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Classical converse theorems in Lyapunov's second method
Advances in computational Lyapunov analysis using sum-of-squares programming
1. | Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, United Kingdom, United Kingdom |
References:
[1] |
A. A. Ahmadi, M. Krstic and P. A. Parrilo, A globally asymptotically stable polynomial vector field with no polynomial Lyapunov function,, in Decision and Control and European Control Conference (CDC-ECC), (2011), 7579.
doi: 10.1109/CDC.2011.6161499. |
[2] |
J. Anderson, Dynamical System Decomposition and Analysis Using Convex Optimization,, PhD thesis, (2012). Google Scholar |
[3] |
G. Blekherman, P. A. Parrilo and R. R. Thomas, Semidefinite Optimization and Convex Algebraic Geometry,, SIAM, (2013).
|
[4] |
J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry,, Springer-Verlag, (1998).
doi: 10.1007/BFb0084605. |
[5] |
S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory,, Society for Industrial and Applied Mathematics, (1994).
doi: 10.1137/1.9781611970777. |
[6] |
S. Boyd, L. El Ghaoul, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory,, Vol. 15, (1987).
|
[7] |
S. Boyd and L. Vandenberghe, Convex Optimization,, Cambridge University Press, (2004).
doi: 10.1017/CBO9780511804441. |
[8] |
G. Chesi, Estimating the domain of attraction for uncertain polynomial systems,, Automatica, 40 (2004), 1981.
doi: 10.1016/j.automatica.2004.06.014. |
[9] |
G. Chesi, A. Garulli, A. Tesi and A. Vicino, Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems,, Springer, (2009).
doi: 10.1007/978-1-84882-781-3. |
[10] |
D. Cox, J. Little and D. O'Shea, Ideals, Varietis, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra,, Springer, (1997).
|
[11] |
P. A. Giesl and S. F. Hafstein, Revised CPA method to compute Lyapunov functions for nonlinear systems,, Journal of Mathematical Analysis and Applications, 410 (2014), 292.
doi: 10.1016/j.jmaa.2013.08.014. |
[12] |
M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 1.21., , (2011). Google Scholar |
[13] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Applied Mathematical Sciences, (1993).
doi: 10.1007/978-1-4612-4342-7. |
[14] |
E. J. Hancock and A. Papachristodoulou, Generalised absolute stability and sum of squares,, Automatica, 49 (2013), 960.
doi: 10.1016/j.automatica.2013.01.006. |
[15] |
D. Henrion and J. B. Lasserre, GloptiPoly: Global optimization over polynomials with Matlab and SeDuMi,, ACM Transactions on Mathematical Software (TOMS), 29 (2003), 165.
doi: 10.1145/779359.779363. |
[16] |
Y. Huang and A. Jadbabaie, Nonlinear H control: An enhanced quasi-LPV approach,, in Proceedings of the IFAC World Congress, (1999), 85. Google Scholar |
[17] |
A. Isidori and A. Astolfi, Disturbance attenuation and $H_{\infty}$-control via measurement feedback in nonlinear systems,, IEEE Transactions on Automatic Control, 37 (1992), 1283.
doi: 10.1109/9.159566. |
[18] |
H. K. Khalil, Nonlinear Systems,, Prentice-Hall, (2000). Google Scholar |
[19] |
V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations,, Kluwer Academic Publishers, (1999).
doi: 10.1007/978-94-017-1965-0. |
[20] |
J. Lasserre, D. Henrion, C. Prieur and E. Trelat, Nonlinear optimal control via occupation measures and LMI-relaxations,, SIAM Journal on Control and Optimization, 47 (2008), 1643.
doi: 10.1137/070685051. |
[21] |
J. Löfberg, Yalmip: A toolbox for modeling and optimization in MATLAB,, in Proceedings of the CACSD Conference, (2004). Google Scholar |
[22] |
W. M. Lu and J. C. Doyle, $H_{\infty}$ control of nonlinear systems: A convex characterization,, IEEE Transactions on Automatic Control, 40 (1995), 1668.
