Quantitative Local Analysis of Nonlinear Systems

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Quantitative Local Analysis of Nonlinear Systems

• NASA NRA Grant/Cooperative Agreement NNX08AC80A
• Analytical Validation Tools for Safety Critical
  Systems
• Dr. Christine Belcastro, Technical Monitor,
  01/01/2008-12/31/2010
• AFOSR FA9550-05-1-0266
• Analysis tools for Certification of Flight
  Control Laws
• 05/05/2005-04/30/2008
• Colleagues
• Univ of Minnesota Pete Seiler, Abhijit
  Chakraborty, Gary Balas
• UC Berkeley Ufuk Topcu, Erin Summers, Tim
  Wheeler, Andy Packard
• Barron Associates Alec Bateman
• www.cds.caltech.edu/utopcu/LangleyWorkshop.html

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• www.cds.caltech.edu/utopcu/LangleyWorkshop.html

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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, Senior systems
engineer, large US corporation.
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Why psychological?

• VV needs a conclusion about physical system using
  model-based analysis leap-of-faith arises from
• 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
• Goal
• Make the leap smaller with quantitative nonlinear
  analysis

while addressing these
Improve these
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Role of linearized analysis

• Linear analysis provides a quick answer to a
  related, but different question
• Q How much gain and time-delay variation can be
  accommodated without undue performance
  degradation?
• A (answers a different question) Heres a
  scatter plot of margins at 1000 equilbrium trim
  conditions throughout envelope
• Why does linear analysis have impact in nonlinear
  problems?
• Domain-specific expertise exists to interpret
  linear analysis and assess relevance
• Speed, scalable Fast, defensible answers on
  high-dimensional systems
• Extend validity of the linearized analysis
• Infinitesimal ? local (with certified estimates)
• Address uncertainty

Heres a scatter plot of guaranteed
region-of-attraction estimates, in the presence
of 40 unmodeled dynamics at plant input, and 3
parametric variations, at 1000 trim conditions
throughout the envelope
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Overview

• Numerical tools to quantify/certify dynamic
  behavior
• Locally, near equilibrium points
• Analysis considered
• Region-of-attraction, input/output gain,
  reachability, establishing local 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

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Nonlinear Analysis

• Autonomous dynamics
• equilibrium point
• uncertain initial condition,
• Question do all solutions converge to
• Driven dynamics
• equilibrium point
• uncertain inputs, ,
• Question how large can get?
• Uncertain dynamics
• Unknown, constant parameters,
• Unmodeled dynamics
• Same questions

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Region-of-Attraction and Reachability

• Dynamics, equilibrium point
• p Analyst-defined function whose
  (well-understood) sub-level sets are to be in
  region-of-attraction

Given a differential equation
and a positive definite function p, how large
can get, knowing
Local DIE Conditions on
Conclusion on ODE
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Solution Approach

• S-procedure to (conservatively) enforce set
  containments in Rn
• Sum-of-squares to (conservatively) enforce
  nonnegativity of h Rn ? R
• Easy (semidefinite program) to check if a given
  polynomial is SOS
• Apply S-procedure/SOS to Analysis set-containment
  conditions. For (e.g.) reachability, minimize ß
  (R fixed, by choice of si and V) such that
• SDP iteration Initialize V, then
• Optimize objective by changing S-procedure
  multipliers
• Recenter V
• Iterate on (a) and (b)
• Initialization of V is important (in a
  complicated fashion) for the iteration to work
• Simulation of system dynamics yields convex
  constraints which contain all (if any) feasible
  Lyapunov function candidates

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Quantitative improvement on linearized analysis

• Consider dynamics
• where matrix A is Hurwitz, and
• function f23 consists of 2nd and 3rd degree
  polynomials, f23(0)0

These SOS/S-procedure formulations are always
feasible using quadratic V
A nonempty region-of-attraction is certified

