Input-to-State Stability presents the dominating stability paradigm in nonlinear control theory that revolutionized our view on stabilization of nonlinear systems, design of robust nonlinear observers, and stability of nonlinear interconnected control systems.
The applications of input-to-state stability (ISS) are manifold and include mechatronics, aerospace engineering, and systems biology. Although the book concentrates on the ISS theory of finite-dimensional systems, it emphasizes the importance of a more general view of infinite-dimensional ISS theory. This permits the analysis of more general system classes and provides new perspectives on and a better understanding of the classical ISS theory for ordinary differential equations (ODEs). Features of the book include:
• a comprehensive overview of the theoretical basis of ISS;
• a description of the central applications of ISS in nonlinear control theory;
• a detailed discussion of the role of small-gain methods in the stability of nonlinear networks; and
• an in-depth comparison of ISS for finite- and infinite-dimensional systems.The book also provides a short overview of the ISS theory for other systems classes (partial differential equations, hybrid, impulsive, and time-delay systems) and surveys the available results for the important stability properties that are related to ISS.
The reader should have a basic knowledge of analysis, Lebesgue integration theory, linear algebra, and the theory of ODEs but requires no prior knowledge of dynamical systems or stability theory. The author introduces all the necessary ideas within the book.
Input-to-State Stability will interest researchers and graduate students studying nonlinear control from either a mathematical or engineering background. It is intended for active readers and contains numerous exercises of varying difficulty, which are integral to the text, complementing and widening the material developed in the monograph.
Author(s): Andrii Mironchenko
Series: Communications and Control Engineering
Publisher: Springer
Year: 2023
Language: English
Pages: 416
City: Cham
Preface
Contents
Abbreviations and Symbols
Abbreviations
Symbols
Sets and Numbers
Various Notation
Sequence Spaces
Function Spaces
Comparison Functions
1 Ordinary Differential Equations with Measurable Inputs
1.1 Ordinary Differential Equations with Inputs
1.2 Existence and Uniqueness Theory
1.3 Boundedness of Reachability Sets
1.4 Regularity of the Flow
1.5 Uniform Crossing Times
1.6 Lipschitz and Absolutely Continuous Functions
1.7 Spaces of Measurable and Integrable Functions
1.8 Concluding Remarks
1.9 Exercises
References
2 Input-to-State Stability
2.1 Basic Definitions and Results
2.2 ISS Lyapunov Functions
2.2.1 Direct ISS Lyapunov Theorems
2.2.2 Scalings of ISS Lyapunov Functions
2.2.3 Continuously Differentiable ISS Lyapunov Functions
2.2.4 Example: Robust Stabilization of a Nonlinear Oscillator
2.2.5 Lyapunov Criterion for ISS of Linear Systems
2.2.6 Example: Stability Analysis of Neural Networks
2.3 Local Input-to-State Stability
2.4 Asymptotic Properties for Control Systems
2.4.1 Stability
2.4.2 Asymptotic Gains
2.4.3 Limit Property
2.5 ISS Superposition Theorems
2.6 Converse ISS Lyapunov Theorem
2.7 ISS, Exponential ISS, and Nonlinear Changes of Coordinates
2.8 Integral Characterization of ISS
2.9 Semiglobal Input-to-State Stability
2.10 Input-to-State Stable Monotone Control Systems
2.11 Input-to-State Stability, Dissipativity, and Passivity
2.12 ISS and Regularity of the Right-Hand Side
2.13 Concluding Remarks
2.14 Exercises
References
3 Networks of Input-to-State Stable Systems
3.1 Interconnections and Gain Operators
3.2 Small-Gain Theorem for Input-to-State Stability of Networks
3.2.1 Small-Gain Theorem for Uniform Global Stability of Networks
3.2.2 Small-Gain Theorem for Asymptotic Gain Property
3.2.3 Semimaximum Formulation of the ISS Small-Gain Theorem
3.2.4 Small-Gain Theorem in the Maximum Formulation
3.2.5 Interconnections of Two Systems
3.3 Cascade Interconnections
3.4 Example: Global Stabilization of a Rigid Body
3.5 Lyapunov-Based Small-Gain Theorems
