Lyapunov Based Analysis and Controller Synthesis for ...pack/library/WloszekPhDThesis.pdf · 1...

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

The dissertation of Zachary William Jarvis-Wloszek is approved:

Chair Date

Date

Date

University of California, Berkeley

Fall 2003

Lyapunov Based Analysis and Controller Synthesis for Polynomial

Systems using Sum-of-Squares Optimization

Copyright Fall 2003

by


1

Abstract

Lyapunov Based Analysis and Controller Synthesis for Polynomial Systems using

Sum-of-Squares Optimization

by


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

2

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

iii

To Sarah for her support and encouragement

iv

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

v

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

vi

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.

1

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

2

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.

3

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

4

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

5

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

6

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

7

• Local Asymptotic Stabilization: §4.4.3

4. Output Feedback Controller Design

• Semi-global Exponential Stabilization: §3.5.2

• Local Asymptotic Stabilization: §4.5.3

8

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

9

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.

10

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

11

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

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.

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

14

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

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

16

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.

17

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)

18

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+

19

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,

20

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.

21

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

22

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

23

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

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].

25

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].

26

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.

27

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)

28

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.

29

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.

30

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

31

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

32

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.

33

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

34

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

35

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.

36

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

37

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.

38

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.

39

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.

40

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.

41

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.

42

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

43

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

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)


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

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

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

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

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,

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

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

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.

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.

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.


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

54

V (0) = 0

maxV

γ

s.t.

V − l ∈ Σn+nξ

−

γV +∇V ∗

f + gC + gDh

A + Bh

∈ Σn+nξ

(3.20)


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

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].

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

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

58

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

59

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.

60

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].

61

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

62

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

63


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

64

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

65

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

66

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

67

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.


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 ∈

68

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.


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.

69

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)

70

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.

71

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)

72

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.


V (i=0) and pick the maximum degree of the Lyapunov function, the SOS multipliers and

73

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.


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


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. 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) :

74

• 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



dV = dl1


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

75

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.

76

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.

77

−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.

78

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

79

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

80

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

81

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].

82

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

83

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

84

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

85

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.

86

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

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.

88

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


β

s.t.

−(β − p)s2 + V s3 + V (β − p)s4 − l1 ∈ 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

,

89

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


Set K(i) = K, s(i)3 = s3 and s


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


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.


and s4 ∈ Σn,ds4

maxs2,s3,s4

c

s.t.


90




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.







deg(ps6) ≥ dl2

deg(ps6) ≥ max deg (∇V ∗(f + gK)s7) ,deg (∇V ∗(f + gK)ps8)

91




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

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.


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.

93

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


94

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.


• 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.





95


dV = dl1


deg(ps6) ≥ dV


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)

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

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

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

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

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

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

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.

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.


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.


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


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.

104


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

105

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.


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) :

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.







deg(ps6) ≥ dl2

deg(ps6) ≥ deg

∇V ∗

f + gC + gDh

A + Bh

s7

deg(ps6) ≥ deg

∇V ∗

f + gC + gDh

A + Bh

ps8

107




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

108

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.


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

109

β(i=0) = 0.


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.


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

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.


• 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.





111


dV = dl1


deg(ps6) ≥ dV


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

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.


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

113


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

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.


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


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

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.

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.

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.

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

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

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

121

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

122

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.

123

The proof of Theorem 11 is conceptually identical to the proof of Theorem 10,

with (V f − V ) standing in for V .


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.

124

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.


, s4 ∈

Σn,ds4, s6 ∈ Σn,ds6

, s7 ∈ Σn,ds7and s8 ∈ Σn,ds8


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


2. Set s3 = s(i)3 , s4 = s

(i)4 , s7 = s

(i)7 and s8 = s


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



125

and s4 ∈ Σn,ds4

maxs2,s3,s4

c

s.t.





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.






126


deg(ps6) ≥ dl2

deg(ps6) ≥ maxdeg((V f − V )s7),deg((V f − V )ps8)




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.

127

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.


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


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)

128

Set β(i) = β and V (i) = V . If β(i) − β(i−1) is less than a specified tolerance goto step


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.


• 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.






dV = dl1


deg(ps6) ≥ dV


deg(V s8) ≥ deg((V f − V )s9)

deg(V s8) = dl2

129

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,

130

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-

131

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.

132

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

133

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

134

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.

135

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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.

141

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.

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