• Consider dynamics
• where matrix A is Hurwitz, and
• f2, g2, h2 quadratic, f3 cubic
• with f2(0,0)f3(0)h2(0)0, and

For some Rgt0,

• Consider dynamics
• where matrix A is Hurwitz, and
• functions b bilinear, q quadratic

For some Rgt0,
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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.,
• Solution approaches SOS conditions to verify
  containments
• Parametrize V, parametrize multipliers, solve
• Ad-hoc iterative, based on linear SDPs
• Bilinear SDP solvers
• Behavior Initial point can have big effect on
  end result, e.g.,
• Unable to reach a feasible point
• Convergence to local optimum (or less)

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ROA Simulations constrain suitable V

• Consider a simpler question. Fix ß, 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
• Other constraints???

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Convex 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
• V1 on convergent trajectories
• V0 on all trajectories
• V decreasing on convergent trajectories
• Quad(V) is a Lyapunov function for Linear(f)
• V1 on divergent trajectories
• If convex constraints yield empty set, then V
  parametrization cannot certify

Basis functions, eg., all degree 4 Hermite
polynomials
Sample this set to get candidate V
HitRun (Smith, 1984, Lovasz, 1999, Tempo,
Calafiore, Dabbene
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Uncertain Systems Parameter-Independent V

• Start with 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, d appears only
  implicitly through the vertices
• SOS analysis is only in x variables
• Simulations are incorporated as before (vary
  initial condition and d)
• 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 ?
Solve separately
?1
?2
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Much better BB 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
• Computation yields a binary tree
• decomposes parameter space
• certificates at each leaf

BTree(k).Analysis Analysis.ParameterDomai
n Analysis.VertexDynamics
Analysis.LyapunovCertificate
Analysis.SOSCertificates
Analysis.CertifiedVolume BTree(k).Children

• Nonconvex parameter-space, and/or coupled
  parameters
• cover with union of polytopes, and refine

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4-state aircraft example w/uncertainty

• Treat as 3 parameters
• Affine dependence
• 2-dimensional manifold in R3
• Cover with polytope in R3
• Solve

• Aircraft Short period longitudinal model, pitch
  axis, with 1-state linear controller
• Spherical shape factor
• 9-processor Branch--Bound
• Divide worst region into 9, improve polytope cover

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Unmodeled dynamics Local small-gain theorem
M
Local, gain constraint (1) on

• Implies Starting from x(0)0,
• for all

? causal, globally stable,
also satisfies DIE

• This gives

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Unmodeled dynamics Local small-gain theorem
Local, gain constraint (1) on
M
? causal, globally stable,

• Then

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4-state aircraft example w/uncertainty
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Adaptive System reachability example analysis

• Model-reference adaptive systems
• Example 2-state P, 2-state ref. model, 3
  adaptive parameters
• Insert additional disturbance (d)
• 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
Reachability analysis certifies that for all
(r,d) with then for all
t, There are particular r and d
satisfying causing e to achieve
at some time t.
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F/A-18 Falling Leaf Mode

• The US Navy has lost many F/A-18 A/B/C/D Hornet
  aircraft due to an out-of-control flight
  departure phenomenon described as the falling
  leaf mode

• Can require 15,000-20,000 ft to recover
• Administrative action by NAVAIR to prevent
  further losses
• Revised control law implemented, deployed in
  2003/4, F/A-18E/F
• uses ailerons to damp sideslip

Heller, David and Holmberg, Falling Leaf Motion
Suppression in the F/A-18 Hornet with Revised
Flight Control Software," AIAA-2004-542, 42nd
AIAA Aerospace Sciences Meeting, Jan 2004, Reno,
NV.
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Baseline/Revised Control Architecture (simplified)
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Baseline vs Revised Analysis

