A polynomial approach for analysis and optimal control of switched nonlinear systems

Acknowledgements

Résumé

Summary

List of Figures

1. Introduction
1.1. Introductory remarks and motivation  
1.2. Contributions, literature review, and outline  
1.2.1. For stability analysis  
1.2.2. For the optimal control problem  
1.2.3. For the piecewise linear model and control of a bioreactor  

2. A Polynomial approach for stability analysis of switched systems
2.1. Definitions and preliminaries  
2.1.1. Basic concepts  
2.1.2. Stability analysis under arbitrary switching and dissipativity  
2.2. An Equivalent polynomial representation
2.3. Results in stability analysis for polynomial constrained dynamical systems  
2.3.1. The sum of squares decomposition  
2.3.2. Numerical example of a polynomial switched system  
2.4. A generalization for nonlinear switched systems
2.4.1. The recasting process for stability analysis  
2.4.2. Example of a non-polynomial switched system

3. On optimal control of switched systems using a polynomial approach
3.1. Definitions and preliminaries  
3.1.1. Switched systems and its optimal control problem  
3.1.2. Maximum principle and necessary conditions
3.1.3. Relaxation and young measures
3.2. An equivalent polynomial optimal control problem
3.2.1. Equivalent representations  
3.2.2. Equivalent optimal control problem  
3.3. Relaxation of the equivalent optimal polynomial problem  
3.3.1. SDP relaxation of the optimal control problem
3.3.2. Switched optimization algorithm
3.3.3. Numerical example: the art stein circle  
3.4. Extension results to more general nonlinear optimal control problems  
3.4.1. The recasting process  
3.4.2. SDP relaxation  
3.4.3. Numerical example: swinging up a pendulum

4. Piecewise-linear approach to nonlinear cellular growth control
4.1. Process description  
4.1.1. Nonlinear model
4.2. The biological CPWL model  
4.2.1. Orthonormal canonical piecewise linear functions
4.2.2. Analysis of CPWL approximation: error estimation
4.2.3. Cellular growth CPWL model  
4.3. Simulation Results  
4.3.1. Model simulation results: nonlinear vs CPWL
4.3.2. Transient analysis of cells concentration
4.4. Probing feed controller  
4.4.1. Feedback algorithm  

5. Conclusions and future work
5.1. Summary of contributions  
5.2. Future research directions  

A. Mathematical background
A.1. Brief introduction to measure theory and integration  
A.1.1. Systems of sets
A.1.2. Measures
A.1.3. Integration  
A.2. Some results on probability theory
A.2.1. Some facts about young measures
A.2.2. the problern of moments
A.3. Basics of convex optirnization
A.3.1. Convex sets
A.3.2. Convex functions
A.3.3. Convex optimization
A.4. Optimization over polynomials using the method of moments  
A.4.1. Convergent semi-definite relaxations

References