lyapunov based analysis and controller synthesis for ...pack/library/wloszekphdthesis.pdf · 1...
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Lyapunov Based Analysis and Controller Synthesis for PolynomialSystems using Sum-of-Squares Optimization
by
Zachary William Jarvis-Wloszek
B.S.E. (Princeton University) 1999M.S. (University of California, Berkeley) 2001
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, ChairProfessor J. Karl Hedrick
Professor Laurent El Ghaoui
Fall 2003
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The dissertation of Zachary William Jarvis-Wloszek is approved:
Chair Date
Date
Date
University of California, Berkeley
Fall 2003
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Lyapunov Based Analysis and Controller Synthesis for Polynomial
Systems using Sum-of-Squares Optimization
Copyright Fall 2003
by
Zachary William Jarvis-Wloszek
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Abstract
Lyapunov Based Analysis and Controller Synthesis for Polynomial Systems using
Sum-of-Squares Optimization
by
Zachary William Jarvis-Wloszek
Doctor of Philosophy in Engineering-Mechanical Engineering
University of California, Berkeley
Professor Andrew K. Packard, Chair
This thesis considers a Lyapunov based approach to analysis and controller syn-
thesis for systems whose dynamics are described by polynomials. We restrict the candidate
Lyapunov functions as well as the controllers to be polynomials, so that the conditions in
the Lyapunov theorem involve only polynomials. The Positivstellensatz delineates the ex-
act manner to ascertain (ie. “certify”) if the theorem’s conditions hold. For computational
reasons we further restrict the choice of certificates to those, which, with fixed Lyapunov
functions and controllers, can be checked using sum-of-squares optimization. Following
these steps, we pose convex or coordinatewise convex (convex in one variable when the
others are held fixed) iterative algorithms to search for Lyapunov functions and controllers.
We provide a basic review of polynomials, the Positivstellensatz and the sum-of-
squares optimization results, which gives the necessary background to follow the subsequent
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developments that lead to our proposed algorithms. First, we consider global stability by
constructing convex algorithms to search for Lyapunov functions that demonstrate semi-
global exponential stability. We then extend these algorithms in a coordinatewise convex
form for both state and output feedback controller design. Additionally, we include a
convex procedure to quantify a system’s performance by bounding the induced norm from
disturbances to outputs. Examples are included for illustration.
Since we do not always desire global results, we provide two algorithmic approaches
to prove local stability. These approaches are coordinatewise convex and estimate the size
of the system’s region of attraction by finding the largest level set of a Lyapunov function on
which the stability theorem’s conditions hold. An example provides a graphical comparison
of the two approaches. As with the global case, we then extend these algorithms to allow
for state and output feedback controller design. Also, we derive bounds for the largest
peak disturbance under which an invariant set remains invariant, and bounds for the local
induced gain from disturbances to outputs on the set.
Additionally, we extend the two local asymptotic stability algorithms to discrete
time polynomial systems. Unfortunately, the structure of the local asymptotic stability
Lyapunov theorem in discrete time does not allow for controller design using our iterative
approach.
Professor Andrew K. PackardDissertation Committee Chair
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To Sarah for her support and encouragement
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Contents
1 Introduction 11.1 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Summary of Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Polynomial Background 82.1 Sum-of-Squares Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1 Properties of the Set of SOS Polynomials . . . . . . . . . . . . . . . 102.1.2 Computational Aspects of SOS Polynomials . . . . . . . . . . . . . . 11
2.2 The Positivstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.2 Theorems Related to the Positivstellensatz . . . . . . . . . . . . . . 23
2.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Global Analysis and Controller Synthesis 273.1 Stability Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1.2 Stability Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Convex Stability Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.1 Stability Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3 Disturbance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.4 State Feedback Controller Design . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4.1 Iterative State Feedback Design Algorithms . . . . . . . . . . . . . . 403.4.2 State Feedback Design Example: Non-Holonomic System . . . . . . 453.4.3 State Feedback Design Example: Nonlinear Spring-Mass System . . 46
3.5 Output Feedback Controller Design . . . . . . . . . . . . . . . . . . . . . . . 493.5.1 Iterative Output Feedback Design Algorithms . . . . . . . . . . . . . 513.5.2 Output Feedback Design Example . . . . . . . . . . . . . . . . . . . 55
3.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
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4 Local Stability and Controller Synthesis 604.1 Local Stability Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.2 Convex Stability Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.1 Expanding D Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 634.2.2 Expanding Interior Algorithm . . . . . . . . . . . . . . . . . . . . . . 704.2.3 Estimating the Region of Attraction Example . . . . . . . . . . . . . 74
4.3 Disturbance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.3.1 Reachable Set Bounds under Unit Energy Disturbances . . . . . . . 784.3.2 Set Invariance under Peak Bounded Disturbances . . . . . . . . . . . 824.3.3 Induced L2 → L2 Gain . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4 State Feedback Controller Design . . . . . . . . . . . . . . . . . . . . . . . . 864.4.1 Expanding D Algorithm for State Feedback Design . . . . . . . . . . 864.4.2 Expanding Interior Algorithm for State Feedback Design . . . . . . 914.4.3 State Feedback Design Example . . . . . . . . . . . . . . . . . . . . 95
4.5 Output Feedback Controller Design . . . . . . . . . . . . . . . . . . . . . . . 1004.5.1 Expanding D Algorithm for Output Feedback Design . . . . . . . . 1014.5.2 Expanding Interior Algorithm for Output Feedback Design . . . . . 1074.5.3 Output Feedback Design Example . . . . . . . . . . . . . . . . . . . 111
4.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5 Discrete Time Containment & Stability 1175.1 Set Invariance for Discrete Time Systems . . . . . . . . . . . . . . . . . . . 118
5.1.1 Set Invariance Example . . . . . . . . . . . . . . . . . . . . . . . . . 1185.1.2 Set Invariance under Disturbances . . . . . . . . . . . . . . . . . . . 1195.1.3 Set Invariance under Disturbances Example . . . . . . . . . . . . . . 120
5.2 Discrete Time Stability Background . . . . . . . . . . . . . . . . . . . . . . 1215.3 Convex Stability Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.3.1 Expanding D Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 1235.3.2 Expanding Interior Algorithm . . . . . . . . . . . . . . . . . . . . . . 1265.3.3 Estimating the Region of Attraction Example . . . . . . . . . . . . . 129
5.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6 Conclusions and Recommendations 132
Bibliography 135
A Semidefinite Programming 140
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Acknowledgements
I would like to thank everyone at Berkeley who has made my time here interesting and
exciting. Things would not have been the same without all my friends and fellow graduate
students, especially Ryan White, Derek Caveney and the BCCI lab.
I would also like to thank Pete Seiler for writing the polynomial software class
that I used in almost every line of my code; he has always been a great source of helpful
questions, comments, and one line counter examples.
Additionally, I would like to thank my committee for taking a genuine interest in
my project. Particular thanks go to Andy Packard for giving me the space to go off and
take my own apporach to exploring these topics.
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Chapter 1
Introduction
In this thesis, we consider the problems of stability analysis and controller synthesis
for nonlinear systems with polynomial dynamics. Our approach uses the primary tool of
nonlinear control, Lyapunov functions. The standard method of analysis consists of seeing
if one can pick a candidate Lyapunov function such that when it is combined with the
system, they meet the Lyapunov theorem requirements, which are detailed as needed at
the beginnings of Chapters 3-5. The synthesis approach is similar in that it searches for
a candidate Lyapunov function as well as a controller that will make the system fit the
assumptions of a Lyapunov theorem. Any Lyapunov function that admits such a controller
is called a Control Lyapunov Function (CLF).
Given a CLF, there are many ways to design a controller to meet the Lyapunov
assumptions and many of the important design advances are detailed in [17]. Most of these
techniques are based on backstepping, which is a control design procedure that uses the
control input to sequentially counteract the system’s linear and nonlinear dynamics. This
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approach is often quite successful, but it does rely on the system’s dynamics having special
structure. Additional transformations can be introduced to widen the types of structure for
which the backstepping concept will work, however, some structure is still required.
We approach the candidate Lyapunov function and controller search from a dif-
ferent angle: optimization. Recent advances in convex optimization, [21], when paired with
the important Positivstellensatz from real algebraic geometry, [3], allow us to use convex
optimization to search for polynomials that meet certain conditions. If we restrict the Lya-
punov functions and controllers to be polynomials this allows us to formulate the conditions
of the Lyapunov theorems as the constraints of a convex optimization problem.
The ideas behind the optimization approach to Lyapunov analysis and synthesis
are not new, since they are at the heart of the Linear Matrix Inequality (LMI) approach
to linear systems, which is well illustrated in [5]. We use the new optimization results to
create algorithms that are either convex or stepwise convex to design Lyapunov functions
as well as state and output feedback controllers. These algorithms extend the early results
of [20] and [13].
1.1 Thesis Overview
This thesis is centered on the questions of proving stability and designing sta-
bilizing controllers for polynomial systems. Our approach is based on classic Lyapunov
function results that can be turned into computationally tractable optimization problems
using recent theoretical results. In this section we give an outline of this thesis and topics
encountered along the way.
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Chapter 2 provides background material to introduce the important polynomial
properties and concepts that are exploited for analysis and synthesis in the later chapters.
First, we introduce the basic polynomial definitions that allow us to define sum-of-squares
polynomials, which allows us to introduce the very important Theorem 2, from [21]. This
theorem makes the essential link between existence of certain polynomials and convex opti-
mization, and when coupled with Theorem 4, the Positivstellensatz, provides our motivation
for working with polynomial systems. After introducing these theoretical results we apply
them to a series of examples and briefly present a few interesting extensions.
Chapter 3 considers the problems of proving global stability for polynomial sys-
tems. After a review of pertinent elements of Lyapunov stability theory, we pose lemmas
for global asymptotic stability and semi-global exponential stability of polynomial systems.
Upon investigation of these, we find that we can construct a pair of algorithms to solve
the semi-global exponential stability problem whose feasibility properties are superior to
those of the global asymptotic stability problem. We then extend these stability algorithms
to consider both state and output feedback controller design problems and provide a test
system to illustrate their utilization. Additionally, we provide a convex method to bound a
system’s induced L2 → L2 gain from disturbance to output.
Chapter 4 expands on the global stability results of the previous chapter to study
local stability. We provide two complementary algorithms to search for Lyapunov functions
to demonstrate local asymptotic stability and approximate the system’s region of attraction.
After demonstrating the aspects of the two algorithmic approaches, we look at techniques
to do local disturbance analysis, including finding the maximum peak disturbance such that
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region remains invariant and a local induced L2 → L2 gain bound. We then extend the
stability algorithms, as in the global case, to both state and output feedback controller de-
sign. We apply the synthesis algorithms to find simple controllers that stabilize increasingly
difficult versions of the test system from Chapter 3.
Chapter 5 looks at applying the polynomial techniques developed in Chapter 2 to
the discrete time polynomial systems. The discrete time formulation allows us to pose simple
invariant set problems in a straight forward manner without using Lyapunov functions. We
then consider the discrete time Lyapunov function framework and find applicable versions
of the local stability algorithms from Chapter 4. Unfortunately, due to the nature of the
discrete time Lyapunov theorem, we can not extend the local stability algorithms to do any
form of controller synthesis.
Chapter 6 presents conclusions and gives recommendations for future work.
1.2 Thesis Contributions
This thesis makes several contributions in the areas of stability analysis and con-
troller synthesis for polynomial systems. These contributions are discussed below.
1. Nonlinear Stability Analysis: We provide convex algorithms to construct poly-
nomial Lyapunov functions for polynomial dynamic systems that prove either global
or local stability. The global algorithms presented in Chapter 3 construct Lyapunov
functions that make the system semi-globally exponentially stable, while the two ap-
proaches to local stability in both Chapter 4 and 5 yield stepwise convex algorithms
to prove that the system is locally asymptotically stable. As well as constructing
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Lyapunov functions, these algorithms provide estimates of the region of attraction.
The local algorithms are applied to both continuous time and discrete time polyno-
mial systems, and importantly, if the system’s linearization is stable, then the local
algorithms will always be feasible.
2. Nonlinear Disturbance Analysis: We investigate the effects of external distur-
bances on polynomial systems by constructing a number of convex Lyapunov based
algorithms to bounds their effects. In Chapter 3 we provide a convex bounding pro-
cedure for the global induced L2 → L2 gain from disturbances to system outputs. In
Chapter 4, we extend this bound so that we can apply it on local invariant regions of
state space to quantify local system performance. Additionally, we provide bounds for
the largest peak value that a disturbance can have so that a given set remains invari-
ant, as well as, bounds for a system’s reachable set under unit energy disturbances,
which previously appeared in [13]. We construct a non-Lyapunov based disturbance
peak bound for set invariance of discrete time systems in Chapter 5.
3. Nonlinear Controller Synthesis: We construct stepwise convex algorithms to de-
sign stabilizing polynomial controllers for polynomial systems for both the global and
the local cases. In Chapter 3, the global synthesis algorithms find both state and out-
put feedback controllers to make the closed loop system semi-globally exponentially
stable. As in the local stability case, we use two different approaches to design both
state and output feedback locally stabilizing controllers in Chapter 4, as well as esti-
mate their domain of attraction. One of the algorithms for state feedback controller
previously appeared in [13]. Additionally, the local algorithms are feasible as long the
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system’s linearization is controllable, however, the local approaches are not applicable
to discrete time controller design.
1.3 Summary of Examples
Examples of the algorithms and techniques constructed in this thesis are provided
for the following topics in the sections indicated. Additionally, all the software necessary to
run the examples listed below is provided at http://jagger.berkeley.edu/~zachary.
1. Stability Analysis
• Semi-global Exponential Stability: §3.2.1
• Local Asymptotic Stability: §4.2.3
• Discrete Time Set Invariance §5.1.1
• Discrete Time Local Asymptotic Stability: §5.3.3
2. Disturbance Analysis
• Global Induced L2 → L2 Gain Bound: §3.4.3 and §3.5.2.
• Local Induced L2 → L2 Gain Bound: §4.4.3
• Bounding Maximum Disturbance Peak for Set Invariance: §4.4.3
• Discrete Time Set Invariance under Disturbances: §5.1.3
3. State Feedback Controller Design
• Semi-global Exponential Stabilization: §3.4.2 and §3.4.3
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• Local Asymptotic Stabilization: §4.4.3
4. Output Feedback Controller Design
• Semi-global Exponential Stabilization: §3.5.2
• Local Asymptotic Stabilization: §4.5.3
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Chapter 2
Polynomial Background
The properties of real polynomials and especially one very important subset of
them form the basis for the results of later chapters; this chapter provides the necessary
background for the later results as well as pointers to references for more in-depth treatments
of the well developed fields that these topics touch upon.
First, let R denote the real numbers and Z+ denote the set of nonnegative integers,
0, 1, . . .. Using this notation we can make the formal definitions that will be used in almost
every result.
Definition 1 (Monomial) Every α ∈ Zn+ defines a function mα : Rn → R, called a
monomial. Given a specific α ∈ Zn+, the monomial mα maps x ∈ Rn into mα(x) = xα :=
xα11 xα2
2 · · ·xαnn . The degree of a monomial is defined as deg mα :=
∑ni=1 αi.
Definition 2 (Polynomial) A polynomial p is defined as a linear combination of a finite
set of monomials mαjkj=1. Given a set of scalar reals, cjk
j=1 ∈ R, a polynomial p is
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defined as:
p :=k∑
j=1
cjmαj
or in terms of its action on x ∈ Rn
p(x) =k∑
j=1
cjmαj (x) =k∑
j=1
cjxαj
Using the definition of degree for a monomial, the degree of p is defined as deg p :=
maxj(deg mαj ).
The set of polynomials with real coefficients and common independent variables,
say, x1, . . . , xn, is often denoted as R[x1, . . . , xn] to emphasize that these polynomials form
a ring. To eliminate reference to a particular set of independent variables, we will denote
the set of all polynomials in n variables with real coefficients as Rn, with the assumption
that if p ∈ Rn and f ∈ Rn then p and f are functions of the same independent variables.
Additionally, define a subset of Rn, Rn,d := p ∈ Rn|deg p ≤ d; this is just the
set of all polynomials in n variables that have maximum degree d. If all the monomials of
polynomial p are of the same degree, say d, then p is called homogeneous and it obeys the
relation p(λx) = λdp(x) for any scalar λ.
Another subset of Rn is the set of positive semidefinite (PSD) polynomials, which
are nonnegative on all of Rn. This set is defined as Pn := p ∈ Rn|p(x) ≥ 0,∀x ∈ Rn.
Also define Pn,d := Pn ∩Rn,d.
Following standard notation for the real numbers, we will define any of these sets
raised to a integer power, m, to denote an m-vector whose elements are drawn from the
indicated set; as an example, Rmn denotes an m-vector of polynomials in n variables.
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2.1 Sum-of-Squares Polynomials
A very important subset of the polynomials are the Sum-of-Squares (SOS) poly-
nomials. Let Σn be the set of all SOS polynomials in n variables, which is defined as
Σn :=
s ∈ Rn
∣∣∣∣∣ ∃M < ∞,∃piMi=1 ⊂ Rn such that s =
M∑i=1
p2i
The SOS polynomials take their name from the fact that they can be represented as sums
of squares of other polynomials. Additionally, define Σn,d = Σn ∩Rn,d.
2.1.1 Properties of the Set of SOS Polynomials
Since every s ∈ Σn is a sum of squared polynomials, it is clear that s(x) ≥ 0,
∀x ∈ Rn, which implies that Σn ⊆ Pn. An interesting question is whether the set of SOS
polynomials is equal to or strictly contained in the set of positive semidefinite polynomials.
Hilbert showed that, when restricted to homogeneous polynomials, there are only
three cases of n, d such that Σn,d = Pn,d. These results can be translated to general
polynomials, see §3.2 in [21], to prove that Σn,d = Pn,d only for
• Polynomials in one variable, n = 1.
• Quadratic polynomials, d = 2.
• Quartics in two variables, n = 2, d = 4.
Thus, in general Σn,d ⊂ Pn,d. Hilbert’s method to construct a polynomial in
Pn \ Σn is very complicated and he did not use it to demonstrate any examples. In [29],
Reznick gives an overview of the technique used, its relation to Hilbert’s 17th problem, as
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well as, a series of examples derived by more modern methods. One of the first examples
exhibited dates from 1965 and is the Motzkin polynomial, M(x, y, z) given below, from [19].
M(x, y, z) = x4y2 + x2y4 + z6 − 3x2y2z2
This polynomial can be shown to be positive semidefinite using the arithmetic-geometric
inequality a+b+c3 ≥ (abc)
13 with (a, b, c) = (x4y2, x2y4, z6). By methods to be described
later, §2.1.2, it can be shown to not be SOS.
2.1.2 Computational Aspects of SOS Polynomials
Working with polynomials in Pn can be difficult since there is no full parameter-
ization of the set, nor, in general, are there efficient tests to check if a given polynomial is
in the set. However, given the number of variables, n, and degree of polynomials, d, we
can form a full parameterization of Σn,d, which directly leads to an efficient semidefinite
programming test to check if a polynomial is SOS (see Appendix A for a brief overview of
semidefinite programming).
A full parameterization of fixed degree SOS polynomials
First we note that SOS polynomials must always be of even degree, so we will
consider the parameterization of the set Σn,2d for some n, d ∈ Z+. The following lemma
provides the starting point for the parameterization.
Lemma 1 If s ∈ Σn,2d, then there exist pi ∈ Rn,d, i = 1, . . . ,M , for some finite M such
that
s =M∑i=1
p2i
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12
This lemma is a restricted version of Theorem 1 in [28], which gives tighter restrictions on
the pi’s when s is known.
Using Lemma 1, we can pose a full parameterization, often referred to as the “Gram
matrix” approach, [7]. First, define zn,d to be the vector of all monomials in n variables
of degree less than or equal to d ordered in the following manner. Given α, β ∈ Zn+, xα
precedes xβ if deg xα < deg β or if deg xα = deg β and the first entry of α−β that is strictly
negative is preceded by a strictly positive entry. As an example, with n = 2, d = 2
z2,2(x) :=
1
x1
x2
x21
x1x2
x22
For a general pair of n and d, zn,d(x) will be a
(n+d
d
)-vector.
With the definition of zn,d(x) it is possible to characterize a polynomial, p ∈ Rn,2d,
as
p(x) = z∗n,d(x)Qzn,d(x)
where Q is the “Gram” matrix. The idea of representing polynomials as quadratic forms
of vectors of monomials predates [7] by many years. The earliest quadratic representation,
dating from 1968, [4], started a framework which was used to find homogeneous polynomial
Lyapunov functions in [36]. The following result, which first appeared as Proposition 2.3 in
[7], generalizes the earlier works and establishes when a polynomial is SOS.
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13
Theorem 1 Fix p ∈ Rn,2d. p ∈ Σn,2d if and only if there exists a Q 0 such that
p(x) = zn,d(x)∗Qzn,d(x).
Proof:
⇒ If p ∈ Σn,2d then via Lemma 1 we know that there exist pi ∈ Rn,d, i = 1, . . . ,M such
that p =∑M
i=1 p2i . Writing each of these polynomials as pi(x) = q∗i zn,d(x) with qi a real
vector of appropriate dimension, we have
p(x) =M∑i=1
(q∗i zn,d(x)
)2
=M∑i=1
z∗n,d(x)qiq∗i zn,d(x)
= z∗n,d(x)
(M∑i=1
qiq∗i
)zn,d(x)
= z∗n,d(x)Qzn,d(x)
From its construction it is clear that Q 0.
⇐ Since Q 0, we can factor Q =∑r
i=1 qiq∗i where r is the rank of Q. Then
reversing the argument above
p(x) = z∗n,d(x)
(r∑
i=1
qiq∗i
)zn,d(x)
=r∑
i=1
z∗n,d(x)qiq∗i zn,d(x)
=r∑
i=1
(q∗i zn,d(x)
)2
(a)=
r∑i=1
p2i (x)
where (a) comes from defining pi(x) := q∗i zn,d(x).
2
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Corollary 1 The number of terms for an SOS decomposition can be chosen to be the num-
ber of elements in zn,d or fewer.
This theorem gives necessary and sufficient conditions for a polynomial to be
SOS, however for p ∈ Σn,2d there are, in general, many symmetric Q such that p(x) =
z∗n,d(x)Qzn,d(x) and some are not positive semidefinite as the following example shows.
Example 1 Take p ∈ R2,4 to be such that p(x) = x41 + x2
1x22 + x4
2. Both Q1 and Q2 below
are such that z∗2,2(x)Qiz2,2(x) = p(x).
