Analysis and Control of Polynomial Dynamic Models with Biological Applications - 1st Edition - ISBN: 9780128154953, 9780128154960

Analysis and Control of Polynomial Dynamic Models with Biological Applications

1st Edition

Authors: Gabor Szederkenyi Attila Magyar Katalin Hangos
eBook ISBN: 9780128154960
Paperback ISBN: 9780128154953
Imprint: Academic Press
Published Date: 21st March 2018
Page Count: 184
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Analysis and Control of Polynomial Dynamic Models with Biological Applications synthesizes three mathematical background areas (graphs, matrices and optimization) to solve problems in the biological sciences (in particular, dynamic analysis and controller design of QP and polynomial systems arising from predator-prey and biochemical models). The book puts a significant emphasis on applications, focusing on quasi-polynomial (QP, or generalized Lotka-Volterra) and kinetic systems (also called biochemical reaction networks or simply CRNs) since they are universal descriptors for smooth nonlinear systems and can represent all important dynamical phenomena that are present in biological (and also in general) dynamical systems.

Key Features

  • Describes and illustrates the relationship between the dynamical, algebraic and structural features of the quasi-polynomial (QP) and kinetic models
  • Shows the applicability of kinetic and QP representation in biological modeling and control through examples and case studies
  • Emphasizes the importance and applicability of quantitative models in understanding and influencing natural phenomena


The broad community of graduate students and researchers in science and engineering, interested in exploring nonlinear phenomena from a system’s theory perspective. Additionally, chemical/biochemical engineers or applied mathematicians developing research in biological phenomena as well as theoretical biologists with interests in mathematical methods

Table of Contents

1. Introduction
1. Dynamic models for describing biological phenomena
2. Kinetic systems
2.1. Chemical reaction networks with mass action law
2.2. Chemical reaction networks with rational functions as reaction rates
3. QP models
3.1. The original Lotka-Volterra equations
3.2. Generalized Lotka-Volterra equations
2. Basic Notions
1. General nonlinear system representation in the form of ODEs
1.1. Autonomous polynomial and quasi-polynomial systems
1.2. Positive polynomial systems
2. Formal introduction of the QP model form
2.1. The QP model form
2.2. LV systems
2.3. Extension with input term
3. Introduction of kinetic models with mass action and rational reaction rates
3.1. General notions for reaction networks
3.2. Reaction networks with mass action kinetics
3.3. Kinetic realizability and structural non-uniqueness of mass action type reaction networks
3.4. Reaction networks with rational function kinetics
3.5. Extension with input term
4. Basic relations between kinetic and QP models
4.1. Representing kinetic models with mass action reaction rates as QP models
4.2. LV models as kinetic systems
3. Model Transformations and Equivalence Classes
1. Affine and linear positive diagonal transformations
1.1. Affine transformations and their special cases for positive polynomial systems
1.2. Positive diagonal transformation of QP systems
1.3. Positive diagonal transformation of CRNs: linear conjugacy
2. Nonlinear diagonal transformations
2.1. X-factorable transformation
2.2. State-dependent time-rescaling
3. Quasi-monomial transformation and the corresponding equivalence classes of QP systems
3.1. The quasi-monomial transformation (QM-transformation)
3.2. The Lotka-Volterra (LV) form and the invariants
4. Embedding transformations and the relationship between classes of positive polynomial systems
4.1. Embedding smooth nonlinear models into QP form (QP-embedding)
4.2. Embedding rational functions into polynomial form (CRN-embedding)
4.3. The generality and relationship between classes of positive polynomial systems
4. Model analysis
1. Stability analysis of QP models
1.1. Local stability analysis of QP and LV models
1.2. Global stability analysis through the solution of linear matrix inequalities
1.3. Extension with time-reparametrization
2. Stability of kinetic systems
2.1. The deficiency zero and deficiency one theorems
3. Invariants (first integrals) for QP and kinetic systems
3.1. Linear first integrals of kinetic systems and their relations to conservation
3.2. Invariants of QP systems and their computation
4. Relations between the Lyapunov functions of QP and kinetic models
4.1. The physical-chemical origin of the natural Lyapunov function of kinetic models
4.2. Relationship with the logarithmic Lyapunov function of QP models
5. Computational analysis of the structure of kinetic systems
5.1. Computational model for determining linearly conjugate realizations of kinetic systems
5.2. Dense and sparse realizations and their properties
5.3. Computing linearly conjugate realizations with preferred properties
5.4. Computing all possible graph structures of a kinetic system
5.5. Computation of linearly conjugate bio-CRNs
5.6. Computation-oriented representation of uncertain kinetic models and their realizations
5. Stabilizing Feedback Control Design 931. Stabilizing control of QP systems by using optimization
1.1. LQ control of QP systems based on their locally linearized dynamics
1.2. Stabilizing control of QP systems by solving bilinear matrix inequalities
2. Stabilizing state feedback control of nonnegative polynomial systems using special CRN realizations of the closed loop system
2.1. The controller design problem
2.2. Feedback computation in the complex balanced closed loop case
2.3. Feedback computation in the weakly reversible closed loop with zero deficiency case
3. Robustness issues and robust design for the stabilizing control of polynomial systems
3.1. Handling the parametric uncertainty of stabilizing control of polynomial systems in the complex balanced closed loop case
6. Case studies
1. Optimization-based structural analysis and design of reaction networks
1.1. Computing all mathematically possible structures of a G1=S transition model in budding yeast
1.2. Analysis of a 5-node-repressilator with auto activation
1.3. Different realizations of an oscillating rational system
1.4. Linearly conjugate structures of the model
2. Computational distinguishability analysis of an uncertain kinetic model
2.1. Biological background
2.2. Kinetic representation
2.3. Analysis of the uncertain model
3. Stability analysis and stabilizing control of fermentation processes in QP form
3.1. Zero dynamics of the simple fermentation process
3.2. Partially actuated fermentation example in QP-form
3.3. Fully actuated fermentation example in QP-form
3.4. Feedback design for a simple fermentation process
3.5. Model parameters 139
A. Notations and abbreviations
A.1. Notations
B. Mathematical tools
B.1. Directed graphs
B.2. Matrices of key importance
B.3. Basics of the applied computational tools
B.3.1.Linear programming
B.3.2.Mixed integer linear programming and propositional logic
B.3.3.Linear and bilinear matrix inequalities
B.4. Basic notions from systems and control theory
B.4.1.Input-affine nonlinear state space models and their realizations
B.4.2.Stability analysis
B.4.3.Stabilizing feedback controllers
B.4.4.Input-output linearization via state feedback
B.4.5.Uncertainty description in dynamic models


