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

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5.4 Lower bounds for β∆ ∗ with ∂(V ) = 2 (green solid curve with “×” marker) and ∂(V ) = 4 (blue solid curve with “⋄” marker) and β nc (red solid curve with “◦” marker) computed at the centers of the cells generated by the B&B Algorithm for the ∂(V ) = 4 run. Dashed curves are for (computed values of) β {δ} where δ is the center of the cell with the smallest lower bound at the corresponding step of the B&B refinement procedure for ∂(V ) = 2 (green curve with “×” marker) and ∂(V ) = 4 (blue curve with “⋄” marker). . . . . 96 5.5 Final partition generated by the B&B algorithm for the ∂(V ) = 4 run. . . . 97 5.6 Controlled short period aircraft dynamics with uncertain first order linear time-invariant dynamics (δ p := (δ 1 , δ 2 )). . . . . . . . . . . . . . . . . . . . . 97 6.1 Bounds on reachable sets due to disturbance w with ‖w‖ 2 2 ≤ R2 for Example 1 without delay (blue curve with dots: before refinement, green curve with ×: after refinement, red curve with ⋄: lower bound). . . . . . . . . . . . . . 115 6.2 Bounds on reachable sets due to disturbance w with ‖w‖ 2 2 ≤ R2 for Example 2 (blue curve with dots: before refinement, green curve with ×: after refinement, red curve with ⋄: lower bound, and black circles: failed PENBMI runs). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.3 Bounds on reachable sets due to disturbance w with ‖w‖ 2 2 ≤ R2 for Example 2 (blue curve with dots: before refinement, green curve with ×: after refinement, red curve with ⋄: lower bound, and black circles: failed PENBMI runs). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 vii
6.4 Upper bounds for ∂(V ) = 2 (with ⋄) and ∂(V ) = 4 (with ×) before the refinement (blue curves) and after the refinement (green curves) along with the lower bounds (red curve). . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.5 Feedback interconnection of ∆ and M. . . . . . . . . . . . . . . . . . . . . . 119 6.6 Controlled short period aircraft dynamics with unmodeled dynamics (δ p := (δ 1 , δ 2 )). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 viii
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: 4.1 Invariant subsets of ROA report
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 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
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[64] J. E. Tierno, R. M. Murray, J.
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