Quantitative Local Analysis of Nonlinear Systems - University of ...

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List of Symbols R R n R n×m Z Z + C 1 M ≽ 0 (x ≽ 0) The real numbers Real n-vectors Real n-by-m matrices The ring of integers Positive integers The set of continuously differentiable real valued functions on R n M symmetric and is positive semidefinite (for x ∈ R n , entries of x are non-negative) M ≻ 0 (x ≻ 0) M is symmetric and positive definite (for x ∈ R n , entries of x are positive) R[x] Σ[x] ∂(π) Ω f,η The set of polynomials (of certain finite degree) in x with real coefficients The set of sum-of-squares polynomials in R[x] The the degree of π ∈ R[x] The η-sublevel set of f, i.e., for η ∈ R and f : R n → R Ω f,γ := {x ∈ R n : f(x) ≤ η}, xi
Acknowledgements It has been a short stay at Berkeley and I have enjoyed every minute of it. I would like to take this opportunity to thank the following people. I would like to express my gratitude to Andy Packard for making my studies at Berkeley possible and being a role model with his passion for research. I want to extend heartfelt thanks to Laurent El Ghaoui for long discussions on optimization and his support and guidance. Special thanks to Kameshwar Poolla for being a great mentor and his interest in my academic progression. I also want to thank Pete Seiler for valuable discussions and for providing me a summer internship opportunity at the Honeywell Labs. Thanks to all at the BCCI for their friendship, support, and creating a pleasant working environment. Finally, I would like to acknowledge the financial support from the Air Force Office of Scientific Research under grant/contract number FA9550-05-1-0266 and thank the program managers Sharon Heise and Scott Wells. xii
Page 1 and 2: Quantitative Local Analysis of Nonl
Page 3 and 4: Quantitative Local Analysis of Nonl
Page 5 and 6: technique is adapted both to region
Page 7 and 8: Contents Contents ii List of Figure
Page 9 and 10: 5.1 Setup and Estimation of the Rob
Page 11 and 12: 4.1 Invariant subsets of ROA report
Page 13 and 14: 6.4 Upper bounds for ∂(V ) = 2 (w
Page 15: 4.4 Optimal values of β in the pro
Page 19 and 20: specifically sum-of-squares program
Page 21 and 22: ections. Finally, it is shown that,
Page 23 and 24: 1.2 Summary of Examples Listed belo
Page 25 and 26: Chapter 2 Background The goal in th
Page 27 and 28: Affine SDPs are convex optimization
Page 29 and 30: and LP constraints, namely for c
Page 31 and 32: Theorem 2.2.1. A polynomial p, in x
Page 33 and 34: containment constraint {x ∈ R n :
Page 35 and 36: ⊳ Example 2.3.1. We now give a pr
Page 37 and 38: The main difference between these c
Page 39 and 40: Chapter 3 Simulation-Aided Region-o
Page 41 and 42: of decision variable space, so that
Page 43 and 44: Using simple generalizations of the
Page 45 and 46: where ∇V (c(t))f(c(t)) < 0, l 1 (
Page 47 and 48: It is well-known that Y lin is conv
Page 49 and 50: Φ T 3 α = b 3 Φ T 2 α = b 2 α
Page 51 and 52: definite l 2 ∈ R[x], define γ
Page 53 and 54: Table 3.1. Parameters used in and r
Page 55 and 56: 3 2 1 0 −1 −2 −3 −2 −1 0
Page 57 and 58: Table 3.2. Volume ratios for (E 1 )
Page 59 and 60: 3.3.3 Controlled Short Period Aircr
Page 61 and 62: 1.5 1 0.5 x 3 0 −0.5 −1 −1.5
Page 63 and 64: sampling some region and imposing
Page 65 and 66: vi. Further optimize initializing C
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where H can be chosen positive defi
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3.6 Appendix Problems 3.2.1 and 3.2
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and p be convex. Define β ∗ a :=
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SOS relaxations [63, 58, 59, 57, 73
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Note that entries of ϕ do not have
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introduces extra complexity [35] an
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δ(= ϕ + α) as long as ϕ l (x)
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and V (0) = 0 are sufficient condit
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• For each δ ∈ E ∆ , compute
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The result is straightforward to ex
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1.5 1 0.5 x 2 0 −0.5 −1 −1.5
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3 2 1 x 2 0 −1 −2 −3 −2 −
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Table 4.5. Optimal value of β in t
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4.7 Appendix Parameters for the unc
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eter space) and covering the graph
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functions, and δ takes values in a
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(SDP), with 3 “types” of decisi
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5.2 Polynomial Parametric Uncertain
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and max σ li ∈S li ,a l ,b l Vol
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5.3 Branch-and-Bound Type Refinemen
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- compute L D k and update U D k;
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Finally, the following continuity r
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1 β lp β nc β 0.5 0 1 lower boun
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15 β10 "quasi−upper" and lower b
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5.5 Chapter Summary We extended the
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Chapter 6 Reachability and Local Ga
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then G R 2 ⊆ Ω V,R 2. ⊳ Given
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where V poly and S’s are prescrib
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Lower bound for the reachable set:
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Consequently, this strengthens the
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Of course, this set could be empty,
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ii. If all trajectories stay in Ω
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Given K current , solve: [1/R∗, 2
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20 15 β 10 5 0 0 5 10 15 20 R 2 Fi
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5 4 γ 3 2 1 0 1 2 3 4 5 6 R Figure
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Proof. Let ẋ 2 (t) = Ax 2 (t) + B
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Proof. Define S := V + Q. d dt S(x
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✲ ẋ 4 = A c x 4 + B c y v = C c
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Chapter 7 Conclusions This thesis i
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ehavior of the actual system. The i
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[8] H. H. Bauschke and J. M. Borwei
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editors, System Theory: Modeling, A
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[45] A. Paice and F. Wirth. Robustn
Page 153 and 154:
[64] J. E. Tierno, R. M. Murray, J.
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