1
x1
x2
x21
x1x2
x22
∗
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
︸ ︷︷ ︸
Q1
1
x1
x2
x21
x1x2
x22
=
1
x1
x2
x21
x1x2
x22
∗
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 1 0 1
0 0 0 0 −1 0
0 0 0 1 0 1
︸ ︷︷ ︸
Q2
1
x1
x2
x21
x1x2
x22
= p(x)
Note that Q1 0 while Q2 6 0. Q1’s positive semidefiniteness shows that p ∈ Σ2,4, while
Q2 6 0 shows nothing.
An LMI Test for SOS
Theorem 1 gives the complete parameterization of the SOS polynomials for a given
number of variables and fixed degree, however it does not give a method to check if a given
polynomial is SOS.
In an effort to understand the set of matrices Q that make z∗n,d(x)Qzn,d(x) = p(x)
for some p ∈ Rn,2d, pick the standard basis for symmetric matrices, Ei, of the appropriate
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15
size,(n+d
d
)×(n+d
d
). Working out z∗n,d(x) (
∑qiEi) zn,d(x) and equating coefficients with
p(x) shows that set of matrices that make the equality hold are an affine subspace of the
symmetric matrices as was shown in [22].
Given p ∈ Rn,2d, let Q0 be any symmetric matrix such that
z∗n,d(x)Q0zn,d(x) = p(x)
and let Qinq
i=1 be the set of symmetric matrices such that
z∗n,d(x)Qizn,d(x) = 0
With this setup, we can define the affine subspace of symmetric matrices related to p as
Qp := Q|z∗n,d(x)Qzn,d = p(x) =
Q0 +
nq∑i=1
λiQi
∣∣∣λi ∈ R, i = 1, . . . , nq
The following example illustrates the general procedure for finding the set of Qi’s that define
the subspace.
Example 2 For p ∈ R2,4 find some symmetric Q0 such that z∗2,2(x)Q0z2,2(x) = p(x).
Picking the standard basis for symmetric matrices, we write z∗2,2(x)Qz2,2(x) = 0 as
1
x1
x2
x21
x1x2
x22
∗
q1 q2 q3 q4 q5 q6
q2 q7 q8 q9 q10 q11
q3 q8 q12 q13 q14 q15
q4 q9 q13 q16 q17 q18
q5 q10 q14 q17 q19 q20
q6 q11 q15 q18 q20 q21
1
x1
x2
x21
x1x2
x22
= 0
equating terms we have
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Monomial Coefficient1 q1
x1 2q2
x2 2q3
x21 q7 + 2q4
x1x2 2q8 + 2q5
x22 q12 + 2q6
x31 2q9
x21x2 2q10 + 2q13
x1x22 2q11 + 2q14
x32 2q15
x41 q16
x31x2 2q17
x21x
22 q19 + 2q18
x1x32 2q20
x42 q21
and since each coefficient must be identically zero, we can find the subspace of matrices such
that z∗2,2(x)Qz2,2(x) = 0 to be
Q|z∗2,2(x)Qz2,2(x) = 0 = λ1 (2E7 − E4)︸ ︷︷ ︸Q1
+λ2 (E8 − E5)︸ ︷︷ ︸Q2
+λ3 (2E12 − E6)︸ ︷︷ ︸Q3
+λ4 (E10 − E13)︸ ︷︷ ︸Q4
+λ5 (E11 − E14)︸ ︷︷ ︸Q5
+λ6 (2E19 − E18)︸ ︷︷ ︸Q6
|λ1, . . . , λ6 ∈ R
= 6∑
i=1
λiQi|λi ∈ R
which shows how to find Qp for a polynomial with fixed number of variables and degree.
In [22], Powers and Wormann got as far as finding the affine subspace, which
allowed them to give the equivalence p ∈ Σn,2d iff ∃Q ∈ Qp such that Q 0. However, they
did not recognize that checking if there existed λi’s to make Q0 +∑
λiQi 0 was convex
and just an LMI feasibility problem, instead they proposed a less efficient search method
using quantifier elimination. In [21], Parrilo realized that the existence of a Q 0, can be
solved as an LMI and gave the following theorem.
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Theorem 2 ([21], Theorem 3.3) Given p ∈ Rn,2d, find the relevant affine subspace Qp =
Q0 +∑
i λiQi|λi ∈ R. p ∈ Σn,2d iff the following LMI is feasible
∃λi
s.t. Q0 +∑
λiQi 0
Proof:
From Theorem 1, we know that p ∈ Σn,2d iff there exists a Q 0 such that z∗2n,d(x)Qzn,d(x) =
p(x), so we only need search over Qp, which is exactly the LMI given.
2
Parrilo also introduced the following important extension that can be proved in a
similar manner to Theorem 2.
Theorem 3 ([21], §3.2) Given a finite set pimi=0 ∈ Rn, the existence of aim
i=1 ∈ R
such that
p0 +m∑
i=1
aipi ∈ Σn
is an LMI feasibility problem.
This theorem is very useful since it allows to answer questions like the following example.
Example 3 Given p0, p1 ∈ Rn, does there exist k ∈ Rn such that
p0 + kp1 ∈ Σn (2.1)
To answer this question, write k as a linear combination of its monomials mj, k =∑sj=1 ajmj. Rewrite (2.1) using this decomposition
p0 + kp1 = p0 +s∑
j=1
aj(mjp1)
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which, since (mjp1) ∈ Rn, can be checked by Theorem 3.
A software package, SOSTOOLS [23, 24], was written to aid in solving the LMIs
that result from Theorem 3. This package sets up the LMIs from the polynomial problems,
does some smart preprocessing to reduce problem size and uses Sturm’s SeDuMi semidefinite
programming solver, [33], to solve the LMIs.
Additional computational gains can be had by exploiting polynomial symmetries,
[9], and using the Newton polytope algorithm presented in [7] to reduce the number of
monomials in the Gram matrix formulation, which makes the resulting LMIs smaller with
fewer free parameters. These computational improvements are set to appear in the next
release of SOSTOOLS.
2.2 The Positivstellensatz
Having introduced SOS polynomials it is now possible to make the algebraic defi-
nitions that are necessary to present one of the seminal theorems of real algebraic geometry,
which generalizes many known results including the S-Procedure, as shown in §2.2.1.
Definition 3 Given g1, . . . , gt ∈ Rn, the Multiplicative Monoid generated by gj’s is
the set of all finite products of gj’s, including the empty product, defined to be 1. It is
denoted as M(g1, . . . , gt). For completeness define M(φ) := 1.
An example: M(g1, g2) = gk11 gk2
2 | k1, k2 ∈ Z+
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Definition 4 Given f1, . . . , fs ∈ Rn, the Cone generated by fi’s is
P(f1, . . . , fs) :=
s0 +∑
sibi | si ∈ Σn, bi ∈M(f1, . . . , fs)
For completeness note that P(φ) := Σn.
Remembering that if f ∈ Rn, s ∈ Σn, then sf2 ∈ Σn, allows us to write any cone as a sum
of 2s terms. Note that this reduction in the number of free SOS polynomials need not be
beneficial.
An example: P(f1, f2) = s0 + s1f1 + s2f2 + s3f1f2 | s0, . . . , s3 ∈ Σn
Definition 5 Given h1, . . . , hu ∈ Rn, the Ideal generated by hk’s is
I(h1, . . . , hu) :=∑
hkpk | pk ∈ Rn
For completeness note that I(φ) := 0.
With these definitions we can state the following theorem which is a version of the
original theorem in [31] restricted to Rn.
Theorem 4 (Positivstellensatz [3, Theorem 4.2.2] ) Given sets of polynomials f1, . . . , fs,
g1, . . . , gt, and h1, . . . , hu in Rn, the following are equivalent:
1. The set x ∈ Rn
∣∣∣∣∣∣∣∣∣∣∣∣
f1(x) ≥ 0, . . . , fs(x) ≥ 0,
g1(x) 6= 0, . . . , gt(x) 6= 0,
h1(x) = 0, . . . , hu(x) = 0
is empty,
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2. There exist polynomials f ∈ P(f1, . . . , fs), g ∈ M(g1, . . . , gt), h ∈ I(h1, . . . , hu) such
that
f + g2 + h = 0
2.2.1 Examples
To reinforce the usefulness of the Positivstellensatz (P-satz), consider the range of
the following examples that become convex and thus tractable when the P-satz is combined
with the results of Theorem 3.
Positivstellensatz Certificates
The LMI based tests for SOS polynomials from Theorem 3 can be used to prove
that the set emptiness condition from the P-satz holds, by finding specific f , g, and h such
that f +g2+h = 0. These f , g, and h are known as P-satz certificates since they certify that
the equality holds. The following theorem states precisely how semidefinite programming
can be used to search for certificates.
Theorem 5 (Theorem 4.8, [21]) Given polynomials f1, . . . , fs, g1, . . . , gt, and h1, . . . , hu
in Rn, if the set
x ∈ Rn|fi(x) ≥ 0, gj(x) 6= 0, hk(x) = 0, i = 1, . . . , s, j = 1, . . . , t, k = 1, . . . , u
is empty then the search for bounded degree Positivstellensatz refutations can be done us-
ing semidefinite programming. If the degree bound is chosen large enough the semidefinite
programs will be feasible and give the refutation certificates.
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The proof of this theorem involves writing out all of the terms of P(f1, . . . , fs),
M(g1, . . . , gt), and I(h1, . . . , hu) to form the equality constraint, f + g2 + h = 0. For a
fixed degree d, set the term g ∈M(g1, . . . , gt) such that it is of degree greater than or equal
to d/2, then pick each of the free polynomials in f and h such that they have degree at
least d. Now run the LMI with the equality constraint as well as the SOS constraints on
the free polynomials in f . If you search over all g for each d, then you eventually find the
Positivstellensatz certificates.
An LMI test for Pn
Using the P-satz, we can now test to see if a polynomial p ∈ Rn is in Pn. If p ∈ Pn,
then ∀x ∈ Rn, p(x) ≥ 0. Equivalently x ∈ Rn|p(x) < 0 is empty, or in the P-satz format
x ∈ Rn| − p(x) ≥ 0, p(x) 6= 0 is empty
This condition holds iff ∃f ∈ P(−p) and g ∈M(p) such that f+g2 = 0. Using the definition
of the cone and the monoid, p ∈ Pn iff ∃s0, s1 ∈ Σn and k ∈ Z+ such that
s0 − ps1 + p2k = 0
If we fix k and the degree of s1 to be d, we can rewrite the conditions above as p ∈ Pn iff
∃s1 such that
s1 ∈ Σn,d
ps1 − p2k ∈ Σn,d (2.2)
with d = max(2k deg p, d+deg p). For fixed k and d we know that, via Theorem 3, checking
the conditions in (2.2) is just an LMI, so for fixed k and d we have an LMI sufficient
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condition for a polynomial to be PSD.
The S-Procedure
What does the familiar S-procedure look like in the Positivstellensatz formalism?
Given symmetric n×n matrices Aimi=0, the S-procedure states: if there exist nonnegative
scalars λimi=1 such that A0 −
∑mi=1 λiAi 0, then
m⋂i=1
x ∈ Rn|x∗Aix ≥ 0 ⊂ x ∈ Rn|x∗A0x ≥ 0
Rephrased as a set emptiness question, we would like to know if
W := x ∈ Rn|x∗A1x ≥ 0, . . . , x∗Amx ≥ 0, −x∗A0x ≥ 0, x∗A0x 6= 0
is empty?
If the λi exist, define Q := A0 −∑m
i=1 λiAi. By assumption Q 0 and thus
x∗Qx ∈ Σn. Define g(x) := x∗A0x ∈M(x∗A0x) as well as
f(x) := (x∗Qx)(−x∗A0x) +m∑
i=1
λi(−x∗A0x)(x∗Aix)
By their non-negativity each λi ∈ Σn and because x∗Qx ∈ Σn we know that the function
f(x) is in the cone P (x∗A1x, . . . , x∗Amx,−x∗A0x). An easy rearrangement gives f+g2 = 0,
which illustrates that f and g are Positivstellensatz certificates that prove that W is empty.
A generalized S-Procedure
The S-procedure given above can be generalized to deal with non-quadratic func-
tions and non-scalar weights in the following way. Given pimi=0 ∈ Rn, if there exist
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simi=1 ∈ Σn such that
p0 −m∑
i=1
sipi = q
with q ∈ Σn, which for fixed degree si’s can be checked with Theorem 3, then
m⋂i=1
x ∈ Rn|pi(x) ≥ 0 ⊂ x ∈ Rn|p0(x) ≥ 0
The related set emptiness question asks if
W := x ∈ Rn|p1(x) ≥ 0, . . . , pm(x) ≥ 0,−p0(x) ≥ 0, p0(x) 6= 0
is empty. Similar to the standard S-procedure approach, define g := p0 ∈M(p0) as well as
f := −qp0 −n∑
i=1
sip0pi
Since q as well as the si’s are SOS, f ∈ P(p1, . . . , pm,−p0). Verifying f + g2 = 0,
f + g2 = −qp0 −n∑
i=1
sip0pi + p20
= −
(p0 −
m∑i=1
sipi
)p0 −
n∑i=1
sip0pi + p20
= 0
illustrating that f and g provide certificates that the set W is empty.
2.2.2 Theorems Related to the Positivstellensatz
The multidimensional moment problem, which considers when a sequence of num-
bers are the moments of some nonnegative Borel measure on Rn, has a long relation with
SOS polynomials, [1]. Many interesting results about SOS polynomials have been gener-
ated from this approach and two theorems that are especially interesting to polynomial
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24
optimization are presented below. The setup is as follows; let fimi=1 ∈ Rn and define
K := x|f1(x) ≥ 0, · · · , fm(x) ≥ 0.
Theorem 6 (Corollary 3, [30]) If K is compact and p ∈ Rn is such that p(x) > 0 for
all x ∈ K, then p ∈ P(f1, · · · , fm).
This tells us that if a polynomial is positive on K then it is in the cone generated by the poly-
nomials that describe K. With one additional assumption this result can be strengthened
further
Theorem 7 (Lemma 4.1, [25]) Let K be compact. p ∈ Rn such that p(x) > 0 for all
x ∈ K belongs to the set
s0 + f1s1 + · · ·+ fmsm|s0, · · · , sm ∈ Σn
if and only if there is a polynomial g in the set with the property that g−1[0,∞) is compact
in Rn.
These theorems can be used to define certificate searches with fewer terms than
the P-satz would require, however, these smaller searches can require polynomials of much
higher degree. In [32], Stengle provides a simple example that requires unbounded degree
polynomial certificates using Theorem 7, but has degree four certificates if the P-satz is
used, as shown in [6].
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Applications to Polynomial Optimization
Consider the following optimization problem
minx∈Rn
f0(x)
s.t. f1(x) ≥ 0
...
fm(x) ≥ 0 (2.3)
with fimi=0 ∈ Rn and no assumed convexity. Let the optimum value be f? > −∞ with
f0(x?) = f?.
Lasserre, [18], noticed that if the feasible region is compact we can always satisfy
the requirements of Theorem 7 by adding an additional constraint on norm of x, fm+1(x) :=
a − ‖x‖2 ≥ 0, since f−1m+1[0,∞) is clearly compact. Additionally, if γ is a lower bound on
f?, then f0(x) > γ at any feasible point, which implies that f0(x) − γ > 0 for x|f1(x) ≥
0, · · · , fm+1(x) ≥ 0. Using the theorem, we can rewrite the optimization (2.3) as
max γ
s.t. f0 − γ = s0 + f1s1 + · · ·+ fm+1sm+1
s0, · · ·, sm+1 ∈ Σn (2.4)
which Theorem 3 shows to be an LMI, as long as the degree of the SOS polynomials is
fixed, however, the degree for which the equality constraint in (2.4) holds is unknown.
By increasing the maximum degree of the si’s, this approach allows for a series of convex
relaxations to the nonconvex problem (2.3), which are shown to monotonically converge to
f? in [18]. A software package to carry out this algorithm is described in [12].
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2.3 Chapter Summary
In this chapter we provided the polynomial background for all of the results in
the following chapters. Most importantly, we illustrated the convex optimization approach
to checking if a given polynomial is a SOS polynomial. Additionally, we introduced the
Positivstellensatz and illustrated some of its applications, which we will expand on in the
following chapters.
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Chapter 3
Global Analysis and Controller
Synthesis
If we consider the system
x(t) = f(x(t)) (3.1)
for x(t) ∈ Rn with f ∈ Rnn as well as f(0) = 0, we can pose many global system theoretic
questions about its behavior as searches for SOS polynomials. However, first we need a few
definitions that will allow us to make the Lyapunov based stability arguments that will be
at the heart of the polynomial searches.
Define the flow of the system (3.1) starting from a point x0 ∈ Rn and evolving
forward for t time units to be φt(x0). Additionally for a differentiable scalar function V ,
defined on the same state space as (3.1), define its derivative with respect to time, V , as
the dot product between its gradient, ∇V , and f ,
V (x) := ∇V (x)∗f(x)
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3.1 Stability Background
Using a polynomial as the Lyapunov function, V , we will be able to prove global
asymptotic stability as well as semi-global exponential stability of the system (3.1) by
checking semialgebraic conditions on polynomials. However, we will first explicitly define
all of the terminology and prove the conditions that we will later exploit to design Lyapunov
functions.
3.1.1 Definitions
Definition 6 (Stability) The system (3.1) is stable about x = 0 if for every ε > 0 there
exists δε > 0 such that if ‖x0‖ < δε, then ‖φt(x0)‖ < ε for all t ≥ 0.
Definition 7 (Asymptotic Stability) The system (3.1) is asymptotically stable about
x = 0 if it is stable about x = 0 and, additionally, there exists h > 0 such that if ‖x0‖ < h,
then limt→∞
‖φt(x0)‖ = 0. Furthermore, if ∀x0 ∈ Rn, limt→∞
‖φt(x0)‖ = 0 then the system is
globally asymptotically stable.
Definition 8 (Exponential Stability) The system (3.1) is exponentially stable about x =
0 if there exist m, c, h > 0 such that
‖φt(x0)‖ ≤ me−ct‖x0‖
for all ‖x0‖ < h and t ≥ 0; c is also referred to as a convergence rate for the system.
Furthermore, if for every h > 0 there exist m and c that depend on h and validate
the inequality, then the system is semi-globally exponentially stable. If one fixed pair of m
and c validate the inequality for all h > 0, then the system is globally exponentially stable.
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Note that both semi-global and global exponential stability imply global asymptotic stabil-
ity. Additionally, we need to define the following classes of functions.
Definition 9 (K∞ Functions) A function σ : R → R is called a K∞ function if it is
continuous, strictly increasing, and has the properties σ(0) = 0 and σ(ξ) →∞ as ξ →∞.
Definition 10 (Positive Definite Functions) A function ρ : Rn → R is called positive
definite if it is continuous, has the property ρ(0) = 0, and there exists some K∞ function σ
such that
σ(‖x‖) ≤ ρ(x)
for all x ∈ Rn.
3.1.2 Stability Theorems
With the definitions above we can state and prove the standard Lyapunov theorem
for global asymptotic stability as well as an extension for semi-global exponential stability.
Theorem 8 (Lyapunov) The system (3.1) is globally asymptotically stable about its equi-
librium point if there exists a positive definite function V : Rn → R+ such that −V is also
positive definite.
Proof:
The proof below follows along the lines of the Lyapunov theorem proof in [16, §3.1].
By definition there exist K∞ functions α, β such that α(‖x‖) ≤ V (x),∀x ∈ Rn and
β(‖x‖) ≤ −V (x),∀x ∈ Rn. We will first prove stability followed by asymptotic convergence
to the fixed point.
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Given any ε > 0 pick δε such that
sup‖x‖<δε
V (x) < α(ε)
which, by the continuity of V , always exists. Pick any point x0 such that ‖x0‖ < δε. At
this point
α(‖x0‖) ≤ V (x0) < α(ε)
which implies that ‖x0‖ < ε. From our assumptions −V is positive definite which makes
V (φT (x0)) ≤ V (x0)
for all T ≥ 0. Bounding this inequality from both sides for ‖x0‖ < δε we have
α(‖φT (x0)‖) ≤ V (φT (x0)) ≤ V (x0) < α(ε)
for all T ≥ 0, thus ‖φT (x0)‖ < ε for all T ≥ 0, proving that the system is stable.
To show asymptotic convergence to the fixed point, x = 0, for any given x0 ∈ Rn
and ε > 0, we need to find a T > 0 such that ‖φt(x0)‖ < ε for all t ≥ T . If x0 = 0 then there
is nothing to prove, so we can assume that x0 6= 0. Assume that there exists an x0 6= 0 and
ε > 0 such that ‖φt(x0)‖ ≥ ε for all t > 0. By integration we know that
V (φt(x0)) = V (x0) +∫ t
0V (φτ (x0)) dτ
Using the positive definiteness of V as well as the fact that ‖φt(x0)‖ ≥ ε we can bound the
expression above from below,
α(ε) ≤ α(‖φt(x0)‖) ≤ V (φt(x0)) = V (x0) +∫ t
0V (φτ (x0)) dτ
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Additionally using the positive definiteness of −V and the fact that ‖φt(x0)‖ ≥ ε we can
bound V (φt(x0)) from above,
V (φt(x0)) = V (x0) +∫ t
0V (φτ (x0)) dτ ≤ V (x0)−
∫ t
0β(‖φτ (x0)‖) dτ ≤ V (x0)− tβ(ε)
End-to-end we now have
α(ε) ≤ V (x0)− tβ(ε)
for all t > 0. However, if t ≥ V (x0)/β(ε) this implies that α(ε) ≤ 0, which contradicts ε > 0.
2
Building from Theorem 8, we can now prove the following theorem for semi-global
exponential stability.
Theorem 9 If there exists a function V , such that V (x) ≥ α‖x‖dd ∀x ∈ Rn, where α > 0
and d is an integer greater than one, as well as a γ > 0 such that
V (x) ≤ −γV (x)
for all x ∈ Rn, then the system (3.1) is semi-globally exponentially stable about its fixed
point, x = 0, with a convergence rate of γ/d.
Proof:
By definition α‖x‖dd ≤ V (x), ∀x ∈ Rn. Clearly the function α(·)d is in class K∞, and
γα‖x‖dd ≤ γV (x) ≤ −V (x), for all x ∈ Rn, which shows that −V is positive definite and
therefore the system (3.1) is globally asymptotically stable about x = 0.
