1. Local Gain Analysis of Nonlinear Systems
Support from AFOSR
FA , April 05-November 06
Authors
Weehong Tan, Tim Wheeler, Andy Packard
Mechanical Engineering, UC Berkeley
Acknowledgements
Thanks to Ufuk Topcu, Gary Balas and Pete Seiler; PENOPT
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. Quantitative Nonlinear Analysis
System properties
Region of attraction estimation
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

3. 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

4. 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…

5. 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

6. 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.

7. 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

8. Reachability of with inputs
Given a differential equation
and a positive definite function p, how large can get, knowing
Special Case:
Controllability grammian gives
where

9. Reachability of with inputsIf
then
Conditions on
Conclusion on ODE
Simple Psatz certification
BMI

10. Reachability of with inputs
Example: 16
Decision variables 14 12 10
Upper Bound b 8 6
Linearized 4 2 2 4 6 8 10 R 2

11. Reachability of with inputs
Choose T:
Conditions for stationarity
Tierno, et.al, 1996
repeat
adjust scalar so
Note: If f is linear, and p is a p.d. quadratic form, then the iteration is the correct power iteration for the maximum.

12. Same example, with a Lower bound on Reachability16 14 12
Upper Bound 10 b
Lower Bnd 8 6
Linearized 4 2 2 4 6 8 10 2 R

13. Upper Bound: Refinement
Replace
with
Then
generally, hk<1 will work
generally, greater than R2

14. Upper Bound Refinement: Derivation
Suppose from x=0, w leads to
at some t>0.
Then
Equivalently:

15. Solving the upper bound refinement
Replace
with
Hold V fixed from first solution, use m partitions, solve m SDPs
(enforced by)

16. Effect of Refinement Refined Upper Bound Upper Bound Lower Bnd16 14
Refined Upper Bound 12
Upper Bound 10 b
Lower Bnd 8 6
Linearized 4
Using worst-case input from linear analysis 2 2 4 6 8 10 R 2

17. gain: Adaptive control example
Plant: with unknown (=2)
Controller:
Properties: Global convergence x1 to 0, x2 to θ-dependent equilibrium point, and (in this case)
Add input disturbance, compute gain from C P
How does adaptation gain affect this?
“Adaptive nonlinear control without overparametrization,” Krstic, Kanellakopoulos , Kokotovic, Systems and Control Letters, vol. 19, pp , 1992

18. elementary sufficient condition
gain of If
then
elementary sufficient condition
Iteration (as before) for stationary points, to yield lower bounds

19. Adaptive control C P Compute/Bound
from equilibrium, for two values of adaptation gain, Γ=1, 4.
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
H∞ norm of the linearization
For small w, large adaptation gain gives better worst-case disturbance attenuation.
But for large w, the situation is reversed…
Trend implied by linearized analysis invalid for large inputs.
Γ=1
Γ=4
0.25

20. Problems, difficulties, risks
Dimensionality:
For general problems, it seems unlikely to move beyond cubic vector fields and quartic V. These result in “tolerable” bilinear SDPs for state dimension < 10.
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
BMI nature of local analysis