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

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specific expression. Similarly, we do not provide certifying Lyapunov functions and multipliers. All missing data and results for all examples in this thesis are available at http://jagger.me.berkeley.edu/~utopcu/dissertation. 2.5 Chapter Summary We presented a summary of background material focusing on the aspects needed for the development in the subsequent chapters. Mainly, this material coupled with Lyapunov-type theorems introduced in respective chapters will be used to translate system analysis questions to numerical optimization problems. A brief overview of semidefinite programming centered on distinguishing properties of affine and bilinear semidefinite programs was followed by the introduction of sum-of-squares polynomials and sum-of-squares programming. Finally, a generalization of the S-procedure, a central tool for handling set containment conditions, and its link to the Positivstellensatz were discussed. 21
Chapter 3 Simulation-Aided Region-of-Attraction Analysis For a dynamical system, the region-of-attraction (ROA) of a locally asymptotically stable equilibrium point is an invariant set such that all trajectories emanating from points in this set converge to the equilibrium point. For nonlinear dynamics, research has focused on determining invariant subsets of the ROA because a closed form characterization of the exact ROA may be too complicated. This potential complication is avoided because of several reasons: (i) Usually there is no systematic (numerical) procedure for computing the exact shape of the ROA. (ii) A complicated characterization of the ROA has limited value for post-analysis purposes. Among all other methods those based on Lyapunov functions are dominant in the literature [24, 27, 75, 21, 20, 46, 58, 59, 32, 63, 62]. These methods compute a Lyapunov function as a local stability certificate and sublevel sets of this Lyapunov function, in which the function decreases along the flow, provide invariant subsets of the ROA. 22
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: The main difference between these c
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
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
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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
Page 105 and 106:
5.3 Branch-and-Bound Type Refinemen
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- compute L D k and update U D k;
Page 109 and 110:
Finally, the following continuity r
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1 β lp β nc β 0.5 0 1 lower boun
Page 113 and 114:
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
Page 151 and 152:
[45] A. Paice and F. Wirth. Robustn
Page 153 and 154:
[64] J. E. Tierno, R. M. Murray, J.
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