in my
shoulder at the base of my neck, and the other, more dangerous
because internal, on the head itself."
"Let me see."
She came and stood above him as he put his hand to the collar of
his doublet, and, unfastening a tie, he slipped it down a little and
showed her at the spring of his neck Werner von Orseln's thrust.
"And the other," she said, covering it up with a little shudder, "that
on the head, where is it?"
The youth blushed, but answered valiantly enough.
"It never was an open wound, and so is a little difficult to find. Here,
where my hand is, above my brow."
"Hold up your head," said the Princess. "On which side was it? On
the right? Strange, I cannot find it. You are too far beneath me. The
light falls not aright. Ah, that is better!"

She kneeled down in front of him and examined each side of his
head with interest, making as she did so, many little exclamations of
pity and remorse.
"I think it must be nearer the brow," she said at last; "hold up your
head—look at me."
Von Lynar looked at the Princess. Their position was one as
charming as it was dangerous. They were kneeling opposite to one
another, their faces, drawn together by the interest of the surgical
examination, had approached very close. The dark eyes looked
squarely into the blue. With stuff so inflammable, fire and tow in
such immediate conjunction, who knows what conflagration might
have ensued had Von Lynar's eyes continued thus to dwell on those
of the Princess?
But the young man's gaze passed over her shoulder. Behind
Margaret of Courtland he saw a man standing at the door with his
hand still on the latch. A dark frown overspread his face. The
Princess, instantly conscious that the interest had gone out of the
situation, followed the direction of Von Lynar's eyes. She rose to her
feet as the young Dane also had done a moment before.
Maurice recognised the man who stood by the door as the same
whom he had seen on the ground in the yew-tree walk when he and
Joan of the Sword Hand had faced the howling mob of the city. For
the second time Prince Wasp had interfered with the amusements of
the Princess Margaret.
That lady looked haughtily at the intruder.

"The lady looked haughtily at the
intruder."
"To what," she said, "am I so fortunate as to owe the unexpected
honour of this visit?"
"I came to pay my respects to your Highness," said Prince Wasp,
bowing low. "I did not know that the Princess was amusing herself.
It is my ill-fortune, not my fault, that I interrupted at a point so full
of interest."
It was the truth. The point was decidedly interesting, and therein lay
the sting of the situation, as probably the Wasp knew full well.
"You are at liberty to leave me now," said the Princess, falling back
on a certain haughty dignity which she kept in reserve behind her

headlong impulsiveness.
"I obey, madam," he replied; "but first I have a message from the
Prince your brother. He asks you to be good enough to accompany
his bride to the minster to-morrow. He has been ill all day with his
old trouble, and so cannot wait in person upon his betrothed. He
must abide in solitude for this day at least. Your Highness is
apparently more fortunate!"
The purpose of the insult was plain; but the Princess Margaret
restrained herself, not, however, hating the insulter less.
"I pray you, Prince Ivan," she said, "return to my brother and tell
him that his commands are ever an honour, and shall be obeyed to
the letter."
She bowed in dignified dismissal. Prince Wasp swept his plumed hat
along the floor with the profundity of his retiring salutation, and in
the same moment he flashed out his sting.
"I leave your Highness with less regret because I perceive that
solitude has its compensations!" he said.
The pair were left alone, but all things seemed altered now.
Margaret of Courtland was silent and distrait. Von Lynar had a frown
upon his brow, and his eyes were very dark and angry.
"Next time I must kill the fellow!" he muttered. He took the hand of
the Princess and respectfully kissed it.
"I am your servant," he said; "I will do your bidding in all things, in
life or in death. If I have forgotten anything, in aught been remiss,
believe me that it was fate and not I. I will never presume, never
count on your friendship past your desire, never recall your ancient
goodness. I am but a poor soldier, yet at least I can faithfully keep
my word."
The Princess withdrew her hand as if she had been somewhat
fatigued.

"Do not be afraid," she said a little bitterly, "I shall not forget. I have
not been wounded in the head! Only in the heart!" she added, as
she turned away.

CHAPTER XIV
AT THE HIGH ALTAR
When Maurice von Lynar reached the open air he stood for full five
minutes, light-headed in the rush of the city traffic. The loud
iteration of rejoicing sounded heartless and even impertinent in his
ear. The world had changed for the young Dane since the Count von
Löen had been summoned by the Princess Margaret.
He cast his mind back over the interview, but failed to disentangle
anything definite. It was a maze of impressions out of which grew
the certainty that, safely to play his difficult part, he must obtain the
whole confidence of the Duchess Joan.
He looked about for the Prince of Muscovy, but failed to see him.
Though not anxious about the result, he was rather glad, for he did
not want another quarrel on his hands till after the wedding. He
would see the Princess Margaret there. If he played his cards well
with the bride, he might even be sent for to escort her.
So he made his way to the magnificent suite of apartments where
the Duchess was lodged. The Prince had ordered everything with
great consideration. Her own horsemen patrolled the front of the
palace, and the Courtland guards were for the time being wholly
withdrawn.
It seemed strange that Joan of the Sword Hand, who not so long
ago had led many a dashing foray and been the foremost in many a
brisk encounter, should be a bride! It could not be that once he had
imagined her the fairest woman under the sun, and himself, for her
sake, the most miserable of men. Thus do lovers deceive themselves

