1. Development of Analysis Tools for Certification of Flight Control Laws
FA , April 05-November 06
Participants
UCB: Ufuk Topcu, Weehong Tan, Tim Wheeler, Andy Packard
Honeywell: Pete Seiler
UMN: Gary Balas
Website
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 or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

2. Tools for Quantitative, Local Nonlinear Analysis
Focus over the past 15 months
Region of attraction estimation
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
Constraints provided by simulation

3. Examples presented last yearC P
0.5 1
1.5 2
0.3
0.35
0.4
0.45
0.55
Adaptive Control, G
= 1 and
= 4 R
L2 to L2 gain
0.25
Γ=4
Γ=1 2 4 6 8 10 12 14 16
Lower Bnd
Upper Bound
Linearized R b
Refined Upper Bound
Using worst-case input from linear analysis
Composite (2 Vi)

4. 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,
denotes set of all sum-of-square polynomials in n variables

5. 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. 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…

6. Semidefinite program: feasibility
Sum-of-Squares as SDP
For a polynomial f, in n real-variables, and of degree 2d
Each Mi 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

7. (s,q) wrt n and 2d 2d n 2 4 6 8 3 10 27 15 75 20
126 35
465 5 50
420 70
1990 7 28
196 84
2646
210
19152 9 45
540
165
10692
495
109890 11 66
1210
286
33033
1001
457743

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. 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

10. Common features of analysis
These analysis all involve search over a nonconvex set of certifying Lyapunov functions, roughly
The SOS relaxations are nonconvex as well, e.g.,
They are “solved” via
PENBMI, commercial BMI solver from PENOPT
Ad-hoc iteration on linear SDPs
Examples were nonconvex problems in ~100s of variables

11. Last year: What we didn’t show…
Obtaining results was challenging….
restart
Run Iteration
Answer not what is needed or expected
restart
YES!
Run PENBMI
diverge
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

12. 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

13. 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

14. Region of Attraction: Bilinear SOS
Maximize  (positive-definite V ) so that
Choose “small” positive definite functions
BMIs
Products of decision variables

15. 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 bilinear SOS formulation yield this?
Can the BMI solver find this solution?
nth order system
cubic vector field
known ROA
Yes
Basically, Yes, Fast (n<10)
1000’s of random examples, n=2-10; two restarts of PENBMI, always successful

16. Region of Attraction Consider a simpler question. Fix β, is
Ad-hoc solution:
run N sims, starting from samples in
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…

17. Outer bound on certifying Lyapunov functions
After simulations
Collection of convergent trajectories starting in
divergent trajectories starting in
Linearly parametrize V, namely
The necessary conditions on V are convex constraints on
V≤1 on convergent trajectories
V≥0 on all trajectories
V decreasing on convergent trajectories
Quad(V) is a Lyapunov function for Linear(f)
V≥1 on divergent trajectories

18. Hit & Run: Uniformly sample convex set in Rn
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
An SOS program, for
Smith, 1984 Operations Research
Lovasz, 1999 Math Programming
Tempo, Calafiore, Dabbene, Springer

19. Assessing V: Checking containments
Each candidate V certifies a region of attraction
Generally, this is solved in two steps
SOS optimization (s8, s9) to maximize the level-set condition on V
SOS optimization (s6) to maximize the condition on p & V
PENBMI and iteration initialized with these as well

20. Assessing V: Checking containments
Alternate conditions, this is solved in two steps
SOS optimization (p1) to maximize the level-set condition on V
SOS optimization (p2) to maximize the condition on p & V
Under the assumption that is negative definite near 0, these confirm
SDP, no bisection
SDP, no bisection

21. Employing simulation Simulate Sample Vouter Seed Iteration Seed PENBMI
Assess ROA using V
Seed Iteration
Seed PENBMI

22. Examples considered
vanDerPol’s

23. Examples considered (cont’d)
Aircraft: Pitch axis, 2-state dynamic inversion controller
Short period longitudinal model,

24. Results: Van der Pol’s oscillator
Quadratic shape factor: βmax ≈1.04
Sims performed from
Assess achieved β from 50 samples of outer bound
Now seed PENBMI with these samples

25. Van der Pol’s Summary Unseeded PENBMI Degree(V)=4 Degree(V)=6 RunTime
30-45(-300) seconds
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
5 seconds
Seed/Run PENBMI
7 seconds
16 seconds
TOTAL
35 seconds
63 seconds
Additional Point (H&R)
0.1 seconds
0.2 seconds
BestAnswer, β=
0.930
1.034
Percentage
100

26. Level Sets
The level sets

27. What did the aircraft analysis entail/yield/require?
Several Analysis
Unseeded calls to PENBMI
Sim-based initialization for iteration
Did not run a seeded PENBMI
Alternate initialization for iteration
Separate extensive simulations to find divergent trajectories
Simple form for shape factor
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)

28. Results: quadratic+cubic+quartic V
There is a divergent trajectory starting from
Simulation-based algorithm
Iteration from “random” starting point
Take P from
Direct unseeded call to PENBMI yields (after 38 hours)
All initial conditions in are in ROA
divergent trajectory from
4000 simulations
5 minutes
Form LP/ConvexP
3 minutes
Get a feasable point
Assess answer with V
2 minutes
Iterate from V
3 minutes/iteration, 6 iters
TOTAL
33 minutes
30 iterations

29. 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

30. 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
extends (conceptually) easily. If
Then
If such a V exists, then from x(0)=0, with there are
upper and lower bounds on V, and
upper bounds on

31. 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

32. 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