Structural Control:
Overview and Fundamentals
Akira Nishitani
Vice President & Professor
WASEDA University, Tokyo, Japan
anix@waseda.jp
Outline
1. Introduction for WASEDA and Myself
2. Introduction for Structural Control
3. Some keywords for structural control
4. Brief view of active structural control
5. Components of control system
6. Semiactive structural control
7. Smart damping or smart dampers
Continued
Outline (Cont’d)
8. Significance of nonlinearity or artificially-added
nonlinearity in structural control
9. Semiactive variable slip-force level dampers
10. Future directions
Appendix LQ control and LQG control
■ 1. Introduction for:
Waseda Univ. and myself
About Waseda
Waseda University since 1882
Waseda University since 1882
早稲田大学
Waseda University:
- Second oldest private university in
Japan, founded in 1882.
- 125th Anniversary in 2007.
- the first private university in Japan
that established engineering school.
- Waseda Department of Architecture
is the second oldest in Japan.
Data of Waseda University:
- Number of students: 50,000
- Number of students in School of
Science and Engineering: 7,000
- More than 100,000 application
forms submitted to the Admission
Center every year
About myself.
Myself :
- PhD at Columbia, 1980
- Vice-President, Waseda Univ.
since 2006.
- Professor of Structural Engineering
in Dept. of Architecture,
since 1993.
Myself (Cont’d) :
- Have been doing researches related to
smart structures technology including
active/semiactive structural control
for nearly 20 years.
- Have been involved to the activity of
IASCM [ International Association for
Structural Control and Monitoring ]
since its establishment in 1994.
Myself (Cont’d) :
- Have been the Chairperson of
the JSPS [Japan Society for Promotion of
Science] 157th Committee on
Structural Response Control
since April 2007.
- Currently, Vice-President, JAEE [Japan
Association of Earthquake Engineering].
■ 2. Introduction for:
Structural Control
Structural Control:
▲ Active control
▲ Passive control
Structural Control:
▲ Active control
▲ Passive control
With or without
Energy supply
With or without
Control computer
Structural Control:
▲ Active control
▲ Passive control
With
supply
Energy
With
Control computer
Structural Control:
▲ Active control
▲ Passive control
Without
Energy supply
Without
Control computer
Structural Control:
▲ Active control
- Full-active control
- Semi-active or Semiactive control
- Hybrid control
▲ Passive control
- Base Isolation
- Passive damper-based control
Structural Control:
▲ The idea of
seismic structural control:
not a totally new idea.
▲ The basic principles
for seismic response control:
presented in Japan in 1960.
Seismic Response Control Principles:
1. Reduce the effect of seismic excitation.
2. Prevent a structure from exhibiting the
resonance vibration.
3. Transfer the vibration energy of a main
structure to the secondary oscillator.
4. Put additional damping effect to a structure.
5. Add a control force to a structure.
These ideas were proposed
by Kobori and Minai in 1960.
Professor Takuji Kobori
They proposed the idea of:
Seismic-Response-Controlled
Structures or
制震構造.
Seismic-response-controlled
structure
Building
Nonlinear
mechanism
Nonlinear
mechanism
Nonlinear
mechanism
Nonlinear
mechanism
Seismic Response Control Principles:
1. Reduce the effect of seismic excitation.
Base Isolation
2. Prevent a structure from exhibiting the
resonance vibration.
Base Isolation
3. Transfer the vibration energy of a structure
to the secondary oscillator.
TMD Control
4. Put additional damping effect to a structure.
Passive damper control
5. Add a control force to a plant. AMD Control
Japan has been leading the
world in terms of the
practical applications of
structural control schemes.
Practical Applications in Japan:
# of Buildings:
Base isolation: over 2,000
Passive dampers: over 300
Active control: over 40
■ Keywords for
structural control.
- TMD
- AMD
- Smart damper
- Semiactive damper
- Controllable damper
- LQ control
- LQG control
- Feedback control
- Feed-forward control
- TMD: Tuned Mass Damper
- AMD: Active Mass Damper
- Smart damper
- Semiactive damper
- Controllable damper
- LQ control
- LQG control
- Feedback control
- Feed-forward control
- TMD: Tuned Mass Damper
- AMD: Active Mass Damper
- Smart damper
- Semiactive damper
- Controllable damper
- LQ control
- LQG control
- Feedback control
- Feed-forward control
There are many kinds of
‘smart’ expressions such as
‘smart’ cars,
‘smart’ dampers,
‘smart’ structures,
‘smart’ medicine, etc.
Indeed,
“The Merriam-Webster
Paperback Dictionary”
gives a modern interpretation
of ‘smart.’
Containing a microprocessor
of limited calculating
capability.
With the names such as
‘smart structures,’
‘intelligent structures,’
‘dynamic intelligent buildings,’ etc.,
civil structures have been getting
more and more human beings-like
characteristics.
■ 4. Overview of
active structural
control:
- In 1989,
a real building with active control
technology applied was completed
in Tokyo, Japan.
- This was the first
full scale implementation of active or
computer-based response control
in the world.
Professor Takuji Kobori
The name of the building:
Kyobashi Seiwa Building
(Currently,
Kyobashi Center Building)
Kyobashi Center Building
- This building employed an AMD
system.
- AMD is one of the typical active
control devices or actuators for
buildings.
AMD
AMD
- AMD is a mass of weight
installed into the top floor or
near top floor,
which is manipulated by
a control computer
based on the response data.
The inertial force
resulting from AMD movement
Control force
Structure
responding to
Seismic or wind excitation
AMD
Driving Force
AMD
Building
AMD
Driving Force u
Mass of AMD
m
AMD
Building
Mass of Building M
x
AMD
xa
k
X
K
xg
building or
main structure
The equation of motion of a structural
system with AMD integrated is:
m