when the new has come to obliterate the old. Some can even
persuade themselves that the old never had any existence.
The young Dane found the Duchess walking up and down on the
noble promenade which faces the river to the west. For the water
curved in a spacious elbow about the city of Courtland, and the
summer palace was placed in the angle.
Maurice von Lynar stood awhile respectfully waiting for the Duchess
to recognise him. Werner, John of Thorn, or any of her Kernsberg
captains would have gone directly up to her. But this youth had been
trained in another school.
"Joan of Hohenstein stood, looking out upon the
river."
Joan of Hohenstein stood a while without moving, looking out upon
the river. She thought with a kind of troubled shyness of the morrow,
oft dreamed of, long expected. She saw the man whom she was not
known ever to have seen—the noble young man of the tournament,
the gracious Prince of the summer parlour, courteous and dignified
alike to the poor secretary of embassy and to his sister the Princess

Margaret of Courtland. Surely there never was any one like him—
proudly thought this girl, as she looked across the river at the rich
plain studded with far-smiling farms and fields just waking to life
after their long winter sleep.
"Ah, Von Lynar, my brave Dane, what good wind blows you here?"
she cried. "I declare I was longing for some one to talk to." A
consciousness of need which had only just come to her.
"I have seen the Princess Margaret," said the youth slowly, "and I
think that she must mistake me for some other person. She spoke
things most strange to me to hear. But fearing I might meddle with
affairs wherewith I had no concern, I forebore to correct her."
The eyes of the Duchess danced. A load seemed suddenly lifted off
her mind.
"Was she very angry?" she queried.
"Very!" returned Von Lynar, smiling in recognition of her smile.
"What said the Princess?"
"First she would have it that my name and style were those of the
Count Von Löen. Then she reproached me fiercely because I denied
it. After that she spoke of certain foreign customs she had been
taught, recalled walks through corridors and rose gardens with me,
till my head swam and I knew not what to answer."
Joan of the Sword Hand laughed a merry peal.
"The Count von Löen, did she say?" she meditated. "Well, so you are
the Count von Löen. I create you the Count von Löen now. I give
you the title. It is mine to give. By to-morrow I shall have done with
all these things. And since as the Count von Löen I drank the wine,
it is fair that you, who have to pay the reckoning, should be the
Count von Löen also."

"My family is noble, and I am the sole heir—that is, alive," said
Maurice, a little drily. To his mind the grandson of Count von Lynar,
of the order of the Dannebrog, had no need of any other distinction.
"But I give you also therewith the estates which pertain to the title.
They are situated on the borders of Reichenau. I am so happy to-
night that I would like to make all the world happy. I am sorry for all
the folk I have injured!"
"Love changes all things," said the Dane sententiously.
The Duchess looked at him quickly.
"You are in love—with the Princess Margaret?" she said.
The youth blushed a deep crimson, which flooded his neck and dyed
his dusky skin.
"Poor Maurice!" she said, touching his bowed head with her hand,
"your troubles will not be to seek."
"My lady," said the youth, "I fear not trouble. I have promised to
serve the Princess in all things. She has been very kind to me. She
has forgiven me all."
"So—you are anxious to change your allegiance," said the Duchess.
"It is as well that I have already made you Count von Löen, and so
in a manner bound you to me, or you would be going off into
another's service with all my secrets in your keeping. Not that it will
matter very much—after to-morrow!" she added, with a glance at
the wing of the palace which held the summer parlour. "But how did
you manage to appease her? That is no mean feat. She is an
imperious lady and quick of understanding."
Then Maurice von Lynar told his mistress of his most allowable
falsehoods, and begged her not to undeceive the Princess, for that
he would rather bear all that she might put upon him than that she
should know he had lied to her.

"Do not be afraid," said the Duchess, laughing, "it was I who tangled
the skein. So far you have unravelled it very well. The least I can do
is to leave you to unwind it to the end, my brave Count von Löen."
So they parted, the Duchess to her apartment, and the young man
to pace up and down the stone-flagged promenade all night,
thinking of the distracting whimsies of the Princess Margaret, of the
hopelessness of his love, and, most of all, of how daintily exquisite
and altogether desirable was her beauty of face, of figure, of temper,
of everything!
For the Sparhawk was not a lover to make reservations.
The morning of the great day dawned cool and grey. A sunshade of
misty cloud overspread the city and tempered the heat. It had come
up with the morning wind from the Baltic, and by eight the ships at
the quays, and the tall beflagged festal masts in the streets through
which the procession was to pass, ran clear up into it and were lost,
so that the standards and pennons on their tops could not be seen
any more than if they had been amongst the stars.
The streets were completely lined with the folk of the city of
Courtland as the Princess Margaret, with the Sparhawk and his
company of lances clattering behind her, rode to the entrance of the
palace where abode the bride-elect.
"Who is that youth?" asked Margaret of Courtland of Joan, as they
came out together; she looked at the Dane—"he at the head of your
first troops? He looks like your brother."
"He has often been taken for such!" said the bride. "He is called the
Count von Löen!"
The Princess did not reply, and as the two fair women came out arm
in arm, a sudden glint of sunlight broke through the leaden clouds

