Nonnegative and Compartmental Dynamical Systems

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