State Variables

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13 Νοε 2013 (πριν από 3 χρόνια και 9 μήνες)

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State Variables

Outline

• State variables.

• State
-
space representation.

• Linear state
-
space equations.

• Nonlinear state
-
space equations.

• Linearization of state
-
space equations.

2

Input
-
output Description


The description is valid for

a) time
-
varying systems:
a
i

,
c
j

,
explicit functions of
time.

b) multi
-
input
-
multi
-
output (MIMO) systems:
l
input
-
output

differential equations,
l
= # of outputs.

c) nonlinear systems: differential equations include

nonlinear terms.

3

State Variables

To solve the differential equation we need

(1) The system input
u
(
t
)
for the period of interest.

(2) A set of constant initial conditions.


Minimal
set of initial conditions: incomplete

knowledge of the set prevents complete solution

but additional initial conditions are not needed to

obtain the solution.

• Initial conditions provide a summary of the

History
of the system up to the initial time
.


4

Definitions

System State
:
minimal set of numbers
{
x
i
(
t
),

i
= 1,2,...,
n
},
needed together with the input

u
(
t
),
t


[
t
0
,
t
f
)
to uniquely determine the

behavior of the system in the interval
[
t
0
,
t
f
].

n
= order of the system.

State Variables
:
As
t
increases, the state of

the system evolves and each of the

numbers
x
i
(
t
)
becomes a time variable.

State Vector
:
vector of state variables

5

Notation


Column vector
bolded



Row
vector
bolded and transposed
x
T
.


6

Definitions

State

Space
:

n
-
dimensional

vector

space

where

{
x
i
(
t
),

i

=

1
,
2
,
...
,
n
}

represent

the

coordinate

axes

State

plane
:

state

space

for

a

2
nd

order

system


Phase

plane
:

special

case

where

the

state

variables

are

proportional

to

the

derivatives

of

the

output
.


Phase variables
: state variables in phase
plane.
State
trajectories
: Curves in state
space


State portrait
: plot of state trajectories in the plane

(
phase portrait
for the phase plane).

7

Example 7.1

• State for equation of motion of a point

mass
m
driven by a force
f


y
= displacement of the point mass.




2


system is second order

8

Example 7.1 State Equations

State
variables



State
vector
2

Phase Variables:
2nd = derivative of the first.

Two first order differential equations



1
. First equation: from definitions of state
variables.

2. Second equation: from equation of motion.

9

Solution of State Equations

Solve the 1st order differential equations then substitute in


y
=
x
1

2 differential equations + algebraic expression are

equivalent to the 2nd order differential equation.

Feedback Control Law
2nd order
underdamped

system


u
/
m
= −
3
x

9
x

1. Solution depends only on initial conditions.

2. Obtain phase portrait using MATLAB command
lsim
,

3. Time is an implicit parameter.

4. Arrows indicate the direction of increasing time.

5. Choice of state variables is not unique.

10

Phase Portrait

11

State Equations


Set

of

first

order

equations

governing

the

state

variables

obtained

from

the

input
-
output

differential

equation

and

the

definitions

of

the

state

variables
.



In

general,

n

state

equations

for

a

n
th

order

system
.



The

form

of

the

state

equations

depends

on

the

nature

of

the

system

(equations

are

time
-
varying

for

time
-
varying

systems
,

nonlinear

for

nonlinear

systems,

etc
.
)



State

equations

for

linear

time
-
invariant

systems

can

also

be

obtained

from

their

transfer

functions
.

12

Output Equation



Algebraic

equation

expressing

the

output

in

terms

of

the

state

variables
.



Multi
-
output

systems
:

a

scalar

output

equation

is

needed

to

define

each

output
.



Substitute

from

solution

of

state

equation

to

obtain

output
.

13

State
-
space Representation



Representation

for

the

system

described

by

a

differential

equation

in

terms

of

state

and

output

equations
.




Linear

Systems
:

More

convenient

to

write

state

(output)

equations

as

a

single

matrix

equation

14

Example 7.2


The state
-
space equations for the system
of Ex.
7.1


15

General Form for Linear Systems

16

State Space in MATLAB

17

Linear Vs. Nonlinear State
-
Space

Example

7
.
3
:

The

following

are

examples

of

state
-
space

equations

for

linear

systems

a
)

3
rd

order

2
-
input
-
2
-
output

(MIMO)

LTI

18

Example 7.3 (b)

2
nd

order

2
-
output
-
1
-
input

(SIMO)

linear

time
-
varying

19

1. Zero direct
D,
constant
B
and
C
.

2. Time
-
varying system:
A
has entries that are functions of
t
.


Example 7.4: Nonlinear System

Obtain

a

state
-
space

representation

for

the

s
-
D
.
O
.
F
.

robotic

manipulator

from

the

equations

of

motion

with

output

q
.

20

Solution

order 2
s
(need 2
s
initial conditions to solve
completely. State
Variables

21

Example 7.5

Write

the

state
-
space

equations

for

the

2
-

D
.
O
.
F
.

anthropomorphic

manipulator
.

22

Equations of Motion

23

Solution

24

Nonlinear State
-
space Equations

f
(.) (
n
×
1)
and
g
(.) (
l
×
1)
= vectors of
functions
satisfying mathematical
conditions
to guarantee the existence and

uniqueness of solution.

affine linear in the control
: often encountered
in practice

(includes equations of robotic manipulators)

25

Linearization of State Equations



Approximate

nonlinear

state

equations

by

linear

state

equations

for

small

ranges

of

the

control

and

state

variables
.



The

linear

equations

are

based

on

the

first

order

approximation
.

26

x
0
constant,
Δ
x

=
x
-

x
0 =
perturbation
x
0
.

Approximation Error of order
Δ
2
x

Acceptable for small perturbations.

Function of
n
Variables

27

Nonlinear State
-
space Equations

28

Perturbations
Abt
’ Equilibrium
(
x
0
,
u
0
)

29

Output Equation

30

Linearized State
-
Space

Equations

31

Jacobians

(drop "
Δ
"
s)

32

Example 7.6

Motion of nonlinear spring
-
mass
-
damper.

y
=
displacement
f
=
applied force

m
=
mass of 1 Kg

b
(
y
) =
nonlinear damper constant

k
(
y
) =
nonlinear spring force.

Find the equilibrium position corresponding

to a force
f
0
in terms of the spring force,

then linearize the equation of motion about

this equilibrium.

33

Solution

Equilibrium

of

the

system

with

a

force

f
0

(set

all

the

time

derivatives

equal

to

zero

and

solve

for

y
)

Equilibrium

is

at

zero

velocity

and

the

position

y
0
.

34

Linearize about the equilibrium

• Entries of state matrix: constants whose

values depend on the equilibrium.

• Originally linear terms do not change with

linearization.

35