and fell upon them, glorifying the white dress of the one, and the
blue and gold apparel of the other.
The bells of the minster clanged a changeful thunder of brazen
acclaim as the bride set out for the first time (so they told each
other on the streets) to see her promised husband.
"'Twas well we did not so manage our affairs, Hans," said a
fishmonger's wife, touching her husband's arm archly.
"Yea, wife," returned the seller of fish; "whatever thou beest, at
least I cannot deny that I took thee with my eyes open!"
They reached the Rathhaus, and the clamour grew louder than ever.
Presently they were at the cathedral and making them ready to
dismount. The bells in the towers above burst forth into yet more
frantic jubilation. The cannons roared from the ramparts.
The Princess Margaret had delayed a little, either taking longer to
her attiring, or, perhaps, gossiping with the bride. So that when the
shouts in the wide Minster Place announced their arrival, all was in
readiness within the crowded church, and the bridegroom had gone
in well-nigh half an hour before them. But that was in accord with
the best traditions.
Very like a Princess and a great lady looked Joan of Hohenstein as
she went up the aisle, with Margaret of Courtland by her side. She
kept her eyes on the ground, for she meant to look at no one and
behold nothing till she should see—that which she longed to look
upon.
Suddenly she was conscious that they had stopped in the middle of
a vast silence. The candles upon the great altar threw down a
golden lustre. Joan saw the irregular shining of them on her white
bridal dress, and wondered that it should be so bright.
There was a hush over all the assembly, the silence of a great
multitude all intent upon one thing.

"My brother, the Prince of Courtland!" said the voice of the Princess
Margaret.
Slowly Joan raised her eyes—pride and happiness at war with a kind
of glorious shame upon her face.
But that one look altered all things.
She stood fixed, aghast, turned to stone as she gazed. She could
neither speak nor think. That which she saw almost struck her dead
with horror.
The man whom his sister introduced as the Prince of Courtland was
not the knight of the tournament. He was not the young prince of
the summer palace. He was a man much older, more meagre of
body, grey-headed, with an odd sidelong expression in his eyes. His
shoulders were bent, and he carried himself like a man prematurely
old.
And there, behind the altar-railing, clad in the scarlet of a prince of
the Church, and wearing the mitre of a bishop, stood the husband of
her heart's deepest thoughts, the man who had never been out of
her mind all these weary months. He held a service book in his
hand, and stood ready to marry Joan of Hohenstein to another.
The man who was called Prince of Courtland came forward to take
her hand; but Joan stood with her arms firmly at her sides. The
terrible nature of her mistake flashed upon her and grew in horror
with every moment. Fate seemed to laugh suddenly and mockingly
in her face. Destiny shut her in.
"Are you the Prince of Courtland?" she asked; and at the sound of
her voice, unwontedly clear in the great church, even the organ
appeared to still itself. All listened intently, though only a few heard
the conversation.
"I have that honour," bowed the man with the bent shoulders.

"Then, as God lives, I will never marry you!" cried Joan, all her soul
in the disgust of her voice.
"Be not disdainful, my lady," said the bridegroom mildly; "I will be
your humble slave. You shall have a palace and an establishment of
your own, an it like you. The marriage was your father's desire, and
hath the sanction of the Emperor. It is as necessary for your State as
for mine."
Then, while the people waited in a kind of palpitating uncertainty,
the Princess Margaret whispered to the bride, who stood with a face
ashen pale as her own white dress.
Sometimes she looked at the Prince of Courtland, and then
immediately averted her eyes. But never, after the first glance, did
Joan permit them to stray to the face of him who stood behind the
altar railings with his service book in his hand.
"Well," she said finally, "I will marry this man, since it is my fate. Let
the ceremony proceed!"
"I thank you, gracious lady," said the Prince, taking her hand and
leading his bride to the altar. "You will never regret it."
"No, but you will!" muttered his groomsman, the Prince Ivan of
Muscovy.
The full rich tones of the prince bishop rose and fell through the
crowded minster as Joan of Hohenstein was married to his elder
brother, and with the closing words of the episcopal benediction an
awe fell upon the multitude. They felt that they were in the presence
of great unknown forces, the action and interaction of which might
lead no man knew whither.
At the close of the service, Joan, now Princess of Courtland, leaned
over and whispered a word to her chosen captain, Maurice von
Lynar, an action noticed by few. The young man started and gazed
into her face; but, immediately commanding his emotion, he nodded
and disappeared by a side door.