For x 6= 0, V (x) > 0 which allows us to write one of the assumptions as
V (x)V (x)
≤ −γ
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or
d
dtlog(V (x)) ≤ −γ
which can be integrated over [0, t] starting from x0 to give
log(V (φt(x0))) ≤ log(V (x0))− γt
which gives the exponential bound
V (φt(x0)) ≤ V (x0)e−γt
that proves that V (φt(x0)) decays exponentially with rate γ. Since α‖φt(x0)‖dd ≤ V (φt(x0))
for all t > 0, we can make the following bound
‖φt(x0)‖dd ≤
(V (x0)α‖x0‖d
d
)e−γt‖x0‖d
d
or
‖φt(x0)‖d ≤ me−ct‖x0‖d
with m =(
V (x0)
α‖x0‖dd
) 1d and c = γ/d. Since m depends on x0 and the inequality holds for
all x0 ∈ Rn the system is semi-globally exponentially stable. Interestingly the convergence
rate can be chosen to be γ/d for all x0.
2
Remark 1 If the system (3.1) has equilibrium points away from x = 0, then the system
can not be semi-globally exponentially stable nor can it be globally asymptotically stable.
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3.2 Convex Stability Tests
With the previous section’s Lyapunov background, we can now look to design-
ing Lyapunov functions for given dynamic systems. Our approach is to design Lyapunov
functions using the assumptions of Theorems 8 and 9 as constraints of an optimization
that searches over a class of candidate Lyapunov functions. This optimization approach is
extensively used and is usually designed so that the resulting optimization is convex. [5]
presents a survey of convex optimization results relating to analysis for linear systems with
quadratic Lyapunov functions. These ideas are extended for smooth nonlinear systems with
linearly parameterized non-quadratic Lyapunov functions in [15]. Additionally a convex op-
timization approach using the concept of a dual to the Lyapunov function is provided in
[27].
Our approach is to consider polynomial systems and restrict the set of candidate
Lyapunov functions to be polynomials. By doing so, we can formulate the following convex
stability tests.
Lemma 2 Given the system (3.1) and fixed positive definite functions l1, l2 ∈ Rn, the
system is globally asymptotically stable if there exists V ∈ Rn with V (0) = 0 such that
V − l1 ∈ Σn
−(∇V ∗f + l2) ∈ Σn.
Proof:
From Theorem 3, it is clear that the conditions that V − l1 and −(∇V ∗f + l2) are SOS
polynomials can be checked as LMIs. If a V is found that meets these conditions, the
positive definiteness of l1 and l2 insure that both V and −V are positive definite, which
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meets the assumptions of Theorem 8 making the system globally asymptotically stable.
2
Note that if the dynamics were chosen to be linear, f(x) = Ax and the Lyapunov function
to be quadratic V (x) = x∗Px then the lemma’s SOS conditions can be simplified to the
LMI conditions P 0 and A∗P + PA ≺ 0.
A set of sufficient conditions which are very similar to Lemma 2 appear in [20],
where they are used to prove non-asymptotic stability for systems like (3.1) with additional
state and control equality and inequality constraints.
Looking to Theorem 9, we can formulate a similar set of polynomial conditions,
however due to a term that is bilinear in γ and V , we can not check the assumptions with
a single LMI.
Lemma 3 Given the system (3.1) and the fixed positive definite function l(x) = ‖x‖dd with
d an integer greater than one, the system is semi-globally exponentially stable if there exists
γ > 0 and V ∈ Rn with V (0) = 0 such that
V − l ∈ Σn
−(γV +∇V ∗f) ∈ Σn
which when V has fixed degree can be checked by performing a linesearch on γ and solving
the resulting LMI at each point. Additionally, γ/d is a rate of convergence for the system.
Proof:
The proof follows along the same lines as the proof for Lemma 2 by establishing that the
assumptions for Theorem 9 are met.
In general the value for d in this and other related lemmas will be picked to be
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the highest even degree in V .
2
Again, if the dynamics are linear and the Lyapunov function is quadratic the SOS
conditions of Lemma 3 collapse into simple LMIs, which are remain bilinear in γ, P 0
and A∗P + PA −γP .
3.2.1 Stability Examples
Consider the following systemx1
x2
=
−x2 − x31
x1 − x32
︸ ︷︷ ︸
f(x)
(3.2)
where it is clear that f(0) = 0. The linearization about the origin has eigenvalues of ±j, so
it is not even possible to verify that the nonlinear system (3.2) is stable.
If we look to construct a Lyapunov function to demonstrate stability, a simple
quadratic Lyapunov function V (x) = ‖x‖22 will work. This V is clearly positive definite,
and if we compute its time derivative we find
V = ∇V ∗f
= 2x1(−x2 − x31) + 2x2(x1 − x3
2)
= −2(x41 + x4
2)
which clearly makes −V positive definite. The definiteness of V and −V satisfy the assump-
tions of Theorem 8, so it is clear that the system (3.2) is globally asymptotically stable.
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d γmax γ/d
2 0 04 .0052 .00136 .0431 .00728 .1094 .013710 .1472 .0147
Table 3.1: Results of applying Lemma 3 to system (3.2)
However, it is unclear if the system is semi-globally exponentially stable, so we will use
Lemma 3 to set up a linesearch on a sum of squares optimization problem to construct a
Lyapunov function to show that it is.
If we follow the approach given in Lemma 3 for d = 2, 4, 6, 8, 10 we can construct
the table of maximal γ values given in Table 3.1. The table shows that for d = 4 we can
demonstrate semi-global exponential stability, since γmax > 0, and that the state decays
with rate γ/d = .0013. As the degree of the Lyapunov function is increased, the maximum
feasible value for γ increases, with γ/d increasing as well. For d > 10, the numerical errors
in solving the resulting LMIs cause the linesearch to become erratic and return lower values
for γmax.
3.3 Disturbance Analysis
In the previous section, we considered two algorithmic approaches to construct
Lyapunov functions that demonstrate specific types of stability. Now, we will consider the
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effect of an external disturbance, w(t) on the following system
x = f(x) + gw(x)w
y = h(x)(3.3)
with x(t) ∈ Rn, w(t) ∈ Rnw , y(t) ∈ Rp, f ∈ Rnn, f(0) = 0, gw ∈ Rn×nw
n , and h ∈ Rpn with
h(0) = 0. We will follow the Lyapunov approach used in [5, §6.3.2] and presented in the
following lemma to find the induced L2 → L2 gain from w(t) to y(t).
Lemma 4 For a system whose dynamics are given by (3.3) with initial condition x0 = 0,
if there exists a positive definite function V : Rn → R+ such that
V (x,w) + h(x)∗h(x)− αw∗w ≤ 0 (3.4)
for all x ∈ Rn and w ∈ Rnw , then the induced L2 → L2 gain from w(t) to y(t) is less than
or equal to√
α.
Proof:
If we integrate the inequality (3.4) from 0 to T ≥ 0 we have
V (φT (x0))− V (x0) +∫ T
0
(h(x(t))∗h(x(t))− αw(t)∗w(t)
)dt ≤ 0
and since V (x0) = 0 and V (φT (x0)) ≥ 0 we get
‖y(t)‖2‖w(t)‖2
≤√
α
which completes the proof.
2
Note that if we assume w(t) := 0 then the inequality (3.4) shows that −V is positive
semi-definite and thus that the system is stable.
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We can now use the SOS relaxations from the stability lemmas to pose the following
lemma that provides our best estimate of the induced gain from w(t) to y(t).
Lemma 5 The best estimate of the induced L2 → L2 gain from w(t) to y(t) for the system
(3.3) is√
α where α comes from the solution of the following optimization. Fix the degree
of the Lyapunov function to be dV . Let l ∈ Rn be a fixed positive definite polynomial and
V ∈ Rn,dVwith V (0) = 0
minV
α
s.t.
V − l ∈ Σn
−(∇V (x)∗(f(x) + gw(x)w) + h(x)∗h(x)− αw∗w
)∈ Σn+nw
Proof:
The proof follows from the proof of Lemma 4.
2
Using this lemma we can apply SOS programming to quantify a system’s ability
to reject disturbances. However, this approach can run into feasibility problems, since
it requires that −V (x,w) have terms to counteract the −h(x)∗h(x) in the second SOS
constraint. Examples of the use of this lemma are provided to analyze the disturbance
rejection capabilities of the controllers that are designed as examples in §3.4.3 and §3.5.2.
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3.4 State Feedback Controller Design
We will now move from analysis to controller synthesis using Lyapunov techniques.
The Lyapunov based approach to controller synthesis has produced many notable results
and design procedures, and much of their history is given in [17]. Most of these results
center on designing a controller from a provided control Lyapunov function, while in our
approach we use optimization to find both the controller and the Lyapunov function. A
related approach using convex optimization to design a controller with a dual to the standard
Lyapunov theorem is shown in [26].
As shown in section 3.2, sufficient conditions for the assumptions of Theorems 8
and 9 can be checked as LMIs or linesearches on LMIs. The existence of efficient tests for
stability analysis brings us to apply similar techniques for controller synthesis. Consider
now the system
x = f(x) + g(x)u (3.5)
for x ∈ Rn with f ∈ Rnn, f(0) = 0 and u ∈ Rm with g ∈ Rn×m
n . If we allow u to be generated
by a state feedback controller K ∈ Rmn with K(0) = 0, we get the following closed loop
system
x = f(x) + g(x)K(x) (3.6)
where K is still unknown.
Now we can look for conditions on K such that we can find a Lyapunov equation
that meets the assumptions for Theorems 8 and 9. The analog for Lemmas 2 and 3 for the
system (3.6) are as follows.
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Lemma 6 Given fixed positive definite functions l1, l2 ∈ Rn, if there exists V ∈ Rn with
V (0) = 0 and K ∈ Rmn with K(0) = 0 such that
V − l1 ∈ Σn
−(∇V ∗(f + gK) + l2) ∈ Σn
then the system (3.5) is globally asymptotically stabilized by the control law u = K(x).
Lemma 7 Given the fixed positive definite function l(x) = ‖x‖dd with d an integer greater
than one, if there exists γ > 0, K ∈ Rmn with K(0) = 0 and V ∈ Rn with V (0) = 0 such
that
V − l ∈ Σn
−(γV +∇V ∗(f + gK)) ∈ Σn
then the system (3.5) is semi-globally exponentially stabilized by the control law u = K(x).
Additionally, γ/d is a rate of convergence for the closed loop system.
The proofs for these lemmas follow exactly along the lines of the proofs of Lemmas
2 and 3. However, since both lemmas have conditions that are bilinear in the monomials of
V and K, the SOS conditions of Lemma 6 nor those of 7 will have to be checked iteratively.
3.4.1 Iterative State Feedback Design Algorithms
Since, in general, Lemmas 6 and 7 are not amenable to the semidefinite program-
ming based approach of Theorem 3, we will need to employ an iterative approach that
solves the lemmas’ SOS conditions in V and K by holding one of these polynomials fixed
while adjusting the other. Even though each step in the iteration will be convex, the overall
problem will remain non-convex. In some cases iterative design procedures of this type can
be shown to converge to answers away from the global optimum as the example in [8] shows.
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However, there is one special case when the SOS conditions in Lemmas 6 and 7
can be checked directly with semidefinite programming. This case is when the dynamics
and controller are linear and the Lyapunov function is quadratic. In this case we can use a
nonlinear change of coordinates from [2] which is often referred to as the “feedback trick,”
to yield LMI conditions for Lemma 6 and a linesearch on LMI conditions for Lemma 7.
Consider the general case for the SOS conditions of Lemma 6; the second SOS
condition is bilinear in V and K as noted above, which indicates that if either is fixed it
is an LMI feasibility problem to find the other. The problem with this approach is that if
V is fixed and the resulting convex search for K fails, then there is no way to redesign V
from the search on K. The same pitfall occurs if K is held fixed and the search is for V .
This approach gives a single shot at finding a controller for a given Lyapunov function or
Lyapunov function for a fixed controller, but it can not be extended to search for both.
The conditions for Lemma 7 have better feasibility properties that allow us to
propose a pair of iterative design procedures to establish semi-global exponential stability.
The procedures either require either a candidate controller or Lyapunov function to began
their iterations. First we will consider an algorithm that starts the iterative search from
a candidate controller polynomial, the K variant, and then we will look at starting the
iterative search from the candidate Lyapunov function, the V variant. Since we can not
guarantee a feasible starting point for either algorithm, the K and the V variants can be
considered two different algorithms.
Algorithm 1 (State Feedback: Candidate K Variant) An iterative search to satisfy
the SOS conditions of Lemma 7 starting from a candidate controller.
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Let i be the iteration index and set i = 1. Denote the candidate controller K(i=0),
and pick the maximum degree of the controller and Lyapunov polynomials, dK and dV
respectively.
1. Fix the controller polynomial K = K(i−1), set l(x) = ‖x‖dVdV
and solve the following
linesearch on γ where V ∈ Rn,dVwith V (0) = 0
maxV
γ
s.t.
V − l ∈ Σn
−(γV +∇V ∗(f + gK)) ∈ Σn
(3.7)
Set V (i) = V . If γmax > 0, then the system (3.6) is semi-globally exponentially stable
with controller K(i−1), else if −∞ < γmax ≤ 0 goto step 2. If γmax = −∞, the
iteration is infeasible from the candidate controller K(i−1), and no stability properties
of the system (3.6) can be inferred.
2. Fix the Lyapunov function V = V (i) and solve the semidefinite programming problem
where K ∈ Rmn,dK
with K(0) = 0
maxK
γ
s.t.
−(γV +∇V ∗(f + gK)) ∈ Σn
(3.8)
Set K(i) = K. If γmax > 0, then the system (3.6) is semi-globally exponentially stable
with controller K(i). If γmax ≤ 0, increment i and loop back to step 1.
If we desire to start from a candidate Lyapunov function instead of a candidate
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controller we can follow the V variant algorithm, which is a trivial reordering of the steps
above, but as noted in the remark after the algorithm, it is subtly different.
Algorithm 2 (State Feedback: Candidate V Variant) An iterative search to satisfy
the SOS conditions of Lemma 7 starting from a positive definite candidate Lyapunov func-
tion.
Let i be the iteration index and set i = 1. Denote the candidate Lyapunov function
V (i=0), and pick the maximum degree of the controller and Lyapunov polynomials, dK and
dV respectively.
1. Fix the Lyapunov function V = V (i−1) and solve the semidefinite programming prob-
lem where K ∈ Rmn,dK
with K(0) = 0
maxK
γ
s.t.
−(γV +∇V ∗(f + gK)) ∈ Σn
(3.9)
Set K(i) = K. If γmax > 0, then the system (3.6) is semi-globally exponentially stable
with controller K(i), if −∞ < γmax ≤ 0 goto to step 2. If γmax = −∞, the iteration
is infeasible starting from the candidate Lyapunov function V (i−1), and no stability
properties of the system (3.6) can be inferred.
2. Fix the controller polynomial K = K(i) and set l(x) = ‖x‖dVdV
. Solve the following
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44
linesearch on γ where V ∈ Rn,dVwith V (0) = 0
maxV
γ
s.t.
V − l ∈ Σn
−(γV +∇V ∗(f + gK)) ∈ Σn
(3.10)
Set V (i) = V . If γmax > 0, then the system (3.6) is semi-globally exponentially stable
with controller K(i), else if γmax ≤ 0, increment i loop back to step 1.
Remark 2 (Properties of the State Feedback Algorithms) :
• If −∞ < γmax in step 1 of either algorithm, then the rest of the iteration’s searches
will be feasible, however, this does not mean that a γmax > 0 will necessarily be found.
• The degree of the Lyapunov function, dV should always be picked to be even, since it
needs to be positive definite.
• Since deg f ≥ 1, deg(∇V ∗(f + gK)) ≥ deg V . This implies that the value of dK needs
to be picked so that the degree of ∇V ∗(f +gK) is even to insure that −(γV +∇V ∗(f +
gK)) is SOS.
• In general the V variant algorithm tends to be feasible when the K variant is not.
A theoretical rational for this heuristic is as follows. Given a candidate controller,
the “likelihood” that it semi-globally exponentially stabilizes the nonlinear system is
low, so unless it does it is impossible to find a Lyapunov function. On the other
hand, given a positive definite candidate Lyapunov function, if the system can be
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45
semi-globally exponentially stabilized, then finding a controller that does so and works
with the candidate Lyapunov function is more likely.
The polynomial controller design step of the V variant algorithm can also be
viewed as treating the candidate Lyapunov function as a form of control Lyapunov function
(CLF), see [17] for background. However, our controller design approach contrasts with
the standard CLF back-stepping procedure by using convex optimization to search for any
polynomial controller of fixed degree instead of matching the controller to counteract the
system’s dynamics.
3.4.2 State Feedback Design Example: Non-Holonomic System
Consider the bilinear system from example 2 in [34],x1
x2
=
3x1 + 4x2
−20x1 + 10x2
u
Clearly this system fits the format of (3.5) with f = 0, so we can use the algorithms
designed in the previous section to design a semi-globally exponentially stabilizing state
feedback controller.
Since the system has f = 0, we can not find our candidate V and K by solving
the control design problem on the system’s linearization. We will instead choose to design
a controller by starting Algorithm 2 with
V (x) = x21 + x2
2
and setting dV = dK = 2. From this candidate Lyapunov function, we find a controller and
Lyapunov function pair that achieves γmax = 0.0078 after 2 iterations. The controller that
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46
the algorithm designs has both quadratic and linear terms, but the linear terms are both
smaller in magnitude than 10−6. If we zero out these tiny terms we find that we still meet
the required SOS conditions. and we are left with the reduced controller
K(x) =1
100
(− 0.426x2
1 + 2.275x1x2 − 1.404x22
)With this controller we can not use the SOS conditions in Lemma 5 to find the
induced L2 gain from disturbances to the system’s states, h(x) = x, since the closed loop
system is a homogeneous cubic polynomial. This makes V a homogeneous quartic poly-
nomial, which lacks the necessary quadratic terms to counteract the −h(x)∗h(x) = −x∗x
term, and thus makes the SOS conditions always infeasible.
3.4.3 State Feedback Design Example: Nonlinear Spring-Mass System
We will design a state feedback controller following Algorithms 1 and 2 for the
spring-mass system given below where x1, x3 represent the displacement of m1,m2 respec-
tively, the k’s identify the springs, d is a anti-damper, and u is a forcing input.
k1
AAA
AAA
m1
- x1
- u k2
AAA
AAA
d
m2
- x3
For simplicity we will consider only unit masses, m1 = m2 = 1. Let k1 be a
stiffening spring that generates the displacement dependent force
Fk1(x) = x1 +110
x31
and let the forces generated by k2 and d be linear in the relative displacement and velocity
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47
respectively with d chosen to provide negative damping to make the system unstable
Fk2 = x3 − x1
Fd = − 110
(x4 − x2).
We can now write the system’s dynamics as
x1
x2
x3
x4
=
x2
−(x1 + 110x3
1) + (x3 − x1)− 110(x4 − x2)
x4
−(x3 − x1) + 110(x4 − x2)
︸ ︷︷ ︸
f(x)
+
0
1
0
0
︸︷︷︸g(x)
u (3.11)
The linearization of the system (3.11) has eigenvalues with positive real part, so the uncon-
trolled system is unstable.
We want to design a state feedback controller, u = K(x), to semi-globally expo-
nentially stabilize the system
x = f(x) + g(x)u
where f , g are defined in (3.11). However, we need to start with either a candidate controller
or Lyapunov function, and we will find both by solving the linearized version of the problem
to find a linear controller and a quadratic Lyapunov function.
Letting Af and Bg be the linearizations of f and g about the point x = 0 the
linearized system is
x = Afx + Bgu (3.12)
It is possible to use the “feedback trick” from [2] to pose the problem of finding a linear
static state feedback controller, u = Klin(x) = Kx, for system (3.12) with a quadratic
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48
Lyapunov function that demonstrates semi-global exponential stability, Vquad(x) = x∗Px,
as the following linesearch with LMI constraints
max γ
s.t.
Q 0
−γQ QA∗f + AfQ + L∗B∗g + BgL
(3.13)
where P = Q−1 and K = LP .
Using Klin as found by (3.13) as the candidate controller to start Algorithm 1 with
dV = 2 or dV = 4 makes the SOS conditions (3.7) infeasible, thus γmax = −∞. This implies
that the K variant of the algorithm fails to tell us anything about the possibility of semi-
globally exponentially stabilizing the unstable system (3.11). However if we start Algorithm
2 with Vquad as the candidate Lyapunov function and fix dK = 3, then a controller is found
in the first optimization (3.9) that makes γmax = 1.2311. However, if we start Algorithm
2 with dK = 1, it fails which shows that we need to go to dK = 3 to achieve semi-global
exponential stability.
The success of the V variant while the K variant fails is not that strange when we
consider what the SOS optimizations were trying to find. In the K case, we were looking for
a Lyapunov function that demonstrates that the nonlinear system with the linear controller
wrapped around it is semi-global exponentially stable. Whereas, in the V case we were
using the quadratic V in the manner of a control Lyapunov function and designing a third
order controller to make the necessary SOS condition hold.
The controller found in the V case is a degree three polynomial in 4 variables, so it
has 34 terms. Since no part of the optimization tries to reduce the number of nonzero terms,
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49
all of them are used. If all the terms whose coefficients are less than 10−6 in absolute value
are set to zero the number of used terms drops to 24 and the reduced controller continues
to make the system globally exponentially stable with the same value of γmax.
Performance of the State Feedback Controller
Now that we have designed a polynomial state feedback controller for the example
system, we can analyze its performance by using Lemma 5 to compute a bound on the
induced L2 gain from disturbances to the states by selecting h(x) = x. If we allow a
disturbance to enter the system through the control channel, gw = g, and keep dV = 2,
then following Lemma 5 we get the following performance bound
‖x(t)‖2‖w(t)‖2
≤ 0.686
which shows that the system is non-expansive.
3.5 Output Feedback Controller Design
We can expand the results for state feedback controllers by allowing the controller
to be a dynamic system and by limiting the information that it receives. In many cases, the
existing output feedback controller design schemes (ie. back stepping [17]) formulate the
problem by making the Lyapunov function depend only on the system’s outputs, however,
we will continue to require that the Lyapunov function depend on all the system’s and the
controller’s states.
Again, we will be searching for ways to validate the assumptions of Theorem 9
using an iterative procedure to check SOS conditions. As in the state feedback case, we will
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50
not consider the globally asymptotic stability case, since its set up will again suffer from
the infeasiblity properties that were illustrated in the previous section.
Define the system to be controlled as
x = f(x) + g(x)u
y = h(x)(3.14)
for x ∈ Rn with f ∈ Rnn, f(0) = 0, u ∈ Rm, g ∈ Rn×m
n with y ∈ Rp, h ∈ Rpn and h(0) = 0.
