1. Quantitative, Local Analysis for Nonlinear Systems
AFOSR: FA , April 2005-April 08
Participants
UCB: Ufuk Topcu, Weehong Tan, Packard, Tim Wheeler
UMN: Gary Balas
Honeywell: Pete Seiler
More info
Copyright 2008, Packard, Topcu, 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. Numerical tools to quantify/certify dynamic behavior
August 2008 Summary
Numerical tools to quantify/certify dynamic behavior
Locally, near equilibrium points
Analysis considered
Region-of-attraction, input/output gain, reachability, hard IQCs
Methodology
Enforce Lyapunov/Dissipation inequalities locally, on sublevel sets
Set containments via S-procedure and SOS constraints
Bilinear semidefinite programs
“always” feasible
Simulation aids nonconvex proof/certificate search
Address model uncertainty
Parametric Uncertainty
Parameter-independent Lyapunov/Storage Fcn
Branch-&-Bound
Dynamic Uncertainty
Local small-gain theorems
Polytope of vector fields
Model valid only over known set

3. Uncertain ICs: Estimating Region of Attraction
Dynamics, equilibrium point
If there exists positive-definite V with
then (Lyapunov) for the ODE, the set is invariant, and in the region-of-attraction.
Strategy: Grow estimate via optimization over V, enforcing containments with S-procedure/SOS conditions
Challenge: set of certifying Lyapunov functions is not convex.

4. Estimating Region of Attraction
Dynamics, equilibrium point
p: Analyst-defined function whose (well-understood) sub-level sets are to be in region-of-attraction
By choice of positive-definite V, maximize  so that:
Known domain of model validity

5. Checking Set Containments: S-proc & SOS
Consider sets defined via inequalities, using given polynomials pA and pB as Question: Does the set containment constraint hold?
Certified by
If there exists an SOS ( ) polynomial such
that
then
Linearly parametrize class to search (for s) over as
Find so
SDP in η and variables which parametrize Σ

6. Region of Attraction: Bilinear SOS
Maximize  (positive-definite V ) so that
Choose “small” positive definite functions
Remark: If l2 is quadratic, then feasibility implies that the linearized dynamics are exponentially stable.
BMIs
SDP iteration or direct BMI solver
Similar expressions for local L2 gain and reachability
Products of decision variables

7. Summary: Estimating Region of Attraction w/ simulation
Dynamics, equilibrium point
Find positive-definite V, with
Then is invariant, and in the region of attraction of , denoted
Given a “shape” function p
QUESTION: Fix β>0, is
Pragmatic solution:
run N sims, starting from samples in
If any diverge, then unambiguously “no”
If all converge, then maybe “yes”, perhaps Lyapunov analysis to prove/certify it
How can we use the simulation data to aid in the nonconvex search for a certifying V?
nonconvex constraint on V
Simulations yield
Collection of convergent trajectories starting in
divergent trajectories starting in
Necessary cond: If V exists to verify, need
V≤1 and decreasing on convergent trajectories
V≥0 on all trajectories
Quad(V) is a Lyapunov function for Linear(f)
V≥1 on the divergent trajectories
Linearly parametrize
Each candidate V certifies some ROA
Assess in 2 steps, using positivity & sum-of-squares (SOS) optimization to enforce subsets
SOS optimization (s1, s2) to maximize the level-set condition on V
SOS optimization (s3) to maximize condition on p, V
Furthermore, coordinatewise solvers are initialized with these, and further optimization (adjusting V too) be performed.
Convex constraints in Lyapunov function coefficient-space
Sample this convex outer-bound for candidate Lyapunov fcns
Necessary conditions are convex constraints on

8. gain of If there exists positive-definite V with bounded, and then
Strategy: enforcing containments with S-procedure/SOS
Fix R, shrink estimate of γ, or…
Fix γ, grow estimate of R
via optimization over V. Use simulation to outer-bound V
Exploit constraints on w if available

9. Uncertain Inputs: Reachability
Given a differential equation and a positive definite function p, how large can get, knowing If
Then
Strategy: Shrink estimate (β) via optimization over V, enforcing containments with S-procedure/SOS conditions. Use simulation and exploit constraints on w if available.
Conditions on
Conclusion on ODE

10. Reachability: Refining the bound
Positive-definite V, with bounded, satisfying
Then
Refinement: Suppose there exists a function h, with 0<h≤1, with
Define

11. (s,q) dependence on n and 2d4 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

12. Quantitative improvement on linearized analysis
Consider dynamics
with
matrix A is Hurwitz, and
function f23 consists of 2nd and 3rd degree polynomials, f23(0)=0
Standard analysis:
“there is an open ball around origin in region of attraction”
Here…
For any quadratic, pos-def l2
These are feasible with some quadratic pos-def l1, using

13. Quantitative improvement to linearized analysis
Consider dynamics
with
f2, g2, h2 quadratic, f3 cubic, with f2(0,0)=f3(0)=h2(0)=0
matrix A is Hurwitz, and
Here…
In fact, DIE can be strengthened to include positive-definite term
These are always feasible, using

14. Quantitative improvement to linearized analysis
Consider dynamics
with
functions b bilinear, q quadratic
matrix A is Hurwitz
Then (eg) the SOS-based local DIE for reachability
is always feasible using
Marginally-stable linearization at x=0. Common structure of certain adaptive systems

