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

More documents

Recommendations

Info

y bounded polytopes. Conservatism (due to parameter-independent Lyapunov functions) is simply reduced by partitioning the uncertainty set using a branch-and-bound type refinement procedure. The approach offers the following advantages: (i) The parameterdependent Lyapunov functions achieved by uncertainty-space partitioning do not require an a priori parametrization of the Lyapunov function in the uncertain parameters. (ii) It leads to optimization problems with smaller semidefiniteness constraints since uncertain parameters do not explicitly appear in the constraints. Although the size of the semidefinite programming constraints does not increase with the number of uncertain parameters, their number does and the problem becomes challenging as the number of uncertain parameters increases. (iii) A sequential implementation for computing suboptimal solutions which decouples these constraints into smaller, independent problems arises naturally. This is suitable for trivial parallel computing offering a major advantage over approaches utilizing parameter-dependent Lyapunov functions. Most of the results of this chapter are reported in [71, 72]. In Chapter 6, we analyze reachability properties and local input/output gains of systems with polynomial vector fields. Upper bounds for the reachable set and nonlinear system gains are characterized using Lyapunov/storage functions and computed solving bilinear sum-of-squares programming problems. A procedure to refine the upper bounds by transforming polynomial Lyapunov/storage functions to non-polynomial Lyapunov functions is developed. The simulation-aided analysis methodology is adapted to reachability and local gain analysis. Finally, a local small-gain theorem is proposed and applied to the robust region-of-attraction analysis for systems with unmodeled dynamics. Parts of the material presented in this chapter is reported in [69] 5
1.2 Summary of Examples Listed below are the examples in this thesis. For all examples, the problem data and results (mainly Lyapunov functions and corresponding multipliers) are available at http://jagger.me.berkeley.edu/~utopcu/dissertation. 1) Simulation-aided region-of-attraction analysis • Van der Pol dynamics §3.3.1 • Examples from [20, 75, 33, 32] § 3.3.2 • Controlled short period aircraft dynamics §3.3.3 • Pendubot dynamics §3.3.4 • Closed-loop dynamics with nonlinear observer based controller §3.3.5 2) Region-of-attraction analysis for uncertain nonlinear systems • Uncertain Van der Pol dynamics and examples from [18, 78] §4.5.1 • Uncertain controlled short period aircraft dynamics § 4.5.2 3) Extensions of robust region-of-attraction analysis • An example from [19] § 5.4.1 • Uncertain controlled short period aircraft dynamics § 5.4.2 • Uncertain controlled short period aircraft dynamics with first-order uncertain dynamics §5.4.3 4) Local reachability and gain analysis • Reachability analysis for an example from the literature §6.4 6
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: ections. Finally, it is shown that,
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
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 −
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
Page 99 and 100:
(SDP), with 3 “types” of decisi
Page 101 and 102:
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
Page 107 and 108:
- 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
Page 119 and 120:
then G R 2 ⊆ Ω V,R 2. ⊳ Given
Page 121 and 122:
where V poly and S’s are prescrib
Page 123 and 124:
Lower bound for the reachable set:
Page 125 and 126:
Consequently, this strengthens the
Page 127 and 128:
Of course, this set could be empty,
Page 129 and 130:
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
Page 135 and 136:
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
Page 143 and 144:
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.
show all

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

Create successful ePaper yourself

Delete template?

Save as template?