Lecture Notes in Control and Information Sciences 389
Editors: M. Thoma, F. Allgöwer, M. Morari Rafael Bru and Sergio Romero-Vivó (Eds.)
Positive Systems Proceedings of the third Multidisciplinary International Symposium on Positive Systems: Theory and Applications (POSTA 2009) Valencia, Spain, September 2–4, 2009
ABC Series Advisory Board P. Fleming, P. Kokotovic, A.B. Kurzhanski, H. Kwakernaak, A. Rantzer, J.N. Tsitsiklis
Editors Rafael Bru Instituto Universitario de Matemática Multidisciplinar, Universidad Politécnica de Valencia, Camí de Vera s/n, 46022 Valencia Spain E-mail: [email protected]
Sergio Romero-Vivó Instituto Universitario de Matemática Multidisciplinar, Universidad Politécnica de Valencia, Camí de Vera s/n, 46022 Valencia Spain E-mail: [email protected]
ISBN 978-3-642-02893-9 e-ISBN 978-3-642-02894-6
DOI 10.1007/978-3-642-02894-6
Lecture Notes in Control and Information Sciences ISSN 0170-8643
Library of Congress Control Number: Applied for
c 2009 Springer-Verlag Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India.
Printed in acid-free paper 543210 springer.com Preface
Nowadays, the researchers into Control Theory and its Applications as well as Matrix Analysis are well aware and recognize the importance of Positive Sys- tems. This volume contains the proceedings of the “Third Multidisciplinary Symposium on Positive Systems: Theory and Applications (POSTA09)” held in Valencia, Spain, September 2–4, 2009. At present, this is the only world congress whose main topic is focused on this field. After this third event, we think that we have established the basis of a regular triennial event supported in the task of the previous organizing committees and hope to have met the requirements of their expectations. This POSTA09 meeting has been organized by members of the “Univer- sitat Polit`ecnica de Val`encia”,who have taken their doctor’s degree in the subjects of the conference. We are grateful to all of them for their enthu- siasm, effort and especially their collaboration, without the help of which the outcome would have never been the same. Also, we highly appreciate Christian Commault for having his experience at our disposal. Besides that, we wish to thank the International Programme Committee components and additional referees for their outstanding work in the review process of the contributions. We are happy to say that their constructive suggestions and positive comments have improved the quality of the presentations. We are very much obliged to the following organizations for having fi- nancially backed this congress: “Ministerio de Educaci´on y Ciencia”, “Con- selleria d’Educaci´o de la Generalitat Valenciana”, “Universitat Polit`ecnica de Val`encia (UPV)” and “Sociedad Espa˜nola de Matem´atica Aplicada (SEMA)”. We extend our gratitude to the “International Linear Algebra Society (ILAS)”, “Institut Universitari de Matem`atica Multidisciplin`aria (IMM)” and the “Algebra´ Lineal, An´alisis Matricial y Aplicaciones (ALAMA)” network for having endorsed this meeting. Finally, we would like to thank all of the par- ticipants for their attendance in spite of these hard times. Their presence here reasserts the success of this conference and we hope to see them successively in the future.
Valencia, Rafael Bru September 2009 Sergio Romero-Viv´o Organization
Program Committee
Chairman
Rafael Bru Universitat Polit`ecnica de Val`encia, Spain
Members
Georges Bastin Universit´e Catholique de Louvain, Belgium Luca Benvenuti Universit`a di Roma “La Sapienza”, Italy Vincent Blondel Universit´e Catholique de Louvain, Belgium Rafael Cant´o Universitat Polit`ecnica de Val`encia, Spain Carmen Coll Universitat Polit`ecnica de Val`encia, Spain Bart De Moor Katholieke Universiteit Leuven, Belgium Alberto De Santis Universit`a di Roma “La Sapienza”, Italy Elena De Santis Universit`a dell’Aquila, Italy Lorenzo Farina Universit`a di Roma “La Sapienza”, Italy Stephane Gaubert INRIA, Ecole Polytechnique, France Alessandro Giua Universit`a di Cagliari, Italy Jean-Luc Gouz´e INRIA Sophia Antipolis, France Diederich Hinrichsen Universit¨at Bremen, Germany Tadeusz Kaczorek Warsaw University of Technology, Poland Ulrich Krause Universit¨at Bremen, Germany Volker Mehrmann Technische Universit¨at Berlin, Germany Ventsi Rumchev Curtin University of Technology, Australia Maria Pia Saccomani Universit`a di Padova, Italy Elena S´anchez Universitat Polit`ecnica de Val`encia, Spain Jan H. van Schuppen CWI, Amsterdam, The Netherlands Robert Shorten The Hamilton Institute, Co. Kildare, Ireland Anton A. Stoorvogel University of Twente, The Netherlands VIII Organization
Ana M. Urbano Universitat Polit`ecnica de Val`encia, Spain Maria Elena Valcher Universit`a di Padova, Italy Paul Van Dooren Universit´e Catholique de Louvain, Belgium Joseph Winkin Universit´e Notre-Dame de la Paix, Belgium
Organizing Committee
Bego˜na Cant´o Universitat Polit`ecnica de Val`encia, Spain Rafael Cant´o Universitat Polit`ecnica de Val`encia, Spain Beatriz Ricarte Universitat Polit`ecnica de Val`encia, Spain Sergio Romero-Viv´o Universitat Polit`ecnica de Val`encia, Spain
Additional Referees
Esteban Bailo Maya Mincheva Gregory Batt Francisco Pedroche Bego˜na Cant´o Juan Manuel Pe˜na Bart De Schutter Harish Pillai Zong Woo “Victor” Geem Beatriz Ricarte Josep Gelonch Sergio Romero-Viv´o Bernd Heidergott Bartek Roszak Julien Hendrickx Boris Shapiro On´esimo Hern´andez-Lerma HalL.Smith Alexandros Karatzoglou Juan Ramon Torregrosa Jerzy Klamka Elena Virnik Thomas G. Kurtz Yimin Wei Hongwei Lin Eva Zerz James H. Liu Contents
Plenary Talks Reputation Systems and Nonnegativity ...... 3 Cristobald de Kerchove, Paul Van Dooren Lyapunov Exponents and Uniform Weak Normally Repelling Invariant Sets ...... 17 Paul Leonard Salceanu, Hal L. Smith Reachability Analysis for Different Classes of Positive Systems ...... 29 Maria Elena Valcher
Invited Sessions On the Positive LQ-Problem for Linear Discrete Time Systems ...... 45 Charlotte Beauthier, Joseph J. Winkin The Importance of Being Positive: Admissible Dynamics for Positive Systems ...... 55 Luca Benvenuti, Lorenzo Farina Detectability, Observability, and Asymptotic Reconstructability of Positive Systems ...... 63 Tobias Damm, Cristina Ethington Stability Radii of Interconnected Positive Systems with Uncertain Couplings ...... 71 Diederich Hinrichsen X Contents
Linear Operators Preserving the Set of Positive (Nonnegative) Polynomials ...... 83 Olga M. Katkova, Anna M. Vishnyakova
Convergence to Consensus by General Averaging ...... 91 Dirk A. Lorenz, Jan Lorenz
Stability and D-stability for Switched Positive Systems ...... 101 Oliver Mason, Vahid S. Bokharaie, Robert Shorten On Positivity and Stability of Linear Volterra-Stieltjes Differential Systems ...... 111 Pham Huu Anh Ngoc
Eigenvalue Localization for Totally Positive Matrices ...... 123 Juan Manuel Pe˜na Positivity Preserving Model Reduction ...... 131 Timo Reis, Elena Virnik The Minimum Energy Problem for Positive Discrete-Time Linear Systems with Fixed Final State ...... 141 Ventsi Rumchev, Siti Chotijah A Rollout Algorithm for Multichain Markov Decision Processes with Average Cost ...... 151 Tao Sun, Qianchuan Zhao, Peter B. Luh Analysis of Degenerate Chemical Reaction Networks ...... 163 Markus Uhr, Hans-Michael Kaltenbach, Carsten Conradi, J¨org Stelling k-Switching Reachability Sets of Continuous-Time Positive Switched Systems...... 173 Maria Elena Valcher
Contributed Papers Inverse-Positive Matrices with Checkerboard Pattern ...... 185 Manuel F. Abad, Mar´ıa T. Gass´o, Juan R. Torregrosa Some Remarks on Links between Positive Invariance, Monotonicity, Strong Lumpability and Coherency in Max-Plus Algebra ...... 195 Mourad Ahmane, Laurent Truffet Contents XI
Stability Analysis and Synthesis for Linear Positive Systems with Time-Varying Delays ...... 205 Mustapha Ait Rami Linear Programming Approach for 2-D Stabilization and Positivity ...... 217 Mohammed Alfidi, Abdelaziz Hmamed, Fernando Tadeo An Algorithmic Approach to Orders of Magnitude in a Biochemical System ...... 233 Eric Benoˆıt, Jean-Luc Gouz´e Structural Identifiability of Linear Singular Dynamic Systems ...... 243 Bego˜na Cant´o, Carmen Coll, Elena S´anchez On Positivity of Discrete-Time Singular Systems and the Realization Problem ...... 251 Rafael Cant´o, Beatriz Ricarte, Ana M. Urbano Multi-Point Iterative Methods for Systems of Nonlinear Equations...... 259 Alicia Cordero, Jos´eL.Hueso,EulaliaMart´ınez, Juan R. Torregrosa Identifiability of Nonaccessible Nonlinear Systems ...... 269 Leontina D’Angi , Maria Pia Saccomani, Stefania Audoly, Giuseppina Bellu Trajectory Tracking Control of a Timed Event Graph with Specifications Defined by a P-time Event Graph ...... 279 Philippe Declerck, Abdelhak Guezzi Tropical Scaling of Polynomial Matrices ...... 291 St´ephane Gaubert, Meisam Sharify Scrutinizing Changes in the Water Demand Behavior ...... 305 Manuel Herrera, Rafael P´erez-Garc´ıa, Joaqu´ın Izquierdo, Idel Montalvo Characterization of Matrices with Nonnegative Group-Projector ...... 315 Alicia Herrero, Francisco J. Ram´ırez, N´estor Thome Robust Design of Water Supply Systems through Evolutionary Optimization ...... 321 Joaqu´ın Izquierdo, Idel Montalvo, Rafael P´erez-Garc´ıa, Manuel Herrera XII Contents
Applications of Linear Co-positive Lyapunov Functions for Switched Linear Positive Systems ...... 331 Florian Knorn, Oliver Mason, Robert Shorten A Problem in Positive Systems Stability Arising in Topology Control ...... 339 Florian Knorn, Rade Stanojevic, Martin Corless, Robert Shorten
Control of Uncertain (min,+)-Linear Systems...... 349 Euriell Le Corronc, Bertrand Cottenceau, Laurent Hardouin On a Class of Stochastic Models of Cell Biology: Periodicity and Controllability ...... 359 Ivo Marek Implementation of 2D Strongly Autonomous Behaviors by Full and Partial Interconnections ...... 369 Diego Napp Avelli, Paula Rocha Ordering of Matrices for Iterative Aggregation - Disaggregation Methods ...... 379 Ivana Pultarov´a The Positive Servomechanism Problem under LQcR Control ...... 387 Bartek Roszak, Edward J. Davison
Author Index ...... 397 Reputation Systems and Nonnegativity
Cristobald de Kerchove and Paul Van Dooren
Abstract. We present a voting system that is based on an iterative method that as- signs a reputation to n + m items, n objects and m raters, applying some filter to the votes. Each rater evaluates a subset of objects leading to an n×m rating matrix with a given sparsity pattern. From this rating matrix a formula is defined for the reputa- tion of raters and objects. We propose a natural and intuitive nonlinear formula and also provide an iterative algorithm that linearly converges to the unique vector of reputations and this for any rating matrix. In contrast to classical outliers detection, no evaluation is discarded in this method but each one is taken into account with different weights for the reputations of the objects. The complexity of one iteration step is linear in the number of evaluations, making our algorithm efficient for large data set.
