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

Quantitative Local Analysis of Nonlinear Systems
by
Ufuk Topcu
B.S. (Bogazici University, Istanbul) 2003
M.S. (University of California, Irvine) 2005
A dissertation submitted in partial satisfaction
of the requirements for the degree of
Doctor of Philosophy
in
Engineering - Mechanical Engineering
in the
GRADUATE DIVISION
of the
UNIVERSITY OF CALIFORNIA, BERKELEY
Committee in charge:
Professor Andrew K. Packard, Chair
Professor Kameshwar Poolla
Professor Laurent El Ghaoui
Fall 2008

The dissertation of Ufuk Topcu is approved.
Chair
Date
Date
Date
University of California, Berkeley
Fall 2008

Quantitative Local Analysis of Nonlinear Systems
Copyright c○ 2008
by
Ufuk Topcu

Abstract
Quantitative Local Analysis of Nonlinear Systems
by
Ufuk Topcu
Doctor of Philosophy in Engineering - Mechanical Engineering
University of California, Berkeley
Professor Andrew K. Packard, Chair
This thesis investigates quantitative methods for local robustness and performance
analysis of nonlinear dynamical systems with polynomial vector fields. We propose measures
to quantify systems’ robustness against uncertainties in initial conditions (regions-ofattraction)
and external disturbances (local reachability/gain analysis). S-procedure and
sum-of-squares relaxations are used to translate Lyapunov-type characterizations to sumof-squares
optimization problems. These problems are typically bilinear/nonconvex (due to
local analysis rather than global) and their size grows rapidly with state/uncertainty space
dimension.
Our approach is based on exploiting system theoretic interpretations of these optimization
problems to reduce their complexity. We propose a methodology incorporating simulation
data in formal proof construction enabling more reliable and efficient search for robustness
and performance certificates compared to the direct use of general purpose solvers. This
1

technique is adapted both to region-of-attraction and reachability analysis. We extend the
analysis to uncertain systems by taking an intentionally simplistic and potentially conservative
route, namely employing parameter-independent, rather than parameter-dependent,
certificates. The conservatism is simply reduced by a branch-and-bound type refinement
procedure.
The main thrust of these methods is their suitability for parallel computing
achieved by decomposing otherwise challenging problems into relatively tractable smaller
ones. We demonstrate proposed methods on several small/medium size examples in each
chapter and apply each method to a benchmark example with an uncertain short period
pitch axis model of an aircraft.
Additional practical issues leading to a more rigorous basis for the proposed methodology
as well as promising further research topics are also addressed. We show that stability
of linearized dynamics is not only necessary but also sufficient for the feasibility of the
formulations in region-of-attraction analysis. Furthermore, we generalize an upper bound
refinement procedure in local reachability/gain analysis which effectively generates nonpolynomial
certificates from polynomial ones. Finally, broader applicability of optimizationbased
tools stringently depends on the availability of scalable/hierarchial algorithms. As
an initial step toward this direction, we propose a local small-gain theorem and apply to
stability region analysis in the presence of unmodeled dynamics.
Professor Andrew K. Packard
Dissertation Committee Chair
2

To my parents Züleyha and Fevzi
for their support and patience,
and
to my wife Zeynep
for sharing the joy of life with me.
i

Contents
Contents
ii
List of Figures
v
List of Tables
ix
1 Introduction 1
1.1 Thesis Overview and Contributions . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Summary of Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Background 8
2.1 Semidefinite and Linear Programming . . . . . . . . . . . . . . . . . . . . . 9
2.2 Sum-of-Squares Polynomials and Sum-of-Squares Programming . . . . . . . 12
2.3 Generalized S-procedure and Positivstellensatz . . . . . . . . . . . . . . . . 15
2.4 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
ii

3 Simulation-Aided Region-of-Attraction Analysis 22
3.1 Characterization of Invariant Subsets of ROA and Bilinear SOS Problem . . 24
3.2 Relaxation of the Bilinear SOS Problem Using Simulation Data . . . . . . . 27
3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4 Critique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 Sanity Check: Does Linear Stability Imply Existence of SOS Certificates . 48
3.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.7 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Local Stability Analysis for Uncertain Nonlinear Systems 55
4.1 Setup and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Computation of Robustly Invariant Sets . . . . . . . . . . . . . . . . . . . . 60
4.3 Implementation Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4 Sanity Check: Does Robust Stability Imply Existence of SOS Certificates . 67
4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5 Extensions of the Robust Region-of-Attraction Analysis: Refinements
and Non-affine Uncertainty Dependence 77
iii

5.1 Setup and Estimation of the Robust ROA of Systems with Affine Parametric
Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 Polynomial Parametric Uncertainty . . . . . . . . . . . . . . . . . . . . . . . 84
5.3 Branch-and-Bound Type Refinement in the Parameter Space . . . . . . . . 88
5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6 Reachability and Local Gain Analysis for Nonlinear Dynamical Systems100
6.1 Upper and Lower Bounds for the Reachable Set and Local Input-Output Gains101
6.2 Upper Bound Refinement for Reachability and L 2 → L 2 Gain Analysis . . . 107
6.3 Simulation-Based Relaxation for the Bilinear SOS Problem in Reachability
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.5 Region-of-Attraction Analysis for Systems with Unmodeled Dynamics using
a Local Small-Gain Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7 Conclusions 126
Bibliography 129
iv

List of Figures
3.1 Sets Y and B and points generated by H&R algorithm. Φ j and b j denote
the j th column of Φ and j th component of b. . . . . . . . . . . . . . . . . . . 32
3.2 Histograms of β ∗ L before CW Opt (black bars) and β∗ L
after CW Opt (white
bars) for ∂(V ) = 2 (top), 4 (middle), and 6 (bottom). . . . . . . . . . . . . 37
3.3 The invariant subsets of the ROA (dot: ∂(V ) = 2, dash: ∂(V ) = 4, and solid:
∂(V ) = 6 (indistinguishable from the outermost curve for the limit cycle)). . 38
3.4 Invariant subset of the ROA for (E 6 ) reported in [32] (solid surface) and that
computed using the sequential procedure from section 3.3.2 (dotted surface). 41
3.5 Invariant subset of the ROA for (E 7 ) reported in [75] (thick solid curve) and
that computed using the sequential procedure from section 3.3.2 (Ω V,γ ∗) (thin
solid curve), and system trajectories (dash-dot curves). . . . . . . . . . . . . 41
3.6 Histograms of β ∗ L before CW Opt (black bars) and β∗ L
after CW Opt (white
bars) for ∂(V ) = 2 (top) and 4 (bottom). . . . . . . . . . . . . . . . . . . . 43
3.7 A slice of the invariant subset of the ROA (solid curve) and initial conditions
(with x 2 = 0 and x 4 = 0) for diverging trajectories (dots). . . . . . . . . . . 44
v

4.1 Invariant subsets of ROA reported in [18] (black curve) and those computed
solving the problem in (4.13) with ∂(V ) = 2 (blue curve) and ∂(V ) = 4 (green
curve) along with initial conditions (red stars) for some divergent trajectories
of the system corresponding to α = 1. . . . . . . . . . . . . . . . . . . . . . 70
4.2 Invariant subsets of ROA with ∂(V ) = 4 (green curve) and ∂(V ) = 6 (blue
curve) along with the unstable limit cycle (red curves) of the system corresponding
to α = −1.0, −0.8, . . . , 0.8, 1.0. . . . . . . . . . . . . . . . . . . . . 72
4.3 Invariant subsets of ROA with ∂(V ) = 2 (green) and ∂(V ) = 4 (blue) along
with initial conditions (red stars) for divergent trajectories. . . . . . . . . . 73
5.1 Polytopic cover for {(ζ, ψ) ∈ R 2 : ζ ∈ [0, 1], ψ = ζ 2 } with 2 cells (red) and
4 cells (yellow). Black curve is the set {(ζ, ψ) ∈ R 2 : ζ ∈ [0, 1], ψ = ζ 2 }. . 93
5.2 Curves with “◦” are for the lower bounds obtained by directly solving (5.6)
with D taken as the vertices of the corresponding cell, curves with “⋄” are for
the lower bounds obtained by applying the sequential procedure from section
4.3 by taking ∆ sample (in the first step) as the center of the corresponding
cell, and curves with “×” (in the top figure only) and “⋆” show β nc and β lp ,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3 Estimates of the robust ROA: from [19] (black), using the branch-and-bound
based method for ∂(V ) = 2 (red) and ∂(V ) = 4 (green). . . . . . . . . . . . 95
vi

5.4 Lower bounds for β∆ ∗ with ∂(V ) = 2 (green solid curve with “×” marker)
and ∂(V ) = 4 (blue solid curve with “⋄” marker) and β nc (red solid curve
with “◦” marker) computed at the centers of the cells generated by the B&B
Algorithm for the ∂(V ) = 4 run. Dashed curves are for (computed values of)
β {δ} where δ is the center of the cell with the smallest lower bound at the
corresponding step of the B&B refinement procedure for ∂(V ) = 2 (green
curve with “×” marker) and ∂(V ) = 4 (blue curve with “⋄” marker). . . . . 96
5.5 Final partition generated by the B&B algorithm for the ∂(V ) = 4 run. . . . 97
5.6 Controlled short period aircraft dynamics with uncertain first order linear
time-invariant dynamics (δ p := (δ 1 , δ 2 )). . . . . . . . . . . . . . . . . . . . . 97
6.1 Bounds on reachable sets due to disturbance w with ‖w‖ 2 2 ≤ R2 for Example
1 without delay (blue curve with dots: before refinement, green curve with
×: after refinement, red curve with ⋄: lower bound). . . . . . . . . . . . . . 115
6.2 Bounds on reachable sets due to disturbance w with ‖w‖ 2 2 ≤ R2 for Example
2 (blue curve with dots: before refinement, green curve with ×: after
refinement, red curve with ⋄: lower bound, and black circles: failed PENBMI
runs). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3 Bounds on reachable sets due to disturbance w with ‖w‖ 2 2 ≤ R2 for Example
2 (blue curve with dots: before refinement, green curve with ×: after
refinement, red curve with ⋄: lower bound, and black circles: failed PENBMI
runs). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
vii

6.4 Upper bounds for ∂(V ) = 2 (with ⋄) and ∂(V ) = 4 (with ×) before the
refinement (blue curves) and after the refinement (green curves) along with
the lower bounds (red curve). . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.5 Feedback interconnection of ∆ and M. . . . . . . . . . . . . . . . . . . . . . 119
6.6 Controlled short period aircraft dynamics with unmodeled dynamics (δ p :=
(δ 1 , δ 2 )). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
viii

List of Tables
3.1 Parameters used in and results of SimLF G and CW Opt algorithms. . . . . 36
3.2 Volume ratios for (E 1 )-(E 7 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 Results of SimLF G and CW Opt algorithms. Upper bounds are established
by a separate run of SimLF G algorithm with N conv = 3000.
The upper
bound for ∂(V ) = 4 is by a divergent trajectory whereas as the upper bound
is by the infeasibility of (3.6), (3.11), and (3.12) for the given β value. Representative
computation times are on 2.0 GHz desktop PC. . . . . . . . . . 43
4.1 N SDP (left columns) and N decision (right columns) for different values of n
and 2d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Number of decision variables in (4.13) (top entry in each cell of the table)
and the number of decision variables in (4.15) (bottom entry in each cell of
the table) for ∂(V ) = 2, ∂(s 2δ ) = 2, and ∂(s 3δ ) = 0. . . . . . . . . . . . . . 67
4.3 Optimal values of β in the problem (4.13) with different values of µ and
∂(V ) = 2 and 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
ix

4.4 Optimal values of β in the problem (4.13) with different values of µ and
∂(V ) = 4 and 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5 Optimal value of β in the first step, β sample , and β subopt with µ for ∂(V ) = 2
and 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.1 Computed (sub)optimal values of β with p(x) = x T x (with ∂(V ) = 2 /
∂(V ) = 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
x

List of Symbols
R
R n
R n×m
Z
Z +
C 1
M ≽ 0 (x ≽ 0)
The real numbers
Real n-vectors
Real n-by-m matrices
The ring of integers
Positive integers
The set of continuously differentiable real valued functions on R n
M symmetric and is positive semidefinite
(for x ∈ R n , entries of x are non-negative)
M ≻ 0 (x ≻ 0)
M is symmetric and positive definite
(for x ∈ R n , entries of x are positive)
R[x]
Σ[x]
∂(π)
Ω f,η
The set of polynomials (of certain finite degree) in x with real coefficients
The set of sum-of-squares polynomials in R[x]
The the degree of π ∈ R[x]
The η-sublevel set of f, i.e., for η ∈ R and f : R n → R
Ω f,γ := {x ∈ R n
: f(x) ≤ η},
xi

Acknowledgements
It has been a short stay at Berkeley and I have enjoyed every minute of it.
I would like to take this opportunity to thank the following people.
I would like to express my gratitude to Andy Packard for making my studies at Berkeley
possible and being a role model with his passion for research. I want to extend heartfelt
thanks to Laurent El Ghaoui for long discussions on optimization and his support and
guidance. Special thanks to Kameshwar Poolla for being a great mentor and his interest in
my academic progression. I also want to thank Pete Seiler for valuable discussions and for
providing me a summer internship opportunity at the Honeywell Labs.
Thanks to all at the BCCI for their friendship, support, and creating a pleasant working
environment.
Finally, I would like to acknowledge the financial support from the Air Force Office of
Scientific Research under grant/contract number FA9550-05-1-0266 and thank the program
managers Sharon Heise and Scott Wells.
xii

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

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

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

• Reachability analysis for Pendubot dynamics §6.4
• L 2 → L 2 gain analysis for a system with adaptive controller §6.4
• Robust region-of-attraction analysis of controlled short period aircraft dynamics
with unmodeled dynamics §6.5.1
7

Chapter 2
Background
The goal in the following chapters is to develop computational tools for estimating
certain robustness and performance measures (e.g. regions-of-attraction and input-output
gains) for nonlinear systems. The general strategy for computing these measures consists
of three main steps:
i. Characterize these measures using Lyapunov-type functions that satisfy certain conditions
which can be translated to set containment (or set emptiness) questions.
ii. Obtain “S-procedure” type sufficient conditions for the set containment constraints.
iii. Search for Lyapunov-type certificates and S-procedure multipliers by solving numerical
optimization (or feasibility) problems.
In the following sections of this chapter, we present a summary of background material
needed to carry out this plan.
8

2.1 Semidefinite and Linear Programming
2.1.1 Semidefinite Programming
A semidefinite program (SDP) 1 is an optimization problem with linear objective and
matrix semidefiniteness constraint. Formally, for c ∈ R n and the symmetric matrix valued
map F : R n → R m×m , a SDP can be written as
We mainly deal with two types of SDPs:
min c T x subject to
x∈R n
F (x) ≽ 0.
• Linear (affine) SDPs: For symmetric matrices F 0 , F 1 , . . . , F n ∈ R m×m , the map F
takes the form F (x) = F 0 + ∑ n
i=1 x iF i . In this case, the constraint F (x) ≽ 0 is called
a linear matrix inequality (LMI).
• Bilinear SDPs: For symmetric matrices F 0 , F i , and F ij in R m×m with i = 1, . . . , n
and j = 1, . . . , m, the map F takes the form
n∑ n∑ m∑
F (x) = F (y, z) = F 0 + y i F i + y i z j F ij . (2.1)
i=1 i=1 j=1
In this case, the constraint F (x) ≽ 0 is called a bilinear matrix inequality (BMI).
Remarks 2.1.1. Although the term “semidefinite program” is generally used for problems
with affine objective function and LMI constraints [15], we will use it here to mean optimization
problems with affine objective function and general matrix inequality constraints
and specify its type if needed.
⊳
1 We will use the abbreviation SDP to mean both semidefinite program and semidefinite programming.
Similarly, LP will be used for linear program and linear programming.
9

Affine SDPs are convex optimization problems, considered to be computationally
tractable with polynomial time algorithms [44, 15, 80].
There are several reliable and
relatively efficient solvers for affine SDPs including SeDuMi [56], DSDP [10], and SDPT3
[65].
2
On the other hand, bilinear SDPs are nonconvex and NP-hard in general [66].
Consequently, the state-of-the-art of the solvers for bilinear SDPs is far behind that for the
linear ones. We now review several strategies to solve bilinear SDPs.
2.1.2 Solution Strategies for Problems with Bilinear Matrix Inequalities
Optimization problems with BMIs provide an effective framework for many problems
in controls, e.g. µ-synthesis [6] and static output feedback [34]. Although there is no
general purpose efficient BMI solver, several methods have been proposed. Global optimization
schemes based on the branch-and-bound algorithm [2] and generalized Benders
decomposition are discussed in [30] and [11], respectively.
See also [43] for a discussion
of methodological, structural, and computational aspect of problem with BMI constraints.
Recently PENBMI, a solver for bilinear SDPs, was introduced [39]. It is a local optimizer
and its behavior (speed of convergence, quality of the local optimal point, etc.) depends on
the point from which the optimization starts. On the other hand, note that although the
function F in (2.1) is not affine in y and z jointly, it is affine in y when z is fixed and the
constraint becomes a LMI in y and vice versa. This observation suggests a simple strategy
to attack bilinear SDPs: first set z to some candidate solution, say ¯z, and optimize over y
by solving
ȳ := argmin y
⎡
c T ⎢
⎣
y
¯z
⎤
⎥
⎦
subject to F (y, ¯z) ≽ 0,
2 As these solvers have specific formats to describe SDPs, parsers such as YALMIP [41] and CVX [31] are
extremely useful in setting up SDPs in these specific formats.
10

then set y to ȳ and optimize over z by solving
⎡ ⎤
¯z = argmin z c T ȳ
⎢ ⎥
⎣ ⎦ subject to F (ȳ, z) ≽ 0,
z
and alternate between these two problems as long as there is satisfactory improvement in
the solution. This two-way iterative search, which we call coordinate-wise affine search, is of
course a local search scheme and generated candidate solutions highly depend on the initial
point from which the search starts (it may not reach the optimal solution or may require a
large number of iterations to tightly approximate the optimal solution) [43]. Nevertheless,
it is practically attractive since it only requires an affine SDP solver and it has been widely
used by controls community (for example, D − K iteration in µ-synthesis [6] is based on
alternating between the controller K and the D-scales). Moreover, our experience suggests
that, coupled with efficient methods for generating high quality initial points (see chapter
3), coordinate-wise affine search can be efficiently used to compute suboptimal solutions for
problems with BMI constraints. Consequently, we implement coordinate-wise affine search
schemes throughout this thesis and provide implementation details in the corresponding
chapters.
2.1.3 Linear Programming
A linear program (LP) is an optimization problem with linear objective and affine constraints.
Formally, for c ∈ R n , A ∈ R m×n , and b ∈ R m , a linear program can be written
as
min c T x subject to
x∈R n
Ax ≽ b.
Several optimization problems in the following chapters will have both SDP constraints
11

and LP constraints, namely for c ∈ R n , A ∈ R m×n , b ∈ R m , symmetric matrix valued map
F : R n → R N SDP ×N SDP
,
min c T x subject to
x∈R n
Ax ≽ b
(2.2)
F (x) ≽ 0.
Finally, note that constraints in (2.2) can equivalently be written as
⎡
⎤
F (x)
a T 1 x − b 1
. ≽ 0,
.. ⎢
⎥
⎣
⎦
a T mx − b m
which is another (larger) SDP constraint (here a T i and b i denote the i-th row of A and b,
respectively). Therefore, the optimization in (2.2) can be solved as a SDP.
Both SDP and LP have been extensively studied and there are excellent references on
the topic (including but not limited to) [9, 15, 79, 28, 42]. It is worth mentioning that the
state-of-the-art of the solvers for LPs are far beyond that for SDPs [80, 9, 3].
2.2 Sum-of-Squares Polynomials and Sum-of-Squares Programming
Restricting our attention to systems with polynomial vectors fields and searching for
unknown functions in pre-specified finite-dimensional subspaces of polynomials, we will
formulate the search for robustness and performance certificates as optimization problems
with polynomial nonnegativity constraints. However, verifying the global nonnegativity of
a multivariate polynomial is an hard problem [49]. On the other hand, if the polynomial can
12

e represented as a sum of squares of finitely many polynomials, i.e., it is a sum-of-squares
(SOS) polynomial, then it trivially follows that the polynomial is globally nonnegative. An
appealing property of sum-of-squares polynomials is that checking whether a polynomial
is sum-of-squares can be formulated as a SDP feasibility problem. Consequently, in the
following sections, the strategy for dealing with global polynomial nonnegativity constraints
will be replacing nonnegativity constraints by sum-of-squares conditions.
We now review useful facts about SOS polynomials. Formally, a polynomial p in x ∈ R n
is said to be SOS if, for p 1 , . . . , p M ∈ R[x] it can be decomposed in the form
p(x) =
M∑
p i (x) 2 . (2.3)
i=1
Obviously, a SOS polynomial is globally nonnegative. Therefore, the set Σ[x] of SOS polynomials
in x (of some fixed degree 3 ), defined as
{
Σ[x] := s ∈ R[x] : ∃M < ∞, p 1 , . . . , p M ∈ R[x] such that s(x) =
}
M∑
p i (x) 2 ,
i=1
is a subset of the set of globally nonnegative polynomials. In fact, Σ[x] is a strict subset
of the set of globally nonnegative polynomials except for univariate polynomials, quadratic
polynomials and quartic polynomials in two variables [53].
Let p be a polynomial in x of degree m and z(x) be a vector of monomials in x up
to degree min{m e ∈ Z : m/2 ≤ m e }. Then, for some symmetric matrix Q, p can be
decomposed as
p(x) = z(x) T Qz(x). (2.4)
Based on this fact, the following theorem provides a characterization of SOS polynomials.
3 We do not specify the degree of polynomials in Σ[x] in notation unless it leads to confusion and use
Σ[x] to denote the set of sum-of-squares polynomials in x ∈ R n of some fixed degree to be inferred from the
context.
13

Theorem 2.2.1. A polynomial p, in x ∈ R n of degree 2d, is SOS if and only if there exists
Q ≽ such that p(x) = z(x) T Qz(x) where z is as defined above.
⊳
A proof of Theorem 2.2.1 can be found in [22]. Here, we highlight a few useful observations
that lead to the proof. Consider that p is SOS, i.e., there exist an integer M > 0
and polynomials p 1 , . . . , p M
such that p(x) = ∑ M
i=1 p i(x) 2 . Then, each p i is of degree d
and therefore can be represented as α T i z(x) for some vector α i. Consequently, the matrix
Q in the theorem statement can be taken as Q = ∑ M
i=1 α iα T i which is clearly positive
semidefinite. On the other hand, if p can be represented as p(x) = z(x) T Qz(x) with Q positive
semidefinite, then p(x) can be written as p(x) = ∑ M
i=1 p i(x) 2 , where, for i = 1, . . . , M,
p i (x) = √ λ i q T i z(x), λ 1, . . . , λ M are eigenvalues of Q and q i ’s are corresponding eigenvectors.
Another important observation is that for a given polynomial p, the entries of z may
not be algebraically independent. Therefore, there may be multiple symmetric Q such that
(2.4) holds. This is most easily demonstrated by an example.
Example 2.2.1. Let x ∈ R 2 , z(x) = [ x 2 1 x 1x 2 x 2 2] T , Qp ∈ R 3×3 be a symmetric matrix,
and p(x) = z(x) T Q p z(x). Then, for any λ ∈ R
⎛
⎞
0 0 λ
p(x) = z(x) T Q p z(x) = z(x) T Q p z(x) + z(x) T 0 2λ 0
z(x)
⎜
⎟
⎝
⎠
−λ 0 0
} {{ }
Q h (λ)
since z(x) T Q h (λ)z(x) = 0 for all λ ∈ R which follows from the relation x 2 1 x2 2 = (x 1x 2 ) 2 .
This degree of freedom of varying λ without violating p(x) = z(x) T Q(λ)z(x), where Q(λ) :=
Q p +Q h (λ), leads to a procedure to search for positive semidefinite Q(λ) by choice of λ. The
search for λ such that Q(λ) ≽ 0 is an affine SDP: Find λ ∈ R such that Q(λ) ≽ 0. In fact
14

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

The Positivstellensatz, a central theorem from real algebraic geometry, provides generalizations
of Lemma 2.3.2. For its statement, a few definitions are needed.
Definition 2.3.1. Given {g 1 , . . . , g t } ∈ R[x], the multiplicative monoid generated by g j ’s is
the set of all finite products of g j ’s, including 1 (i.e. the empty product). It is denoted as
M(g 1 , . . . , g t ). For completeness define M(∅) := 1.
⊳
Definition 2.3.2. Given {f 1 , . . . , f r } ∈ R[x], the cone generated by f i ’s is
{
}
m∑
P(f 1 , . . . , f r ) := s 0 + s i b i : m ∈ Z + , s i ∈ Σ[x], b i ∈ M(f 1 , . . . , f r ) .
i=1
⊳
Definition 2.3.3. Given {h 1 , . . . , h u } ∈ R[x], the ideal generated by h k ’s is
{∑ }
I(h 1 , . . . , h u ) := hk p k : p k ∈ R[x] .
⊳
With these definitions, we can state the following theorem from [12, Theorem 4.2.2]:
Theorem 2.3.1 (Positivstellensatz). Given polynomials {f 1 , . . . , f r }, {g 1 , . . . , g t }, and
{h 1 , . . . , h u } in R[x], the following are equivalent:
i. The set below is empty:
⎧
⎪⎨
x ∈ R n :
⎪⎩
f 1 (x) ≥ 0, . . . , f r (x) ≥ 0,
g 1 (x) ≠ 0, . . . , g t (x) ≠ 0,
h 1 (x) = 0, . . . , h u (x) = 0
⎫
⎪⎬
⎪⎭
ii. There exist polynomials f ∈ P(f 1 , . . . , f r ), g ∈ M(g 1 , . . . , g t ), h ∈ I(h 1 , . . . , h u ) such
that
f + g 2 + h = 0.
17