0
0   x   k
     
M   X   k
 k  x 
m
 u 
       xg  

k  K  X 
M 
 u
m

0
m   xa   k
     
M   X   k
0   xa 
m
 u 
       xg  

K  X 
M 
 u
The equation of motion of a structural
system with AMD integrated:
m

0
m   xa   k
     
M   X   k
0   xa 
m 
 u 
       xg  

K  X 
M 
 u
From the first raw ,
 )  kx   m x  u
m ( xa  X
a
g
  x )   kx  u
m ( xa  X
g
a
  x )   u  kx
 m ( xa  X
g
a
Combining the above with the second raw of (1),
  KX   Mx  m ( x  X
  x )
MX
g
a
g
(1)
The equation of motion of a structural
system with AMD integrated:
  KX   Mx  m ( x  X
  x )
MX
g
a
g
  KX  ( M  m ) x  mx
( M  m) X
g
a
AMD
xa
x
xg
As a result,
since the birth of the world’s first
active-controlled building,
now more than 40 buildings
in Japan have installed
a variety of active control schemes.
Full-scale active control implementations:
Kyobashi Seiwa Bldg., 1989
Bidg. #21, Kajima Technical Research Institute, 1990
Sendagaya INTES, 1992
Applause Tower, 1992
Osaka ORC 200, 1992
Kansai Airport Control Tower, 1992
Long Term Credit Bank, 1993
Ando Nishikicho Bldg., 1993
Porte Kanazawa, 1994
Shinjuku Park Tower, 1994
RIHGA Royal Hotel, 1994
MHI Yokohama Bldg., 1994
Hikarigaoka J City, 1994
Hamamatsu ACT City, 1994
Riverside Sumida, 1994
Hotel Ocean 45, 1994
Osaka WTC Bldg., 1995
Full-scale active control implementations(cont.):
Dowa Kasai Phoenix Tower, 1995
Rinku Gate Tower, 1995
Hirobe Miyake Bldg, 1995
Plaza Ichihara, 1995
HERBIS Osaka, 1997
Nisseki Yokohama Bldg., 1997
Itoyama Tower, 1997
Otis Elevator Test Tower, 1998
Bunka Gakuen, 1998
Oita Oasis Hiroba 21, 1998
Odakyu Southern Tower, 1998
Kajima Shizuoka Bldg., 1998
Sotetsu Bldg., 1998
Century Park Tower, 1999
Sosokan, Keio Univ., 2000
Gifu Regional Office, Chubu Power Electric Company, 2001
However,
most of these implementations were
mainly aimed at the response control
against small/moderate seismic or
strong wind excitation.
The ultimate goal of active control:
 To enhance the structural safety
against severe seismic events.
 Need to establish
such a control scheme as to
achieve the final goal of
active structural control.
Reference:
A. Nishitani and Y. Inoue (2001).
“Overview of the application of
active/semiactive control in Japan,”
Earthquake Engineering & Structural
Dynamics, Vol. 30(11), pp.1565-1574.
Active structural control:
- The full-scale active control
implementation to a civil structure
has opened the door to ‘modern’
earthquake engineering or ‘modern’
structural engineering.
- Structural engineering is now
integrating more and more modern,
advanced and IT-related technologies.
■ 5. Components of
Control System:
- How is a control system
composed?
From the point of view of
system control engineering, …..
Control System:
- Plant structure whose
responses are controlled
- Sensors
- Control computer (Controller)
- Control actuator
Control System:
Control
Input
Seismic Input
Plant
Actuator
Controller
Sensors
Seismic Structural Control:
1. Reduce the effect of seismic excitation
which a plant is subjected to.
2. Prevent a plant from exhibiting
the resonance vibration.
3. Transfer the vibration energy of a plant
to a control-actuator.
4. Put additional damping effect to a plant.
5. Add a control force to a plant
through an actuator or actuators.
Passive Control System:
✓ Plant structure whose
■
responses are controlled
■ Sensors
■ Control computer (Controller)
✓ Control actuator
■
Base Isolation:
✓ Plant structure whose
■
responses are controlled
■ Sensors
■ Control computer (Controller)
✓ Control actuator
■
Passive Damper Control:
1. Reduce the effect of seismic excitation.
2. Prevent a plant from exhibiting
the resonance vibration.
