1. Stability Region Analysis using composite Lyapunov functions and bilinear SOS programming Support from AFOSR FA9550-05-1-0266, April 05-November 06 Authors Weehong Tan, Andy Packard Mechanical Engineering, UC Berkeley Acknowledgements Thanks to Ufuk Topcu, Gary Balas and Pete Seiler; PENOPT Website http://jagger.me.berkeley.edu/~pack/certify Copyright 2006, Packard, Tan, Wheeler, Seiler and Balas. This work is licensed under the Creative Commons Attribution-ShareAlike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/2.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.http://creativecommons.org/licenses/by-sa/2.0/

2. Quantitative Nonlinear Analysis Initial focus –Region of attraction estimation –Attractive invariant sets – induced norms for –finite-dimensional nonlinear systems, with polynomial vector fields parameter uncertainty (also polynomial) Main Tools: –Lyapunov/HJI formulation –Sum-of-squares proofs to ensure nonnegativity and set containment –Semidefinite programming (SDP), Bilinear Matrix Inequalities Optimization interface: YALMIP and SOSTOOLS SDP solvers: Sedumi BMIs: using PENBMI (academic license from www.penopt.com)www.penopt.com

3. Estimating Region of Attraction Dynamics, equilibrium point User-defined function p, whose sub-level sets are to be in region-of-attraction By choice of positive-definite V, maximize  so that

4. Convexity of Analysis In a global stability analysis, the certifying Lyapunov functions are themselves a convex set. In local analysis, the condition holds on sublevel sets This set of certifying Lyapunov functions is not convex. Example:

5. Estimating Region of Attraction Dynamics, equilibrium point User-defined function p, whose sub-level sets are to be in region-of-attraction By choice of positive-definite V, maximize  so that

6. Sum-of-Squares Sum-of-squares decompositions will be the main tool to decide set containment conditions, and certify nonnegativity. A polynomial f, in n real-variables is a sum-of-squares if it can be expressed as a sum-of-squares of other polys, Notation set of all sum-of-square polynomials in n variables set of all polynomials in n variables

7. Sum-of-Squares Decomposition For a polynomial f, in n real-variables, and of degree 2d The entries of z are not algebraically independent –e.g. x 1 2 x 2 2 = (x 1 x 2 ) 2 –M is not unique (for a specified f) The set of matrices, M, which yield f, is an affine subspace –one particular + all homogeneous –Particular solution depends on f –all homogeneous solutions depend only on n & d. Searching this affine subspace for a p.s.d element is a SDP…

8. Sum-of-Squares as SDP For a polynomial f, in n real-variables, and of degree 2d Each M i is s×s, where Using the Newton polytope method, both s and q can often be reduced, depending on the terms present in f. Semidefinite program: feasibility

9. (s,q) dependence on n and 2d 2d n 2468 2306610271575 3401020 12635465 450155035420701990 6702819684264621019152 8904554016510692495109890 10110661210286330331001457743

10. Synthesizing Sum-of-Squares as SDP Given: polynomials Decide if an affine combination of them can be made a sum-of-squares. This is also an SDP.

11. Synthesizing Sum-of-Squares as Bilinear SDP Given: polynomials A problem that will arise in this talk is: find such that This is a nonconvex SDP, namely a bilinear matrix inequality

12. Psatz Given: polynomials Goal: Decide if the set is empty. Φ is empty if and only if such that

13. Region of Attraction By choice of positive-definite V, maximize  so that Simple Psatz: “ small ” positive definite functions Products of decision variables BMIs PENBMI from PENOPT

14. Sanity check For a positive definite matrix B, Proof: Consider p.d. quadratic shape factor The best obtainable result is the “largest” value such that That containment easy to characterize: Questions: –Can the formulation we wrote yield this? –Can the BMI solver find this solution? n th order system cubic vector field known ROA Yes Basically, Yes 100 ’ s of random examples, n=2-8; 3 restarts of PENBMI, always successful

15. Example: Van der Pol: ROA Classical 2-d system Features: –Unstable limit cycle around origin –One equilibrium point: stable, at origin –Here, we use an elliptical shape factor Except for n V =4 case, the results are comparable to Papachristodoulou (2005) and Wloszek (2003)

16. Region of Attraction: pointwise-max If V 1 and V 2 are positive definite, and and Then proves asymptotic stability of on

17. Region of Attraction with pointwise-max Use Psatz to get a sufficient condition for using V of the form

18. ROA with Pointwise-Max Lyapunov functions Original (single V) Composite (2 V i )

19. Pointwise max of 6 th degree V 1,V 2

20. Is V 2, by itself, a decent Lyapunov function? –Sub-Level set looks similar to result, –But, derivative on sublevel set is not negative

21. Different shape factor Nearly the same results.

22. ROA: 3 rd order example Example (from Davison, Kurak): Solutions diverge from these initial conditions,i.e these initial conditions are not in the ROA

23. Problems, difficulties, risks Dimensionality: –For general problems, it seems unlikely to move beyond cubic vector fields and (pointwise-max) quadratic V. These result in “tolerable” SDPs for state dimension < 15. –Theory may lead to reduced complexity in specific instances of problems (sparsity, Newton polytope reduction, symmetries) Solvers (SDP): numerical accuracy, conditioning Connecting the Lyapunov-type questions to MilSpec-type measures –Decay rates –Damping ratios –Oscillation frequencies –Time-to-double BMI nature of local analysis

24. Other avenues Quantitative analysis around locally unstable equilibrium points (eg., reversed VanderPol) –ROA to a set, but not to a point –Reachability from locally unstable eq. point Appropriate question/analysis when equilibrium point depends on uncertain values –Dependence of eq.point on uncertainty, relation to nominal –ROA to the uncertain eq. point Other induced norms Megretski/Rantzer-like IQC formalism –Known nonlinear system –Unknown which satisfies various IQCs