⊳
Example 2.3.1. We now give a proof of Lemma 2.3.2 to demonstrate the use of the Positivstellensatz.
Note that the set containment constraint (2.7) holds if and only if
{x | g 1 (x) ≥ 0, . . . , g m (x) ≥ 0, −g 0 (x) ≥ 0, g 0 (x) ≠ 0} = ∅. (2.9)
Theorem 2.3.1, applied to (2.9), gives that (2.9) holds if and only if there exist s (·) ∈ Σ[x]
and k ∈ Z + such that
s + s 0 (−g 0 ) +
m∑
s i g i +
i=1
m∑
s 0i (−g 0 )g i +
i=1
m∑ m∑
m∏
s ij g i g j + · · · + s 0...m (−g 0 ) g i + g0 2k = 0. (2.10)
i=1 j=i
Setting k = 1 and all s (·) = 0 except s 0 , s 01 , . . . , s 0m , we have a sufficient condition
⎡
⎤
m∑
− g 0
⎣s 0 + s 0j g j − g 0
⎦ = 0. (2.11)
j=1
i=1
Since g 0 is not identically zero, (2.8) follows from (2.11) by renaming s 0i as s i .
⊳
Lemma 2.3.3. Let g ∈ R[x] be positive definite, h ∈ R[x], γ > 0, s 1 , s 2 ∈ Σ[x], l ∈ R[x] be
positive definite and satisfy l(0) = 0. Suppose that
− [(γ − g)s 1 + hs 2 + l] ∈ Σ[x] (2.12)
holds. Then, it follows that
{x ∈ R n : g(x) ≤ γ, x ≠ 0} ⊂ {x ∈ R n : h(x) < 0} (2.13)
⊳.
Proof. Note that (2.13) holds if and only if
{x ∈ R n : γ − g(x) ≥ 0, l(x) ≠ 0, h(x) ≥ 0} = ∅
18

and the Positivstellensatz, applied to the last condition, yields (2.12) after appropriate
simplifications [37].
An independent proof of Lemma 2.3.3 can be obtained without using the Positivstellensatz.
Let x ∈ {x ∈ R n : g(x) ≤ γ, x ≠ 0}. Suppose that s 2 (x) = 0. Noting that g(x) ≤ γ
and l(x) > 0 and substituting into (2.12) lead to a contradiction. Hence, s 2 (x) > 0 for all
x ∈ Ω g,γ \{0}. Now, suppose (2.12) holds. Then, h(x)s 2 (x) ≤ −l(x). By the first part,
h(x) ≤ −l(x)/s 2 (x). Consequently, h(x) < 0.
2.3.1 Affine versus Bilinear Sufficient Conditions
Let A 1 , A 2 : R[x] → R[x] be affine maps on R[x], f 1 ∈ R[x] and f 2 ∈ R[x], and consider
the constraint
{x ∈ R n : A 1 (f 1 (x)) ≥ 0} ⊆ {x ∈ R n : A 2 (f 2 (x)) ≥ 0} . (2.14)
We will use the generalized S-procedure to handle mainly two types of questions:
Question 1: Given f 1 ∈ R[x] and f 2 ∈ R[x], does (2.14) hold
Question 2: Given f 2 ∈ R[x], does there exist f 1 ∈ R[x] such that (2.14) (possibly along
with other constraints on f 1 ) holds
Then, S-procedure based sufficient conditions are
Sufficient condition for Question 1: Existence of s 2 ∈ Σ[x] such that
A 2 (f 2 (x)) − A 1 (f 1 (x))s 2 (x) ∈ Σ[x]. (2.15)
Sufficient condition for Question 2: Existence of s 2 ∈ Σ[x] and f 1 ∈ R[x] such that (possibly
along with other constraints on f 1 )
A 2 (f 2 (x)) − A 1 (f 1 (x))s 2 (x) ∈ Σ[x]. (2.16)
19

The main difference between these conditions is that the constraint (2.16) is bilinear
in its decision variables in f 1 and s 2 whereas the constraint (2.15) is affine in its decision
variables in s 2 (f 1 is fixed in this case and does not contain any decision variables). Consequently,
S-procedure based sufficient conditions lead to affine SDPs for Question 1 and
bilinear SDPs for Question 2.
2.4 Preliminary Remarks
• Throughout this thesis, we only consider causal nonlinear input-output systems with
no time delay represented by ordinary differential equations of the form
ẋ(t) = f(x(t), w(t))
z(t) = h(x(t)),
(2.17)
where x is the state vector, w denotes the input/disturbance, and z denotes the output.
Occasionally, we will drop the time variable t in the notation and use
ẋ = f(x, w)
z = h(x)
(2.18)
in short.
• In several places, a relationship between an algebraic condition on some real variables
and input/output/state properties of a dynamical system is claimed. In nearly all of
these types of statements, we use same symbol for a particular real variable in the
algebraic statement as well as the corresponding signal in the dynamical system. This
could be a source of confusion, so care on the reader’s part is required.
• In the examples in the following chapters, we occasionally do not provide exact numerical
values used for computations. We either state the form or an approximation of the
20

specific expression. Similarly, we do not provide certifying Lyapunov functions and
multipliers. All missing data and results for all examples in this thesis are available
at http://jagger.me.berkeley.edu/~utopcu/dissertation.
2.5 Chapter Summary
We presented 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 was 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 were discussed.
21

Chapter 3
Simulation-Aided
Region-of-Attraction Analysis
For a dynamical system, the region-of-attraction (ROA) of a locally asymptotically stable
equilibrium point is an invariant set such that all trajectories emanating from points in
this set converge to the equilibrium point. For nonlinear dynamics, research has focused
on determining invariant subsets of the ROA because a closed form characterization of the
exact ROA may be too complicated.
This potential complication is avoided because of
several reasons: (i) Usually there is no systematic (numerical) procedure for computing the
exact shape of the ROA. (ii) A complicated characterization of the ROA has limited value
for post-analysis purposes. Among all other methods those based on Lyapunov functions
are dominant in the literature [24, 27, 75, 21, 20, 46, 58, 59, 32, 63, 62]. These methods
compute a Lyapunov function as a local stability certificate and sublevel sets of this Lyapunov
function, in which the function decreases along the flow, provide invariant subsets of
the ROA.
22

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

then for all x 0 ∈ Ω V,γ , the solution of (3.1) exists, satisfies φ(t; x 0 ) ∈ Ω V,γ for all t ≥ 0,
and lim t→∞ φ(t; x 0 ) = 0, i.e., Ω V,γ is an invariant subset of R 0 .
⊳
In order to enlarge the computed invariant subset of the ROA by choice of V, we define
a variable sized region Ω p,β , where p ∈ R[x] is a fixed positive definite convex polynomial,
and maximize β while imposing the constraint Ω p,β ⊆ Ω V,γ along with the constraints
(3.2)-(3.4). This can be written as
β ∗ (V) :=
max β
β>0,V ∈V
subject to
(3.2) − (3.4), Ω p,β ⊆ Ω V,γ .
(3.5)
Here V denotes the set of candidate Lyapunov functions over which the maximum is computed,
for example all continuously differentiable functions.
Remarks 3.1.1. The objective in (3.5) is to compute a tight inner estimate of the robust
ROA by choice of V . The β-sublevel sets of the fixed shape factor p are used to scalarize
this objective. However, note that for two fixed functions V 1 and V 2 satisfying constraints
of (3.5) with β equal to β 1 and β 2 respectively, even when 0 < β 1 < β 2 are largest positive
scalars such that Ω p,β1 ⊆ Ω V1 and Ω p,β2 ⊆ Ω V2 hold, it is not necessarily true that Ω V1 ⊆ Ω V2 .
In literature, mainly for quadratic Lyapunov functions, the volume of the computed sublevel
sets is used as the objective function (e.g. in [20]). Our choice of using the shape factor
p instead of the volume in the optimization objective is that it may be possible to choose p
to reflect the intent of the analyst or utilize prior knowledge about the system (see section
3.3.2 for the latter). ⊳
The problem in (3.5) is an infinite dimensional problem. In order to make it amenable
to numerical optimization (specifically SOS optimization), we restrict V to be all polynomials
of some fixed degree and use SOS sufficient conditions for polynomial nonnegativity.
25

Using simple generalizations of the S-procedure (Lemmas 2.3.2 and 2.3.3), we obtain sufficient
conditions for set containment constraints. Specifically, let l 1 and l 2 be a positive
definite polynomials (typically ɛx T x for some small real number ɛ). Then, since l 1 is radially
unbounded, the constraint
V − l 1 ∈ Σ[x] (3.6)
and V (0) = 0 are sufficient conditions for (3.2) and (3.3). By Lemma 2.3.2, if s 1 ∈ Σ[x],
then
− [(β − p)s 1 + (V − γ)] ∈ Σ[x] (3.7)
implies the set containment Ω p,β ⊆ Ω V,γ , and by Lemma 2.3.3, if s 2 , s 3 ∈ Σ[x], then
− [(γ − V )s 2 + ∇V fs 3 + l 2 ] ∈ Σ[x] (3.8)
is a sufficient condition for (3.4). Using these sufficient conditions, a lower bound on β ∗ (V)
can be defined as
βB ∗ (V, S) := max
β
V ∈V,β,s i ∈S i
(3.6) − (3.8),
subject to
(3.9)
V (0) = 0, and β > 0.
Here, the sets V and S i are prescribed finite-dimensional subsets of polynomials. Although
β ∗ B
depends on these subspaces, it will not always be explicitly notated. Note that since
the conditions (3.6)-(3.8) are only sufficient conditions,
β ∗ B(V, S) ≤ β ∗ (V) ≤ β ∗ (C 1 ).
The optimization problem in (3.9) is bilinear because of the product terms βs 1 in (3.7) and
V s 2 and ∇V fs 3 in (3.8). However, the problem has more structure than a general BMI
problem. If V is fixed, the problem becomes affine in S = {s 1 , s 2 , s 3 } and vice versa. In
section 3.2, we will construct a convex outer bound on the set of feasible V and sample from
26

this outer bound set to obtain candidate V ’s, and then solve (3.9) for S, holding V fixed.
This “qualifies” V as a certificate, and then further BMI optimization (e.g. coordinate-wise
affine search) is executed, using this (V, S) pair as an initial seed.
3.2 Relaxation of the Bilinear SOS Problem Using Simulation
Data
The usefulness of simulation in understanding the ROA for a given system is undeniable.
Faced with the task of performing a stability analysis (e.g., “for a given p, is Ω p,β contained
in the ROA”), a pragmatic, fruitful and wise approach begins with a linearized analysis
and at least a modest amount of simulation runs. Certainly, just one divergent trajectory
starting in Ω p,β certifies that Ω p,β ⊄ R 0 . Conversely, a large collection of only convergent
trajectories hints to the likelihood that indeed Ω p,β ⊂ R 0 . Suppose this latter condition is
true, let C be the set of N conv trajectories c converging to the origin with initial conditions
in Ω p,β . In the course of simulation runs, divergent trajectories d whose initial conditions
are not in Ω p,β may also get discovered, so let the set of d’s be denoted by D and N div be
the number of elements of D. Although C and D depend on β and the manner in which
Ω p,β is sampled, this is not explicitly notated.
With β and γ fixed, the set of Lyapunov functions which certify that Ω p,β ⊂ R 0 , using
conditions (3.6)-(3.8), is simply
{V ∈ R[x] : (3.6) − (3.8) hold for some s i ∈ Σ[x]} .
Of course, this set could be empty, but it must be contained in the convex set
{V ∈ R[x] : (3.10) holds},
27

where
∇V (c(t))f(c(t)) < 0,
l 1 (c(t)) ≤ V (c(t)), and V (c(0)) ≤ γ,
(3.10)
γ + δ ≤ V (d(t)),
for all c ∈ C, d ∈ D, and t ≥ 0, where δ is a fixed (small) positive constant. Informally,
these conditions simply say that any V which verifies that Ω p,β ⊂ R 0 using the conditions
(3.6)-(3.8) must, on the trajectories starting in Ω p,β , be decreasing and take on values
between 0 and γ. Moreover, V must be greater than γ on divergent trajectories. In fact,
with the exception of the strengthened lower bound on V (beyond mere positivity), the
conditions in (3.10) are even necessary conditions for any V ∈ C 1 which verify Ω p,β ⊂ R 0
using conditions (3.2)-(3.4).
3.2.1 Affine Relaxation Using Simulation Data
Let V be linearly parameterized as
V := {V ∈ R[x] : V (x) = z(x) T α},
where α ∈ R n b
and z is n b -dimensional vector of polynomials in x. Given z(x), constraints
in (3.10) can be viewed as constraints on α ∈ R n b
yielding the convex set
{α ∈ R n b
: (3.10) holds for V = z(x) T α}.
For each c ∈ C, d ∈ D, let T c and T d be finite subsets of the interval [0, ∞) including the
origin. A polytopic outer bound for this set described by finitely many constraints is
Y sim := {α ∈ R n b
: (3.11) holds},
28

where
[∇z(c(τ c ))f(c(τ c ))] T α < 0,
l 1 (c(τ c )) ≤ z(c(τ c )) T α, and z(c(0)) T α ≤ γ,
(3.11)
z(d(τ d )) T α ≥ γ + δ
for all c ∈ C, τ c ∈ T c , d ∈ D, and τ d ∈ T d . Note that z(c(0)) T α ≤ γ in (3.11) provides
necessary conditions for Ω p,β ⊆ Ω V,γ since c(0) ∈ Ω p,β for all c ∈ C. In practice, we
replace the strict inequality in (3.11) by [∇z(c(τ c ))f(c(τ c ))] T α ≤ −l 3 (c(τ c )), where l 3 is a
fixed, positive definite polynomial imposing a bound on the rate of decay of V along the
trajectories. More compactly and for future reference, express the inequalities represented
by (3.11) with this modification as Φ T α ≼ b.
The constraint that ∇V f be negative on a sublevel set of V implies that ∇V f is negative
on a neighborhood of the origin. While a large number of sample points from the trajectories
will approximately enforce this, in some cases (e.g. exponentially stable linearization) it
is easy to analytically express as a constraint on the low order terms of the polynomial
Lyapunov function. For instance, assume V has a positive-definite quadratic part, and that
separate eigenvalue analysis has established that the linearization of (3.1) at the origin, i.e.,
ẋ = ∇f(0)x, is asymptotically stable. Define
L(Q) := (∇f(0)) T Q + Q (∇f(0)) ,
where Q T = Q ≻ 0 is such that x T Qx is the quadratic part of V . Then, if (3.8) holds, it
must be that
L(Q) ≺ 0. (3.12)
Let
Y lin := {α ∈ R n b
: Q = Q T ≻ 0 and (3.12) holds}.
29

It is well-known that Y lin is convex [15].
Again, in practice, (3.12) is replaced by the
condition L(Q) ≼ −ɛI, for some small real number ɛ. Furthermore, define
Y SOS := {α ∈ R n b
: (3.6) holds}.
By [49], Y SOS is convex. Since Y sim , Y lin and Y SOS are convex,
Y := Y sim ∩ Y lin ∩ Y SOS
is a convex set in R n b. Equations (3.11) and (3.12) constitute a set of necessary conditions
for (3.6)-(3.8); thus, we have
Y ⊇ B := {α ∈ R n b
: ∃s 2 , s 3 ∈ Σ[x] such that (3.6) − (3.8) hold}.
This inclusion is depicted in Figure 3.1 for an illustrative example. Since (3.8) is not jointly
convex in V and the multipliers the shaded region B in Figure 3.1, may not be a convex set
and even may not be connected.
Remarks 3.2.1. Note that although we do not state explicitly, the first constraint in (3.11)
provides necessary conditions for the set containment Ω p,β ⊆ Ω V,γ since c i (0) ∈ Ω p,β for all
c i ∈ C.
⊳
A point in Y can be computed solving an affine (feasibility) SDP with the constraints
(3.6), (3.11) and (3.12). An arbitrary point in Y may or may not be in B. However, if we
generate a collection A := {α (k) } N V −1
k=0
of N V points distributed approximately uniformly
in Y, it may be that some of the points are in B. A point generation algorithm is discussed
in the next section.
A requirement of the algorithm is that Y is convex and bounded.
Convexity has already been established. If Y determined by the constraints (3.6), (3.11)
and (3.12) is not bounded, constraints on the absolute value of the components of α will be
imposed.
30

3.2.2 Hit-and-Run Random Point Generation Algorithm
The following random point generation algorithm is used to generate points in Y [61]:
Hit-and-run random point generation (H&R) algorithm: Given a positive integer N V
and α (0) ∈ Y, set k = 0
i. Generate a random vector ζ (k) = y (k) /‖y (k) ‖, where y (k) ∼ N (0, I nb ).
ii. Compute the minimum value of t (k) ≤ 0 and the maximum value of t (k) ≥ 0 such that
α (k) + t (k) ζ (k) ∈ Y and α (k) + t (k) ζ (k) ∈ Y.
iii. Pick w (k) from a uniform distribution on [0, 1];
iv. α (k+1) = w (k) (α (k) + t (k) ζ (k) ) + (1 − w (k) )(α (k) + t (k) ζ (k) ).
v. k = k + 1.
vi. If k = N V , return A. Otherwise, go to (i).
⊳
Figure 3.1 shows the initial point (big dot), an arbitrary point satisfying (3.6), (3.11) and
(3.12), and the points generated by H&R algorithm (small dots) along with the random
directions from step (i) (dashed line segments) for the illustrative example. Y is convex
and points are generated as convex combinations of points in Y (step (iv)); therefore, the
algorithm generates N V
points in Y. Since Y is compact, for sufficiently large N V , points
of A become approximately uniformly distributed in Y [61].
Step (ii) of the H&R algorithm involves solving a linear SOS optimization problem and
a LMI problem
t (k)
SOS :=
max t≥0 t s.t. z(x) T (α (k) + tζ (k) ) ∈ Σ[x],
t (k)
lin := max t≥0 t s.t. L(Q (k) + tΛ (k) ) ≼ −ɛI,
(3.13)
31

Φ T 3 α = b 3
Φ T 2 α = b 2
α (0) α (5)
α (1)
α (3)
Φ T 1
= α
b 1
α (4) α (2)
Φ T 4 α = b 4
Figure 3.1. Sets Y and B and points generated by H&R
column of Φ and j th component of b.
algorithm. Φ j and b j denote the j th
where Q (k) = Q (k)T ∈ R n×n and Λ (k) = Λ (k)T ∈ R n×n are such that x T Q (k) x and x T Λ (k) x
are the quadratic parts of z(x) T α (k) and z(x) T ζ (k) , respectively. t (k) and t (k) are determined
by the elementary expression
t (k) := min
t (k) := max
{max j
{
}
0, b j−Φ T j α(k)
, t (k)
Φ T j ζ(k)
SOS , t(k) lin
{ }
{min j 0, b j−Φ T j α(k)
, t (k)
Φ T j ζ(k) SOS , t(k) lin
}
,
}
,
where t (k)
SOS
and t(k)
lin are computed replacing max t≥0 with min t≤0 in (3.13).
3.2.3 Algorithms
Since a feasible value of β is not known a priori, an iterative strategy to simulate and
collect convergent and divergent trajectories is necessary. This process when coupled with
the H&R algorithm constitutes the Lyapunov function candidate generation.
32

Simulation and Lyapunov function generation (SimLF G) algorithm:
Given positive
definite convex p ∈ R[x], a vector of polynomials z(x) and constants β SIM , N conv , N V ,
β shrink ∈ (0, 1), and empty sets C and D, set γ = 1, N more = N conv , N div = 0.
i. Integrate (3.1) from N more initial conditions in the set {x ∈ R n : p(x) = β SIM }.
ii. If there is no diverging trajectory, add the trajectories to C and go to (iii). Otherwise,
add the divergent trajectories to D and the convergent trajectories to C, let N d
denote the number of diverging trajectories found in the last run of (i) and set N div
to N div + N d . Set β SIM to the minimum of β shrink β SIM and the minimum value of p
along the diverging trajectories. Set N more to N more − N d , and go to (i).
iii. At this point C has N conv elements. For each i = 1, . . . , N conv , let τ i satisfy c i (τ) ∈
Ω p,βSIM for all τ ≥ τ i . Eliminate times in T i that are less than τ i .
iv. Find a feasible point for (3.6), (3.11), and (3.12).
If (3.6), (3.11), and (3.12) are
infeasible, set β SIM = β shrink β SIM , and go to (iii). Otherwise, go to (v).
v. Generate N V Lyapunov function candidates using H&R algorithm, and return β SIM
and Lyapunov function candidates.
⊳
The suitability of a Lyapunov function candidate is assessed by solving two optimization
problems. Both problems require bisection and each bisection step involves a linear
SOS problem. Alternative linear formulations appear in section 3.6. These do not require
bisection, but generally involve higher degree polynomial expressions.
Optimization Problem 3.2.1. Given V ∈ R[x] (from SimLF G algorithm) and positive
33

definite l 2 ∈ R[x], define
γ ∗ L :=
max γ subject to s 2 , s 3 ∈ Σ[x], γ > 0,
γ>0,s 2 ∈S 2 ,s 3 ∈S 3
− [(γ − V )s 2 + ∇V fs 3 + l 2 ] ∈ Σ[x].
(3.14)
⊳
If Problem 3.2.1 is feasible, then γL ∗ > 0 and define
Optimization Problem 3.2.2. Given V ∈ R[x], p ∈ R[x], and γ ∗ L , solve
β ∗ L :=
max β subject to s 1 ∈ Σ[x], β > 0,
β>0,s 1 ∈S 1
− [(β − p)s 1 − (V − γ ∗ L )] ∈ Σ[x]. (3.15)
⊳
Although γ ∗ L and β∗ L depend on S 1, S 2 , and S 3 , this is not explicitly notated.
Assuming Problem 3.2.1 is feasible, it is true that
Ω p,β ∗
L
\{0} ⊆ Ω V,γ ∗
L
\{0} ⊂ {x ∈ R n : ∇V (x)f(x) < 0},
so V certifies that Ω p,β ∗
L
⊂ R 0 . Solutions to Problems 3.2.1 and 3.2.2 provide a feasible
point for the problem in (3.9). This feasible point can be further improved by solving the
problem in (3.9) using iterative coordinate-wise affine optimization schemes, one of which
is given next.
Coordinate-wise optimization (CW Opt) algorithm: Given V ∈ R[x], positive definite
l 1 , l 2 ∈ R[x], a constant ε iter , and maximum number of iterations N iter , set k = 0
i. Solve Problems 3.2.1 and 3.2.2.
ii. Given s 1 , s 2 , s 3 , and γL ∗ from step (i), set γ in (3.7)-(3.8) to γ∗ L
, solve (3.9) for V and
β, and set β ∗ L = β∗ B . 34

iii. If k = N iter or the increase in βL ∗ between successive applications of (ii) is less than
ε iter , return V, γL ∗ , and β∗ L
. Otherwise, set k to k + 1 and go to (i).
⊳
3.2.4 Discussion of Algorithms
The algorithms (SimLF G, Problems 3.2.1 and 3.2.2, and CW Opt) yield lower bounds
on β ∗ (C 1 ), as they produce a Lyapunov function which certifies that a particular value of
β satisfies Ω p,β ∈ R 0 . Upper bounds (i.e., values of β that are not certifiable) may also be
obtained. More specifically, diverging trajectories found in the course of simulation runs
provide upper bounds on β ∗ (C 1 ) while inconsistency of the constraints (3.6), (3.11), and
(3.12) provide upper bounds on βB ∗ . A diverging trajectory with the initial condition x 0
satisfying p(x 0 ) = β proves that Ω p,β cannot be a subset of the ROA, i.e., β ∗ (C 1 ) < β.
Furthermore, restricting Lyapunov function candidates to
V z := { z(x) T α : α ∈ R n }
b
has additional implications. Infeasibility of any of the constraints (3.6), (3.11), and (3.12)
for some value of β (recall (3.11) implicitly depends on β) verifies
β ∗ B (V z , S) ≤ β ∗ (V z ) < β,
regardless of the subspaces constituting S (see section 3.4.1 for an example). Moreover,
the gap between the value of β proven unachievable and what we actually certify, namely
a lower bound to βB ∗ (V z, S) , can be used as a measure of suboptimality introduced due to
the finiteness of the degree of the multipliers and the fact that the bilinear search and the
coordinate-wise linear search are only local optimization schemes.
35

Table 3.1. Parameters used in and results of SimLF G and CW Opt algorithms.
∂(V ) = 2 ∂(V ) = 4 ∂(V ) = 6
β SIM (initial) 4.0 4.0 4.0
β shrink 0.95 0.95 0.95
N V 50 50 50
N conv 100 200 500
N div 5 6 18
β ∗ L
β ∗ L
(before CW Opt) 1.33 1.75 1.05
(after CW Opt) 1.57 2.14 2.34
3.3 Examples
In the examples, l i (x) = 10 −6 x T x for i = 1, 2, 3.
3.3.1 Van der Pol Dynamics
The Van der Pol dynamics
ẋ 1 = −x 2 ,
ẋ 2 = x 1 + (x 2 1 − 1)x 2
have a stable equilibrium point at the origin and an unstable limit cycle. The limit cycle is
the boundary of the ROA. We applied SimLF G and CW Opt algorithms with p(x) = x T x.
Parameters used in and results of these runs are shown in Table 3.1. Figure 3.2 shows values
β ∗ L
before and after applying CW Opt algorithm.
Figure 3.3 shows computed invariant
subsets of the ROA.
We also performed further optimization initializing PENBMI with V ′ s generated by the
36

50
40
30
20
10
0
1.2 1.4 1.6 1.8 2 2.2 2.4 2.6
β
50
40
30
20
10
0
1.2 1.4 1.6 1.8 2 2.2 2.4 2.6
β
50
40
30
20
10
0
1.2 1.4 1.6 1.8 2 2.2 2.4 2.6
β
Figure 3.2. Histograms of βL ∗ before CW Opt (black bars) and β∗ L
after CW Opt (white bars)
for ∂(V ) = 2 (top), 4 (middle), and 6 (bottom).
37

