Abstract
Exciting recent developments at the interface of optimization and control have shown that several fundamental problems in dynamics and control, such as stability, collision avoidance, robust performance, and controller synthesis can be addressed by a synergy of classical tools from Lyapunov theory and modern computational techniques from algebraic optimization. In this chapter, we give a brief overview of our recent research efforts (with various coauthors) to (i) enhance the scalability of the algorithms in this field, and (ii) understand their worst case performance guarantees as well as fundamental limitations. The topics covered include the concepts of “dsos and sdsos optimization”, path-complete and non-monotonic Lyapunov functions, and some lower bounds and complexity results for Lyapunov analysis of polynomial vector fields and hybrid systems. In each case, our relevant papers are tersely surveyed and the challenges/opportunities that lie ahead are stated.
This chapter is a revised and expanded version of the conference paper in [13], which was presented as a tutorial talk at the 53rd annual Conference on Decision and Control.
Amir Ali Ahmadi—His research is partially supported by an NSF CAREER Award, an AFOSR Young Investigator Program Award, and a Google Research Award.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
The familiar reader may safely skip this section. For a more comprehensive introductary exposition, see: https://blogs.princeton.edu/imabandit/guest-posts/.
- 2.
Here, we are assuming a strictly feasible solution to the SDP. Indeed a strictly feasible solution to (7) is required to get the strict inequalities in (6). Luckily, unless the SDP has an empty interior, a strictly feasible solution will automatically be returned by the interior point solver. See the discussion in [1, p. 41].
- 3.
Once again, strict feasibility of the constraints in (8) is required to rule out trivial solutions and lead to the strict inequalities that we would like to impose on V.
- 4.
- 5.
A homogeneous polynomial vector field is one where all monomials have the same degree. Linear systems are an example.
References
A.A. Ahmadi, Non-monotonic Lyapunov functions for stability of nonlinear and switched systems: theory and computation. Master’s Thesis, Massachusetts Institute of Technology, June 2008. http://aaa.lids.mit.edu/publications
A.A. Ahmadi, Algebraic relaxations and hardness results in polynomial optimization and Lyapunov analysis. Ph.D. Thesis, Massachusetts Institute of Technology, September 2011. http://aaa.lids.mit.edu/publications
A.A. Ahmadi, On the difficulty of deciding asymptotic stability of cubic homogeneous vector fields. in Proceedings of the American Control Conference (2012)
A.A. Ahmadi, R. Jungers, SOS-convex Lyapunov functions with applications to nonlinear switched systems. in Proceedings of the IEEE Conference on Decision and Control (2013)
A.A. Ahmadi, R. Jungers, On complexity of Lyapunov functions for switched linear systems. in Proceedings of the 19th World Congress of the International Federation of Automatic Control (2014)
A.A. Ahmadi, A. Majumdar, Some applications of polynomial optimization in operations research and real-time decision making (2014). Under review
A.A. Ahmadi, A. Majumdar, DSOS and SDSOS: more tractable alternatives to sum of squares and semidefinite optimization (in preparation, 2016)
A.A. Ahmadi, P.A. Parrilo, Non-monotonic Lyapunov functions for stability of discrete time nonlinear and switched systems. in Proceedings of the 47 \(^{th}\,\) IEEE Conference on Decision and Control (2008)
A.A. Ahmadi, P.A. Parrilo, Converse results on existence of sum of squares Lyapunov functions. in Proceedings of the 50 \(^{th}\) IEEE Conference on Decision and Control (2011)
A.A. Ahmadi, P.A. Parrilo, On higher order derivatives of Lyapunov functions. in Proceedings of the 2011 American Control Conference (2011)
A.A. Ahmadi, P.A. Parrilo, A complete characterization of the gap between convexity and sos-convexity. SIAM J. Optim. 23(2), 811–833 (2013)
A.A. Ahmadi, P.A. Parrilo, Stability of polynomial differential equations: complexity and converse Lyapunov questions. Under review. Preprint available at http://arxiv.org/abs/1308.6833, 2013
A.A. Ahmadi, P.A. Parrilo, Towards scalable algorithms with formal guarantees for Lyapunov analysis of control systems via algebraic optimization. in The Proceedings of the 53rd Annual Conference on Decision and Control (2014), pp. 2272–2281