3. Transfer the vibration energy of a plant
to a control-actuator.
4. Put additional damping effect to a plant.
5. Add a control force to a plant.
TMD Control:
1. Reduce the effect of seismic excitation.
2. Prevent a plant from exhibiting
the resonance vibration.
3. Transfer the vibration energy of a plant
to a control-actuator.
4. Put additional damping effect to a plant.
5. Add a control force to a plant.
Base Isolation:
1. Reduce the effect of seismic excitation.
2. Prevent a plant from exhibiting
the resonance vibration.
3. Transfer the vibration energy of a plant
to a control-actuator.
4. Put additional damping effect to a plant.
5. Add a control force to a plant.
Active Control System:
✓ Plant structure whose
■
responses are controlled
✓ Sensors
■
✓ Control computer (Controller)
■
✓ Control actuator
■
AMD Control:
1. Reduce the effect of seismic excitation.
2. Prevent a plant from exhibiting
the resonance vibration.
3. Transfer the vibration energy of a plant
to a secondary vibration system.
4. Put additional damping effect to a plant.
5. Add a control force to a plant.
Theoretically,
There are two kinds of active
control schemes: ……..
Theoretically,
There are two kinds of active
control schemes:
Feedback control
and
Feed-forward control.
External input such as seismic excitation
Control
Input
Actuator
Plant
Sensors
Output
Controller
External input such as seismic excitation
Control
Input
Actuator
Plant
Sensors
Output
Controller
Feedback Control
External input such as seismic excitation
Control
Input
Plant
Sensors
Output
Controller+
Actuator
Feedback Control
External input such as seismic excitation
Plant
Response
Control
Input
Controller
Feedback Control
External input excitation
H(s)
Response
Control
Input
G(s)
Feedback Control
External input excitation
Plant transfer function
H(s)
Control
Input
Response
Feedback gain
G(s)
Feedback Control
External input excitation
Plant transfer function
H(s)
Control
Input
Response
Feedback gain
G(s)
Feedback Control
Controller+
Actuator
Sensors
External
input such
as seismic
excitation
Plant
Control
Input
Response
G(s)
External
input
excitation
H(s)
Control
Input
Response
G(s)
External
input
excitation
H(s)
Control
Input
Response
Feed-forward Control
■ 6. Semiactive Structural
Control:
- What is semiactive control?
- How is semiactive control
conducted?
Semiactive control:
Combines the beneficial
features of both of passive
and active control systems.
Semiactive control:
Passive control:
No energy supply to a control
actuator needed.
Active control:
Flexibility, Adaptability,
Efficient performance.
Semiactive control:
- Less energy
- More efficiency
- Better performance
Control System:
- Plant structure whose
responses are controlled
- Sensors
- Control computer (Controller)
- Control actuator
Control System:
Seismic Input
Plant
Actuator
Controller
Sensors
Semiactive control:
There are two major ways
defining or characterizing
semiactive control concept.
The most general definition:
Semiactive control is ……
The most general definition:
Semiactive control is conducted
by changing or controlling
a part of charactersitics of
control actuator
only at appropriate time instants.
The most general definition:
Semiactive control is conducted
by changing or controlling
a part of charactersitics of
control actuator
only at appropriate time instants.
 Adaptive characteristics.
This definition leads to:
- Large power not needed.
- Required power not dependent
of the magnitude of
seismic excitation.
The second significant point:
Semiactive control operation
does not inject mechanical energy
into a plant structure or control
device or actuator.