If we allow u to be generated by an unknown nξ-state dynamic output feedback controller
of the form
ξ = A(ξ) + B(ξ)y
u = C(ξ) + D(ξ)y(3.15)
for ξ ∈ Rnξ with A ∈ Rnξnξ , A(0) = 0, B ∈ Rnξ×p
nξ , C ∈ Rmnξ
, C(0) = 0 and D ∈ Rm×pnξ . With
this controller structure, the closed loop system becomes
x = f(x) + g(x)(C(ξ) + D(ξ)h(x))
ξ = A(ξ) + B(ξ)h(x).(3.16)
By the assumptions on f, h, A,C, we know that the combined system has a fixed point at
[x; ξ] = [0; 0], and we can pose the following analog of Lemma 3 to test the closed loop
system (3.16) for semi-global exponential stability.
Lemma 8 Let nξ be a fixed positive integer, and l([x; ξ]) = ‖[x; ξ]‖dd with d some integer
greater than one. If there exists γ > 0, A ∈ Rnξnξ , A(0) = 0, B ∈ Rnξ×p
nξ , C ∈ Rmnξ
, C(0) = 0,
D ∈ Rm×pnξ and V ∈ Rn+nξ
with V (0) = 0 such that
V − l ∈ Σn+nξ
−
γV +∇V ∗
f + gC + gDh
A + Bh
∈ Σn+nξ
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51
then the system (3.14) is semi-globally exponentially stabilized by the controller (3.15). Ad-
ditionally, γ/d is a rate of convergence for the closed loop system.
As before this lemma can be proved along the lines of Lemma 2 and again it is
bilinear in the elements of the controller system and the Lyapunov function. However, unlike
the earlier lemmas, when the systems are linear and the Lyapunov function is quadratic
we can not use the “feedback trick” to check the conditions in Lemma 8 with a single
semidefinite program, although, by the separation theorem, we could design separately an
observer and a state feedback controller.
3.5.1 Iterative Output Feedback Design Algorithms
In their stated forms, the SOS conditions of Lemma 8 do not fit into the set up for
Theorem 3 since they are bilinear in the controller system and the Lyapunov function. To
get around this problem we will propose two iterative approaches that are rather similar to
Algorithms 1 and 2 that also start from the candidate controller, the A,B, C, D variant, or
the Lyapunov function, the V variant. As in the state feedback case, since neither algorithm
can be guaranteed to be feasible the V and the A,B, C, D variants can be considered two
different algorithms.
Algorithm 3 (Output Feedback: Candidate A,B, C, D Variant) An iterative search
to satisfy the SOS conditions of Lemma 8 starting from a candidate controller.
Let i be the iteration index and set i = 1. Denote the candidate controller A(i=0),
B(i=0), C(i=0), D(i=0), and pick the maximum degree of the controller and Lyapunov poly-
nomials, dA, dB, dC , dD and dV respectively.
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52
1. Fix the controller polynomials A = A(i−1), B = B(i−1), C = C(i−1), D = D(i−1), and
set l([x; ξ]) = ‖[x; ξ]‖dVdV
. Solve the following linesearch on γ where V ∈ Rn+nξ,dVwith
V (0) = 0
maxV
γ
s.t.
V − l ∈ Σn+nξ
−
γV +∇V ∗
f + gC + gDh
A + Bh
∈ Σn+nξ
(3.17)
Set V (i) = V . If γmax > 0, then the system (3.14) is semi-globally exponentially
stable with controller A(i−1), B(i−1), C(i−1), D(i−1), else if −∞ < γmax ≤ 0 goto step
2. If γmax = −∞, the iteration is infeasible starting from the candidate controller
A(i−1), B(i−1), C(i−1), D(i−1), and no stability properties of the system (3.14) can be
inferred.
2. Fix the Lyapunov function V = V (i), and solve the semidefinite programming problem
where A ∈ Rnξ
nξ,dAwith A(0) = 0, B ∈ Rnξ×p
nξ,dB, C ∈ Rm
nξ,dC, C(0) = 0, and D ∈ Rm×p
nξ,dD
maxA,B,C,D
γ
s.t.
−
γV +∇V ∗
f + gC + gDh
A + Bh
∈ Σn+nξ
(3.18)
Set A(i) = A,B(i) = B,C(i) = C,D(i) = D. If γmax > 0, then the system (3.14)
is semi-globally exponentially stable with controller A(i), B(i), C(i), D(i), if γmax ≤ 0,
increment i loop back to step 1.
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53
As with the state feedback case, if we want to start from a candidate Lyapunov
function instead of a candidate controller we can reorder the steps above to follow the V
variant of the algorithm.
Algorithm 4 (State Feedback: Candidate V Variant) An iterative search to satisfy
the SOS conditions of Lemma 8 starting from a positive definite candidate Lyapunov func-
tion.
Let i be the iteration index and set i = 1. Denote the candidate Lyapunov function
V (i=0), and pick the maximum degree of the controller and Lyapunov polynomials, dA, dB,
dC , dD and dV respectively.
1. Fix the Lyapunov function V = V (i−1), solve the semidefinite programming problem
where A ∈ Rnξ
nξ,dAwith A(0) = 0, B ∈ Rnξ×p
nξ,dB, C ∈ Rm
nξ,dC, C(0) = 0, and D ∈ Rm×p
nξ,dD
maxA,B,C,D
γ
s.t.
−
γV +∇V ∗
f + gC + gDh
A + Bh
∈ Σn+nξ
(3.19)
Set A(i) = A,B(i) = B,C(i) = C,D(i) = D. If γmax > 0, then the system (3.14) is
semi-globally exponentially stable with controller A(i), B(i), C(i), D(i), if −∞ < γmax ≤
0 goto to step 2. If γmax = −∞, the iteration is infeasible starting from the candidate
Lyapunov function V , and no stability properties of the system (3.14) can be inferred.
2. Fix the controller polynomials A = A(i), B = B(i), C = C(i), D = D(i) and set
l([x; ξ]) = ‖[x; ξ]‖dVdV
. Solve the following linesearch on γ where V ∈ Rn+nξ,dVwith
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V (0) = 0
maxV
γ
s.t.
V − l ∈ Σn+nξ
−
γV +∇V ∗
f + gC + gDh
A + Bh
∈ Σn+nξ
(3.20)
Set V (i) = V . If γmax > 0, then the system (3.14) is semi-globally exponentially stable
with controller A(i), B(i), C(i), D(i), else if γmax ≤ 0, increment i and loop back to step
1.
Remark 3 (Properties of the Output Feedback Algorithms) :
• If −∞ < γmax in step 1 of either algorithm, then the rest of the iteration’s searches
will be feasible, however, this does not mean that a γmax > 0 will necessarily be found.
• The degree of the Lyapunov function, dV should always be picked to be even, since it
needs to be positive definite.
• Since deg f ≥ 1,
deg
∇V ∗
f + gC + gDh
A + Bh
≥ deg V
This implies that the values of dA, dB, dC , dD need to be chosen so that the degree of
∇V ∗[f + gC + gDh;A + Bh] is even. If it were odd, then the SOS conditions required
for stability could never be satisfied.
• In the output feedback case we can not employ the “feedback trick” so we can not
start the algorithm from its linear analog. Due to this, we must find other candidate
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55
controller and Lyapunov polynomials, and the selection of these greatly influences the
feasibility of the two algorithm variants. If a good controller starting point can found,
then Algorithm 3 will tend to be feasible more often than Algorithm 4. Conversely, if
a good candidate Lyapunov function can be found, then the V variant will tend to be
feasible more often.
3.5.2 Output Feedback Design Example
Consider again the spring mass system used in §3.4.3, whose dynamics are given
by (3.11). If we make the system’s output
y = h(x) =
x1
x2
the problem fits the form of (3.14), so we can try to design an output feedback controller
for the system using Algorithms 3 and 4.
First, we need to find a candidate controller and a candidate Lyapunov function
so that we can compare the A,B, C, D and the V variants. Unlike the state feedback case,
we can not just linearize the problem and solve the resulting LMIs, so we will first construct
a robust linear controller and its Lyapunov function to start the algorithms.
We choose to start the algorithms with a robust linear controller in the hope
that its robustness will compensate for the system’s nonlinearities which can be viewed as
uncertainties. In this light, we will maximize our robustness against plant uncertainty by
picking the candidate controller (A,B, C, D) to minimize the H∞ gain from d to e for the
block diagram shown in Figure 3.1, which will give optimal robustness as measured by the
gap metric [10].
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56
d1 - c -6
e1
Af
Ch 0
Bg- e2
?c d2A
C
B
D
6
Figure 3.1: The Gap Metric Block Diagram
If nξ = n, we can use standard software to find the H∞ controller (A,B, C, D)
that minimizes the gain from d to e in the block diagram above. If this controller achieves
a gain from d to e of β then if we denote the closed loop system Ac, Bc, Cc, Dc the positive
definite matrix X that makesA∗cX + XAc XBc C∗
c
B∗c X −βI D∗
c
Cc Dc −βI
≺ 0 (3.21)
provides a quadratic Lyapunov function [x; ξ]∗X[x; ξ] to use as a candidate Lyapunov func-
tion. If nξ 6= n, then we can devise a candidate controller and Lyapunov function as follows.
First find a random nξ state controller that stabilizes the linearized system (Af , Bg, Ch, 0).
Since the closed loop system (Ac, Bc, Cc, Dc) is linear in the controller (A,B, C, D), we can
start with the random stabilizing controller and iterate between adjusting the controller
and the Lyapunov function to minimize β subject to X 0 and the LMI constraint (3.21)
With the candidate controller and Lyapunov function found by the above method
we can use Algorithms 3 and 4. If we first search for a full order controller, nξ = 4, we
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57
can find a linear controller, dA = dC = 1 and dB = dD = 0, that semi-globally exponen-
tially stabilizes the nonlinear system using both the V and the A,B, C, D variants of the
algorithm, as shown by Table 3.2.
A,B, C, D variant V variantV (1), A(0), B(0), C(0), D(0) γ = −8.88 V (0), A(1), B(1), C(1), D(1) γ = −28.63V (1), A(1), B(1), C(1), D(1) γ = −6.41 V (1), A(1), B(1), C(1), D(1) γ = −12.46V (2), A(1), B(1), C(1), D(1) γ = −0.09 V (1), A(2), B(2), C(2), D(2) γ = −3.08V (2), A(2), B(2), C(2), D(2) γ = +0.12 V (2), A(2), B(2), C(2), D(2) γ = −0.09
– V (2), A(3), B(3), C(3), D(3) γ = +0.06
Table 3.2: The results of Algorithms 3&4 on (3.11) with h = [x1;x2] and nξ = 4.
If we reduce the number of controller states so that nξ = 2, we can still find a linear
controller that semi-globally exponentially stabilizes the system. However, only Algorithm
3, the A,B, C, D variant, works while the V variant is infeasible. The final two state linear
controller and closed loop Lyapunov function are below
A B
C D
=
1.37 −225.10 −240.10 −13.94
190.19 −2040.73 −2119.78 −122.79
715.60 3009.76 −3654.87 −3778.65
and
X =
2.67 −0.17 −0.69 0.21 0.65 −0.59
−0.17 1.39 −0.34 0.39 1.45 2.56
−0.69 −0.34 1.17 −0.23 0.07 −0.48
0.21 0.39 −0.23 1.35 −0.09 0.56
0.65 1.45 0.07 −0.09 93.84 88.98
−0.59 2.56 −0.48 0.56 88.98 87.15
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while the progress of the iteration is shown in Table 3.3
A,B, C, D variant nξ = 2V (1), A(0), B(0), C(0), D(0) γ = −9.57V (1), A(1), B(1), C(1), D(1) γ = −7.62V (2), A(1), B(1), C(1), D(1) γ = −0.10V (2), A(2), B(2), C(2), D(2) γ = −0.10V (3), A(2), B(2), C(2), D(2) γ = −0.09V (3), A(3), B(3), C(3), D(3) γ = +1.45
Table 3.3: Progress of Algorithm 3 on (3.11) with h = [x1;x2] and nξ = 2.
However, if we reduce the number of controller states again by setting nξ = 1, we
can not find a linear controller to semi-globally exponentially stabilize the system.
Also, if we set h = [x3;x4], which separates the control and the observations,
both of the algorithms turn out to be infeasible for the full order controller case when the
controller is allowed to be linear or cubic with a quadratic or quartic Lyapunov function.
Performance of the Output Feedback Controllers
As with the state feedback design example, we can now look at the performance
of the controllers derived above, by bounding the system’s induced L2 gain. We will again
allow the disturbance to enter in the control channel making gw = g. However, we found
that, in these examples, using the Lyapunov function that the controller design algorithms
returned found the smallest values for√
α, the bound on the induced L2 gain from w
to y. The results are presented in Table 3.4, which shows that the performance of the
controller designed by the V variant algorithm is vastly worse in this metric than either of
the A,B, C, D variant controllers. It is interesting to note that the achieved values for γmax
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seem to have little or no relation to the performance bound. However, as expected, the full
order controller exhibits better performance than the reduced order controller.
Algorithm nξ√
α
4 4 45.6063 4 0.0433 2 0.111
Table 3.4: Performance of the controllers designed in the example.
3.6 Chapter Summary
In this chapter we investigated convex optimization algorithms to construct Lya-
punov functions that demonstrate either global asymptotic or semi-global exponential stabil-
ity. As a measure of a system’s performance, we provided a convex optimization procedure
to bound the induced L2 gain from disturbances to output signals.
We then applied the semi-global exponential stability results to derive iterative al-
gorithms to design both state and output feedback controllers. These synthesis approaches
were tested on examples and the resulting controllers’ performance were bounded by com-
puting a bound on the system’s induced L2 gain from a disturbance in the control channel
to the system’s output.
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Chapter 4
Local Stability and Controller
Synthesis
If we again consider the system (3.1)
x = f(x)
for x ∈ Rn with f ∈ Rnn as well as f(0) = 0, we can pose local system theoretic questions
as searches for SOS polynomials in addition to the global ones considered in the previous
chapter. Again, we will first make Lyapunov stability arguments and adapt them into SOS
programming problems. Additionally, we will continue to use the shorthand V := ∇V ∗f .
4.1 Local Stability Background
Using the stability definitions from §3.1.1 we can state the basic local stability
Lyapunov function result based on [16, Theorem 3.1].
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Theorem 10 Let D ⊂ Rn be a domain containing the equilibrium point x = 0 of the system
(3.1). Let V : D → R be a continuously differentiable function such that
V (0) = 0
V (x) > 0 on D \ 0
and
V := ∇V ∗f < 0 on D \ 0
then the system (3.1) is asymptotically stable about x = 0. Moreover, any region Ωβ :=
x ∈ Rn|V (x) ≤ β such that Ωβ ⊆ D describes an positively invariant region contained in
the equilibrium point’s domain of attraction.
Proof:
Given ε > 0, choose r ∈ (0, ε) such that
Br := x ∈ Rn|‖x‖ ≤ r ⊂ D
Let a = min‖x‖=r V (x). Then, since V (x) > 0 for x 6= 0, a > 0. Take b ∈ (0, a) and let
Ωb := x ∈ Br|V (x) ≤ b. Since 0 < b < a, the containment Ωb ⊂ Br holds. Since Ωb ⊂ D,
we know that V (x) ≤ 0 on all of Ωb. Thus, for x0 ∈ Ωb
V (φT (x0)) ≤ V (x0) ≤ b
for all T ≥ 0. Since V is continuous and V (0) = 0 there must exist δε such that
Bδε := x ∈ Rn|‖x‖ ≤ δε ⊂ Ωb ⊂ Br
End to end, we now have,
x0 ∈ Bδε ⇒ x0 ∈ Ωb ⇒ φT (x0) ∈ Ωb ⇒ φT (x0) ∈ Br
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thus ‖x0‖ < δε implies ‖φT (x0)‖ < ε for all T ≥ 0, proving stability.
To prove asymptotic stability, we need to show that every initial condition x0 ∈ Br
converges to the origin, which is that for every c > 0 there exists a Tc > 0 such that
‖φt(x0)‖ < c for all t > Tc. From earlier we know that for every c > 0 there exists d > 0
such that Ωd ⊂ Bc, which makes it sufficient to show that V (φt(x0)) → 0 as t →∞. Since
V (φt(x0)) is monotonically decreasing and bounded below by zero,
V (φt(x0)) → v ≥ 0
as t →∞. To show that v = 0, suppose that v > 0. By continuity of V there exists w > 0
such that Bw ⊂ Ωv. The limit V (φt(x0)) → v > 0 implies that the state trajectory must
remain outside of Bw for all time. Let −α = maxw≤‖x‖≤r V (x), which exists because V is
continuous over this compact set. Clearly −α < 0. Integrating V from any x0 ∈ Br we
have
V (φt(x0)) = V (x0) +∫ t
0V (φτ (x0)) dτ ≤ V (x0)− αt
The right hand side will eventually become negative, contradicting the assumption that
v > 0.
Consider now the set Ωβ . Since Ωβ ⊂ D, if x ∈ Ωβ \ 0 then V (x) > 0 and
V (x) < 0. This shows, following the arguments above, that V (φt(x)) → 0 and thus
‖φt(x)‖ → 0 as t → ∞ for any x ∈ Ωβ \ 0. It follows that Ωβ is contained in the
equilibrium point’s domain of attraction.
2
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4.2 Convex Stability Tests
Using the Lyapunov conditions for local asymptotic stability in Theorem 10 we
can construct two SOS programming based algorithms to prove a fixed point’s stability.
Additionally, these algorithms can be used to find and maximize the size of certain invariant
subsets of its region of attraction. Unlike Zubov’s theorem, [11], which finds the exact region
of attraction, our approach allows us to use convex optimization to begin to understand the
size and shape of the region of attraction by finding invariant subsets.
The first algorithm presented below works to find the largest estimate of the fixed
point’s region of attraction by expanding the set D and then finding the largest level set of
the resulting Lyapunov function that is contained in D, which in line with Theorem 10 is
invariant and is contained in the region of attraction. The second algorithm approaches the
problem from somewhat of a converse point of view. It expands a region that is contained
in a level set of the Lyapunov function on which all of the Lyapunov conditions hold.
4.2.1 Expanding D Algorithm
To fit the assumptions of Theorem 10 into an SOS programming framework we
will first restrict V ∈ Rn with V (0) = 0, as in the global case. Additionally we will describe
D with a semi-algebraic set
D := x ∈ Rn|p(x) ≤ β
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with p ∈ Σn, positive definite, and β ≥ 0 to insure that D is connected and contains the
origin. Now the requirements of Theorem 10 for asymptotic stability become
x ∈ Rn|p(x) ≤ β \ 0 ⊆ x ∈ Rn|V (x) < 0
x ∈ Rn|p(x) ≤ β \ 0 ⊆ x ∈ Rn|V (x) > 0
If we can find a V to satisfy these conditions for a fixed p and any value of β > 0, then the
system (3.1) is asymptotically stable about the fixed point x = 0. However, using this set
up, the largest invariant set we can demonstrate that converges to the origin is the largest
level set of V that is contained in D. In order to find this largest estimate of the region of
attraction, we will satisfy the Lyapunov conditions above for the largest D by fixing p and
maximizing β subject to the Lyapunov conditions. In this approach, the level sets of p give
the shape of the regions over which we will be checking the Lyapunov conditions.
We pose the following optimization to search for V using the set emptiness form
of the set containment constraints above
maxV ∈Rn,V (0)=0
β
s.t.
x ∈ Rn|p(x) ≤ β, x 6= 0, V (x) ≤ 0 = φ
x ∈ Rn|p(x) ≤ β, x 6= 0, V (x) ≥ 0 = φ
These conditions are not yet semi-algebraic as they contain the non-polynomial constraint
x 6= 0. We can get around this problem by using l1, l2 ∈ Σn, positive definite, and replacing
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x 6= 0 with l(x) 6= 0 to get
maxV ∈Rn,V (0)=0
β
s.t.
x ∈ Rn|p(x) ≤ β, l1(x) 6= 0, V (x) ≤ 0 = φ
x ∈ Rn|p(x) ≤ β, l2(x) 6= 0, V (x) ≥ 0 = φ
Invoking the Positivstellensatz (Theorem 4), we can rewrite the constraints to form the
equivalent optimization
maxV ∈Rn,V (0)=0
β
s.t.
s1 + (β − p)s2 − V s3 − V (β − p)s4 + l2k11 = 0
s5 + (β − p)s6 + V s7 + V (β − p)s8 + l2k22 = 0
where s1, . . . , s8 ∈ Σn, k1, k2 ∈ Z+ and f , p are given. These constraints can not be checked
with SOS programming in this general form. However, if we specify convenient values for
the k’s, the s’s and fix p we can use an iterative SOS programming approach to find the
Lyapunov function that maximizes the size of D.
To limit the degree of the problem, we pick k1 = k2 = 1. Additionally, since
the product of two SOS polynomials is SOS, we can further simplify the problem by by
replacing s1, . . . , s4 with s1l1, . . . , s4l1 as well as s5, . . . , s8 with s5l2, . . . , s8l2. We can then
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factor out the l terms to get the following optimization with SOS constraints
maxV ∈Rn,V (0)=0,si∈Σn
β
s.t.
−(β − p)s2 + V s3 + V (β − p)s4 − l1 ∈ Σn
−(β − p)s6 − V s7 − V (β − p)s8 − l2 ∈ Σn
(4.1)
If this optimization is feasible, then V shows the system to be asymptotically stable with
the Lyapunov requirements holding on all of D. However, these constraints are bilinear
in V and the s’s, so we will have to solve the problem iteratively. An algorithm to solve
the optimization (4.1) and find the largest estimate of the fixed point’s region of attraction
within D is given in Algorithm 5 which follows the discussion of finding the largest invariant
set contained in D
Having found the largest D = x ∈ Rn|p(x) ≤ β over which the Lyapunov
conditions hold we can now formulate a search for the largest level set of V which it contains
as
max c
s.t.
x ∈ Rn|V (x) ≤ c ⊆ x ∈ Rn|p(x) ≤ β
where as opposed to (4.1) both β and V are fixed. We begin the transformation to an SOS
programming problem by forming empty set constraint version
max c
s.t.
x ∈ Rn|V (x) ≤ c, p(x) > β = φ
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which is equivalent to
max c
s.t.
x ∈ Rn|V (x) ≤ c, p(x) ≥ β, p(x) 6= β = φ
Now we use the Positivstellensatz to transform the constraint, and pick k = 1 for the monoid
to get the SOS programming problem
maxs2,s3,s4∈Σn
c
s.t.
−s2(c− V )− s3(p− β)− s4(p− β)(c− V )− (p− β)2 ∈ Σn
(4.2)
which can be solved with a bisection on c.
