1. Development of Analysis Tools for Certification of Flight Control Laws FA9550-05-1-0266, April 05-November 05 Participants UCB: Weehong Tan, Tim Wheeler, Andy Packard, Ufuk Topcu Honeywell: Pete Seiler UMN: Gary Balas Website http://jagger.me.berkeley.edu/~pack/certify Copyright 2005, 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. Validation/Verification/Certification (VVC) Control Law VVC - Verification: assure that the flight control system fulfills the design requirements. - Validation: assure that the developed flight control system satisfies user needs under defined operating conditions. - Certification: applicant demonstrates compliance of the design to the certifying authority. Current practice: Partially guided by MilSpec –Linearized analyses Closed-loop: Time domain Open-loop: Frequency domain –Numerous nonlinear sims. –Strategies/Process to manage/distill all of this data into a actionable conclusion. “as much a psychological exercise as it is a mathematical analysis”, Anonymous, Pratt-Whitney systems engineer.

3. Why psychological? VV needs a conclusion about physical system using model-based analysis… leap-of-faith Inadequacy in model –Known unknowns –Unknown unknowns –Gross simplification Inadequacy in analysis to resolve issue –Inability to precisely answer question –Relevance of question to issue at hand Initial Goal: –Make the leap smaller with quantitative nonlinear analysis –Generate suite of examples to build experience Improve these while addressing these

4. Quantitative Nonlinear Analysis Initial focus –Region of attraction estimation – 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

5. Estimating Region of Attraction Dynamics, equilibrium point User-defined function 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. 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 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

8. 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.

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

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

11. 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:

12. 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 -2.5-2-1.5-0.500.511.522.5 -3 -2 0 1 2 3 x 1 x 2 ROA for Van der Pol nV = 2,  = 0.59 nV = 4,  = 0.66 nV = 6,  = 0.78

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

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

15. ROA with Pointwise-Max Lyapunov functions -2.5-2-1.5-0.500.511.522.5 -3 -2 0 1 3 x 1 x 2 ROA for Van der Pol nV = 6,  = 0.78 2 x nV = 2,  = 0.75 2 x nV = 4,  = 0.93 2 2 x nV = 6,  = 1

16. Different Shape factor -2.5-2-1.5-0.500.511.522.5 -3 -2 0 1 3 x 1 x 2 ROA for Van der Pol 2

17. Reachability of with inputs If then Simple Psatz certification

18. Reachability of with inputs Example: 0246810 0 2 4 6 8 12 14 16 Linearized R 2  Upper Bound

19. Reachability of with inputs Choose T: Conditions for stationarity adjust scalar so Tierno, et.al, 1996 Note: If f is linear, and p is a p.d. quadratic form, then the iteration is the correct power iteration for the maximum. repeat

20. Reachability of with inputs Lower bound 0246810 0 2 4 6 8 12 14 16 Lower Bnd Upper Bound Linearized R 2 

21. Refinement Replace with Then generally, h k <1 will work generally, greater than R 2

22. Refinement 0246810 0 2 4 6 8 12 14 16 Lower Bnd Upper Bound Linearized R 2  Refined Upper Bound Using worst-case input from linear analysis

23. gain: Adaptive control example Plant: with unknown (=2) Controller: Properties: Global convergence x 1 to 0, x 2 to θ-dependent equilibrium point, and (in this case) Add input disturbance, compute gain from “ Adaptive nonlinear control without overparametrization, ” Krstic, Kanellakopoulos, Kokotovic, Systems and Control Letters, vol. 19, pp. 177-185, 1992 C P How does adaptation gain affect this?

24. gain of If then elementary sufficient condition Iteration (as before) for stationary points, to yield lower bounds

25. 0.511.52 0.3 0.35 0.4 0.45 0.5 0.55 Adaptive Control,  = 1 and  = 4 R L2 to L2 gain Adaptive control Compute/Bound for two values of adaptation gain, Γ=1, 4. C P H ∞ norm of the linearization For small, large adaptation gain gives better worst-case disturbance attenuation. But for large, the situation is reversed… Trend implied by linearized analysis invalid for large inputs. 0 0.25 Γ=4 Γ=1

26. Region of Attraction for uncertain system Uncertain Dynamics Apriori constraint on uncertainty Consider an equilibrium point that does not depend on Choose V to maximize  so that:

27. ROA: Uncertain 2-D Van der Pol -2.5-2-1.5-0.500.511.522.5 -4 -3 -2 0 1 2 3 4 x 1 x 2 ROA for Uncertain Van der Pol V(x,  ), nV = 4,  = 0.6 V(x), nV = 4,  = 0.54

28. ROA: 3 rd order example Example (from Davison, Kurak): Solutions diverge from these initial conditions

29. SDP Solvers: Issues An “old” robustness analysis problem that is written as an SDP is “Routine” since 1988, although the best SDP solvers today often fail on such problems. Example: –5-state, all scalar signals (taken from 2005 ACC, Hu, et. al.) –Sedumi is unable to find a feasible point –SDPT3 is unable to find a feasible point –LMIlab finds “optimal” (upper/lower bounds on inf) value Other numerical inconsistencies exist as well… Work remains.

30. 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 BMI nature of local analysis

31. 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

32. Iterative Stability Region Estimation Algorithm [Chiang/Thorp, 1989 IEEE TAC] 1.Construct a local Lyapunov function, V 0 (x) and find the largest c such that dV 0 /dt<0 for x in S V0 (c) \ x s 2.Choose  and iteratively update the Lyapunov Function: V k+1 (x) = V k ( x+  f(x) ) Notation Let x s be a stable equilibrium point of dx/dt = f(x). Let S V (c) denote the connected component containing x s of the set {x: V(x)≤c}. Main Result [Chiang/Thorp, 1989 IEEE TAC] For a finite number of iterations, there exists  M such that for 0<  <  M, 1. S Vk (c) is contained in the stability region for each k. 2. S Vk (c)  S Vk+1 (c) Combining the Chiang/Thorp Iteration with SOS Techniques The initial Lyapunov function and stability region estimate (steps 1 and 2 of the algorithm) can be found using SOS techniques. The iteration can then be applied to further improve the stability region estimate.