1 Introduction
Many measures of reputation have been proposed under the names of reputation, voting, ranking or trust systems and they deal with various contexts ranging from the classification of football teams to the reliability of each individual in peer to peer systems. Surprisingly enough, the most used method for reputation on the Web amounts simply to average the votes. In that case, the reputation is, for instance, the average of scores represented by 5 stars in YouTube, or the percentage of positive transactions in eBay. Therefore such a method trusts evenly each rater of the sys- tem. Besides this method, many other algorithms exploit the structure of networks generated by the votes: raters and evaluated items are nodes connected by votes. A great part of these methods use efficient eigenvector based techniques or trust
Cristobald de Kerchove and Paul Van Dooren Universit«e catholique de Louvain (UCL), Department of Applied Mathematics, Avenue Georges Lemaˆõtre, 4 B-1348 Louvain-la-Neuve Belgium, e-mail: [email protected],[email protected]
R. Bru and S. Romero-Viv«o (Eds.): Positive Systems, LNCIS 389, pp. 3Ð16. springerlink.com c Springer-Verlag Berlin Heidelberg 2009 4 C. de Kerchove and P. Van Dooren propagation over the network to obtain the reputation of every node [7, 9, 13Ð17]. They can be interpreted as a distribution of some reputation flow over the network where reputations satisfy some transitivity: you have a high reputation if you have several incoming links coming from nodes with a high reputation. The average method, the eigenvector based techniques and trust propagation may suffer from noise in the data and bias from dishonest raters. For this reason, they are sometimes accompanied by statistical methods for spam detection [10, 19], like in the context of web pages trying to boost their PageRank scores by adding artificial incoming links [2, 8]. Detected spam is then simply removed from the data. This describes the three main strategies for voting systems: simple methods averaging votes where raters are evenly trusted, eigenvector based techniques and trust propagation where reputations directly depend on reputations of the neighbours, and finally statistical measures to classify and possibly remove some of the items. Concerning the Iterative Filtering (IF) systems which we introduce here, we will make the following assumption: Raters diverging often from other raters’ opinion are less taken into account. We label this the IF-property and will formally define it later on. This property is at the heart of the filtering process and implies that all votes are taken into account, but with a continuous validation scale, in contrast with the di- rect deletion of outliers. Moreover, the weight of each rater depends on the distance between his votes and the reputation of the objects he evaluates: typically weights of random raters and outliers decrease during the iterative filtering. The main criticism one can have about the IF-property is that it discriminates “marginal” evaluators, i.e., raters who vote differently from the average opinion for many objects. How- ever, IF systems may have different basin of attraction, each corresponding to a group of people with a coherent opinion. Votes, raters and objects can appear, disappear or change making the system dy- namical. This is for example the case when we consider a stream of news like in [5]: news sources and articles are ranked according to their publications over time. Nowadays, most sites driven by raters involve dynamical opinions. For instance, the blogs, the site Digg and the site Flickr are good places to exchange and discuss ideas, remarks and votes about various topics ranging from political election to photos and videos. We will see that IF systems allow to consider evolving voting matrices and then provide time varying reputations.
2 Iterative Filtering Systems
We first consider the case where the votes are fixed, i.e., the voting matrix does not change over time, and all objects are evaluated by all raters, i.e., the voting matrix is full. With these assumptions, we present the main properties of IF systems and then we restrict ourselves to the natural case of quadratic IF systems where the reputations are given by a linear combination of the votes and the weights of the raters are based on the Euclidean distance between the reputations and the votes. Reputation Systems and Nonnegativity 5
n×m n ∈ Ê Let X ∈ Ê be the voting matrix, r be the reputation vector of the objects ∈ m and w Ê be the weight vector of the raters. The entry Xij is the vote to object th i given by rater j and the vector x j,the j column of X, represents the votes of rater j: X =[x1 ...xm]. The bipartite graph formed by the objects, the raters and their votes is represented by the n × m adjacency matrix A,i.e.,Aij = 1 if object i is evaluated by rater j,and 0 otherwise. For the sake of simplicity, we assume in this section that every object has been evaluated by all raters
Aij = 1foralli, j.(1)
The general case, where the bipartite graph is not complete, will be handled later. The belief divergence d j of rater j is the normalized distance between his votes and the reputation vector r (for a particular choice of norm) 1 d = x − r2. (2) j n j Let us already remark that when the bipartite graph is not complete, i.e., Eq. (1) is not satisfied, then the number of votes varies from one rater to another. Therefore the normalization of the belief divergence d j in Eq. (2) will change depending on this number. Before introducing IF systems, we define the two basic functions of these systems:
n m
→ Ê ( )= , (1) the reputation function F : Ê : F w r that gives the reputation vector depending on the weights of the raters and implicitly on the voting matrix X;
m n
→ Ê ( )= , (2) The filtering function G : Ê : G d w that gives the weight vector for the raters depending on the belief divergence d of each rater defined in Eq. (2). We formalize the so-called IF-property described in the introduction that claims that raters diverging often from the opinion of other raters are less taken into ac- count. We will make the reasonable assumption that raters with identical belief di- vergence receive equal weights. Hence, we can write ⎡ ⎤ g(d1) ⎢ ⎥ ( )= . . G d ⎣ . ⎦ (3) g(dm)
We call the scalar function g the discriminant function associated with G. 6 C. de Kerchove and P. Van Dooren
A filtering function G satisfies the IF-property if its associated discriminant func- tion g is positive and decreases with d. Therefore, the IF-property merely implies that a decrease in belief divergence d j for any rater j corresponds to a larger weight w j. Eq. (3) indicates that every rater has the same discriminant function g,butwe could also consider personalized functions g j penalizing differently the raters. In [4] three choices of function g are shown to have interesting properties
− g(d)=d k, (4) g(d)=e−kd, (5) g(d)=1 − kd. (6)
All discriminant function g are positive and decrease with d for positive k and there- fore satisfy the IF-property. However k must be small enough to keep g positive in Eq. (6) and hence to avoid negative weights.
Definition 1. IF systems are systems of equations in the reputations rt of the objects and the weights wt of the raters that evolve over discrete time t according to the voting matrix X
+ rt 1 = F(wt ), (7) 1 wt+1 = G(dt+1) with dt+1 = x − rt+12 (8) j n j for j = 1,...,m and some initial vector of weights w0.
Definition 1 does not imply any convergence properties, nor robustness to initial conditions. The system (7-8) can have several converging solutions and it allows the existence of cycles in the iterative processes. The fixed points of (7-8) satisfy
∗ ∗ r = F(w ), (9) 1 w∗ = G(d∗) with d∗ = x − r∗2 (10) j n j for j = 1,...,m. Let us remark that IF systems can be interpreted as a particular iterative search method to find the stable fixed points of Eq. (9-10). IF systems are a simple iterative scheme for this system with the advantage to be easily extended to take into account dynamical voting matrices Xt with t ≥ 0. In this paper, we focus on IF systems where we fix the reputation function F ap- pearing in Eq. (7,9) and the norm . given in the definition of the belief divergence in Eq. (2).
Definition 2. Quadratic IF systems are IF systems where the reputation function F and the belief divergence are respectively given by Reputation Systems and Nonnegativity 7
w F(w)=X , (11) 1T w 1 ◦ d = (XT − 1rT ) 21, (12) n where 1 is the vector of all 1’s and (XT − 1rT )◦2 is the componentwise product (XT − 1rT ) ◦ (XT − 1rT ).
In that definition, the reputation function F(w) is naturally given by taking the weighted average of the votes and the belief divergence d (given in the matrix form) is defined using the Euclidian norm. Therefore Eq. (12) are quadratic equations in r and amount to consider an estimate of the variances of the votes for every rater according to a given reputation vector r. For any positive vector w, the reputation vector r then belongs to the polytope
m m P = { ∈ n | = λ λ = λ ≥ }. r Ê r ∑ jx j with ∑ j 1and j 0 (13) j=1 j=1
From Eq. (11), the iterations and the fixed point in Eq. (7,9) are given by quadratic equations in r and w
rt+1(1T wt)=Xwt, (14) ∗ ∗ ∗ r (1T w )=Xw . (15)
The next theorem establishes the correspondence between the iterations of quad- ratic IF systems and some steepest descent methods minimizing some energy func- tion. The fixed points in Eq. (14,15) are then the stationary points of that energy function.
Theorem 1. (see [4]) The fixed points of quadratic IF systems with integrable discriminant function g, are the stationary points of the energy function m d j(r) E(r)= ∑ g(u)du, (16) j=1 0 where d j is the belief divergence of rater j that depends on r. Moreover one itera- tion step in quadratic IF systems corresponds to a steepest descent direction with a particular step size t+1 t t t r = r − α ∇rE(r ), (17) with αt = n . 2(1T wt )
3 Iterative Filtering with Affine Discriminant Function
We look at the quadratic IF system with the discriminant function g defined in Eq. (6) where the iterations are given by 8 C. de Kerchove and P. Van Dooren
wt rt+1 = F(wt )=X , (18) 1T wt wt+1 = G(dt+1)=1 − k dt+1, (19) starting with equal weights w0 = 1. By substituting w, the fixed point of the system is given by a system of cubic equations in r∗
k (X − r∗1T )(1 − (XT − 1(r∗)T )◦21)=0, (20) n with r∗ in the polytope P defined in Eq. (13). Theorem 2 claims that r∗ is unique in P if k is such that the weights are strictly positive for all vectors of reputations r ∈ P. This result uses the associated energy function that we define for affine IF systems.
3.1 The Energy Function
The energy function in Eq. (16) associated with system (18,19) is given by 1 E(r)=− wT w + constant, (21) 2k where w depends on r according the function G(r). We will see later that this energy t function decreases with the iterations, i.e., (E(r ))t≥0 decreases, and under some assumption on k, it converges to the unique minimum. The iterations in system (18,19) can be written as a particular minimization step on the function E, + 1 rt 1 = argmin − G(r)T G(rt) . r 2k Therefore, we have for all t that (wt+1)T (wt ) ≥ (wt )T (wt ).
3.2 Uniqueness
The following theorem proves that the stable point of quadratic IF systems with g defined in Eq. (6) is unique, under some condition on parameter k. This result fol- lows directly from the energy function E that is a fourth-order polynomial equation.
