Control of Piecewise Smooth Systems: Generalized Absolute Stability and Applications to Supercavitating Vehicles

Many systems in engineering applications are modeled as piecewise smooth systems. The piecewise smoothness presents great challenges for stability analysis and control synthesis for these systems. Over the years, the theory of absolute stability has been one of the few tools developed by control theory researchers to meet these challenges. For systems in which the nonlinearity is known to be bounded within certain sectors, many stability and control problems can be addressed using results from absolute stability theory. During the last few decades, many important advances have been made in the study of the absolute stability. In these studies, it is commonly assumed that the sector bound for the system nonlinearity is \textit{symmetric} with respect to the origin in state space. However, in many practical engineering systems, the nonlinearity does not satisfy such a symmetry assumption. To study stability and control problems for these systems, in this work the author studies generali…

Author: Lin, Guojian

Source: University of Maryland

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1 Introduction
1.1 Piecewise Smooth Systems
1.2 Literature on Absolute Stability
1.3 Generalization of Classical Absolute Stability
1.4 Organization of Dissertation
2 Preliminary Material
2.1 Absolute Stability
2.1.1 Definition
2.1.2 Frequency Domain Results
2.1.3 Time Domain Results
2.2 Linear Programming
2.3 Backstepping Control
3 Classical Absolute Stability for Second-Order Systems
3.1 Orginal Theory and Refinement: Leonov’s Result
3.2 Original Theory and Refinement: Margaliot and Langholz’s Result
3.3 Equivalence of the Two Results
3.4 Example
4 Generalized Absolute Stability for Second-Order Systems
4.1 Motivation for the Generalization
4.2 Problem Statement
4.3 Duality
4.4 Solution to Problem 4.1: Generalization of Sector Bounds
4.5 Solution to Problem 4.2: Further Generalization of Sector Bounds
5 Generalized Absolute Stability for Finite-Order Systems
5.1 Problem Statement
5.2 Existence of Piecewise Linear Lyapunov Functions
5.3 Construction of Piecewise Linear Lyapunov Functions (PWLLF)
5.4 Computational Issues
6 Supercavitating Vehicles
6.1 Supercavitation
6.2 Dive-Plane Model
6.3 System Dynamics
6.3.1 Time-Domain Simulations
6.3.2 Equilibrium Point Analysis
6.3.3 Limit Cycle Prediction
6.4 Challenges in Stabilization and Control
iv6.5 Control System Design
6.5.1 Linear Feedback Control
6.5.2 Switching Control
6.6 Actuator Saturation
6.7 Bifurcations in Supercavitating Vehicles
6.7.1 Range of the Cavitation Number
6.7.2 Bifurcation Results
6.7.3 Bifurcation Control
7 Application of Generalized Absolute Stability Results
7.1 Application of The Results on Second-Order Systems
7.2 Application of The Results on General Finite-Order Systems
7.2.1 A Second-Order System
7.2.2 A Third-Order System
7.2.3 The Fourth-order Supercavitating Vehicle System
8 Conclusions and Suggestions for Future Work
8.1 Conclusions
8.2 Suggested Future Work
A Appendix
A.1 State Transformation for The Study of Absolute Stability

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