3
2
1
0
−1
−2
−3
−2 −1 0 1 2
x 2
x 1
Figure 3.3. The invariant subsets of the ROA (dot: ∂(V ) = 2, dash: ∂(V ) = 4, and solid:
∂(V ) = 6 (indistinguishable from the outermost curve for the limit cycle)).
SimLF G algorithm. Practically, every seeded PENBMI run terminated with the same βB
∗
value which is the largest known (at least by us) value of β for which (3.9) is feasible with
the prescribed families of Lyapunov functions and multipliers. In addition, we performed
10 unseeded PENBMI runs for ∂(V ) = 4 and 6. Of these runs 90% and 50%, respectively,
terminated successfully (with an optimal value of β equal to that from the seeded PENBMI
runs).
Moreover, unseeded PENBMI runs took longer computation times than seeded
PENBMI runs. For comparison, seeded PENBMI runs took 3 − 8 and 11 − 24 seconds for
∂(V ) = 4 and 6, respectively, on a desktop PC, whereas they took 50 − 250 and 1000 − 2500
seconds, respectively, for unseeded PENBMI runs.
Remarks 3.3.1. It is worth mentioning that the invariant subsets of the ROA certified by
∂(V ) = 4 and 6 Lyapunov functions are larger than those in [58, 59]. A reason for this
38

improvement is that we use higher degree multipliers in (3.9)-(3.8). This demonstrates the
benefit of using higher degree multipliers when computationally tolerable.
3.3.2 Examples from the Literature
We present results obtained using the method from the previous section for the systems
in (3.16)-(3.22). (E 1 )-(E 3 ) are from [20], (E 4 ) and (E 7 ) are from [75], and (E 5 ) and (E 6 ) are
from [33] and [32], respectively. Since the dynamics in (E 1 )-(E 7 ) have no physical meaning
and there is no p given, we applied SimLF G algorithm sequentially: Apply SimLF G
algorithm with p(x) = x T x and N V = 1 for ∂(V ) = 2. Call the quadratic Lyapunov function
obtained ˆV . Set p to ˆV and apply SimLF G algorithm with this p and N V = 1 for ∂(V ) = 4.
For (E 5 )-(E 7 ), we further applied CW Opt algorithm with N iter = 10. Table 3.2 shows the
ratio of the volume of the invariant subset of the ROA obtained using this procedure to
that reported in the corresponding references. Empirical volumes of sublevel sets of V are
computed by randomly sampling a hypercube containing the sublevel set. Values in Table
3.2 are volumes normalized by π and 4π/3 for 2 and 3 dimensional problems, respectively.
For (E 4 ), (E 6 ), and (E 7 ), we also empirically verified that the invariant subsets of the ROA
reported in the corresponding references are contained in those computed by this sequential
procedure. Figure 3.4 and Figure 3.5 show the invariant subsets of the ROA for (E 6 ) and
(E 7 ) computed here and reported in [32] and [75], respectively.
⎧
⎪⎨
(E 1 ) :
⎪⎩
ẋ 1 = x 2 ,
ẋ 2 = −2x 1 − 3x 2 + x 2 1 x 2.
(3.16)
39

Table 3.2. Volume ratios for (E 1 )-(E 7 ).
example volume ratio example volume ratio
(E 1 ) 16.7/10.2 (E 2 ) 0.99/0.85
(E 3 ) 37.2/23.5 (E 4 ) 1.00/0.28
(E 5 ) 62.3 /7.3 (E 6 ) 35.0/15.3
(E 7 ) 1.44/0.70
⎧
⎪⎨
(E 2 ) :
⎪⎩
⎧
⎪⎨
(E 3 ) :
ẋ 1 = x 2 ,
(3.17)
ẋ 2 = −2x 1 − x 2 + x 1 x 2 2 − x5 1 + x 1x 4 2 + x5 2 .
ẋ 1 = x 2 ,
ẋ 2 = x 3 ,
(3.18)
⎪⎩
⎧
⎪⎨
(E 4 ) :
ẋ 3 = −4x 1 − 3x 2 − 3x 3 + x 2 1 x 2 + x 2 1 x 3.
ẋ 1 = −x 2 ,
ẋ 2 = −x 3 ,
(3.19)
⎧
⎪⎨
(E 7 ) :
⎪⎩
⎪⎩ ẋ 3 = −0.915x 1 + (1 − 0.915x 2 1 )x 2 − x 3 .
⎧
ẋ 1 = x 2 + 2x 2 x 3 ,
⎪⎨
(E 5 ) : ẋ 2 = x 3 ,
⎪⎩ ẋ 3 = −0.5x 1 − 2x 2 − x 3 .
⎧
ẋ 1 = −x 1 + x 2 x 2 3
⎪⎨
,
(E 6 ) : ẋ 2 = −x 2 + x 1 x 2 ,
⎪⎩ ẋ 3 = −x 3 .
ẋ 1 = −0.42x 1 − 1.05x 2 − 2.3x 2 1 − 0.5x 1x 2 − x 3 1 ,
ẋ 2 = 1.98x 1 + x 1 x 2 .
(3.20)
(3.21)
(3.22)
40

2
x 3
0
−2
−2
0
2
−4 −2 0 2 4 6
x 1
x 2
Figure 3.4. Invariant subset of the ROA for (E 6 ) reported in [32] (solid surface) and that computed
using the sequential procedure from section 3.3.2 (dotted surface).
1
0.5
0
−0.5
−1
−1.5
x 2
−2
−1.5 −1 −0.5 0 0.5 1 1.5 2
x 1
Figure 3.5. Invariant subset of the ROA for (E 7 ) reported in [75] (thick solid curve) and that
computed using the sequential procedure from section 3.3.2 (Ω V,γ ∗) (thin solid curve), and system
trajectories (dash-dot curves).
41

3.3.3 Controlled Short Period Aircraft Dynamics
Consider a cubic polynomial approximation of the short period pitch-axis model of an
aircraft 1 ⎡
ẋ p = ⎢
⎣
c 1 (x p )
q 2 (x p )
⎤ ⎡
+
⎥ ⎢
⎦ ⎣
l T b x p
b 2
0
⎤
u,
⎥
⎦
x 1
where x p = [x 1 x 2 x 3 ] T , x 1 , x 2 , and x 3 denote the pitch rate, the angle of attack, and
the pitch angle, respectively, c 1 is a cubic polynomial, q 2 is a quadratic polynomial, l 12
and l b are vectors in R 3 , b 2 ∈ R. The control input, elevator deflection, is determined by
ẋ 4 = −0.864y 1 − 0.321y 2 and u = 2x 4 , where x 4 is the controller state and the plant output
y = [x 1 x 3 ] T . Define x := [ x T ] T
p x 4 . We applied SimLF G and CW Opt algorithms with
p(x) = x T x, N V = 1, β SIM = 15.0, β shrink = 0.85. Results of these runs are shown in Table
3.3. In summary, the following bound for β ∗ (C 1 ) has been established
16.31 ≤ β ∗ (C 1 ) < 17.32.
This example will be repeatedly used in the subsequent chapters to demonstrate proposed
methods. Therefore, we also provide typical computation times for this example in Table
3.3. Additionally, we applied SimLF G and CW Opt algorithms with N V = 50. Figure 3.6
shows values β ∗ L
before and after applying CW Opt algorithm.
3.3.4 Pendubot Dynamics
The pendubot is an underactuated two-link pendulum with torque action only on the
first link. Third order polynomial approximation of the closed-loop dynamics (with a linear
1 The vector field is obtained by setting δ 1 = 1.52 and δ 20 in the model with uncertainty in section 4.5.2.
42

Table 3.3. Results of SimLF G and CW Opt algorithms. Upper bounds are established by
a separate run of SimLF G algorithm with N conv = 3000. The upper bound for ∂(V ) = 4 is
by a divergent trajectory whereas as the upper bound is by the infeasibility of (3.6), (3.11),
and (3.12) for the given β value. Representative computation times are on 2.0 GHz desktop
PC.
∂(V ) = 2 ∂(V ) = 4
N conv 300 500
β ∗ L
β ∗ L
(before CW Opt) 8.26 9.82
(after CW Opt) 9.34 16.31
upper bound 13.79 17.32
typical computation time 40 seconds 6 minutes
50
40
30
20
10
0
5 10 15
β
50
40
30
20
10
0
5 10 15
β
Figure 3.6. Histograms of βL ∗ before CW Opt (black bars) and β∗ L
after CW Opt (white bars)
for ∂(V ) = 2 (top) and 4 (bottom).
43

1.5
1
0.5
x 3
0
−0.5
−1
−1.5
−1.5 −1 −0.5 0 0.5 1 1.5
x 1
Figure 3.7. A slice of the invariant subset of the ROA (solid curve) and initial conditions
(with x 2 = 0 and x 4 = 0) for diverging trajectories (dots).
quadratic regulator to balance the pendubot about its upright position) is
ẋ 1 = x 2 ,
ẋ 2 = 782x 1 + 135x 2 + 689x 3 + 90x 4 ,
ẋ 3 = x 4
ẋ 4 = 279x 1 x 2 3 − 1425x 1 − 257x 2 + 273x 3 3 − 1249x 3 − 171x 4 .
Here, x 1 and x 3 are angular positions of the first link and the second link (relative to the
first link). We applied SimLF G algorithm sequentially as described in section 3.3.2 and
CW Opt algorithm with 10 iterations and obtained β ∗ L = 1.69.
Conversely, we found a
diverging trajectory with the initial condition ¯x with p(¯x) = 1.95 proving that
1.69 ≤ β ∗ (C 1 ) < 1.95.
Figure 3.7 shows the x 2 = 0 and x 4 = 0 slice of the invariant subset of the ROA along with
initial conditions (with x 2 = 0 and x 4 = 0) for some diverging trajectories.
44

3.3.5 Closed-loop Dynamics with Nonlinear Observer Based Controller
For the dynamics
ẋ 1 = u,
ẋ 2 = −x 1 + x 3 1 /6 − u,
y = x 2 ,
where x 1 and x 2 are the states, u is the control input and y is the output, an observer
L with polynomial vector field ż = L(y, z) with ∂(L) = 3 and a control law in the form
u = −145.9z 1 + 12.3z 2 , where z 1 and z 2 are the observer states, were computed in [57]. The
application of SimLF G algorithm with ∂(V ) = 2 and p from [57] and CW Opt algorithm
with N iter = 4 lead to β ∗ L = 0.32.
We also applied CW Opt algorithm (initialized with
the quadratic V found in the first application) with ∂(V ) = 4 and N iter = 6 and obtained
β ∗ L
= 0.52. Conversely, we found a diverging trajectory with the initial condition (¯x, ¯z)
satisfying p(¯x, ¯z) = 0.54 proving that 0.52 ≤ β ∗ (C 1 ) < 0.54.
3.4 Critique
3.4.1 Sampling vs. Simulating
A common question we get is “why simulate to get the sample points – just sample
some region, and impose ∇V (x)f(x) < 0 there.” There are a few answers to this. Intuitively,
even running a few simulations gives insight into the system behavior. Engineers commonly
use simulation to assess rough measures of stability robustness and ROA. Moreover, as
converse Lyapunov theorems [76] implicitly define a certifying Lyapunov function in terms
of the flow, it makes sense to sample the flow when looking for a Lyapunov function of a
specific form. Furthermore, we have the following observation demonstrating that merely
45

sampling some region and imposing ∇V (x)f(x) < 0 there carries misleading information.
Consider the Van der Pol dynamics with p(x) = x T x and let S β denote a finite sample of
Ω p,β . It can be shown that the set of quadratic positive definite functions V that satisfy
S 1.8 \{0} ⊂ {x ∈ R n : ∇V (x)f(x) < 0} (3.23)
is nonempty. In fact, for V (x) = 0.32x 2 1 − 0.25x 1x 2 + 0.31x 2 2 , (3.23) is satisfied (actually
for all x ∈ Ω p,1.8 , ∇V (x)f(x) ≤ −l 3 (x)). This naively suggests to draw samples from the
set of quadratic positive definite functions satisfying (3.23) in order to try to prove that
Ω p,1.8 ⊂ R 0 . However, simulations reveal a contradicting fact: Using trajectories with initial
conditions in S 1.8 for ∂(V ) = 2, i.e., with z(x) = [x 2 1 , x 1x 2 , x 2 2 ]T , constraints (3.6), (3.11)
(with γ = 1), and (3.12) turn out to be infeasible. This verifies that no quadratic Lyapunov
function can prove Ω p,1.8 ⊂ R 0 using conditions (3.6)-(3.8), with the additional constraint
that ˙V (x) ≤ −10 −6 x T x on all trajectories starting in Ω p,1.8 .
Recall though, that using
quartic Lyapunov functions we know β ∗ (V z , S) ≥ 2.14. By these observations, we have the
following series of inclusions for the subsets of the positive definite quadratic polynomials
{V : V certifies Ω p,β ⊂ R 0 using (3.6) − (3.8)}
⊂ {V : ∇V (c s (τ))f(c s (τ)) < 0 ∀τ, ∀s ∈ S β }
⊂ {V : ∇V (s)f(s) < 0 ∀ s ∈ S β },
where c s denotes the trajectory with the initial condition s ∈ S β . Therefore, merely sampling
instead of using simulations leads to a larger outer set from which the samples for
V are taken in step (v) of SimLF G algorithm and it is less likely to find a function that
certifies that Ω p,β ⊂ R 0 .
46

3.4.2 Extensions and Limitations
The number of decision variables N decision and the size N SDP of the matrix in the SDP
for checking existence of a SOS decomposition for a degree 2d polynomial in n variables
grows polynomially with n if d is fixed and vice versa [49]. However, N SDP and N decision get
practically intractable for the state-of-the-art SDP solvers even for moderate variables for
fixed d. Moreover, using higher degree Lyapunov functions and/or higher degree multipliers
as well as higher degree vector fields makes problems larger, and, in fact, the growth of the
problem size with the simultaneous increase in n and d is exponential.
Based on these
limitations, we believe that systems with cubic vector field are likely to be in the scope
of the analysis tools based on SOS programming up to 6-7 states with quartic Lyapunov
functions and 15-16 states with quadratic Lyapunov functions. It is worth re-emphasizing
that bilinearity introduces extra complexity as seen in sections 3.3.1 and 3.3.3.
H&R, SimLF G and CW Opt algorithms become more efficient using parallel computing.
Steps of the proposed analysis method can be summarized as follows: Given a set G
of initial conditions
i. Generate a random sample of initial conditions in G;
ii. Simulate the system starting from these initial conditions;
iii. Solve the convex feasibility problem with constraints (3.6), (3.11), and (3.12) (i.e.,
find a feasible point in Y);
iv. Sample Y for different Lyapunov function candidates;
v. Assess these Lyapunov function candidates;
47

vi. Further optimize initializing CW Opt from qualified Lyapunov function candidates
drawn from Y.
Note that H&R algorithm consequently steps (i) and (iv) can be trivially parallelized.
Similarly, simulations from different initial conditions, evaluating appropriate functions on
these simulation trajectories, and assessing different Lyapunov function candidates are steps
suitable for parallel computing. Moreover, step (iii) is only a feasibility problem with a large
number of LP constraints and a few small LMIs (as opposed to an optimization problem)
and may be solved utilizing a parallel implementation of alternating projection method [8].
Based on the state-of-the-art of the solvers for SDPs [9], the bottleneck in simulationaided
ROA analysis is the size of a single SDP that needs to be solved on a single processor.
An interesting complement for the simulation-aided methodology may be exploiting parallel
solvers for SDPs. Although not mature enough, parallel algorithms for SDPs have been
reported (see, for example, [36, 10] and references therein).
3.5 Sanity Check: Does Linear Stability Imply Existence of
SOS Certificates
Recall that any feasible solution for the optimization problem (3.9) with constraints
V ∈ R[x], V (0) = 0, s 1 ∈ Σ[x], s 2 ∈ Σ[x], s 3 ∈ Σ[x], γ > 0, β > 0
V − l 1 ∈ Σ[x]
− [(β − p)s 1 + (V − γ)] ∈ Σ[x]
− [(γ − V )s 2 + ∇V fs 3 + l 2 ] ∈ Σ[x],
(3.24a)
(3.24b)
(3.24c)
(3.24d)
48

characterizes an invariant subset of the ROA. When l 1 and l 2 have positive definite quadratic
parts, a necessary condition for these constraints is that the linearized dynamics is asymptotically
stable (which, in fact, is equivalent to the local asymptotic stability of the nonlinear
dynamics). In this section, we focus on systems governed by ordinary differential equations
of the form
ẋ = f(x) = Ax + f 2 (x) + f 3 (x), (3.25)
where f 2 and f 3 are vectors of quadratic and cubic polynomials, respectively, and A ∈ R n×n ,
and prove that asymptotic stability of the linearized dynamics is also sufficient for the
feasibility of the constraints (3.24) (for sufficiently small γ > 0).
This result supports
the claim that the local nonlinear analysis proposed in this chapter extends the classical
linearization based analysis for nonlinear systems which only assures the existence of an
infinitesimally small ROA but does not quantify its size [76].
To this end, we need the
following preliminary results.
Let z(x) be a vector of degree 2 monomials in x with no repetition and n z denote the
length of z(x). Let L i ∈ R n×nz be such that L i z(x) = x i x.
Fact 3.5.1. Let Q ∈ R n×n be symmetric, f(x) = x T Qx, and g(x) = c 1 x 2 1 + c 2x 2 2 +
. . . + c n x 2 n. Then, f(x)g(x) can be written in the form f(x)g(x) = z(x) T Hz(x) =
z(x) T [ c 1 L T 1 QL 1 + . . . + c n L T n QL n
]
z(x).
⊳
Proof. f(x)g(x) = x T Qx(c 1 x 2 1 + . . . + c nx 2 n) = ∑ n
i=1 c ix 2 i (xT Qx). Then, x 2 i (xT Qx) =
(x i x) T Q(x i x) = z(x) T L T i QL iz(x), and the result follows.
Lemma 3.5.1. Let Q ∈ R n×n be positive definite and define f(x) = x T Qx. Let c 1 , . . . , c n >
0 and g(x) = c 1 x 2 1 + c 2x 2 2 + . . . + c nx 2 n. Then, f(x)g(x) can be decomposed as
f(x)g(x) = z(x) T Hz(x),
49

where H can be chosen positive definite.
⊳
Proof. By Fact 3.5.1, there is a decomposition of f(x)g(x) in the form
f(x)g(x) = z(x) T Hz(z) = z(x) T [ c 1 L T 1 QL 1 + . . . + c n L T n QL n
]
z(x).
Let v ∈ R nz
be nonzero. Then
v T Hv
= v T [ c 1 L T 1 QL 1 + . . . + c n L T ]
n QL n v
⎡
⎤
c 1 Q
= v T L T . .. Lv,
⎢
⎥
⎣
⎦
c n Q
= v T Hv
where L := [ L T 1 . . . LT n
] T . Note that L has full column rank since every entry of z(x) can
be written as x j x k for some 1 ≤ j, k ≤ n; consequently, H ≻ 0.
Proposition 3.5.1. Let f be an n-vector of cubic polynomials in x satisfying f(0) = 0, and
let P ≻ 0, R 1 ≻ 0, R 2 ≻ 0, p(x) := x T P x, l 1 (x) := x T R 1 x, and l 2 (x) := x T R 2 x. If there
exists Q ≻ 0 such that A T Q + QA ≺ 0, then the constraints in (3.24) are feasible.
⊳
Proof. The proof is constructive. Let z(x) be as defined above, ˜Q ≻ 0 satisfy A T ˜Q + ˜QA ≼
−2R 2 and ˜Q ≽ R 1 (such ˜Q can be obtained by properly scaling Q). Let ɛ := λ min (R 2 ),
V (x) := x T ˜Qx, and H ≻ 0 be such that (x T x)V (x) = z(x) T Hz(x) (which exists by Lemma
3.5.1). Let M 2 ∈ R n×nz and symmetric M 3 ∈ R nz×nz satisfy
∇V f 2 (x)
∇V f 3 (x)
= x T M 2 z(x)
= z(x) T M 3 z(x).
50

Define
s 1 (x)
:=
λmax( ˜Q)
λ min (P )
c 2 := λmax(M 3+ 1 2ɛ M T 2 M 2)
λ min (H)
s 2 (x)
:= c 2 x T x
γ := ɛ
2c 2
β
:= γ
2s 1
s 3 (x) := 1
Clearly, s 1 ∈ Σ[x], s 2 ∈ Σ[x], and s 3 ∈ Σ[x]. Note that V (x)−l 1 (x) = x T (˜(Q)−R 1 )x ∈ Σ[x],
since ˜Q − R 1 ≽ 0.
b 1 (x) := − [(γ − V )s 2 + ∇V fs 3 + l 2 ]
⎡ ⎤T ⎡
⎤ ⎡
x
−γc 2 I − R 2 − (A T ˜Q + ˜QA) −M2 /2
= ⎢ ⎥ ⎢
⎥ ⎢
⎣ ⎦ ⎣
⎦ ⎣
z(x)
−M2 T /2
c 2H − M 3
} {{ }
B 1
⎡
⎤
B 1 ≽
⎢
⎣
ɛ
2 I −M 2/2
−M T 2 /2 cH − M 3
⎥
⎦ ≽ 0
x
z(x)
⎤
⎥
⎦
by the Schur’s complement formula. Consequently, b 1 (x) ∈ Σ[x]. Finally,
− [(β − p)s 1 + (V − γ)] =
⎡
⎢
⎣
1
x
⎤
⎥
⎦
T ⎡
⎢
⎣
−βs 1 + γ 0
⎤ ⎡
⎥ ⎢
⎦ ⎣
0 s 1 P − ˜Q
} {{ }
B 2
1
x
⎤
⎥
⎦ ,
where B 2 ≽ 0 and consequently b 2 ∈ Σ[x].
51

3.6 Appendix
Problems 3.2.1 and 3.2.2 compute lower bounds on the largest value of γ and β such
that, for given V and p, Ω V,γ \{0} ⊂ {x ∈ R n : ∇V (x)f(x) < 0} and Ω p,β ⊂ Ω V,γ . We
propose alternative formulations, that do not require line search, to compute similar lower
bounds. Labeled γa ∗ and βa, ∗ these are generally different than γL ∗ and β∗ L
. For h, g ∈ R[x]
and a positive integer d, define
Optimization Problem 3.6.1.
Optimization Problem 3.6.2.
µ o (h, g) := inf x≠0 h(x) subject to
g(x) = 0,
µ ∗ (h, g, d) := sup µ subject to
µ>0,r∈R[x]
(h − µ) ( x 2d
1 + · · · + ) x2d n − gr ∈ Σ[x].
Lemma 3.6.1. µ ∗ (h, g, d) ≤ µ o (h, g).
⊳
Proof. If there is no µ ∈ R for which there exists r ∈ R[x] such that (h −
µ) ( x 2d
1 + · · · + ) x2d n − gr ∈ Σ[x] holds, then µ ∗ (h, g, d) = ∞. Therefore, let µ ∈ R and
assume that there exists such that (h − µ) ( x 2d
1 + · · · + ) x2d n − gr ∈ Σ[x] holds. Then,
µ ≤ h(x) whenever g(x) = 0. Consequently, µ ∗ (h, g, d) ≤ h(x) whenever g(x) = 0 and
µ ∗ (h, g, d) ≤ µ o (h, g).
Lemma 3.6.1. Let g, h : R n → R be continuous, h be positive definite, g(0) = 0, and
g(x) < 0 for all nonzero x ∈ O, a neighborhood of the origin. Define γ o := µ o (h, g).
Then, the connected component of {x ∈ R n
: h(x) < γ o } containing the origin is a subset
of {x ∈ R n : g(x) < 0} ∪ {0}. ⊳
52

Proof. Suppose not and let x ≠ 0 be in the connected component of {x ∈ R n : h(x) < γ o }
containing the origin but g(x) ≥ 0. Then, there exists a continuous function ϑ : [0, 1] → R n
such that ϑ(0) = 0, ϑ(1) = x, and h(ϑ(t)) < γ o for all t ∈ [0, 1].
Since g(0) = 0 and
g(x) < 0 for all nonzero x ∈ O, there exists 0 < ɛ < 1 such that g(ϑ(ɛ)) < 0. Since x
is not in {x ∈ R n
: g(x) < 0} , g(ϑ(1)) ≥ 0. Since g and ϑ are continuous, there exists
t ∗ ∈ (0, 1] such that g(ϑ(t ∗ )) = 0, which implies h(ϑ(t ∗ )) ≥ γ o . This contradiction leads to
x ∈ {x ∈ R n : g(x) < 0}.
Corollary 3.6.1. Let V ∈ R[x] be a positive definite C 1 function and satisfy (3.12) and
V (0) = 0. Then, for all γ such that 0 < γ < µ o (V, ∇V f), the connected component of Ω V,γ
containing the origin is an invariant subset of the ROA.
⊳
Proof. Since the quadratic part of V is a Lyapunov function for the linearized system,
there exists a neighborhood O of the origin such that ∇V (x)f(x) < 0 for all nonzero
x ∈ O. By Lemma (3.6.1), the connected component of Ω V,γ containing the origin, a subset
of the connected component of {x ∈ R n : V (x) < µ o (V, ∇V f)} containing the origin,
is contained in {x ∈ R n
: ∇V (x)f(x) < 0} ∪ {0}. Corollary (3.6.1) follows from regular
Lyapunov arguments [76].
Corollary 3.6.2. For some positive integer d 1 , define γ ∗ a := µ ∗ (V, ∇V f, d 1 ). Then, if
γ < γ ∗ a for some positive integer d 1 , then the connected component of Ω V,γ containing the
origin is an invariant subset of the ROA.
⊳
Proof. By Lemma 3.6.1 we have γ < γ ∗ a ≤ µ o (V, ∇V f). Then, result follows from Corollary
(3.6.1).
Corollary 3.6.3. Let 0 < γ < γ ∗ a, d 2 be a positive integer, V, p ∈ R[x] be positive definite
53

and p be convex. Define β ∗ a := µ ∗ (p, V − γ, d 2 ). Then for any β < β ∗ a, Ω p,β ⊂ Ω V,γ and
Ω p,β ⊂ R 0 .
⊳
3.7 Chapter Summary
We proposed a method for computing invariant subsets of the region-of-attraction for
asymptotically stable equilibrium points of dynamical systems with polynomial vector fields.
We used 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 developed 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 reported promising results in all these directions.
54