Theorem 2. (see [4]) The system (18,19) has a unique fixed point r∗ in P if
−1 k < min d∞ . r∈P Reputation Systems and Nonnegativity 9
−4.5
−5.5 −.5 The Energy function The Energy function
−7 −1 0 0
0.5 0 0.5 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1 1 (a) (b)
0
−.4 −0.5 The Energy function The Energy function
−.8 −1 0 0
0.5 0 0.5 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1 1 (c) (d)
Fig. 1 Four energy functions with two objects and increasing values of k. We have in the unit square: (a) a unique minimum; (b) a unique minimum but other stationary points are close to the boundary; (c) a unique minimum and other stationary points; (d) a unique maximum.
3.3 Convergence of the Method
We analyze the convergence of system (18,19) that reaches the minimum of the energy function E in P.Letrt and rt+1 be two subsequent points of the iterations given by some search method. Then the next point rt+1 is obtained by choosing a vector v and a scalar γ such that
rt+1 = rt + γv. (22)
This corresponds to some line search on the scalar energy function
e(y)=E(rt + yv) (23) that is a polynomial of degree 4. We have that e(0) is the energy at rt and e(γ) is the energy at rt+1. Finally it is useful for the sequel to define the scalar that minimizes e given by β = arg min E(rt + yv). (24) y with rt +yv∈P 10 C. de Kerchove and P. Van Dooren
System (18,19) provides a steepest descent method with a particular step size. The direction v and the scalar γ in Eq. (22) are n v = −∇ E(rt) and γ = αt = , r 2(1T wt ) so that we recover Eq. (17). This particular step size αt can be compared to the step size β that minimizes E in the same direction given in Eq. (24). We have that the particular step size αt is generally smaller than β in numeric simulations, meaning that the step stops before reaching the minimum of the energy function E in the t direction v. The sequence (E(r ))t≥0 can be shown to decrease so that we have the following convergence result.
Theorem 3. (see [4]) The steepest descent method given by system (18,19) con- verges to the unique fixed point in P if
−1 k < min d∞ . r∈P There exist greater values of k such that the minimum of E remains unique and the previous methods converge to this minimum. By increasing k, we allow the maxima of E to appear in the polytope P, see Fig. 1(c). Then, we need to verify during the iterations if (rt ) remains in the basin of attraction of E. Theorem 4. (see [4]) If the energy function E in Eq. (21) has a minimum, then the system (18,19) is locally convergent and its asymptotic rate of convergence is linear. Let us remark that for a singular matrix X, the rate of convergence will be faster. In particular, when X is a rank 1 matrix, we have X = r∗1T (every object receives m identical votes from the raters) and the method converges in one step. When we take greater values of k maxima of the function E may appear in P.Howeverif the sequence (1T wt ) remains positive, the sequence (E(rt )) remains decreasing and converges to a stationary point of E. In order to avoid saddle points and maxima, we need to avoid to reach the minimum. The idea of increasing k is to make the discriminant function g more penalizing and therefore to have a better separation between honest and dishonest raters. We refer to [4] for more details on this.
4 Sparsity Pattern and Dynamical Votes
This section extends some previous results to the case where the voting matrix has some sparsity pattern, that is when an object is not evaluated by all raters. Moreover we analyze dynamical voting matrices representing votes that evolve over time. Reputation Systems and Nonnegativity 11
4.1 Sparsity Pattern
In general, the structure of real data is sparse. We hardly find a set of raters and objects with a vote for all possible pairs. An absence of vote for object i from rater j will imply that the entry (i, j) of the matrix X is equal to zero, that is, by using the adjacency matrix A, if Aij = 0, then Xij = 0. These entries must not be considered as votes but instead as missing values. There- fore the previous equations presented in matrix form require some modifications that will include the adjacency matrix A. We write the new equations and their im- plications using the order of the previous section. Let us already mention that some theorems will be simply stated without proof. Whenever their extensions with an adjacency matrix A are straightforward. The belief divergence for IF systems in Eq. (2) becomes 1 d j = x j − a j ◦ r, n j
th th where a j is the j column of the adjacency matrix A and n j is the j entry of the vector n containing the numbers of votes given to each item, i.e.,
n = AT 1.
On the other hand, the scalar n remains the total number of objects, i.e., the number of rows in A. Therefore, when A is full, then n = n1. Eq. (11-12) for quadratic IF systems can be replaced by the following ones: the reputation function, that remains the weighted average of the votes, is given in ma- trix form by [Xw] F(w)= , [Aw] [·] where [·] is the componentwise division. Let us remark that every entry of Aw must be strictly positive. This means that every object is evaluated by at least one rater with nonzero weight. Then all possible vectors of reputations r are include in the polytope
m m Pø = { ∈ n | = λ λ = λ ≥ }. r Ê ri ∑ jx j with ∑ j aij 1and j 0 j=1 j=1
The third equation (12) for the belief divergence with the Euclidian norm is changed into
(XT − AT ◦ 1rT )◦21 d = . (25) [AT 1] 12 C. de Kerchove and P. Van Dooren
With these modifications, the iterations and the fixed point in Eq. (7,9) are given by quadratic equations in r and w
(A ◦ rt+11T )wt = Xwt (26) ∗ ∗ ∗ (A ◦ r 1T )w = Xw . (27)
Hence we expect an energy function to exist and Theorem 1 is generalized by the following theorem.
Theorem 5. (see [4]) The fixed points of quadratic IF systems with integrable discriminant function g, are the singular points of the energy function
( ) 1 m d j r E(r)= ∑ n j g(u)du, (28) n j=1 0 where d j is the belief divergence of rater j that depends on r. Moreover one iteration step in quadratic IF systems corresponds to a dilated steepest descent direction with a particular step size t+1 t t t r = r − α ◦ ∇rE(r ) (29) αt = n [1] with 2 [Awt ] . The number of votes n gives somehow a weight of importance for the mini- j d j ( ) mization of the surface 0 g u du. Therefore a rater with more votes receives more attention in the minimization process.
4.2 Affine Quadratic IF Systems
The system for the discriminant function g(d)=1 − kd is given by
+ [Xw] rt 1 = F(wt )= , (30) [Aw] wt+1 = G(dt+1)=1 − k dt+1, (31) with the belief divergence defined in Eq. (25). The energy function is given by 1 E(r)=− wT [w ◦ n]+constant, (32) 2kn where w depends on r according to the function G(r). Theorem 2 remains valid for the system (30-31) and the arguments are similar. The steepest descent method adapted to the system (30-31) converges with the prop- erty that the sequence (E(rt )) decreases. The proofs are closely related to the ones presented in Theorems 3. Reputation Systems and Nonnegativity 13
Fig. 2 Trajectory of reputations (circles) for a 5-periodic voting matrix
Theorem 6. (see [4]) The steepest descent method given by system (30,31) con- verges to the unique fixed point in Pø if
−1 k < min d∞ . r∈Pø The choice of k can be made larger to better separate honest from dishonest raters. Theorem 4 remains valid with a few modifications in its proof to take into account the adjacency matrix A.
Theorem 7. (see [4]) If the energy function E in Eq. (32) has a minimum, then (30,31) is locally convergent and its asymptotic rate of convergence is linear.
This section shows that most of the earlier analysis can still be applied when we introduce a sparsity pattern in the voting matrix.
4.3 Dynamical Votes
We consider in this section the case of time-varying votes. Formally, we have dis- crete sequences t t (X )t≥0, (A )t≥0 14 C. de Kerchove and P. Van Dooren of voting matrices and adjacency matrices evolving over time t. Hence the IF sys- t+1 tem (7,8) takes into account the new voting matrix X in the functions Ft+1 and Gt+1 that become time-dependent:
t+1 t r = Ft+1(w ), (33) t+1 t+1 w = Gt+1(d ). (34)
The system (30,31) for dynamical voting matrices is then given by (30,31)
t+1 t t+1 t X w r = F + (w )= , (35) t 1 [At+1wt] t+1 t+1 t+1 w = Gt+1(d )=1 − k d , (36) with the belief divergence dt+1 defined as in Eq. (25) after replacing X and r by Xt+1 t+1 t and r . We already now that for subsequent constant matrices X with T1 ≤ t ≤ T2, the iterations on rt and wt of system (35,36) tend to fixed vectors r∗ and w∗ provided that k is not too large. In [4] we give stronger results for the case of 2-periodic voting sequences.
5 Concluding Remarks
The general definition of Iterative Filtering systems provides a new framework to analyze and evaluate voting systems. We emphasized the need for a differentiation of trusts between the raters unlike what is usually done on the Web. The originality of the approach lies in the continuous validation scale for the votes. Next, we as- sumed that the set of raters is characterized by various possible behaviors including raters who are clumsy or partly dishonest. However, the outliers being in obvious disagreement with the other votes remain detectable by the system as shown in the simulations in the cases of alliances, random votes and spammers. Our paper focuses on the subclass of quadratic IF systems and we show the ex- istence of an energy function that allows us to link a steepest descent to each step of the iteration. It then follows that the system minimizes the belief divergence ac- cording to some norm defined from the choice of the discriminant function. This method was illustrated in [4] using two data sets: (i) the votes of 43 countries during the final of the EuroVision 2008 and (ii) the votes of 943 movie lovers in the website of MovieLens. It was shown that the IF method penalizes certain types of votes. In the first set of data, this yielded a difference in the ranking used by Euro- vision and the ranking obtained by our method, in the sense that countries trading votes with e.g. neighboring countries, would get a smaller weight. The second set of data was used to verify the desired property mentioned in the introduction: raters diverging often from other raters’ opinion are less taken into account. We see two application areas of voting systems: first, the general definition of IF systems offers the possibility to analyze various systems depending on the context Reputation Systems and Nonnegativity 15 and the objectives we aim for; second, the experimental tests and the comparisons are crucial to validate the desired properties (including dynamical properties) and to discuss the choice of the IF systems.
Acknowledgements. This paper presents research results of the Belgian Programme on In- teruniversity Attraction Poles, initiated by the Belgian Federal Science Policy Office, and a grant Action de Recherche Concert«ee (ARC) of the Communaut«eFranc¸aise de Belgique. The scientific responsibility rests with its authors.
