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
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