3.5.1 Small-Gain Theorem for Homogeneous and Subadditive Gain Operators
3.5.2 Examples on the Max-Formulation of the Small-Gain Theorem
3.5.3 Interconnections of Linear Systems
3.6 Tightness of Small-Gain Conditions
3.7 Concluding Remarks
3.8 Exercises
References
4 Integral Input-to-State Stability
4.1 Basic Properties of Integrally ISS Systems
4.2 iISS Lyapunov Functions
4.3 Characterization of 0-GAS Property
4.4 Lyapunov-Based Characterizations of iISS Property
4.5 Example: A Robotic Manipulator
4.6 Integral ISS Superposition Theorems
4.7 Integral ISS Versus ISS
4.8 Strong Integral Input-to-State Stability
4.9 Cascade Interconnections Revisited
4.10 Relationships Between ISS-Like Notions
4.11 Bilinear Systems
4.12 Small-Gain Theorems for Couplings of Strongly iISS Systems
4.13 Concluding Remarks
4.14 Exercises
References
5 Robust Nonlinear Control and Observation
5.1 Input-to-state Stabilization
5.2 ISS Feedback Redesign
5.3 ISS Control Lyapunov Functions
5.4 ISS Backstepping
5.5 Global Stabilization of Axial Compressor Model. Gain Assignment Technique
5.6 Event-based Control
5.7 Outputs and Output Feedback
5.8 Robust Nonlinear Observers
5.9 Observers and Dynamic Feedback for Linear Systems
5.10 Observers for Nonlinear Systems
5.11 Concluding Remarks and Extensions
5.11.1 Concluding Remarks
5.11.2 Obstructions on the Way to Stabilization
5.11.3 Further Control Techniques & ISS
5.12 Exercises
References
6 Input-to-State Stability of Infinite Networks
6.1 General Control Systems
6.2 Infinite Networks of ODE Systems
6.3 Input-to-State Stability
6.4 ISS Superposition Theorems
6.5 ISS Lyapunov Functions
6.6 Small-Gain Theorem for Infinite Networks
6.6.1 The Gain Operator and Its Properties
6.6.2 Positive Operators and Their Spectra
6.6.3 Spectral Radius of the Gain Operator
6.6.4 Small-Gain Theorem for Infinite Networks
6.6.5 Necessity of the Required Assumptions and Tightness of the Small-Gain Result
6.7 Examples
6.7.1 A Linear Spatially Invariant System
6.7.2 A Nonlinear Multidimensional Spatially Invariant System
6.7.3 A Road Traffic Model
6.8 Concluding Remarks
6.9 Exercises
References
7 Conclusion and Outlook
7.1 Brief Overview of Infinite-Dimensional ISS Theory
7.1.1 Fundamental Properties of ISS Systems: General Systems
7.1.2 ISS of Linear and Bilinear Boundary Control Systems
7.1.3 Lyapunov Methods for ISS Analysis of PDE Systems
7.1.4 Small-Gain Theorems for the Stability of Networks
7.1.5 Infinite Networks
7.1.6 ISS of Time-Delay Systems
7.1.7 Applications
7.2 Input-to-State Stability of Other Classes of Systems
7.2.1 Time-Varying Systems
7.2.2 Discrete-Time Systems
7.2.3 Impulsive Systems
7.2.4 Hybrid Systems
7.3 ISS-Like Stability Notions
7.3.1 Input-to-Output Stability (IOS)
7.3.2 Incremental Input-to-State Stability (Incremental ISS)
7.3.3 Finite-Time Input-to-State Stability (FTISS)
7.3.4 Input-to-State Dynamical Stability (ISDS)
7.3.5 Input-to-State Practical Stability (ISpS)
References
Appendix A Comparison Functions and Comparison Principles
A.1 Comparison Functions
A.1.1 Elementary Properties of Comparison Functions
A.1.2 Sontag's mathcalKL-Lemma
A.1.3 Positive Definite Functions
A.1.4 Approximations, Upper and Lower Bounds
A.2 Marginal Functions
A.3 Dini Derivatives
A.4 Comparison Principles
A.5 Brouwer's Theorem and Its Corollaries
A.6 Concluding Remarks
A.7 Exercises
References
Appendix B Stability
B.1 Forward Completeness and Stability
B.2 Attractivity and Asymptotic Stability
B.3 Uniform Attractivity and Uniform Asymptotic Stability
B.4 Lyapunov Functions
B.5 Converse Lyapunov Theorem
B.6 Systems with Compact Sets of Input Values
B.7 UGAS of Systems Without Inputs
B.8 Concluding Remarks
B.9 Exercises
References
Appendix C Nonlinear Monotone Discrete-Time Systems
C.1 Basic Notions
C.2 Linear Monotone Systems
C.3 Small-Gain Conditions
C.4 Sets of Decay
C.5 Asymptotic Stability of Induced Systems
C.6 Paths of Strict Decay
C.7 Gain Operators
C.8 Max-Preserving Operators
C.9 Homogeneous Subadditive Operators
C.9.1 Computational Issues
C.10 Summary of Properties of Nonlinear Monotone Operators and Induced Systems
C.11 Concluding Remarks
C.12 Exercises
References
Index