References
1. Akerloff, G.: The Market for Lemons: Quality Uncertainty and the Market Mechanism. Quaterly Journal of Economics 84, 488Ð500 (1970) 2. Baeza-Yates, R., Castillo, C., L«opez, V.: PageRank Increase under Different Collusion Topologies. In: First International Workshop on Adversarial Information Retrieval on the Web (2005), http://airweb.cse.lehigh.edu/2005/baeza-yates.pdf 3. de Kerchove, C., Van Dooren, P.: Reputation Systems and Optimization. Siam News (March 14, 2008) 4. de Kerchove, C., Van Dooren, P.: Iterative Filtering in Reputation Systems (submitted, 2009) 5. Del Corso, G.M., Gull«õ, A., Romani, F.: Ranking a stream of news. In: Proceedings of the 14th international conference on World Wide Web (2005) 6. Ginsburgh, V., Noury, A.: Cultural Voting. The Eurovision Song Contest. Mimeo (2004) 7. Guha, R., Kumar, R., Raghavan, P., Tomkins, A.: Propagation of Trust and Distrust. In: Proceedings of the 13th International Conference on World Wide Web, pp. 403Ð412 (2004) 8. Gy¬ongyi, Z., Garcia-Molina, H.: Link spam alliances. In: VLDB 2005: Proceedings of the 31st international conference on Very large data bases, pp. 517Ð528 (2005) 9. Kamvar, S., Schlosser, M., Garcia-molina, H.: The Eigentrust Algorithm for Reputation Management in P2P Networks. In: Proceedings of the 12th International Conference on World Wide Web, pp. 640Ð651 (2003) 10. Kotsovinos, E., Zerfos, P., Piratla, N.M., Cameron, N., Agarwal, S.: Jiminy: A Scal- able Incentive-Based Architecture for Improving Rating Quality. In: St¿len, K., Wins- borough, W.H., Martinelli, F., Massacci, F. (eds.) iTrust 2006. LNCS, vol. 3986, pp. 221Ð235. Springer, Heidelberg (2006) 11. Laureti, P., Moret, L., Zhang, Y.-C., Yu, Y.-K.: Information Filtering via Iterative Refine- ment. EuroPhysic Letter 75, 1006Ð1012 (2006) 12. McLachlan, G., Krishnan, T.: The EM algorithm and extensions. John Wiley & Sons, New York (1996) 13. Mui, L., Mohtashemi, M., Halberstadt, A.: A Computational Model of Trust and Repu- tation. In: Proceedings of the 35th Annual Hawaii International Conference, pp. 2431Ð 2439 (2002) 14. O’Donovan, J., Smyth, B.: Trust in recommender systems. In: Proceedings of the 10th International Conference on Intelligent User Interfaces, pp. 167Ð174 (2005) 15. Page, L., Brin, S., Motwani, R., Winograd, T.: The PageRank Citation Ranking: Bringing Order to the Web. Stanford Digital Library Technologies Project (1998) 16 C. de Kerchove and P. Van Dooren
16. Richardson, M., Agrawal, R., Domingos, P.: Trust Management for the Semantic Web. In: Fensel, D., Sycara, K., Mylopoulos, J. (eds.) ISWC 2003. LNCS, vol. 2870, pp. 351Ð 368. Springer, Heidelberg (2003) 17. Theodorakopoulos, G., Baras, J.: On Trust Models and Trust Evaluation Metrics for Ad Hoc Neworks. IEEE Journal on Selected Areas in Communications 24(2), 318Ð328 (2006) 18. Yu, Y.-K., Zhang, Y.-C., Laureti, P., Moret, L.: Decoding information from noisy, redun- dant, and intentionally distorted sources. Physica A 371(2), 732Ð744 (2006) 19. Zhang, S., Ouyang, Y., Ford, J., Make, F.: Analysis of a Lowdimensional Linear Model under Recommendation Attacks. In: Proceedings of the 29th annual International ACM SIGIR conference on Research and development in information retrieval, pp. 517Ð524 (2006) Lyapunov Exponents and Uniform Weak Normally Repelling Invariant Sets
Paul Leonard Salceanu and Hal L. Smith
Abstract. Let M be a compact invariant set contained in a boundary hyperplane of n the positive orthant of Ê for a discrete or continuous time dynamical system defined on the positive orthant. Using elementary arguments, we show that M is uniformly weakly repelling in directions normal to the boundary in which M resides provided all normal Lyapunov exponents are positive. This result is useful in establishing uniform persistence of the dynamics.
1 Introduction
Dynamical systems models in population biology are typically defined on the non- negative cone in Euclidean space. In order to establish persistence of some or all components (species) in the model, it is often necessary to show that a compact invariant set on the boundary of the cone is an isolated invariant set and that it is repelling, at least in some directions normal to M.See[4,5,7,15,19,20]forre- cent work in the theory of persistence, sometimes called permanence. In this paper, building on the work of [4, 15] and [11, 14], we show that Lyapunov exponents can be used to establish the requisite repelling properties for both discrete and contin- uous time systems. This is well known when M is a fixed point or periodic orbit but not so when the dynamics on M is more complicated. We use only elementary arguments rather than appealing to the multiplicative ergodic theorem [1, 2, 4, 15]. This extends our earlier work in [12Ð14] which covered only the discrete case. The use of Lyapunov exponents in the study of biological models was pioneered by Metz [8], Metz et. al. [9], who proposed that the dominant Lyapunov exponent gives the best measure of invasion fitness, and by Rand et. al. [10] who used it to characterize the invasion “speed” of a rare species. See also the more recent review
Paul Leonard Salceanu and Hal L. Smith School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA, e-mail: [email protected], e-mail: [email protected]
R. Bru and S. Romero-Viv«o (Eds.): Positive Systems, LNCIS 389, pp. 17Ð27. springerlink.com c Springer-Verlag Berlin Heidelberg 2009 18 P.L. Salceanu and H.L. Smith by Ferriere and Gatto [3] which deals with computational aspects. Roughly, a pos- itive dominant Lyapunov exponent corresponding to a potential invading species in the environment set by a resident species attractor implies that the invader can successfully invade. Our results will give a mathematically rigorous interpretation of this for the nonlinear dynamics. Ashwin et al. [2] use “normal” Lyapunov ex- ponents and invariant measures to answer the following question: if f : M → M is a smooth map on a smooth finite dimensional manifold, N is a lower dimensional submanifold for which f (N) ⊆ N,andA ⊆ N is an attractor for f|N ,isA an attractor for f , or it is an unstable saddle?
2MainResults
Due to our need to use both subscripts and occasionally superscripts for sequences, =( (1) (m) T m | | = | (i)| we adopt the notation x x ,...,x ) ∈ Ê .Let x ∑ x denote the norm
m m m m m
( , ) ∈ Ê ⊂ Ê Ê = { ∈ Ê on Ê and d z M for the distance of z to M . Denote by + x :
(i) m m
≥ ∈ Ê =( ) x ≥ 0,∀i} the nonnegative cone in Ê . We write x 0whenx +;ifA aij is an n × n matrix, then A ≥ 0ifaij ≥ 0foralli, j. Observe that |x + y| = |x| + |y|
, ∈ m m = Ê for vectors x y Ê+.WeletZ+ +. We consider the discrete dynamical system
zn+1 = F(zn), z0 ∈ Z+ (1) and the continuous dynamical system
z (t)=F(z(t)), z0 := z(0) ∈ Z+ (2) on the nonnegative cone Z+. It is assumed that (1) and (2) generate a semi-dynamical system on Z+. In case m ∈ , (i) = of (1), F : Z+ → Z+ is continuous; in case (2), F : Z+ → Ê satisfies z Z+ z 0 ⇒ F(i)(z) ≥ 0 and sufficient regularity properties such that solutions of (2) exist and are unique. We assume that m = p + q, p,q ≥ 1andthatZ+ is decomposed as follows
=( , ) ∈ m p q = Ê × Ê z x y Z+ ≡ Ê+ + +
Compatible with this decomposition, assume that F(z)=(f (z),g(z)).Define
X = {z =(x,y) ∈ Z+ : y = 0}.
We assume: X and Z+ \ X are positively invariant sets. (3) Positive invariance of X for both (1) and (2) means that g(x,0)=0, (x,0) ∈ X. If F satisfies additional smoothness hypotheses, then it would follow from posi- tive invariance of X that (1) and (2) can be expressed as Lyapunov Exponents and Normal Repellers 19 x + = f (z ) n 1 n (4) yn+1 = A(zn)yn and, respectively as x = f (z) (5) y = A(z)y where the matrix function z → A(z) is continuous and satisfies:
A(z) ≥ 0, z ∈ X (6) in case of (4), and Aij(z) ≥ 0, i = j, z ∈ X (7) in case of (5). Rather than assume the required smoothness of F, we simply assume
hereafter that (6) holds for (4) and that (7) holds for (5).
Ê ∈ Ì Let Ì+ denote either + or +. When we write t +, that means we consider both discrete and continuous cases. To make the notation more general, let also zt , yt
etc. denote z(t), y(t) etc., when t ∈ Ê+. ( φ( )) ∈ + Let t t∈Ì+ be the dynamical system generated by (4) (for t )orby(5)
∈ + O ( ) = {φ( , ) ∈ Ì } (for t Ê+), respectively. Let z : t z : t + , which we will refer to as the positive orbit through z.LetI denote the identity matrix. Let P(n,z) and P(t,z) denote the fundamental matrix solutions for
un+1 = A(φ(n,z))un (8) and for v (t)=A(φ(t,z))v(t). (9) They satisfy:
P(n + 1,z)=A(φ(n,z))P(n,z), P(0,z)=I (10) for discrete time and d P(t,z)=A(φ(t,z))P(t,z), P(0,z)=I (11) dt for continuous time. In either case, it follows from (6), (7) that . P(t,z) ≥ 0,∀ z ∈ X, ∀t ∈ Ì+ (12)
Let M ⊆ X be compact and positively invariant set. We envision that in typical applications, M will be an invariant set in the interior of the face X of the cone Z+. In this paper, we will focus on the behavior of solutions near M in Z+ \ X. Following Arnold [1], P(n,z) (or P(t,z)) is a matrix co-cycle generated by (8) (or by (9)). It is trivial to check that P has the following (co-cycle) property:
( , ∀ , ∈ Ì . P t2,φ(t1,z))P(t1,z)=P(t1 +t2,z), ∀ z ∈ + t1 t2 + (13)
20 P.L. Salceanu and H.L. Smith ∈ Ê Hereafter, when we take t ∈ +, we refer to (4), and when we take t +, we refer to (5). ∈ q Following[1,2,6],foranyz M and η ∈ Ê we define the normal Lyapunov exponent λ(z,η) as
1 | ( , ) . λ(z,η)=limsup ln P t z η|, t ∈ Ì+ (14)
t→∞ t
λ( ,η)=λ( , η), ∀ ∈ Ê \{ } As noted in [1], λ(z,η) ∈{−∞}∪ Ê and z z a a 0 .We q +( ,η) ∈ ( ,η) only consider the case that η ∈ Ê+ because in that case z 0 Z+ and 0 represents a normal vector to M at z. The co-cycle property (13) can be used to show that λ(z,η)=λ(φ(s,z),P(s,z)η), s ≥ 0. (15) Definition 1. We call the compact positively invariant set M a uniformly weak normally repelling set if there exists ε > 0suchthat
limsupd(φ(t,z),M) > ε, ∀z ∈ Z+ \ X. t→∞
Equivalently, in view of (3), there exists a neighborhood V of M in Z+ such that
∀z ∈ V \ X, ∃ t = t(z) > 0, φ(t,z) ∈/ V.
