Development of Analysis Tools for Certification of Flight Control Laws FA9550-05-1-0266, April 05-November 06 Participants UCB: Ufuk Topcu, Weehong Tan, Tim Wheeler, Andy Packard Honeywell: Pete Seiler UMN: Gary Balas 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. Tools for Quantitative, Local Nonlinear Analysis Focus over the past 15 months – Region of attraction estimation – L2 L2 induced norms – L2 L induced norms for – Locally stable, 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 (BMI) • Optimization interface: YALMIP and SOSTOOLS • SDP solvers: Sedumi • BMIs: using PENBMI (academic license from www.penopt.com) – Constraints provided by simulation Examples presented last year w C ˆ u P x2 x1 0.5 L2 to L2 gain x1 x2 ( x1 1) x2 u Γ=1 0.45 G 1ˆ x1 x1 1 x2 x1 ˆx1 1 1 ˆ x1 1 u x2 x1 ˆx1 1 x1 1 ˆ x2 ˆ x1 1 ˆx1 1 Adaptive Control, G = 1 and G= 4 0.55 0.4 0.35 0.3 0.25 0 Vi q 1 q 2 2 13 38 4 57 120 6 166 338 max w 2 R Γ=4 0.5 1 x1 2 w2 1.5 R 16 x1 x1 1 x22 x2 14 x x 1 x 2 w 2 2 1 12 Composite (2 Vi) 10 b p( x) Cx Upper Bound 2 8 C 8 4 4 4 6 Refined Upper Bound 1 2 Lower Bnd Linearized 4 2 0 0 2 4 Using worst-case input from linear analysis 6 8 10 2 R 2 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, p f g 2j j 1 n denotes set of all sum-of-square polynomials in n variables Sum-of-Squares Decomposition For a polynomial f, in n real-variables, and of degree 2d f n M 0 such that f z T Mz where z [1, x1 , x2 ,, xn , x1 x2 ,, xnd ]T . The entries of z are not algebraically independent – e.g. x12x22 = (x1x2)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 an SDP… Sum-of-Squares as SDP For a polynomial f, in n real-variables, and of degree 2d f n q R q such that M 0 i M i 0 i 1 Semidefinite program: feasibility Each Mi is s×s, where n d s d 2 n d 1 n d n 2d q 2 d d 2d Using the Newton polytope method, both s and q can often be reduced, depending on the terms present in f. (s,q) wrt n and 2d n d s d 2d n n d 2 n d n 2d 1 q 2 d d 2d 2 4 6 8 2 3 0 6 6 10 27 15 75 3 4 0 10 20 20 126 35 465 4 5 0 15 50 35 420 70 1990 6 7 0 28 196 84 2646 210 19152 8 9 0 45 540 165 10692 495 109890 10 11 0 66 1210 286 33033 1001 457743 Synthesizing Sum-of-Squares as SDP Given: polynomials f 0 , f1 , , f m Decide if an affine combination of them can be made a sum-of-squares. This is also an SDP. m R m with f 0 k f k n k 1 qm R q m with M 0 i M i 0 i 1 Synthesizing Sum-of-Squares as Bilinear SDP Given: polynomials f 0 , f1 , , f m g 0 , g1 , , g m h0 , h1 , , hm A problem that will arise in this talk is: find such that R m , R m m m f 0 k f k g 0 k g k h0 k hk n k 1 k 1 k 1 m This is a nonconvex SDP, namely a bilinear matrix inequality Common features of analysis These analysis all involve search over a nonconvex set of certifying Lyapunov functions, roughly x : x 0,V ( x) 1 x : V f 0 The SOS relaxations are nonconvex as well, e.g., find V ,V (0) 0, and s8 , s9 n subject to 1 V s8 s9V f l2 n They are “solved” via – PENBMI, commercial BMI solver from PENOPT – Ad-hoc iteration on linear SDPs – Examples were nonconvex problems in ~100s of variables Last year: What we didn’t show… Obtaining results was challenging…. restart Run PENBMI diverge Answer not what is needed or expected restart Run Iteration YES! YES! Answer not what is needed or expected By contrast: – Today’s results better, reliable and naturally obtained For now – Restrict attention to region-of-attraction estimation Estimating Region of Attraction Nonconvex problem, nonconvex relaxation. Solution approaches: SOS conditions to verify containments – Parametrize V, parametrize multipliers, solve… • Bilinear SDP solvers • Ad-hoc iterative, based on linear SDPs Behavior: – Initial point has big effect on end result, e.g., • Unable to reach a feasible point • Convergence to local optimum What are prospects for generating “good” initial points? – Easily computable – Promising results Estimating Region of Attraction Dynamics, equilibrium point x f ( x), f (x) 0 V 1 p3 User-defined function whose sub-level sets are to be in region-of-attraction x : p( x) b ROA x p2 p 1 x By choice of positive-definite V, maximize b so that