The great organ swelled out. The marriage procession was re-
formed. The prince-bishop had retired to his sacristy to change his
robes. The new Princess of Courtland came down the aisle on the
arm of her husband.
Then the bells almost turned over in their fury of jubilation, and
every cannon in the city bellowed out. The people shouted
themselves hoarse, and the line of Courtland troops who kept the
people back had great difficulty in restraining the enthusiasm which
threatened to break all bounds and involve the married pair in a
whirling tumult of acclaim.
In the centre of the Minster Place the four hundred lances of the
Kernsberg escort had formed up, a serried mass of beautiful well-
groomed horses, stalwart men, and shining spears, from each of
which the pennon of their mistress fluttered in the light wind.
"Ha! there they come at last! See them on the steps!" The shouts
rang out, and the people flung their headgear wildly into the air. The
line of Courtland foot saluted, but no cheer came from the array of
Kernsberg lances.
"They are sorry to lose her—and small wonder. Well, she is ours
now!" the people cried, congratulating one another as they shook
hands and the wine gurgled out of the pigskins into innumerable
thirsty mouths.
On the steps of the minster, after they had descended more than
half-way, the new Princess of Courtland turned upon her lord. Her
hand slipped from his arm, which hung a moment crooked and
empty before it dropped to his side. His mouth was a little open with
surprise. Prince Louis knew that he was wedding a wilful dame, but
he had not been prepared for this.
"Now, my lord," said the Princess Joan, loud and clear. "I have
married you. The bond of heritage-brotherhood is fulfilled. I have
obeyed my father to the letter. I have obeyed the Emperor. I have
done all. Now be it known to you and to all men that I will neither

live with you nor yet in your city. I am your wife in name. You shall
never be my husband in aught else. I bid you farewell, Prince of
Courtland. Joan of Hohenstein may marry where she is bidden, but
she loves where she will."
The horse upon which she had come to the minster stood waiting.
There was the Sparhawk ready to help her into the saddle.
Ere one of the wedding guests could move to prevent her, before the
Prince of Courtland could cry an order or decide what to do, Joan of
the Sword Hand had placed herself at the head of her four hundred
lances, and was riding through the shouting streets towards the
Plassenburg gate.
The people cheered as she went by, clearing the way that she might
not be annoyed. They thought it part of the day's show, and voted
the Kernsbergers a gallant band, well set up and right bravely
arrayed.
So they passed through the gate in safety. The noble portal was all
aflutter with colour, the arms of Hohenstein and Courtland being
quartered together on a great wooden plaque over the main
entrance.
As soon as they were clear the Princess Joan turned in her saddle
and spake to the four hundred behind her.
"We ride back to Kernsberg," she cried. "Joan of the Sword Hand is
wed, but not yet won. If they would keep her they must first catch
her. Are you with me, lads of the hills?"
Then came back a unanimous shout of "Aye—to the death!" from
four hundred throats.
"Then give me a sword and put the horses to their speed. We ride
for home. Let them catch us who can!"
And this was the true fashion of the marrying of Joan of the Sword
Hand, Duchess of Hohenstein, to the Prince Louis of Courtland, by

his brother Conrad, Cardinal and Prince of Holy Church.

CHAPTER XV
WHAT JOAN LEFT BEHIND
After the departure of his bride, the Prince of Courtland stood on the
steps of the minster, dazed and foundered by the shame which had
so suddenly befallen him. Beneath him the people seethed
tumultuously, their holiday ribands and maypole dresses making as
gay a swirl of colour as when one looks at the sun through the
facets of a cut Venetian glass. Prince Louis's weak and fretful face
worked with emotion. His bird-like hands clawed uncertainly at his
sword-hilt, wandering off over the golden pouches that tasselled his
baldric till they rested on the sheath of the poignard he wore.
"Bid the gates be shut, Prince!" The whisper came over his shoulder
from a young man who had been standing all the time twisting his
moustache. "Bid your horsemen bit and bridle. The plain is fair
before you. It is a long way to Kernsberg. I have a hundred
Muscovites at your service, all well mounted—ten thousand behind
them over the frontier if these are not enough! Let no wench in the
world put this shame upon a reigning Prince of Courtland on his
wedding-day!"
Thus Ivan of Muscovy, attired in silk, banded of black and gold,
counselled the disdained Prince Louis, who stood pushing upward
with two fingers the point of his thin greyish beard and gnawing the
straggling ends between his teeth.
"I say, 'To horse and ride, man!' Will you dare tell this folk of yours
that you are disdained, slighted at the very church door by your
wedded wife, cast off and trodden in the mire like a bursten glove?
Can you afford to proclaim yourself the scorn of Germany? How it
will run, that news! To Plassenburg first, where the Executioner's

