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Strategies for convergence of region of attraction estimation algorithms
When estimating the region of attraction of nonlinear systems using sum-of-squares, convergence and volume determination in iterative approaches is a nontrivial problem. We seek to review different strategies that are proposed in the literature based on representative benchmark problems.
In recent years, applications of sum-of-squares methods for aerospace systems have grown in popularity. A common problem is the estimation of a nonlinear region of attraction based on LaSalle’s theorem and Lyapunov functions, which can be stated as
(1): max Vol(ℛ) s.t. d/dt V < 0 for all x ∈ ℛ
Volume determination of an inner estimate ℛ, as well as convergence towards the true region of attraction, are nontrivial problems. Several strategies have been proposed in the literature and demonstrated for exemplary systems. Yet, a real comparison of those strategies is difficult as examples vary from academic to practically relevant between selected work.
**Problem formulation**
A common approach to region of attraction estimation is the V-s-iteration which solves (1) by iterating over alternating, convex or quasi-convex optimization problems. Here, volume determination and convergence are realized by additional constraints, which depend on the strategies considered in the literature. We seek to review these methods based on representative benchmark problems.
**Tasks**
- Literature review of region of attraction estimation algorithms
- Implementation within the toolbox-in-development bisosprob
- Definition of benchmark problems to compare different strategies
- optionally: Proposal and evaluation of novel strategy
**Requirements**
Have, or be willing to acquire,
- good understanding of nonlinear system theory and Lyapunov methods
- some knowledge of convex and nonconvex optimization
- experience with programming, ideally with Matlab
**Literature**
- Genesio, R., Tartaglia, M., & Vicino, A. (1985). On the Estimation of Asymptotic Stability Regions: State of the Art and New Proposals. _IEEE Transactions on Automatic Control_, 30(8), 747–755. https://doi.org/10.1109/TAC.1985.1104057
- Chakraborty, A., Seiler, P., & Balas, G. J. (2011). Nonlinear region of attraction analysis for flight control verification and validation. _Control Engineering Practice_, 19(4), 335–345. https://doi.org/j.conengprac.2010.12.001
- Khodadadi, L., Samadi, B., & Khaloozadeh, H. (2014). Estimation of region of attraction for polynomial nonlinear systems: A numerical method. _ISA Transactions_, 53(1), 25–32. https://doi.org/10.1016/j.isatra.2013.08.005
In recent years, applications of sum-of-squares methods for aerospace systems have grown in popularity. A common problem is the estimation of a nonlinear region of attraction based on LaSalle’s theorem and Lyapunov functions, which can be stated as
(1): max Vol(ℛ) s.t. d/dt V < 0 for all x ∈ ℛ
Volume determination of an inner estimate ℛ, as well as convergence towards the true region of attraction, are nontrivial problems. Several strategies have been proposed in the literature and demonstrated for exemplary systems. Yet, a real comparison of those strategies is difficult as examples vary from academic to practically relevant between selected work.
**Problem formulation**
A common approach to region of attraction estimation is the V-s-iteration which solves (1) by iterating over alternating, convex or quasi-convex optimization problems. Here, volume determination and convergence are realized by additional constraints, which depend on the strategies considered in the literature. We seek to review these methods based on representative benchmark problems.
**Tasks**
- Literature review of region of attraction estimation algorithms
- Implementation within the toolbox-in-development bisosprob
- Definition of benchmark problems to compare different strategies
- optionally: Proposal and evaluation of novel strategy
**Requirements**
Have, or be willing to acquire,
- good understanding of nonlinear system theory and Lyapunov methods
- some knowledge of convex and nonconvex optimization
- experience with programming, ideally with Matlab
**Literature**
- Genesio, R., Tartaglia, M., & Vicino, A. (1985). On the Estimation of Asymptotic Stability Regions: State of the Art and New Proposals. _IEEE Transactions on Automatic Control_, 30(8), 747–755. https://doi.org/10.1109/TAC.1985.1104057
- Chakraborty, A., Seiler, P., & Balas, G. J. (2011). Nonlinear region of attraction analysis for flight control verification and validation. _Control Engineering Practice_, 19(4), 335–345. https://doi.org/j.conengprac.2010.12.001
- Khodadadi, L., Samadi, B., & Khaloozadeh, H. (2014). Estimation of region of attraction for polynomial nonlinear systems: A numerical method. _ISA Transactions_, 53(1), 25–32. https://doi.org/10.1016/j.isatra.2013.08.005
Review, implement, and compare different strategies for convergence of region of attraction estimation algorithms based on iterations of sum-of-squares problems based appropriate benchmark problems.
The thesis can be written in German or English. Publication of results is envisaged.
Review, implement, and compare different strategies for convergence of region of attraction estimation algorithms based on iterations of sum-of-squares problems based appropriate benchmark problems.
The thesis can be written in German or English. Publication of results is envisaged.
Dr. Torbjørn Cunis,
Institute of Flight Mechanics and Control at the
University of Stuttgart
Pfaffenwaldring 27,
70589 Stuttgart
Tel: 0711 685-66617
Dr. Torbjørn Cunis, Institute of Flight Mechanics and Control at the University of Stuttgart