x:V(x) 1 is bounded x : p( x) b x:V(x) 1 dV x : x x ,V ( x) 1 x: f 0 dx dV f 0 dx Region of Attraction: Bilinear SOS V 1 p3 p2 Maximize b (positive-definite V ) so that x : V(x) 1 is bounded x : p( x) b x : V(x) 1 x : x x,V ( x) 1 x : V f p 1 x 0 Choose “small” positive definite functions dV f 0 dx l1 ,l2 max b over V , V ( x ) 0, s6 , s8 , s9 n subject to V l1 n BMIs b p s6 V 1 n 1 V s8 s9V f l2 n Products of decision variables Sanity check For a positive definite matrix B, x x xBx xROA x : xBx 1 Proof: V x : xBx 0 V 2V V 1. nth order system cubic vector field known ROA Consider p.d. quadratic shape factor p( x) xT Rx The best obtainable result is the “largest” value bmax such that x : xRx b x : xBx 1 That containment easy to characterize: 12 bmax max R BR 12 Questions: Yes – Can the bilinear SOS formulation yield this? – Can the BMI solver find this solution? Basically, Yes, Fast (n<10) 1000’s of random examples, n=2-10; two restarts of PENBMI, always successful 1 Region of Attraction Consider a simpler question. Fix β, is Pb : x : p( x) b ROA x ? Ad-hoc solution: – run N sims, starting from samples in x : p( x) b • If any diverge, then “no” • If all converge, then maybe “yes”, and perhaps the Lyapunov analysis can prove it In this case, how can we use the simulation data? Necessary condition: If V exists to verify, it must be – ≤1 on all trajectories – ≥0 on all trajectories – Decreasing on all trajectories – and possibly some more… x : p( x) b x:V(x) 1bounded x : x x ,V ( x) 1 x: dV f 0 dx Outer bound on certifying Lyapunov functions After simulations – Collection of convergent trajectories starting in c – divergent trajectories starting in Pb Linearly parametrize V, namely Pb Nb V ( x ) k k ( x ) k 1 The necessary conditions on V are convex constraints on R N b V≤1 on convergent trajectories V≥0 on all trajectories V ( x) l1 ( x) n ( x) V decreasing on convergent trajectories Quad(V) is a Lyapunov function for Linear(f) V≥1 on divergent trajectories Hit & Run: Uniformly sample convex set in Rn 1. 2. 3. 4. 5. Start with an interior point, w Pick a direction v in Rn, N(0,I) Find tmin and tmax such that w+tv just in set Pick μ, uniformly in [tmin tmax] NextX = x + μv In Lyapunov coefficient space, get samples: – Assess the ROA that they certify, or… – Use as a seed for • PENBMI, and/or iteration Finding [tmin tmax] involves – Several simple 1-d linear inequalities – A linear matrix inequality for AT P PA 0 – An SOS program, for V l1 n Smith, 1984 Operations Research Lovasz, 1999 Math Programming Tempo, Calafiore, Dabbene, Springer Assessing V: Checking containments Each candidate V certifies a region of attraction b cert ,V : max b such that satisfying x : p( x) b x : V ( x) x : V ( x) 0 Generally, this is solved in two steps – SOS optimization (s8, s9) to maximize the level-set condition on V V s8 s9V f l2 n – SOS optimization (s6) to maximize the condition on p & V b p s6 V n PENBMI and iteration initialized with these as well Assessing V: Checking containments Alternate conditions, this is solved in two steps – SOS optimization (p1) to maximize the level-set condition on V V i1 xi2d n 1 p1V f n SDP, no bisection – SOS optimization (p2) to maximize the condition on p & V b p x n 2d2 i 1 i Under the assumption that 0, these confirm V p2 n SDP, no bisection V f is negative definite near b cert ,V : max b such that satisfying x : p( x) b x : x 0,V ( x) cc ,0 x : V ( x) 0 Employing simulation Simulate Sample Vouter Assess ROA using V Seed Iteration Seed PENBMI Examples considered vanDerPol’s x1 x2 x2 x1 x12 1x2 Limit cycle for VDP 3 2 x2 1 0 -1 -2 -3 -2.5 -2 -1.5 -1 -0.5 0 x1 0.5 1 1.5 2 2.5 Examples considered (cont’d) Aircraft: Pitch axis, 2-state dynamic inversion controller x1 x2 x22 x2 x3 x2 x4 x2 x5 x23 2 x2 x5 x5 x Ax 0 0 0 q x 1 2 A 0 0 0 0 0 0 0 0 Short period longitudinal model, Results: Van der Pol’s oscillator Quadratic shape factor: βmax ≈1.04 Sims performed from x : p( x) 0.6 Assess achieved β from 50 samples of outer bound th 4nd 26 order V V order 12 number of points 10 8 6 4 2 0 0.3 0.4 0.5 b LP LP Now seed PENBMI with these samples 0.6 0.7 0.8 Van