Son will smile triumphantly to his witch woman, and straightway
send off a messenger to tickle the well-larded ribs of his friend the
Margraf George with the rare jest."
The Prince Louis appeared to be moved by the Wasp's words. He
turned about to the nearest knight-in-waiting.
"Let us to horse—every man of us!" he said. "Bid that the steeds be
brought instantly."
The banded Wasp had further counsels to give.
"Give out that you go to meet the Princess at a rendezvous. For a
pleasantry between yourselves, you have resolved to spend the
honeymoon at a distant hunting-lodge. Quick! Not half a dozen of all
the company caught the true import of her words. You will tame her
yet. She will founder her horses in a single day's ride, while you have
relays along the road at every castle, at every farm-house, and your
borders are fifty good miles away."
Beneath, in the square, the court jesters leaped and laughed,
turning somersaults and making a flying skirt, like that of a morrice
dancer, out of the long, flapping points of their parti-coloured
blouses. The streets in front of the cathedral were alive with
musicians, mostly in little bands of three, a harper with his harp of
fourteen strings, his companion playing industriously upon a Flute-
English, and with these two their 'prentice or servitor, who
accompanied them with shrill iterance of whistle, while both his
hands busied themselves with the merry tuck of tabour.
In this incessant merrymaking the people soon forgot their
astonishment at the sudden disappearance of the bride. There was,
indeed, no understanding these great folk. But it was a fine day for a
feast—the pretext a good one. And so the lasses and lads joked as
they danced in the lower vaults of the town house, from which the
barrels had been cleared for the occasion.

"If thou and I were thus wedded, Grete, would you ride one way
and I the other? Nay, God wot, lass! I am but a tanner's 'prentice,
but I'd abide beside thee, as close as bark by hide that lies three
years in the same tan-pit—aye, an' that I would, lass!"
Then Gretchen bridled. "I would not marry thee, nor yet lie near or
far, Hans; thou art but a boy, feckless and skill-less save to pole
about thy stinking skins—faugh!"
"Nay, try me, Grete! Is not this kiss as sweet as any civet-scented
fop could give?"
At the command of the Prince the trumpets rang out again the call
of "Boot-and-saddle!" from the steps of the cathedral. At the sound
the grooms, who were here and there in the press, hasted to find
and caparison the horses of their lords. Meanwhile, on the wide
steps the Prince Louis fretted, dinting his nails restlessly into his
palms and shaking with anger and disappointment till his deep
sleeves vibrated like scarlet flames in a veering wind.
Suddenly there passed a wave over the people who crowded the
spacious Dom Platz of Courtland. The turmoil stilled itself
unconsciously. The many-headed parti-coloured throng of women's
tall coifs, gay fluttering ribands, men's velvet caps, gallants' white
feathers that shifted like the permutations of a kaleidoscope, all at
once fixed itself into a sea of white faces, from which presently
arose a forest of arms flourishing kerchiefs and tossing caps. To this
succeeded a deep mouth-roar of burgherish welcome such as the
reigning Prince had never heard raised in his own honour.
"Conrad—Prince Conrad! God bless our Prince-Cardinal!"
The legitimate ruler of Courtland, standing where Joan had left him,
with his slim-waisted Muscovite mentor behind him, half-turned to
look. And there on the highest place stood his brother in the scarlet
of his new dignity as it had come from the Pope himself, his red
biretta held in his hand, and his fair and noble head erect as he
looked over the folk to where on the slope above the city gates he

could still see the sun glint and sparkle on the cuirasses and
lanceheads of the four hundred riders of Kernsberg.
But even as the Prince of Courtland looked back at his brother, the
whisper of the tempter smote his ear.
"Had Prince Conrad been in your place, and you behind the altar
rails, think you that the Duchess Joan would have fled so cavalierly?"
By this time the young Cardinal had descended till he stood on the
other side of the Prince from Ivan of Muscovy.
"You take horse to follow your bride?" he queried, smiling. "Is it a
fashion of Kernsberg brides thus to steal away?" For he could see
the grooms bringing horses into the square, and the guards beating
the people back with the butts of their spears to make room for the
mounting of the Prince's cavalcade.
"Hark—he flouts you!" came the whisper over the bridegroom's
shoulder; "I warrant he knew of this before."
"You have done your priest's work, brother," said Louis coldly, "e'en
permit me to go about that of a prince and a husband in my own
way."
The Cardinal bowed low, but with great self-command held his
peace, whereat Louis of Courtland broke out in a sudden overboiling
fury.
"This is your doing!" he cried; "I know it well. From her first coming
my bride had set herself to scorn me. My sister knew it. You knew it.
You smile as at a jest. The Pope's favour has turned your head. You
would have all—the love of my wife, the rule of my folk, as well as
the acclaim of these city swine. Listen—'The good Prince Conrad!
God save the noble Prince!' It is worth while living for favour such as
this."
"Brother of mine," said the young man gently, "as you know well, I
never set eyes upon the noble Lady Joan before. Never spoke word

