Stability and Boundary Stabilization of 1-D Hyperbolic Systems (Progress in Nonlinear Differential Equations and Their Applications Book 88)

This monograph explores the modeling of conservation and balance laws of one-dimensional hyperbolic systems using partial differential equations. It presents typical examples of hyperbolic systems for a wide range of physical engineering applications, allowing readers to understand the concepts in whichever setting is most familiar to them. With these examples, it also illustrates how control boundary conditions may be defined for the most commonly used control devices.The authors begin with the simple case of systems of two linear conservation laws and then consider the stability of systems under more general boundary conditions that may be differential, nonlinear, or switching. They then extend their discussion to the case of nonlinear conservation laws and demonstrate the use of Lyapunov functions in this type of analysis. Systems of balance laws are considered next, starting with the linear variety before they move on to more general cases of nonlinear ones. They go on to show how the problem of boundary stabilization of systems of two balance laws by both full-state and dynamic output feedback in observer-controller form is solved by using a “backstepping” method, in which the gains of the feedback laws are solutions of an associated system of linear hyperbolic PDEs. The final chapter presents a case study on the control of navigable rivers to emphasize the main technological features that may occur in real live applications of boundary feedback control.__Stability and Boundary Stabilization of 1-D Hyperbolic Systems__ will be of interest to graduate students and researchers in applied mathematics and control engineering. The wide range of applications it discusses will help it to have as broad an appeal within these groups as possible. Preface 6 Acknowledgements 10 Contents 12 1 Hyperbolic Systems of Balance Laws 16 1.1 Definitions and Notations 16 1.1.1 Riemann Coordinates and Characteristic Form 18 1.1.2 Steady State and Linearization 19 1.1.3 Riemann Coordinates Around the Steady State 19 1.1.4 Conservation Laws and Riemann Invariants 20 1.1.5 Stability, Boundary Stabilization, and the Associated Cauchy Problem 21 1.1.5.1 The Cauchy Problem in Riemann Coordinates 22 1.1.5.2 The Well-Posedness of the General Cauchy Problem for Strictly Hyperbolic Systems 23 1.2 Telegrapher Equations 25 1.3 Raman Amplifiers 27 1.4 Saint-Venant Equations for Open Channels 28 1.4.1 Boundary Conditions 30 1.4.2 Steady State and Linearization 31 1.4.3 The General Model 32 1.5 Saint-Venant-Exner Equations 33 1.6 Rigid Pipes and Heat Exchangers 34 1.6.1 The Shower Control Problem 36 1.6.2 The Water Hammer Problem 37 1.6.3 Heat Exchangers 38 1.7 Plug Flow Chemical Reactors 39 1.8 Euler Equations for Gas Pipes 41 1.8.1 Isentropic Equations 42 1.8.2 Steady State and Linearization 43 1.8.3 Musical Wind Instruments 44 1.9 Fluid Flow in Elastic Tubes 45 1.10 Aw-Rascle Equations for Road Traffic 46 1.10.1 Ramp Metering 48 1.11 Kac-Goldstein Equations for Chemotaxis 48 1.12 Age-Dependent SIR Epidemiologic Equations 50 1.12.1 Steady State 51 1.13 Chromatography 53 1.13.1 SMB Chromatography 54 1.14 Scalar Conservation Laws 58 1.15 Physical Networks of Hyperbolic Systems 60 1.15.1 Networks of Electrical Lines 61 1.15.2 Chains of Density-Velocity Systems 62 1.15.2.1 Gas Pipe Lines 62 1.15.2.2 Navigable Rivers and Irrigation Channels 63 1.15.3 Genetic Regulatory Networks 65 1.16 References and Further Reading 67 2 Systems of Two Linear Conservation Laws 70 2.1 Stability Conditions 70 2.1.1 Exponential Stability for the L∞-Norm 72 2.1.2 Exponential Lyapunov Stability for the L2-Norm 74 2.1.3 A Note on the Proofs of Stability in L2-Norm 79 2.1.4 Frequency Domain Stability 79 2.1.5 Example: Stability of a Lossless Electrical Line 80 2.2 Boundary Control of Density-Flow Systems 82 2.2.1 Feedback Stabilization with Two Local Controls 83 2.2.2 Dead-Beat Control 84 2.2.3 Feedback-Feedforward Stabilization with a Single Control 84 2.2.4 Proportional-Integral Control 85 2.2.4.1 Stability Analysis in the Frequency Domain 87 2.2.4.2 Lyapunov Stability Analysis 92 2.3 The Nonuniform Case 96 2.4 Conclusions 98 3 Systems of Linear Conservation Laws 99 3.1 Exponential Stability for the L2-Norm 100 3.1.1 Dissipative Boundary Conditions 102 3.2 Exponential Stability for the C0-Norm: Analysis in the Frequency Domain 103 3.2.1 A Simple Illustrative Example 106 3.2.2 Robust Stability 108 3.2.3 Comparison of the Two Stability Conditions 109 3.3 The Rate of Convergence 110 3.3.1 Application to a System of Two