Development of Analysis Tools for Certification of Flight Control Laws FA , April 05-November 05 Participants UCB: Weehong Tan, Tim Wheeler, Andy Packard, Ufuk Topcu Honeywell: Pete Seiler UMN: Gary Balas Website 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 or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

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.

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

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

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

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

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

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.

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

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

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:

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 x 1 x 2 ROA for Van der Pol nV = 2,  = 0.59 nV = 4,  = 0.66 nV = 6,  = 0.78

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

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

ROA with Pointwise-Max Lyapunov functions x 1 x 2 ROA for Van der Pol nV = 6,  = x nV = 2,  = x nV = 4,  = x nV = 6,  = 1

Different Shape factor x 1 x 2 ROA for Van der Pol 2

Reachability of with inputs If then Simple Psatz certification

Reachability of with inputs Example: Linearized R 2  Upper Bound

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

Reachability of with inputs Lower bound Lower Bnd Upper Bound Linearized R 2 

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

Refinement Lower Bnd Upper Bound Linearized R 2  Refined Upper Bound Using worst-case input from linear analysis

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 , 1992 C P How does adaptation gain affect this?

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

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 Γ=4 Γ=1

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:

ROA: Uncertain 2-D Van der Pol x 1 x 2 ROA for Uncertain Van der Pol V(x,  ), nV = 4,  = 0.6 V(x), nV = 4,  = 0.54

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

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.

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

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

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.