Local Gain Analysis of Nonlinear Systems Support from AFOSR FA9550-05-1-0266, 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 http://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 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 www.penopt.com)

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

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…

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

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.

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

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

Reachability of with inputs If then Conditions on Conclusion on ODE Simple Psatz certification BMI

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

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.

Same example, with a Lower bound on Reachability 16 14 12 Upper Bound 10 b Lower Bnd 8 6 Linearized 4 2 2 4 6 8 10 2 R

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

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

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

Effect of Refinement Refined Upper Bound Upper Bound Lower Bnd 16 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

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. 177-185, 1992

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

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

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