15. Uncertain Systems: Parameter-Independent V
For simplicity, take affine parameter uncertainty
Solve earlier conditions, but enforcing
at the vertex values of f.
Then is invariant, and in the Robust ROA of .
Advantages: a robust ROA, and
V is only a function of x, δ appears only implicitly through the vertices
SOS analysis is only in x variables
Simulations are incorporated as before (vary initial condition and δ)
Limitations
Conservative with regard to uncertainty
Conclusions apply to time-varying parameters, hence…
often conclusions are too weak for time-invariant parameters
polytope in Rm
Subdivide Δ
Solve separately Δ2 Δ1

16. Much better: B&B in Uncertainty Space
Of course, growth is still exponential in parameters… but
kth local problem uses Vk(x)
Solve conservative problem over subdomain
Local problems are decoupled
Trivial parallelization δ1 δ2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1
1 subdivisions
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1
3 subdivisions
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 d 2
0.1
0.2
0.3
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0.7
0.8
0.9 1
2 subdivisions
Non-affine uncertainty, for example
Treat as 2 parms on 1-d curve.
Cover with polytope. Solve.
Refine to union of polytopes.
Solve on each polytope.
Intersect ROAs → Robust ROA

17. Generalization of covering manifold
Given:
polynomial p(δ) in many real variables,
Domain , typically a polytope
Find a polytope that covers the manifold
Tradeoff between number of vertices, and
Excess “volume” in polytope
One approach:
Find “tightest” affine upper and lower bounds over H
Enforce with S-procedure
linear function of c0, c

18. Generalization of covering manifold
Partition H, repeat
For multivariable p,
Bound, on H (above and below), each component of p with affine functions, c, d, (e.g, using S-procedure). Then, a covering polytope (Amato, Garofalo, Gliemo) is
with 2m+k easily computed vertices.

19. Unmodeled dynamics Δ C P Fixed structure, parametric Operator P C P C
Parametrize fixed-structure system representing unmodeled dynamics
Compute certified invariant subset of region-of-attraction valid for all parameter values
Just more parameters
Operator
Introduce perturbation inputs/outputs
Compute L2→L2 gain across channel, valid over some L2 ball of inputs.
Apply a local small-gain theorem P C xc xp xδ P C xc xp Pδ C Δ
Local L2→L2 gain for system with parametric uncertainty. Same approach as ROA.

20. Unmodeled dynamics: operatorΔ Δ C P M
Local induced gain constraint (<1) on M
Starting from x(0)=0,
for all
Δ causal, globally stable,
Facts:

21. 4-state aircraft example
Aircraft: Short period longitudinal model, pitch axis, with 1-state linear controller
Simple form for shape factor:
Eliminate parameter uncertainty
Different Lyapunov function structures
Quadratic (βcert=8.6)
pointwise-max quadratics (βcert=8.6)
Quadratic+Quartic (βcert=12.2)
Fully quartic (quadratic + cubic + quartic)
βcert=15.3
Other approaches have deficiencies
Directly use commercial BMI solver (PENBMI)
βcert=15.2, but…
6 hours…
SDP iteration from “random” starting point
Use P from
Initialize
30 iterations, βcert=8.6
4000 simulations
3 minutes
Form LP/ConvexP
0.5 minutes
Get a feasable point
1 minute
Assess answer with V
SDP Iterate from V
0.5 min/iteration, 10 iters
TOTAL
10 minutes
discover divergent trajectories
proving
Divergent initial condition
Certified set of convergent initial conditions
Disk in 4-d state space, centered at equilibrium point

22. 4-state aircraft example w/uncertainty
Aircraft: Short period longitudinal model, pitch axis, with 1-state linear controller
Same form for shape factor:
Ad-hoc 9-processor B&B
Divide worst region into 9
Not-uncertain results
Quadratic (βcert=8.6)
Fully quartic (βcert=15.3)
Divergent IC,

23. 4-state aircraft example w/uncertaintyΔ
.75 C
1.25 Pδ
Results
Nominal
with δM, δCG
8.6 (15.3)
5.1 (7.5)
With Δ (as operator)
4.2 (6.7)
2.4 (4.1)
With Δ (1st order, γ, β)
4.9
2.8 (4.6)

24. Adaptive System: reachability example analysis
Model-reference adaptive systems
Example: 2-state P, 2-state ref. model, 3 adaptive parameters
Insert additional disturbance (d) not handled by theory
Bound worst-case effect of external signals (r,d) on tracking error (e)
Initial conditions: r
Reference model -
Adaptive control
plant e
Quadratic vector field, marginally stable linearization
Input/output gain analysis certifies that for all signals r and d satisfying , error e satisfies for all t.
There are particular r and d satisfying
causing e to achieve at some time t. -2 -1 1 2 E1 E2

25. Adaptive System: L2 gain example
plant
Reference model
Adaptive control - 5 4 3 g 2 1 2 4 6 R R

26. Wrapup/Perspective Tools that handle (cubic, in x, vector field)
15 states, 3 parameters, unmodeled dynamics, analyze with ∂(V)=2
7 states, 3 parameters, unmodeled dynamics, analyze with ∂(V)=4
Certified answers, not clear that these are appropriate for design choices
Sproc/SOS/DIE more quantitative than linearization
Linearized analysis: quadratic storage functions, infinitesimal sublevel sets
SOS/S-procedure always works
Can this scale up to large, complex systems analysis (e.g., adaptive flight controls) where “certificates” are desired?
Proofs of behavior with certificates
Extensive simulation
Probabilistic guarantees?
and linearized analysis