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About the Author

Gabor Szederkenyi

Gábor Szederkényi received the M.Eng degree in computer engineering (University of Veszprém, 1998), his PhD in information sciences (University of Veszprém, 2002), and the DSc title in engineering sciences (Hungarian Academy of Sciences, 2013). Currently, he is a full professor at PPKE and the head of the Analysis and Control of Dynamical Systems research group. His main research interest is the computational analysis and control of nonlinear systems with special emphasis on reaction networks and kinetic models. He is the co-author of one book, several book chapters, more than 40 journal papers, and more than 60 conference papers on the theory and applicaton of the analysis and control of nonlinear systems. His education record includes BSc and MSc level courses on linear systems theory, nonlinear control and its application in robotics and in biological systems.

The history of the scientific cooperation of the proposed three authors dates back to 2003. Since then, they have published more than 25 joint scientific papers in international journals and conference proceedings mostly related to the topic of the proposed book. The scientific background of the three authors are really complementary: Prof. Katalin Hangos is an internationally known expert in the modelling and control of thermodynamical and (bio)chemical systems, Dr. Attila Magyar has significant experience in the analysis, application and control of quasi-polynomial systems, while Prof. Gábor Szederkényi has results on the optimization-based structural analysis and synthesis of kinetic systems. Moreover, all three authors have had continuous education and supervising experience both on the MSc and PhD levels at different universities.

Affiliations and Expertise

Professor, Faculty of Information Technology and Bionics, Pázmány Péter Catholic University (PPKE), Budapest, Hungary

Attila Magyar

Attila Magyar received his MSc in computer science (University of Pannonia, 2004), his PhD in computer science (University of Pannonia, 2008), respectively. He is working at the Department of Electrical Engineering and Information Systems at University of Pannonia He is the member of the Research Laboratory of Intelligent Control Systems of the Faculty of Information Technology. His main research interests lie in nonlinear control, system identification and robotics.

Affiliations and Expertise

Ass. Professor, Department of Electrical Engineering and Information Systems, University of Pannonia, Vesyprém, Hungary

Katalin Hangos

Katalin Hangos received her MSc in chemistry (ELTE TTK, 1976), her BSc in computer science (ELTE TTK, 1980), DSc (1993), and habilitations (chemical engineering, 1994, engineering informatics, 2000), respectively. She is the Research and Education Head of the Department of Electrical Engineering and Information Systems at University of Pannonia and the Head of the Process Control Research Group of the Computer and Automation Research Institute of Hung. Acad. Sci. She is also the Head of the Research Laboratory of Intelligent Control Systems of the Faculty of Information Technology. She is one of the very few female professors in process systems and control, who has a strong interdisciplinary background in systems and control theory and computer science, as well.

Her main research interest lies in dynamic modelling of process systems for control and diagnostic purposes. She is a co-author of more than 100 papers on various aspects of modelling and control of process systems with nonlinear, stochastic, Petri net, qualitative and graph theory based models.

Affiliations and Expertise

Res. Professor, Process Control Research Group, Institute for Computer Science and Control of Hungarian Academy of Sciences, Budapest, Hungary , and Professor, Department of Electrical Engineering and Information Systems, University of Pannonia, Veszprém, Hungary

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