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

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Chapter 1 Introduction 1.1 Thesis Overview and Contributions The objective of this thesis is to develop quantitative robustness and performance analysis tools for nonlinear dynamical systems. Nonlinear systems possess local properties that are not global. For example, an asymptotically stable equilibrium point may not be globally attractive or input-output properties may radically vary for different ranges of disturbance levels. Therefore, we emphasize local analysis rather than global and focus on the following measures of robustness: (i) inner estimates of the region-of-attraction of an equilibrium point; (ii) outer estimates of reachable sets under bounded disturbances; (iii) upper bounds for local input-output gains. In each case, we account for modeling uncertainties and extend the applicability of proposed tools to uncertain systems. Using Lyapunov/storage function type characterizations for these robustness measures and S-procedure type relaxations, analysis questions are translated to verification of global nonnegativity of functions satisfying certain properties. Polynomial optimization (more 1
specifically sum-of-squares programming) provides an effective framework for this verification. Therefore, we restrict our attention to systems with polynomial vector fields and polynomial robustness certificates and obtain bilinear sum-of-squares programming problems with two challenging features: nonconvexity and rapid growth of the size with increasing state and/or uncertainty space dimension. Our approach is based on exploiting system theoretic interpretations of these problems to reduce their complexity. For example, we propose a methodology incorporating simulations in formal proof generation by constructing convex outer-bounds on the set of feasible Lyapunov/storage functions. Lyapunov/storage function candidates drawn from this outer-bound set either directly qualify as robustness certificates or can be used as initial seeds for further bilinear search with more efficient and reliable performance. Another example comes from the realization that, in the presence of parametric uncertainties, the dependence of the constraints in the resulting optimization problems on the two groups of indeterminate variables, namely states and uncertain parameters, is not the same. In order to take advantage of this difference, we choose not to reflect the dependence of robustness properties in the form of the Lyapunov/storage function candidates and employ parameter-independent certificates. The conservatism due to this choice is simply reduced by sub-partitioning the uncertainty set by a branch-and-bound type refinement procedure. A common feature of proposed techniques is that most of the computation is suitable for “embarrassingly parallel” computation [1]. The goal of this work is to lay out a feasible path toward computationally efficient and reliable schemes for analyzing the behavior of nonlinear systems with 15 states, 5 uncertain parameters, and cubic polynomial vector fields. Heading toward this goal, restrictions and choices (on the system description or the form of certificates) we make throughout this 2
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: Acknowledgements It has been a shor
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 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
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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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