to her, held no communication by word or pen."
"Von Dessauer—his secretary!" whispered Ivan, dropping the
suggestion carefully over his shoulder like poison distilled into a cup.
"You were constantly with the old fox Dessauer, the envoy of
Plassenburg—who came from Kernsberg, bringing with him that slim
secretary. By my faith, now, when I think of it, Prince Ivan told me
last night he was as like this madcap girl as pea to pea—some fly-
blown base-born brother, doubtless!"
Conrad shook his head. His brother had doubtless gone momentarily
distract with his troubles.
"Nay, deny it not! And smile not either—lest I spoil the symmetry of
that face for your monkish mummery and processions. Aye, if I have
to lie under ten years' interdict for it from your friend the most Holy
Pope of Rome!"
"Do not forget there is another Church in my country, which will lay
no interdict upon you, Prince Louis," laughed Ivan of Muscovy. "But
to horse—to horse—we lose time!"
"Brother," said the Cardinal, laying his hand on Louis's arm, "on my
word as a knight—as a Prince of the Church—I knew nothing of the
matter. I cannot even guess what has led you thus to accuse me!"
The Princess Margaret came at that moment out of the cathedral
and ran impetuously to her favourite brother.
He put out his hand. She took it, and instead of kissing his bishop's
ring, as in strict etiquette she ought to have done, she cried out,
"Conrad, do you know what that glorious wench has done? Dared
her husband's authority at the church door, leaped into the saddle,
whistled up her men, cried to all these Courtland gallants, 'Catch me
who can!' And lo! at this moment she is riding straight for Kernsberg,
and now our Louis must catch her. A glorious wedding! I would I
were by her side. Brother Louis, you need not frown, I am nowise
affrighted at your glooms! This is a bride worth fighting for. No

puling cloister-maid this that dares not raise her eyes higher than
her bridegroom's knee! Were I a man, by my faith, I would never eat
or drink, neither pray nor sain me, till I had tamed the darling and
brought her to my wrist like a falcon to a lure!"
"So, then, madam, you knew of this?" said her elder brother,
glowering upon her from beneath his heavy brows.
"Nay!" trilled the gay Princess, "I only wish I had. Then I, too, would
have been riding with them—such a jest as never was, it would have
been. Goodbye, my poor forsaken brother! Joy be with you on this
your bridal journey. Take Prince Ivan with you, and Conrad and I will
keep the kingdom against your return, with your prize gentled on
your wrist."
So smiling and kissing her hand the Princess Margaret waved her
brother and Prince Ivan off. The Prince of Courtland neither looked
at her nor answered. But the Muscovite turned often in his saddle as
if to carry with him the picture she made of saucy countenance and
dainty figure as she stood looking up into the face of the Cardinal
Prince Conrad.
"What in Heaven's name is the meaning of all this—I do not
understand in the least?" he was saying.
"Haste you and unrobe, Brother Con," she said; "this grandeur of
yours daunts me. Then, in the summer parlour, I will tell you all!"

CHAPTER XVI
PRINCE WASP'S COMPACT
"I cannot go back to Courtland dishonoured," said Prince Louis to
Ivan of Muscovy, as they stood on the green bank looking down on
the rushing river, broad and brown, which had so lately been the
Fords of Alla. The river had risen almost as it seemed upon the very
heels of the four hundred horsemen of Kernsberg, and the ironclad
knights and men-at-arms who followed the Prince of Courtland could
not face the yeasty swirl of the flood.

"They stood ... looking down at
the rushing river."
Prince Ivan, left to himself, would have dared it.
"What is a little brown water?" he cried. "Let the men leave their
armour on this side and swim their horses through. We do it fifty
times a month in Muscovy in the springtime. And what are your hill-
fed brooks to the full-bosomed rivers of the Great Plain?"
"It is just because they are hill-fed that we know them and will not
risk our lives. The Alla has come down out of the mountains of
Hohenstein. For four-and-twenty hours nothing without wing may
pass and repass. Yet an hour earlier and our Duchess had been
trapped on the hither side even as we. But now she will sit and
laugh up there in Kernsberg. And—I cannot go back to Courtland
without a bride!"
Prince Ivan stood a moment silent. Then his eyes glanced over his
companion with a certain severe and amused curiosity. From foot to
head they scanned him, beginning at the shoes of red Cordovan
leather, following upwards to the great tassel he wore at his
poignard; then came the golden girdle about his waist, the flowered
needlework at his wrists and neck, and the scrutiny ended with the
flat red cap on his head, from which a white feather nodded over his
left eye.
Then the gaze of Prince Ivan returned again slowly to the pointed
red shoes of Cordovan leather.
If there was anything so contemptuous as that eye-blink in the open
scorn of all the burghers of Courtland, Prince Louis was to be
excused for any hesitation he might show in facing his subjects.
The matter of Prince Wasp's meditation ran somewhat thuswise:
"Thou man, fashioned from a scullion's nail-paring, and cocked upon
a horse, what can I make of thee? Thou, to have a country, a crown,
a wife! Gudgeon eats stickleback, jack-pike eats gudgeon and grows

