Positive 1D and 2D Systems
Published on 23. October 2012
Book
Paperback/Softback
XIII, 431 pages
978-1-4471-1097-2 (ISBN)
Description
In the last decade a dynamic development in positive systems has been observed. Roughly speaking, positive systems are systems whose inputs, state variables and outputs take only nonnegative values. Examples of positive systems are industrial processes involving chemical reactors, heat exchangers and distillation columns, storage systems, compartmental systems, water and atmospheric pollution models. A variety of models having positive linear system behaviour can be found in engineering, management science, economics, social sciences, biology and medicine, etc. The basic mathematical tools for analysis and synthesis of linear systems are linear spaces and the theory of linear operators. Positive linear systems are defined on cones and not on linear spaces. This is why the theory of positive systems is more complicated and less advanced. The theory of positive systems has some elements in common with theories of linear and non-linear systems. Schematically the relationship between the theories of linear, non-linear and positive systems is shown in the following figure Figure 1.
More details
Series
Edition
Softcover reprint of the original 1st ed. 2002
Language
English
Place of publication
London
United Kingdom
Target group
Professional and scholarly
Research
Illustrations
XIII, 431 p.
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 25 mm
Weight
674 gr
ISBN-13
978-1-4471-1097-2 (9781447110972)
DOI
10.1007/978-1-4471-0221-2
Schweitzer Classification
Other editions
Additional editions
Tadeusz Kaczorek
Positive 1D and 2D Systems
Book
09/2001
1st Edition
Springer
€96.00
Article exhausted; check different version
Person
Tadeusz Kaczorek, received the MSc., PhD and DSc degrees from Electrical Engineering of Warsaw University of Technology in 1956, 1962 and 1964, respectively. Between 1968 and 1969 he was the dean of Electrical Engineering Faculty and from 1970until 1973 he was the prorector of Warsaw University of Technology. He has been a full Professor since 1974. In 1996 he was elected as a full member of Polish Academy of Sciences. From 1988 to 1991 he was the director of the Research Centre of Polish Academy of Sciences in Rome. In June 1999 he was elected as a full member of the Academy of Engineering in Poland. In May 2004 he was elected the honorary member of the Hungarian Academy of Sciences. He holds honorary doctorates from the University of Zielona Góra (2002), the Technical University of Lublin (2004), the Technical University of Szczecin (2004) and Warsaw University of Technology (2004).
His research interests cover the theory of systems and the automatic control systems, particularly singular multidimensional systems, positive multidimensional systems and singular positive 1- and 2-dimensional systems. He has researched the field of singular and positive 2-dimensional linear systems. He has published 18 books (5 in English) and over 700 scientific papers (in journals like IEEE Transactions on Automatic Control, IEEE Transactions on Neural Networks, Multidimensional Systems and Signal Processing, International Journal of Control, Bull. Pol. Acad. Sciences, etc.) and proceedings of conferences. He has presented more than 100 invited papers to international conferences and world congresses and has given invited lectures in more than 50 universities in the US, Canada, UK, Germany, Italy, France, Japan, Greece etc. He has been a member of many international committees and programme committees.
Professor Kaczorek has supervised over 60 Ph.D. theses. More than 20 of this PhD students have become professors inthe US, UK and Japan. He is Editor-in-Chief of Bulletin of the Polish Academy of Sciences, Techn. Sciences and editorial member of about ten international journals.
Content
1. Positive matrices and graphs.- 1.1 Generalised permutation matrix, nonnegative matrix, positive and strictly positive matrices.- 1.2 Reducible and irreducible matrices.- 1.3 The Collatz - Wielandt function.- 1.4 Maximum eigenvalue of a nonnegative matrix.- 1.5 Bounds on the maximal eigenvalue and eigenvector of a positive matrix.- 1.6 Dominating positive matrices of complex matrices.- 1.7 Oscillatory and primitive matrices.- 1.8 The canonical Frobenius form of a cyclic matrix.- 1.9 Metzler matrix.- 1.10 M-matrices.- 1.11 Totally nonnegative (positive) matrices.- 1.12 Graphs of positive systems.- 1.13 Graphs of reducible, irreducible, cyclic and primitive systems.- Problems.- References.- 2. Continuous-ime and discrete-ime positive systems.- 2.1 Externally positive systems.- 2.2 Internally positive systemst.- 2.3 Compartmental systems.- 2.4 Stability of positive systems.- 2.5 Input-output stability.- 2.6 Weakly positive systems.- 2.7 Componentwise asymptotic stability and exponental stability of positive systems.- 2.8 Externally and internally positive singular systems.- 2.9 Composite positive linear systems.- 2.10 Eigenvalue assignment problem for positive linear systems.- Problems.- References.- 3. Reachability, controllability and observability of positive systems.- 3.1 discrete-time systems.- 3.2 continuous-time systems.- 3.3 Controllability of positive systems.- 3.4 Minimum energy control of positive systems.- 3.5 Reachability and controllability of weakly positive systems with state feedbacks.- 3.6 Observability of discrete-time positive systems.- 3.7 Reachability and controllability of weakly positive systems.- Problems.- References.- 4. Realisation problem of positive 1D systems.- 4.1 Basic notions and formulation of realisation problem.- 4.2 Existence andcomputation of positive realisations.- 4.3 Existence and computation of positive realisations of multi-input multi-output systems.- 4.4 Existence and computation of positive realisations of weakly positive multi-input multi-output systems.- 4.5 Positive realisations in canonical forms of singular linear.- Problems.- References.- 5. 2D models of positive linear systems.- 5.1 Internally positive Roesser model.- 5.2 Externally positive Roesser model.- 5.3 Internally positive general model.- 5.4 Externally positive general model.- 5.5 Positive Fornasini-Marchesini models and relationships between models.- 5.6 Positive models of continuous-discrete systems.- 5.7 Positive generalised Roesser model.- Problems.- References.- 6 Controllability and minimum energy control of positive 2D systems.- 6.1 Reachability, controllability and observability of positive Roesser model.- 6.2 Reachability, controllability and observability of the positive general model.- 6.3 Minimum energy control of positive 2D systems.- 6.4 Reachability and minimum energy control of positive 2D continuous-discrete systems.- Problems.- References.- 7. Realisation problem for positive 2D systems.- 7.1 Formulation of realisation problem for positive Roesser model.- 7.2 Existence of positive realisations.- 7.3 Positive realisations in canonical form of the Roesser model.- 7.4 Determination of the positive Roesser model by the use of state variables diagram.- 7.5 Determination of a positive 2D general model for a given transfer matrix.- 7.6 Positive realisation problem for singular 2D Roesser model.- 7.7 Concluding remarks and open problems.- Problems.- References.- Appendix A Oeterminantal Sylvester equality.- Appendix B Computation of fundamental matrices of linear systems.- Appendix C Solutions of 20 linear discrete models.- Appendix D Transformations of matrices to their canonical forms and lemmas for 1D singular systems.