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the reverse implication also holds: If there is no λ such that Q(λ) ≽ 0, then p is not SOS [49]. ⊳ Theorem 2.2.2. [48, 49] The existence of a SOS decomposition of a polynomial in n variables of degree 2d can be decided by solving a feasibility SDP. ⊳ A useful corollary of Theorem 2.2.2 is that if the polynomial p contains decision variables, checking whether p is SOS for some choice of these decision variables is also a SDP. More precisely, if p is a polynomial in x parameterized by α ∈ R m , then the search for α such that p(x, α) ∈ Σ[x] is a SDP feasibility problem. If p is affine in α, then this is an affine SDP. By a SOS program (or SOS programming problem), we mean an optimization problem with linear objective and SOS constraints. If the constraints are affine (bilinear) in the decision variables , then the problem in an affine (bilinear) SOS programming problem. Finally, recall that SOS programming problems can be translated to SDPs and there are specialized software packages for this translation, namely SOSTOOLS (only for affine SOS programs) [51] and YALMIP (for both affine and bilinear SOS constraints) [41]. 2.3 Generalized S-procedure and Positivstellensatz We now discuss algebraic sufficient conditions for set containment constraints used throughout this thesis. S-procedure is widely used in robust control theory to obtain linear matrix inequality based sufficient conditions for set containment questions involving quadratic function [13, 26]: for quadratic functions q 0 , q 1 , . . . , q m of the form q i (x) = [x 1]Q i [x 1] T , for i = 0, . . . , m, with symmetric matrices Q i ∈ R (n+1)×(n+1) , does the set 15
containment constraint {x ∈ R n : q 1 (x) ≥ 0, . . . , q m (x) ≥ 0} ⊆ {x ∈ R n : q 0 (x) ≥ 0} (2.5) hold A (possibly conservative) certificate for this containment is the existence of nonnegative real numbers such that Q 0 − τ 1 Q 1 − . . . − τ 2 Q 2 ≽ 0, which is a LMI. We now state a straightforward generalization of the S-procedure to the case where the quadratic functions are replaced by general scalar valued functions. Lemma 2.3.1. Given scalar valued functions g 0 , g 1 , · · · , g m : R n → R, if there exist positive semidefinite functions s 1 , · · · , s m such that g 0 (x) − then m∑ s i (x)g i (x) ≥ 0 for all x ∈ R n , (2.6) i=1 {x ∈ R n : g 1 (x), . . . , g m (x) ≥ 0} ⊆ {x ∈ R n : g 0 (x) ≥ 0} . (2.7) ⊳ Lemma 2.3.1 provides algebraic sufficient conditions (nonnegativity of the multipliers s i and (2.6)) for the set containment constraints (2.7). However, these algebraic conditions require verifying the global nonnegativity of certain scalar valued functions. In order to circumvent this difficulty, we now specialize Lemma 2.3.1 to the case where g 0 , g 1 , · · · , g m polynomials and replace nonnegativity conditions by SOS conditions that are suitable for numerical verification. Lemma 2.3.2 (Generalized S-procedure). Given g 0 , g 1 , · · · , g m ∈ R[x], if there exist s 1 , · · · , s m ∈ Σ[x] such that then (2.7) holds. m∑ g 0 − s i g i ∈ Σ[x], (2.8) i=1 ⊳ 16
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: Theorem 2.2.1. A polynomial p, in x
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
Page 79 and 80: δ(= ϕ + α) as long as ϕ l (x)
Page 81 and 82: 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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