doi: 10.1109/9.412643. |
[23] |
A. Papachristodoulou, J. Anderson, G. Valmorbida, S. Prajna, P. Seiler and P. A. Parrilo, SOSTOOLS: Sum of Squares Optimization Toolbox for MATLAB,, , (2013). Google Scholar |
[24] |
A. Papachristodoulou, M. M. Peet and S. Lall, Analysis of polynomial systems with time delays via the sum of squares decomposition,, IEEE Transactions on Automatic Control, 54 (2009), 1058.
doi: 10.1109/TAC.2009.2017168. |
[25] |
A. Papachristodoulou and S. Prajna, Analysis of non-polynomial systems using the sum of squares decomposition,, in Positive Polynomials in Control, 312 (2005), 23.
|
[26] |
P. A. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization,, PhD thesis, (2000). Google Scholar |
[27] |
P. A. Parrilo, Semidefinite programming relaxations for semialgebraic problems,, Mathematical Programming, 96 (2003), 293.
doi: 10.1007/s10107-003-0387-5. |
[28] |
P. A. Parrilo and B. Sturmfels, Minimizing polynomials functions,, , (2001).
|
[29] |
M. M. Peet, Exponentially stable nonlinear systems have polynomial Lyapunov functions on bounded regions,, Automatic Control, 54 (2009), 979.
doi: 10.1109/TAC.2009.2017116. |
[30] |
M. M. Peet and A. Papachristodoulou, A converse sum of squares Lyapunov result with a degree bound,, IEEE Transactions on Automatic Control, 57 (2012), 2281.
doi: 10.1109/TAC.2012.2190163. |
[31] |
M. M. Peet, A. Papachristodoulou and S. Lall, Positive forms and stability of linear time-delay systems,, SIAM J. Control Optim., 47 (2008), 3237.
doi: 10.1137/070706999. |
[32] |
S. Prajna, A. Papachristodoulou and F. Wu, Nonlinear control synthesis by sum of squares optimization: A Lyapunov-based approach,, in 5th Asian Control Conference, (2004), 157. Google Scholar |
[33] |
S. Prajna, P. A. Parrilo and A. Rantzer, Nonlinear control synthesis by convex optimization,, IEEE Transactions on Automatic Control, 49 (2004), 310.
doi: 10.1109/TAC.2003.823000. |
[34] |
J. F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,, Optimization Methods and Software, 11/12 (1999), 625.
doi: 10.1080/10556789908805766. |
[35] |
W. Tan, Nonlinear Control Analysis and Synthesis using Sum-of-Squares Programming,, PhD thesis, (2006). Google Scholar |
[36] |
B. Tibken and Y. Fan, Computing the domain of attraction for polynomial systems via BMI optimization method,, in Proceedings of the American Control Conference, (2006), 117.
doi: 10.1109/ACC.2006.1655340. |
[37] |
M. J. Todd, Semidefinite optimization,, Acta Numerica 2001, 10 (2001), 515.
doi: 10.1017/S0962492901000071. |
[38] |
K. C. Toh, M. J. Todd and R. H. Tütüncü, SDPT3 - a Matlab software package for semidefinite programming, version 1.3,, Optimization Methods and Software, 11 (1999), 545.
doi: 10.1080/10556789908805762. |
[39] |
U. Topcu, A. Packard, P. Seiler and G. J. Balas, Robust region-of-attraction estimation,, IEEE Transactions on Automatic Control, 55 (2010), 137.
doi: 10.1109/TAC.2009.2033751. |
[40] |
G. Valmorbida and J. Anderson, Region of attraction analysis via invariant sets,, in Proc. of the American Control Conference, (2014), 3591.
doi: 10.1109/ACC.2014.6859263. |
[41] |
L. Vandenberghe and S. Boyd, Semidefinite programming,, SIAM Review, 38 (1996), 49.
doi: 10.1137/1038003. |
[42] |
Q. Zheng and F. Wu, Nonlinear output feedback $H_{\infty}$ control for polynomial nonlinear systems,, in Proceedings of the 2008 American Control Conference, (2008), 1196. Google Scholar |
[43] |
Q. Zheng and F. Wu, Generalized nonlinear $H_{\infty}$ synthesis condition with its numerically efficient solution,, International Journal of Robust and Nonlinear Control, 21 (2011), 2079.
