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

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List of Tables 3.1 Parameters used in and results of SimLF G and CW Opt algorithms. . . . . 36 3.2 Volume ratios for (E 1 )-(E 7 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3 Results of SimLF G and CW Opt algorithms. Upper bounds are established by a separate run of SimLF G algorithm with N conv = 3000. The upper bound for ∂(V ) = 4 is by a divergent trajectory whereas as the upper bound is by the infeasibility of (3.6), (3.11), and (3.12) for the given β value. Representative computation times are on 2.0 GHz desktop PC. . . . . . . . . . 43 4.1 N SDP (left columns) and N decision (right columns) for different values of n and 2d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Number of decision variables in (4.13) (top entry in each cell of the table) and the number of decision variables in (4.15) (bottom entry in each cell of the table) for ∂(V ) = 2, ∂(s 2δ ) = 2, and ∂(s 3δ ) = 0. . . . . . . . . . . . . . 67 4.3 Optimal values of β in the problem (4.13) with different values of µ and ∂(V ) = 2 and 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ix
4.4 Optimal values of β in the problem (4.13) with different values of µ and ∂(V ) = 4 and 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.5 Optimal value of β in the first step, β sample , and β subopt with µ for ∂(V ) = 2 and 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.1 Computed (sub)optimal values of β with p(x) = x T x (with ∂(V ) = 2 / ∂(V ) = 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 x
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: 6.4 Upper bounds for ∂(V ) = 2 (w
Page 17 and 18: Acknowledgements It has been a shor
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
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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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