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

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Simulation and Lyapunov function generation (SimLF G) algorithm: Given positive definite convex p ∈ R[x], a vector of polynomials z(x) and constants β SIM , N conv , N V , β shrink ∈ (0, 1), and empty sets C and D, set γ = 1, N more = N conv , N div = 0. i. Integrate (3.1) from N more initial conditions in the set {x ∈ R n : p(x) = β SIM }. ii. If there is no diverging trajectory, add the trajectories to C and go to (iii). Otherwise, add the divergent trajectories to D and the convergent trajectories to C, let N d denote the number of diverging trajectories found in the last run of (i) and set N div to N div + N d . Set β SIM to the minimum of β shrink β SIM and the minimum value of p along the diverging trajectories. Set N more to N more − N d , and go to (i). iii. At this point C has N conv elements. For each i = 1, . . . , N conv , let τ i satisfy c i (τ) ∈ Ω p,βSIM for all τ ≥ τ i . Eliminate times in T i that are less than τ i . iv. Find a feasible point for (3.6), (3.11), and (3.12). If (3.6), (3.11), and (3.12) are infeasible, set β SIM = β shrink β SIM , and go to (iii). Otherwise, go to (v). v. Generate N V Lyapunov function candidates using H&R algorithm, and return β SIM and Lyapunov function candidates. ⊳ The suitability of a Lyapunov function candidate is assessed by solving two optimization problems. Both problems require bisection and each bisection step involves a linear SOS problem. Alternative linear formulations appear in section 3.6. These do not require bisection, but generally involve higher degree polynomial expressions. Optimization Problem 3.2.1. Given V ∈ R[x] (from SimLF G algorithm) and positive 33
definite l 2 ∈ R[x], define γ ∗ L := max γ subject to s 2 , s 3 ∈ Σ[x], γ > 0, γ>0,s 2 ∈S 2 ,s 3 ∈S 3 − [(γ − V )s 2 + ∇V fs 3 + l 2 ] ∈ Σ[x]. (3.14) ⊳ If Problem 3.2.1 is feasible, then γL ∗ > 0 and define Optimization Problem 3.2.2. Given V ∈ R[x], p ∈ R[x], and γ ∗ L , solve β ∗ L := max β subject to s 1 ∈ Σ[x], β > 0, β>0,s 1 ∈S 1 − [(β − p)s 1 − (V − γ ∗ L )] ∈ Σ[x]. (3.15) ⊳ Although γ ∗ L and β∗ L depend on S 1, S 2 , and S 3 , this is not explicitly notated. Assuming Problem 3.2.1 is feasible, it is true that Ω p,β ∗ L \{0} ⊆ Ω V,γ ∗ L \{0} ⊂ {x ∈ R n : ∇V (x)f(x) < 0}, so V certifies that Ω p,β ∗ L ⊂ R 0 . Solutions to Problems 3.2.1 and 3.2.2 provide a feasible point for the problem in (3.9). This feasible point can be further improved by solving the problem in (3.9) using iterative coordinate-wise affine optimization schemes, one of which is given next. Coordinate-wise optimization (CW Opt) algorithm: Given V ∈ R[x], positive definite l 1 , l 2 ∈ R[x], a constant ε iter , and maximum number of iterations N iter , set k = 0 i. Solve Problems 3.2.1 and 3.2.2. ii. Given s 1 , s 2 , s 3 , and γL ∗ from step (i), set γ in (3.7)-(3.8) to γ∗ L , solve (3.9) for V and β, and set β ∗ L = β∗ B . 34
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 and 16: 4.4 Optimal values of β in the pro
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: Φ T 3 α = b 3 Φ T 2 α = b 2 α
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
Page 67 and 68: where H can be chosen positive defi
Page 69 and 70: 3.6 Appendix Problems 3.2.1 and 3.2
Page 71 and 72: and p be convex. Define β ∗ a :=
Page 73 and 74: SOS relaxations [63, 58, 59, 57, 73
Page 75 and 76: Note that entries of ϕ do not have
Page 77 and 78: introduces extra complexity [35] an
Page 79 and 80: δ(= ϕ + α) as long as ϕ l (x)
Page 81 and 82: and V (0) = 0 are sufficient condit
Page 83 and 84: • For each δ ∈ E ∆ , compute
Page 85 and 86: The result is straightforward to ex
Page 87 and 88: 1.5 1 0.5 x 2 0 −0.5 −1 −1.5
Page 89 and 90: 3 2 1 x 2 0 −1 −2 −3 −2 −
Page 91 and 92: Table 4.5. Optimal value of β in t
Page 93 and 94: 4.7 Appendix Parameters for the unc
Page 95 and 96: eter space) and covering the graph
Page 97 and 98: functions, and δ takes values in a
Page 99 and 100: (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
Page 105 and 106:
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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