We can now combine the optimization to maximize the size of D (4.1) with the level
set maximization (4.2) to form the following algorithm for estimating the asymptotically
stable fixed point’s domain of attraction.
Algorithm 5 (Expanding D) An iterative search to expand the region D in Theorem 10
starting from a candidate Lyapunov function.
Let i be the iteration index and set i = 1. Denote the candidate Lyapunov function
V (i=0) and pick the maximum degree of the Lyapunov function, the SOS multipliers and the
l polynomials to be dV , ds2 , ds3 , ds4 , ds6 , ds7 , ds8 and dl1 , dl2 respectively. Pick the maximum
degrees for the the level set maximization problem (4.2) and denote them ds2 , ds3 and ds4.
Fix lk(x) = ε∑n
j=1 xdlkj for k = 1, 2 and some ε > 0. Additionally set β(i=0) = 0.
1. Set V = V (i−1) and solve the linesearch on β where s2 ∈ Σn,ds2, s3 ∈ Σn,ds3
, s4 ∈
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Σn,ds4, s6 ∈ Σn,ds6
, s7 ∈ Σn,ds7and s8 ∈ Σn,ds8
maxs2,s3,s4,s6,s7,s8
β
s.t.
−(β − p)s2 + V s3 + V (β − p)s4 − l1 ∈ Σn
−(β − p)s6 − V s7 − V (β − p)s8 − l2 ∈ Σn
(4.3)
Set s(i)3 = s3, s
(i)4 = s4, s
(i)7 = s7 and s
(i)8 = s8. Continue to step 2.
2. Set s3 = s(i)3 , s4 = s
(i)4 , s7 = s
(i)7 and s8 = s
(i)8 . Solve the linesearch on β where
V ∈ Rn,dVwith V (0) = 0, s2 ∈ Σn,ds2
and s6 ∈ Σn,ds6
maxV,s2,s6
β
s.t.
−(β − p)s2 + V s3 + V (β − p)s4 − l1 ∈ Σn
−(β − p)s6 − V s7 − V (β − p)s8 − l2 ∈ Σn
(4.4)
Set β(i) = β and V (i) = V . If β(i) − β(i−1) is less than a specified tolerance go to step
3, else increment i and go to step 1.
3. Set V = V (i) and β = β(i). Solve the linesearch on c where s2 ∈ Σn,ds2, s3 ∈ Σn,ds3
and s4 ∈ Σn,ds4
maxs2,s3,s4
c
s.t.
−s2(c− V )− s3(p− β)− s4(p− β)(c− V )− (p− β)2 ∈ Σn
Set cmax = c. The set x ∈ Rn|V (x) ≤ cmax is the resulting estimate of the region
of attraction, for it is positively invariant, contained within D and all of its points
converge to the fixed point x = 0.
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Remark 4 (Properties of the expanding D algorithm) :
• If the algorithm is started from a feasible point, which is a candidate Lyapunov function
such that there exist si’s that make (4.1) feasible, then the algorithm will always re-
main feasible. If the system’s linearization is stable, then the linearization’s quadratic
Lyapunov function will always work, since it will stabilize the nonlinear system near
to the origin.
• Note that V need not be positive definite, since it is required to be positive only on
D \ 0, which is the first constraint in (4.1). However, to be positive over a region
about the origin V must have no linear terms and dV should be even. Clearly if V
were chosen to be positive definite, then this constraint can just become V − l1 ∈ Σn.
• Since the s’s, l’s and the s’s are SOS, they must be of even degree. Additionally, the
degrees need to be chosen so that following relations hold:
For the first SOS constraint
maxdeg(ps2),deg(V s3) ≥ dl1
maxdeg(ps2),deg(V s3) ≥ deg(V ps4)
For the second SOS constraint
deg(ps6) ≥ dl2
deg(ps6) ≥ maxdeg(V s7),deg(V ps8)
For the c maximization constraint
maxdeg(s2V ),deg(s4pV ) ≥ deg(p2)
maxdeg(s2V ),deg(s4pV ) ≥ deg(s3p)
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4.2.2 Expanding Interior Algorithm
The previous algorithm looks to Theorem 10 and searches for estimates of the
region of attraction by enlarging D. However, since D is described by the level sets of p
and the estimate of the fixed point’s region of attraction is the largest level set of V that is
contained in D, it is possible to have a large D that contains a much smaller largest level
set of V . This algorithm takes the other approach by requiring that the set whose size
is maximized is contained within an invariant region. This algorithm originally appeared
in the context of state feedback controller design in [13] which also appears in a modified
version in §4.4.2.
Consider again the system (3.1). If we define a variable sized region
Pβ := x ∈ Rn|p(x) ≤ β
for p ∈ Σn and positive definite, we can estimate the region of attraction by maximizing β
subject to the constraint that all of the points in Pβ converge to the origin under the flow
of the system. Following Theorem 10, if we define
D := x ∈ Rn|V (x) ≤ 1
for some as of yet unknown candidate Lyapunov function, then Pβ must be contained D.
Additionally for the theorem’s other assumptions
x ∈ Rn|V (x) ≤ 1 \ 0 ⊆ x ∈ Rn|V (x) < 0
and, since V and thus D is unknown, the only effective way to ensure that V is positive on
D \ 0 it to require that V be positive everywhere away from x = 0.
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The problem of finding the best estimate of the region of attraction in this frame-
work can be written with set emptiness constraints as
maxV ∈Rn,V (0)=0
β
s.t.
x ∈ Rn|V (x) ≤ 0, x 6= 0 = φ
x ∈ Rn|p(x) ≤ β, V (x) ≥ 1, V (x) 6= 1 = φ
x ∈ Rn|V (x) ≤ 1, V (x) ≥ 0, x 6= 0 = φ
(4.5)
which, as with the expanding D version, is not semi-algebraic due to the x 6= 0 constraints.
If we use the same work around as before by replacing these constraints with l1(x) 6= 0 and
l2(x) 6= 0 with l1, l2 ∈ Σn, positive definite, we have
maxV ∈Rn,V (0)=0
β
s.t.
x ∈ Rn|V (x) ≤ 0, l1(x) 6= 0 = φ
x ∈ Rn|p(x) ≤ β, V (x) ≥ 1, V (x) 6= 1 = φ
x ∈ Rn|V (x) ≤ 1, V (x) ≥ 0, l1(x) 6= 0 = φ
(4.6)
Applying the Positivstellensatz (Theorem 4) the optimization becomes
maxV ∈ Rn, V (0) = 0, k1, k2, k3 ∈ Z+
s1, . . . , s10 ∈ Σn
β
s.t.
s1 − V s2 + l2k11 = 0
s3 + (β − p)s4 + (V − 1)s5 + (β − p)(V − 1)s6 + (V − 1)2k2 = 0
s7 + (1− V )s8 + V s9 + (1− V )V s10 + l2k32 = 0
(4.7)
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which is not amenable to SOS programming unless we pick convenient values for some of
the s’s and the k’s. To keep the degree of the problem down, we pick k1 = k2 = k3 = 1.
To simplify the first constraint, we pick s2 = l1 and factor l1 out of s1. The second
constraint has the term (V −1)2 which is quadratic in the coefficients of V and thus inhibits
optimization over V , so set s3 = s4 = 0 and factor out a (V −1) term. The third constraint
has the term (1− V )V s10 which is quadratic in the coefficients of V so we set s10 = 0 and
factor out l2. These selections allow us to write the optimization in a form that is suitable
for an iterative algorithm.
maxV ∈ Rn, V (0) = 0
s6, s8, s9 ∈ Σn
β
s.t.
V − l1 ∈ Σn
−(
(β − p)s6 + (V − 1))∈ Σn
−(
(1− V )s8 + V s9 + l2
)∈ Σn
(4.8)
We now can propose an algorithm to find an estimate of the region of attraction
for the fixed point, x = 0 of the system (3.1) by iteratively solving the SOS constrained
optimization (4.8).
Algorithm 6 (Expanding Interior) An iterative search to expand the region Pβ and thus
the region D := x ∈ Rn|V (x) ≤ 1 in Theorem 10 starting from a positive definite candidate
Lyapunov function.
Let i be the iteration index and set i = 1. Denote the candidate Lyapunov function
V (i=0) and pick the maximum degree of the Lyapunov function, the SOS multipliers and
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the l polynomials to be dV , ds6 , ds8 , ds9 and dl1 , dl2 respectively. Fix lk(x) = ε∑n
j=1 xdlkj for
k = 1, 2 and some ε > 0. Additionally set β(i=0) = 0.
1. Set V = V (i−1) and solve the linesearch on β where s6 ∈ Σn,ds6, s8 ∈ Σn,ds8
and
s9 ∈ Σn,ds9
maxs6,s8,s9
β
s.t.
−(
(β − p)s6 + (V − 1))∈ Σn
−(
(1− V )s8 + V s9 + l2
)∈ Σn
(4.9)
Set s(i)8 = s8 and s
(i)9 = s9. Continue to step 2.
2. Set s8 = s(i)8 and s9 = s
(i)9 . Solve the linesearch on β where V ∈ Rn,dV
with V (0) = 0
and s6 ∈ Σn,ds6
maxV,s6
β
s.t.
V − l1 ∈ Σn
−(
(β − p)s6 + (V − 1))∈ Σn
−(
(1− V )s8 + V s9 + l2
)∈ Σn
(4.10)
Set β(i) = β and V (i) = V . If β(i) − β(i−1) is less than a specified tolerance goto step
3, else increment i and go to step 1.
3. The set x ∈ Rn|V (i)(x) ≤ 1 contains Pβ(i) and is the largest estimate of the fixed
point’s region of attraction.
Remark 5 (Properties of the expanding interior algorithm) :
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• If the algorithm is started from a feasible point, which is a Lyapunov function such
that there exist si’s that make (4.8) feasible, then the algorithm will always remain
feasible. If the system’s linearization is stable, then the linearization’s quadratic Lya-
punov function will always work, since it will stabilize the nonlinear system near to
the origin.
• Note that, unlike Algorithm 5, V must be positive definite, so it should have no linear
terms and dV should be even.
• Since the s’s, and the l’s are SOS, they must be of even degree. Additionally, the
degrees need to be chosen so that following relations hold:
For the first SOS constraint
dV = dl1
For the second SOS constraint
deg(ps6) ≥ dV
For the third SOS constraint
deg(V s8) ≥ deg(V s9)
deg(V s8) = dl2
4.2.3 Estimating the Region of Attraction Example
To compare the effectiveness of Algorithm 5 (Expanding D) and Algorithm 6
(Expanding Interior) consider estimating the region of attraction for the following system,
a damped pendulum where the sine term has been replaced with the first two terms of its
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Taylor series expansion
x1 = x2
x2 = − 810x2 −
(x1 −
x316
) (4.11)
The system has three equilibrium points: (0, 0) and (±√
6, 0). Looking at the linearizations
of the system about each equilibrium point we find, as expected, that the first is a stable
sink while the other are saddles.
To use either Algorithm 5 or 6 we need to pick p and find a suitable starting
candidate Lyapunov function. In order to demonstrate the way the two algorithms work,
we will pick a completely uninspired polynomial to expand its level sets:
p(x) = x21 + x2
2
Additionally we will pick the uninspired candidate Lyapunov function V (x) = x∗Px where
P comes from the LMI feasibility problem
P 0
A∗fP + PAf ≺ 0
with Af the linearization of the system (4.11). Solving this feasibility problem, we find
P =
99.63 25.33
25.33 81.98
which we will use to form the candidate Lyapunov function.
Setting the both algorithm’s stopping tolerance to β(i) − β(i−1) = .01 and setting
the degrees as follows we can compare the two algorithms. For the expanding D algorithm,
set dV = 2, ds2 = ds3 = ds6 = 2, ds4 = ds7 = ds8 = 0 and dl1 = dl2 = 4. For the expanding
interior algorithm set dV = 2, ds8 = 2, ds6 = ds9 = 0, dl1 = 2 and dl2 = 4.
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We can see the progress of the two algorithms in Table 4.1 which shows the radius
of the region being expanded (√
β in both cases). Note that, D contains the estimate of
the region of attraction while the estimate contains Pβ.
Iteration Index D Radius Pβ Radius1 2.42 0.852 2.42 1.163 - 1.594 - 1.845 - 2.056 - 2.05
Table 4.1: Results of applying Algorithms 5 and 6 to (4.11).
The results of running both algorithms are shown in Figure 4.1. The large dots
indicate the system’s fixed points and the thin lines connecting them are the system’s stable
and unstable manifolds. The dashed lines show the maximal D from the expanding D algo-
rithm and the maximal Pβ from the expanding interior algorithm. The thick-lined ellipses
show the two estimates of the region of attraction; the smaller ellipse that is contained in
D comes from Algorithm 5, while the larger ellipse that contains Pβ comes from Algorithm
6.
Even though the Lyapunov function is quadratic and the SOS multipliers are of
low degree, we can look at Figure 4.1 and tell that, for both algorithms, the largest value of
β possible has been found for the given p. In the expanding D case, the region D borders the
system’s saddle points where f , and thus V , is zero. Where as, in the expanding interior
case, Pβ comes right up to two of the system’s stable manifolds, by which the region of
attraction must be bounded.
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−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
x1
x 2
D
Pβ
Figure 4.1: Phase plot of the system (4.11) with the largest D and Pβ as well as theirellipsoidal estimates of the region of attraction that they respectively circumscribe andinscribe.
Note that in Figure 4.1 the region D crosses the two of the system’s stable mani-
folds. Clearly the region of attraction can not cross any of the system’s stable or unstable
manifolds, so all of the area of D outside the manifolds can not be part of the estimate of
the region of attraction. This skews the Lyapunov function that Algorithm 5 designs, since
it is expanding D over areas where the level sets can not follow. Additionally, it seems
unrealistic to assume that, for a general polynomial system, the region where V < 0 is
described by the a level set of a low degree polynomial.
Clearly both of these algorithms would perform better if p were chosen to have level
sets that more closely align with the shape of the region of attraction, but the uninspired
choice of p here indicates what will happen in higher dimensions where we are unable to
visualize the region of attraction.
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4.3 Disturbance Analysis
The previous local SOS applications were centered on proving stability; we will now
shift gears to consider the local effect of external disturbances on a polynomial system. A
wealth of interesting results of this type for linear systems are posed in an LMI framework in
[5]. Using SOS programming we can, in general, formulate parallel versions for polynomial
systems.
In most cases the generalization from linear systems and LMIs to polynomial
systems and SOS programs is straight forward, as is shown in the example below in §4.3.1.
However, it is very important to point out that the LMI tests given in [5] provide conditions
for the system to have a given characteristic globally, which as we have seen in the stability
and controller synthesis cases may be impossible or just undesired. So, in general, it will
make sense to add a few set containment constraints to make the result hold only on some
subset of the whole space.
4.3.1 Reachable Set Bounds under Unit Energy Disturbances
The original problem in [5, §6.1.1] is, given the linear system
x = Ax + Bww
with x ∈ Rn and w ∈ Rnw , can we bound the set of points that are reachable from x0 = 0
in T time units if the disturbance is constrained to have unit energy,∫ T0 w(t)∗w(t) dt ≤ 1.
The bounding approach used is Lyapunov function based. Let the Lyapunov function be
quadratic
V (x) = x∗Px
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with P an as of yet unknown positive definite matrix. Then, if
V (x) = ∇V ∗(Ax + Bww) ≤ w∗w
for all x,w, we can integrate both sides from 0 to T to find
V (x(T ))− V (x0) ≤∫ T
0w(t)∗w(t) dt ≤ 1
which, since x0 = 0 implies that the set x ∈ Rn|x∗Px ≤ 1 contains x(T ). The positive
definiteness of V and the inequality bound on V can be, written in an LMI setup as P 0
and A∗P + PA PBw
B∗wP −I
0
These conditions provide a set that contains the reachable set from x0 = 0 with unit energy
disturbances, and, with a cost function dependent on the eigenvalues of P , we could optimize
the bound for tightness.
Approach for Polynomial Systems
We can now follow the reasoning above, as originally done in [13], to mimic the
linear system’s reachable set bound for the following polynomial system,
x = f(x) + gw(x)w (4.12)
with x(t) ∈ Rn, w(t) ∈ Rnw , f ∈ Rnn, f(0) = 0 and gw ∈ Rn×nw
n . The two central constraints
of the Lyapunov based approach are that V ∈ Rn must be positive definite, and that
∇V (x)∗(f(x) + gw(x)w) ≤ w∗w
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for all x,w. If we can find a V that satisfies these constraints, we know that the set
Ω1 := x ∈ Rn|V (x) ≤ 1 bounds the unit energy reachable set. At this point, the
transition of the problem from linear to polynomial system is complete and we would just
need to write the constraints above as SOS conditions.
However, in this set up the V constraint is required to hold for all x,w, which
seems extreme since φt(x0) remains in Ω1 for all t ∈ [0, T ]. Additionally, we have no easy
way to minimize the size of the bounding set.
To get around the problems presented above, we will introduce a fixed, user chosen,
function p ∈ Σn that is positive definite and make the following definition
Pβ := x ∈ Rn|p(x) ≤ β
of a set that will contain Ω1. Additionally, since the state trajectory always remains in Ω1,
we will only require that the V inequality hold for (x,w) ∈ Ω1×Rnw . Now we can pose the
following optimization to minimize the size of set that bounds the unit energy reachable
set, Ω1, where the constraints mentioned above have been turned into their set emptiness
forms
minV
β
s.t.
x ∈ Rn|V (x) ≤ 0, l(x) 6= 0 = φ
x ∈ Rn|V (x) ≤ 1, p(x) ≥ β, p(x) 6= β = φx ∈ Rn
x ∈ Rnw
∣∣∣∣∣∣∣∣∣∣∣∣
V (x) ≤ 1,
∇V (x)∗(f(x) + gw(x)w) ≥ w∗w,
∇V (x)∗(f(x) + gw(x)w) 6= w∗w
= φ
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with l ∈ Σn, positive definite. Using the Positivstellensatz, this becomes
minV
β
s.t.
s1 − V s2 + l2k1 = 0
s3 + (1− V )s4 + (p− β)s5 + (1− V )(p− β)s6 + (p− β)2k2 = 0s7 +
(∇V (x)∗(f(x) + gw(x)w)− w∗w
)s8 + (1− V )s9
+(1− V )(∇V (x)∗(f(x) + gw(x)w)− w∗w
)s10
+(∇V (x)∗(f(x) + gw(x)w)− w∗w
)2k3
= 0
with s1, . . . , s6 ∈ Σn, s7, . . . , s10 ∈ Σn+nw and k1, k2, k3 ∈ Z+. To begin the transition
from set emptiness constraints into SOS constraints, pick k1 = k2 = k3 = 1. For the first
constraint, set s2 = l and factor out an l. For the second constraint, we can just rearrange it.
And for the third constraint, we pick s7 = s9 = 0 and factor out a(∇V ∗(f + gww)−w∗w
).
In all this gives us the optimization
minV
β
s.t.
V − l ∈ Σn
−((1− V )s4 + (p− β)s5 + (1− V )(p− β)s6 + (p− β)2
)∈ Σn
−((1− V )s10 +
(∇V ∗(f + gww)− w∗w
))∈ Σn+nw
(4.13)
which we can solve iteratively by alternating linesearches on the si’s and V . For further
details and a numeric example see [13, §III].
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4.3.2 Set Invariance under Peak Bounded Disturbances
Considering again a polynomial system subject to disturbances as in (4.12),
x = f(x) + gw(x)w
we can now look at finding the maximum peak disturbance value such that a given set re-
mains invariant under these bounded disturbances and the action of the system’s dynamics.
Let the peak of w be bounded by
‖w(t)‖∞ ≤ √γ
and define the invariant set as
Ω1 := x ∈ Rn|V (x) ≤ 1
for some fixed V ∈ Rn, positive definite. We know that if V (x,w) ≤ 0 on the boundary of
Ω1 for all w meeting the peak bound, then the flow of the system from any point in Ω1 can
not ever leave Ω1, which makes it invariant. In set containment terms we can write this
relationship as
x ∈ Rn, w ∈ Rnw |V (x) = 1 ∩ x ∈ Rn, w ∈ Rnw |w∗w ≤ γ
⊆ x ∈ Rn, w ∈ Rnw |V (x,w) ≤ 0 (4.14)
which can be rewritten in set emptiness form as
x ∈ Rn, w ∈ Rnw |V (x)− 1 = 0, γ − w∗w ≥ 0, V (x,w) ≥ 0, V (x,w) 6= 0 = φ
Employing the Positivstellensatz, this becomes
s0 + s1(γ − w∗w) + s2(V ) + s3(γ − w∗w)(V ) + (V )2k + q(V − 1) = 0
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with k ∈ Z+, q ∈ Rn+nw and s0, s1, s2, s3 ∈ Σn+nw .
Using our standard approach of k = 1, we can write the following SOS constraint
that guarantees invariance under bounded w,
−s1(γ − w∗w)− s2(V )− s3(γ − w∗w)(V )− V 2 − q(V − 1) ∈ Σn+nw (4.15)
Notice that this SOS condition has terms that are not linear in the monomials of V , and
thus there is no way to use our convex optimization approach to adjust V while checking
this condition. Since (4.15) in linear in γ we can search for the maximum peak disturbance
for which the set is invariant, by searching over q and the si’s to maximize γ subject to
(4.15). We will need to have the following degree relationship hold to make (4.15) possibly
feasible
max(
deg(s1) + 2,deg(s2V ),deg(qV ))≥ max
(deg(s3V ) + 2, 2 deg(V )
)If we set x0 = 0, then the invariant set Ω1 bounds the system’s reachable set
under disturbances with peak less than γ. This bound is similar, but less stringent, than
the bound for linear systems given in [5].
In §4.4.3 we design two state feedback controllers to make the largest region pos-
sible attracted to the origin, and then we provide an example of the technique above by
finding the largest peak disturbance that these controllers can reject.
Effect of ‖w(t)‖∞ on ‖x(t)‖∞
Using the bounded peak disturbances techniques above to find the largest distur-
bance peak value for which Ω1 is invariant, we can then bound the peak size of the system’s
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state to get a relationship that is similar to the induced L∞ → L∞ norm from disturbance
to state for this invariant set.
For a given V , we solve the optimization to find the largest γ such that (4.15) is
feasible. Then we can bound the size of the state by optimizing to find the smallest α such
that
Ω1 = x ∈ Rn|V (x) ≤ 1 ⊆ x ∈ Rn|x∗x ≤ α
This containment constraint is easily solved with a generalized S-procedure following from
§2.2.1. From this point we know that the following implication holds
‖w(t)‖∞ ≤ √γ ⇒ ‖x(t)‖∞ ≤√
α
as long as x0 ∈ Ω1, which provides our induced norm-like bound.