doi: 10.1002/rnc.1682. |
show all references
References:
[1] |
A. A. Ahmadi, M. Krstic and P. A. Parrilo, A globally asymptotically stable polynomial vector field with no polynomial Lyapunov function,, in Decision and Control and European Control Conference (CDC-ECC), (2011), 7579.
doi: 10.1109/CDC.2011.6161499. |
[2] |
J. Anderson, Dynamical System Decomposition and Analysis Using Convex Optimization,, PhD thesis, (2012). Google Scholar |
[3] |
G. Blekherman, P. A. Parrilo and R. R. Thomas, Semidefinite Optimization and Convex Algebraic Geometry,, SIAM, (2013).
|
[4] |
J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry,, Springer-Verlag, (1998).
doi: 10.1007/BFb0084605. |
[5] |
S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory,, Society for Industrial and Applied Mathematics, (1994).
doi: 10.1137/1.9781611970777. |
[6] |
S. Boyd, L. El Ghaoul, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory,, Vol. 15, (1987).
|
[7] |
S. Boyd and L. Vandenberghe, Convex Optimization,, Cambridge University Press, (2004).
doi: 10.1017/CBO9780511804441. |
[8] |
G. Chesi, Estimating the domain of attraction for uncertain polynomial systems,, Automatica, 40 (2004), 1981.
doi: 10.1016/j.automatica.2004.06.014. |
[9] |
G. Chesi, A. Garulli, A. Tesi and A. Vicino, Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems,, Springer, (2009).
doi: 10.1007/978-1-84882-781-3. |
[10] |
D. Cox, J. Little and D. O'Shea, Ideals, Varietis, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra,, Springer, (1997).
|
[11] |
P. A. Giesl and S. F. Hafstein, Revised CPA method to compute Lyapunov functions for nonlinear systems,, Journal of Mathematical Analysis and Applications, 410 (2014), 292.
doi: 10.1016/j.jmaa.2013.08.014. |
[12] |
M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 1.21., , (2011). Google Scholar |
[13] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Applied Mathematical Sciences, (1993).
doi: 10.1007/978-1-4612-4342-7. |
[14] |
E. J. Hancock and A. Papachristodoulou, Generalised absolute stability and sum of squares,, Automatica, 49 (2013), 960.
doi: 10.1016/j.automatica.2013.01.006. |
[15] |
D. Henrion and J. B. Lasserre, GloptiPoly: Global optimization over polynomials with Matlab and SeDuMi,, ACM Transactions on Mathematical Software (TOMS), 29 (2003), 165.
doi: 10.1145/779359.779363. |
[16] |
Y. Huang and A. Jadbabaie, Nonlinear H control: An enhanced quasi-LPV approach,, in Proceedings of the IFAC World Congress, (1999), 85. Google Scholar |
[17] |
A. Isidori and A. Astolfi, Disturbance attenuation and $H_{\infty}$-control via measurement feedback in nonlinear systems,, IEEE Transactions on Automatic Control, 37 (1992), 1283.
doi: 10.1109/9.159566. |
[18] |
H. K. Khalil, Nonlinear Systems,, Prentice-Hall, (2000). Google Scholar |
[19] |
V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations,, Kluwer Academic Publishers, (1999).
doi: 10.1007/978-94-017-1965-0. |
[20] |
J. Lasserre, D. Henrion, C. Prieur and E. Trelat, Nonlinear optimal control via occupation measures and LMI-relaxations,, SIAM Journal on Control and Optimization, 47 (2008), 1643.
doi: 10.1137/070685051. |
[21] |
J. Löfberg, Yalmip: A toolbox for modeling and optimization in MATLAB,, in Proceedings of the CACSD Conference, (2004). Google Scholar |
[22] |
W. M. Lu and J. C. Doyle, $H_{\infty}$ control of nonlinear systems: A convex characterization,, IEEE Transactions on Automatic Control, 40 (1995), 1668.