fat, till at last the sturgeon in his armour eats him. I will fatten this
jack. I will feed him like the gudgeons of Kernsberg and Hohenstein,
then take him with a dainty lure indeed, black-tipped, with sleeves
gay as cranes' wings, and answering to the name of 'my lady Joan.'
But wait—I must be wary, and have a care lest I shadow his water."
So saying within his heart, Prince Wasp became exceedingly
thoughtful and of a demure countenance.
"My lord," he said, "this day's work will not go well down in
Courtland, I fear me!"
Prince Louis moved uneasily, keeping his regard steadily upon the
brown turmoil of the Alla swirling beneath, whereas the eyes of Ivan
were never removed from his friend's meagre face.
"Your true Courtlander is more than half a Muscovite," mused Prince
Wasp, as if thinking aloud; "he wishes not to be argued with. He
wants a master, and he will not love one who permits himself to be
choused of a wife upon his wedding-day!"
Prince Louis started quickly as the Wasp's sting pricked him.
"And pray, Prince Ivan," he said, "what could I have done that I left
undone? Speak plainly, since you are so prodigal of smiles
suppressed, so witty with covert words and shoulder-tappings!"
"My Louis," said Prince Wasp, laying his hand upon the arm of his
companion with an affectation of tenderness. "I flout you not—I
mock you not. And if I speak harshly, it is only that I love not to see
you in your turn flouted, mocked, scorned, made light of before your
own people!"
"I believe it, Ivan; pardon the heat of my hasty temper!" said the
Prince of Courtland. The watchful Muscovite pursued his advantage,
narrowing his eyes that he might the better note every change on
the face of the man whom he held in his toils. He went on, with a
certain resigned sadness in his voice—

"Ever since I came first to Courtland with the not dishonourable
hope of carrying back to my father a princess of your house, none
have been so amiable together as you and I. We have been even as
David and Jonathan."
The Prince Louis put out a hand, which apparently Ivan did not see,
for he continued without taking it.
"Yet what have I gained either of solid good or even of the lighter
but not less agreeable matter of my lady's favour? So far as your
sister is concerned, I have wasted my time. If I consider the union
of our peoples, already one in heart, your brother works against us
both; the Princess Margaret despises me, Prince Conrad thwarts us.
He would bind us in chains and carry us tinkling to the feet of his
pagan master in Rome!"
"I think not so," answered Prince Louis—"I cannot think so of my
brother, with all his faults. Conrad is a brave soldier, a good knight—
though, as is the custom of our house, it is his lot to be no more
than a prince-bishop!"
The Wasp laughed a little hard laugh, clear and inhuman as the snap
and rattle of Spanish castanets.
"Louis, my good friend, your simplicity, your lack of guile, do you
wrong most grievous! You judge others as you yourself are. Do you
not see that Conrad your brother must pay for his red hat? He must
earn his cardinalate. Papa Sixtus gives nothing for nothing.
Courtland must pay Peter's pence, must become monkish land. On
every flake of stockfish, every grain of sturgeon roe, every ounce of
marled amber, your Holy Father must levy his sacred dues. And the
clear ambition of your brother is to make you chief cat's-paw
pontifical upon the Baltic shore. Consider it, good Louis."
And the Prince of Muscovy twirled his moustache and smiled
condescendingly between his fingers. Then, as if he thought
suddenly of something else and made a new calculation, he laughed
a laugh, quick and short as the barking of a dog.

"Ha!" he cried, "truly we order things better in my country. I have
brothers, one, two, three. They are grand dukes, highnesses very
serene. One of them has this province, another this sinecure, yet
another waits on my father. My father dies—and I—well, I am in my
father's place. What will my brothers do with their serene highnesses
then? They will take each one the clearest road and the shortest for
the frontier, or by the Holy Icon of Moscow, there would very
speedily be certain new tablets in the funeral vault of my fathers."
The Prince of Courtland started.
"This thing I could never imagine of Conrad my brother. He loves
me. At heart he ever cared but for his books, and now that he is a
priest he hath forsworn knighthood, and tournaments, and wars."
"Poor Louis," said Ivan sadly, "not to see that once a soldier always a
soldier. But 'tis a good fault, this generous blindness of the eyes. He
hath already the love of your people. He has won already the voice
that speaks from every altar and presbytery. The power to loose and
bind men's consciences is in his hand. In a little, when he has
bartered away your power for his cardinal's hat, he may be made a
greater than yourself, an elector of the empire, the right-hand man
of Papa Sixtus, as his uncle Adrian was before him. Then indeed
your Courtland will underlie the tinkle of Peter's keys!"
"I am sure that Conrad would do nothing against his fatherland or to
the hurt of his prince and brother!" said Prince Louis, but he spoke
in a wavering voice, like one more than half convinced.
"Again," continued Ivan, without heeding him, "there is your wife. I
am sure that if he had been the prince and you the priest—well, she
had not slept this night in the Castle of Kernsberg!"
"Ivan, if you love me, be silent," cried the tortured Prince of
Courtland, setting his hand to his brow. "This is the mere idle
dreaming of a fool. How learned you these things? I mean how did
the thoughts enter into your mind?"

