Nonnegative and Compartmental Dynamical Systems

Wassim M. Haddad is professor in the School of Aerospace Engineering at Georgia Institute of Technology. VijaySekhar Chellaboina holds a PhD in aerospace engineering from Georgia Tech and works for Tata Consultancy Services in Hyderabad, India. Haddad and Chellaboina have coauthored six books, including "Nonlinear Dynamical Systems and Control" (Princeton). Qing Hui is assistant professor in the Department of Mechanical Engineering at Texas Tech University.

• Preface xv CHAPTER 1. Introduction 1 CHAPTER 2. Stability Theory for Nonnegative Dynamical Systems 7 2.1 Introduction 7 2.2 Lyapunov Stability Theory for Nonnegative Dynamical Systems 7 2.3 Invariant Set Stability Theorems 16 2.4 Semistability of Nonnegative Dynamical Systems 21 2.5 Stability Theory for Linear Nonnegative Dynamical Systems 30 2.6 Nonlinear Compartmental Dynamical Systems 43 2.7 Compartmental Systems in Biology, Ecology, Epidemiology, and Pharmacology 49 2.8 Discrete-Time Lyapunov Stability Theory for Nonnegative Dynamical Systems 61 2.9 Discrete-Time Invariant Set Theorems and Semistability Theorems 64 2.10 Stability Theory for Discrete-Time Linear Nonnegative Dynamical Systems 69 2.11 Discrete-Time Nonlinear Compartmental Dynamical Systems 83 CHAPTER 3. Stability Theory for Nonnegative and Compartmental Dynamical Systems with Time Delay 89 3.1 Introduction 89 3.2 Lyapunov Stability Theory for Time-Delay Nonnegative Dynamical Systems 90 3.3 Invariant Set Stability Theorems 93 3.4 Stability Theory for Continuous-Time Nonnegative Dynamical Systems with Time Delay 97 3.5 Discrete-Time Lyapunov Stability Theory for Time-Delay Nonnegative Dynamical Systems 103 3.6 Stability Theory for Discrete-Time Nonnegative Dynamical Systems with Time Delay 106 CHAPTER 4. Nonoscillation and Monotonicity of Solutions of Nonnegative Dynamical Systems 111 4.1 Introduction 111 4.2 Nonoscillation and Monotonicity of Linear Nonnegative Dynamical Systems 112 4.3 Mammillary Systems 119 4.4 Monotonicity of Nonlinear Nonnegative Dynamical Systems 123 4.5 Monotonicity of Discrete-Time Linear Nonnegative Dynamical Systems 127 4.6 Monotonicity of Discrete-Time Nonlinear Nonnegative Dynamical Systems 132 4.7 Monotonicity of Nonnegative Dynamical Systems with Time Delay 135 CHAPTER 5. Dissipativity Theory for Nonnegative Dynamical Systems 143 5.1 Introduction 143 5.2 Dissipativity Theory for Nonnegative Dynamical Systems 145 5.3 Feedback Interconnections of Nonnegative Dynamical Systems 153 5.4 Dissipativity Theory for Nonlinear Nonnegative Dynamical Systems 158 5.5 Feedback Interconnections of Nonnegative Nonlinear Dynamical Systems 164 5.6 Dissipativity Theory for Discrete-Time Nonnegative Dynamical Systems 166 5.7 Specialization to Discrete-Time Linear Nonnegative Dynamical Systems 173 5.8 Feedback Interconnections of Discrete-Time Nonnegative Dynamical Systems 177 5.9 Dissipativity Theory for Nonnegative Dynamical Systems with Time Delay 183 5.10 Feedback Interconnections of Nonnegative Dynamical Systems with Time Delay 188 5.11 Dissipativity Theory for Discrete-Time Nonnegative Dynamical Systems with Time Delay 191 5.12 Feedback Interconnections of Discrete-Time Nonnegative Dynamical Systems with Time Delay 194 CHAPTER 6. Hybrid Nonnegative and Compartmental Dynamical Systems 197 6.1 Introduction 197 6.2 Stability Theory for Nonlinear Hybrid Nonnegative Dynamical Systems 199 6.3 Hybrid Compartmental Dynamical Systems 203 6.4 Dissipativity Theory for Hybrid Nonnegative Dynamical Systems 207 6.5 Specialization to Linear Impulsive Dynamical Systems 215 6.6 Feedback Interconnections of Nonlinear Hybrid Nonnegative Dynamical Systems 217 CHAPTER 7. System Thermodynamics, Irreversibility, and Time Asymmetry 223 7.1 Introduction 223 7.2 Dynamical System Model 226 7.3 Reversibility, Irreversibility, Recoverability, and Irrecoverability 228 7.4 Reversible Dynamical Systems, Volume-Preserving Flows, and Poincar'e Recurrence 233 7.5 System Thermodynamics 243 7.6 Entropy and Irreversibility 247 7.7 Semistability and the Entropic Arrow of Time 254 7.8 Monotonicity of System Energies in Thermodynamic Processes 259 CHAPTER 8. Finite-Time Thermodynamics 263 8.1 Introduction 263 8.2 Finite-Time Semistability of Nonlinear Nonnegative Dynamical Systems 264 8.3 Homogeneity and Finite-Time Semistability 268 8.4 Finite-Time Energy Equipartition in Thermodynamic Systems 275 CHAPTER 9. Modeling and Analysis of Mass-Action Kinetics 281 9.1 Introduction 281 9.2 Reaction Networks 282 9.3 The Law of Mass Action and the Kinetic Equations 284 9.4 Nonnegativity of Solutions 288 9.5 Realization of Mass-Action Kinetics 290 9.6 Reducibility of the Kinetic Equations 293 9.7 Stability Analysis of Linear and Nonlinear Kinetics 297 9.8 The Zero-Deficiency Theorem 