The second significant point:
Semiactive control operation
does not inject mechanical energy
into a plant structure or control
device or actuator.
 It has much less potential
to destabilize the structure.
In typical semiactive control:
Actuator: Damper
Controlled characteristics such as
the damping coefficient,
the magnitude of relief load, etc.,
of the damper are controlled.
This kind of dampers are ……..
Typical semiactive control:
Actuator: Damper
Ccontrolled characteristics such as
the damping coefficient,
the magnitude of relief load, etc.
of the damper are controlled.
This kind of dampers are
called ‘controllable’ dampers.
Then, for example,
consider a type of semiactive control
in which the damping coefficients
of installed viscous dampers are
controlled.
Then, for example,
consider a type of semiactive control
in which the damping coefficients
of installed viscous dampers are
controlled.
 This change would not have any effect
on the structure which is not subject to
any other external input excitation.
On the contrary,
the movement of AMD could
make an entire structure vibrate
even in case of no other external
input excitation.
On the contrary,
the movement of AMD would
make an entire structure vibrate
even in case of no other external
input excitation.
 This is very significant
difference between
full-active and semi-active control.
AMD
Power
AMD
Building
Controlled dampers
 Smart dampers
One of smart control schemes
 Control scheme based on
“smart” or “controlled” dampers
■ 7. Smart damping
or Smart Dampers
Vibration Control
- Buildings
- Motor vehicle suspensions
z
Car Body or
Building
Spring
Damper
xg
- Computer control of
of suspension systems
in 1980s.
- Computer control
of buildings in 1989.
z
Car Body
Spring
Damper
xg
- Ride Comfort
 Absolute movement
of car body = 0
- Driving Stability
 Movement of car body
= Movement of ground
Trade-off between
ride comfort and
driving stability
Spring
Damper
Variable
Transfer function
from xg to z
z (ω )
x g (ω )
Low damping
High damping
1
0
2
For better ride comfort,
smaller absolute accelerations.
 High damping is not appropriate
for the high-frequency region.
 Constant damping is not appropriate.
Skyhook damper
z
xg
Skyhook damper
Csh
z
C
xg
Skyhook damper
Csh
z
C
. .
xg
.
C (z-xg) = Csh z
Skyhook damper
Csh
z
C
. .
.
C (z-xg) = Csh z
.
. .
C = Csh [z / (z-xg)]
xg
Pioneering Implementations
of Smart Damping:
• Kajima Shizuoka Building
• Keio University Soso-kan Building
• Chubu Electric Power (CEP)
Gifu Regional Office Building
Kajima Shizuoka
Building
- Kajima Shizuoka Building
The World’s first smart damping
or semiactive variable damping
implementation to a building.
Variable damping system in
Kajima Shizuoka Bldg.:
The damping coefficients
of oil-dampers is controlled
so that LQG-based optimal control
force should be provided
in terms of damping force.
Keio Univ. Soso-kan Building
- Keio Univ. Soso-kan Building
The world’s first smart baseisolated building or
building with base isolation
integrating semiactively-controlled
variable damping system.
CEP Gifu Regional
Office Building
- CEP Gifu Regional Office Building：
The world’s first building
employing
an autonomous-decentralized
semiactive smart damping system.
Autonomous-decentralized
control system
A-D Control System:
Seismic Input
Act.
Plant
Act.
Act.
Sensors
Controller
Controller
Sensors