Chapter 4
Local Stability Analysis for
Uncertain Nonlinear Systems
We consider the problem of computing invariant subsets of the region-of-attraction
(ROA) for uncertain systems with polynomial nominal vector fields and local polynomial
uncertainty description. The literature on ROA analysis for systems with uncertain dynamics
includes a generalization of Zubov’s method [17] and an iterative algorithm that
asymptotically gives the robust ROA for with time-varying perturbations [45]. Systems with
parametric uncertainties are considered in [18, 57, 74]. The focus in [18] is on computing
the largest sublevel set of a given Lyapunov function that can be certified to be an invariant
subset of the ROA. [57, 74] propose parameter-dependent Lyapunov functions which
lead to potentially less conservative results (compared to parameter-independent Lyapunov
functions) at the expense of increased computational complexity. Similar to other problems
in local analysis of dynamical systems based on Lyapunov arguments and S-procedure and
55

SOS relaxations [63, 58, 59, 57, 73, 60], our formulation leads to optimization problems
with bilinear matrix inequality (BMI) constraints.
The uncertainty description detailed in section 4.1 contains two types: uncertain components
in the vector field that obey local polynomial bounds and/or uncertain parameters
appearing affinely and multiplying polynomial terms. Using this description, we develop
a bilinear SDP to compute robustly invariant subsets of the ROA. The number of BMIs
(and consequently the number of variables) in this problem increases exponentially with
the sum of the number of components of the vector field containing uncertainty with polynomial
bounds and the number of uncertain parameters. One (suboptimal) way to deal
with this difficulty is first to compute a Lyapunov function for a particular system (imposing
extra robustness constraints) and then determine the largest sublevel set in which the
computed Lyapunov function serves as a local stability certificate for the whole family of
systems. Once a Lyapunov function is determined for the system in the first step, second
step involves solving smaller decoupled linear SDPs. Therefore, this sequential procedure is
well suited for parallel computation leading to relatively efficient numerical implementation.
Moreover, the method proposed in chapter 3 (see also [73, 70]), which uses simulation to aid
in the nonconvex search for Lyapunov functions, extends easily to the robust ROA analysis
using simulation data for finitely many systems from the family of possible systems (e.g.
systems corresponding to the vertices of the uncertainty polytope). For the examples in
this chapter, we implement this generalization of the simulation-aided ROA analysis method
along with the sequential suboptimal solutions technique.
56

4.1 Setup and Motivation
We now introduce the uncertainty description used in the rest of the chapter and explain
its usefulness in ROA analysis based on computing Lyapunov functions using SOS
programming. Consider the system governed by
ẋ(t) = f(x(t)) = f 0 (x(t)) + ϕ(x(t)) + ψ(x(t)), (4.1)
where f 0 , ϕ, ψ : R n → R n are locally Lipschitz. Assume that f 0 is known, ϕ ∈ ∆ ϕ , and
ψ ∈ ∆ ψ , where
∆ ϕ := {ϕ : ϕ l (x) ≼ ϕ(x) ≼ ϕ u (x) ∀x ∈ G},
∆ ψ := {ψ : ψ(x) = Ψ(x)α ∀x ∈ G, α l ≼ α ≼ α u }.
Here, G is a given subset of R n containing the origin, ϕ l and ϕ u are n dimensional vectors
of known polynomials satisfying
ϕ l (x) ≼ 0 ≼ ϕ u (x) for all x ∈ G,
α, α l , α u ∈ R N , and Ψ is a matrix of known polynomials. Let ϕ i , ϕ l,i , ϕ u,i , α i , α l,i , and α u,i
denote i-th entry of ϕ, ϕ l , ϕ u , α, α l , and α u , respectively. Define 1
∆ := ∆ ϕ + ∆ ψ .
We assume that f 0 (0) = 0, ϕ(0) = 0 for all ϕ ∈ ∆ ϕ (,i.e., ϕ l (0) = 0, and ϕ u (0) = 0), and
ψ(0) = 0 for all ψ ∈ ∆ ψ (,i.e., Ψ(0) = 0), which assures that all systems in (4.1) have a
common equilibrium point at the origin. In order to be able to use SOS programming, we
restrict our attention to the case where f 0 , ϕ l , ϕ u , and Ψ have only polynomial entries and
G is defined as
G := {x ∈ R n
: g(x) ≽ 0, g i ∈ R[x], i = 1, . . . , m}.
1 For subsets X 1 and X 2 of a vector space X , X 1 + X 2 := {x 1 + x 2 : x 1 ∈ X 1, x 2 ∈ X 2}.
57

Note that entries of ϕ do not have to be polynomial but have to satisfy local polynomial
bounds.
Motivation for this kind of system description stems from the following sources:
i. Perturbations as in (4.1) may be due to modeling errors, aging, disturbances and
uncertainties due to environment which may be present in any realistic problem.
Prior knowledge about the system may provide local bounds on the entries of ϕ
and/or bounds for the parametric uncertainties α. Moreover, uncertainties that do
not change system order can always be represented as in (4.1) (see p.339 in [38]).
ii. Analysis of dynamical systems using SOS programming is often limited to systems
with polynomial or rational vector field. In [47], a procedure for re-casting non-rational
vector fields into rational ones at the expense of increasing the state dimension is
proposed. Another way to deal with non-polynomial vector fields is to approximate
the vector field by its Taylor series expansion about the origin. For practical purposes
only finite number of terms can be used. Finite-term approximations are relatively
accurate in a restricted region containing the origin. However, they are not exact. On
the other hand, it may be possible to represent terms, for which the error between
the exact vector field and its finite-term approximation obey local polynomial bounds,
using ϕ in (4.1).
iii. SOS programming can be used to analyze systems with polynomial vector fields. The
number N decision of decision variables
⎡⎛
⎞
N decision = 1 n + d
⎢⎜
2 ⎣⎝
d
⎟
⎠
2
⎛
+ ⎜
⎝
n + d
d
⎞⎤
⎛
⎟
⎠⎥
⎦ − ⎜
⎝
n + 2d
d
⎞
⎟
⎠
58

Table 4.1. N SDP (left columns) and N decision (right columns) for different values of n and
2d.
2d
n 4 6 8 10
2 6 6 10 27 15 75 21 165
5 21 105 56 1134 126 6714 252 2e4
9 55 825 220 1e4 715 2e5 ⋆ ⋆
14 120 4200 680 ⋆ ⋆ ⋆ ⋆ ⋆
16 153 6936 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
and the size N SDP
N SDP =
⎛
⎜
⎝
n + d
of the matrix in the SDP for checking existence of a SOS decomposition for a degree
2d polynomial in n variables grows polynomially with n if d is fixed and vice versa
[49]. However, N SDP and N decision get practically intractable for the state-of-the-art
SDP solvers even for moderate values of n for fixed d (see Table 2, where solid lines
in the table represent a fuzzy boundary between tractable and intractable SDPs).
Moreover, using higher degree Lyapunov functions and/or higher degree multipliers
(used in the sufficient conditions for certain set containment constraints in section
4.2) as well as higher degree vector fields increases the problem size, and, in fact, the
growth of the problem size with the simultaneous increase in n and d is exponential.
Therefore, in order to be able to use SOS programming, one may have to simplify the
dynamics by truncating higher degree terms in the vector field. In this case, ϕ l and
ϕ u provide local bounds on the truncated terms. This is discussed further at the end
of section (4.2). It is also worth mentioning that bilinearity, a common feature of the
optimization problems for local analysis using Lyapunov arguments (see section 4.2),
d
⎞
⎟
⎠
59

introduces extra complexity [35] and therefore a further necessity for simplifying the
system dynamics.
In summary, representation in (4.1) and definitions of ∆ ϕ and ∆ ψ are motivated by uncertainties
introduced due to incapabilities in modeling and/or analysis.
4.2 Computation of Robustly Invariant Sets
In this section, we will develop tools for computing invariant subsets of the robust ROA.
The robust ROA is the intersection of the ROAs for all possible systems governed by (4.1)
and formally defined, assuming that the origin is an asymptotically equilibrium point of
(4.1) for all δ ∈ ∆, as
Definition 4.2.1. The robust ROA R r 0
of the origin for systems governed by (4.1) is
R r 0 := ⋂
δ∈∆
{x ∈ R n : lim
t→∞
φ(t; x 0 , δ) = 0},
where φ(t; x 0 , δ) denotes the solution of (4.1) with the initial condition x 0 .
⊳
The robust ROA is an open and connected subset of R n containing the origin and
is invariant under the flow of all possible systems described by (4.1) [45].
We focus on
computing invariant subsets of the robust ROA characterized by sublevel sets of appropriate
Lyapunov functions. Since the uncertainty description for ϕ and ψ holds only for x ∈ G,
we will also require the computed invariant set to be a subset of G. To this end, we modify
Lemma 3.1.1 such that a stricter version of condition (3.4) holds for (4.1) for all δ ∈ ∆
(,i.e., for all ϕ ∈ ∆ ϕ and ψ ∈ ∆ ψ ). Namely, for µ ≥ 0, we replace (3.4) by
Ω V \ {0} ⊂ {x ∈ R n : ∇V (x)f(x) < −µV (x)} (4.2)
60

Remarks 4.2.1. With nonzero µ, (4.2) not only assures that trajectories starting in Ω V stay
in Ω V and converge to the origin but also imposes a bound on the rate of exponential decay of
V certifying the convergence and provides an implicit threshold for the level of disturbances
that could drive the system out of Ω V . Therefore, one may consider the stability property
implied by (4.2) with nonzero µ to be more desirable in practice. With this in mind, all
subsequent derivations contain the µV term. The relaxed condition in equation (3.4) can be
recovered by setting µ = 0.
⊳
Proposition 4.2.1. For µ ≥ 0, if there exists a continuously differentiable function
V : R n → R such that, for all δ ∈ ∆,
V (0) = 0 and V (x) > 0 for all x ≠ 0, (4.3)
Ω V is bounded, (4.4)
Ω V ⊆ G, and (4.5)
Ω V \ {0} ⊂ {x ∈ R n : ∇V (x)(f 0 (x) + δ(x)) < −µV (x)} , (4.6)
hold, then for all x ∈ Ω V and for all δ ∈ ∆, the solution of (4.1) exists, satisfies φ(t; x 0 , δ) ∈
Ω V for all t ≥ 0, and lim t→∞ φ(t; x 0 , δ) = 0, i.e., Ω V is an invariant subset of R r 0 . ⊳
Proof. Proposition 4.2.1 follows from Lemma 3.1.1. Indeed, for any given system ẋ =
f 0 (x) + δ(x), (4.6) assures that (4.2) is satisfied. Then, for any fixed δ ∈ ∆ and for all
x ∈ Ω V , φ(t; x 0 , δ) exists and satisfies φ(t; x 0 , δ) ∈ Ω V for all t ≥ 0, lim t→∞ φ(t; x 0 , δ) = 0,
and Ω V is an invariant subset of {x ∈ R n
subset of R r 0 .
: φ(t; x 0 , δ) → 0}. Therefore, Ω V is an invariant
Remarks 4.2.2. Proposition 4.2.1 is conservative because Ω V
is invariant for both timeinvariant
and time-varying perturbations.
In fact, the conclusion holds for time-varying
61

δ(= ϕ + α) as long as ϕ l (x) ≼ ϕ(x, t) ≼ ϕ u (x) and α l ≼ α(t) ≼ α u for all x ∈ G and
t ≥ 0. Recall that, in the uncertain linear system literature, e.g. [7], the notion of quadratic
stability is similar, where a single quadratic Lyapunov function proves the stability of an
entire family of uncertain linear systems.
⊳
Note that ∆ has infinitely many elements; therefore, there are infinitely many constraints
in (4.6). Now, define
E ϕ := {ϕ : ϕ i ∈ R[x] and ϕ i is equal to ϕ l,i or ϕ u,i },
E ψ := {ψ : ψ(x) = Ψ(x)α, α i is equal to α l,i or α u,i },
and
E ∆ := E ϕ + E ψ .
E ∆ , a finite subset of ∆, can be used to transform condition (4.6) to a finite set of constraints
that are more suitable for numerical verification:
Proposition 4.2.2. If
Ω V \ {0} ⊆ {x ∈ R n : ∇V (x)(f 0 (x) + δ(x)) < −µV (x)} (4.7)
holds for all δ ∈ E ∆ , then (4.6) holds for all δ ∈ ∆.
⊳
Proof. Let ˜x ∈ Ω V
be nonzero and δ ∈ ∆. Then, ˜x ∈ G by (4.5); therefore, there exist
l 1 , · · · , l n , k 1 , . . . , k N (depending on ˜x) with 0 ≤ l i ≤ 1 and 0 ≤ k i ≤ 1 such that
δ(˜x) = Lϕ l (˜x) + (I − L)ϕ u (˜x) + Ψ(˜x)(Kα l + (I − K)α u ),
where L and K are diagonal with L ii = l i and K ii = k i . Hence, there exist nonnegative
numbers ν δ (determined from l’s and k’s) for δ ∈ E ∆ with ∑ δ∈E ∆
ν δ = 1 such that δ(˜x) =
62

∑
δ∈E ∆
ν δ δ(˜x). Consequently, by
∇V (˜x)(f 0 (˜x) + δ(˜x))
= ∇V (˜x)(f 0 (˜x) + ∑ δ∈E ∆
ν δ δ(˜x))
= ∑ δ∈E ∆
ν δ ∇V (˜x)(f 0 (˜x) + δ(˜x))
< − ∑ δ∈E ∆
ν δ µV (˜x)
= −µV (˜x),
the condition (4.6) holds.
In order to enlarge the computed invariant subset of the robust ROA, we define a
variable sized region Ω p,β = {x ∈ R n : p(x) ≤ β} , where p ∈ R[x] is a fixed, positive
definite, convex polynomial, and maximize β while imposing constraints (4.3)-(4.4), (4.5),
(4.7), and Ω p,β ⊆ Ω V . This can be written as an optimization problem.
β ∗ (V) :=
max
V ∈V,β>0
β subject to (4.8a)
V (0) = 0 and V (x) > 0 for all x ≠ 0, Ω V is bounded, (4.8b)
Ω p,β = {x ∈ R n : p(x) ≤ β} ⊆ Ω V , Ω V ⊆ G, and
Ω V \ {0} ⊆ {x ∈ R n : ∇V (x)(f 0 (x) + δ(x)) < −µV (x)} ∀δ ∈ E ∆ .
(4.8c)
(4.8d)
Here, V denotes the set of candidate Lyapunov functions over which the maximum is defined
(e.g. V may be equal to C 1 ).
In order to make the problem in (4.8) amenable to numerical optimization (specifically
SOS programming), we restrict V to be a polynomial in x of fixed degree and use SOS
sufficient condition for polynomial nonnegativity. Using Lemmas 2.3.2 and 2.3.3, we obtain
sufficient conditions for set containment constraints. Specifically, let l 1 and l 2 be a positive
definite polynomials (typically ɛx T x with some (small) real number ɛ). Then, since l 1 is
radially unbounded, the constraint
V − l 1 ∈ Σ[x] (4.9)
63

and V (0) = 0 are sufficient conditions for the constraints in (4.8b). By Lemma 2.3.2, if
s 1 ∈ Σ[x] and s 4k ∈ Σ[x] for k = 1, . . . , m, then
− [(β − p)s 1 + (V − 1)] ∈ Σ[x] (4.10)
g k − (1 − V )s 4k ∈ Σ[x], k = 1, . . . , m, (4.11)
imply the first and second constraints in (4.8c), respectively. By Lemma 2.3.3, if s 2δ , s 3δ ∈
Σ[x] for δ ∈ E ∆ , then
− [(1 − V )s 2δ + (∇V (f 0 + δ) + µV )s 3δ + l 2 ] ∈ Σ[x] (4.12)
is a sufficient condition for the feasibility of the constraint in (4.8d). Using these sufficient
conditions, a lower bound on β ∗ (V) can be defined as an optimization:
Proposition 4.2.3. Let β ∗ B
be defined as
β ∗ B (V poly, S) :=
max β subject to
V,β,s 1 ,s 2δ ,s 3δ ,s 4k
(4.9) − (4.12),
(4.13)
s 1 ∈ Σ[x], s 2δ ∈ Σ[x], s 3δ ∈ Σ[x], s 4k ∈ Σ[x],
V (0) = 0, V ∈ V poly , s 1 ∈ S 1 , s 2δ ∈ S 2δ , s 3δ ∈ S 3δ , s 4k ∈ S 4k , and β > 0. Here, V poly ⊂ V
and S’s are prescribed finite-dimensional subsets of R[x]. Then, β ∗ B (V poly, S) ≤ β ∗ (V poly ). ⊳
4.3 Implementation Issues
The optimization problem in (4.13) provides a recipe to compute subsets of R n that are
invariant under the flow of all possible systems described by (4.1). The number of constraints
in (4.13) (consequently the number of decision variables since each new constraint includes
new variables) increases exponentially with N and n − ¯n where ¯n is defined as the number
64

of entries of the vectors ϕ l and ϕ u satisfying ϕ l (x) = ϕ u (x) = 0 for all x ∈ G. Namely,
there are 2 (n−¯n)+N SOS conditions in (4.13) due to the constraint in (4.12). Revisiting the
discussion in item (iii) at the end of section 4.1, we note that covering the high degree vector
field with low degree uncertainty reduces the dimension of the SOS constraints but increases
(exponentially, depending on n − ¯n) the number of constraints. Consequently, the utility of
this approach will depend on n − ¯n and is problem dependent. Example (3) in section 4.5.1
illustrates this technique. This difficulty can be partially alleviated by accepting suboptimal
solutions for (4.13) in a sequential manner:
4.3.1 Sequential Suboptimal Solution Technique:
• Fix a finite sample ∆ sample of ∆ (for example ∆ sample can be chosen as the center of
∆) and solve
max β subject to
V,β,s 1 ,s 2δ ,s 3δ ,s 4k
V − l 1 ∈ Σ[x],
− [(β − p)s 1 + (V − 1)] ∈ Σ[x]
g k − (1 − V )s 4k ∈ Σ[x], k = 1, . . . , m,
(4.14)
− [(1 − V )s 2δ + (∇V (f 0 + δ) + µV )s 3δ + l 2 ] ∈ Σ[x] for δ ∈ ∆ sample ,
s 1 ∈ Σ[x], s 2δ ∈ Σ[x], s 3δ ∈ Σ[x], s 4k ∈ Σ[x],
V (0) = 0, V ∈ V poly , s 1 ∈ S 1 , s 2δ ∈ S 2δ , s 3δ ∈ S 3δ , s 4k ∈ S 4k , and β > 0. Let V sample
denote the optimizing V in (4.14).
65

• For each δ ∈ E ∆ , compute
γ δ :=
max γ subject to
γ>0,s 2δ ∈S 2δ ,s 3δ ∈S 3δ
−[(γ − V sample )s 2δ + (∇V sample (f 0 + δ) + µV sample )s 3δ + l 2 ] ∈ Σ[x],
s 2δ ∈ Σ[x], s 3δ ∈ Σ[x],
(4.15)
and define
γ subopt := min {γ δ : δ ∈ E ∆ } .
At this point, Ω Vsample ,γ subopt
is an invariant subset of the robust ROA.
• Determine the largest sublevel set Ω p,β subopt of p contained in Ω Vsample ,γsubopt by solving
β subopt := max
s 1 ∈S 1 ,β
β
subject to
−[(β − p)s 1 + V sample − γ subopt ] ∈ Σ[x]
s 1 ∈ Σ[x].
⊳
While the sequential procedure sacrifices optimality (i.e., β subopt ≤ βB ∗ ), it has practical
computational advantages. The constraints in (4.12) decouple in the problem (4.15). In
fact, for each δ ∈ E ∆ , the problem in (4.15) contains only a single constraint from (4.12).
Table 4.3.1 shows the number of decision variables in (4.13) (top entry in each cell) and the
number of decision variables in (4.15) (bottom entry in each cell) for ∂(V ) = 2, ∂(s 2δ ) = 2,
and ∂(s 3δ ) = 0. Despite a rapid increase in the number of decision variables in (4.13) with
n − ¯n + N, the number of decision variables in (4.15) does not vary with n − ¯n + N. Of
course, the number of decoupled small problems that have to be solved in the second step
of the sequential procedure still grows exponentially with n − ¯n + N. However, problems in
(4.15) can be solved independently for different δ ∈ E ∆ and therefore computations can be
66

Table 4.2. Number of decision variables in (4.13) (top entry in each cell of the table) and the
number of decision variables in (4.15) (bottom entry in each cell of the table) for ∂(V ) = 2,
∂(s 2δ ) = 2, and ∂(s 3δ ) = 0.
n
❳❳ n − ¯n + N ❳ ❳ ❳❳
2 6 10 14
1
24 458 2588 8718
10 218 1266 4306
3
164 3510 20312 69002
10 218 1266 4306
5
324
10
6998
218
40568
1266
137898
4306
trivially parallelized. Advantages of this decoupling may be better appreciated by noting
that one of the main difficulties in solving large-scale SDPs is the memory requirements of
the interior-point type algorithms [9]. Consequently, it is possible to perform ROA analysis
for systems with relatively reasonable number of states and/or uncertain parameters using
the proposed suboptimal solution technique.
4.4 Sanity Check: Does Robust Stability Imply Existence of
SOS Certificates
The solution of the optimization problem in (4.13) provides a characterization for invariants
subsets of the robust ROA. When the constraints in (4.13) are feasible with l 1
and l 2 having positive definite quadratic parts, then the linearization of the system (4.1) is
robustly stable with a common Lyapunov function. In this section, we show, for systems
with cubic vector fields, that if the linearization is robustly stable with common quadratic
Lyapunov function then there exist feasible solutions for the constraints in (4.13). For notational
simplicity, the development is only carried out for the parametric uncertainty case.
67

The result is straightforward to extend to the more general uncertainty description in (4.1)
with φ l and φ u vectors of cubic polynomials.
Let ∆ be a bounded polytope in R N and for α ∈ ∆ consider the uncertain nonlinear
dynamics 2 ẋ = f 0 (x) + ∑ N
i=1 α if i (x), (4.16)
where f 0 , f 1 , . . . , f m are vectors of cubic polynomials.
The vector field in (4.16) can be
re-written in the form
ẋ = f α (x) = A α x + f 2,α (x) + f 3,α (x), (4.17)
where
A α
f 2,α
:= ∇f 0 (0) + ∑ N
i=1 α i∇f i (0)
:= f 0,2 (x) + ∑ N
i=1 α if i,2 (x)
f 3,α
:= f 0,3 (x) + ∑ N
i=1 α if i,3 (x),
and for i = 0, . . . , m, f i,2 and f i,3 are quadratic and cubic polynomial parts of f i .
Proposition 4.4.1. Let f 0 , . . . , f N be cubic polynomials in x satisfying f 0 (0) = . . . =
f N (0) = 0, P ≻ 0, R 1 ≻ 0, R 2 ≻ 0, p(x) = x T P x, l 1 (x) = x T R 1 x, and l 2 (x) = x T R 2 x. For
δ ∈ ∆, let A δ be such that A δ x is the linear (in x) part of f 0 (x) + ∑ N
i=1 δ if i (x). If there
exists Q ≻ 0 satisfying A T δ Q + QA δ ≺ 0 for all δ ∈ E ∆ , then the constraints in ((4.13)) are
feasible.
⊳
Proof. Let z(x) be a vector of degree 2 monomials in x with no repetition and n z denote
the length of z(x). Let ˜Q ≻ 0 satisfy A δ ˜Q + ˜QAδ ≼ −2R 2 , for all δ ∈ E ∆ , and ˜Q ≽ R 1
(such ˜Q can be obtained by properly scaling Q).
Let ɛ = λ min (R 2 ), V (x) := x T ˜Qx,
and H be a positive definite Gram matrix for (x T x)V (x). Let M 2δ ∈ R n×nz , and M 3δ ∈
2 Note that, unlike previous sections, ∆ denotes a polytope in R N .
68

R nz×nz
be such that x T M 2δ z(x), and z(x) T M 3δ z(x) are cubic and quartic (in x) parts of
∇V (f 0 (x) + ∑ N
i=1 δ if i (x)), respectively. Define s 1 (x) = λ max ( ˜Q)/λ min (P ), s 2δ (x) = c 2δ x T x
with c 2δ = λ max
(
M3δ + 1 2ɛ M T 2δ M 2δ)
/λmin (H) , s 3δ (x) = 1, γ = min {ɛ/(2c 2δ ) : δ ∈ E ∆ } ,
and β = γ/(2s 1 ). Then, V − l 1 and − [(β − p)s 1 + (V − γ)] are SOS since they are positive
semidefinite quadratic polynomials. For δ ∈ E ∆ , b δ (x) = − [ (γ − V )s 2δ + ∂V
∂x f δs 3δ + l 2
]
=
[
x T z(x) T ] B δ
[
x T z(x) T ] T , where
B δ =
⎡
⎢
⎣
⎤
−γc 2δ I − R 2 − (A T ˜Q δ
+ ˜QA δ ) −M 2δ /2
⎥
⎦
−M2δ T /2
c 2δH − M 3δ
is positive semidefinite and consequently b δ ∈ Σ[x].
4.5 Examples
In the following examples, p(x) = x T x (except for example (2) in section 4.5.1), l 1 (x) =
10 −6 x T x and l 2 (x) = 10 −6 x T x. All computations use the generalization of the simulationbased
ROA analysis method from chapter 3.
4.5.1 Examples from the Literature
(1) Consider the following system from [18]:
ẋ 1 = x 2
ẋ 2 = −x 2 + α(−x 1 + x 3 1 ),
where α ∈ [1, 3] is a parametric uncertainty. We solved problem (4.13) with ∂(V ) = 2
and ∂(V ) = 4 for µ = 0, 0.01, 0.05, 0.1, 0.15, and 0.2. Note that µ = 0.244 is an upper
bound for the value of µ for which the problem in (4.13) can be feasible. 3 Figure 4.1
3 An upper bound ¯µ for the value of µ, for which the problem in (4.13) can be feasible, can be established
computing the maximum value of µ for which the linearized uncertain dynamics are quadratically stable.
69