"I learned the matter from the Princess Margaret, who in the brief
space of a day became your wife's confidante!"
"Did Margaret tell it you?"
The Prince Ivan laughed a short, self-depreciatory laugh.
"Nay, truly," he said, smiling sadly, "you and I are in one despite,
Louis. Your wife scorns you—me, my sweetheart. Did Margaret tell
me? Nay, verily! Yet I learned it, nevertheless, even more certainly
because she denied it so vehemently. But, after all, I daresay all will
end for the best."
"How so?" demanded Prince Louis haughtily.
"Why, I have heard that your Papa at Rome will do aught for money.
Doubtless he will dissolve this marriage, which indeed is no more
than one in name. He has done more than that already for his own
nephews. He will absolve your brother from his vows. Then you can
be the monk and he the king. There will be a new marriage, at
which doubtless you shall hold the service book and he the lady's
hand. Then we shall have no ridings back to Kernsberg, with four
hundred lances, at a word from a girl's scornful mouth. And the Alla
down there may rise or fall at its pleasure, and neither hurt nor
hinder any!"
The Prince of Courtland turned an angry countenance upon his
friend, but the keen-witted Muscovite looked so kindly and yet so
sadly upon him that after awhile the severity of his face relaxed as it
had been against his will, and with a quick gesture he added, "I
believe you love me, Ivan, though indeed your words are no better
than red-hot pincers in my heart."
"Love you, Louis?" cried Prince Ivan. "I love you better than any
brother I have, though they will never live to thwart me as yours
thwarts you—better even than my father, for you do not keep me
out of my inheritance!"
Then in a gayer tone he went on.

"I love you so much that I will pledge my father's whole army to
help you, first to win your wife, next to take Hohenstein, Kernsberg,
and Marienfeld. And after that, if you are still ambitious, why—to
Plassenburg and the Wolfmark, which now the Executioner's Son
holds. That would make a noble kingdom to offer a fair and wilful
queen."
"And for this you ask?"
"Only your love, Louis—only your love! And, if it please you, the
alliance with that Princess of your honourable house, of whom we
spoke just now!"
"My sister Margaret, you mean? I will do what I can, Ivan, but she
also is wilful. You know she is wilful! I cannot compel her love!"
The Prince Ivan laughed.
"I am not so complaisant as you, Louis, nor yet so modest. Give me
my bride on the day Joan of the Sword Hand sleeps in the palace of
Courtland as its princess, and I will take my chance of winning our
Margaret's love!"

CHAPTER XVII
WOMAN'S WILFULNESS
Joan rode on, silent, a furlong before her men.
Behind her sulked Maurice von Lynar. Had any been there to note,
their faces were now strangely alike in feature, and yet more
curiously unlike in expression. Joan gazed forward into the distance
like a soul dead and about to be reborn, planning a new life. Maurice
von Lynar looked more like a naughty schoolboy whom some tyrant
Fate, rod-wielding, has compelled to obey against his will.
Yet, in spite of expression, it was Maurice von Lynar who was
planning the future. Joan's heart was yet too sore. Her tree of life
had, as it were, been cut off close to the ground. She could not go
back to the old so soon after her blissful year of dreams. There was
to be no new life for her. She could not take up the old. But Maurice
—his thoughts were all for the Princess Margaret, of the ripple of her
golden hair, of her pretty wilful words and ways, of that dimple on
her chin, and, above all, of her threat to seek him out if—but it was
not possible that she could mean that. And yet she looked as though
she might make good her words. Was it possible? He posed himself
with this question, and for half an hour rode on oblivious of all else.
"Eh?" he said at last, half conscious that some one had been
speaking to him from an infinite distance. "Eh? Did you speak,
Captain von Orseln?"
Von Orseln grunted out a little laugh, almost silently, indeed, and
expressed more by a heave of his shoulders than by any alteration
of his features.

"Speak, indeed? As if I had not been speaking these five minutes.
Well nigh had I stuck my poignard in your ribs to teach you to mind
your superior officer. What think you of this business?"
"Think?" the Sparhawk's disappointment burst out. "Think? Why, 'tis
past all thinking. Courtland is shut to us for twenty years."
"Well," laughed Von Orseln, "who cares for that? Castle Kernsberg is
good enough for me, so we can hold it."
"Hold it?" cried Maurice, with a kind of joy in his face; "do you think
they will come after us?"
Von Orseln nodded approval of his spirit.
"Yes, little man, yes," he said; "if you have been fretting to come to
blows with the Courtlanders you are in good case to be satisfied. I
would we had only these lumpish Baltic jacks to fear."
Even as they talked Castle Kernsberg floated up like a cloud before
them above the blue and misty plain, long before they could
distinguish the walls and hundred gables of the town beneath.
But no word spoke Joan till that purple shadow had taken shape as
stately stone and lime, and she could discern her own red lion flying
abreast of the banner of Louis of Courtland upon the topmost
pinnacle of the round tower.
Then on a little mound without the town she halted and faced about.
Von Orseln halted the troop with a backward wave of the hand.
"Men of Hohenstein," said the Duchess, in a clear, far-reaching alto,
"you have followed me, asking no word of why or wherefore. I have
told you nothing, yet is an explanation due to you."
There came the sound as of a hoarse unanimous muttering among
the soldiers. Joan looked at Von Orseln as a sign for him to interpret
it.

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Saturated Switching Systems 1st Edition Abdellah Benzaouia Auth

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