Monotone dynamical systems, graphs, and stability of large scale interconnected systems [Elektronische Ressource] / von Björn Sebastian Rüffer

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• Mathematics

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• Publié par: universitat_bremen
• Publié le: 01 janvier 2008
• Nombre de lectures: 19
• Poids de l'ouvrage: 1 Mo

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Bj¨ orn S. Ruffer¨
Monotone dynamical systems, graphs, and stability of
large-scale interconnected systemsMonotone dynamical systems,
graphs, and stability of large-scale
interconnected systems
von Bjorn¨ Sebastian Ruffer¨
Dissertation zur Erlangung des Grades eines Doktors
der Naturwissenschaften
– Dr. rer. nat. –
Vorgelegt im Fachbereich 3
(Mathematik & Informatik)
der Universit¨ at Bremen
im August 2007Datum des Promotionskolloquiums: 19. September 2007
Gutachter: Prof. Dr. Fabian Wirth (Universit¨ at Wurzburg)¨
Prof. Dr. Lars Grune¨ (Universit¨ at Bayreuth)To my parents, Antje and Herbert.Contents
Introduction 1
1. Preliminaries 9
1.1.Comparisonfunctions...................... 9
1.2.Orderandmonotonicity 12
1.2.1.Order........................... 12
1.2.2.Monotonicity....................... 14
1.2.3.Matrixinducedoperators................ 16
1.3.Graphsandprojections 18
1.4.Dynamicalsystems........................ 22
1.5. Sets of decay and the “no joint increase” condition . . . . . . 25
1.6.Notesandreferences 31
2. Monotone dynamics 33
2.1.Fromattractivitytoorder.................... 34
2.2.Fromordertoatractivity 38
2.3.Pathconstruction......................... 44
2.4.Dichotomyintwodimensions.................. 63
2.5. Monotone inequalities, solutions, and bounds on solutions . . 66
2.6.AlocalversionofTheorem2.5.1................ 74
2.7.Notesandreferences....................... 76
3. Input-to-output stability for interconnected systems 79
3.1.Stabilityconcepts 83
3.1.1. Trajectory estimates based stability concepts . . . . . 84
3.1.2.Lyapunovfunctionbasedconcepts........... 85
3.2.Problemstatement........................ 87
3.2.1.IOpSestimatesforthesubsystems 87
3.3.Smalgaintheorems....................... 93
3.3.1.AG,GS,andISS..................... 94
3.3.2.ISSforlinearsystems..................104
3.3.3.Localtrajectorybasedestimates............105
3.3.4.IOpSLyapunovfunctions................108
vContents
3.4.Notesandreferences.......................115
4. Implementation and Application 119
4.1.ComputationalMethods.....................120
4.1.1. Testing the no joint increase condition locally . . . . . 121
4.1.2. Finding a point in Ω∩ S ................121
r
4.1.3. Constructing D on [0,w]forw ∈ Ω .........133
4.1.4. σ on [0,w]forw ∈ Ω ..........133
4.2.Applicationtoformationcontrol134
A. Nonsmooth analysis 143
viIntroduction
This work is devoted to the quest for sufficient criteria for the stability of large
scale nonlinearly interconnected control systems such that these criteria can
be stated in a ‘closed’ form.
We have been inspired by the linear case: Given n linear one dimensional
control systems
n

x˙ =−a x + b x + d u,i =1,...,n,i i i ij j i i
j=1
where x ,a,b ,d ∈ R for i,j =1,...,nsuch that a > 0 for all i=1,...,n.i i ij i i
The state variables x of the individual systems are one dimensional, so wei
can write the n single equations in vector notation to obtain
⎛ ⎞
−a b b ... b
1 12 13 1n
⎜ ⎟b −a b ... b
21 2 23 2n
⎜ ⎟
x˙ = x + Du, (†)
⎜ ⎟..
⎝ ⎠.
b ... ... b −a
1n n,n−1 n
wherewewritex =( x ,...,x ) , u =( u ,...,u ),and
1 n 1 n
D =diag(d ,...,d ). Denoting the matrix in (†)byA,thenitiswell
1 n
known that the origin is globally asymptotically stable for the compound
system (†) with inputs u set to zero, if and only if A is a Hurwitz matrix,
i.e., all eigenvalues of A lie in the left open complex half plane.
Observethatthisisamatrixstabilitycriterion.Inthiscasethematrix
A encodes all information that we have about the systems, excluding infor-
mation about external inputs u. Also this criterion is not only sufficient but
also necessary.
If the interconnected systems themselves are nonlinear and the intercon-
nection structure, i.e., how individual systems influence each other, is also
nonlinear, then of course we cannot directly obtain a matrix to investigate
from the individual systems themselves. Still, under some circumstances sta-
bility properties of equilibriums of the compound nonlinear system might be
accessible through different methods. Either we could try to prove stability
1properties for the large-scale system directly, or otherwise we could decom-
pose the compound system into smaller components, assess these individually
and somehow aggregate the obtained results to establish stability of the com-
pound system. Such a divide and conquer method could be, e.g, to assess
simple interconnections like cascades and simple feedback loops between the
individual systems and so, in an iterative manner, aggregate several stable
systems to one stable system in each step, until only one stable system is left.
Of course, by applying such a method we are able to prove stability of the
compound system for certain classes of systems and stability concepts. At
this stage the concept of input-to-state stability comes into the game. This
is a nonlinear c for robust stability. For this stability concept it is
known, that cascades of stable systems and, under a small gain condition,
feedback loops of stable systems are stable.
Figure 0.1.: A cascade and a feedback interconnection of two input-output
systems.
Now we ask for a criterion similar to the one in the linear case, roughly
stating something similar to “A is Hurwitz, so the network must be stable”,
but for general interconnection types.
Input-to-state stability in essence gives us a matrix of nonlinear functions
that describe worst case bounds on the interactions of the interconnected
systems. We state and prove conditions on that matrix that ensure stability
of the network. In order to do so, we need something equivalent to ma-
trix theory, but for ‘nonlinear’ matrices. Therefore the content of this work
comprises two subjects. The first one deals with monotone operators on the
npositive orthant in R . We may think of these operators as the matrices just
mentioned. The second subject is a stability analysis of large-scale intercon-
nections of dynamical systems, where we heavily depend on the “nonlinear
matrix theory”, i.e., the theory of monotone operators that we develop first.
Monotone Operators
n nThe first subject to be studied concerns monotone operators T : R → R ,
+ +
n nwhere R =( R ) , R =[0 ,∞). We investigate a condition that implies
+ +
+
foraclassofsuchoperatorsthat
2