1.5
1
0.5
x 2
0
−0.5
−1
−1.5
−1.5 −1 −0.5 0 0.5 1 1.5
x 1
Figure 4.1. Invariant subsets of ROA reported in [18] (black curve) and those computed
solving the problem in (4.13) with ∂(V ) = 2 (blue curve) and ∂(V ) = 4 (green curve) along
with initial conditions (red stars) for some divergent trajectories of the system corresponding
to α = 1.
Table 4.3. Optimal values of β in the problem (4.13) with different values of µ and ∂(V ) = 2
and 4.
❍ ❍❍❍❍ µ
∂(V ) ❍
0 0.01 0.05 0.1 0.15 0.2
2 0.623 0.603 0.494 0.404 0.277 0.137
4 0.771 0.763 0.742 0.720 0.698 0.676
shows the invariant subset of the robust ROA reported in [18] (black curve) and those
computed here with ∂(V ) = 2 (blue curve) and ∂(V ) = 4 (green curve) for µ = 0 along
with two points (red stars) that are initial conditions for divergent trajectories of the
system corresponding to α = 1. Table 4.3 shows the optimal values of β in the problem
(4.13) with ∂(V ) = 2 and 4 for different values of µ.
Here, ¯µ is defined as
where A δ := ∇ x(f 0 + δ)| x=0
.
¯µ := max µ subject to
µ≥0,P =P T ≽0
A T δ P + P A δ + µP ≺ 0, for all δ ∈ E ∆ ,
70

Table 4.4. Optimal values of β in the problem (4.13) with different values of µ and ∂(V ) = 4
and 6.
❍ ❍❍❍❍ µ
∂(V ) ❍
0 0.01 0.05 0.1 0.2 0.5 0.75
4 0.773 0.767 0.741 0.708 0.640 0.517 0.406
6 0.826 0.820 0.803 0.787 0.750 0.651 0.573
(2) Consider the system
ẋ 1 = −x 2 + 0.2αx 2
ẋ 2 = x 1 + (x 2 1 − 1)x 2,
where α ∈ [−1, 1] [57]. For easy comparison with the results in [57], let p(x) = 0.378x 2 1 −
0.274x 1 x 2 + 0.278x 2 2 and µ = 0. In [57], it was shown that Ω p,0.545 (with a single
parameter independent quartic V ), Ω p,0.772 (with pointwise maximum of two parameter
independent quartic V ’s), Ω p,0.600 (with a single parameter dependent quartic (in state)
V ), Ω p,0.806 (with pointwise maximum of two parameter dependent quartic (in state)
V ’s) are contained in the robust ROA. On the other hand, the solution of problem (4.13)
with ∂(V ) = 4 and ∂(V ) = 6 certifies that Ω p,0.773 and Ω p,0.826 are subsets of the robust
ROA, respectively. Figure 4.2 shows invariant subsets of the robust ROA computed
using ∂(V ) = 4 (green curve) and ∂(V ) = 6 (blue curve) along with the unstable limit
cycle (red curves) of the system corresponding to α = −1.0, −0.8, . . . , 0.8, 1.0. In order
to demonstrate the effect of the parameter µ on the size of the invariant subsets of
the robust ROA verifiable solving the optimization problem in (4.13), the analysis is
repeated with µ = 0.01, 0.05, 0.1., 0.2, 0.5, and 0.75. Note that µ = 0.769 is an upper
bound for the value of µ for which the problem in (4.13) can be feasible. Table 4.4
shows the optimal values of β in the problem (4.13) with ∂(V ) = 4 and 6 for different
values of µ.
71

3
2
1
x 2
0
−1
−2
−3
−2 −1 0 1 2
x 1
Figure 4.2. Invariant subsets of ROA with ∂(V ) = 4 (green curve) and ∂(V ) = 6 (blue
curve) along with the unstable limit cycle (red curves) of the system corresponding to
α = −1.0, −0.8, . . . , 0.8, 1.0.
(3) Consider the system governed by
⎡
ẋ =
⎢
⎣
−2x 1 + x 2 + x 3 1 + 1.58x3 2
−x 1 − x 2 + 0.13x 3 2 + 0.66x2 1 x 2
⎤
⎥
⎦ + ϕ(x), (4.18)
where ϕ satisfies the bounds
−0.76x 2 2 ≤ ϕ 1(x) ≤ 0.76x 2 2
−0.19(x 2 1 + x2 2 ) ≤ ϕ 2(x) ≤ 0.19(x 2 1 + x2 2 )
in the set G = {x ∈ R 2
: g(x) = x T x ≤ 2.1}. Figure 4.3 shows invariant subsets of
the robust ROA computed with ∂(V ) = 2 (green) and ∂(V ) = 4 (blue) along with two
points (red stars) that are initial conditions for divergent trajectories.
4.5.2 Controlled Short Period Aircraft Dynamics
We apply the robust ROA analysis for controlled short period aircraft dynamics with
two parametric uncertainties. Let x 1 , x 2 , and x 3 denote the pitch rate, the angle of attack,
72

1
0.5
x 2
0
−0.5
−1
−1.5 −1 −0.5 0 0.5 1 1.5
x 1
Figure 4.3. Invariant subsets of ROA with ∂(V ) = 2 (green) and ∂(V ) = 4 (blue) along
with initial conditions (red stars) for divergent trajectories.
and the pitch angle, respectively. Then, the uncertain dynamics are
⎡
⎤ ⎡
c 01 (x p ) + δ 1 c 11 (x p ) + δ1 2 q 31(x p )
l T b x p + b 11 + b 12 δ 1
ẋ p =
q 02 (x p ) + δ 1 l T 12
⎢
x p + δ 2 q 22 (x p )
+
b 21 + b 22 δ 2
⎥ ⎢
⎣
⎦ ⎣
0
x 1
⎤
u, (4.19)
⎥
⎦
where x p = [x 1 x 2 x 3 ] T , c 01 and c 11 are cubic polynomials, q 02 , q 22 , and q 31 are quadratic
polynomials, l 12 and l b are vectors in R 3 , b 11 , b 12 , b 21 , and b 22 are real scalars, and u,
the elevator deflection, is the control input (see the Appendix for the values of the missing
parameters). δ 1 ∈ [0.99, 2.05] models variations in the center of gravity in the longitudinal
direction and δ 2 ∈ [−0.1, 0.1] models variations in the mass. The control input is determined
by ẋ 4 = −0.864y 1 + −0.321y 2 and u = 2x 4 , where x 4 is the controller state and the plant
output y = [x 1 x 3 ] T . Define x := [ x T p
x 4
] T . The dependence of the vector field on δ1 is
not affine (because of the δ 2 1
term) and the method proposed in this chapter is not directly
73

Table 4.5. Optimal value of β in the first step, β sample , and β subopt with µ for ∂(V ) = 2 and
4.
∂(V ) = 2 ∂(V ) = 4
β sample 9.01 16.11
β subopt 4.12 0.43
applicable. Therefore, let us replace δ 2 1 by an artificial parameter δ 3 and cover the graph
{
(δ1 , δ 2 , δ 3 ) ∈ R 3 : δ 3 = δ 2 1, δ 1 ∈ [0.99, 2.05], δ 2 ∈ [−0.1, 0.1] }
by the polytope 4
{
(δ1 , δ 2 , δ 3 ) ∈ R 3 : 3.04δ 1 − 2.03 ≤ δ 3 ≤ 3.04δ 1 − 2.31, δ 1 ∈ [0.99, 2.05], δ 2 ∈ [−0.1, 0.1] } .
We implemented the sequential suboptimal solution technique with µ = 0 for ∂(V ) = 2 and
∂(V ) = 4 using ∆ sample = (1.52, 0, 1.52 2 ) (in the first step). Table 4.5 shows optimal value
of β in the first step, call β sample , and β subopt (computed in the final step). To further assess
the suboptimality of the results, we performed 500 simulations for the uncertain system
setting the uncertain parameters to the vertex values and found a diverging trajectory with
the initial condition x 0 satisfying p(x 0 ) = 8.51 (for (δ 1 , δ 2 , δ 3 ) = (2.05, 0.1, 3.92). The gap
between the value β subopt and the values β sample and p at the initial point of this divergent
trajectory may be due to the finite dimensional parametrization for V, the issues mentioned
in Remark 4.2.2, the fact that we only use sufficient conditions and/or suboptimality of
the sequential implementation used for this example. Extensions proposed in the following
chapter aim at partially reducing this suboptimality.
4 This covering polytope is obtained using the method proposed in section 5.2.
74

4.6 Chapter Summary
We proposed a method to compute provably invariant subsets of the region-of-attraction
for the asymptotically stable equilibrium points of uncertain nonlinear dynamical systems.
We considered 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 region-of-attraction. We demonstrated the method on three
examples from the literature and a controlled short period aircraft dynamics example.
As pointed out throughout the chapter, the methodology developed in this chapter is
restrictive and conservative due to the following:
• Affine dependence of the vector field on the uncertainty is assumed;
• Only polytopic uncertainty sets are allowed;
• A common Lyapunov function is to certify the stability of an entire family of systems.
These restrictive assumptions/choices have been made mainly due to computational considerations
and tools developed in this chapter provide the infrastructure for the extensions
in the following chapter:
• Extension to non-polytopic uncertainty sets and non-affine uncertainty dependence
• Reduction of conservatism by a branch-and-bound type refinement procedure in the
”uncertainty space” using tools of this chapter repeatedly for subsets of the uncertainty
set.
75

4.7 Appendix
Parameters for the uncertain controlled short period aircraft dynamics:
c 01 (x p ) = −0.24366x 3 2 + 0.082272x 1x 2 + 0.30492x 2 2 + 0.015426x 2x 3 − 3.1883x 1
− 2.7258x 2 − 0.59781x 3
l b = [0
− 0.041136 0] T
b 11 = 1.594150
q 02 (x p ) = −0.054444x 2 2 + 0.10889x 2x 3 − 0.054444x 2 3 + 0.91136x 1 − 0.64516x 2 − 0.016621x 3
b 21 = 0.0443215
c 11 (x p ) = 0.30765x 3 2 + 0.099232x2 2 + 0.12404x 1 + 0.90912x 2 + 0.023258x 3
b 12 = −0.06202
l 12 = [0 0.00045754 0] T
q 22 (x p ) = −0.054444x 2 2 + 0.10889x 2x 3 − 0.054444x 2 3 − 0.6445x 2 − 0.016621x 3
b 22 = 0.044321
76

Chapter 5
Extensions of the Robust
Region-of-Attraction Analysis:
Refinements and Non-affine
Uncertainty Dependence
In this chapter, we extend the applicability of the method proposed in the previous
chapter for estimating the robust ROA for uncertain nonlinear dynamical systems in two
directions:
i. The attention in chapter 4 was restricted to affine dependence on uncertain parameters.
Here, we propose an extension to non-affine (in particular polynomial) parameter
dependence based on replacing non-affine appearances of uncertain parameters in the
vector field by artificial parameters (increasing the dimension of the uncertain param-
77

eter space) and covering the graph of non-affine functions of uncertain parameters by
bounded polytopes in the appended uncertain parameter space.
ii. In chapter 4, we characterized invariant subsets of the robust ROA using parameterindependent
Lyapunov functions, i.e., a common Lyapunov function is to certify the
computed invariant subset of the ROA over the entire parameter uncertainty set.
Similar to quadratic stability analysis [7], where a single quadratic Lyapunov function
proves the stability of an entire family of uncertain linear systems, this characterization
is conservative, i.e., it leads to conservative inner estimates of the robust ROA. In order
to reduce the conservatism, we now propose a branch-and-bound type refinement procedure
where the uncertainty set is partitioned and a different parameter-independent
Lyapunov function is computed for each cell of the partition.
The refinement procedure computes lower and upper bounds on a “measure” of the size
of the computed invariant subset of the robust ROA (detailed in section 5.1) at each iteration.
The gap between lower and upper bounds decreases as the partition gets finer and
localizes the optimal value of this measure (which would be achieved if a different Lyapunov
function could be computed for every singleton in the uncertainty set). As another remedy
for the conservativeness of parameter-independent Lyapunov functions, polynomially
parameter-dependent Lyapunov functions are proposed in [18, 57]. Although SOS optimization
can be used with parameter-dependent Lyapunov functions, the ensuing optimization
problem is harder than that for parameter-independent Lyapunov functions mainly because
uncertain parameters are treated as new independent variables in addition to state variables.
Moreover, choosing a polynomially parameter-dependent basis for the Lyapunov
function may be less intuitive than the parameter-independent case. Motivated by these
78

difficulties, we restrict our attention to parameter-independent Lyapunov functions and
polytopic uncertainty sets (non-polytopic uncertainty sets can be handled by using a polytopic
cover). This approach offers two main advantages: (i) It is potentially more flexible
than using parameter-dependent Lyapunov functions since it does not require an a priori
parametrization of the Lyapunov function in the uncertain parameters. It simply reduces
the conservatism by partitioning the uncertainty set (see section 5.4.1). (ii) It leads to optimization
problems with smaller semidefiniteness constraints since uncertain parameters do
not explicitly appear in the constraints (see section 5.1). 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. We partially alleviate this difficulty by accepting suboptimal solutions
obtained using the sequential implementation from section 4.3. Recall that this implementation
is suitable for parallel computing offering a major advantage over approaches utilizing
parameter-dependent Lyapunov functions.
5.1 Setup and Estimation of the Robust ROA of Systems
with Affine Parametric Uncertainty
Consider the system governed by
ẋ(t) = f(x, δ) = f 0 (x(t)) +
m m∑
∑ pu
δ i f i (x(t)) + g j (δ)f m+j (x(t)), (5.1)
i=1
j=1
where f 0 , f 1 , . . . , f m , f m+1 . . . , f m+mpu
: R n → R n are vector valued polynomial functions
satisfying f 0 (0) = . . . = f m+mpu (0) = 0, and g 1 , . . . , g mpu
are scalar valued continuous
79

functions, and δ takes values in a bounded polytope ∆. 1 Note that for the case m pu = 0
results from chapter 4 apply. Next, we review the results of chapter 4 for this special case
as a basis for the extension of these results for the case where g 1 , . . . , g mpu
are in R[δ].
For a subset D ⊂ R m , define
M D := ⋂
{x ∈ R n : ∇V (x)f(x, δ) < 0} .
δ∈D
Proposition 5.1.1. If there exists a continuously differentiable function V : R n → R
such that
V (0) = 0 and V (x) > 0 for all x ≠ 0, (5.2)
Ω V = {x ∈ R n : V (x) ≤ 1} is bounded, and (5.3)
Ω V \{0} ⊂ M ∆ , (5.4)
then for all x 0 ∈ Ω V
and for all δ ∈ ∆, the solution of (5.1) ϕ(t; x 0 , δ) exists, satisfies
ϕ(t; x 0 , δ) ∈ Ω V for all t ≥ 0, and lim t→∞ ϕ(t; x 0 , δ) = 0, i.e., Ω V is an invariant subset of
the robust ROA.
⊳
Proposition 5.1.2. For the vector field in (5.1) with m pu = 0, M ∆ = M E∆ .
⊳
Proof. Proposition 5.1.2 follows from the more general result in Proposition 4.2.2.
Similar to previous chapters, we introduce a fixed shape factor p (a positive definite,
and convex polynomial) and maximize β sublevel set subject to conditions in Proposition
1 Unlike the previous chapter, we only consider the parametric uncertainty case in this chapter and ∆ is
a polytope in R m .
80

5.1.1 and Ω p,β ⊆ Ω V :
β opt
∆
(V) := max
V ∈V,β>0
β subject to (5.5a)
V (0) = 0 and V (x) > 0 for all x ≠ 0,
(5.5b)
Ω V = {x ∈ R n : V (x) ≤ 1} is bounded, (5.5c)
Ω p,β ⊆ Ω V ,
Ω V \ {0} ⊂ M E∆ .
(5.5d)
(5.5e)
Finally, using the generalized S-procedure and Positivstellensatz, we relax this problem into
a SOS programming problem: Let l 1 and l 2 be positive definite polynomials (typically ɛx T x
with some (small) real number ɛ). Let V poly ⊆ V, R[x], S 1 , S 2 , and S 3 be prescribed finitedimensional
subsets of R[x], and denote S = (S 1 , S 2 , S 3 ). For a polytopic subset D of ∆,
define β D (V poly , S) as
β D (V poly , S) :=
max
V ∈V poly ,β,s 1 ∈S 1 ,s 2δ ∈S 2 ,s 3δ ∈S 3
β subject to
s 1 ∈ Σ[x], s 2δ ∈ Σ[x], s 3δ ∈ Σ[x], for all δ ∈ E D , (5.6a)
β > 0, V (0) = 0, V ∈ V poly ,
V − l 1 ∈ Σ[x],
− [(β − p)s 1 + (V − 1)] ∈ Σ[x], and
(5.6b)
(5.6c)
(5.6d)
− [(1 − V )s 2δ + ∇V (f 0 + ∑ m
i=1 δ if i )s 3δ + l 2 ] ∈ Σ[x], for all δ ∈ E D . (5.6e)
The feasibility of the constraints is (5.6) is sufficient for the feasibility of the constraints
in (5.5). Therefore, β ∆ (V poly , S) ≤ β opt
∆
(V). For simplicity, we will omit the dependence of
β ∆ on V poly and S and the dependence of β opt
∆
needed).
on V in notation hereafter (unless explicitly
The optimization in (5.6) is naturally converted to a bilinear semidefinite program
81

(SDP), with 3 “types” of decision variables: the free parameters in V, the free parameters
in the s polynomials, and the free parameters introduced by the SOS constraints.
The
SDP is bilinear in the free parameters in V and multipliers s, as evidenced by the product
terms (e.g. V s 2δ , ∇V fs 3δ , etc). We have made significant pragmatic progress in obtaining
high-quality solutions to (5.6), using simulation to first derive a convex outer-bound on
the set of feasible V parameters (see chapter 3 and [70]) and then drawing samples from
that set to initialize the nonconvex search from a viable starting point. Nevertheless, the
nonconvexity of the feasible set is not to be taken lightly, and any numerical attempt to
compute β D should be itself treated as a lower bound of β D . If the optimization yields a
feasible point, the achieved objective value may be lower than the optimal objective value.
For this reason, upper bounds on the optimal objective value are also useful.
5.1.1 Upper Bounds
Trajectories of (5.1) that do not converge to the origin provide upper bounds for β opt
∆
consequently for β ∆ . Let β nc be a value of p attained on such a non-convergent trajectory.
Since every trajectory entering an invariant subset of the robust ROA has to converge to the
origin, Ω p,β nc cannot be a subset of the robust ROA; hence, the inequality β ∆ ≤ β opt
∆
holds.
and
< βnc
In order to establish another upper bound, fix β > 0. If there exists V ∈ V certifying
that Ω p,β is in the robust ROA through (5.5b)-(5.5e), then V has to be (i) positive for all
nonzero x ∈ R n , (ii) less than or equal to 1 on and decreasing along every trajectory of (5.1)
starting in Ω p,β and converging to the origin. Therefore, if no V ∈ V satisfies properties (i)
and (ii), then one can conclude that there is no V ∈ V satisfying (5.5b)-(5.5e) (i.e., certifying
that Ω p,β is in the robust ROA through (5.5b)-(5.5e)). Consequently, such a value of β, call
82

β lp , provides an upper bound on β opt
∆
(V). In the case V = {V : V
(x) = αT z(x)}, where
z(x) is a basis vector for V (e.g. a vector of polynomials) and α is a vector of real scalars
(decision variables in V ), constraints on V become affine constraints on α. When these
conditions are imposed for finitely many x (on convergent trajectories starting in Ω p,β ) and
finitely many δ ∈ ∆, checking their feasibility yields a linear programming problem with
the constraints on α
0 < z(x) T α ≤ 1,
)
m∑
α T ∇z(x)
(f T 0 (x) + δ i f i (x) < 0.
i=1
(5.7)
In practice, we replace strict inequalities in (5.7) by non-strict inequalities ɛ 1 x T x ≤ z(x) T α
and α T ∇z(x) T f(x) ≤ −ɛ 2 x T x where ɛ 1 and ɛ 2 are two small positive constants. Effectively,
the set of α satisfying (5.7) is an outer bound for the set of α such that V (x) = z(x) T α
certifies that Ω p,β is in the robust ROA through (5.5b)-(5.5e). This outer bound becomes
tighter as the number of constraints in (5.7) increases and/or additional necessary conditions,
such as necessary conditions from divergent trajectories and due to the stability of
the linearized dynamics (this latter case would lead a convex feasibility problem instead
of mere linear programming problem), are imposed. See section 3.4 and [70] for a related
discussion.
Remarks 5.1.1. For any subset of D of ∆, β D provides another upper bound for β ∆ .
However, computing β D requires to solve a nonconvex optimization problem.
practically, β D can not be used as a reliable upper bound.
Therefore,
⊳
83

5.2 Polynomial Parametric Uncertainty
We extend the development in section 5.1 to systems of the form (5.1) with m pu = 1
and a polynomial g 1 . We later propose a generalization for m pu ≥ 1.
Replacing g 1 (δ) by an artificial parameter φ, the dynamics in (5.1) can be written as
m∑
ẋ(t) = f 0 (x(t)) + δ i f i (x(t)) + φf m+1 (x(t)). (5.8)
Our approach is based on covering the graph of g
{
(ζ, g(ζ)) ∈ R m+1 : ζ ∈ ∆ }
i=1
by a bounded polytope Γ ⊂ R m+1 . Then, the dependence of the vector field in (5.8) on
the parameters (δ, φ) is affine and (δ, φ) takes values in the bounded polytope Γ. Therefore,
results from section 5.1 are applicable for the system in (5.8) by replacing ∆ by Γ.
A polytope covering the graph of g can be obtained by bounding g by an affine function
a T l δ + b l from below and by another affine function a T u δ + b u from above over the set ∆ :
⎧
⎫
ζ ∈ ∆,
⎪⎨
⎪⎬
Γ(a l , a u , b l , b u ) := (ζ, ψ) ∈ R m+1 : a T l ζ + b l ≤ ψ .
⎪⎩
a T u ζ + b u ≥ ψ
⎪⎭
Here, a l , a u ∈ R m and b l , b u ∈ R. Then, the volume, Volume(Γ(a l , a u , b l , b u )), of the polytope
Γ is a linear function of its arguments a l , a u , b l , and b u :
Volume(Γ(a l , a u , b l , b u ))
∫
:= (a u − a l ) T ζ + (b u − b l ) dζ l . . . dζ m
∆
∫
∫
= (a u − a l ) T ζdζ + (b u − b l ) dζ.
The polytope with smallest volume among such covering polytopes can be computed
∆
∆
84

through the optimization
Volume ∗ :=
min
a l ,a u,b l ,b u
Volume(Γ(a l , a u , b l , b u )) subject to
g(δ) − (a T l δ + b l) ≥ 0, ∀δ ∈ ∆,
(5.9)
g(δ) − (a T u δ + b u ) ≤ 0, ∀δ ∈ ∆.
Using Lemma 2.3.1, an upper bound for Volume ∗ can be computed by a linear SOS optimization
problem. To this end, let affine functions h i , i = 1, . . . , N, provide an inequality
description for ∆, i.e.,
∆ = {ζ ∈ R m : h i (ζ) ≥ 0, i = 1, . . . , N} .
Proposition 5.2.1. The value of the optimization problem
min
a l ,a u,b l ,b u,σ ui ∈S ui ,σ li ∈S li
Volume(Γ(a l , a u , b l , b u )) subject to
− g(δ) + (a T u δ + b u ) − ∑ N
i=1 σ ui(δ)h i (δ) ∈ Σ[δ], (5.10a)
g(δ) − (a T l δ + b l) − ∑ N
i=1 σ li(δ)h i (δ) ∈ Σ[δ], (5.10b)
σ ui ∈ Σ[δ], σ li ∈ Σ[δ] , i = 1, . . . , N,
is an upper bound for Volume ∗ . Here S’s are finite dimensional subsets of R[δ].
⊳
Remarks 5.2.1.
i. Note that Volume(Γ(a l , a u , b l , b u )) = Volume(Γ(0, a u , 0, b u )) − Volume(Γ(0, a l , 0, b l )),
and therefore the optimizing values of the variables a l , a u , b l , and b u in Proposition
5.2.1 can equivalently be computed by two smaller optimization problems:
min
σ ui ∈S ui ,a u,b u
Volume(Γ(0, a u , 0, b u )) subject to
(5.10a) and σ ui ∈ Σ[δ], i = 1, . . . , N,
85

and
max
σ li ∈S li ,a l ,b l
Volume(Γ(0, a l , 0, b l )) subject to
(5.10b) and σ li ∈ Σ[δ], i = 1, . . . , N.
ii. Higher order relaxations for semialgebraic set containment based on Positivstellensatz
[49] can be used for less conservative results at the expense of increased computational
cost.
⊳
In case m pu ≥ 1, m + 1 dimensional polytopes Γ 1 , . . . , Γ mpu covering graphs of
g 1 , . . . , g mpu , respectively, can be determined using the procedure proposed in this section
repeatedly. Then, a polytope covering the graph of (g 1 , . . . , g mpu ) can be constructed as the
intersection
˜Γ := ˜Γ i ∩ . . . ∩ ˜Γ mpu ,
where, for i = 1, . . . , m pu ,
˜Γ i := { (ζ, ψ) ∈ R m+mpu : (ζ, ψ i ) ∈ Γ i
}
.
The following proposition characterizes the extreme points of ˜Γ. For a similar result see [4].
Proposition 5.2.2. For i = 1, . . . , m pu , let a T li δ + b li and a T ui δ + b ui be affine functions
bounding g i over ∆ from below and above, respectively. Then, the set of vertices of ˜Γ is
E˜Γ
:= ⋃ {
(ζ, ψ1 , . . . , ψ mpu ) ∈ R m+mpu : ψ i = a T }
αiζ + b αi , α ∈ {l, u}, i = 1, . . . , m pu .
ζ∈E ∆
⊳
Proof. For j = 0, 1, 2, let θ [j] =
(
ζ [j] , ψ [j]
1 , . . . , ψ[j] m pu
)
∈ E˜Γ
be such that θ [0] = λθ [1] + (1 −
λ)θ [2] for some λ ∈ (0, 1). Since ζ [0] = λζ [1] + (1 − λ)ζ [2] , ζ [0] ∈ E ∆ , and ζ [1] , ζ [2] ∈ ∆, it
follows that ζ [0] = ζ [1] = ζ [2] (by the definition of vertices (extreme points)) [54]. Now, fix
86

an arbitrary i ∈ {1, . . . , m pu }. If ψ [0]
i
= a T li ζ[0] + b li , then
ψ [0]
i
= a T li ζ[0] + b li = λψ [1]
i
+ (1 − λ)ψ [2]
i
.
Since θ [1] , θ [2] ∈ ˜Γ, we have that ψ [1]
i
≥ a T li ζ[0] + b li and ψ [2]
i
≥ a T li ζ[0] + b li . If any of last two
inequalities is strict, we reach a contradiction
a T li ζ[0] + b li = ψ [0]
i
= λψ [1]
i
+ (1 − λ)ψ [2]
i
> a T li ζ[0] + b li ,
since λ ∈ (0, 1). Hence, ψ [0]
i
= ψ [1]
i
= ψ [2]
i
. Same reasoning can be repeated when ψ [0]
i
=
a T ui ζ[0] +b ui . Since i ∈ {1, . . . , m pu } is arbitrary, θ [0] = θ [1] = θ [2] and θ [0] is an extreme point
of ˜Γ [54]. Now, let ˜θ = (˜ζ, ˜ψ 1 , . . . , ˜ψ mpu ) /∈ E˜Γ.
Then, two cases are possible:
i. ˜θ /∈ ˜Γ : ˜θ cannot be a vertex of ˜Γ since ˜Γ is a convex polytope.
ii. ˜θ = (˜ζ, ˜ψ1 , . . . , ˜ψ mpu ) ∈ ˜Γ: either ˜ζ is in the interior of ∆ and consequently ˜θ cannot
be a vertex of ˜Γ or ˜ζ ∈ E ∆ and all inequalities a T ˜ζ li
+ b li ≤ ˜ψ i ≤ a T ˜ζ ui + b ui are actually
strict. In the latter case, there exists λ ∈ (0, 1) such that ˜θ = (˜ζ, ˜ψ 1 , . . . , λ(a T ˜ζ li
+b li )+
(1 − λ)(a T ˜ζ ui + b ui ), . . . , ˜ψ mpu ) and therefore is not an extreme point of ˜Γ.
Remarks 5.2.2.
i. Proposition 5.2.2 gives one specific procedure to cover the graph of a polynomial function
by a convex polytope. Further research that advances graph covering strategies and
quantifies the trade-off between the number of vertices and the volume of the covering
polytope would be applicable to the robust ROA problem.
ii. Proposition can be used with bounded non-polynomial functions g 1 , . . . , g mpu
as affine upper and lower bounds are provided.
as long
⊳
87