We stress that M may be an attractor relative to the dynamics restricted to the positively invariant set X but we are concerned with the behavior of solutions near M in the positively invariant set Z+ \ X. In [14], we used the terminology “M a uni- formly weak repeller” for the definition above; we believe the current terminology gives a more accurate description. The adjective “uniform” reflects that ε is inde- pendent of z; “weak” reflects that limit superior, rather than limit inferior, appears in the definition; “normal” indicates that we are only interested in the behavior of solutions in Z+ \ X. First, we give a lemma adapted from [11, 14] that gives an alternative formulation for the “positivity” of Lyapunov exponents. Let = { q |η| = }. U η ∈ Ê+ : 1
Lemma 1. Let K ⊂ X be compact. Assume that ∀ ( , \{ }τ = τ( ,η) | (τ, )η| > . z η) ∈ K ×U, ∃ Ì+ 0 z such that P z 1 (16)
Then ∃ c > 1, ∃ V a bounded neighborhood of K in Z+, such that if L ⊆ Visa positively invariant set, then L ⊂ X and ∀ ( , , ν → ∞, | (ν , )η| > p, ∀ ≥ . z η) ∈ L ×U, ∃ (νp)p ⊆ Ì+ p P p z c p 1 (17)
If, in addition, K is positively invariant, then (16) is equivalent to
λ(z,η) > 0, ∀ (z,η) ∈ K ×U. (18) Lyapunov Exponents and Normal Repellers 21
Proof. Let W = K ×U (so W is compact) andw ˆ =(zˆ,ηˆ ) ∈ K ×U.From(16)we \{ } | (τ, )η| > have that there exists τˆ = τˆ(zˆ,ηˆ ) ∈ Ì+ 0 such that P ˆ zˆ ˆ 1. The function (z,η) →|P(τˆ,z)η| being continuous, there exist δwˆ > 0, cwˆ > 1suchthat
| (τˆ, )η| > , ∀ =( ,η) ∈ ( ) = { ∈ + × || − | < δ }. P z cwˆ w z Bδwˆ wˆ : w˜ Z U w˜ wˆ wˆ (19)
Since W is compact, there exists a finite set {w1,...,wk}⊆W such that W ⊂ C := k i i ∪ = Bδ (w ), where for every i = 1,...,k, δ i is the quantity corresponding to w , i 1 wi w coming from (19) (i.e.,foreveryi = 1,...,k, (19) is satisfied withw ˆ replaced by i τ = τ( i), δ = δ , = ,..., = w ). To simplify notation, let i : w i : wi i 1 k. Also, let c : mincwi i (hence c > 1) and τ = maxτi. Thus, from (19) we have that i | (τ , )η| > , ∀ =( ,η) ∈ ( i), ∀ = ,..., . P i z c w z Bδi w i 1 k (20)
Now let V ⊂ Z+ be a bounded neighborhood of K such that V × U ⊆ C and let L ⊆V be positively invariant. We prove that L ⊂ X arguing by contradiction: suppose L \ X = 0./ Let a =(ax,ay) ∈ L \ X.Since|ay| > 0, we can define α := ay/|ay|.Note that α ∈ U. We will show that ∃ ( , ν → ∞, | (ν , )α| > p, ∀ ≥ . νp)p ⊂ Ì+ p such that P p a c p 1 (21) by inductively constructing the sequence (νp)p. Thus, there exists i ∈{1,...,k} such ( ,α) ∈ ( i) | (ν , )α| > ν = τ that a Bδi w . Then, from (20) we have P 1 a c,where 1 i.Now p suppose |P(νp,a)α| > c for some p ≥ 1. Let α˜ = P(νp,a)α/|P(νp,a)α|.Since (2) (2) L\X is positively invariant, φ(νp,a) ∈ L\X, hence φ (νp,a) > 0, where φ (t,z) denotes the vector formed with the last q components of φ(t,z).So
1 1 (2) P(νp,a)α = P(νp,a)ay = φ (νp,a) > 0. |ay| |ay|
α ∈ ∈{ ,..., } (φ(ν , ),α) ∈ ( j) Thus, ˜ U. There exists j 1 k such that p a ˜ Bδ j w .Then again, from (20) we have
p+1 |P(τ j,φ(νp,a))α˜ | > c, which implies |P(τ j,φ(νp,a))P(νp,a)α| > c .
p+1 This means, using (13), that |P(νp+1,a)α| > c , where we define νp+1 = νp +τ j. Note that, by construction, νp → ∞ as p → ∞. Hence (21) holds. Then, we have that
(2) p |φ (νp,a)| = |P(νp,a)ay| > c |ay|, ∀ p ≥ 1,
(2) which implies that |φ (νp,a)|→∞ as p → ∞. But this is a contradiction to L being bounded. Hence, L ⊂ X. Now (17) can be proved identically as for (21), using that L ⊂ X is positively ( , ) ≥ , ∀ ∈ , ∀ ∈ invariant and that P t z 0 z X t Ì+. 22 P.L. Salceanu and H.L. Smith
Now assume that K is also positively invariant. The implication (18) ⇒ (16) is trivial. For the converse, using (17) and the fact that νp ≤ pτ, ∀ p ≥ 1, we have, for all (z,η) ∈ K ×U,that
1/νp p/νp 1/τ 1 1 |P(νp,z)η| > c ≥ c ⇒ ln|P(νp,z)η| > lnc, ∀ p ≥ 1. νp τ Hence 1 1 λ(z,η)=limsup ln(|P(t,z)η|) ≥ lnc > 0. t→∞ t τ This completes our proof.
In the next result we establish sufficient conditions for M to be a uniformly weak normally repelling set. Let
Ω(M)=∪z∈Mω(z), (22) where ω(z) represents the omega limit set of z.
Theorem 1. Let M ⊂ X be a nonempty compact and positively invariant. M is a uniformly weak normally repelling set if
λ(z,η) > 0, ∀ (z,η) ∈ M ×U. (23)
If ∀ ( , , ( , )η = z η) ∈ M ×U, ∀t ∈ Ì+ P t z 0 (24) and λ(z,η) > 0, ∀(z,η) ∈ Ω(M) ×U (25) then (23) holds.
Proof. First we show that (23) implies that M is a uniformly weak normally repelling set. For this, we argue by contradiction: suppose M is not a uniformly m weak normally repelling set. So, there exists a sequence (z˜ )m ⊆ Z+ \ X such that
limsupd(φ(t,z˜m),M) < 1/m, ∀ m ≥ 1. t→∞ ( ≥ Hence there exists a sequence τm)m ⊂ Ì+ such that, for each m 1, we have
m d(φ(t,z˜ ),M) < 1/m, ∀t ≥ τm. (26)
m m m m Let z =(x ,y )=φ(τm,z˜ ). Using the positive invariance of Z+ \X,wehavethat
ym > 0, ∀ m ≥ 1. (27)
From the semiflow property of φ and from (26) we get ( m , ≥ . d φ(t,z ),M) < 1/m, ∀t ∈ Ì+ m 1 (28) Lyapunov Exponents and Normal Repellers 23
Using (23), we obtain from Lemma 1 (applied with K = M) that there exists V a bounded neighborhood of M in Z+, having the property that any positively invariant ∈ = { ∈ set contained in V is a subset of X. Then there exists m Æ such that Bm : z Z+|d(z,M) ≤ 1/m} is contained in V.ThesetL = {φ(n,zm)|n ≥ 0} is positively invariant and, according to (28), it is contained in Bm.Also(see(27))L \ X = 0./ But this is a contradiction, according to Lemma 1. Hence, M is a uniformly weak repeller. Now we prove the final assertion. Let (a,α) ∈ M × U.Using2) and the fact that ω(a) ⊂ X is compact and invariant, we can again apply Lemma 1, now with K = ω(a).SoletVa be a neighborhood of ω(a) and c > 1 as in the above mentioned φ( , ) ∈ lemma. Since φ(t,a) → ω(a) as t → ∞, there exists τa ∈ Ì+ such that t a Va, ∀t ≥ τa.LetL = {φ(t,a)|t ≥ τa}.ThenL is a positively invariant set contained in Va.Letα˜ = P(Na,a)α/|P(Na,a)α|. Note that α˜ is well defined, due to (24), and that α˜ ∈ U. So, from (17), there exists a sequence νp → ∞ such that |P(νp,φ(τa,a))α˜ | > cp, ∀ p ≥ 1. Thus, using (13) we get
p |P(νp + τa,a)α| > c |P(τa,a)α|, ∀ p ≥ 1.
p We can find a p large enough such that to have c |P(τa,a)α| > 1. So, we proved that ∀ ( , \{ } | (τ, )η| > , z η) ∈ M ×U, ∃ τ ∈ Ì+ 0 such that P z 1 which is equivalent to (23), by Lemma 1. This completes our proof. Note that (24) is automatically satisfied in the continuous case. In the discrete case, it is equivalent to A(z)η = 0, ∀ z ∈ M, ∀ η ∈ U. As it will be seen below, when the matrix A(z) satisfies stronger positivity con- ditions, then the Lyapunov exponents are independent of the unit vector η.Let ||A|| = sup{|Aξ | : |ξ | = 1} denote the norm of an n×n matrix. For matrices A,B,we write A ≤ B if aij ≤ bij, ∀i, j; inequality A 0 means all entries of A are positive.
Proposition 1. Let z ∈ X have compact orbit closure O+(z). In the discrete case, assume that + ∃N, P(n,z0) 0, n ≥ N, z0 ∈ O (z). (29) In the continuous case, assume that
+ A(z0) is irreducible, z0 ∈ O (z). (30)
Then 1 λ(z,η)=limsup ln||P(t,z)||, ∀ η ∈ U. (31) t→∞ t In particular, if (29), respectively (30), holds for each z ∈ M, then (31) holds for every z ∈ M. ∈ ( ) = ( , ) Proof. First we give the proof for the discrete case (t Æ). Let P n : P n z .
