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Using sum-of-squares (SOS) relaxations for polynomial nonnegativity [49], it is possible to search for polynomial Lyapunov functions for systems with polynomial and/or rational dynamics using semidefinite programming [46, 58, 59, 32]. However, the SOS relaxation for the problem of computing invariant subsets of the ROA leads to nonconvex optimization problems with bilinear matrix inequality constraints (namely bilinear SOS problems). By contrast, simulating a nonlinear system of moderate size, except those governed by stiff differential equations, is computationally efficient. Therefore, extensive simulation is a tool used in real applications. Although the information from simulations is inconclusive, i.e., cannot be used to find provably invariant subsets of the ROA, it provides insight into the system behavior. For example, if, using Lyapunov arguments, a function certifies that a set P is in the ROA, then that function must be positive and decreasing on any solution trajectory initiating in P. Using a finite number of points on finitely many convergent trajectories and a linear parametrization of the Lyapunov function V, those constraints become affine, and the feasible polytope (in V -coefficient space) is a convex outer bound on the set of coefficients of valid Lyapunov functions. It is intuitive that drawing samples from this set to seed the bilinear SDP solvers may improve the performance of the solvers. In fact, if there are a large number of simulation trajectories, samples from the set often are suitable Lyapunov functions (without further optimization) themselves. Information from simulations is also used in [52] and [55] for computing approximate Lyapunov functions. Effectively, we are relaxing the bilinear problem (using a very specific system theoretic interpretation of the problem) to a linear problem, and the true feasible set is a subset of the linear problem’s feasible set. By contrast, a general relaxation for bilinear problems based on replacing bilinear terms by new variables and nonconvex equality constraints by convex inequality constraints is proposed in [14]. This general relaxation increases the dimension 23
of decision variable space, so that the true feasible set is a low dimensional manifold in the relaxed feasible space. There may be efficient manners to correctly “project” solutions to the relaxed problem into appropriate solutions to the original problem, but we do not pursue this. 3.1 Characterization of Invariant Subsets of ROA and Bilinear SOS Problem Consider the autonomous nonlinear dynamical system ẋ(t) = f(x(t)), (3.1) where x(t) ∈ R n is the state vector and f : R n → R n is such that f(0) = 0, i.e., the origin is an equilibrium point of (3.1), and f is locally Lipschitz. Let φ(t; x 0 ) denote the solution to (3.1) at time t with the initial condition x(0) = x 0 . If the origin is asymptotically stable but not globally attractive, one often wants to know which trajectories converge to the origin as time approaches ∞. The region-of-attraction R 0 of the origin for the system (3.1) is R 0 := { } x 0 ∈ R n : lim φ(t; x 0 ) = 0 . t→∞ A modification of a similar result in [76] provides a characterization of invariant subsets of the ROA in terms of sublevel sets of appropriately chosen Lyapunov functions. Lemma 3.1.1. Let γ ∈ R be positive. If there exists a C 1 function V : R n → R such that Ω V,γ is bounded, and (3.2) V (0) = 0 and V (x) > 0 for all x ∈ R n (3.3) Ω V,γ \ {0} ⊂ {x ∈ R n : ∇V (x)f(x) < 0} , (3.4) 24
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: Chapter 3 Simulation-Aided Region-o
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 −
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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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