5.3 Branch-and-Bound Type Refinement in the Parameter
Space
The optimization problem in (5.6), when applied with D = ∆, provides a method for
computing invariant subsets of the robust ROA characterized by a single Lyapunov function.
Therefore, results by (5.6) may be conservative: certified invariant subset may be too small
relative the robust ROA. On the other hand, a less conservative estimate of the robust ROA
can be obtained by solving (5.6) for each δ ∈ ∆ with D = {δ}: β∆ ∗ , where, for a subset
D ⊆ ∆, β ∗ D
is defined as
β
∗ D := min
δ∈D β {δ}, (5.11)
is greater than or equal to β ∆ . 2 However, computing β ∗ ∆
requires solving an optimization
problem for each δ ∈ ∆, and consequently is impractical.
In the following, we propose
an informal “branch-and-bound” type procedure for computing lower and upper bounds
for β∆ ∗ , i.e., localizing the value of β∗ ∆
. The method is based on computing a different
Lyapunov function for each cell of a finite partition of ∆. Therefore, it potentially leads
to less conservative estimates of the robust ROA compared to directly solving (5.6) with
D = ∆ and more conservative estimates compared to Ω p,β ∗
∆
.
5.3.1 Branch-and-Bound Algorithm
Branch-and-bound is an algorithmic method for global optimization. The method is
based on two steps: first the search region is covered by smaller subregions (branching) and
then upper and lower bounds for the objective function restricted to each subregion are
computed (bounding) [40]. A formal B&B application converges to the global optimum
2 Note that for a singleton {δ}, E {δ} = {δ}.
88

if the difference between the upper and lower bounds uniformly converges to zero as the
“size” of the subregions goes to zero. Without specific convergence guarantees, these steps
are repeated refining the partition of the search region until the gap between the upper
and lower bounds gets smaller than a prescribed tolerance or a maximum number of subpartitioning
is reached.
For a subset D ⊆ ∆, let L(D) and U(D) be lower and upper bounds for β ∗ D . For a
partition D of ∆, define
L D := min D∈D
U D := min D∈D
L(D)
U(D).
Then, it follows that
L D ≤ β ∗ ∆ ≤ U D .
Using L D and U D , the following branch-and-bound type refinement can be implemented for
localizing β ∗ ∆ [5]
Branch-and-Bound (B&B) Algorithm:
Given an initial partition D 0 of ∆, positive
integer N iter , and positive scalar ɛ > 0,
• k ← 0;
• compute L D k and U D k;
• while k ≤ N iter and U D k − L D k > ɛ
– k ← k + 1;
– pick ˜D ∈ D k−1 such that β ˜D
= L D k−1;
– split ˜D into ˜D I and ˜D II ;
– form D k from D k−1 by removing ˜D and adding ˜D I and ˜D II ;
89

– compute L D k and update U D k;
• return D exit := D k and corresponding lower and upper bounds.
⊳
Next, we discuss the generation of lower and upper bounds used in the B&B Algorithm.
5.3.2 Computing Lower Bounds
A lower bound L D k for β ∗ ∆ associated with the partition Dk can be obtained as L D k :=
min D∈D k
β D , since, for each δ ∈ ∆, there exists D ∈ D k such that δ ∈ D. Furthermore,
for every D ∈ D k , β ∆ ≤ β D since D ⊆ ∆. Then, taking the minimum of β D over D ∈ D k ,
we obtain the inequality β ∆ ≤ L D k. Consequently, the inequality
β ∆ ≤ L D k ≤ β ∗ ∆
holds.
Remarks 5.3.1. If the problem in (5.6) is infeasible for some D ∈ D k , then L D k = 0.
Therefore, further refinement of the partition is needed in order to get a nonzero lower
bound.
⊳
5.3.3 Computing Upper Bounds
Divergent trajectories and infeasibility of certain necessary conditions for the constraints
in (5.11) provide upper bounds for β ∗ ∆ . βnc , smallest value of p attained on a non-convergent
simulation trajectory, is an upper bound for β ∗ ∆ . βlp , as defined in section 5.1.1, provides an
upper bound for β∆ ∗ , but only when constraints in (5.7) are imposed for a singleton δ ∈ ∆
(see (5.11)).
90

Remarks 5.3.2. The inequalities L D k ≤ L D k+1 ≤ β∆ ∗ hold for k = 0, 1, 2, . . . since for each
D ∈ D k+1 there exists D ′ ∈ D k such that D ⊆ D ′ . When U D k
is defined as the smallest
value of the upper bounds found up to the k-th step of B&B Algorithm, the inequality
β∆ ∗ ≤ U D k+1 ≤ U Dk holds. Therefore, the B&B Algorithm generates a sequence of upper
and lower bounds for β ∗ ∆
that satisfy
L D k ≤ L D k+1 ≤ β ∗ ∆ ≤ U D k+1 ≤ U D k for k = 0, 1, 2, ...,
and the value of β ∗ ∆
is better localized as k increases.
⊳
5.3.4 Implementation Issues
The B&B type refinement procedure with the lower bound computed through the problem
in (5.6) provides a pragmatic approach for robust local stability analysis. However, as
noted in section 4.3 the number of constraints in (5.6) and consequently the number of decision
variables increase exponentially with m+m pu because (5.6e) contains a SOS constraint
for each vertex value of the uncertainty polytope. The increase in the problem size may
render the problem in (5.6) computationally challenging for even modest values of m+m pu .
To partially alleviate this difficulty, we use the sequential suboptimal solution technique
proposed in section 4.3 for lower bound computation along with a generalization of the
simulation-aided technique from chapter 3 in the first step.
Remarks 5.3.3. Let D ⊆ ∆ and D sample ⊂ D be a singleton. Then, the value β Dsample −
β subopt
D
is always nonnegative and can be interpreted as a measure of potential improvement
in the lower bound for β ∗ ∆
from further sub-dividing D in the B&B refinement procedure.
Therefore, it may be used as a stopping criterion in the B&B algorithm.
However, we
re-emphasize that β Dsample is computed solving a nonconvex optimization problem. ⊳
91

Finally, the following continuity result (based on a stricter version of Proposition 5.1.1
from chapter 4 - namely Propositions 4.2.2 and 4.2.1- which replaces ∇V (x)f(x) < 0 by
∇V (x)f(x) < µV (x) for some positive scalar µ) is a first step toward a convergence analysis
of the implementation of the B&B algorithm for estimating the robust ROA.
Lemma 5.3.1. Let µ > 0 and 0 < γ < 1 be given. Let V 0 be a positive definite quadratic
polynomial, s 20 ∈ Σ[x] be a quadratic polynomial, and s 30 > 0 such that
b(x) := − [(1 − V 0 )s 20 + (∇f 0 + µV 0 )s 30 + l 2 ] ∈ Σ[x].
Then, there exists ε > 0 and α > 0 such that
[
(
)
]
m∑
b δ (x) := − (γ − V 0 )(s 20 + αx T x) + (∇ f 0 + δ i f i + µ 2 V 0)s 30 + l 2 ∈ Σ[x]
i=1
for all δ ∈ R m with δ i ∈ [−ε, ε].
⊳
Proof. b δ (x) can be written as
b δ (x)
= b(x) + ˜b δ (x)
= b(x) + [ (1 − γ)s 20 (x) − γαx T x + αV 0 (x)x T x − ∑ m
i=1 δ i∇V 0 (x)f i (x)s 30 + µ 2 V (x)s 30]
.
Using ideas from the proofs of Lemmas 3.5.1 and Proposition 3.5.1, it can be shown that
there exist α > 0 and ε > 0 such that ˜b δ ∈ Σ[x] for all δ ∈ R m with δ i ∈ [−ε, ε].
5.4 Examples
In all examples, l 1 (x) = l 2 (x) = 10 −6 x T x and p(x) = x T x.
92

1
0.8
0.6
φ
0.4
0.2
0
0 0.2 0.4
δ
0.6 0.8 1
Figure 5.1. Polytopic cover for {(ζ, ψ) ∈ R 2 : ζ ∈ [0, 1], ψ = ζ 2 } with 2 cells (red) and 4
cells (yellow). Black curve is the set {(ζ, ψ) ∈ R 2 : ζ ∈ [0, 1], ψ = ζ 2 }.
5.4.1 An Example From the Literature
Consider the system [19] governed by
ẋ =
⎡
⎢
⎣
⎤ ⎡
−x 1
⎥
⎦ + ⎢
⎣
3x 1 − 2x 2
⎤ ⎡
−6x 2 + x 2 2 + x3 1
⎥
⎦ δ + ⎢
⎣
−10x 1 + 6x 2 + x 1 x 2
⎤
4x 2 − x 2 2
⎥
⎦ δ2 ,
12x 1 − 4x 2
where the uncertain scalar parameter δ takes values in [0, 1]. Following the procedure from
section 5.2, we replaced δ 2 by an artificial parameter φ. With the initial partition {[0, 1]},
we applied the refinement procedure for ∂(V ) = 2 and ∂(V ) = 4. Figure 5.1 shows the
polytopic cover for {(ζ, ψ) ∈ R 2 : ζ ∈ [0, 1], ψ = ζ 2 } with 2 cells (red) and 4 cells
(yellow). Upper and lower bounds for β[0,1] ∗ are shown in Figure 5.2 (top for ∂(V ) = 2
and bottom for ∂(V ) = 4). Curves with “◦” are for the lower bounds obtained by directly
solving (5.6) with D taken as the vertices of the corresponding cell, curves with “⋄” are for
the lower bounds obtained by applying the sequential procedure from section 4.3 by taking
∆ sample (in the first step) as the center of the corresponding cell, and curves with “×” (in
the top figure only) and “⋆” show β nc and β lp , respectively.
93

1
β lp
β nc
β
0.5
0
1
lower bounds
10 20 30 40
number of iterations
50
β nc
β
0.5
0
lower
bounds
20 40 60
number of iterations
Figure 5.2. Curves with “◦” are for the lower bounds obtained by directly solving (5.6) with
D taken as the vertices of the corresponding cell, curves with “⋄” are for the lower bounds
obtained by applying the sequential procedure from section 4.3 by taking ∆ sample (in the
first step) as the center of the corresponding cell, and curves with “×” (in the top figure
only) and “⋆” show β nc and β lp , respectively.
94

1.5
1
x 2
0.5
0
−0.5
−1
−1 −0.5 0 0.5 1
x 1
Figure 5.3. Estimates of the robust ROA: from [19] (black), using the branch-and-bound
based method for ∂(V ) = 2 (red) and ∂(V ) = 4 (green).
5.4.2 Controlled Short Period Aircraft Dynamics
We apply the branch-and-bound type refinement procedure for the uncertain controlled
short period pitch axis model of an aircraft used in Example 4.5.2 with ∂(V ) = 2 and
∂(V ) = 4 using the two-step implementation from section 5.3.4 on a computer cluster with
9 processors: after the first B&B iteration, the cell with the smallest lower bound for β∆ ∗ is
subdivided into 3 subcells and cells with 2-nd, 3-rd, and 4-th smallest lower bounds for β ∗ ∆
are sub-divided into 2 subcells. Figure 5.4 shows the lower bounds for β ∗ ∆
with ∂(V ) = 2
(green solid curve with “×” marker) and ∂(V ) = 4 (blue solid curve with “⋄” marker) and
β nc (red solid curve with “◦” marker) computed at the centers of the cells generated by
the B&B Algorithm for the ∂(V ) = 4 run. Dashed curves are for (computed values of)
β {δ} where δ is the center of the cell with the smallest lower bound at the corresponding
step of the B&B refinement procedure for ∂(V ) = 2 (green curve with “×” marker) and
∂(V ) = 4 (blue curve with “⋄” marker).
Figure 5.5 shows the final partition generated
by the B&B algorithm for the ∂(V ) = 4 run. β nc , smallest value of p attained on non-
95

15
β10
"quasi−upper" and lower bounds for ∂(V) = 4
β nc
5
"quasi−upper" and lower bounds for ∂(V) = 2
0
0 1 2 3 4 5 6
number of B&B steps
Figure 5.4. Lower bounds for β∆ ∗ with ∂(V ) = 2 (green solid curve with “×” marker) and
∂(V ) = 4 (blue solid curve with “⋄” marker) and β nc (red solid curve with “◦” marker)
computed at the centers of the cells generated by the B&B Algorithm for the ∂(V ) = 4
run. Dashed curves are for (computed values of) β {δ} where δ is the center of the cell with
the smallest lower bound at the corresponding step of the B&B refinement procedure for
∂(V ) = 2 (green curve with “×” marker) and ∂(V ) = 4 (blue curve with “⋄” marker).
convergent simulation trajectories, is 8.56 and obtained for (δ 1 , δ 2 ) = (2.039, −0.099) and
the initial condition (0.17, 2.65, −0.10, 1.24).
5.4.3 Controlled Short Period Aircraft Dynamics with Unmodeled Dynamics
Consider the closed-loop dynamics in Figure 5.6 where uncertain first-order dynamics
are introduced between the controller output (v) and the plant input (u) from section 5.4.2
u(s) = 1.25 + G(s, δ 3 , δ 4 )v(s) =
[
]
s − δ 4
1.25 + 0.75δ 3 v(s). (5.12)
s + δ 4
Here, δ 3 ∈ [−1, 1] and δ 4 ∈ [10 −2 , 10 2 ] are uncertain parameters and G(s, δ 3 , δ 4 ) is introduced
to examine the effect of unmodeled dynamics on the ROA. Let ẋ 5 = −δ 4 x 5 − δ 4 v and
u = 1.5δ 3 x 5 + (1.25 + 0.75δ 3 )v be a realization of G and x = [ x T p x 4 x 5
] T
denote the
96

0.1
0.05
δ 2
0
−0.05
−0.1
1 1.2 1.4 1.6 1.8 2
δ 1
Figure 5.5. Final partition generated by the B&B algorithm for the ∂(V ) = 4 run.
✲ 0.75δ3
s−δ 4
s+δ 4
✲
ẋ 4 = A c x 4 + B c y
v = C c x 4
v
✲ 1.25
✲• ❄u
ẋ
✲ p = f p (x p , δ p ) + B(x p , δ p )u y
+
y = [x 1 x 3 ] T
Figure 5.6. Controlled short period aircraft dynamics with uncertain first order linear timeinvariant
dynamics (δ p := (δ 1 , δ 2 )).
state of the closed-loop dynamics. We apply the B&B based refinement procedure with
∂(V ) = 2 for two cases: (i) (δ 1 , δ 2 ) set to nominal values (1.52, 0), (ii) δ 1 and δ 2 are treated
as uncertain parameters with the bounds δ 1 ∈ [0.99, 2.05] and δ 2 ∈ [−0.1, 0.1]. Note that in
case (ii) the uncertain parameters appear in the vector field as δ 1 , δ 2 , δ 3 , δ 4 , δ 1 δ 3 , δ 2 δ 3 , and
δ1 2. Consequently, the covering polytopes are in R7 with up to 128 (distinct) vertices. In
case (i), Ω p,4.90 is shown to be in the robust ROA, whereas, in case (ii), Ω p,2.80 is certified
to be in the robust ROA.
97

5.5 Chapter Summary
We extended the applicability of the method proposed in the previous chapter to compute
invariant subsets of the region-of-attraction for the asymptotically stable equilibrium
points of polynomial dynamical systems with bounded parametric uncertainty:
• Systems with non-affine uncertainty dependence can now be handled.
• Conservatism associated with using a common Lyapunov function for an entire family
of uncertain systems has been reduced by a branch-and-bound type refinement
procedure in the uncertain parameter space.
The approach put forth here offers some advantages:
• The parameter-dependent Lyapunov functions achieved by uncertainty-space partitioning
do not require an a priori parametrization of the Lyapunov function in the
uncertain parameters. Conservatism (due to parameter-independent Lyapunov functions)
is simply reduced by partitioning the uncertainty set.
• It leads to optimization problems with smaller semidefiniteness constraints since uncertain
parameters do not explicitly appear in the constraints (see section 5.1). 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.
• A sequential implementation (similar to section 4.3) for computing suboptimal solutions
which decouples these constraints into smaller, independent problems arises
98

naturally. This is suitable for trivial parallel computing offering a major advantage
over approaches utilizing parameter-dependent Lyapunov functions.
99

Chapter 6
Reachability and Local Gain
Analysis for Nonlinear Dynamical
Systems
We consider the problem of computing upper bounds for reachable sets and local inputto-output
gains of nonlinear dynamical systems with polynomial vector fields. Similar problems
were studied in [37, 60, 50, 23, 77]. Following [37, 60], we characterize upper bounds
of reachable sets and local input-output gains due to bounded L 2 and/or L ∞ disturbances
using Lyapunov/storage functions and formulate bilinear SOS programming problems to
estimate these bounds (with certificates).
Motivated by the difficulties associated with
bilinear programming, we adapt the simulation data based relaxations (see chapter 3) to
formal proof construction for reachability and local gain analysis. We extend the upper
bound refinement procedure proposed in [60] in the context of reachability to L 2 → L 2
gain analysis and provide a general proof. Finally, we propose a local small-gain theorem
100

which enables (robust) region-of-attraction analysis based on Lyapunov/storage functions
constructed for individual systems in an interconnection of systems.
6.1 Upper and Lower Bounds for the Reachable Set and Local
Input-Output Gains
Consider the nonlinear dynamical system
ẋ(t) = f(x(t), w(t)), (6.1)
where x(t) ∈ R n , w(t) ∈ R nw , and f is a n-vector with elements in R[(x, w)] such that
f(0, 0) = 0. Let φ(t; x 0 , w) denote the solution to (6.1) at time t with the initial condition
x(0) = x 0 driven by the input/disturbance w. For a piecewise continuous map u : [0, ∞) →
R m , define the L 2 and L ∞ norms, respectively, as
√ ∫ ∞
‖u‖ 2 :=
0
u(t) T u(t)dt
‖u‖ ∞
:= sup t≥0 ‖u(t)‖.
6.1.1 Upper Bounds for the Reachable Set
For ‖w‖ 2 ≤ R, the set G R 2
of points reachable from the origin under (6.1) is defined as
G R 2 := { φ(T ; 0, w) ∈ R n : T ≥ 0, ‖w‖ 2 2 ≤ R 2} .
Lemma 6.1.1 adapted from the Lyapunov-like argument in [13§6.1.1] provides a characterization
of sets containing G R 2 [37, 60].
Lemma 6.1.1. If there exists a continuously differentiable function V such that
V (x) > 0 for all x ∈ R n \{0} with V (0) = 0, and (6.2)
∇V f(x, w) ≤ w T w for all x ∈ Ω V,R 2, w ∈ R nw , (6.3)
101

then G R 2 ⊆ Ω V,R 2.
⊳
Given this characterization for upper bounds for the reachable set, one may ask two
questions: (i) For given R such that ‖w‖ 2 ≤ R, what is a tight upper bound for G R 2, and
(ii) given β > 0 and positive definite function p, what is the largest value of R such that
G R 2 ⊆ Ω p,β We will focus on the second question and consider the optimization
Rreach,opt 2 (V, β) := max
R 2 >0,V ∈V R2 subject to
(6.2), (6.3), and Ω V,R 2 ⊆ Ω p,β ,
(6.4)
where V denotes the set of candidate Lyapunov functions over which the maximum is computed.
In order to make the problem in (6.4) amenable to numerical optimization (specifically
SOS optimization), we restrict V and p to be polynomials of some fixed degree. Using
the generalized S-procedure and SOS sufficient conditions for polynomial nonnegativity, a
lower bound on R reach,opt (V, β) can be determined through a SOS programming problem.
Proposition 6.1.1. Let β > 0, l 1 be a positive definite polynomial satisfying l 1 (0) = 0,
R reach be defined by
R 2 reach (V poly, S, β) :=
max R 2 subject to (6.5)
V ∈V poly ,R 2 >0,s 1 ∈S 1 ,s 2 ∈S 2
V (0) = 0, s 1 ∈ Σ[x], and s 2 ∈ Σ[(x, w)], (6.6)
V − l 1 ∈ Σ[x], (6.7)
(β − p) − (R 2 − V )s 1 ∈ Σ[x], (6.8)
− [ (R 2 − V )s 2 + ∇V f(x, w) − w T w ] ∈ Σ[(x, w)], (6.9)
where V poly ⊂ V and S 1 are prescribed finite-dimensional subsets of R[x]. Then,
R reach (V poly , S 1 , β) ≤ R reach,opt (V, β).
⊳
102

6.1.2 Upper Bounds for the Local L 2 → L 2 Gain
Consider the dynamical system governed by
ẋ(t) = f(x(t), w(t)),
z(t) = h(x(t)),
(6.10)
where x, w, and f are as before and h is an n z -vector of polynomials such that h(0) = 0.
We use the following lemma characterizing an upper bound for the w to z induced L 2 → L 2
gain of the system (6.10).
Lemma 6.1.2. [60] If there exists a real scalar γ > 0 and a continuously differentiable
function V such that
V (0) = 0 and V (x) ≥ 0, (6.11)
∇V f(x, w) ≤ w T w − γ −2 z T z ∀x ∈ Ω V,R 2 and w ∈ R nw , (6.12)
then the system in (6.10) with x(0) = 0 satisfies ‖z‖ 2 ≤ γR whenever ‖w‖ 2 ≤ R.
⊳
Similar to the reachability analysis, one may ask two related questions: (i) For given R
such that ‖w‖ 2 ≤ R, what is a tight upper bound for the L 2 → L 2 gain, and (ii) given γ,
what is the largest value of R such that ‖z‖ 2 ≤ γR whenever ‖w‖ 2 ≤ R and x(0) = 0 We
use Proposition 6.1.2 to address the second question (the first one can be similarly handled).
Proposition 6.1.2. For given γ > 0, let R L2
be defined by
R 2 L 2
(V poly , S, γ) :=
max R 2 subject to (6.13)
V ∈V poly ,R 2 >0,s 1 ∈S 1
V (0) = 0, s 1 ∈ Σ[(x, w)], (6.14)
V ∈ Σ[x], (6.15)
− [ (R 2 − V )s 1 + ∇V f(x, w) − w T w + γ −2 z T z ] ∈ Σ[(x, w)], (6.16)
103

where V poly and S’s are prescribed finite-dimensional subsets of R[x]. Then, R L2 (V poly , S, γ)
is a lower bound for the largest value of R such that ‖z‖ 2 ≤ γR whenever ‖w‖ 2 ≤ R and
x(0) = 0.
⊳
Remarks 6.1.1. Incorporating L ∞ norm bounds on the disturbance (in addition to L 2
norm bounds) in reachability and L 2 → L 2 gain analysis only requires a straightforward
application of the generalized S-procedure.
For example, for ρ > 0, if the conditions in
Proposition 6.1.1, except that (6.9) is replaced by
− [ (R 2 − V )s 2 + ∇V f(x, w) − w T w ] − s 3 (ρ − w T w) ∈ Σ[(x, w)],
where s 3 ∈ Σ[(x, w)], hold, then for x(0) = 0 and all T ≥ 0,
x(T ) ∈ {φ(t; 0, w) ∈ R n : ‖w‖ 2 ≤ R, ‖w‖ ∞ ≤ ρ, t ≥ 0} .
Similarly, when the conditions of Proposition 6.1.2 holds except that (6.16) is replaced by
− [ (R 2 − V )s 1 + ∇V f(x, w) − w T w + γ −2 z T z ] − s 2 (ρ − w T w) ∈ Σ[(x, w)],
where s 2 ∈ Σ[(x, w)], then, for x(0) = 0, ‖w‖ ∞ ≤ ρ and ‖w‖ 2 ≤ R imply that ‖z‖ 2 ≤ γR. ⊳
6.1.3 Upper Bounds for the Local L ∞ → L ∞ Gain
gain.
The following result, reminiscent of [77, 50], provides an upper bound for L ∞ → L ∞
Lemma 6.1.1. For ρ > 0 and ɛ > 0, if there exist a real scalar γ > 0 and a continuously
differentiable function V such that V (0) < 1 and
∇V f(x, w) ≤ −ɛ ∀x ∈ {x ∈ R n : V (x) = 1} , ∀w ∈ W, (6.17)
Ω V = {x ∈ R n : V (x) ≤ 1} ⊆ {x ∈ R n : |h(x)| ≤ ργ} , (6.18)
104

where W := {w ∈ R nw : ‖u‖ ∞ ≤ ρ} , then for the system in (6.10) with x(0) ∈ Ω V ,
‖z‖ ∞ ≤ γρ whenever ‖w‖ ∞ ≤ ρ.
⊳
By the generalized S-procedure,
−ɛ − ∇V f(x, w) − r(1 − V ) − ∑ n w
i=1
s i (ρ 2 − wi 2 ) ∈ Σ[(x, w)], (6.19)
(γ 2 ρ 2 − h 2 ) − s 0 (1 − V ) ∈ Σ[x], (6.20)
s 0 ∈ Σ[x], s 1 , . . . , s nw ∈ Σ[(x, w)], (6.21)
r ∈ R[x], V ∈ R[x], V (0) < 1 (6.22)
are sufficient for the conditions in Lemma 6.1.1.
Remarks 6.1.2. Let E W be the set of extreme points of W. When f(x, w) is affine in w
(for fixed x),
−ɛ − ∇V f(x, w) − r w (1 − V ) ∈ Σ[x] for all w ∈ E W ,
where r w ∈ R[x], is also a sufficient condition for (6.17) which does not depend explicitly on
w but there is one SOS condition for each vertex. This condition is suitable for adaptations
of the sequential suboptimal solution strategy introduced in section 4.3.
⊳
6.1.4 Lower Bounds
We now discuss lower bounds on reachable sets and local input-output gains. These
lower bounds will be used to assess the suboptimality of upper bound computations.
105