A.A. Ahmadi, M. Krstic, P.A. Parrilo, A globally asymptotically stable polynomial vector field with no polynomial Lyapunov function. in Proceedings of the 50 \(^{th}\) IEEE Conference on Decision and Control (2011)
A.A. Ahmadi, R. Jungers, P.A. Parrilo, M. Roozbehani, Analysis of the joint spectral radius via Lyapunov functions on path-complete graphs, Hybrid Systems: Computation and Control 2011, Lecture Notes in Computer Science (Springer, 2011)
A.A. Ahmadi, R.M. Jungers, P.A. Parrilo, M. Roozbehani, When is a set of LMIs a sufficient condition for stability? in Proceedings of the IFAC Symposium on Robust Control Design (2012)
A.A. Ahmadi, A. Majumdar, R. Tedrake, Complexity of ten decision problems in continuous time dynamical systems. in Proceedings of the American Control Conference (2013)
A.A. Ahmadi, R. Jungers, P.A. Parrilo, M. Roozbehani, Joint spectral radius and path-complete graph Lyapunov functions. SIAM J. Optim. Control 52(1), 687 (2014)
V.I. Arnold. Problems of present day mathematics, XVII (Dynamical systems and differential equations). Proc. Symp. Pure Math., 28(59) (1976)
A. Ataei-Esfahani, Q. Wang, Nonlinear control design of a hypersonic aircraft using sum-of-squares methods. in Proceedings of the American Control Conference (IEEE, 2007), pp. 5278–5283
A. Bacciotti, L. Rosier, Liapunov Functions and Stability in Control Theory (Springer, Heidelberg, 2005)
A.J. Barry, A. Majumdar, R. Tedrake, Safety verification of reactive controllers for UAV flight in cluttered environments using barrier certificates. in Proceedings of the IEEE International Conference on Robotics and Automation (IEEE, 2012), pp. 484–490
D. Bertsimas, D.A. Iancu, P.A. Parrilo, A hierarchy of near-optimal policies for multistage adaptive optimization. IEEE Trans. Autom. Control 56(12), 2809–2824 (2011)
A.R. Butz, Higher order derivatives of Liapunov functions. IEEE Trans. Autom. Control AC–14, 111–112 (1969)
A. Chakraborty, P. Seiler, G.J. Balas, Susceptibility of F/A-18 flight controllers to the falling-leaf mode: nonlinear analysis. J. Guid. Control Dyn. 34(1), 73–85 (2011)
G. Chesi, D. Henrion (eds.), Special issue on positive polynomials in control. IEEE Trans. Autom. Control 54(5), 935 (2009)
CPLEX. V12. 2: Users manual for CPLEX. International Business Machines Corporation, 46(53), 157 (2010)
J. Daafouz, J. Bernussou, Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties. Syst. Control Lett. 43(5), 355–359 (2001)
A.C. Doherty, P.A. Parrilo, F.M. Spedalieri, Distinguishing separable and entangled states. Phys. Rev. Lett. 88(18), 187904 (2002)
R. Goebel, A.R. Teel, T. Hu, Z. Lin, Conjugate convex Lyapunov functions for dual linear differential inclusions. IEEE Trans. Autom. Control 51(4), 661–666 (2006)
N. Gvozdenović, M. Laurent, Semidefinite bounds for the stability number of a graph via sums of squares of polynomials. Math. Program. 110(1), 145–173 (2007)
J. Harrison, Verifying nonlinear real formulas via sums of squares, Theorem Proving in Higher Order Logics (Springer, Heidelberg, 2007), pp. 102–118
D. Henrion, A. Garulli (eds.), Positive Polynomials in Control, vol. 312 (Lecture Notes in Control and Information Sciences (Springer, Heidelberg, 2005)
D. Henrion, M. Korda, Convex computation of the region of attraction of polynomial control systems. IEEE Trans. Autom. Control 59(2), 297–312 (2014)
D. Hilbert, Über die Darstellung Definiter Formen als Summe von Formenquadraten. Math. Ann. 32, (1888)
C. Hillar, L.-H. Lim, Most tensor problems are NP-hard. arXiv preprint arXiv:0911.1393, 2009
J.E. Hopcroft, R. Motwani, J.D. Ullman, Introduction to Automata Theory, Languages, and Computation (Addison Wesley, Boston, 2001)
T. Hu, Z. Lin, Absolute stability analysis of discrete-time systems with composite quadratic Lyapunov functions. IEEE Trans. Autom. Control 50(6), 781–797 (2005)
T. Hu, L. Ma, Z. Li, On several composite quadratic Lyapunov functions for switched systems. in Proceedings of the 45 \(^{th}\,\) IEEE Conference on Decision and Control (2006)
T. Hu, L. Ma, Z. Lin, Stabilization of switched systems via composite quadratic functions. IEEE Trans. Autom. Control 53(11), 2571–2585 (2008)
Z. Jarvis-Wloszek, R. Feeley, W. Tan, K. Sun, A. Packard, Some controls applications of sum of squares programming. in Proceedings of the 42 \(^{th}\,\) IEEE Conference on Decision and Control (2003), pp. 4676–4681