der Pol’s Summary Unseeded PENBMI Degree(V)=4 Degree(V)=6 RunTime 30-45(-300) seconds 900-3000 seconds BestAnswer, β= 0.928 1.034 Percentage 90 30 Seeded PENBMI Degree(V)=4 Degree(V)=6 Simulations 10 seconds (100) 20 seconds (200) Form LP/ConvexP 1 second 2 seconds Get feasible point 10 seconds 20 seconds Associate multipliers 2 seconds 5 seconds Seed/Run PENBMI 7 seconds 16 seconds TOTAL 35 seconds 63 seconds Additional Point (H&R) 0.1 seconds 0.2 seconds Associate multipliers 2 seconds 5 seconds Seed/Run PENBMI 7 seconds 16 seconds BestAnswer, β= 0.930 1.034 Percentage 100 100 Level Sets x : V ( x) 1 Level set of V for VDP 3 2 1 x2 The level sets 0 -1 nV = 2 nV = 4 -2 n =6 V -3 -2.5 -2 -1.5 -1 -0.5 0 x1 0.5 1 1.5 2 2.5 What did the aircraft analysis entail/yield/require? Several Analysis max b over V , V ( x ) 0, s6 , s8 , s9 n – Unseeded calls to PENBMI subject to V l1 n – Sim-based initialization for iteration b p s6 V 1 n • Did not run a seeded PENBMI 1 V s8 s9V f l2 n – Alternate initialization for iteration – Separate extensive simulations to find divergent trajectories Simple form for shape factor p( x) : xT x Different Lyapunov function structures – Quadratic (8.6, 8.6) – pointwise-max quadratics (8.6) – Quadratic+Quartic (12.2, 12.2) – Fully quartic (quadratic + cubic + quartic) Results: quadratic+cubic+quartic V There is a divergent trajectory starting from p( x0 ) 16.1 Simulation-based algorithm divergent trajectory from 4000 simulations 5 minutes Form LP/ConvexP 3 minutes Get a feasable point 5 minutes Assess answer with V 2 minutes Iterate from V 3 minutes/iteration, 6 iters TOTAL 33 minutes p( x0 ) 16.9 x : p( x) 14.6 ROA Iteration from “random” starting point AT P PA I 5 T V ( x) x Px 0.001i 1 xi4 Take P from 30 iterations x : p( x) 8.5 ROA Direct unseeded call to PENBMI yields (after 38 hours) – All initial conditions in x : p( x) 15.2 are in ROA What’s possible? Assuming no breakthroughs in – SDP/BMI solvers – exploiting problem structure Then, reliable and time-tolerable analysis for systems with – Cubic vector fields – State dimension between 10 and 15, pointwise-max quadratic Lyapunov functions – State dimension ≤6, quartic Lyapunov functions How should this be viewed? – Linearized analysis is effectively • Infinitesimal analysis of dynamics with quadratic Lyapunov fcns – So, the proposed method extends both the degree of approximation of the dynamics, and the richness of the Lyapunov function Extensions Using simulation data to impose necessary conditions on Lyapunov/storage functions that prove – Local, input/output gain bounds – Local state reachability – Attractive invariant sets – ROA for uncertain systems x f ( x, w), z h( x) extends (conceptually) easily. If V f x, w wT w 2 z T z on V x R 2 ; V (0) 0, V 0 Then x(0) 0 V ( x(t )) R 2 , z 2 R w2R If such a V exists, then from x(0)=0, with w 2 R there are – upper and lower bounds on V, and – upper bounds on V Some things to pursue Principal component analysis on the Lyapunov coefficient space (manifold discovery, Coifman) – Superficially, this won’t help much, since most of the variables in the SOS optimization come from the nonuniqueness of the Gram matrix, and are a function of the degree and order. McEnneay’s curse-of-dimensionality free computing More inner-loop airframe closed-loop analysis Wang, Lall, West (2005 Allerton) level set advection methods – Approximately integrate polynomial level set backwards, preserving polynomial structure. Iteration of linear SDPs Effect of number of simulations, employing backward sims Alternate iterations Time delays, other robustness metrics Problems, difficulties, risks Dimensionality: – Theory leads to reduced complexity in specific instances of problems (sparsity, Newton polytope reduction, symmetries) Solvers (SDP): numerical accuracy, conditioning Connecting the input/output gains to other measures of robustness and performance – Decay rates – Damping ratios – Oscillation frequencies – Including time-delays BMI nature of local analysis, though the simulation-based approach to seed BMI is promising
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