As with the previous disturbance analysis technique, we demonstrate this one in
§4.4.3 on the designed state feedback controllers.
4.3.3 Induced L2 → L2 Gain
Consider the disturbance driven system with outputs (3.3),
x = f(x) + gw(x)w
y = h(x)
with x(t) ∈ Rn, w(t) ∈ Rnw , y(t) ∈ Rp, f ∈ Rnn, f(0) = 0, gw ∈ Rn×nw
n , and h ∈ Rpn with
h(0) = 0. In §3.3 we studied this system’s global induced L2 → L2 gain from w to y, and
now we look to bound this gain on some invariant region.
For a region, Ω1 = x ∈ Rn|V (x) ≤ 1 as in §4.3.2, that is invariant under
disturbances with ‖w(t)‖∞ ≤ √γ, we can bound the induced L2 → L2 gain from w to y on
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this invariant set by finding a positive definite H ∈ Rn and β ≥ 0 such that the following
set containment holds
x ∈ Rn, w ∈ Rnw |w∗w ≤ γ ∩ x ∈ Rn, w ∈ Rnw |V (x) ≤ 1
⊆ x ∈ Rn, w ∈ Rnw |H(x,w) + h(x)∗h(x)− βw∗w ≤ 0 (4.16)
If we can find a β, H pair to make (4.16) hold, then we can follow the steps from §3.3 to
show that
‖y(t)‖2‖w(t)‖2
≤√
β
provided that x0 = 0 and ‖w(t)‖∞ ≤ √γ.
We can search for the tightest bound on the induced norm by employing a gener-
alized S-procedure to satisfy (4.16) and solving the following optimization
minH∈Rn
β
s.t.
H − l ∈ Σn
−(H(x,w) + h(x)∗h(x)− βw∗w
)− s1(γ − w∗w)− s2(1− V ) ∈ Σn+nw
(4.17)
with s1, s2 ∈ Σn+nw and l ∈ Σn, positive definite.
In an effort to make the optimization (4.17) feasible we will pick the degrees of s1
and s2 so that
deg(s1) + 2 ≥ deg(H + h∗h)
and
deg(s2) + deg(V ) ≥ deg(H + h∗h)
Once more, we will exhibit this technique to study the performance of the state
feedback controllers that we design in §4.4.3.
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4.4 State Feedback Controller Design
In the previous section, we found two SOS programming based approaches to prove
asymptotic stability for a polynomial system and estimate the domain of attraction of the
fixed point at the origin. We now expand these approaches for state feedback controller
design.
Consider again the system (3.5)
x = f(x) + g(x)u
for x ∈ Rn with f ∈ Rnn, f(0) = 0 and u ∈ Rm with g ∈ Rn×m
n . If we allow u to be
generated by a state feedback controller K ∈ Rmn with K(0) = 0, we get the closed loop
system (3.6)
x = f(x) + g(x)K(x)
where K is still unknown. We can now look to find state feedback controller design algo-
rithms that parallel Algorithms 5 and 6.
4.4.1 Expanding D Algorithm for State Feedback Design
We will follow the steps of the development of the expanding D algorithm in §4.2.1
to develop a state feedback controller design algorithm. As before we restrict V ∈ Rn with
V (0) = 0 and describe D with a semi-algebraic set
D := x ∈ Rn|p(x) ≤ β
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87
with p ∈ Σn, positive definite, and β ≥ 0. Now the requirements of Theorem 10 for
asymptotic stability for the closed loop system (3.6) become
x ∈ Rn|p(x) ≤ β \ 0 ⊆ x ∈ Rn|V (x) > 0
x ∈ Rn|p(x) ≤ β \ 0 ⊆ x ∈ Rn|∇V (x)∗(f(x) + g(x)K(x) < 0
If we can find V,K to satisfy these conditions for a fixed p and any value of β > 0, then the
system (3.1) is asymptotically stable about the fixed point x = 0.
As in Algorithm 5, the largest invariant set we can demonstrate that converges to
the origin is the largest level set of V that is contained in D. Following the manipulations
that got us to SOS constraints in stability case we find that maximizing D under the set
containments above is equivalent to
maxV ∈ Rn, V (0) = 0
K ∈ Rmn , K(0) = 0
si ∈ Σn
β
s.t.
−(β − p)s2 + V s3 + V (β − p)s4 − l1 ∈ Σn
−(β − p)s6 −∇V ∗(f + gK)s7 −∇V ∗(f + gK)(β − p)s8 − l2 ∈ Σn
(4.18)
Again, since the SOS conditions are trilinear in V,K and the s’s they will have
to be checked iteratively. Also, as before, the largest estimate of the region of attraction
is the largest invariant set contained in D and is found using the same optimization as in
§4.2.1. The following algorithm uses a nested iterative approach to maximize the size of
D by adjusting V , K and the s’s and then finds largest level set of V contained in D to
estimate the region of attraction.
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Algorithm 7 (Expanding D State Feedback Design) An iterative search to expand
the region D in Theorem 10 starting from a candidate Lyapunov function and a candidate
controller.
Let i be the iteration index and set i = 1. Denote the candidate Lyapunov func-
tion V (i=0) and the candidate controller K(i=0). Pick the maximum degree of the Lya-
punov function, the controller, the SOS multipliers and the l polynomials to be dV , dK ,
ds2 , ds3 , ds4 , ds6 , ds7 , ds8 and dl1 , dl2 respectively. Pick the maximum degrees for the the level
set maximization problem (4.2) and denote them ds2 , ds3 and ds4. Fix lk(x) = ε∑n
j=1 xdlkj
for k = 1, 2 and some ε > 0. Additionally set β(i=0) = 0.
1. SOS Weight Iteration:
Set V = V (i−1) and K = K(i−1) and solve the linesearch on β where s2 ∈ Σn,ds2,
s3 ∈ Σn,ds3, s4 ∈ Σn,ds4
, s6 ∈ Σn,ds6, s7 ∈ Σn,ds7
and s8 ∈ Σn,ds8
maxs2,s3,s4,s6,s4,s8
β
s.t.
−(β − p)s2 + V s3 + V (β − p)s4 − l1 ∈ Σn
−(β − p)s6 −∇V ∗(f + gK)s7 −∇V ∗(f + gK)(β − p)s8 − l2 ∈ Σn
Set s(i)7 = s7, s
(i)8 = s8 and continue to step 2.
2. Controller Iteration:
Set V = V (i−1), s7 = s(i)7 and s8 = s
(i)8 . Solve the linesearch on β where s2 ∈ Σn,ds2
,
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s3 ∈ Σn,ds3, s4 ∈ Σn,ds4
, s6 ∈ Σn,ds6and K ∈ Rm
n,dKwith K(0) = 0
maxK,s2,s3,s4,s6
β
s.t.
−(β − p)s2 + V s3 + V (β − p)s4 − l1 ∈ Σn
−(β − p)s6 −∇V ∗(f + gK)s7 −∇V ∗(f + gK)(β − p)s8 − l2 ∈ Σn
Set K(i) = K, s(i)3 = s3 and s
(i)4 = s4. Continue to step 3.
3. Lyapunov function iteration
Set K = K(i), s3 = s(i)3 , s4 = s
(i)4 , s7 = s
(i)7 and s8 = s
(i)8 . Solve the linesearch on β
where s2 ∈ Σn,ds2, s6 ∈ Σn,ds6
and V ∈ Rn,dVwith V (0) = 0
maxV,s2,s6
β
s.t.
−(β − p)s2 + V s3 + V (β − p)s4 − l1 ∈ Σn
−(β − p)s6 −∇V ∗(f + gK)s7 −∇V ∗(f + gK)(β − p)s8 − l2 ∈ Σn
Set V (i) = V and continue to step 4.
4. If β(i)− β(i−1) is less than a specified tolerance goto step 5, else increment i and goto
step 1.
5. Set V = V (i) and β = β(i). Solve the linesearch on c where s2 ∈ Σn,ds2, s3 ∈ Σn,ds3
and s4 ∈ Σn,ds4
maxs2,s3,s4
c
s.t.
−s2(c− V )− s3(p− β)− s4(p− β)(c− V )− (p− β)2 ∈ Σn
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Set cmax = c. The set x ∈ Rn|V (x) ≤ cmax is the resulting estimate of the region
of attraction, for it is positively invariant, contained within D and all of its points
converge to the fixed point x = 0.
Remark 6 (Properties of the expanding D state feedback algorithm) :
• As with the stability version of the album, if the algorithm is started from a feasible
point it will always remain feasible. If the system’s linearization is controllable, then
a linear controller that stabilizes the linearized system and the resulting quadratic
Lyapunov function will always work, since they will stabilize the nonlinear system
near to the origin. Since we can be guaranteed of a feasible starting point, we do not
need to consider K and V variants of the algorithm.
• As with the stability version, V need not be positive definite, since it is required to be
positive only on D \0. However to be positive over a region about the origin V must
have no linear terms and dV should be even. Clearly if V were chosen to be positive
definite, then the first constraint in (4.18) can just become V − l1 ∈ Σn.
• Since the s’s, l’s and the s’s are SOS, they must be of even degree. Additionally, the
degrees need to be chosen so that following relations hold:
For the first SOS constraint
maxdeg(ps2),deg(V s3) ≥ dl1
maxdeg(ps2),deg(V s3) ≥ deg(V ps4)
For the second SOS constraint
deg(ps6) ≥ dl2
deg(ps6) ≥ max deg (∇V ∗(f + gK)s7) ,deg (∇V ∗(f + gK)ps8)
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For the c maximization constraint
maxdeg(s2V ),deg(s4pV ) ≥ deg(p2)
maxdeg(s2V ),deg(s4pV ) ≥ deg(s3p)
4.4.2 Expanding Interior Algorithm for State Feedback Design
As with the expanding D algorithm, we can now adapt the expanding interior
algorithm to search for state feedback controllers to stabilize (3.5) as was proposed in [13].
As with the stability version, we define a variable sized region
Pβ := x ∈ Rn|p(x) ≤ β
for p ∈ Σn and positive definite. We then define
D := x ∈ Rn|V (x) ≤ 1
for some as of yet unknown candidate Lyapunov function. Following the developments for
the stability case we know that we need
x ∈ Rn|p(x) ≤ β ⊆ x ∈ Rn|V (x) ≤ 1
x ∈ Rn|V (x) ≤ 1 \ 0 ⊆ x ∈ Rn|∇V ∗(x)(f(x) + g(x)K(x)
)< 0
x ∈ Rn|V (x) ≤ 0, x 6= 0 = φ
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92
with V (0) = 0 to satisfy the assumptions of Theorem 10. Maximizing the size of Pβ subject
to these set emptiness and containment constraints can be reformulated as
maxV ∈ Rn, V (0) = 0
K ∈ Rmn , K(0) = 0
s6, s8, s9 ∈ Σn
β
s.t.
V − l1 ∈ Σn
−(
(β − p)s6 + (V − 1))∈ Σn
−(
(1− V )s8 +∇V ∗(f + gK)s9 + l2)∈ Σn
(4.19)
We now can propose an algorithm to design a controller to enlarge our estimate of
the region of attraction for the fixed point, x = 0, of (3.5) by solving the SOS constrained
optimization (4.19) with a nested iteration. For the controller synthesis part of the iteration
a slight modification is made to the third constraint of (4.19) by introducing an intermediate
variable α and expanding the Lyapunov level set x ∈ Rn|V (x) < α. After the controller
is designed, the Lyapunov function constraint is rescaled by α.
Algorithm 8 (Expanding Interior State Feedback Design) An iterative search to ex-
pand the region Pβ and thus the region D := x ∈ Rn|V (x) ≤ 1 in Theorem 10 starting
from a positive definite candidate Lyapunov function and a candidate state feedback con-
troller.
Let i be the iteration index and set i = 1. Denote the candidate Lyapunov function
V (i=0) and the candidate state feedback controller K(i=0). Pick the maximum degree of the
Lyapunov function, the controller, the SOS multipliers and the l polynomials to be dV , dK ,
ds6 , ds8 , ds9 and dl1 , dl2 respectively. Fix lk(x) = ε∑n
j=1 xdlkj for k = 1, 2 and some ε > 0.
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Additionally set l(i=0)2 = l2, and β(i=0) = 0.
1. SOS Weight iteration:
Set V = V (i−1), K = K(i−1) and l2 = l(i−1)2 . Solve the linesearch on α where s8 ∈
Σn,ds8and s9 ∈ Σn,ds9
maxs8,s9
α
s.t.
−((α− V )s8 +∇V ∗
(f + gK
)s9 + l2
)∈ Σn
Set s(i)9 = s9 and continue to step 2.
2. Controller iteration:
Set V = V (i−1), l2 = l(i−1)2 , and s9 = s
(i)9 . Solve the linesearch on α where s8 ∈ Σn,ds8
and K ∈ Rmn,dK
with K(0) = 0
maxK,s8
α
s.t.
−((α− V )s8 +∇V ∗
(f + gK
)s9 + l2
)∈ Σn
Set K(i) = K, s(i)8 = s8 and l
(i)2 = l
(i−1)2 /α. Continue to step 2.
3. Lyapunov function iteration:
Set K = K(i), s8 = s(i)8 , s9 = s
(i)9 and l2 = l
(i)2 . Solve the linesearch on β where
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s6 ∈ Σn,ds6and V ∈ Rn,dV
with V (0) = 0
maxV,s6
β
s.t.
V − l1 ∈ Σn
−(
(β − p)s6 + (V − 1))∈ Σn
−(
(1− V )s8 + V s9 + l2
)∈ Σn
Set V (i) = V and β(i) = β. Continue to step 4.
4. If β(i)−β(i−1) is less than a specified tolerance, the set x ∈ Rn|V (i)(x) ≤ 1 contains
Pβ(i) and is the largest estimate of the fixed point’s region of attraction, else increment
i and goto step 1.
Remark 7 (Properties of the expanding interior algorithm) :
• If the algorithm is started from a feasible point, then it will always remain feasible.
If the system’s linearization is controllable, then any linear controller that stabilizes
the linearized system and the corresponding quadratic Lyapunov function will always
work, since they will stabilize the nonlinear system near to the origin. Again, this
feasibility property lets us not consider both a V and a K variant.
• Note that, unlike Algorithm 5, V must be positive definite, so it should have no linear
terms and dV should be even.
• Since the s’s, and the l’s are SOS, they must be of even degree. Additionally, the
degrees need to be chosen so that following relations hold:
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95
For the first SOS constraint
dV = dl1
For the second SOS constraint
deg(ps6) ≥ dV
For the third SOS constraint
deg(V s8) ≥ deg(∇V ∗(f + gK)s9
)deg(V s8) = dl2
4.4.3 State Feedback Design Example
Consider the spring-mass-damper system from §3.4.3, and recall that it was pos-
sible to design a degree three polynomial state feedback controller to make the system
semi-globally exponentially stable. However, no lower degree controller could be found that
met the stability requirements. Also, if the linear spring were replaced with a nonlinear
spring that behaved the same as the one that fixed m1 to the wall, then no polynomial
controller can be found to make the system semi-globally exponentially stable for dK ≤ 5
and dV ≤ 6.
We now look to find a local controller to demonstrate stability of the spring-mass-
damper system with two nonlinear springs
x1
x2
x3
x4
=
x2
−(x1 + 110x3
1) +((x3 − x1) + 1
10(x3 − x1)3)− 1
10(x4 − x2)
x4
−((x3 − x1) + 1
10(x3 − x1)3)
+ 110(x4 − x2)
︸ ︷︷ ︸
f(x)
+
0
1
0
0
︸︷︷︸g(x)
u (4.20)
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96
As before, the linearization of the system is unstable and we will use the linear controller
and quadratic Lyapunov function from the “feedback trick” LMI (3.13) as the candidate
controller and Lyapunov function. To show an additional use for Algorithms 7 & 8 we will
also tack on the constraint that the system model (4.20) is only valid when ‖x‖ ≤ 10.
From the perspective of the expanding D algorithm, this model validity constraint
can be handled by making D := x11 + x2
2 + x23 + x2
4 and setting the upper bound on the
β linesearches to be 100. With this choice of D, we know that the largest level set of the
resulting Lyapunov function will be within the region where the model is valid, so any
initial condition within this level set will converge to the origin and always remain within
the region where the model is valid.
To handle the model validity constraint within the context of the expanding interior
algorithm, set p = x11 + x2
2 + x23 + x2
4 and the upper bound on β in all the linesearches to
100. Unlike the expanding D case, this does not yet take care of the constraint. When the
algorithm finishes we can do a bisection on a generalized S-procedure to find the largest
value r such that
x ∈ Rn|V (x) ≤ r ⊆ x ∈ Rn|x21 + x2
2 + x23 + x2
4 ≤ 100
Then all initial conditions in the set where V is less than or equal to r converge to the origin
and always remain in the region where the model is valid.
Using these approaches we can solve for a state feedback controller for (4.20). We
use Algorithm 7 with dV = 2, dK = 1, ds2 = ds3 = 2, ds4 = 0, ds6 = 4, ds7 = ds8 = 0,
dl1 = dl2 = 4, with ds2 = ds3 = 2 and ds4 = 0. For Algorithm 8 we use dV = 2, dK = 1,
ds6 = ds8 = 2, ds9 = 0, with dl1 = 2 and dl2 = 4. For both algorithms we set the tolerance
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97
to 0.01. After 4 iterations Algorithm 7 converges to β = 100 with cmax = 15.10, while
Algorithm 8 reaches the β tolerance limit after 24 iterations at β = 14.96 with r = 10.01.
For clarity, let the controller and Lyapunov function found by the expanding D
algorithm be Kd and Vd. Likewise, let the results from the expanding interior algorithm be
Ki and Vi. To summarize the results, under the linear controller
Kd(x) =[−62.58 −26.31 36.08 −27.06
]
x1
x2
x3
x4
the system (4.20) is asymptotically stable and the region Ωd := x ∈ Rn|Vd(x) ≤ cmax
with
Vd(x) =
x1
x2
x3
x4
∗
4.82 0.72 −1.83 2.47
0.72 0.27 −0.40 0.34
−1.83 −0.40 2.54 −0.31
2.47 0.34 −0.31 3.02
x1
x2
x3
x4
is an invariant set contained in the fixed point’s region of attraction as well as the set where
the model is valid, x ∈ Rn|‖x‖ ≤ 10. Additionally under the controller
Ki(x) =[−35.57 −7.70 22.40 −14.56
]
x1
x2
x3
x4
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98
the system (4.20) is asymptotically stable and the region Ωi := x ∈ Rn|Vi(x) ≤ r with
Vi(x) =
x1
x2
x3
x4
∗
4.33 0.35 −2.19 1.89
0.35 0.14 −0.25 0.19
−2.19 −0.25 2.45 −0.63
1.89 0.19 −0.63 2.36
x1
x2
x3
x4
is an invariant set contained in the fixed point’s region of attraction as well as the set where
the model is valid, x ∈ Rn|‖x‖ ≤ 10.
Comparing the two controllers we find that the volume of Ωd is 779.74, while Ωi
has smaller volume of 535.75. Interestingly, Ωd does not contain Ωi, and r would have
to scaled back by 30% to make Ωi fit inside of Ωd. Since Ωd has a larger volume, Kd is
the preferred controller, unless there is specific information about the initial condition that
shows that it is in Ωi \ Ωd.
Disturbance Rejection Qualities of the Controllers
Using the disturbance analysis techniques of §4.3, we can now quantify the perfor-
mance of the controllers Kd and Ki designed above. The approaches of §4.3.2 and §4.3.3
allow us to find the largest peak disturbance under which Ωd and Ωi are invariant, the
peak norm of the state that this implies, and the L2 → L2 gains from a disturbance to the
system’s states on these invariant regions.
For all these analyses we will assume that the disturbance enters additively in the
control channel, which makes gw(x) = g(x). Additionally, since we are considering a state
feedback system, we will set h(x) = x for the induced L2 → L2 gain.
To compute,√
γ, the bound on ‖w(t)‖ under which Ωd and Ωi are invariant, the
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99
Kd Ki√γ 18.57 4.42√α 10.07 10.00√β 0.06 0.27
Table 4.2: State Feedback Controller Disturbance Rejection Results.
condition (4.15) becomes, at minimum, a search for SOS polynomials of degree 8 in 5
variables, since n = 4, nw = 1 and deg(V ) = 4. Polynomials in Σ5,8 have 1287 coefficients.
To work around searching for polynomials with so many coefficients, we will set
q = V 2q and si = V 2si for i = 1, 2, 3. This allows us to factor out a V 2 term to get the
following condition
−s1(γ − w∗w)− s2(V )− s3(γ − w∗w)(V )− 1− q(V − 1) ∈ Σn+nw
which, if we pick deg(s1) = 4, deg(s2) = 2, deg(s3) = 0 and deg(q) = 4, becomes a search
for a polynomial of degree 6 in 5 variables. Polynomials in Σ5,6 have 462 terms, which is a
very good improvement in problem size.
Using this approach to reduce the problem size of (4.15) and picking deg H = 2
and deg s1 = deg s2 = 0 in (4.17), we get the results of the three disturbance rejection
analyses that are shown in Table 4.2. Note that the bounds are as follows, the respective
invariant set is invariant under
‖w(t)‖∞ ≤ √γ
and
‖w(t)‖∞ ≤ √γ ⇒ ‖x(t)‖∞ ≤√
α
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100
as well as, if ‖w(t)‖∞ ≤ √γ then
‖x(t)‖2‖w(t)‖2
≤√
β
The results shown in the table illustrate that under all three criterion considered, the
controller designed by the expanding D algorithm out performs the controller designed by
the expanding interior algorithm. The superior disturbance rejection together with the fact
that Ωd is much larger than Ωi point to Kd being the preferred controller for this example.
4.5 Output Feedback Controller Design
As in the global case, we can now expand the state feedback results by allowing
the controller to be dynamic. As with the local asymptotic stability and state feedback
controller design problems, we will provide an expanding D approach as well as an expanding
interior approach to design a controller to demonstrate the largest estimate of the region of
attraction.
Following the global case, define the same system to be controlled (3.14)
x = f(x) + g(x)u
y = h(x)
for x ∈ Rn with f ∈ Rnn, f(0) = 0, u ∈ Rm, g ∈ Rn×m
n with y ∈ Rp, h ∈ Rpn and h(0) = 0.
u is generated by an unknown nξ-state dynamic output feedback controller (3.15)
ξ = A(ξ) + B(ξ)y
u = C(ξ) + D(ξ)y
for ξ ∈ Rnξ with A ∈ Rnξnξ , A(0) = 0, B ∈ Rnξ×p
nξ , C ∈ Rmnξ
, C(0) = 0 and D ∈ Rm×pnξ . With
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101
this controller structure, the closed loop system becomes (3.16)
x = f(x) + g(x)(C(ξ) + D(ξ)h(x))
ξ = A(ξ) + B(ξ)h(x).