301 CHAPTER 10. Semistability and State Equipartition of Nonnegative Dynamical Systems 315 10.1 Introduction 315 10.2 Semistability and State Equipartitioning 315 10.3 Semistability and Equipartition of Linear Compartmental Systems with Time Delay 324 10.4 Semistability and Equipartition of Nonlinear Compartmental Systems with Time Delay 328 CHAPTER 11. Robustness of Nonnegative Dynamical Systems 343 11.1 Introduction 343 11.2 Nominal System Model 343 11.3 Semistability and Homogeneous Dynamical Systems 347 11.4 Uncertain System Model 348 CHAPTER 12. Modeling and Control for Clinical Pharmacology 359 12.1 Introduction 359 12.2 Pharmacokinetic Models 360 12.3 State Space Models 361 12.4 Drug Action, Effect, and Interaction 362 12.5 Pharmacokinetic Parameter Estimation 363 12.6 Pharmacodynamic Models 365 12.7 Open-Loop Drug Dosing 366 12.8 Closed-Loop Drug Dosing 368 12.9 Closed-Loop Control of Cardiovascular Function 369 12.10 Closed-Loop Control of Anesthesia 371 12.11 Electroencephalograph-Based Control 372 12.12 Bispectral Index-Based Control 373 12.13 Pharmacokinetic and Pharmacodynamic Models for Drug Distribution 374 12.14 Challenges and Opportunities in Pharmacological Control 377 CHAPTER 13. Optimal Fixed-Structure Control for Nonnegative Systems 379 13.1 Introduction 379 13.2 Optimal Zero Set-Point Regulation for Nonnegative Dynamical Systems 379 13.3 Optimal Nonzero Set-Point Regulation for Nonnegative Dynamical Systems 383 13.4 Suboptimal Control for Nonnegative Dynamical Systems 390 13.5 Optimal Fixed-Structure Control for Nonnegative Dynamical Systems 392 13.6 Nonnegative Control for Nonnegative Dynamical Systems 394 13.7 Optimal Fixed-Structure Control for General Anesthesia 396 CHAPTER 14. H2 Suboptimal Control for Nonnegative Dynamical Systems Using Linear Matrix Inequalities 405 14.1 Introduction 405 14.2 H2 Suboptimal Control for Nonnegative Dynamical Systems 405 14.3 Suboptimal Estimation for Nonnegative Dynamical Systems 413 14.4 H2 Suboptimal Dynamic Controller Design for Nonnegative Dynamical Systems 418 CHAPTER 15. Adaptive Control for Nonnegative Systems 425 15.1 Introduction 425 15.2 Adaptive Control for Linear Nonnegative Uncertain Dynamical Systems 427 15.3 Adaptive Control for Linear Nonnegative Dynamical Systems with Nonnegative Control 435 15.4 Adaptive Control for General Anesthesia: Linear Model 438 15.5 Adaptive Control for Nonlinear Nonnegative Uncertain Dynamical Systems 444 15.6 Adaptive Control for General Anesthesia: Nonlinear Model 451 15.7 Adaptive Control for Nonlinear Nonnegative Uncertain Dynamical Systems 454 15.8 Adaptive Control for Linear Nonnegative Uncertain Dynamical Systems with Time Delay 463 15.9 Adaptive Control for Linear Nonnegative Dynamical Systems with Nonnegative Control and Time Delay 476 15.10 Adaptive Control for Nonnegative Systems with Time Delay and Actuator Amplitude Constraints 480 15.11 Adaptive Control for General Anesthesia: Linear Model with Time Delay and Actuator Constraints 484 CHAPTER 16. Adaptive Disturbance Rejection Control for Compartmental Systems 491 16.1 Introduction 491 16.2 Compartmental Systems with Exogenous Disturbances 492 16.3 Adaptive Control for Linear Compartmental Uncertain Systems with Exogenous Disturbances 493 16.4 Adaptive Control for Linear Compartmental Dynamical Systems with L2 Disturbances 502 16.5 Adaptive Control for Automated Anesthesia with Hemorrhage and Hemodilution Effects 512 CHAPTER 17. Limit Cycle Stability Analysis and Control for Respiratory Compartmental Models 523 17.1 Introduction 523 17.2 Ultrametric Matrices, Periodic Orbits, and Poincar'e Maps 524 17.3 Compartmental Modeling of Lung Dynamics: Dichotomy Architecture 528 17.4 State Space Multicompartment Lung Model 531 17.5 Limit Cycle Analysis of the Multicompartment Lung Model 534 17.6 A Regular Dichotomy Model 538 17.7 A General Tree Structure Model 541 17.8 Direct Adaptive Control for Switched Linear Time-Varying Systems 545 17.9 Adaptive Control for a Multicompartment Lung Model 548 CHAPTER 18. Identification of Stable Nonnegative and Compartmental Systems 553 18.1 Introduction 553 18.2 State Reconstruction 554 18.3 Constrained Optimization for Subspace Identification of Stable Nonnegative Systems 557 18.4 Constrained Optimization for Subspace Identification of Compartmental Systems 566 18.5 Illustrative Numerical Examples 567 CHAPTER 19. Conclusion 571 Bibliography 573 Index 599

Wassim Haddad, Winner of the 2014 Pendray Aerospace Literature Award, American Institute of Aeronautics and Astronautics "[T]his monograph may be interesting for mathematicians and control engineers, on the one hand, and biologists, clinicians and economists, on the other, and it may stimulate cooperation among these communities."--Andrzej Swierniak, Mathematical Reviews