R. Jungers, The Joint Spectral Radius: Theory and Applications, vol. 385 (Lecture Notes in Control and Information Sciences (Springer, Heidelberg, 2009)
H. Khalil, Nonlinear Systems, 3rd edn. (Prentice Hall, New Jersey, 2002)
J.B. Lasserre, Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001)
J.B. Lasserre, D. Henrion, C. Prieur, E. Trélat, Nonlinear optimal control via occupation measures and LMI-relaxations. SIAM J. Control Optim. 47(4), 1643–1666 (2008)
J.W. Lee, G.E. Dullerud, Uniform stabilization of discrete-time switched and Markovian jump linear systems. Automatica 42(2), 205–218 (2006)
J.W. Lee, P.P. Khargonekar, Detectability and stabilizability of discrete-time switched linear systems. IEEE Trans. Autom. Control 54(3), 424–437 (2009)
D. Lind, B. Marcus, An Introduction to Symbolic Dynamics and Coding (Cambridge University Press, Cambridge, 1995)
J. Löfberg, YALMIP: a toolbox for modeling and optimization in MATLAB. in Proceedings of the CACSD Conference (2004). http://control.ee.ethz.ch/~joloef/yalmip.php
A. Magnani, S. Lall, S. Boyd, Tractable fitting with convex polynomials via sum of squares. in Proceedings of the 44 \(^{th}\,\) IEEE Conference on Decision and Control (2005)
A. Majumdar, A.A. Ahmadi, R. Tedrake, Control design along trajectories with sums of squares programming. in Proceedings of the IEEE International Conference on Robotics and Automation (2013)
E. Marianna, M. Laurent, A. Varvitsiotis, Complexity of the positive semidefinite matrix completion problem with a rank constraint, Discrete Geometry and Optimization (Springer, Switzerland, 2013), pp. 105–120
A. Majumdar, A.A. Ahmadi, R. Tedrake, Control and verification of high-dimensional systems via dsos and sdsos optimization. in Submitted to the 53rd IEEE Conference on Decision and Control (2014)
A. Majumdar, R. Vasudevan, M.M. Tobenkin, R. Tedrake, Convex optimization of nonlinear feedback controllers via occupation measures. Int. J. Robot. Res. 33(9), 1209 (2014)
P.A. Parrilo, Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. Ph.D. Thesis, California Institute of Technology, May 2000
P.A. Parrilo, Semidefinite programming relaxations for semialgebraic problems. Math. Program. 96(2, Ser. B), 293–320 (2003)
P.A. Parrilo, Polynomial games and sum of squares optimization. in Proceedings of the 45 \(^{th}\,\) IEEE Conference on Decision and Control (2006)
S. Prajna, A. Jadbabaie, Safety verification of hybrid systems using barrier certificates, Hybrid Systems: Computation and Control (Springer, Heidelberg, 2004), pp. 477–492
S. Prajna, A. Jadbabaie, G.J. Pappas, A framework for worst-case and stochastic safety verification using barrier certificates. IEEE Trans. Autom. Control 52(8), 1415–1428 (2007)
S. Prajna, A. Papachristodoulou, P.A. Parrilo, SOSTOOLS: sum of squares optimization toolbox for MATLAB, May 2002. http://www.cds.caltech.edu/sostools and http://www.mit.edu/~parrilo/sostools
B. Reznick, Uniform denominators in Hilbert’s 17th problem. Math Z. 220(1), 75–97 (1995)
B. Reznick, Some concrete aspects of Hilbert’s 17th problem, Contemporary Mathematics, vol. 253 (American Mathematical Society, Rhode Island, 2000), pp. 251–272
M. Roozbehani, Optimization of Lyapunov invariants in analysis and implementation of safety-critical software systems. Ph.D. Thesis, Massachusetts Institute of Technology, 2008
P. Seiler, G.J. Balas, A.K. Packard, Assessment of aircraft flight controllers using nonlinear robustness analysis techniques, Optimization Based Clearance of Flight Control Laws (Springer, Heidelberg, 2012), pp. 369–397
R. Tae, B. Dumitrescu, L. Vandenberghe, Multidimensional FIR filter design via trigonometric sum-of-squares optimization. IEEE J. Sel. Top. Signal Process. 1(4), 641–650 (2007)
Acknowledgements
We are grateful to Anirudha Majumdar for his contributions to the work presented in Sect. 4 and to Russ Tedrake for the robotics applications and many insightful discussions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Ahmadi, A.A., Parrilo, P.A. (2017). Some Recent Directions in Algebraic Methods for Optimization and Lyapunov Analysis. In: Laumond, JP., Mansard, N., Lasserre, JB. (eds) Geometric and Numerical Foundations of Movements . Springer Tracts in Advanced Robotics, vol 117. Springer, Cham. https://doi.org/10.1007/978-3-319-51547-2_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-51547-2_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-51546-5
Online ISBN: 978-3-319-51547-2
eBook Packages: EngineeringEngineering (R0)