By the assumptions on f, h, A and C, we know that the combined system has a fixed point
at [x; ξ] = [0; 0].
From this point we can adapt Algorithms 7 and 8 to solve for a stabilizing output
feedback controller. Again, as in the global output feedback case, when the systems are
linear and the Lyapunov function is quadratic we can not use the “feedback trick” to solve
a single LMI for a feasible candidate controller and Lyapunov function pair.
4.5.1 Expanding D Algorithm for Output Feedback Design
Following the development of the state feedback expanding D algorithm we can
develop a output feedback controller design algorithm. First we fix the number of states in
the output feedback controller, nξ ∈ Z+, restrict the Lyapunov function V ∈ Rn+nξwith
V (0) = 0 and define D with a semi-algebraic set
D := [x; ξ] ∈ Rn+nξ | p([x; ξ]) ≤ β
with p ∈ Σn+nξ, positive definite, and β ≥ 0.
The requirements of Theorem 10 for asymptotic stability of the closed loop system
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102
(3.16) become
[x; ξ] ∈ Rn+nξ | p([x; ξ]) ≤ β \ 0 ⊆ [x; ξ] ∈ Rn+nξ |V ([x; ξ]) > 0
[x; ξ] ∈ Rn+nξ | p([x; ξ]) ≤ β \ 0 ⊆[x; ξ] ∈ Rn+nξ
∣∣∣∣∣∣∣∣∇V ([x; ξ])∗
f(x) + g(x)(C(ξ) + D(ξ)h(x))
A(ξ) + B(ξ)h(x)
< 0
The system is asymptotically stable about the fixed point [x; ξ] = 0, if any V,A, B, C,D
can be found to satisfy these requirements for a fixed p and any β > 0.
Following Algorithm 5, the largest estimate of the region of attraction that we
can demonstrate is the largest level set of V that is contained in D. Using the same
manipulations as in the state feedback case we get that maximizing the size of the region
D under the set containments above is equivalent to
maxV ∈ Rn+nξ
, V (0) = 0, si ∈ Σn+nξ
A ∈ Rnξnξ
, A(0) = 0, B ∈ Rnξ×pnξ
C ∈ Rmnξ
, C(0) = 0, D ∈ Rm×pnξ
β
s.t.
−(β − p)s2 + V s3 + V (β − p)s4 − l1 ∈ Σn+nξ
−(β − p)s6 −∇V ∗
f + gC + gDh
A + Bh
s7
−∇V ∗
f + gC + gDh
A + Bh
(β − p)s8 − l2 ∈ Σn+nξ
(4.21)
Since the SOS conditions are not linear in the decision variables, we will have to solve the
optimization iteratively.
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103
Algorithm 9 (Expanding D Output Feedback Design) An iterative search to expand
the region D in Theorem 10 starting from a candidate Lyapunov function and a candidate
output feedback controller with nξ states.
Let i be the iteration index and set i = 1. Denote the candidate Lyapunov function
V (i=0) and the candidate controller A(i=0), B(i=0), C(i=0), D(i=0). Pick the maximum degree
of the Lyapunov function, the controller, the SOS multipliers and the l polynomials to be
dV , dA, dB, dC , dD, ds2 , ds3 , ds4 , ds6 , ds7 , ds8 and dl1 , dl2 respectively. Pick the maximum
degrees for the the level set maximization problem (4.2) and denote them ds2 , ds3 and ds4.
Fix lk(x) = ε(∑n
j=1 xdlkj +
∑nξ
j=1 ξdlkj
)for k = 1, 2 and some ε > 0. Additionally set
β(i=0) = 0.
1. SOS Weight Iteration:
Set V = V (i−1), A = A(i−1), B = B(i−1), C = C(i−1) and D = D(i−1). Solve the
linesearch on β where s2 ∈ Σn+nξ,ds2, s3 ∈ Σn+nξ,ds3
, s4 ∈ Σn+nξ,ds4, s6 ∈ Σn+nξ,ds6
,
s7 ∈ Σn+nξ,ds7and s8 ∈ Σn+nξ,ds8
maxs2,s3,s4,s6,s4,s8
β
s.t.
−(β − p)s2 + V s3 + V (β − p)s4 − l1 ∈ Σn+nξ
−(β − p)s6 −∇V ∗
f + gC + gDh
A + Bh
s7
−∇V ∗
f + gC + gDh
A + Bh
(β − p)s8 − l2 ∈ Σn+nξ
Set s(j)7 = s7 and s
(j)8 = s8. Continue to step 2.
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104
2. Controller Iteration:
Set V = V (i−1), s7 = s(i)7 and s8 = s
(i)8 . Solve the linesearch on β where s2 ∈ Σn+nξ,ds2
,
s3 ∈ Σn+nξ,ds3, s4 ∈ Σn+nξ,ds4
, s6 ∈ Σn+nξ,ds6, A ∈ Rnξ
nξ,dA, A(0) = 0, B ∈ Rnξ×p
nξ,dB,
C ∈ Rmnξ,dC
with C(0) = 0 and D ∈ Rm×pnξ,dD
maxA, B, C, D
s2, s3, s4, s6
β
s.t.
−(β − p)s2 + V s3 + V (β − p)s4 − l1 ∈ Σn+nξ
−(β − p)s6 −∇V ∗
f + gC + gDh
A + Bh
s7
−∇V ∗
f + gC + gDh
A + Bh
(β − p)s8 − l2 ∈ Σn+nξ
Set A(i) = A, B(i) = B, C(i) = C, D(i) = D, s(i)3 = s3 and s
(i)4 = s4. Continue to step
3.
3. Lyapunov function iteration
Set A = A(i), B = B(i), C = C(i), D = D(i), s3 = s(i)3 , s4 = s
(i)4 , s7 = s
(i)7
and s8 = s(i)8 . Solve the linesearch on β where s2 ∈ Σn+nξ,ds2
, s6 ∈ Σn+nξ,ds6and
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V ∈ Rn+nξ,dVwith V (0) = 0
maxV,s2,s6
β
s.t.
−(β − p)s2 + V s3 + V (β − p)s4 − l1 ∈ Σn+nξ
−(β − p)s6 −∇V ∗
f + gC + gDh
A + Bh
s7
−∇V ∗
f + gC + gDh
A + Bh
(β − p)s8 − l2 ∈ Σn+nξ
Set β(i) = β and V (i) = V . Continue to step 4.
4. If β(i)− β(i−1) is less than a specified tolerance goto step 5, else increment i and goto
step 1.
5. Set V = V (i) and β = β(i). Solve the linesearch on c where s2 ∈ Σn+nξ,ds2, s3 ∈
Σn+nξ,ds3and s4 ∈ Σn+nξ,ds4
maxs2,s3,s4
c
s.t.
−s2(c− V )− s3(p− β)− s4(p− β)(c− V )− (p− β)2 ∈ Σn
Set cmax = c. The set [x; ξ] ∈ Rn+nξ |V ([x; ξ]) ≤ cmax is the resulting estimate of
the region of attraction, for it is positively invariant, contained within D and all of
its points converge to the fixed point x = 0.
Remark 8 (Properties of the expanding D output feedback algorithm) :
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106
• As with the stability and state feedback versions, if the algorithm is started from a
feasible point it will always remain feasible. If the system’s linearization is controllable,
then a linear output feedback controller that stabilizes the linearized system and the
resulting quadratic Lyapunov function will always work, since they will stabilize the
nonlinear system near to the origin. Again, this allows us to not consider separate
algorithms starting from the candidate controller and from the candidate Lyapunov
function.
• As with the previous versions, V need not be positive definite, since it is required to be
positive only on D \0. However to be positive over a region about the origin V must
have no linear terms and dV should be even. Clearly if V were chosen to be positive
definite, then the first constraint in (4.21) can just become V − l1 ∈ Σn+nξ.
• Since the s’s, l’s and the s’s are SOS, they must be of even degree. Additionally, the
degrees need to be chosen so that following relations hold:
For the first SOS constraint
maxdeg(ps2),deg(V s3) ≥ dl1
maxdeg(ps2),deg(V s3) ≥ deg(V ps4)
For the second SOS constraint
deg(ps6) ≥ dl2
deg(ps6) ≥ deg
∇V ∗
f + gC + gDh
A + Bh
s7
deg(ps6) ≥ deg
∇V ∗
f + gC + gDh
A + Bh
ps8
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For the c maximization constraint
maxdeg(s2V ),deg(s4pV ) ≥ deg(p2)
maxdeg(s2V ),deg(s4pV ) ≥ deg(s3p)
4.5.2 Expanding Interior Algorithm for Output Feedback Design
We now adapt the expanding interior algorithm proposed in [13] to search for
dynamic output feedback controllers to stabilize (3.14). First we fix the dynamic controller’s
number of states, nξ ∈ Z+, and we define a variable sized region
Pβ := [x; ξ] ∈ Rn+nξ | p([x; ξ]) ≤ β
for p ∈ Σn+nξand positive definite. We then define
D := [x; ξ] ∈ Rn+nξ |V ([x; ξ]) ≤ 1
for some as of yet unknown candidate Lyapunov function. Following the developments for
the state feedback case
[x; ξ] ∈ Rn+nξ | p([x; ξ]) ≤ β ⊆ [x; ξ] ∈ Rn+nξ |V ([x; ξ]) ≤ 1
[x; ξ] ∈ Rn+nξ |V ([x; ξ]) ≤ 1 \ 0 ⊆[x; ξ] ∈ Rn+nξ
∣∣∣∣∣∣∣∣∇V ∗([x; ξ])
f(x) + g(x)(C(ξ) + D(ξ)h(x))
A(ξ) + B(ξ)h(x)
< 0
[x; ξ] ∈ Rn+nξ |V ([x; ξ]) ≤ 0, [x; ξ] 6= 0 = φ
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with V (0) = 0. Via the Positivstellensatz, maximizing the size of Pβ subject to the con-
straints above becomes
maxV ∈ Rn+nξ
, V (0) = 0, si ∈ Σn+nξ
A ∈ Rnξnξ
, A(0) = 0, B ∈ Rnξ×pnξ
C ∈ Rmnξ
, C(0) = 0, D ∈ Rm×pnξ
β
s.t.
V − l1 ∈ Σn+nξ
−(
(β − p)s6 + (V − 1))∈ Σn+nξ
−
(1− V )s8 +∇V ∗
f + gC + gDh
A + Bh
s9 + l2
∈ Σn+nξ
(4.22)
We now can propose an algorithm to prove that the fixed point [x; ξ] = 0 is asymp-
totically stable and estimate its region of attraction by iteratively solving (4.22). Again,
for the controller synthesis part of the iteration we will have to introduce the intermediate
value α.
Algorithm 10 (Expanding Interior Output Feedback Design) An iterative search
to expand the region Pβ and thus the region D := x ∈ Rn|V (x) ≤ 1 in Theorem 10 start-
ing from a positive definite candidate Lyapunov function and a nξ state candidate dynamic
output feedback controller.
Let i be the iteration index and set i = 1. Denote the candidate Lyapunov function
V (i=0) and the candidate output feedback controller A(i=0), B(i=0), C(i=0), D(i=0). Pick the
maximum degree of the Lyapunov function, the controller, the SOS multipliers and the l
polynomials to be dV , dA, dB, dC , dD, ds6 , ds8 , ds9 and dl1 , dl2 respectively. Fix lk([x; ξ]) =
ε(∑n
j=1 xdlkj +
∑nξ
j=1 ξdlkj
)for k = 1, 2 and some ε > 0. Additionally set l
(i=0)2 = l2 and
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β(i=0) = 0.
1. SOS Weight Iteration:
Set V = V (i−1), A = A(i−1), B = B(i−1), C = C(i−1), D = D(i−1) and l2 = l(i−1)2 .
Solve the linesearch on α where s8 ∈ Σn+nξ,ds8and s9 ∈ Σn+nξ,ds9
maxs8,s9
α
s.t.
−
(α− V )s8 +∇V ∗
f + gC + gDh
A + Bh
s9 + l2
∈ Σn+nξ
Set s(i)9 = s9. Continue to step 2.
2. Controller Iteration:
Set V = V (i−1), l2 = l(i−1)2 and s9 = s
(i)9 . Solve the linesearch on α where s8 ∈ Σn,dA8
,
A ∈ Rnξ
nξ,dA, A(0) = 0, B ∈ Rnξ×p
nξ,dB, C ∈ Rm
nξ,dC, C(0) = 0 and D ∈ Rm×p
nξ,dD
maxA,B,C,D,s8
α
s.t.
−
(α− V )s8 +∇V ∗
f + gC + gDh
A + Bh
s9 + l2
∈ Σn+nξ
Set A(i) = A, B(i) = B, C(i) = C, D(i) = D, s(i)8 = s8 and l
(i)2 = l
(i−1)2 /α. Continue
to step 3.
3. Lyapunov function iteration:
Set A = A(i), B = B(i), C = C(i), D = D(i), s8 = s(i)8 , s9 = s
(i)9 and l2 = l
(i)2 . Solve
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110
the linesearch on β where s6 ∈ Σn,ds6and V ∈ Rn+nξ,dV
with V (0) = 0
maxV,s6
β
s.t.
V − l1 ∈ Σn+nξ
−(
(β − p)s6 + (V − 1))∈ Σn+nξ
−
(1− V )s8 +∇V ∗
f + gC + gDh
A + Bh
s9 + l2
∈ Σn+nξ
Set V (i) = V and β(i) = β. Continue to step 4.
4. If β(i)−β(i−1) is less than a specified tolerance, the set x ∈ Rn|V (i)(x) ≤ 1 contains
Pβ(i) and is the largest estimate of the fixed point’s region of attraction, else increment
i and goto step 1.
Remark 9 (Properties of the expanding interior algorithm) :
• If the algorithm is started from a feasible point it will always remain feasible. If
the system’s linearization is controllable, then any linear controller that stabilizes the
linearized system and the corresponding quadratic Lyapunov function will always work,
since they will stabilize the nonlinear system near to the origin. Once more, this
feasibility property allows us to consider only a V variant of the algorithm.
• Note that, unlike Algorithm 5, V must be positive definite, so it should have no linear
terms and dV should be even.
• Since the s’s, and the l’s are SOS, they must be of even degree. Additionally, the
degrees need to be chosen so that following relations hold:
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111
For the first SOS constraint
dV = dl1
For the second SOS constraint
deg(ps6) ≥ dV
For the third SOS constraint
deg(V s8) ≥ deg
∇V ∗
f + gC + gDh
A + Bh
s9
deg(V s8) = dl2
4.5.3 Output Feedback Design Example
We can demonstrate Algorithms 9 and 10 with the double nonlinear spring-mass-
damper system from §4.4.3 if we introduce an output function, h. We will consider two
output functions
h1(x) =
x3
x4
and
h2(x) = x4
Both of these output functions provide measurements from the second mass, m2, that is
not directly actuated by u. In the example for semi-global exponential stability output
feedback algorithms, we had to use h(x) = [x1;x2], which makes the observations and
actuation colocated, to make the algorithms feasible.
Using the system dynamics given by (4.20) with either h1 or h2 above we can find
a linear controller and quadratic Lyapunov function candidate pair by employing the linear
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112
gap metric design steps shown in §3.5.2.
In order to keep the size of the computations down, as well as, pose a challenging
control problem, we will fix nξ = 1. For both algorithms and also both output functions we
pick
p([x; ξ]) = x11 + x2
2 + x23 + x2
4 + ξ21
We can now solve for an output feedback controller for (4.20) with each of the
output functions. For Algorithm 9 set dV = 2, dA = dC = 1, dB = dD = 0, ds2 = ds3 = 2,
ds4 = 0, ds6 = 4, ds7 = 2, ds8 = 0, dl1 = dl2 = 4, with ds2 = ds3 = 2 and ds4 = 0. For
Algorithm 10 we use dV = 2, dA = dC = 1, dB = dD = 0, ds6 = ds8 = 2, ds9 = 0, with
dl1 = 2 and dl2 = 4.
In this set up Algorithm 9 will involve searches for SOS polynomials in 5 variables
of degree 4, so we will reduce the computational load by setting the tolerance to 0.1. While,
since Algorithm 10 has SOS weights in 5 variables of degree 2 we set the tolerance to 0.01.
Using h1
After 11 iterations Algorithm 9 converges to β = 11.9 with cmax = 25.9, while
Algorithm 10 reaches the β tolerance limit after 4 iterations at β = 0.22.
For clarity, let the controller and Lyapunov function found by the expanding D
algorithm be Ad, Bd, Cd, Dd and Vd. Likewise, let the results from the expanding interior
algorithm be Ai, Bi, Ci, Di and Vi. To summarize the results, under the linear controller Ad Bd
Cd Dd
=
−1.50 −0.22 −1.55
3.36 1.37 2.20
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113
the system (4.20) is asymptotically stable and the region Ωd := x ∈ Rn|Vd(x) ≤ cmax
with
Vd([x; ξ]) =
x1
x2
x3
x4
ξ1
∗
59.64 −8.65 −60.76 −31.19 −70.42
−8.65 38.02 28.73 34.70 65.70
−60.76 28.73 81.82 59.34 112.81
−31.19 34.70 59.34 67.29 104.82
−70.42 65.70 112.81 104.82 194.25
x1
x2
x3
x4
ξ1
is an invariant set contained in the fixed point’s region of attraction. Additionally under
the controller Ai Bi
Ci Di
=
−0.88 0.70 −1.19
0.99 0.10 1.06
the system (4.20) is asymptotically stable and the region Ωi := x ∈ Rn|Vi(x) ≤ 1 with
Vi(x) =1
100
x1
x2
x3
x4
ξ1
∗
1.27 −0.03 −0.70 −0.09 −0.41
−0.03 1.27 −0.03 1.31 0.90
−0.70 −0.03 0.70 0.19 0.21
−0.09 1.31 0.19 2.86 1.48
−0.41 0.90 0.21 1.48 1.03
x1
x2
x3
x4
ξ1
is an invariant set contained in the fixed point’s region of attraction.
Comparing the two controllers using h1, we find the volume of Ωd is 13.06, while
Ωi has a volume of 22.34 with no containment of Ωd. From this information, it would seem
that the controller (Ai, Bi, Ci, Di) would be preferable, however this comparison is based
on the size of the entire region including the contribution from the controller’s state. In
most control implementations, we would pick the initial condition ξ1 = 0, and if we look
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114
at the sizes of the invariant regions, with this restriction on ξ1’s initial condition we find
something very interesting. The volume of Ωd with ξ1 = 0 is 6.65, while the volume of Ωi
with ξ1 = 0 is only 5.80. This reversal in volumes shows that as long as we are starting the
controller with initial condition ξ1 = 0, we should use the controller (Ad, Bd, Cd, Dd).
Using h2
After 4 iterations Algorithm 9 converges to β = 1.1 with cmax = 0.3, while Algo-
rithm 10 reaches the β tolerance limit after 4 iterations at β = 0.08.
For clarity, let the controller and Lyapunov function found by the expanding D
algorithm be Ad, Bd, Cd, Dd and Vd. Likewise, let the results from the expanding interior
algorithm be Ai, Bi, Ci, Di and Vi. To summarize the results, under the linear controller Ad Bd
Cd Dd
=
−0.66 1.13
−1.09 0.60
the system (4.20) is asymptotically stable and the region Ωd := x ∈ Rn|Vd(x) ≤ cmax
with
Vd([x; ξ]) =
x1
x2
x3
x4
ξ1
∗
4.88 −0.23 −1.85 −0.58 2.61
−0.23 3.05 0.35 1.33 −1.29
−1.85 0.35 1.37 0.50 −1.17
−0.58 1.33 0.50 1.75 −2.48
2.61 −1.29 −1.17 −2.48 2.46
x1
x2
x3
x4
ξ1
is an invariant set contained in the fixed point’s region of attraction. Additionally under
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115
the controller Ai Bi
Ci Di
=
−0.92 −2.23
0.82 0.88
the system (4.20) is asymptotically stable and the region Ωi := x ∈ Rn|Vi(x) ≤ 1 with
Vi(x) =
x1
x2
x3
x4
ξ1
∗
6.67 −0.31 −1.35 −0.73 −2.81
−0.31 5.63 0.70 4.72 2.36
−1.35 0.70 1.78 2.61 2.74
−0.73 4.72 2.61 5.16 2.70
−2.81 2.36 2.74 2.70 2.60
x1
x2
x3
x4
ξ1
is an invariant set contained in the fixed point’s region of attraction.
Comparing the two controllers using h2, we find the volume of Ωd is 0.11, while Ωi
has a volume of 1.58 with no containment of Ωd. Again this would seem to imply that the
controller (Ai, Bi, Ci, Di) would be preferable. When we look at setting ξ1 = 0 to reflect
the choice of controller initial condition, the volume of Ωd with ξ1 = 0 is 0.13, while the
volume of Ωi with ξ1 = 0 is still larger at 0.72 and also it contains the restricted version of
Ωd. Note that these volumes are now in R4 while the original volumes were in R5, so the
volume increase for Ωi is legitimate.
Since Ωi is larger in both the restricted and non-restricted cases, (Ai, Bi, Ci, Di) is
the preferred controller. Note though, that the volumes for the invariant regions using h2
are smaller than those achievable from h1, which in turn are smaller than those in the state
feedback case. Clearly, the more information that is available, the larger the set of initial
conditions for which the controller will asymptotically stabilize the system.
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116
4.6 Chapter Summary
In this chapter we looked at using SOS optimization to do local system analysis
and controller synthesis. We derived two algorithms, expanding D and expanding interior,
that are at the heart of our approach to both stability analysis and controller synthesis.
Additionally they provide ways to optimize the size of positively invariant regions about a
fixed point.
Also, we provided methods to analyze the effects of external disturbances on a
given system, by bounding the system’s unit energy reachable set set, finding the largest
peak disturbance for which a given set is invariant, and bounding the induced L2 → L2
gain from disturbance to output on an invariant set.
We then used the expanding D and expanding interior algorithms to design both
state feedback and output feedback controllers. Using these approaches we design state
feedback controllers for an example system and applied the disturbance analysis techniques
to study the controllers’ ability to reject disturbances. Additionally, we applied our algo-
rithms to design single state output feedback controllers for the same example system with
difficult output functions. As compared to the semi-global exponential stability designs,
these local algorithms can stabilize much more difficult systems, but the results only hold
on smaller regions of the state space.