∀ ∈ , ∃ , ∈ Æ ≤ ≤ − = + We have that n Æ kn pn , with 0 pn N 1, such that n knN pn.Let = ( , ), ∀ ≥ = , ∀ ≥ ˜( )= ··· Bs : P N zsN s 0. Here, z0 z. Hence Bs 0 s 0. Let P n Bkn B0. ( )= ( , ) ˜( ) So P n P pn zknN P n . First, we want to apply Theorem 3.4. in [16] for the 24 P.L. Salceanu and H.L. Smith
+ sequence of matrices B0,...,Bs,....SinceO (z) is compact, it follows that there exist constant matrices C, D 0 such that D ≥ Bs ≥ C, ∀ s ≥ 0. Let δ = min(Cij) i, j and γ = max(Dij). So, the following hold: i, j a) min(Bs)ij ≥ δ > 0, ∀ s ≥ 0; i, j b) max(Bs)ij ≤ γ < ∞. i, j Thus, hypotheses of [16, Theorem 3.4] hold and (see exercise 3.6 in [16]) we have that P˜(n)li → cij > 0asn → ∞, (32) P˜(n)lj i th for some cij independent of l. Denote by P˜(n) the i column of P˜(n). Then (32) implies that |P˜(n)i| lim = cij. (33) n→∞ |P˜(n) j| th Let ei ∈ U be the unit vector whose i component equals one, and the other compo- nents are zero. Then, using (33), we get, for any i ∈{1,...,q},that
λ( , )= 1 | ( , ) ˜( ) |≤ 1( || ( , )||+ z ei limsup ln P pn zknN P n ei limsup ln P pn zknN n→∞ n n→∞ n 1 (34) + ln|P˜(n)i|)=limsup ln|P˜(n)i|. n→∞ n On the other hand,
1 1 i λ(z,ei) ≥ limsup ln|P(knN,z)ei| = limsup ln|P˜(n) | n→∞ knN n→∞ knN n 1 1 (35) = limsup ln|P˜(n)i| = limsup ln|P˜(n)i|. n→∞ knN n n→∞ n Thus, from (34) and (35) we have that
1 i λ(z,ei)=limsup ln|P˜(n) |. (36) n→∞ n
But, for any i, j ∈{1,...,q} we have that 1 1 |P˜(n)i| |P˜(n)i| = |P˜(n) j| limsup ln limsup ln j n→∞ n n→∞ n |P˜(n) | 1 |P˜(n)i| 1 = + |P˜(n) j| limsup ln j ln n→∞ n |P˜(n) | n 1 = limsup ln|P˜(n) j|. n→∞ n Thus, let Lyapunov Exponents and Normal Repellers 25
1 i c = λ(z,ei)=limsup ln|P˜(n) |, ∀ i = 1,...,q. (37) n→∞ n q Let η ∈ U.Thereexistp1,..., pq ∈ [0,1] such that η = ∑ piei. Then, using (37), we i=1 obtain 1 1 q λ(z,η)=limsup ln|P(n)η| = limsup ln| ∑ piP(n)ei| →∞ →∞ n n n n i=1 1 q 1 ≥ limsup ∑ pi ln|P(n)ei| = limsup ln|P(n)ei| = c. (38) →∞ →∞ n n i=1 n n On the other hand, we have
q λ(z,η)=λ(z, ∑ piei) ≤ max λ(z,ei)=c, (39) i= ,..,q i=1 1 where we used the following two properties of Lyapunov exponents (see [1] page 114):
1) λ(z,η1 + η2) ≤ max{λ(z,η1),λ(z,η2)},and \{ } 2) λ(z,aη)=λ(z,η), ∀ a ∈ Ê 0 . From (38) and (39) we obtain λ(z,η)=c. It is clear that
1 c = λ(z,η) ≤ limsup ln||P(n)||. (40) n→∞ n Now, we want to show the opposite inequality. Because all norms on a finite dimen- sional normed linear space are equivalent, there exist constants a,b > 0 such that i a||B||1 ≤||B|| ≤ b||B||1 for all matrices B,where||B||1 = max|B |.Then i 1 || ( )|| ≤ 1 || ( , )||·||˜( )|| limsup ln P n limsup ln P pn zknN P n n→∞ n n→∞ n = 1 || ˜( )|| limsupn→∞ n ln P n 1 (41) ≤ limsup ln(b||P˜(n)||1) n→∞ n 1 1 = limsup ln||P˜(n)||1 = lim ln||P˜(nk)||1, n→∞ n k→∞ nk ( , → ∞ → ∞ ∈{ ,..., } for some sequence nk)k ⊆ Æ nk as k . There exists j 1 q such j that ||P˜(nk)||1 = |P˜(nk) | for infinitely many k s. Hence, there exists a subsequence j (n˜k)k of (nk)k such that ||P˜(n˜k)||1 = |P˜(n˜k) |, ∀ k. Then, from (41) we have that
1 1 j 1 j limsup ln||P(n)|| ≤ lim ln|P˜(n˜k) |≤limsup ln|P˜(n) | = c. (42) n→∞ n k→∞ n˜k n→∞ n 26 P.L. Salceanu and H.L. Smith
1 Now, (40) and (42) imply λ(z,η)=limsup ln||P(n)||.Sinceη ∈U was arbitrarily n→∞ n chosen, the proof for the discrete case is complete. ∈ ∈ Now let us consider the continuous case (t Ê+). Again, considering z X fixed,
( , ) ( ) [·] → we can denote P t z , in short, by P t . Denote by : Ê the greatest integer function. The same argument used to prove (36) leads to 1 λ(z,η)=limsup ln|P([t])η|. (43) t→∞ [t]
1 | ( ) Let n :=[t].Thenλ(z,η)=limsup ln P n η|.LetBn := P(1,zn), ∀ n ∈ Æ.Then n→∞ n P(n)=Bn−1 ···B0. Our hypotheses on matrix A(z) guarantee that Bn 0, ∀ n ≥ 0 (see [17, Theorem 1.1]). Now the same proof as for the discrete case, applied with N = 1 (hence kn = n, pn = 0andP˜(n)=P(n)), carries over and leads to 1 1 limsup ln|P(n)η| = limsup ln||P(n)||. n→∞ n n→∞ n But 1 1 1 limsup ln||P(n)|| = limsup ln||P([t])|| = limsup ln||P(t)|| n→∞ n t→∞ [t] t→∞ t Indeed, the left side is clearly less than or equal to the right hand side and the oppo- site inequality is obtained as in (34). This completes our proof.
Theorem 1 shows that M is a uniformly weak normally repelling set provided λ(z,η) > 0forallz ∈ M and all η ∈ U. Furthermore, under a mild hypothesis, it suffices to show λ(z,η) > 0forz ∈ Ω(M)=∪z∈Mω(z) and η ∈ U. Proposition 1 gives conditions for λ(z,η) to be independent of η ∈ U for all z ∈ M. In this case, using (15), we see that λ(z)=λ(φ(s,z)), s ≥ 0 is constant on forward orbits. We assume hereafter that λ(z)=λ(z,η) depends only on z ∈ M. As a consequence, if the hypotheses of Theorem 1 hold, and if Ω(M) consists of a finite number of periodic orbits Oi, i = 1,2,...,p, then it suffices to show that λ(zi) > 0forsome choice zi ∈ Oi, i = 1,2,...,p. In this case, only finitely many exponents must be computed. See [3, 11, 14] where λ(zi) is related to the spectral radii of a certain Floquet matrix. According to the multiplicative ergodic theorem [1, 6], if M is invariant and there exists an ergodic F-invariant Borel probability measure μ on M,thenλ(z) is a constant on M, almost surely. Unfortunately, for our results almost sure positivity of the Lyapunov exponent does not suffice.
References
1. Arnold, L.: Random Dynamical Systems. Springer, Heidelberg (1998) 2. Ashwin, P., Buescu, J., Stewart, I.: From attractor to chaotic saddle: a tale of transverse instability. Nonlinearity 9, 703Ð737 (1996) Lyapunov Exponents and Normal Repellers 27
3. Ferriere, R., Gatto, M.: Lyapunov Exponents and the Mathematics of Invasion in Oscil- latory or Chaotic Populations. Theor. Population Biol. 48, 126Ð171 (1995) 4. Garay, B.M., Hofbauer, J.: Robust Permanence for Ecological Differential Equations, Minimax, and Discretizations. SIAM J. Math. Anal. 34, 1007Ð1039 (2003) 5. Hirsch, M.W., Smith, H.L., Zhao, X.-Q.: Chain transitivity, attractivity and strong repel- lors for semidynamical systems. J. Dynamics and Diff. Eqns. 13, 107Ð131 (2001) 6. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, New York (1995) 7. Magal, P., Zhao, X.-Q.: Global Attractors and Steady States for Uniformly Persistent Dynamical Systems. SIAM J. Math. Anal. 37, 251Ð275 (2005) 8. Metz, J.A.J.: Fitness. Evol. Ecol. 2, 1599Ð1612 (2008) 9. Metz, J.A.J., Nisbet, R.M., Geritz, S.A.H.: How Should We Define “Fitness” for General Ecological Scenarios? Tree 7, 198Ð202 (1992) 10. Rand, D.A., Wilson, H.B., McGlade, J.M.: Dynamics and Evolution: Evolutionarily Sta- ble Attractors, Invasion Exponents and Phenotype Dynamics. Philosophical Transac- tions: Biological Sciences 343, 261Ð283 (1994) 11. Salceanu, P.L.: Lyapunov exponents and persistence in dynamical systems, with appli- cations to some discrete-time models. Phd. Thesis, Arizona State University (2009) 12. Salceanu, P.L., Smith, H.L.: Persistence in a Discrete-time, Stage-structured Epidemic Model. J. Difference Equ. Appl. (to appear, 2009) 13. Salceanu, P.L., Smith, H.L.: Persistence in a Discrete-time Stage-structured Fungal Dis- ease Model. J. Biol. Dynamics 3, 271Ð285 (2009) 14. Salceanu, P.L., Smith, H.L.: Lyapunov Exponents and Persistence in Discrete Dynamical Systems. Discrete and Continuous Dynamical Systems-B (to appear, 2009) 15. Schreiber, S.J.: Criteria for Cr Robust Permanence. J. Differ. Equations 162, 400Ð426 (2000) 16. Seneta, E.: Non-negative Matrices, an Introduction to Theory and Applications. Halsted Press, New York (1973) 17. Smith, H.L.: Monotone Dynamical Systems: an introduction to the theory of competitive and cooperative systems. Amer. Math. Soc. Surveys and Monograghs 41 (1995) 18. Smith, H.L., Zhao, X.-Q.: Robust Persistence for Semidynamical Systems. Nonlinear Anal. 47, 6169Ð6179 (2001) 19. Thieme, H.R.: Mathematics in Population Biology. Princeton University Press, New Jer- sey (2003) 20. Zhao, X.-Q.: Dynamical Systems in Population Biology. Springer, New York (2003) Reachability Analysis for Different Classes of Positive Systems
Maria Elena Valcher
Abstract. In this survey paper, reachability properties for discrete-time positive systems, two-dimensional discrete state-space models and discrete-time positive switched systems are introduced and characterized. Comparisons among the re- sults obtained in these three settings are presented, thus enlightening which results can be easily extended and what aspects, at present time, are still challenging open problems.
1 Introduction
Since the early seventies, positive systems have been the object of a noteworthy in- terest in the literature. Positive linear systems [14] naturally arise in various fields, such as bioengineering (compartmental models), economic modeling, behavioral science, and stochastic processes (Markov chains or hidden Markov models). Gen- erally speaking, these systems provide the natural framework for modeling physical systems whose describing variables necessarily take nonnegative values. It is clear, however, that apart from the nonnegativity constraint, various additional features may be relevant when capturing the system dynamics. These instances led to the in- troduction of different classes of positive systems, in particular, (one-dimensional) positive systems, two-dimensional (2D) positive systems and switched positive sys- tems. Even though for each of this class of systems several theoretical problems have been thoroughly investigated, a common research topic for all these classes of sys- tems has been the analysis of structural properties and, in particular, of reachability. The aim of this paper is that of providing a brief survey on the reachability charac- terizations obtained within these three settings, with special attention to the discrete time cases (even though similar analyses have been performed in the continuous- time cases). Specifically, reachability of discrete-time positive systems will be the
Maria Elena Valcher Dip. Ingegneria dell’Informazione, Universit`a di Padova, Italy, e-mail: [email protected]
R. Bru and S. Romero-Viv«o (Eds.): Positive Systems, LNCIS 389, pp. 29Ð41. springerlink.com c Springer-Verlag Berlin Heidelberg 2009 30 M.E. Valcher object of Sect. 2, and it has been investigated, just to quote some of the available ref- erences, in [5Ð7, 10Ð13, 24]; reachability of discrete two-dimensional positive sys- tems will be discussed in Sect. 3, and it has been the object of [1, 2, 17, 18, 21, 22]; finally, reachability of discrete-time positive switched systems will be addressed in Sect. 4 (see [9, 20, 26Ð28]).
