Stability Region Analysis using composite Lyapunov functions and bilinear SOS programming

Support from AFOSRFA9550-05-1-0266, April 05-November 06

AuthorsWeehong Tan, Andy PackardMechanical Engineering, UC Berkeley

AcknowledgementsThanks to Ufuk Topcu, Gary Balas and Pete Seiler; PENOPT

Websitehttp://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.

Quantitative Nonlinear Analysis

Initial focus–Region of attraction estimation–Attractive invariant sets– induced norms– 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 www.penopt.com)

22 LL LL2

Estimating Region of Attraction

Dynamics, equilibrium point

User-defined function p, whose sub-level sets are to be in region-of-attraction

By choice of positive-definite V, maximize so that

0fdx

dV

1V

1p

2p

3p

0)(),( xfxfx

x xROA )(:: xpxP

01)(,: f

dx

dVx:xVxxx

1)(: x:V(x)xpx

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:

00: fVVV with

100: VfVVV on with

)(42.0)(58.0)(1.0)(

4.695.1916)()(

21

22

4321

xVxVxVxxV

xxxxVxxf

c

Estimating Region of Attraction

Dynamics, equilibrium point

User-defined function p, whose sub-level sets are to be in region-of-attraction

By choice of positive-definite V, maximize so that

0fdx

dV

1V

1p

2p

3p

0)(),( xfxfx

x xROA )(:: xpxP

01)(,: f

dx

dVx:xVxxx

1)(: x:V(x)xpx

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,

Notation

set of all sum-of-square polynomials in n variables

set of all polynomials in n variables

p

jjgf

1

2

n

nP

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. x12x2

2 = (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 a SDP…

.],,,,,,,1[ where

such that 0

2121Td

nn

Tn

xxxxxxz

MzzfMf

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.

q

iii

qn MMf

10 0 that suchR

d

dn

d

dn

d

dnq

d

dns

2

2

2

12

Semidefinite program: feasibility

(s,q) dependence on n and 2d

2d

n2 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

d

dn

d

dn

d

dnq

d

dns

2

2

2

12

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.

mq

iii

mq

n

m

kkk

m

MM

ff

10

10

0

with

with

R

R

mfff ,,, 10

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

n

m

kkk

m

kkk

m

kkk hhggff

10

10

10

m

m

m

hhh

ggg

fff

,,,

,,,

,,,

10

10

10

mm RR ,

Psatz

Given: polynomials

Goal: Decide if

the set is empty.

Φ is empty if and only if

such that

qpm hhhgggfff ,,,,,,, 111 IMP ,

qpm hhggff ,,,,,,,, 111

0)(,,0)(

,0)(,,0)(,0)(,,0)(:

:

1

1

1

xhxhxgxgxfxfx

q

p

m

02 hgf

RI kkkq pphhh :,,1 miiiim ffbsbsff ,,,:,, 11 MP

pip Zggg i :,,1 M

n

n

n

n

lfVssVVsp

lV

sssxVV

298

6

1

986

11

,,,0)(,max

to subjectover

Region of Attraction

By choice of positive-definite V, maximize so that

0fdx

dV

1V

1p

2p

3p

x

01)(,: fVx:xVxxx

1)(: x:V(x)xpx

Simple Psatz:“small” positive

definite functions

Products of decision variables

BMIsPENBMI from

PENOPT

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 formulation we wrote yield this?–Can the BMI solver find this solution?

.1: VVVBxxxV

1: Bxxx:Rxxx

Rxxxp T)(

1: BxxxxBxxxx0ROA nth order system

cubic vector fieldknown ROA

max

121

21 BRRmaxmax

Yes

Basically, Yes100’s of random examples, n=2-8; 3 restarts of PENBMI, always successful

Example: Van der Pol: ROA

Classical 2-d system

22112

21

1 xxxxxx

Features:–Unstable limit cycle around origin –One equilibrium point: stable, at origin–Here, we use an elliptical shape factor

Except for nV=4 case, the results are comparable to

Papachristodoulou (2005) and Wloszek (2003)

1666

574

132

#decvarsV

12 V

11 V

Region of Attraction: pointwise-max

If V1 and V2 are positive definite, and

and

Then

proves asymptotic stability of on

)()(

1)(on0)(

12

11

xVxV

xVxf

dx

dV

)()(

1)(on0)(

21

22

xVxV

xVxf

dx

dV

)(),(max:)( 21 xVxVxV

)(xfx 1)(: xVx

1V

n

q

ijjjiijiiii

nii

ni

nijiiiii

VVslfVssV

VsplV

ssssVV

,1

0298

6

1

0986

1

1

,,,,0)0(,max

to subjectover

Region of Attraction with pointwise-max

Use Psatz to get a sufficient condition for

using V of the form

01)(,0: fVx:xVxx

1)(: x:V(x)xpx

)(,),(),(max:)( 21 xVxVxVxV qi

ROA with Pointwise-Max Lyapunov functions

3381666

120574

38132

21 qqVi

Original (single V)Composite (2 Vi)

Pointwise max of 6th degree V1,V2

121 1,1 VVV

11 V12 V

AI ,1

BI ,1

01 V

121 1,1 VVV

01 V

02 V

02 V

Is V2 , by itself, a decent Lyapunov function?–Sub-Level set looks similar to result,

–But, derivative on sublevel set is not negative

1)(: 2 xVx

Different shape factor

Nearly the same results.

ROA: 3rd order example

Example (from Davison, Kurak):

322113

32

21

915.01915.0 xxxxx

xx

xx

xxxp T

4.135.80.3

5.88.201.8

0.31.85.12

)(

Solutions diverge from these initial conditions,i.e these initial conditions are not in the ROA

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–Time-to-double

BMI nature of local analysis