Conservation Laws 111 3.4 Differential Linear Boundary Conditions 111 3.4.1 Frequency Domain 112 3.4.2 Lyapunov Approach 112 3.4.3 Example: A Lossless Electrical Line Connecting an Inductive Power Supply to a Capacitive Load 113 3.4.4 Example: A Network of Density-Flow Systems Under PI Control 116 3.4.5 Example: Stability of Genetic Regulatory Networks 120 3.5 The Nonuniform Case 123 3.6 Switching Linear Conservation Laws 124 3.6.1 The Example of SMB Chromatography 125 3.6.1.1 A Simulation Experiment 128 3.7 References and Further Reading 129 4 Systems of Nonlinear Conservation Laws 131 4.1 Dissipative Boundary Conditions for the C1-Norm 133 4.2 Control of Networks of Scalar Conservation Laws 144 4.2.1 Example: Ramp-Metering Control in Road Traffic Networks 146 4.3 Interlude: Solutions Without Shocks 149 4.4 Dissipative Boundary Conditions for the H2-Norm 150 4.4.1 Proof of Theorem 4.11 152 4.5 Stability of General Systems of Nonlinear Conservation Laws in Quasi-Linear Form 157 4.5.1 Stability Condition for the C1-Norm 159 4.5.2 Stability Condition for the Cp-Normfor Any p N{0} 167 4.5.3 Stability Condition for the Hp-Norm for Any p N {0,1} 170 4.6 References and Further Reading 170 5 Systems of Linear Balance Laws 173 5.1 Lyapunov Exponential Stability 174 5.1.1 Example: Feedback Control of an Exothermic Plug Flow Reactor 177 5.2 Linear Systems with Uniform Coefficients 180 5.2.1 Application to a Linearized Saint-Venant-Exner Model 181 5.2.1.1 Steady State and Linearization 182 5.2.1.2 Riemann Coordinates 183 5.2.1.3 Lyapunov Stability 186 5.3 Existence of a Basic Quadratic Control Lyapunov Function for a System of Two Linear Balance Laws 190 5.3.1 Application to the Control of an Open Channel 195 5.4 Boundary Control of Density-Flow Systems 198 5.4.1 Transfer Functions 199 5.4.2 Boundary Feedback Stabilization with TwoLocal Controls 201 5.4.3 Feedback-Feedforward Stabilization witha Single Control 202 5.4.4 Stabilization with Proportional-Integral Control 204 5.5 Proportional-Integral Control in Navigable Rivers 207 5.5.1 Dissipative Boundary Condition 209 5.5.2 Control Error Propagation 209 5.6 Limit of Stabilizability 211 5.7 References and Further Reading 215 6 Quasi-Linear Hyperbolic Systems 216 6.1 Stability of Systems with Uniform Steady States 216 6.2 Stability of General Quasi-Linear Hyperbolic Systems 218 6.2.1 Stability Condition for the H2-Norm for Systems with Positive Characteristic Velocities 219 6.2.2 Stability Condition for the Hp-Norm for Any p N {0,1} 230 6.3 References and Further Reading 231 7 Backstepping Control 232 7.1 Motivation and Problem Statement 232 7.2 Full-State Feedback 233 7.3 Observer Design and Output Feedback 236 7.4 Backstepping Control of Systems of Two Balance Laws 239 7.5 References and Further Reading 240 8 Case Study: Control of Navigable Rivers 242 8.1 Geographic and Technical Data 242 8.2 Modeling and Simulation 243 8.3 Control Implementation 246 8.3.1 Local or Nonlocal Control? 247 8.3.2 Steady State and Set-Point Selection 248 8.3.3 Choice of the Time Step for Digital Control 249 8.4 Control Tuning and Performance 251 8.5 References and Further Reading 253 A Well-Posedness of the Cauchy Problem for Linear Hyperbolic Systems 255 B Well-Posedness of the Cauchy Problem for Quasi-Linear Hyperbolic Systems 267 C Properties and Comparisons of the Functions ρ, ρ2 and ρ∞ 272 C.1 Properties of the Function ρ2 272 C.2 Proof of Theorem 3.12 278 C.3 Proof of Proposition 4.7 290 D Proof of Lemma 4.12 (b) and (c) 292 E Proof of Theorem 5.11 296 F Notations 303 References 305 Index 315 This monograph explores the modeling of conservation and balance laws of one-dimensional hyperbolic systems using partial differential equations. It presents typical examples of hyperbolic systems for a wide range of physical engineering applications, allowing readers to understand the concepts in whichever setting is most familiar to them. With these examples, it also illustrates how control boundary conditions may be defined for the most commonly used control devices. The authors begin with the simple case of systems of two linear conservation laws and then consider the stability of systems under more general boundary conditions that may be differential, nonlinear, or switching. They then extend their discussion to the case of nonlinear conservation laws and demonstrate the use of Lyapunov functions in this type of analysis. Systems of balance laws are considered next, starting with the linear variety before they move on to more general cases of nonlinear ones. They go on to show how the problem of boundary stabilization of systems of two balance laws by both full-state and dynamic output feedback in observer-controller form is solved by using a zbacksteppingy method, in which the gains of the feedback laws are solutions of an associated system of linear hyperbolic PDEs. The final chapter presents a case study on the control of navigable rivers to emphasize the main technological features that may occur in practical applications of boundary feedback control. Stability and Boundary Stabilization of 1-D Hyperbolic Systems will be of interest to graduate students and researchers in applied mathematics and control engineering. The wide range of applications it discusses will help it to have as broad an appeal within these groups as possible.