Notation.The(i, j)th entry of a matrix A is denoted by [A]i, j. In the special case of [ ] a vector v, its ith entry is v i. The symbol Ê+ denotes the semiring of nonnegative
real numbers. A matrix A (in particular, a vector) with entries in Ê+ is said to be nonnegative (A ≥ 0). If A ≥ 0, but at least one entry is positive, A is said to be positive (A > 0). n We let ei denote the ith vector of the canonical basis in Ê (where n is always clear from the context). Given a vector v,thenonzero pattern of v is the set of indices corresponding to its nonzero entries, namely ZP(v) := {i : [v]i = 0}. A vector ∈ n ( )= ( )={ } v Ê+ is an ith monomial vector if ZP v ZP ei i .Amonomial matrix is a nonsingular square positive matrix whose columns are (distinct) monomial vectors. The Hurwitz products of two n × n matrices A1 and A2 are inductively defined [15] as
i j A1 A2 = 0, when either i or j is negative, i 0 = i , ≥ , 0 j = j , ≥ , A1 A2 A1 if i 0 A1 A2 A2 if j 0 i j i−1 j i j−1 A1 A2 = A1(A1 A2)+A2(A1 A2), if i, j > 0.
i j Notice that ∑i+ j= A1 A2 =(A1 + A2) . Basic definitions and results about cones may be found, for instance, in [3, 4]. We K ⊂ n recall here only those facts that will be used within this paper. A set Ê is said to be a cone if αK ⊂ K for all α ≥ 0; a cone is convex if it contains, with any two points, the line segment between them. A cone K is said to be polyhedral if it can be expressed as the set of nonnegative linear combinations of a finite set of generating vectors. This amounts to saying that a positive integer k and an n × k matrix C can be found, such that (s.t.) K coincides with the set of nonnegative combinations of the columns of C. In this case, we adopt the notation K := Cone(C). To efficiently introduce our results, we also need some definitions borrowed from the algebra of non-commutative polynomials [25]. Given the alphabet Ξ = ∗
{ξ1,ξ2,...,ξp}, the free monoid Ξ with base Ξ is the set of all words w =
ξ ξ ···ξ , ∈ Æ,ξ ∈ Ξ. i1 i2 ik k ih The integer k is called the length of w and is de- noted by |w|, while |w|i represents the number of occurrences of ξi in w.If = ξ ξ ···ξ Ξ ∗ w˜ j1 j2 jp is another element of , the product is defined by concatena- = ξ ξ ···ξ ξ ξ ···ξ . ε = tion ww˜ i1 i2 im j1 j2 jp This produces a monoid with 0,/ the empty word, as unit element. Clearly, |ww˜| = |w| + |w˜| and |ε| = 0. ξ ,ξ ,...,ξ 1 2 p is the algebra of polynomials in the noncommuting indetermi- n×n A = { , ,..., } nates ξ1,ξ2,..., ξp. For every family of p matrices in , : A1 A2 Ap , the map ψ defined on {ε,ξ1,ξ2,...,ξp} by the assignments ψ(ε)=In and ψ(ξi)=
n×n ξ ,ξ ,...,ξ Ai, i ∈p, uniquely extends to an algebra morphism of 1 2 p into Reachability Analysis for Different Classes of Positive Systems 31
n×n Ξ ∗ (as an example, ψ(ξ1ξ2)=A1A2 ∈ ). If w is a word in (i.e. a monic mono- ξ ,ξ ,...,ξ ψ ( , ,..., ) mial in 1 2 p ), the -image of w is denoted by w A1 A2 Ap .
2 Reachability of Discrete-Time Positive Systems
A (discrete-time) positive system is a state-space model
x(k + 1)=Ax(k)+Bu(k), k = 0,1,2,..., (1) where x(k) and u(k) denote the n-dimensional state variable and the m-dimensional
∈ n×n n×m ∈ Ê input variable, respectively, at the time instant k, while A Ê+ and B + .Un- der the nonnegativity constraint on the system matrices, the state trajectories of the system are constrained within the positive orthant, provided that the initial condition ( ) ( ), ∈ x 0 and the input sequence u k k +, are nonnegative. Reachability property for this type of systems focuses only on nonnegative states, reached by means of nonnegative inputs, and hence it is defined as follows.
n Definition 1. Given the positive system (1), a state x f ∈ Ê+ is said to be reach- ( ) = , ,..., − able if there exist k f ∈ Æ and a nonnegative input sequence u k , k 0 1 k f 1, that transfers the state of the system from the origin at k = 0tox f at time k = k f . The positive system (1) is monomially reachable if every monomial vector (equiv- alently, every canonical vector ei, i ∈n) is reachable, and reachable if every state n x f ∈ Ê+ is reachable.
It is easily seen that monomial reachability is a necessary and sufficient condition for reachability. Necessity is obvious. On the other hand, if each canonical vector i ei,i ∈n, is reachable at some time ki by means a nonnegative input sequence u (k), then each of them is reachable at k f := maxi ki by means of a suitably right-shifted i i version, say uø (k), of the sequence u (k). Consequently, every positive vector x f can n i( )[ ] be reached at time k f by means of the nonnegative input sequence ∑i=1 uø k x f i. This simple remark allows to convert the reachability problem in the easier mono- mial reachability problem. An algebraic characterization of monomial reachability, and hence of reachability, can be easily obtained by resorting to the reachability matrix of the system.
Definition 2. The reachability matrix at time k of system (1) is
k−1 Rk(A,B) :=[B | AB | ... | A B].
As the expression of the state at time k, starting from the zero initial condition x(0) and under the (nonnegative) soliciting input u(·),isgivenby 32 M.E. Valcher ⎡ ⎤ u(0) ⎢ ( ) ⎥ − ⎢ u 1 ⎥ x(k)=[B | AB | ... | Ak 1B]⎢ . ⎥, ⎣ . ⎦ u(k − 1) it is clear that the monomial vector ei is reachable at time k if and only if the reach- ability matrix Rk(A,B) includes an ith monomial column. Therefore Proposition 1. For the n-dimensional positive system (1), the following facts are equivalent ones: • the system is reachable; • the system is monomially reachable;
• ∈ Æ R ( , ) × there exists k f such that the reachability matrix k f A B includes an n n monomial matrix.
All the results reported up to now are quite straightforward. A quite nontrivial step, instead, was taken by Coxson, Larson and Schneider [10] in proving that if a positive system (1) is reachable then the index k f , in the third item of the previous proposition, can always be chosen equal to the system dimension n.
Proposition 2. The n-dimensional positive system (1) is reachable if and only if the reachability matrix Rn(A,B) includes an n × n monomial matrix. If we define the reachability index of a reachable positive system (1) as the small-
n
∈ Æ R ( , )= Ê est k f such that k f A B +, the previous proposition tells us that the reach- ability index cannot exceed the system dimension. The proof of this result is rather involved and it resorts to the precious graph-theoretic approach to the study of the structural properties of positive systems. Indeed, to every n-dimensional system with m inputs (1) we may associate [7, 8, 29] a digraph (directed graph) D(A,B), with n vertices, indexed by 1,2,...,n,andm sources s1,s2,...,sm. There is an arc ( j,i) from j to i if and only if [A]ij > 0, and there is an arc (s j,i) from the source s j to vertex i if and only if [B]ij > 0. A sequence s j → i0 → i1 →··· → ik−1, starting from the source s j, and passing through the vertices i0,...,ik−1,isans-path from s j to ik−1 (of length k) provided (s j,i0),(i0,i1),...,(ik−2,ik−1) are all arcs of D(A,B). It is easily seen that there is a path of length k from s j to some vertex i if and only if the (i, j)th entry of k−1 A B is positive. Clearly, leaving from some source s j,afterk steps one can reach several distinct vertices. This corresponds to saying that the jth column of Ak−1B has, in general, more than one nonzero entry. We say that an s-path of length k from s j deterministically reaches some vertex i, if no other vertex of the digraph can be reached in k steps starting from s j. If so, we refer to such an s-path as to a deterministic path (of length k)toi. Again, it is obvious that a vertex i can be deterministically reached from some source s j by means of a path of length k if and only if the jth column of Ak−1B is a ith monomial vector. So, we have realized that monomial reachability (and hence reachability) of a positive system (1) can be easily tested by simply verifying that for each vertex Reachability Analysis for Different Classes of Positive Systems 33 i ∈n there is a source and a deterministic path from that source reaching the vertex i.
Example 1. Consider the positive system (1) with ⎡ ⎤ ⎡ ⎤ 01000 00 ⎢ ⎥ ⎢ ⎥ ⎢10000⎥ ⎢10⎥ ⎢ ⎥ ⎢ ⎥ A = ⎢00010⎥ B = ⎢10⎥. ⎣00001⎦ ⎣00⎦ 00000 01
By inspecting the associated digraph D(A,B) one easily sees that, starting from s2, vertex 5 can be reached deterministically in one step, 4 in two steps and 3 in three steps. On the other hand, starting from s1, vertex 1 can be reached deterministically in two steps, while 2 in three steps. So, the system is reachable.
- - 5 4 3 D(A,B) 6
s1 @ § ¤ ? s2 @R 21 ¦6 ¥
Fig. 1 Graph description of the system of Example 1
Coxson and Larson proved [10] (by using a slightly different terminology, though) that if there exists a deterministic path from some source s j to some vertex i, then there exists a deterministic path from s j to i of length not greater than n,the number of vertices. This led to Proposition 2. This graph-theoretic interpretation turned out to be very profitable, as it allowed to derive canonical forms for reachable positive systems. A first result about canon- ical forms was derived in [29]. More refined results were later obtained by Bru and co-workers in [5, 7] (see, also, [6], where the concept of reachability index of a positive system (1) was generalized and characterized in graph-theoretic terms).
3 Reachability of Discrete 2D Positive Systems
A(discrete)two-dimensional (2D) positive system is a 2D state-space model de- scribed by the following first order state-updating equation [15]:
x(h + 1,k + 1)=A x(h,k + 1)+A x(h + 1,k) 1 2 (2) + B1u(h,k + 1)+B2u(h + 1,k), 34 M.E. Valcher where the n-dimensional local states x(·,·) and the m-dimensional inputs u(·,·) take nonnegative values, A1 and A2 are nonnegative n × n matrices, B1 and B2 are nonnegative n × m matrices, and the initial conditions are assigned by specifying the (nonnegative) values of the state vectors on the separation set C0 := {(h,k) : , ∈ , + = }, h k h k 0 namely by assigning all local states of the initial global state X0 := {x(h,k) : (h,k) ∈ C0}. All input sequences involved have supports included
{( , ) ∈ × + ≥ } in the half-plane h k : h k 0 . For this class of systems (even when no positivity constraint is introduced) reach- ability represents a rather articulate concept [15, 16]. This is an immediate conse- quence of the fact that, when defining this concept, we may either refer to the local states or to the global states Xt := {x(h,k) : (h,k) ∈ Ct }, which collect all local C , + = }. states lying on the separation set t := {(h,k) : h,k ∈ h k t
Definition 3. A 2D positive system (2) is said to be • X ∗ n locally reachable if, upon assuming 0 = 0, for every x ∈ Ê+ there exist
( , ) ∈ × + > (·,·) ( , )= h k , h k 0, and a nonnegative input sequence u such that x h k x∗. When so, we will say that x∗ is reachable in h + ksteps; ∗ • globally reachable if, upon assuming X0 = 0, for every global state X with
n
∈ (·,·) entries in Ê+,thereexistN + and a nonnegative input sequence u such that ∗ ∗ the global state XN coincides with X . When so, we will say that X is reachable in N steps. If all local (global) states are reachable, system (2) is locally (globally) reachable, and the smallest number of steps which allows to reach every nonnegative local (global) state represents its local (global) reachability index ILR (IGR).
