Analysis and Control of Polynomial Dynamic Models with Biological Applications

✍ Scribed by Attila Magyar, Gábor Szederkényi, Katalin M. Hangos

• Publisher: Academic Press
• Year: 2018
• Tongue: English
• Leaves: 173
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Cover......Page 1
Analysis and Control of
Polynomial Dynamic Models
with Biological Applications
......Page 3
Copyright......Page 4
Dedication......Page 5
About the Authors......Page 6
Preface......Page 7
Acknowledgments......Page 9
Introduction......Page 10
Dynamic Models for Describing Biological Phenomena......Page 11
Kinetic Systems......Page 12
Chemical Reaction Networks With Mass Action Law......Page 13
Chemical Reaction Networks With Rational Functions as Reaction Rates......Page 14
Original Lotka-Volterra Equations......Page 15
Generalized Lotka-Volterra Equations......Page 16
Model Transformations and Equivalence Classes......Page 46
Affine Transformations and Their Special Cases for Positive Polynomial Systems......Page 47
Positive Diagonal Transformation of CRNs: Linear Conjugacy......Page 48
Linear Conjugacy of Networks With Mass Action Kinetics......Page 49
Linear Conjugacy of CRNs With Rational Reaction Rates......Page 50
X-Factorable Transformation......Page 53
Time-Rescaling Transformation of QP Models......Page 54
Quasimonomial Transformation and the Corresponding Equivalence Classes of QP Systems......Page 56
The Lotka-Volterra (LV) Form and the Invariants......Page 57
Embedding Smooth Nonlinear Models Into QP Form (QP Embedding)......Page 59
Embedding Rational Functions Into Polynomial Form (CRN embedding)......Page 61
Generality and Relationship Between Classes of Positive Polynomial Systems......Page 62
Model Analysis......Page 64
Local Stability Analysis of QP and LVModels......Page 65
Global Stability Analysis Through the Solution of Linear Matrix Inequalities......Page 66
The Time-Reparametrization Problem as a BMI......Page 67
Example......Page 70
Deficiency Zero and Deficiency OneTheorems......Page 71
Linear First Integrals of Kinetic Systems and Their Relations to Conservation......Page 74
Mass Conserving Chemical Reactions......Page 75
Invariants of QP Systems and Their Computation......Page 76
The Physical-Chemical Origin of the Natural Lyapunov Function of Kinetic Models......Page 77
Relationship With the Logarithmic Lyapunov Function of QP Models......Page 78
Computational Analysis of the Structure of Kinetic Systems......Page 79
Computational Model for Determining Linearly Conjugate Realizations of Kinetic Systems......Page 80
Dense Reaction Graphs......Page 81
Sparse Realizations......Page 83
Computing Linearly Conjugate Realizations With Preferred Properties......Page 84
Weakly Reversible Structures......Page 85
Deficiency Zero Realizations......Page 86
Further Structures Computable in an Optimization Framework......Page 88
Computing All Possible Graph Structures of a Kinetic System......Page 89
Computation of Linearly Conjugate Bio-CRNs......Page 94
Computation-Oriented Representation of Uncertain Kinetic Models and Their Realizations......Page 95
Realizations of an Uncertain Kinetic Model......Page 96
Dense and Sparse Realizations......Page 98
Stabilizing Feedback Control Design......Page 100
LQ Control of QP Systems Based on Their Locally Linearized Dynamics......Page 101
LQ Feedback Structure......Page 102
Suboptimal LQ With Diagonal Stability......Page 103
Stabilizing Control of QP Systems by Solving Bilinear Matrix Inequalities......Page 104
Numerical Solution of the Controller Design Problem......Page 105
Stabilizing State Feedback Control of Nonnegative Polynomial Systems Using Special CRN Realizations of the Closed-Loop S......Page 108
Underlying Realization Computation Problem......Page 109
Feedback Computation in the Complex BalancedClosed-Loop Case......Page 110
Feedback Computation in the Weakly Reversible Closed Loop With Zero Deficiency Case......Page 111
Handling the Parametric Uncertainty of Stabilizing Control of Polynomial Systems in the Complex Balanced Closed-Loop C......Page 113
Optimization-Based Structural Analysis and Design of Reaction Networks......Page 116
Computing All Mathematically Possible Structures of a G1/S Transition Model in Budding Yeast......Page 117
Analysis of a Five-Node-Repressilator WithAuto Activation......Page 120
Different Realizations of an Oscillating Rational System......Page 122
Dynamically Equivalent Representation......Page 123
Linearly Conjugate Structures of the Model......Page 125
Kinetic Representation......Page 128
10% Uncertainty for All Parameters......Page 130
20% Uncertainty for All Parameters......Page 131
Constrained Uncertain Model......Page 132
Analysis of the Number of Realizations for Different Degrees of Uncertainty......Page 133
Zero Dynamics of the Simple Fermentation Process......Page 135
Partially Actuated Fermentation Example in QP Form......Page 137
Fully Actuated Fermentation Example in QP Form......Page 139
Feedback Design for a Simple Fermentation Process......Page 141
Notations......Page 144
Directed Graphs......Page 148
Matrices of Key Importance......Page 150
Linear Programming......Page 153
Mixed Integer LP and Propositional Logic......Page 154
Basic Notions From Systems and Control Theory......Page 156
Transformation of States......Page 157
Lyapunov Function, Lyapunov Theorem......Page 158
Stabilizing Feedback Controllers......Page 159
Input-Output Linearization via State Feedback......Page 160
Polytopic Sets......Page 161
Bibliography......Page 163
E......Page 168
L......Page 169
O......Page 170
R......Page 171
Z......Page 172
Back Cover......Page 173