Lower bound for the reachable set:
Fix β > 0 and let R 2 reach (V poly, S, β) be as defined
in (6.5)-(6.9). Then, for any positive T, it follows that
{
max
w
≤ max
w
∫ T
p(x(t)) : x(0) = 0 and
0
{
p(x(t)) : x(0) = 0 and
w(t) T w(t)dt ≤ R 2 reach , 0 ≤ t ≤ T }
∫ ∞
0
w(t) T w(t)dt ≤ R 2 reach , t ≥ 0 }
(6.23)
(6.24)
≤ β. (6.25)
Lower bound for L 2 → L 2 gain:
Fix γ > 0 and let R 2 L 2
(V poly , S, γ) be as defined in
(6.1.2). Then, for any positive T, it follows that
{∫ ∞
max
w
≤ max
w
0
z(t) T z(t)dt :
{∫ ∞
z(t) T z(t)dt :
0
x(0) = 0 and
x(0) = 0 and
∫ T
w(t) T w(t)dt ≤ R 2 L 2
}
0
∫ ∞
0
w(t) T w(t)dt ≤ R 2 L 2
}
(6.26)
(6.27)
≤ γ 2 R 2 L 2
. (6.28)
Remarks 6.1.3.
i. Lower bounds for the L ∞ → L ∞ gain can be similarly defined.
ii. Problems in (6.23) and (6.26) are optimal control problems which can be attacked
by any nonlinear programming solver [42, 29] after appropriate parametrization of
the input and/or the state.
On the other hand, [60] adapted an iterative scheme
from [64] for computing solutions to these optimal control problems based on the first
order conditions for optimality [16]. In the following section, we use this technique to
compute reported lower bounds.
⊳
106

6.2 Upper Bound Refinement for Reachability and L 2 → L 2
Gain Analysis
Conditions in (6.2)-(6.3) provide a relation between L 2 norms of the disturbance signals
and reachable sets due to these disturbance signals in terms of an appropriate Lyapunov
function V .
However, this relation may be conservative because ∇V f(x, w) ≤ w T w is
required to hold everywhere in Ω V,R 2. To reduce this conservatism, a refinement procedure
which, once V is determined solving (6.13)-(6.16), partitions Ω V,R 2
into smaller annular
regions and analyzes V in each subregion separately imposing potentially weaker conditions
[60]. Here, we provide an independent and more general proof and also an extension for
L 2 → L 2 gain analysis. To this end, let g : R → R be a piecewise continuous function
satisfying 0 < g(τ) ≤ 1 for all τ ≥ 0 and suppose that
∇V f(x, w) ≤ g(V (x))w T w, ∀x ∈ Ω V,R 2, ∀w ∈ R nw . (6.29)
Define
and R ext by
Then, R 2 ext ≥ R 2 ,
K(x) :=
R 2 ext :=
∇K(x) =
∫ V (x)
0
∫ R 2
0
1
dτ (6.30)
g(τ)
1
dτ. (6.31)
g(τ)
1
∇V (x),
g(V (x))
{x ∈ R n : V (x) ≤ R 2 } = {x ∈ R n : K(x) ≤ R 2 ext},
and K is a positive semidefinite differentiable function (vanishing only where V vanishes)
satisfying
∇Kf(x, w) ≤ w T w, ∀x ∈ Ω K,R 2
ext
, ∀w ∈ R nw .
107

Consequently, this strengthens the reachability result by certifying a larger bound on the
admissible disturbance signals: if x(0) = 0, then,
‖w‖ 2 ≤ R ext ⇒ x(t) ∈ Ω K,R 2
ext
= Ω V,R 2 ⊆ Ω p,β for all t ≥ 0.
Remarks 6.2.1. Effectively, K is a non-polynomial Lyapunov/storage function constructed
via the polynomial Lyapunov/storage function V.
⊳
For given V and R, search for g satisfying (6.29) can be formulated as a sequence of
SOS programming problems. To this end, we restrict g to be piecewise constant 1 . Let
m > 0 be an integer, define ɛ := R 2 /m, and partition the set Ω R 2 ,V
into m subregions
Ω V,R 2 ,k := {x ∈ R n : (k − 1)ɛ ≤ V (x) ≤ kɛ} for k = 1, . . . , m.
If
∇V f(x, w) ≤ g k w T w for all x ∈ Ω R 2 ,V,k, w ∈ R nw , (6.32)
holds for some g k > 0, then for any k ≤ m, the system in (6.1) with piecewise continuous
w starting from the origin satisfies
∫ T
0
( 1
w T w dt < ɛ + · · · + 1 )
⇒ V (φ(T ; 0, w)) ≤ kɛ.
g 1 g k
Note that (6.32) already holds with g k = 1. Therefore, it may be possible to make ɛ(1/g 1 +
. . . + 1/g k ) (in particular k = m) greater than R 2 k/m (in particular R 2 ) by minimizing g k
such that (6.32) holds. A sufficient condition for (6.32) is the existence of SOS polynomials
s 1k and s 2k such that
g k w T w − ∇V f(x, w) − s 1k (kɛ − V ) − s 2k (V − (k − 1)ɛ) ∈ Σ[(x, w)].
Similar ideas apply to the L 2 → L 2 gain analysis where (6.29) is replaced by
∇V f(x, w) ≤ g(V (x))(w T w − γ −2 z T z), ∀x ∈ Ω V,R 2, ∀w ∈ R nw (6.33)
1 Same procedure can be applied using piecewise polynomial g instead of piecewise constant.
108

Let K and R ext be as defined before. Then, (6.33) is equivalent to
∇Kf(x, w) ≤ w T w − γ −2 z T z, ∀x ∈ Ω K,R 2
ext
, ∀w ∈ R nw ,
and it follows that, starting from x(0) = 0,
‖w‖ 2 ≤ R ext ⇒ ‖z‖ 2 ≤ γR ext .
6.3 Simulation-Based Relaxation for the Bilinear SOS Problem
in Reachability Analysis
The usefulness of using simulation data in local stability analysis was demonstrated in
chapter 3 and [73] where a methodology that incorporates simulation data in formal proof
construction for estimating regions-of-attraction is proposed. In this section, we propose a
similar technique for reachability analysis for nonlinear dynamical systems with polynomial
vector fields. To this end, for given β > 0 and disturbance level R 2 > 0, let us ask the
question whether the reachable set G R 2
is contained in the set Ω p,β . Certainly, just one
disturbance signal w (with ‖w‖ 2 ≤ R) that leads to a system trajectory on which p takes
on a value larger than β certifies that G R 2
Ω p,β . Conversely, a large collection of input
signals w with ‖w‖ 2 2 ≤ R2 that do not drive the system out of Ω p,β hints to the likelihood
that indeed G R 2 ⊆ Ω p,β . In this latter case, let W be a finite collection of signals
⎧
⎫
⎪⎨
w ∈ L 2 [0, ∞), ‖w‖ 2 2
W := (w, x) :
≤ R2 ⎪⎬
.
⎪⎩
p(φ(t; 0, w)) ≤ β, ∀t ≥ 0
⎪⎭
With β and R 2 fixed, the set of Lyapunov functions which certify that G R 2
⊆ Ω p,β , using
conditions (6.7)-(6.9), is simply
{V ∈ R[x] : (6.7) − (6.9) hold for some s i ∈ Σ[x]} .
109

Of course, this set could be empty, but it must be contained in the convex set {V
∈
R[x] : (6.34) holds}, where
l 1 (x(t)) ≤ V (x(t)),
V (x(t)) ≤ R 2 ,
∇V (x(t))f(x(t), w(t)) ≤ w(t) T w(t)
(6.34a)
(6.34b)
(6.34c)
for all (w, x) ∈ W and t ≥ 0.
6.3.1 Affine Relaxation
Let V be linearly parameterized as
V := {V ∈ R[x] : V (x) = ϕ(x) T α},
where α ∈ R n b
and ϕ is n b -dimensional vector of polynomials in x. Given ϕ(x), constraints
in (6.34) can be viewed as constraints on α ∈ R n b
yielding the convex set
{α ∈ R n b
: (6.34) holds for V = ϕ(x) T α}.
For each (w, x) ∈ W, let T w be a finite subset of the interval [0, ∞). A polytopic outer bound
for this set described by finitely many constraints is Y sim := {α ∈ R n b
: (6.35) holds},
where
l 1 (x(t)) ≤ ϕ(x(t)) T α,
ϕ(x(t)) T α ≤ R 2 ,
∇ϕ(x(t)) T αf(x(t), w(t)) ≤ w(t) T w(t)
(6.35a)
(6.35b)
(6.35c)
for all (w, x) ∈ W and t ∈ T w .
110

Additionally, define Y sim and Y SOS as Y lin := {α ∈ R n b
: Q = Q T ≻ 0 and Lin(Q) ≼
0}, where, for some λ > 1,
Lin(Q) :=
⎡
⎢
⎣
∇ x f(0, 0) T Q + Q∇ x f(0, 0) Q∇ w f(0, 0)
∇ w f(0, 0) T Q
Q T = Q ≻ 0 be such that x T Qx is the quadratic part of V, and Y SOS := {α ∈
−λI
⎤
⎥
⎦ ,
R n b
: (6.7) holds}. Since Y sim , Y lin and Y SOS are convex,
Y := Y sim ∩ Y lin ∩ Y SOS
is a convex set in R n b.
Equations (6.35) and LinQ ≤ 0 constitute a set of necessary
conditions for (6.7)-(6.9); thus, we have
Y ⊇ B := {α ∈ R n b
: ∃s 1 ∈ Σ[x], s 2 ∈ Σ[(x, w)] such that (6.7) − (6.9) hold}.
With this definition of the convex outer-bound set Y, the algorithms introduced in section
3.2.3 can be used with minor modifications.
6.3.2 Algorithms
Since an appropriate (feasible and not too conservative) value of R 2 is not known a priori,
an iterative strategy to simulate and collect trajectories is necessary. This process when
coupled with the H&R algorithm constitutes the Lyapunov function candidate generation.
Simulation and Lyapunov function generation (SimLF G) algorithm:
Given positive
definite convex p ∈ R[x], a vector of polynomials ϕ(x), positive constants β, R 2 , N SIM
(integer), N V (integer), σ shrink ∈ (0, 1), and empty set W.
i. Generate N SIM input signals w with ‖w‖ 2 2 ≤ R2 and integrate (6.1) from the origin.
111

ii. If all trajectories stay in Ω p,β , add (w, x) to W and go to step (iii). Otherwise, set R 2
to σ shrink R 2 and go to step (i).
iii. Find a feasible point for (6.7), (6.35) and LinQ ≤ 0. If (6.7), (6.35) and LinQ ≤ 0 are
infeasible, set R 2 = σ shrink R 2 , and go to step (i). Otherwise, go to step (iv).
iv. Generate N V Lyapunov function candidates using H&R algorithm, and return R 2
and Lyapunov function candidates.
⊳
Step (i) of SimLFG algorithm requires to generate input signals w with ‖w‖ 2 2 ≤ R2 . In the
current implementation of this algorithm, we use randomly generated piecewise constant
input signals and signals generated by the lower bound computation using the given shape
factor p and additional randomly generated shape factors. The suitability of a Lyapunov
function candidate is assessed by solving the optimization problem
Affine Problem: Given V ∈ R[x] (from SimLF G algorithm), p ∈ R[x], and β, define
R 2 L :=
max R 2 subject to
R 2 >0,s 1 ∈S 1 ,s 2 ∈S 2
s 1 ∈ Σ[x], s 2 ∈ Σ[(x, w)], R 2 > 0,
(β − p) − (R 2 − V )s 1 ∈ Σ[x],
(6.36)
− [ (R 2 − V )s 2 + ∇V f(x, w) − w T w ] ∈ Σ[(x, w)].
Although RL 2 depends on the allowable degree of s 1 and s 2 , this is not explicitly notated.
Note that for given V and fixed R 2 the problem in (6.36) is affine in the multipliers and
can be solved by (relatively) efficient linear SDP solvers such as [56] via a line search on
the parameter R 2 .
Assuming Affine Problem is feasible, it is true that G R 2
L
⊆ Ω V,R 2
L
⊆ Ω p,β . The solution to
Affine Problem provides a feasible point for the problem in (6.5). This feasible point can be
112

further improved by computing suboptimal solutions for the problem in (6.5) using iterative
coordinate-wise affine optimization schemes based on solving (6.6)-(6.9) alternatingly for V
with fixed multipliers and vice versa. However, as noted in the previous paragraph, this
requires a line search on R 2 at each step. Alternatively, the following coordinate-wise affine
search scheme iterates without line search by a change of variables.
Coordinate-wise optimization (CW Opt) algorithm without line search:
For β > 0, if the
problem in (6.6)-(6.9) has the solution R 2 reach (V poly, S, β), V, s 1 , and s 2 , then
1
R 2 reach (V poly,S,β) =
min
1
subject to (6.37)
K∈V poly ,1/R 2 >0,˜s 1 ∈S 1 ,˜s 2 ∈S 2 R2 ˜s 1 ∈ Σ[x], ˜s 2 ∈ Σ[(x, w)], and K ∈ R[x] (6.38)
K − l 1 /R 2 ∈ Σ[x], (6.39)
(β − p) − (1 − K)˜s 1 ∈ Σ[x], (6.40)
− [ (1 − K)s 2 + ∇Kf(x, w) − 1 R 2 w T w ] ∈ Σ[(x, w)]. (6.41)
In fact, the optimal values of ˜s 1 , ˜s 2 , and K are equal to Rreach 2 s 1, s 2 , and V/Rreach 2 respectively.
This correspondence enables to implement a coordinate-wise affine search algorithm
that does not require line search on R 2 once a feasible solution for (6.6)-(6.9) is obtained.
Indeed, let V current and Rcurrent 2 be such that (6.6)-(6.9) are feasible. Then, a coordinatewise
affine search scheme initialized with K current (x) = V current (x)/Rcurrent 2 for computing
suboptimal solutions for (6.5) is composed of the following optimization problems solved
alternatingly.
113

Given K current , solve:
[1/R∗, 2 ˜s ∗ 1 , ˜s∗ 2 ] = argmin 1
1/R 2 >0,˜s 1 ∈S 1 ,˜s 2 ∈S 2
subject to
R2 ˜s 1 ∈ Σ[x], ˜s 2 ∈ Σ[(x, w)],
(β − p) − (1 − K current )˜s 1 ∈ Σ[x],
(6.42)
− [ (1 − K current )˜s ∗ 2 + ∇K currentf(x, w) − 1 R 2 w T w ] ∈ Σ[(x, w)].
Given ˜s ∗ 1 and ˜s∗ 2 , solve:
[1/R∗, 2 1
K current ] = argmin 1/R 2 >0,K∈V poly
subject to
R2 K ∈ R[x],
K − l 1 /R 2 ∈ Σ[x],
(β − p) − (1 − K)˜s ∗ 1 ∈ Σ[x],
(6.43)
− [ (1 − K)˜s ∗ 2 + ∇Kf(x, w) − 1 w T w ] ∈ Σ[(x, w)].
R 2
Remarks 6.3.1. A coordinate-wise affine search scheme for the bilinear SOS problems
in L 2 → L 2 analysis is as follows: Let V current and Rcurrent 2 be feasible for (6.13)-(6.16)
and define K current = V current /Rcurrent. 2 Then, a coordinate-wise affine search scheme for
computing suboptimal solutions for (6.13)-(6.16) is composed of the following optimization
problems solved alternatingly.
Given K current , solve:
[1/R 2 ∗, ˜s ∗ 1 ] = argmin 1/R 2 >0,˜s 1 ∈S 1
1
R 2
subject to
˜s 1 ∈ Σ[(x, w)]
− [ (1 − K current )˜s 1 + ∇Kf(x, w) − 1 R 2 w T w + 1 R 2 γ −2 z T z ] ∈ Σ[(x, w)]
Given ˜s ∗ 1 , solve: [1/R 2 ∗, K current ] = argmin 1/R 2 >0,K∈V poly
1
R 2
subject to
K ∈ Σ[x]
− [ (1 − K)˜s ∗ 1 + ∇Kf(x, w) − 1 R 2 w T w + 1 R 2 γ −2 z T z ] ∈ Σ[(x, w)].
114

20
15
β
10
5
0
0 5 10 15 20 25 30
R 2
Figure 6.1. Bounds on reachable sets due to disturbance w with ‖w‖ 2 2 ≤ R2 for Example
1 without delay (blue curve with dots: before refinement, green curve with ×: after
refinement, red curve with ⋄: lower bound).
⊳
6.4 Examples
Example 1
(Reachability analysis for a system from the literature) Consider the nonlinear
system from [37, 60]
ẋ 1 = −x 1 + x 2 − x 1 x 2 2 , ẋ 2 = u − x 2 1 x 2 + w
y = x 2 , u = −y.
Let the shape factor be p(x) = x 2 1 +x2 2 . For l 1(x) = 10 −6 x T x, ∂(V ) = 2, ∂(s 1 ) = 0, ∂(s 2 ) = 2
(with no constant and linear terms), N SIM = 100, N V
= 1, ε iter = 0.01, N iter = 20, and
m = 10, we applied SimLFG and CWOpt algorithms and the refinement procedure for
different values of β. Figure 6.1 shows the upper bounds for β versus R 2 before and after
applying the refinement procedure and the lower bounds.
115

20
15
β
10
5
0
0 5 10 15 20
R 2
Figure 6.2. Bounds on reachable sets due to disturbance w with ‖w‖ 2 2 ≤ R2 for Example 2
(blue curve with dots: before refinement, green curve with ×: after refinement, red curve
with ⋄: lower bound, and black circles: failed PENBMI runs).
Next, we introduced t d = 0.4 units of time delay at the input and modeled the delay
using a (balanced) first order Pade approximation
ẋ 3 (t) = −2/t d x 3 (t) + 2/ √ t d y(t)
and replaced the input by
u(t) = 2/ √ t d x 3 (t) + y(t),
where x 3 is the state of delay dynamics, and used p(x) = x 2 1 + x2 2 + 0.01x2 3 . Figure 6.2
shows the upper and lower bounds for β versus R 2 before and after applying the refinement
procedure. We also solved the problem in (6.5)-(6.9) for fixed values of β using PENBMI.
When these runs return a feasible solution, the returned value of R 2 is usually equal or close
to the best known value. But, PENBMI runs often terminate with numerical problems or
infeasible results (although a solution is known to exist using the procedure proposed here).
For these values of β, we put black circles around corresponding dots on the blue solid
curves in Figure 6.2.
116

1.2
1
0.8
β
0.6
0.4
0.2
0
0 0.05 0.1 0.15 0.2
R 2
Figure 6.3. Bounds on reachable sets due to disturbance w with ‖w‖ 2 2 ≤ R2 for Example 2
(blue curve with dots: before refinement, green curve with ×: after refinement, red curve
with ⋄: lower bound, and black circles: failed PENBMI runs).
Example 2
(Reachability analysis for pendubot dynamics)
Consider the pendubot dynamics form chapter 3 with a disturbance signal inserted at
the input channel
ẋ 1 = x 2
ẋ 2 = 45w + 782x 1 + 135x 2 + 689x 3 + 90x 4
ẋ 3 = x 4
ẋ 4 = 279x 1 x 2 3 + 273x3 3 − 85w − 1425x 1 − 257x 2 − 1249x 3 − 171x 4 .
Here, x 1 and x 3 are angular positions of the first link and the second link (relative to the
first link) and w is the scalar disturbance. Figure 6.3 shows the upper (with the parameters
used in Example 1 except for N SIM = 600) and lower bounds for β versus R 2 before and
after applying the refinement procedure. We put black circles around corresponding dots
on the blue solid curves in Figure 6.3 for failed PENBMI runs.
117

5
4
γ
3
2
1
0 1 2 3 4 5 6
R
Figure 6.4. Upper bounds for ∂(V ) = 2 (with ⋄) and ∂(V ) = 4 (with ×) before the
refinement (blue curves) and after the refinement (green curves) along with the lower bounds
(red curve).
Example 3
(L 2 → L 2 gain analysis for a system with adaptive controller) Consider the
system with an adaptive controller
ẋ 1 = −x 1 + w + r + x 1 z x + z r r
ẋ m = −x m + r
ż x = −x 2 1 + x 1x m
ż r = −x 1 r + x m r,
where x 1 is the plant state, x m is the state of the reference model, z x and z m are adaptation
gains, r is the reference input, and w is disturbance. We applied the L 2 → L 2 gain analysis
with the output −1.8x 1 + w + r + x 1 z x + z r r. Figure 6.4 shows lower and upper (before and
after refinement) bounds for ∂(V ) = 2 and ∂(V ) = 4.
118

z
✲
∆
w
M
✛
Figure 6.5. Feedback interconnection of ∆ and M.
6.5 Region-of-Attraction Analysis for Systems with Unmodeled
Dynamics using a Local Small-Gain Theorem
We prove a local small-gain type theorem and use it to analyze the effect of unmodeled
dynamics, satisfying certain gain inequalities, on the regions-of-attraction. Consider the
system interconnection in Figure 6.5. Let
ẋ 1 (t) = f 1 (x 1 (t), w(t))
z(t) = h 1 (x 1 (t))
(6.44)
be a realization of M, where x 1 ∈ R n 1
, f 1 and h 1 are vectors of polynomials satisfying
f 1 (0, 0) = 0 and h 1 (0) = 0.
Proposition 6.5.1. Consider the system interconnection in Figure 6.5 with ∆ a stable
linear time-invariant system satisfying ‖∆‖ ∞ < 1. Let 0 < γ < 1, R > 0, and p be a
positive definite function with p(0) = 0. If there exists a continuously differentiable positive
definite function V satisfying V (0) = 0 and
∇V f 1 (x 1 , w) ≤ w T w − 1 γ 2 zT z − p(x 1 ) for all w ∈ R nw and x 1 ∈ Ω V,R 2, (6.45)
then, for all x 1 (0) ∈ { x 1 ∈ R n 1
: V (x 1 ) ≤ R 2} and ∆ starting from rest, x 1 (t) ∈ Ω V,R 2
and x 1 (t) → 0 as t → ∞.
⊳
119

Proof. Let
ẋ 2 (t) = Ax 2 (t) + Bz(t)
w(t) = Cx 2 (t) + Dz(t)
be a realization of ∆ with x 2 ∈ R n 2
. By Kalman-Yakubovich-Popov Lemma [25], there exist
P ≻ 0 and ɛ > 0 such that
⎡
⎢
⎣
A T P + P A + C T C
B T P + D T C
P B + C T D
−I + D T D
⎤
⎥
⎦ + ɛI ≼ 0 (6.46)
Let Q(x 2 ) = x T 2 P x 2, then, for all x 2 ∈ R n 2
and z ∈ R nz , (6.46) implies that
d
dt Q(x 2) = ∂Q(x 2)
∂x 2
(Ax 2 + Bz) = −ɛx T 2 x 2 − ɛz T z − w T w + z T z
≤ z T z − w T w − ɛx T 2 x 2.
(6.47)
Now, let (x 1 , x 2 ) ∈ Ω V +Q,R 2. By (6.46) and (6.47),
d
dt (V (x 1) + Q(x 2 ))
Integrate (6.48) from 0 to T ≥ 0 to get
≤
(
1 − 1
γ 2 )
z T z − p(x 1 ) − ɛx T 2 x 2
≤ −p(x 1 ) − ɛx T 2 x 2.
(6.48)
V (x 1 (T )) ≤ V (x 1 (T )) + Q(x 2 (T )) ≤ −
∫ T
0
(p(x 1 ) + ɛx T 2 x 2 )dt + V (x 1 (0)) ≤ V (x 1 (0)) ≤ R 2 ,
which implies that, for all t ≥ 0, x 1 (t) ∈ Ω V,R 2 whenever x 1 (0) ∈ Ω V,R 2 and x 2 (0) =
0. Convergence of (x 1 , x 2 ) and, in particular of x 1 , follows from Lemma 3.1.1 using the
inequality in (6.48).
Remarks 6.5.1.
Note that Proposition 6.5.1 only states that Ω V,R 2 is invariant but does not assure the invariance
of the sublevel sets Ω V,ξ for all ξ ≤ R 2 . Hence, V may increase along the trajectories of
the closed-loop system but, if V (x 1 (0)) ≤ R 2 and ∆ starts from rest, then V cannot exceed
R 2 and converges to the origin along every trajectory with x 1 (0) ∈ Ω V,R 2
and x 2 (0) = 0. ⊳
120