doi: 10.1109/9.412643. |
[23] |
A. Papachristodoulou, J. Anderson, G. Valmorbida, S. Prajna, P. Seiler and P. A. Parrilo, SOSTOOLS: Sum of Squares Optimization Toolbox for MATLAB,, , (2013). Google Scholar |
[24] |
A. Papachristodoulou, M. M. Peet and S. Lall, Analysis of polynomial systems with time delays via the sum of squares decomposition,, IEEE Transactions on Automatic Control, 54 (2009), 1058.
doi: 10.1109/TAC.2009.2017168. |
[25] |
A. Papachristodoulou and S. Prajna, Analysis of non-polynomial systems using the sum of squares decomposition,, in Positive Polynomials in Control, 312 (2005), 23.
|
[26] |
P. A. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization,, PhD thesis, (2000). Google Scholar |
[27] |
P. A. Parrilo, Semidefinite programming relaxations for semialgebraic problems,, Mathematical Programming, 96 (2003), 293.
doi: 10.1007/s10107-003-0387-5. |
[28] |
P. A. Parrilo and B. Sturmfels, Minimizing polynomials functions,, , (2001).
|
[29] |
M. M. Peet, Exponentially stable nonlinear systems have polynomial Lyapunov functions on bounded regions,, Automatic Control, 54 (2009), 979.
doi: 10.1109/TAC.2009.2017116. |
[30] |
M. M. Peet and A. Papachristodoulou, A converse sum of squares Lyapunov result with a degree bound,, IEEE Transactions on Automatic Control, 57 (2012), 2281.
doi: 10.1109/TAC.2012.2190163. |
[31] |
M. M. Peet, A. Papachristodoulou and S. Lall, Positive forms and stability of linear time-delay systems,, SIAM J. Control Optim., 47 (2008), 3237.
doi: 10.1137/070706999. |
[32] |
S. Prajna, A. Papachristodoulou and F. Wu, Nonlinear control synthesis by sum of squares optimization: A Lyapunov-based approach,, in 5th Asian Control Conference, (2004), 157. Google Scholar |
[33] |
S. Prajna, P. A. Parrilo and A. Rantzer, Nonlinear control synthesis by convex optimization,, IEEE Transactions on Automatic Control, 49 (2004), 310.
doi: 10.1109/TAC.2003.823000. |
[34] |
J. F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,, Optimization Methods and Software, 11/12 (1999), 625.
doi: 10.1080/10556789908805766. |
[35] |
W. Tan, Nonlinear Control Analysis and Synthesis using Sum-of-Squares Programming,, PhD thesis, (2006). Google Scholar |
[36] |
B. Tibken and Y. Fan, Computing the domain of attraction for polynomial systems via BMI optimization method,, in Proceedings of the American Control Conference, (2006), 117.
doi: 10.1109/ACC.2006.1655340. |
[37] |
M. J. Todd, Semidefinite optimization,, Acta Numerica 2001, 10 (2001), 515.
doi: 10.1017/S0962492901000071. |
[38] |
K. C. Toh, M. J. Todd and R. H. Tütüncü, SDPT3 - a Matlab software package for semidefinite programming, version 1.3,, Optimization Methods and Software, 11 (1999), 545.
doi: 10.1080/10556789908805762. |
[39] |
U. Topcu, A. Packard, P. Seiler and G. J. Balas, Robust region-of-attraction estimation,, IEEE Transactions on Automatic Control, 55 (2010), 137.
doi: 10.1109/TAC.2009.2033751. |
[40] |
G. Valmorbida and J. Anderson, Region of attraction analysis via invariant sets,, in Proc. of the American Control Conference, (2014), 3591.
doi: 10.1109/ACC.2014.6859263. |
[41] |
L. Vandenberghe and S. Boyd, Semidefinite programming,, SIAM Review, 38 (1996), 49.
doi: 10.1137/1038003. |
[42] |
Q. Zheng and F. Wu, Nonlinear output feedback $H_{\infty}$ control for polynomial nonlinear systems,, in Proceedings of the 2008 American Control Conference, (2008), 1196. Google Scholar |
[43] |
Q. Zheng and F. Wu, Generalized nonlinear $H_{\infty}$ synthesis condition with its numerically efficient solution,, International Journal of Robust and Nonlinear Control, 21 (2011), 2079.
doi: 10.1002/rnc.1682. |
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