29
Linearization and Review of Stability ME584 Fall 2010 Linear 1
Linearization and
Review of Stability
ME584
Fall 2010
Linear 1
Lecture Objectives
and Activities• Review of stability
• Importance of linearization
• Linearization of nonlinear systems
• Active learning activities
– Pair-share problems
Linear 2
Definition of StabilityLinear 3
A stable system is a system with a bounded response to
a bounded input
Response to a displacement/initial condition will produce
either a decreasing, neural, or increasing response.
Stability Analysis – 1st Order ODE
0,0
/0
:
:1
1001
001
ifunstableifstableisSystem
aaaa
equationsticCharacteri
ubxadt
dxaOrderst
Linear 4
3/
3026:
1,0;226:
t
o
o
exx
equationsticCharacteri
xuuxxExample
3026:
1,0;226:
equationsticCharacteri
xuuxxExampleo
3/t
oexx
Stability Analysis – 2nd Order ODE
stabilityofdegreestabilityRelative
onoscilliatiNodampedCritically
onoscilliatiNoOverdamped
onOscilliatidUnderdampe
responseStable
orandifunstableisSystem
j
a
a
a
aLet
EquationsticCharacteri
ubxadt
dxa
dt
xdaOrder
nn
nn
nn oo
nd
:
)(:1
)(:1
)(:1
0/
1
02
,2
,1
:
:2
21
2
2,1
22
12
2
0012
2
2
Linear 5
Stability Analysis – State Space (SS)
0det
0)(
,
AI
ifsolutiontrivialNon
xAI
AxxorAkeke
substitutekexLet
Axx
formatspaceState
tt
t
Linear 6
Stability Analysis with SS - Example
Linear 7
Importance of linearization
• Dynamics Analysis
• Control and estimation systems design
Linear 8
Importance of linearization
• Is nature linear or nonlinear? physical systems?
– Many physical systems behave linearly within some range of
variable, but become nonlinear as variables increase without
limit
– Possible to linearize nonlinear systems
• Example: Pendulum
• Tractable analysis with linear model
L
mgKm
)sin(
)sin(, smallFor
Linear 9
)/( LmgKm
Importance of linearization
xofestimateisxwhere
xGu
n•Estimatio
eperformancdesiredachievetoGChoose
xBGAGxBAxBuAxx
matrixcontroltheisGWhere
Gxu
uBmKlg
Awhere
BuAxx
xvariablesstate
lmgKm
ˆ
ˆ
)()(
,
Control•
;1
0;
//
10
][
:model space State•
)/(stable? system thisIs•
Linear 10
Linear Approximation
testingfromprovidedbealsocanuandx
uxfset
uandxforsolveTo
uofvariationoronperturbatismallu
xofvariationoronperturbatismallx
takenisionlinearizatwhichaboutuofvaluemequilibriuu
valuenalvalue/nomistatesteadycalledalso
takenisionlinearizatwhichaboutxofvaluemequilibriux
uuu
xxx
uxfdt
dxx
0),(
,
:*
:*
:
:
*
*
),(
Linear 11
Taylor’s Expansion
matricesJacobiancalledareBandA
u
fBand
x
fA
where
BuAxdt
dx
TOHuu
fx
x
fuxf
dt
dx
dt
dx
xxx
uuxx
uuxx
uuxx
uuxx
***
...**),(*
0
*
Linear 12
Linearization Procedure
***
.3
...
...
...
...
...
...
)()(.
