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thesis are motivated by the trade-off between system complexity, available computational resources, and strength of the proofs. The content and the contributions of respective chapters are outlined below. Chapter 2 presents 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 is 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” are discussed. In Chapter 3, we propose a method for computing invariant subsets of the regionof-attraction for asymptotically stable equilibrium points of dynamical systems with polynomial vector fields. We use polynomial Lyapunov functions as local stability certificates whose certain sublevel sets are invariant subsets of the region-of-attraction. Similar to many local analysis problems, this is a nonconvex problem. Furthermore, its sum-of-squares relaxation leads to a bilinear optimization problem. We develop a method utilizing information from simulations for easily generating Lyapunov function candidates. For a given Lyapunov function candidate, checking its feasibility and assessing the size of the associated invariant subset are affine sum-of-squares optimization problems. Solutions to these problems provide invariant subsets of the region-of-attraction directly and/or they can further be used as seeds for local bilinear search schemes or iterative coordinate-wise affine search schemes for improved performance of these schemes. We report promising results in all these di- 3
ections. Finally, it is shown that, for systems with cubic polynomial vector fields, if the linearized dynamics are exponentially stable, then the bilinear sum-of-squares programming problems (formulated to estimate the region-of-attraction) are always feasible. The material presented in this chapter is partly parallel to [73, 70]. Chapter 4 is dedicated to developing a method to compute provably invariant subsets of the region-of-attraction for asymptotically stable equilibrium points of uncertain nonlinear dynamical systems. We consider polynomial dynamics with perturbations that either obey local polynomial bounds or are described by uncertain parameters multiplying polynomial terms in the vector field. This uncertainty description is motivated by both incapabilities in modeling, as well as bilinearity and dimension of the sum-of-squares programming problems whose solutions provide invariant subsets of the robust region-of-attraction. Finally, we discuss a sequential suboptimal solution technique suitable for parallel computing. The technique proposed in this chapter, coupled with the sequential implementation, is conservative. Yet, its computational complexity is lower than other methods with more elegant formulations based on parameter-dependent Lyapunov functions. Therefore, it provides a relatively feasible infrastructure for the extensions in chapter 5. More importantly, lessons learnt from the study reported in this chapter apply to other analysis (and possibly synthesis) questions with parametric uncertainty. The material presented in this chapter is partly parallel to [68, 67]. In Chapter 5, we extend the applicability of the method proposed in Chapter 4 to handle systems with non-affine uncertainty dependence and to reduce the conservatism associated with using a common Lyapunov function for an entire family of uncertain systems. Non-affine appearances of uncertain parameters in the vector field are replaced by artificial parameters and the graph of non-affine functions of uncertain parameters are covered 4
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: specifically sum-of-squares program
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
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
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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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