-- Provided by publisher This monograph explores the modeling of conservation and balance laws of one-dimensional hyperbolic systems using partial differential equations. It presents typical examples of hyperbolic systems for a wide range of physical engineering applications, allowing readers to understand the concepts in whichever setting is most familiar to them. With these examples, it also illustrates how control boundary conditions may be defined for the most commonly used control devices. The authors begin with the simple case of systems of two linear conservation laws and then consider the stability of systems under more general boundary conditions that may be differential, nonlinear, or switching. They then extend their discussion to the case of nonlinear conservation laws and demonstrate the use of Lyapunov functions in this type of analysis. Systems of balance laws are considered next, starting with the linear variety before they move on to more general cases of nonlinear ones. They go on to show how the problem of boundary stabilization of systems of two balance laws by both full-state and dynamic output feedback in observer-controller form is solved by using a "backstepping" method, in which the gains of the feedback laws are solutions of an associated system of linear hyperbolic PDEs. The final chapter presents a case study on the control of navigable rivers to emphasize the main technological features that may occur in real live applications of boundary feedback control. Stability and Boundary Stabilization of 1-D Hyperbolic Systems will be of interest to graduate students and researchers in applied mathematics and control engineering. The wide range of applications it discusses will help it to have as broad an appeal within these groups as possible. This monograph explores the modeling of conservation and balance laws of one-dimensional hyperbolic systems using partial differential equations. It presents typical examples of hyperbolic systems for a wide range of physical engineering applications, allowing readers to understand the concepts in whichever setting is most familiar to them. With these examples, it also illustrates how control boundary conditions may be defined for the most commonly used control devices. The authors begin with the simple case of systems of two linear conservation laws and then consider the stability of systems under more general boundary conditions that may be differential, nonlinear, or switching. They then extend their discussion to the case of nonlinear conservation laws and demonstrate the use of Lyapunov functions in this type of analysis. Systems of balance laws are considered next, starting with the linear variety before they move on to more general cases of nonlinear ones. They go on to show how the problem of boundary stabilization of systems of two balance laws by both full-state and dynamic output feedback in observer-controller form is solved by using a ĺlbacksteppingĺl method, in which the gains of the feedback laws are solutions of an associated system of linear hyperbolic PDEs. The final chapter presents a case study on the control of navigable rivers to emphasize the main technological features that may occur in practical applications of boundary feedback control. Stability and Boundary Stabilization of 1-D Hyperbolic Systems will be of interest to graduate students and researchers in applied mathematics and control engineering. The wide range of applications it discusses will help it to have as broad an appeal within these groups as possible Front Matter....Pages i-xiv Hyperbolic Systems of Balance Laws....Pages 1-54 Systems of Two Linear Conservation Laws....Pages 55-83 Systems of Linear Conservation Laws....Pages 85-116 Systems of Nonlinear Conservation Laws....Pages 117-158 Systems of Linear Balance Laws....Pages 159-201 Quasi-Linear Hyperbolic Systems....Pages 203-218 Backstepping Control....Pages 219-228 Case Study: Control of Navigable Rivers....Pages 229-241 Back Matter....Pages 243-307