.2
,...,,,,...,,,...,,
0,
.1
,,
2
,
1
,
2
,
2
2
,
1
2
,
1
,
2
1
,
1
1
,,
2
,
1
,
2
,
2
2
,
1
2
,
1
,
2
1
,
1
1
212121
BuAxdt
dxForm
Step
u
f
u
f
u
f
u
f
u
f
u
f
u
f
u
f
u
f
u
fBand
x
f
x
f
x
f
x
f
x
f
x
f
x
f
x
f
x
f
x
fA
mnisBandnnisAAssumeBandAforSolve
Step
fffformanduuuandxxxDefine
uandxforsolvetoxsetspecifiednotareuandxIf
Step
ux
m
n
ux
n
ux
n
ux
m
uxux
ux
m
uxux
uuxx
ux
n
n
ux
n
ux
n
ux
n
uxux
ux
n
uxux
uuxx
nmn
Linear 13
Example
12
2
2
1
2
1
21
sin
)(0,,
0)sin(
xL
gx
m
k
x
x
x
f
ff
inputcontrolnouxxlet
L
mgKm
Linear 14
Step 1
0sin
00
,0
122
221
2
1
xL
gx
m
kx
xxx
x
xxSet
1
x
Linear 15
0
0
0,
2,,0
2
1
1
x
xx
xconsiderusletexamplethisFor
Step 2
uxux
uxux
uxux
uxux
uxux
ux
uuxx
xL
gx
m
k
xx
f
xL
gx
m
k
xx
f
xxx
f
xxx
f
x
f
x
f
xx
f
x
f
x
fA
,12
2
,
2
2
,12
1
,
1
2
,2
2
,
2
1
,2
1
,
1
1
,
2
2
,
1
2
2
2
1
,
1
1
)sin(
)sin(
)(
)(
L
gx
L
gx 01 1
cos
m
k
Linear 16
12
2
2
1
sin xL
gx
m
k
x
f
f
1
0
Step 2 (continued)
0
uuxx
u
fB
Linear 17
Step 3
m
k
L
gassameThe
xm
kx
L
gx
xx
x
x
m
k
L
gx
x
dt
dx
***
**
*
*10
*
**
212
21
2
1
2
1
Linear 18
Pair-Share Exercise
3
2
1
2
3
21
/1000
/1000
/10
200
,
,0
1
mNk
mNk
smg
kgm
where
xkxkmguxm
uforceappliedthe
andmbyspringthecompressesmassthewhere
tpointheaboutsystemtheLinearize
u
Linear 19
Step 1
0
0
1
2
1
3
1
2
1
1
2
2
1
2
1
21
u
x
xx
xm
kx
m
kg
m
u
x
x
x
f
ff
xxandxxLet
Linear 20
Step 2
005.
0
020
10
03
10
,
2
,
1
2
1
21
,2
2
,1
2
,2
1
,1
1
ux
ux
uxux
uxux
u
f
u
f
B
xm
k
m
k
x
f
x
f
x
f
x
f
A
Linear 21
Step 3
*005.
0*
020
10* uxx
Linear 22
Pair-Share Example
A simple robot arm is modeled as
where I is moment of inertia of arm, m is the mass,
and T is the torque that the motor supplies.
We want the motor to hold the arm at five angles:
Find the torque required and determine what will
happen if something hits the arm and slightly alters its
position?
cosmgLTI
oooooo
e225,180,135,90,45,0
m
T
arm
θ
Linear 23
L
Step 1
0.707mgLu,x
mgLu,1x
0.707mgLu,135x
u,90x
0.707mgLu,45x
mgLu,0x
xmgLu
xandxandxxxu
x
xx
xI
mgL
I
u
x
x
x
f
ff
TuxxLet
o
1
o
1
o
1
o
o
1
o
1
e
225
80
0
cos
00set ,at torque thefind To
0
cos
,,
1
1
21
2
1
1
2
2
1
2
1
.
21
Linear 24
Step 2
Iu
f
u
f
B
xI
mgL
x
f
x
f
x
f
x
f
A
ux
ux
uxux
uxux
10
0sin
10
,
2
,
1
1
,2
2
,1
2
,2
1
,1
1
Linear 25
Step 3
I
uxx
I
mgLx
u
Ix
xx
I
mgLx
xx
112
2
1
1
2
1
)(sin
10
0
1sin
0
Linear 26
When Arm is HitLinear 27
12,1
1
2
111
12
sin
0)(sin
)(sin
,
xI
mgL
rootshas
xI
mgL
equationsticCharacteri
I
uxx
I
mgLx
soxx
)
/(707.
,225
)(0,0,180
)(707.
,135
)(,90
)(707.
,45
)(0,0,0
2,11
2,11
2,11
2,11
2,11
2,11
stableneutrally
undampedI
mgLjx
stableneutrallyx
unstableI
mgLx
unstableI
mgLx
unstableI
mgLx
stableneutrallyx
For
o
o
o
o
o
o
Lecture Recap
• Many nonlinear systems behave linearly
with small perturbation
• Linearization procedure
– Establish equilibrium
– Solve for A and B
• Analysis is tractable with linear models
• Next lecture: Stability analysis and
simulation with Matlab
Linear 28
References
• Woods, R. L., and Lawrence, K., Modeling
and Simulation of Dynamic Systems,
Prentice Hall, 1997.
• Palm, W. J., Modeling, Analysis, and
Control of Dynamic Systems
• Close, C. M., Frederick, D. H., Newell, J.
C., Modeling and Analysis of Dynamic
Systems, Third Edition, Wiley, 2002
Linear 29