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117
Chapter 5
Discrete Time Containment &
Stability
Up to this point all of the applications of SOS programming that we have con-
sidered have been for continuous time polynomial systems; now, we will look at similar
problems for discrete time polynomial systems. As opposed to the continuous time case, in
discrete time we can easily make arguments about set invariance without using Lyapunov
functions. However, these set invariance results do not provide any information about the
system’s stability, so we will use a Lyapunov function approach to prove stability. Unfortu-
nately, due to the way the discrete time Lyapunov approach proves stability, we will not be
able to use SOS programming to design controllers for discrete time polynomial systems.
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118
5.1 Set Invariance for Discrete Time Systems
Consider the system
xk+1 = f(xk) (5.1)
with f ∈ Rnn and f(0) = 0. We would like to know if a region of the state space X ⊂ Rn is
invariant under the action of f . We know that X is invariant under f if x ∈ X ⇒ f(x) ∈ X.
If we define X with a semialgebraic set
X := x ∈ Rn|p(x) ≤ β
we can check invariance with the following set containment constraint
x ∈ Rn | p(x) ≤ β ⊆ x ∈ Rn | p(f(x)) ≤ β (5.2)
We can check this constraint with a generalized S-procedure from §2.2.1, which asks if there
exists s ∈ Σn such that
(β − p f)− s(β − p) ∈ Σn (5.3)
To have a chance at making this constraint feasible we will need to pick s such that
deg(sp) ≥ deg(p f)
which insures that the positive term sp has the highest degree. Additionally, we can look
to find the largest invariant region by adding a cost function of β to the feasibility problem
(5.3).
5.1.1 Set Invariance Example
Consider the system
xk+1 = −x4k + x2
k
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119
The system has fixed points at x = 0,−1.325, and its linearization about the origin gives
no information about the system. Fixing p(x) = x2, we would like to find the largest value
of β such that the constraint (5.3) holds. With ds = 6, making both terms in (5.3) of degree
8, we find the largest value of β to be 1.75, which implies that the set
x ∈ R| − 1.322 ≤ x ≤ 1.322
is invariant under the system’s dynamics. Since this region pushes up against the fixed
point, we know that we have indeed expanded the set x ∈ R|p(x) ≤ β as much as is
possible.
5.1.2 Set Invariance under Disturbances
The check above is useful to determine if a region is invariant under a system’s
dynamics, however, it is often more useful to know if the region remains invariant when
a disturbance is introduced into the system. For example, in the receding horizon control
strategy put forward in [14], one of the requirements for it to work is that there must exist
region of state space that is invariant under the action of the dynamics and some set of
disturbances.
We can expand the previous invariance check to include external disturbances by
checking if the system
xk+1 = f(xk, wk) (5.4)
with wk ∈ W ⊂ Rm, f ∈ Rnn+m and f(0, 0) = 0 satisfies
x ∈ X, w ∈ W ⇒ f(x,w) ∈ X
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120
If we describe W with a semialgebraic set
W := w ∈ Rm|q(w) ≤ γ
then we can write an analog of (5.2)
x ∈ Rn, w ∈ Rm|p(x) ≤ β ∩ x ∈ Rn, w ∈ Rm|q(w) ≤ γ
⊆ x ∈ Rn, w ∈ Rm|p(f(x,w)) ≤ β
which, again, can be checked with a generalized S-procedure, from §2.2.1,
∃s1, s2 ∈ Σn+m
s.t.
(β − p f)− s1(β − p)− s2(γ − q) ∈ Σn+m
(5.5)
We can also add a cost function to (5.5) to find the maximum allowable disturbance
(max γ), the smallest invariant set (minβ), or any other combination of these objectives.
However, for feasibility we will need the highest degree SOS term to be positive or
max deg(s1p),deg(s2q) ≥ deg(p f)
This inequality presents us again with potentially very large optimization problems, since
s1, s2 have potentially large degree and are now polynomials in both x and w.
5.1.3 Set Invariance under Disturbances Example
Consider the previous containment example, except now subject it to a disturbance
xk+1 = −x4k + x2
k + wk
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Fixing p(x) = x2, and q(w) = w2, we will fix values of β ∈ [0, 1.75] and for each of these
values we will search for the largest value of γ such that the constraint (5.5) holds.
Additionally, since w enters the dynamics linearly, we can solve for the exact value
of γ for a given β. Noting that x2 ≤ β and w2 ≤ γ are equivalent to |x| ≤√
β and |w| ≤ √γ,
respectively, we know that for |x| ≤√
β the invariance relation becomes
−√
β +√
γ ≤ −x4 + x2 ≤√
β −√γ
Thus we can solve for the exact value of γ with
γ(β) =(
max(√
β − max|x|≤
√β
∣∣−x4 + x2∣∣ , 0))2
We can now compare the SOS programming approach to the exact results. Figure
5.1 gives the exact results with a solid line, while the largest values of γ using the optimiza-
tion approach of (5.5), with ds1 = ds2 = 6, are plotted as stars. From the plot, we can tell
that the SOS approach provides a very good lower bound and that we should pick β ≈ 1.2
to maximize the magnitude of the disturbance that the system can reject while remaining
in its original invariant set.
5.2 Discrete Time Stability Background
Remembering that the system under consideration is (5.1)
xk+1 = f(xk)
with f ∈ Rnn and f(0) = 0, we can form a Lyapunov argument for asymptotic stability
with the following theorem. The discrete time version of the asymptotic stability theorem
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
β
γ
Figure 5.1: The maximal γ for which (5.5) is feasible for given β are plotted as stars, whilethe solid line shows the exact results.
is identical to the continuous time one, Theorem 10, except that the conditions on V have
been replaced with conditions on (V f − V ).
Theorem 11 Let D ⊂ Rn be a domain containing the equilibrium point x = 0 of the system
(5.1). Let V : D → R be a continuously differentiable function such that
V (0) = 0
V (x) > 0 on D \ 0
and
V (f(x))− V (x) < 0 on D \ 0
then the system (5.1) is asymptotically stable about x = 0. Moreover, any region Ωβ :=
x ∈ Rn|V (x) ≤ β such that Ωβ ⊆ D describes an positively invariant region contained in
the equilibrium point’s domain of attraction.
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The proof of Theorem 11 is conceptually identical to the proof of Theorem 10,
with (V f − V ) standing in for V .
5.3 Convex Stability Tests
Since the requirements for asymptotic stability are almost identical in the discrete
and continuous time setups, we can very easily adapt Algorithms 5 and 6 for the system
(5.1).
5.3.1 Expanding D Algorithm
Following §4.2.1 we look to satisfy the assumptions of Theorem 11 by defining
D := x ∈ Rn|p(x) ≤ β
with p ∈ Σn, positive definite, and searching for V ∈ Rn with V (0) = 0 such that
x ∈ Rn|p(x) ≤ β \ 0 ⊆ x ∈ Rn|V (f(x))− V (x) < 0
x ∈ Rn|p(x) ≤ β \ 0 ⊆ x ∈ Rn|V (x) > 0
If we can these conditions for a fixed p and any value of β > 0, then the system (5.1) is
asymptotically stable about the fixed point x = 0.
However, as in the continuous time case, the largest invariant set we can demon-
strate that converges to the origin is the largest level set of V that is contained in D, which
we will search for after expanding D as much as possible. We can follow the reasoning of
§4.2.1 and propose discrete time version of the expanding D algorithm
Algorithm 11 (Expanding D) An iterative search to expand the region D in Theorem
11 starting from a candidate Lyapunov function.
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Let i be the iteration number starting at one. Denote the candidate Lyapunov
function V (i=0) and pick the maximum degree of the Lyapunov function, the SOS multipliers
and the l polynomials to be dV , ds2 , ds3 , ds4 , ds6 , ds7 , ds8 and dl1 , dl2 respectively. Pick the
maximum degrees for the the level set maximization problem (4.2) and denote them ds2 , ds3
and ds4. Fix li(x) = ε∑n
j=1 xdlij for i = 1, 2 and some ε > 0. Additionally set β(i=0) = 0.
1. Set V = V (i−1) and solve the linesearch on β where s2 ∈ Σn,ds2, s3 ∈ Σn,ds3
, s4 ∈
Σn,ds4, s6 ∈ Σn,ds6
, s7 ∈ Σn,ds7and s8 ∈ Σn,ds8
maxs2,s3,s4,s6,s7,s8
β
s.t.
−(β − p)s2 + V s3 + V (β − p)s4 − l1 ∈ Σn
−(β − p)s6 − (V f − V )s7 − (V f − V )(β − p)s8 − l2 ∈ Σn
(5.6)
Set s(i)3 = s3, s
(i)4 = s4, s
(i)7 = s7 and s
(i)8 = s8. Continue to step 2.
2. Set s3 = s(i)3 , s4 = s
(i)4 , s7 = s
(i)7 and s8 = s
(i)8 . Solve the linesearch on β where
V ∈ Rn,dVwith V (0) = 0, s2 ∈ Σn,ds2
and s6 ∈ Σn,ds6
maxV,s2,s6
β
s.t.
−(β − p)s2 + V s3 + V (β − p)s4 − l1 ∈ Σn
−(β − p)s6 − (V f − V )s7 − (V f − V )(β − p)s8 − l2 ∈ Σn
(5.7)
Set β(i) = β and V (i) = V . If β(i) − β(i−1) is less than a specified tolerance go to step
3, else increment i and go to step 1.
3. Set V = V (i) and β = β(i). Solve the linesearch on c where s2 ∈ Σn,ds2, s3 ∈ Σn,ds3
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and s4 ∈ Σn,ds4
maxs2,s3,s4
c
s.t.
−s2(c− V )− s3(p− β)− s4(p− β)(c− V )− (p− β)2 ∈ Σn
Set cmax = c. The set x ∈ Rn|V (x) ≤ cmax is the resulting estimate of the region
of attraction, for it is positively invariant, contained within D and all of its points
converge to the fixed point x = 0.
Remark 10 (Properties of the expanding D algorithm) :
• If the algorithm is started from a feasible point it will always remain feasible. If the
system’s linearization is stable, then the linearization’s quadratic Lyapunov function
will always work, since it will stabilize the nonlinear system near to the origin.
• Note that V need not be positive definite, since it is required to be positive only on
D \ 0. However, to be positive over a region about the origin V must have no linear
terms and dV should be even. Clearly if V were chosen to be positive definite, then
the first constraint in (5.6) and (5.7) can just become V − l1 ∈ Σn.
• Since the s’s, l’s and the s’s are SOS, they must be of even degree. Additionally, the
degrees need to be chosen so that following relations hold:
For the first SOS constraint
maxdeg(ps2),deg(V s3) ≥ dl1
maxdeg(ps2),deg(V s3) ≥ deg(V ps4)
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For the second SOS constraint
deg(ps6) ≥ dl2
deg(ps6) ≥ maxdeg((V f − V )s7),deg((V f − V )ps8)
For the c maximization constraint
maxdeg(s2V ),deg(s4pV ) ≥ deg(p2)
maxdeg(s2V ),deg(s4pV ) ≥ deg(s3p)
5.3.2 Expanding Interior Algorithm
We can also adapt Algorithm 6 for discrete time systems by following the steps
that we used to derive it in §4.2.2. Define a variable sized region
Pβ := x ∈ Rn|p(x) ≤ β
for p ∈ Σn, positive definite, and we estimate the region of attraction by maximizing β
subject to the constraint that all of the points in Pβ converge to the origin under the
system’s dynamics. Following Theorem 11, if we define
D := x ∈ Rn|V (x) ≤ 1
for an unknown candidate Lyapunov function, then Pβ must be contained D. Additionally
we need,
x ∈ Rn|V (x) ≤ 1 \ 0 ⊆ x ∈ Rn|V (f(x))− V (x) < 0
Since V is unknown, to ensure that V is positive on D \ 0 we require that V be positive
everywhere away from x = 0. With this set of requirements, we can propose the discrete
time version of the expanding interior algorithm.
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Algorithm 12 (Expanding Interior) An iterative search to expand the region Pβ and
thus the region D := x ∈ Rn|V (x) ≤ 1 in Theorem 10 starting from a positive definite
candidate Lyapunov function.
Let i be the iteration number starting at one. Denote the candidate Lyapunov
function V (i=0) and pick the maximum degree of the Lyapunov function, the SOS multipliers
and the l polynomials to be dV , ds6 , ds8 , ds9 and dl1 , dl2 respectively. Fix li(x) = ε∑n
j=1 xdlij
for i = 1, 2 and some ε > 0. Additionally set β(i=0) = 0.
1. Set V = V (i−1) and solve the linesearch on β where s6 ∈ Σn,ds6, s8 ∈ Σn,ds8
and
s9 ∈ Σn,ds9
maxs6,s8,s9
β
s.t.
−(
(β − p)s6 + (V − 1))∈ Σn
−(
(1− V )s8 + (V f − V )s9 + l2)∈ Σn
(5.8)
Set s(i)8 = s8 and s
(i)9 = s9. Continue to step 2.
2. Set s8 = s(i)8 and s9 = s
(i)9 . Solve the linesearch on β where V ∈ Rn,dV
with V (0) = 0
and s6 ∈ Σn,ds6
maxV,s6
β
s.t.
V − l1 ∈ Σn
−(
(β − p)s6 + (V − 1))∈ Σn
−(
(1− V )s8 + (V f − V )s9 + l2)∈ Σn
(5.9)
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Set β(i) = β and V (i) = V . If β(i) − β(i−1) is less than a specified tolerance goto step
3, else increment i and go to step 1.
3. The set x ∈ Rn|V (i)(x) ≤ 1 contains Pβ(i) and is the largest estimate of the fixed
point’s region of attraction.
Remark 11 (Properties of the expanding interior algorithm) :
• If the algorithm is started from a feasible point it will always remain feasible. If the
system’s linearization is stable, then the linearization’s quadratic Lyapunov function
will always work, since it will stabilize the nonlinear system near to the origin.
• Note that, unlike Algorithm 11, V must be positive definite, so it should have no linear
terms and dV should be even.
• Since the s’s, and the l’s are SOS, they must be of even degree. Additionally, the
degrees need to be chosen so that following relations hold:
For the first SOS constraint
dV = dl1
For the second SOS constraint
deg(ps6) ≥ dV
For the third SOS constraint
deg(V s8) ≥ deg((V f − V )s9)
deg(V s8) = dl2
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5.3.3 Estimating the Region of Attraction Example
To compare the effectiveness of Algorithm 11 (Expanding D) and Algorithm 12
(Expanding Interior) consider estimating the region of attraction for the following system,
a sampled version of the damped pendulum system (4.11),x1
x2
k+1
=
x1
x2
k
+ ts
x2
− 810x2 −
(x1 −
x316
)
k
(5.10)
where ts is the sampling time. As with the continuous time version, the system has three
equilibrium points: (0, 0) and (±√
6, 0) and the linearization about the first is stable while
the others are not. Following the continuous time version we pick
p(x) = x21 + x2
2
and we use the system’s linearization Af to construct a quadratic Lyapunov function V (x) =
x∗Px where
P 0
A∗fPAf − P ≺ 0
Solving this feasibility problem we find
P =
2.07 0.87
0.87 2.13
which we will use to form the candidate Lyapunov function.
If we set the stopping tolerances to β(i)−β(i−1) = .01, ts = 110 , and set the degrees
as below, we can compare Algorithms 11 and 12 with their continuous time counterparts,
Algorithms 5 and 6 . For the expanding D algorithm, set dV = 2, ds2 = ds3 = ds7 = 2,
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ds4 = ds8 = 0, ds6 = 6, dl1 = 4 and dl2 = 8. For the expanding interior algorithm set
dV = 2, ds8 = 4, ds6 = ds9 = 0, dl1 = 2 and dl2 = 6.
The expanding D algorithm converges to have a radius of√
β = 2.41 after 2
iterations, while the expanding interior algorithm converges so that Pβ has a radius of√
β =
2.05 after 12 iterations. These numbers are essentially the same as in the continuous time
case, which shows that the expanding regions are both as large as possible. Additionally the
invariant regions given by the Lyapunov function level sets that contain and are contained
by Pβ and D, respectively, are visually identical their continuous time analogs shown in
Figure 4.1.
Since the discrete time and continuous time results line up almost perfectly we can
see that the Lyapunov design techniques work in similar manners for both. In the continuous
time case, it was straight forward to extend the Lyapunov design algorithms to include
controller design, and we would like to extend this discrete/continuous stability parallel
to controller design. However, we can not extend these techniques to the discrete time
controller synthesis problem. Since we would be searching for K to make (V (f +gK)−V )
negative on some region, which is not linear in the coefficients of K even when V is fixed,
we would not have an SOS problem.
5.4 Chapter Summary
In this chapter we investigated using SOS optimization techniques to answer sys-
tem theoretic questions for discrete time systems with polynomial system maps. First, we
broke with the continuous time approach by studying set invariance without using a Lya-
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punov function. We illustrated a way to directly show that a given set is invariant under the
system’s dynamics as well as under a bounded disturbance. Simple examples were provided
for both approaches to demonstrate their utility. In the set invariance under disturbances
example we also found the exact solution to the given problem to illustrate the quality of
our approach.
We then extended the local stability algorithms, expanding D and expanding in-
terior, to discrete time systems. We used a sampled version of the continuous time local
stability example to show that the algorithms function in essentially the same manner. Ad-
ditionally, we showed the fundamental limitation of our Lyapunov based SOS optimization
approach when it is applied to discrete time systems, which keeps us from being able to
move from investigating a system’s stability to designing controllers for it.
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Chapter 6
Conclusions and Recommendations
This thesis considered the problem of using a Lyapunov based approach to an-
alyze the stability and performance of polynomial systems and to synthesize polynomial
state feedback and output feedback controllers. The polynomial Lyapunov approach was
made computationally tractable by invoking the Positivstellensatz to form suitable sufficient
conditions that could be iteratively solved with convex optimization.
In Chapter 3, we illustrated a iterative approach to finding Lyapunov functions and
controllers to prove that a closed loop polynomial system was semi-globally exponentially
stable. Additionally, we provided a Lyapunov base approach to bound the system’s induced
L2 gain from disturbance to output. However the global nature of these results somewhat
limits their application, since in many cases the system either has fixed points away from
the origin or is a model of an underlying system that is only valid on a fixed region of state
space.
In light of these restrictions, we developed two different algorithmic approaches
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to do localized analysis and controller synthesis in Chapter 4. The Expanding D approach
works to find the largest estimate of the system’s region of attraction by expanding the re-
gion where the Lyapunov conditions hold, while the Expanding Interior approach searches
for an invariant set that contains an expanding region. The algorithms both have strengths
and weaknesses and these are often complimentary. Also, we expanded our approach to
disturbance analysis by providing bounds on the reachable set under a unit energy dis-
turbance, proving set invariance under peak bounded disturbances and introducing a local
induced L2 gain bound.
We then extended the local continuous time stability results, using the two different
approaches, to discrete time systems in Chapter 5. Due to the nature of the Lyapunov
theorem in discrete time, we were unable to use similar methods to do controller design.
However, due to the discrete time set up we were able to find non-Lyapunov based conditions
to insure that a region of initial conditions was invariant under the system’s dynamics and
a given set of disturbances.
The results presented in this thesis leave several topics open for future research.
Two of them are discussed below
• Integrate Performance Measures into Controller Design:
In this thesis, we have constructed a series of algorithms that find Lyapunov functions
and controllers to demonstrate either global or local stability, as well as techniques
for analyzing a system’s performance under disturbances. However, we have not
integrated these performance measures into the controller design procedures.
If controller design and performance analysis were performed simultaneously, then it
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would be possible to use the design algorithm to optimize some aspect of the system’s
performance, instead of just checking the performance after the controller is designed.
With a joint approach to these problems we could provide a valuable tool that would
allow us to design robust controllers for polynomial systems with built-in performance
guarantees.
• Minimize the Number of Non-Zero Controller Terms
If we were to design a high degree polynomial controller, we would be confronted
with the problem of trying to implement a hugely complex polynomial with a large
number of terms. Are all these terms necessary, or can some of them be set to zero?
Clearly, we can run the design algorithm once including all of the monomials, after
which we can run the algorithm again with all of the monomials that have small
coefficients removed. In the global state feedback controller design example, this
technique worked, but this approach does not guarantee that the given controller has
the fewest number of non-zero terms. If we could find a way to setup the controller
design algorithm to minimize the number of non-zero controller terms, then we could,
possibly greatly, reduce the complexity of the controller implementation.
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Appendix A
Semidefinite Programming
Semidefinite programming (SDP) considers the following optimization problem
minx∈Rn
c∗x
s.t. F (x) := F0 + x1F1 + · · ·+ xnFn 0
with c ∈ Rn, and Fi = F ∗i ∈ Rp×p for i = 0, 1, . . . , n. The most important theoretical prop-
erty of SDP is its convexity, which allows it to be solved numerically with great efficiently
[33]. SDP has many other useful theoretical properties that are covered in detail in the
survey [35].
Often SDP is used to solve the feasibility problem: does there exist x ∈ Rn such
that F (x) 0? The generalized inequality F (x) 0 is linear, or strictly speaking affine,
in x, so the feasibility problem is often referred to as a linear matrix inequality (LMI). One
property of LMIs is that a set of them can be turned into single larger block diagonal LMI.
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Example 4 Consider the LMIs F (x) 0 and G(x) 0 they can be written asG(x)
F (x)
=
G0
F0
+ x1
G1
F1
+ · · ·+ xn
Gn
Fn
0
To illustrate a less obvious use of LMIs consider the following Lyapunov stability argument.
Example 5 Consider the linear system x = Ax with x ∈ Rn and the Lyapunov function
V (x) = x∗Px, with P unknown. If a P can be picked such that V (x) is positive definite and
−V (x) is positive semidefinite, then the system is stable. Noting that V (x) = x∗(A∗P +
PA)x, the problem can be posed as: does there exist a P such that
P 0
−(A∗P + PA) 0
If P is represented on the standard basis as∑m
i=1 piEi, with m := (n + 1)n/2, then the
LMIs above become∑m
i=1 piEi − εI
−(A∗∑m
i=1 piEi +∑m
i=1 piEiA)
0
for any ε > 0. The question of what value to use for ε can also be incorporated to give the
following SDP in the variables ε, p1, . . . , pm
maxp1,...,pm
ε
s.t. p1
E1
−(A∗E1 + E1A)
+ . . . + pm
Em
−(A∗Em + EmA)
+ ε
−I
0
0
which, if the maximum value for ε is strictly positive, gives a matrix P such that the Lya-
punov function V demonstrates the stability of A.