• Is revised better? Yes, several years service
  confirm but can this be ascertained with a
  model-based validation?
• Recall that Baseline underwent validation, yet
  had problems.
• Linearized Analysis at equilibrium and several
  steady turn rates
• Classical loop-at-a-time margins
• Disk margin analysis (Nichols)
• Multivariable input disk-margin
• Diagonal input multiplicative uncertainty
• Full-block input multiplicative uncertainty
• Parametric stability margin (µ ) using physically
  motivated uncertainty in 8 aero coefficients
• Conclusion Both designs have excellent (and
  nearly identical) linearized robustness margins
  trimmed across envelope

Chakraborty , Seiler and Balas, Applications of
Linear and Nonlinear Robustness Analysis
Techniques to the F/A-18 Control Laws, AIAA
Guidance, Navigation and Control Conference,
Chicago IL, August 2009.
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Baseline vs Revised Beyond Linearized Analysis

• Perform region-of-attraction estimate as
  described
• Unfortunately, closed-loop models too complex
  (high dynamic order) for direct approach, at this
  time.
• Model approximation
• reduced state dimension (domain-specific
  simplifications)
• polynomial approximation of closed-loop dynamic
  models

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ROA Results

• Ellipsoidal shape factor, aligned w/ states,
  appropriated scaled
• 5 hours for quartic Lyapunov function certificate
• 100 hours for divergent sims with small initial
  conditions

Chakraborty , Seiler and Balas, Applications of
Linear and Nonlinear Robustness Analysis
Techniques to the F/A-18 Control Laws, AIAA
Guidance, Navigation and Control Conference,
Chicago IL, August 2009.
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Wrapup/Perspective

• Tools (Multipoly, SOSOPT, SeDuMi) 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
• 4 states, 3 parameters, unmodeled dynamics,
  analyze with ?(V)6-8
• Certified answers, however, 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
• Work to scale up to large, complex systems
  analysis (e.g., adaptive flight controls) where
  certificates are desired.

Proofs of behavior with certificates
Extensive simulation
and linearized analysis
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Decomposition for high-order Heterogeneous Systems

• Interconnection of locally stable systems (w/
  Summers)
• M is constant matrix
• Associated with each Ni
• (offset) Stable, linear Gi
• (weight) Stable, linear, min-phase Wi
• The system has local
  L2-gain 1, certified as presented. For
  low-order Ni, coupled with low-order G and W,
  this is done with high-degree V
• (Linear) robustness analysis on an
  interconnection involving M, G, and W-1 yields
  conditions on d, under which gain from d to e is
  bounded
• Hierarchical - easy to include WM, GM, and
  bound local gain of

• Poor-mans IQC theory for locally-stable
  interconnections
• Combinatorial number of ways to split original
  system
• Infinite choices for G and W
• Elements must be stable, so reject decompositions
  based on linearization
• A possible route to answering some questions on
  medium-order systems

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Uncertain Model Invalidation Analysis

• Given time-series data for a collection of
  experiments, with selected features and simple
  measurement uncertainty descriptions

Task prove that regardless of the values chosen
for the parameters, the model below cannot
account for the observed data,
where
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Generalization of covering manifold

• Given
• polynomial p(d) 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
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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 2mk easily computed vertices.

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Sum-of-Squares

• Sum-of-squares (SOS) decompositions (Parrilo)
• certify nonnegativity, and
• (with S-procedure) certify set containment
  conditions
• A polynomial f, in n real-variables is SOS if it
  can be expressed as a sum-of-squares of other
  polys,
• SDP decides SOS For f with degree 2d
• Each Mi is ss, where

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(s,q) dependence on n and 2d
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(s,q) dependence on n and 2d
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Region-of-attraction 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 (ßcert8.6)
• Fully quartic (quadratic cubic quartic)
• ßcert15.3
• Other approaches have deficiencies
• Directly use commercial BMI solver (PENBMI)
• ßcert15.2, but
• 6 hours

Certified set of convergent initial conditions
Disk in 4-d state space, centered at equilibrium
point