Clearly, as in the standard (nonpositive) case, global reachability ensures local reachability, while the converse is not true.
3.1 Local Reachability of 2D Positive Systems
In order to characterize local reachability, we first introduce the reachability matrix in k steps [15] of the 2D positive system (2), i.e. R ( , , , )=[ + 2 k A1 A2 B1 B2 B1 B2 A1B1 A1B2 A2B1 A2B2 A1B1 (A 1 1A )B + A2B ... Ak−1B ] 1 2 1 1 2 2 2 = ( i−1 j ) +( i j−1 ) A1 A2 B1 A1 A2 B2 i, j≥0, 0
Proposition 3. [18] Given a 2D positive system (2), the following facts are equiv- alent ones: • the system is locally reachable; • the system is “locally monomially reachable”, i.e. all monomial vectors can be reached (starting from X0 = 0) by means of nonnegative inputs; • ∈ R ( , , , ) there exists k Æ such that the reachability matrix in k steps, k A1 A2 B1 B2 , includes an n × n monomial submatrix;
•∃ ( × = ( ) ∈ , ∈ , n pairs hi,ki) ∈ + + and n indices j j i m i n s.t. hi−1 ki hi ki−1 ZP (A1 A2)B1e j +(A1 A2)B2e j = {i}.
If so,
ILR = maxi min{hi + ki : ∃ j = j(i) s.t. hi,ki hi−1 ki hi ki−1 (A1 A2)B1e j +(A1 A2)B2e j is an ith monomial vector}.
As for positive systems (1), local reachability of 2D positive systems is a struc- tural property, by this meaning that it only depends on the nonzero patterns of the system matrices and not on the specific values of their nonzero elements. Conse- quently, it can be investigated by resorting to a graph-theoretic approach. To every 2D positive system (2), of size n, with m inputs, we associate a 2D (2) influence digraph D (A1,A2,B1,B2) with n vertices, 1,2,...,n, and m sources s1,s2,...,sm.ThereisanA1-arc (an A2-arc) from j to i if and only if the (i, j)th entry of A1 (of A2) is nonzero. There is a B1-arc (a B2-arc) from s j to i if and only if the (i, j)th entry of B1 (of B2) is nonzero. Example 2. The positive system with a single input described by the matrices ⎛⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎞ 1500 0000 0 2 ⎜⎢0000⎥ ⎢0040⎥ ⎢0⎥ ⎢0⎥⎟ (A ,A ,B ,B )=⎜⎢ ⎥,⎢ ⎥,⎢ ⎥,⎢ ⎥⎟ (3) 1 2 1 2 ⎝⎣1000⎦ ⎣2001⎦ ⎣0⎦ ⎣0⎦⎠ 0000 0000 1 0 corresponds to the 2D digraph, with 4 vertices and a single source, of Fig. 2. A1- arcs and B1-arcs have been represented by means of thick lines, while A2-arcs and B2-arcs by means of thin lines. 36 M.E. Valcher - 4 3 @ @ @R - s 1 2
Fig. 2 2D influence digraph corresponding to (3)
(2) A path p in D (A1,A2,B1,B2) is a sequence of adjacent arcs and, in particular, an s j-path is a path which originates from the source s j.Apathp in a 2D digraph is specified by assigning its vertices and the type of arcs they are connected by. If we denote by |p|1 the number of A1-arcs and B1-arcs and by |p|2 the number of A2-arcs and B2-arcs occurring in p,then[|p|1 |p|2] is the composition of p and |p| = |p|1 + |p|2 its length. Once we have introduced these concepts, local reachability admits an interesting and useful characterization in terms of the 2D influence digraph associated with h −1 k h k −1 the system. Indeed, saying that (A1 i i A2)B1e j +(A1 i i A2)B2e j is an ith monomial vector just means that the set of s j-paths p of composition [|p|1 |p|2]= [hi ki] is not empty and each of them reaches the vertex i alone. If so, we will say that the vertex i is deterministically reached by all s j-paths of composition [hi ki]. As a consequence, the 2D system (2) is locally reachable if and only if for every i ∈n there exists j = j(i) such that the vertex i is deterministically reached by all s j-paths of some composition [hi ki].Moreover,
ILR = maxi min{hi + ki : ∃ j = j(i) s.t. all s j-paths hi,ki of composition [hi ki] deterministically reach i}.
Even though the graph-theoretic approach represents a noteworthy tool in the study of the local reachability index of a locally reachable 2D positive system, unfortu- nately, at present time, no upper bound on the maximum value of the local reach- ability index has been derived yet. Nonetheless, upper bounds for specific classes of systems have been derived in [18] and in [1, 2]. As a result, we can now at least claim [2] that the upper-bound on ILR is not smaller than n n I = + 1 n + 1 − . LR 2 2
3.2 Global Reachability of 2D Positive Systems
When addressing global reachability, it suffices, again, to focus on the reachability of those global states which consist of all zero (local) states except for one of them, Reachability Analysis for Different Classes of Positive Systems 37 which coincides with ei, i ∈n (“global monomial reachability” for the class of 2D positive systems). Moreover, the reachability of such global states can be, again, described in terms of conditions on certain columns of the reachability matrix.
Proposition 4. [18] Given a 2D positive system (2), the following facts are equiv- alent ones: • the system is globally reachable; • the system is “globally monomially reachable”;
•∃ ( × = ( ) ∈ , ∈ , n pairs hi,ki) ∈ + + and n indices j j i m i n s.t. hi−1 ki hi ki−1 ZP (A1 A2)B1e j +(A1 A2)B2e j = {i}, (4) h−1 k h k−1 ZP A1 A2)B1e j +(A1 A2)B2e j = /0, (5)
∀ (h,k) =(hi,ki) with h + k = hi + ki.
Proposition 4 can be interpreted in graph-theoretic terms: the 2D system (2) is globally reachable if and only if for every i ∈n there exists j = j(i) ∈m such that the vertex i is deterministically reached by all s j-paths of a given composition [hi ki], and no s j-path exists, having the same length hi + ki and different composition. = { + ∃ = ( ) Moreover, IGR maxi minhi,ki hi ki : j j i s.t. all s j-paths of composition [hi ki] deterministically reach i and there is no s j-path of length hi + ki and different composition}. In [18] a canonical form for globally reachable positive 2D systems has been obtained, even more it has been proved that the global reachability index of a 2D (globally reachable) system never exceeds the system dimension n, a result which makes global reachability easy to be tested, differently from local reachability.
4 Reachability of Discrete-Time Positive Switched Systems
A discrete-time positive switched system is described, at each time instant k ∈ +, by a first-order difference equation of the following type:
x(k + 1)=Aσ(k)x(k)+Bσ(k)u(k), (6) where x(k) and u(k) denote the n-dimensional state variable and the m-dimensional input variable, respectively, at the time instant k, while σ is a switching sequence, P = ∈ P defined on + and taking values in a finite set p . For each i , the pair ( n×n Ai,Bi) represents a discrete-time positive system (1), which means that Ai ∈ Ê+ n×m and Bi ∈ Ê+ . The definition of reachability for discrete-time positive switched systems may be given by suitably adjusting the definition given in [19, 30], in order to introduce the nonnegativity constraint on the state and input variables. 38 M.E. Valcher
n Definition 4. Given the positive switched system (6), a state x f ∈ Ê+ is said to σ [ , − ] → P be reachable if there exist k f ∈ Æ, a switching sequence : 0 k f 1 and [ , m ( )= an input sequence u : 0 k f − 1] → Ê+ that lead the state trajectory from x 0 0 to x(k)=x f . The positive switched system (6) is monomially reachable if every n monomial vector is reachable, and reachable if every state x f ∈ Ê+ is reachable.
We refer to the cardinality of the discrete time interval [0,k f − 1] as to the length |σ| of the switching sequence σ (in this case, |σ| = k f ). When reachability prop- erty is ensured, a natural goal one may want to pursue is that of determining the maximum number of steps required to reach every nonnegative state.
Definition 5. Given a reachable positive switched system (6), we define its reach- ability index as I := sup n min{k : x is reachable at time k}. R x∈Ê+
As we will see, reachable systems can be found endowed with an infinite IR.This fact represents a significant difference with respect to both standard switched sys- tems and positive systems. Another significant fact, which makes switched positive systems different from the previous ones we considered, is that monomial reacha- bility is no longer sufficient for reachability. This is rather intuitive as, indeed, non- negative combinations of switching sequences do not generally lead to admissible switching sequences (as they do not take values in P). To explore monomial reachability and reachability, it is first convenient to pro- ∈ vide the explicit expression of the state at any time instant k Æ, starting from the initial condition x(0), under the effect of the input sequence u(0),u(1),...,u(k−1), and of the switching sequence σ(0),σ(1),...,σ(k − 1). It turns out (see, for in- stance, [19]) that k−1 k−1 ( )= σ ( )+ σ ( )+···+ ( − ), x k A 1 Bσ(0)u 0 A 2 Bσ(1)u 1 Bσ(k−1)u k 1 (7) where we have resorted to the following shorthand notation: ··· , < k−1 Aσ(k−1)Aσ(k−2) Aσ(l) if l k; σ = A l : In, if l = k.
It is immediately seen that, when the input sequence u(·) is nonnegative, the state at the time instant k belongs to the polyhedral cone generated by the (columns of the) k−1 σ matrices A l Bσ(l−1),asl ranges from 1 to k, namely to the cone generated by the columns of the reachability matrix associated with the switching sequence σ of length k: k−1 k−1 k−1 R (σ) := σ ... σ σ . k Bσ(k−1) A k−1Bσ(k−2) A 2 Bσ(1) A 1 Bσ(0)
When dealing with standard discrete-time switched systems, it has been proved [19] that the system is reachable if and only if there exists a switching sequence σ (of (R n. length say k) such that Im k(σ)) = Ê For positive switched systems, instead, if n , a switching sequence σ of length k exists such that Cone(Rk(σ)) = Ê+ then the Reachability Analysis for Different Classes of Positive Systems 39 system is reachable, but the converse is not true [27]. Even the weaker condition that there exists a finite number of switching sequences of finite lengths, such that the union of the cones generated by the columns of their reachability matrices covers the positive orthant, is only sufficient for the system reachability. σ ,σ ,...,σ Proposition 5. [27] If there exist switching! sequences 1 2 (of lengths