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

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• Reachability analysis for Pendubot dynamics §6.4 • L 2 → L 2 gain analysis for a system with adaptive controller §6.4 • Robust region-of-attraction analysis of controlled short period aircraft dynamics with unmodeled dynamics §6.5.1 7
Chapter 2 Background The goal in the following chapters is to develop computational tools for estimating certain robustness and performance measures (e.g. regions-of-attraction and input-output gains) for nonlinear systems. The general strategy for computing these measures consists of three main steps: i. Characterize these measures using Lyapunov-type functions that satisfy certain conditions which can be translated to set containment (or set emptiness) questions. ii. Obtain “S-procedure” type sufficient conditions for the set containment constraints. iii. Search for Lyapunov-type certificates and S-procedure multipliers by solving numerical optimization (or feasibility) problems. In the following sections of this chapter, we present a summary of background material needed to carry out this plan. 8
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: 1.2 Summary of Examples Listed belo
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
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
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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
Page 85 and 86:
The result is straightforward to ex
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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
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(SDP), with 3 “types” of decisi
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5.2 Polynomial Parametric Uncertain
Page 103 and 104:
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
Page 111 and 112:
1 β lp β nc β 0.5 0 1 lower boun
Page 113 and 114:
15 β10 "quasi−upper" and lower b
Page 115 and 116:
5.5 Chapter Summary We extended the
Page 117 and 118:
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
Page 127 and 128:
Of course, this set could be empty,
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ii. If all trajectories stay in Ω
Page 131 and 132:
Given K current , solve: [1/R∗, 2
Page 133 and 134:
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
Page 137 and 138:
Proof. Let ẋ 2 (t) = Ax 2 (t) + B
Page 139 and 140:
Proof. Define S := V + Q. d dt S(x
Page 141 and 142:
✲ ẋ 4 = A c x 4 + B c y v = C c
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Chapter 7 Conclusions This thesis i
Page 145 and 146:
ehavior of the actual system. The i
Page 147 and 148:
[8] H. H. Bauschke and J. M. Borwei
Page 149 and 150:
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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