Understanding Equilibrium Points & Nonlinear Phenomena in Systems & Control, Slides of Nonlinear Control Systems

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Nonlinear Systems and Control

Lecture # 1

Introduction

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Nonlinear State Model

x

1

f

1

t, x

1

,... , x

n

, u

1

,... , u

p

x

2

f

2

t, x

1

,... , x

n

, u

1

,... , u

p

x

n

f

n

t, x

1

,... , x

n

, u

1

,... , u

p

x

i

denotes the derivative of

x

i

with respect to the time

variable

t

u

1

u

2

u

p

are input variables

x

1

x

2

x

n

the state variables

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x

f

t, x, u

y

h

t, x, u

x

is the state,

u

is the input

y

is the output (

q

-dimensional vector)

Special Cases: Linear systems:

x

A

t

x

B

t

u

y

C

t

x

D

t

u

Unforced state equation:

x

f

t, x

Results from

x

f

t, x, u

with

u

γ

t, x

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Autonomous System:

x

f

x

Time-Invariant System:

x

f

x, u

y

h

x, u

A time-invariant state model has a time-invariance propertywith respect to shifting the initial time from

t

0

to

t

0

a

provided the input waveform is applied from

t

0

a

rather

than

t

0

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A function

f

t, x

is locally Lipschitz in

x

on a domain

(open and connected set)

D

R

n

if it is locally Lipschitz at

every point

x

0

D

When

n

and

f

depends only on

x

f

y

f

x

y

x

L

On a plot of

f

x

versus

x

, a straight line joining any two

points of

f

x

cannot have a slope whose absolute value is

greater than

L

Any function

f

x

that has infinite slope at some point is

not locally Lipschitz at that point

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A discontinuous function is not locally Lipschitz at the pointsof discontinuityThe function

f

x

x

1

/

3

is not locally Lipschitz at

x

since

f

′

x

x

−

2

/

3

a

x

On the other hand, if

f

′

x

is continuous at a point

x

0

then

f

x

is locally Lipschitz at the same point because

continuity of

f

′

x

ensures that

f

′

x

is bounded by a

constant

k

in a neighborhood of

x

0

; which implies that

f

x

satisfies the Lipschitz condition

L

k

More generally, if for

t

J

R

and

x

in a domain

D

R

n

f

t, x

and its partial derivatives

∂f

i

/∂x

j

are

continuous, then

f

t, x

is locally Lipschitz in

x

on

D

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Example:

x

x

2

f

x

x

2

is locally Lipschitz for all

x

x

x

t

t

x

t

as

t

the solution has a

finite escape time

at

t

In general, if

f

t, x

is locally Lipschitz over a domain

D

and the solution of

x

f

t, x

has a finite escape time

t

e

then the solution

x

t

must leave every compact (closed

and bounded) subset of

D

as

t

t

e

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Global Existence and Uniqueness A function

f

t, x

is globally Lipschitz in

x

if

f

t, x

f

t, y

L

x

y

for all

x, y

R

n

with the same Lipschitz constant

L

If

f

t, x

and its partial derivatives

∂f

i

/∂x

j

are continuous

for all

x

R

n

, then

f

t, x

is globally Lipschitz in

x

if and

only if the partial derivatives

∂f

i

/∂x

j

are globally bounded,

uniformly in

t

f

x

x

2

is locally Lipschitz for all

x

but not globally

Lipschitz because

f

′

x

x

is not globally bounded

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Lemma:

Let

f

t, x

be piecewise continuous in

t

and

locally Lipschitz in

x

for all

t

t

0

and all

x

in a domain

D

R

n

. Let

W

be a compact subset of

D

, and suppose

that every solution of

x

f

t, x

x

t

0

x

0

with

x

0

W

lies entirely in

W

. Then, there is a unique

solution that is defined for all

t

t

0

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Example:

x

x

3

f

x

f

x

is locally Lipschitz on

R

, but not globally Lipschitz

because

f

′

x

x

2

is not globally bounded

If, at any instant of time,

x

t

is positive, the derivative

x

t

will be negative. Similarly, if

x

t

is negative, the derivative

x

t

will be positive

Therefore, starting from any initial condition

x

a

, the

solution cannot leave the compact set

x

R

x

a

Thus, the equation has a unique solution for all

t

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A linear system

x

Ax

can have an isolated equilibrium

point at

x

(if A is nonsingular) or a continuum of

equilibrium points in the null space of

A

(if

A

is singular)

It cannot have multiple isolated equilibrium points , for if

x

a

and

x

b

are two equilibrium points, then by linearity any point

on the line

αx

a

α

x

b

connecting

x

a

and

x

b

will be

an equilibrium pointA nonlinear state equation can have multiple isolatedequilibrium points .For example, the state equation

x

1

x

2

x

2

a

sin

x

1

bx

2

has equilibrium points at

x

1

nπ, x

2

for

n

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Linearization A common engineering practice in analyzing a nonlinearsystem is to linearize it about some nominal operating pointand analyze the resulting linear modelWhat are the limitations of linearization?

Since linearization is an approximation in theneighborhood of an operating point, it can only predictthe “local” behavior of the nonlinear system in thevicinity of that point. It cannot predict the “nonlocal” or“global” behavior There are “essentially nonlinear phenomena” that cantake place only in the presence of nonlinearity

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