In the proof Proposition 6.5.1, the linear time-invariance property of ∆ is only used to
establish the inequality (6.47). Therefore, if ∆ is any dynamical system satisfying (6.47),
then the result of Proposition 6.5.1 still applies.
Proposition 6.5.2. Consider the system interconnection in Figure 6.5. Let 0 < γ <
1, R > 0, and p and q be positive definite functions. Let
ẋ 2 (t) = f 2 (x 2 (t), z(t))
w(t) = h 2 (x 2 (t))
(6.49)
be a realization of ∆ with x 2 ∈ R n 2
such that there exists a continuously differentiable
positive definite function Q satisfying Q(0) = 0 and
∇Qf 2 (x 2 , z) ≤ z T z − w T w − q(x 2 ) for all z ∈ R nz and x 2 ∈ R n 2
. (6.50)
If there exists a continuously differentiable positive definite function V satisfying V (0) = 0
satisfying (6.45), then, for x 2 (0) = 0 and all x 1 (0) ∈ Ω V,R 2, (x 1 (t), x 2 (t)) ∈ Ω V +Q,R 2
for
all t ≥ 0 and lim t→∞ (x 1 (t), x 2 (t)) = (0, 0), in particular, x 1 (t) ∈ Ω V,R 2 for all t ≥ 0 and
lim t→∞ x 1 (t) = 0. ⊳
Conditions in Proposition 6.5.2 can be relaxed to obtain the following result.
Proposition 6.5.3. Consider the system interconnection in Figure 6.5. Let 0 < γ < 1, R >
˜R > 0, and p be a positive definite function. Let (6.49) be a realization of ∆ such that there
exists a continuously differentiable positive definite function Q satisfying Q(0) = 0 and
∇Qf 2 (x 2 , z) ≤ z T z − w T w for all z ∈ R nz and x 2 ∈ R n 2
. (6.51)
If there exists a continuously differentiable positive definite function V satisfying V (0) = 0
and (6.45), then, for x 2 (0) = 0 and all x 1 (0) ∈ Ω V, ˜R2, (x 1 (t), x 2 (t)) ∈ Ω V +Q, ˜R2
for all t ≥ 0
and, in particular, x 1 (t) ∈ Ω V, ˜R2
for all t ≥ 0. Moreover, lim t→∞ x 1 (t) = 0. ⊳
121

Proof. Define S := V + Q.
d
dt S(x 1, x 2 ) = d dt (V (x 1) + Q(x 2 )) ≤ −p(x 1 ) ∀x 1 ∈ Ω V,R 2, ∀x 2 ∈ R n 2
.
Let x 1 (0) ∈ Ω V, ˜R2
and x 2 (0) = 0, and integrate this last equation to get
S(x 1 (t), x 2 (t)) − S(x 1 (0), x 2 (0)) ≤ V (x 1 (t)) + Q(x 2 (t)) − V (x 1 (0)) ≤ −
∫ t
0
p(x 1 (τ))τ,
and x 1 (t) ∈ Ω V, ˜R2
and (x 1 (t), x 3 (t)) ∈ Ω S, ˜R2
follow. With x 1 (0) ∈ Ω V, ˜R2
and x 2 (0) = 0, S is
monotonically non-increasing and bounded below (by zero). Therefore, lim t→∞
∫ t
0
Ṡ(τ)dτ
exists and is finite.
Since S is bounded and positive definite, (x 1 (t), x 2 (t)) is uniformly
bounded for t ≥ 0 and (ẋ 1 (t), ẋ 2 (t)) = (f 1 (x 1 (t), x 2 (t)), f 2 (x 1 (t), x 2 (t))) is uniformly
bounded for all t ≥ 0. Hence x 1 (t), x 2 (t) is uniformly continuous on [0, ∞). S is uniformly
continuous in t on [0, ∞) because S(x 1 , x 2 ) is uniformly continuous in (x 1 , x 2 ) on
the compact set Ω S, ˜R2. Therefore, by Barbalat’s Lemma [38], Ṡ(t) → 0 as t goes to zero.
Consequently, p(x 1 (t)) → 0 and x 1 (t) → 0 as t → ∞.
Remarks 6.5.2. Conditions in (6.47), (6.51), and (6.51) are supposed to hold globally in
x 2 . On the other hand, if these conditions are imposed locally, i.e., if they are imposed on
sublevel sets of Q, then results similar to Propositions 6.5.1-6.5.3 can still be derived.
⊳
6.5.1 Controlled Short Period Aircraft Dynamics
We now apply Theorem 6.5.2 to the robust ROA analysis for the aircraft model used
in previous chapters (see section 4.5.2). To this end, let 1 > γ > 0 µ > 0. Let ẋ = f(x, w)
and z = h(x) be the realization of the system from the input w to the output z shown in
Figure 6.6 (where the plant dynamics and the controller dynamics are as in section 4). By
Proposition 6.5.1, for linear time-invariant δ with ‖δ‖ ∞ < 1 and starting from rest, if there
122

exists positive definite V such that V (0) = 0 and
∇V f(x, w) ≤ w T w − 1 γ 2 zT z − µV (x) for all w ∈ R nw and x ∈ Ω V,R 2,
then Ω V,R 2 is invariant and all trajectories of the closed-loop system with x 1 (0) ∈ Ω V,R 2
converge to the origin. In order to enlarge computed Ω V,R 2 by the choice of V, we introduce a
fixed, positive definite, convex polynomial p (as before), impose the constraint Ω p,β ⊆ Ω V,R 2,
and maximize β. This can be written as the following optimization.
max
V ∈V,β>0,R 2 >0
β
subject to
V (x) > 0 for all nonzero x, V (0) = 0,
∇V f(x, w) ≤ w T w − 1
γ 2 z T z − µV (x) ∀ x ∈ Ω V,R 2,
Ω p,β ⊆ Ω V,R 2.
∀ w ∈ R
(6.52)
Using the generalized S-procedure and SOS relaxations, a lower bound for the optimal value
of the optimization (6.52) can be formulated as a bilinear SOS optimization problem. We
computed suboptimal solutions to this problem for two cases with p(x) = x T x, γ = 0.99,
and µ = 10 −6 using a variant of coordinate-wise affine search scheme:
• (no parametric uncertainty in the plant δ 1 = 1.52 and δ 2 = 0) The computed values
of β are 4.24 and 6.67 for ∂(V ) = 2 and ∂(V ) = 4, respectively.
• (with parametric uncertainty in the plant δ 1 ∈ [0.99, 2.05] and δ 2 ∈ [−0.1, 0.1]) We
implemented a branch-and-bound type refinement procedure coupled with a sequential
suboptimal solution technique (similar to that in section 4.3). The computed values
of β are 2.39 and 4.14 for ∂(V ) = 2 and ∂(V ) = 4, respectively.
123

✲
ẋ 4 = A c x 4 + B c y
v = C c x 4
z✲
0.75
✲ 1.25
∆
w
✲• ❄u
ẋ
✲ p = f p (x p , δ p ) + B(x p , δ p )u y
+
y = [x 1 x 3 ] T
Figure 6.6. Controlled short period aircraft dynamics with unmodeled dynamics (δ p :=
(δ 1 , δ 2 )).
6.5.2 Summary of the Results for the Controlled Short Period Aircraft
Dynamics Example
We have used the controlled short period pitch axis model of an aircraft to demonstrate
the methods proposed in chapters 3-6 (section 4.5.2, section 4.5.2, sections 5.4.2-5.4.3, and
section 6.5.1). Namely, we examined the following cases:
i. nominal dynamics (section 4.5.2),
ii. dynamics with parametric uncertainty in the plant (section 5.4.2),
iii. nominal plant dynamics with uncertain first order linear time-invariant dynamics (section
5.4.3),
iv. dynamics with parametric uncertainty in the plant and uncertain first order linear
time-invariant dynamics (section 5.4.3),
v. nominal plant dynamics with unmodeled dynamics satisfying (6.51) (section 6.5.1).
Computed (sub)optimal values of β with p(x) = x T x, where x is the state of the closed-loop
dynamics, are shown in Table 6.1..
124

Table 6.1. Computed (sub)optimal values of β with p(x) = x T x (with ∂(V ) = 2 / ∂(V ) = 4)
.
without plant uncertainty
with plant uncertainty
no unmodeled dynamics 9.38 / 16.11 5.45 / 7.93
with multiplicative uncertainty
δ 3
s−δ 4
s+δ 4
4.90 / – 2.80 / –
with unmodeled dynamics satisfying
(6.51)
4.24 / 6.67 2.39 / 4.14
6.6 Chapter Summary
We analyzed reachability properties and local input/output gains of systems with polynomial
vector fields. Upper bounds for the reachable set and nonlinear system gains were
characterized using Lyapunov/storage functions and computed solving bilinear sum-ofsquares
programming problems. A procedure to refine the upper bounds by transforming
polynomial Lyapunov/storage functions to non-polynomial Lyapunov functions was developed.
The simulation-aided analysis methodology was adapted to reachability analysis. Finally,
a local small-gain theorem was proposed and applied to the robust region-of-attraction
analysis for systems with unmodeled dynamics.
125

Chapter 7
Conclusions
This thesis investigated quantitative methods for local robustness and performance
analysis of nonlinear dynamical systems with polynomial vector fields. We proposed measures
to quantify systems’ robustness against uncertainties in initial conditions (regions-ofattraction)
and external disturbances (local reachability/gain analysis). S-procedure and
sum-of-squares relaxations were used to translate Lyapunov-type characterizations to sumof-squares
optimization problems. These problems are typically bilinear/nonconvex (due to
local analysis rather than global) and their size grows rapidly with state/uncertainty space
dimension.
Our approach was based on exploiting system theoretic interpretations of these optimization
problems to reduce their complexity. We proposed a methodology incorporating
simulation data in formal proof construction enabling more reliable and efficient search
for robustness and performance certificates compared to the direct use of general purpose
solvers. This technique was adapted both to region-of-attraction and reachability analysis.
We extended the analysis to uncertain systems by taking an intentionally simplistic
126

and potentially conservative route, namely employing parameter-independent, rather than
parameter-dependent, certificates. The conservatism was simply reduced by a branch-andbound
type refinement procedure. The main thrust of these methods is their suitability
for parallel computing achieved by decomposing otherwise challenging problems into relatively
tractable smaller ones. We demonstrated proposed methods on several small/medium
size examples in each chapter and applied each method to a benchmark example with an
uncertain short period pitch axis model of an aircraft.
Additional practical issues leading to a more rigorous basis for the proposed methodology
as well as promising further research topics were also addressed. We showed that
stability of linearized dynamics is not only necessary but also sufficient for the feasibility
of the formulations in region-of-attraction analysis. Furthermore, we generalized an upper
bound refinement procedure in local reachability/gain analysis which effectively generates
non-polynomial certificates from polynomial ones.
As an initial step toward developing
optimization-based hierarchical algorithms for nonlinear system analysis, we proposed a local
small-gain theorem and applied to stability region analysis in the presence of unmodeled
dynamics.
Summaries of the material presented in each chapter were provided at the end of respective
chapters. Here, we discuss several promising future research directions.
• In section 6.5, we proposed a local small-gain type theorem which enabled us to perform
robust region-of-attraction analysis in the presence of unmodeled dynamics. This
should be regarded as an initial step toward an analysis strategy where complicated
systems are modeled as an interconnection of smaller nonlinear systems and the inputoutput
relations of these smaller systems are examined to draw conclusions about the
127

ehavior of the actual system. The integral quadratic constraint (IQC) methodology
coupled with local analysis tools proposed in this thesis may provide a feasible path
toward developing algorithms with (relatively) better scalability properties.
• As emphasized at several places throughout this thesis, the main thrust of the proposed
techniques is the suitability of most of the computation to trivial parallelization.
However, we have not examined the possibility of parallelization of the solution of the
semidefinite programs that arise. In fact, the size of a single semidefinite program
that has to be solved on a single processor sets the limits for the applicability of the
proposed algorithms. Therefore, development of parallel semidefinite programming
solvers will certainly find extensive use in optimization-based system analysis and
synthesis.
• Although parallel computing is one of the powerful tools to attack challenging problems
in many fields of engineering and sciences, it has not found a major role in
controls. We demonstrated its effectiveness in a few problems and we believe that
many classical problems in and possibly new interesting problems for controls fall in
the scope of parallel computing.
• Finally, understanding the trade-off between system/model complexity, available computational
resources, and strength of the proofs in the context of nonlinear system
analysis may lead to methodologies with better scalability properties.
128

Bibliography
[1] https://computing.llnl.gov/tutorials/parallel comp/.
[2] http://control.ee.ethz.ch/~joloef/wiki/pmwiki.phpn=Solvers.BMIBNB.
[3] http://www-user.tu-chemnitz.de/~helmberg/semidef.html.
[4] F. Amato, F. Garofalo, and L. Glielmo. Polytopic coverings and robust stability analysis
via Lyapunov quadratic forms. In Variable Structure and Lyapunov Control, pages
269–288. Springer-Verlag, 1994.
[5] V. Balakrishnan, S. Boyd, and S. Balemi. Branch and bound algorithm for computing
the minimum stability degree of parameter-dependent linear systems. Int. J. of Robust
and Nonlinear Control, 1(4):295–317, 1991.
[6] G. Balas, J. Doyle, K. Glover, A. Packard, and R. Smith. µ-Analysis and Synthesis
Toolbox - User’s Guide, 1998.
[7] B.R. Barmish. Necessary and sufficient conditions for quadratic stabilizability of an
uncertain system. Journal of Optimization Theory and Applications, 46(4):399–408,
1985.
129

[8] H. H. Bauschke and J. M. Borwein. On projection algorithms for solving convex
feasibility problems. SIAM Review, 38(3):367–426, 1996.
[9] A. Ben-Tal and A. Nemirovski. Lectures on Modern Convex Optimization: Analysis,
Algorithms, and Engineering Applications. MPS-SIAM Series on Optimization, 2001.
[10] S. Benson and Y. Ye. DSDP: Dual-scaling algorithm for semidefinite programming.
Technical Report ANL/MCS-P851-1000, Argonne National Laboratory, 2001. Available
at http://www-unix.mcs.anl.gov/DSDP/.
[11] E. Beran, L. Vandenberghe, and S. Boyd. A global BMI algorithm based on the
generalized Benders decomposition. 1997. Paper number 934.
[12] J. Bochnak, M. Coste, and M.-F. Roy. Real Algebraic Geometry. Springer-Verlag, 1998.
[13] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in
Systems and Control Theory. SIAM, Philadelphia, 1994.
[14] S. Boyd and L. Vandenberghe. Semidefinite programming relaxations of non-convex
problems in control and combinatorial optimization. In A. Paulraj, V. Roychowdhuri,
and C. Schaper, editors, Communications, Computation, Control and Signal Processing:
A Tribute To Thomas Kailath, pages 279–288. Kluwer, 1997.
[15] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge Univ. Press, 2004.
[16] A. E. Bryson and Y.-C. Ho. John Wiley & Sons Inc., 1979.
[17] F. Camilli, L. Grune, and F. Wirth. A generalization of Zubov’s method to perturbed
systems. SIAM Journal on Control and Optimization, 40(2):496–515, 2001.
130

[18] G. Chesi. Estimating the domain of attraction of uncertain polynomial systems. Automatica,
40:1981–1986, 2004.
[19] G. Chesi. On the estimation of the domain of attraction for uncertain polynomial
systems via lmis. In Proc. Conf. on Decision and Control, pages 881–886, Bahamas,
2004.
[20] G. Chesi, A. Garulli, A. Tesi, and A. Vicino. LMI-based computation of optimal
quadratic Lyapunov functions for odd polynomial systems. Int. J. Robust Nonlinear
Control, 15:35–49, 2005.
[21] H.-D. Chiang and J. S. Thorp. Stability regions of nonlinear dynamical systems: A
constructive methodology. IEEE Transaction on Automatic Control, 34(12):1229–1241,
1989.
[22] M. Choi, T. Lam, and B. Reznick. Sums of squares of real polynomials. In Proceedings
of Symposia in Pure Mathematics, volume 58(2), pages 103–126, 1995.
[23] D. F. Coutinho, M. Fu, A. Trofino, and P. Danes. L 2 -gain analysis and control of
uncertain nonlinear systems with bounded disturbance inputs. International Journal
of Robust and Nonlinear Control, 18:88–110, 2007.
[24] E. J. Davison and E. M. Kurak. A computational method for determining quadratic
Lyapunov functions for nonlinear systems. Automatica, 7:627–636, 1971.
[25] G. E. Dullerud and F. Paganini. Springer, 2000.
[26] E. Feron. Nonconvex quadratic programming, semidefinite relaxations and randomization
algorithms in information and decision systems. In T. Djaferis and I. Schick,
131

editors, System Theory: Modeling, Analysis and Control, pages 255–274. Kluwer Academic
Publishers, 1999.
[27] R. Genesio, M. Tartaglia, and A. Vicino. On the estimation of asymptotic stability
regions: State of the art and new proposals. IEEE Transaction on Automatic Control,
30(8):747–755, 1985.
[28] L. El Ghaoui and S. Niculescu, editors. Advances in linear matrix inequality methods
in control: advances in design and control.
Society for Industrial and Applied
Mathematics, Philadelphia, PA, USA, 2000.
[29] P. E. Gill, W. Murray, M. A. Saunders, and M. H. Wright. User’s guide for NPSOL
(version 4.0): a Fortran package for nonlinear programming. Technical Report 86-2,
Systems Optimization Laboratory, Stanford University, 1986.
[30] K.-C. Goh, M.G. Safonov, and G.P. Papavassilopoulos. A global optimization approach
for the BMI problem. volume 3, pages 2009–2014, 1994.
[31] M. Grant and S. Boyd. CVX: Matlab software for disciplined convex programming
(web page and software). http://stanford.edu/ boyd/cvx.
[32] O. Hachicho and B. Tibken. Estimating domains of attraction of a class of nonlinear
dynamical systems with LMI methods based on the theory of moments. In Proc. Conf.
on Decision and Control, pages 3150–3155, Las Vegas, NV, 2002.
[33] J. Hauser and M.C. Lai. Estimating quadratic stability domains by nonsmooth optimization.
In Proc. American Control Conf., pages 571–576, Chicago, IL, 1992.
[34] D. Henrion, J. Lofberg, M. Kocvara, and M. Stingl. Solving polynomial static output
feedback problems with PENBMI. pages 7581–7586, 2005.
132

[35] D. Henrion and M. Šebek. Overcoming nonconvexity in polynomial robust control
design.
In Proc. Symp. Math. Theory of Networks and Systems, Leuven, Belgium,
2006.
[36] I. D. Ivanov and E. de Klerk. Parallel implementation of a semidefinite programming
solver based on CSDP in a distributed memory cluster. Discussion Paper 2007-20,
Tilburg University, Center for Economic Research, 2007.
[37] Z. Jarvis-Wloszek, R. Feeley, W. Tan, K. Sun, and Andrew Packard. Control applications
of sum of squares programming. In D. Henrion and A. Garulli, editors, Positive
Polynomials in Control, pages 3–22. Springer-Verlag, 2005.
[38] H. K. Khalil. Nonlinear Systems. Prentice Hall, 3 rd edition, 2002.
[39] M. Ko˘cvara and M. Stingl. PENBMI User’s Guide (Version 2.0), available from
http://www.penopt.com. 2005.
[40] E.L. Lawler and D.E.Wood. Branch-and-bound methods: a survey. Operations Research,
14(4):679–719, 1966.
[41] J. Lofberg. Yalmip : A toolbox for modeling and optimization in MATLAB. In
Proceedings of the CACSD Conference, Taipei, Taiwan, 2004.
[42] D. G. Luenberger. Linear and Nonlinear Programming. Springer, 2003.
[43] M. Mesbahi, M. G. Safonov, and G. P. Papavassilopoulos. Bilinearity and complementarity
in robust control. pages 269–292, 2000.
[44] Y. Nesterov and A. Nemirovskii. Interior Point Polynomial Methods in Convex Programming:
Theory and Applications. Society for Industrial and Applied Mathematics,
Philadelphia, 1994.
133

[45] A. Paice and F. Wirth. Robustness analysis of domains of attraction of nonlinear
systems. In Proc. of the Mathematical Theory of Networks and Systems, pages 353–
356, Padova, Italy, 1998.
[46] A. Papachristodoulou. Scalable analysis of nonlinear systems using convex optimization.
Ph.D. dissertation, Caltech, January 2005.
[47] A. Papachristodoulou and S. Prajna. Analysis of non-polynomial systems using the sum
of squares decomposition. In D. Henrion and A. Garulli, editors, Positive Polynomials
in Control, pages 23–43. Springer-Verlag, 2005.
[48] P. Parrilo. Structured semidefinite programs and semialgebraic geometry methods in
robustness and optimization.
Ph.D. dissertation, California Institute of Technology,
May 2000.
[49] P. Parrilo. Semidefinite programming relaxations for semialgebraic problems. Mathematical
Programming Series B, 96(2):293–320, 2003.
[50] S. Prajna. Barrier certificates for nonlinear model validation. volume 3, pages 2884–
2889, 2003.
[51] S. Prajna, A. Papachristodoulou, P. Seiler, and P. A. Parrilo. SOSTOOLS: Sum of
squares optimization toolbox for MATLAB, 2004.
[52] D.V. Prokhorov and L.A. Feldkamp. Application of SVM to Lyapunov function approximation.
In Proc. Int. Joint Conf. on Neural Networks, Washington, DC, 1999.
[53] B. Reznick. Some concrete aspects of hilberts 17th problem. In Contemporary Mathematics,
volume 253, pages 251–272. American Mathematical Society, 2000.
[54] W. Rudin. Functional Analysis. McGraw-Hill Science/, Groningen, 1991.
134

[55] G. Serpen. Search for a Lyapunov function through empirical approximation by artificial
neural nets: Theoretical framework. In Proc. Int. Joint Conf. on Artificial Neural
Networks, pages 735–740, Montreal, Canada, 2005.
[56] J.F. Sturm. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric
cones. Optimization Methods and Software, 11–12:625–653, 1999. Available at
http://sedumi.mcmaster.ca/.
[57] W. Tan. Nonlinear control analysis and synthesis using sum-of-squares programming.
Ph.D. dissertation, UC, Berkeley, May 2006.
[58] W. Tan and A. Packard. Stability region analysis using sum of squares programming.
In Proc. American Control Conf., pages 2297–2302, Minneapolis, MN, 2006.
[59] W. Tan and A. Packard. Stability region analysis using polynomial and composite
polynomial lyapunov functions and sum-of-squares programming. Automatic Control,
IEEE Transactions on, 53(2):565–571, 2008.
[60] W. Tan, A. Packard, and T. Wheeler. Local gain analysis of nonlinear systems. In
Proc. American Control Conf., pages 92–96, Minneapolis, MN, 2006.
[61] R. Tempo, G. Calafiore, and F. Dabbene. Randomized Algorithms for Analysis and
Control of Uncertain Systems. Springer, 2005.
[62] B. Tibken. Estimation of the domain of attraction for polynomial systems via LMIs.
In Proc. Conf. on Decision and Control, pages 3860–3864, Sydney, Australia, 2000.
[63] B. Tibken and Y. Fan. Computing the domain of attraction for polynomial systems
via BMI optimization methods. In Proc. American Control Conf., pages 117–122,
Minneapolis, MN, 2006.
135

[64] J. E. Tierno, R. M. Murray, J. C. Doyle, and I. M. Gregory. Numerically efficient
robustness analysis of trajectory tracking for nonlinear systems.
AIAA Journal of
Guidance, Control, and Dynamics, 20:640–647, 1997.
[65] K. C. Toh, M. J. Todd, and R. Tutuncu. SDPT3 — a Matlab software package
for semidefinite programming. Optimization Methods and Software, 11:545–581, 1999.
Available at http://www.math.nus.edu.sg/~mattohkc/sdpt3.html.
[66] O. Toker and H. Ozbay. On the NP-hardness of solving bilinear matrix inequalities
and simultaneous stabilization with static output feedback. In Proc. American Control
Conf., pages 2525–2526, 1995.
[67] U. Topcu and A. Packard. Local stability analysis for uncertain nonlinear systems,
2007. To appear in IEEE Transaction on Automatica Control.
[68] U. Topcu and A. Packard. Stability region analysis for uncertain nonlinear systems.
In Proc. Conf. on Decision and Control, pages 1693–1698, New Orleans, LA, 2007.
[69] U. Topcu and A. Packard. Simulation-aided reachability and local gain analysis for
nonlinear dynamical systems, 2008. Under review for the Conf. on Decision and Control.
[70] U. Topcu, A. Packard, and P. Seiler. Local stability analysis using simulations and
sum-of-squares programming, 2006. To appear in Automatica, paper number: 6817.
[71] U. Topcu, A. Packard, P. Seiler, and G. Balas. Local stability analysis for uncertain
nonlinear systems using a branch-and-bound algorithm. In Proc. American Control
Conf., pages 3428–3433, Seattle, WA, 2008.
136

[72] U. Topcu, A. Packard, P. Seiler, and G. Balas. Local stability analysis for uncertain
nonlinear systems using a branch-and-bound type algorithm, 2008. Submitted to IEEE
Transaction on Automatica Control.
[73] U. Topcu, A. Packard, P. Seiler, and T. Wheeler. Stability region analysis using simulations
and sum-of-squares programming. In Proc. American Control Conf., pages
6009–6014, New York, NY, 2007.
[74] A. Trofino. Robust stability and domain of attraction of uncertain nonlinear systems.
In Proc. American Control Conf., pages 3707–3711, Chicago, IL, 2000.
[75] A. Vannelli and M. Vidyasagar. Maximal Lyapunov functions and domains of attraction
for autonomous nonlinear systems. Automatica, 21(1):69–80, 1985.
[76] M. Vidyasagar. Nonlinear Systems Analysis. Prentice Hall, 2 nd edition, 1993.
[77] R. Vinter. A characterization of the reachable set for nonlinear control systems. SIAM
Journal on Control and Optimization, 18(6):599–610, 1980.
[78] T-C. Wang, S. Lall, and M. West. Polynomial level-set methods for nonlinear dynamical
systems analysis. In Proc. Allerton Conf. on Communication, Control, and Computing,
Allerton, IL, 2005.
[79] H. Wolkowicz, R. Saigal, and L. Vandenberghe, editors. Handbook of Semidefinite
Programming: Theory, Algorithms, and Applications. Kluwer Academic Publishers,
2000.
[80] S. J. Wright. Primal-dual interior-point methods. Society for Industrial and Applied
Mathematics, Philadelphia, PA, USA, 1997.
137