Controller
Sensors
Autonomous-Decentralized
Control System:
- Each of distributed control systems
is autonomously controlled
by its own local, decentralized
controller, not by only one center
controller.
- Height of a huge, high-rise building
- Width of a huge building with very
wide floors
One central control computer
does not seem appropriate.
Autonomous-decentralized
control system (AD control
system)
A-D Semiactive Damper
Switching Oil Damper with Built-in
Controller
“Switching oil damper with built-in
controller”
-The ‘damper’ is a Maxwell type of
system consisting of
a stiffness element (spring) and
a controllable oil damper element.
Damper
Spring
Cmax
Cmin
Vel
+
K
Disp
By properly choosing the damping coefficient,
2
Cmax
Cmin
1
Cmin
4
Cmax
3
Passive Damper
Hysteresis
Cmax
Cmax
Cmax
Cmax
②
①
①
③
④
②
②
③
③
④
④
- Each damper autonomously
controlled by its own decentralized
controller
 Autonomous-decentralized
control system
-Several newly constructed buildings in
Japan have installed this type of
semiactive damper systems.
-“Switching oil damper with built-in
controller”
The Shi’odome District
The Shi’odome Kajima Tower
The Shi’odome Kajima Tower
Roppongi Tower
Autonomous-decentralized
control system
- Control operation could be conducted
based upon the response information
only in the neighborhood of each
control devise.
Autonomous-decentralized control
+ Artificial Nonlinearity concept
seems appropriate or fitted to
structural control against severe
seismic excitations.
■ 8. Significance of nonlinearity
or artificially-added nonlinearity
in structural control
- Basic concept
- Control effect
- Oil hydraulic dampers
Bi-linear subsystem
Linear
structure
tan-1βK
tan-1αK
tan-1αK
tan-1(α+β)K
γ＝α/（α＋β）
-1
tan
tan-1 K
γK
ΔW
W
Damping
Coefficient =
ΔW/W/(4π)
tan-1γK
tan-1 K
Equivalent viscous damping ratio
= (1-γ)/((1+γ)π)
α=0.7
α=0.8
α=0.9
α=1.0
β
What would happen to a SDOF
structure subjected to seismic
excitation with this algorithm?
Case 1:
α=β=0.5
Case 2:
α<β
α= 0.3; β= 0.7
El Centro 1940 earthquake
2
NS component with 2 m/sec
Response Accelerations
α=β= 0.5 0.5
α=0.3, β= 0.7
①
Response Displacement
α=β= 0.50.5
α= 0.3, β= 0.7
0.7
Damper hystereses
α= β= 0.5 0.5
α=0.3, β= 0.7
As an AD semiactive control
system integrating artificial
nonlinearity philosophy,
Variable slip-force
level dampers
■ 9. Semiactive Variable
Slip-force Level Dampers
- Basic concept
- Control effect
- Oil hydraulic dampers
- Basic concept:
- Semiactive control
- Utilizing artificial nonlinearity
- Autonomous-decentralized
system
Force
slip-force-level
fSlip-levelforce スリ
ップレベル
displacement
D
層間変位
図7 完全弾塑性型
A damper is controlled
so that it begins to slip at the
occurrence of peak velocity.
 - No need for modeling.
- Only local response
information needed.
Damper
ductility
factor = 2
The effectiveness of this scheme:
is analytically measured in
terms of equivalent viscous
damping ratio.
Damper+Structure
tan-1 αK
tan-1(α+β)K
What would happen to a SDOF
structure subjected to seismic
excitation with this algorithm?
Case 1:
α=β=0.5
Case 2:
α<β
α= 0.3; β= 0.7
El Centro 1940 earthquake
2
NS component with 2 m/sec
Response Accelerations
α=β= 0.5 0.5
α=0.3, β= 0.70.7
①
Response Displacement
α=β= 0.50.5
α= 0.3, β= 0.7 0.7
Damper hystereses
α= β= 0.5 0.5
α= 0.3, β= 0.70.7
Case 1: α=β= 0.5
Estimated damping
coefficient = 0.087
Case 2: α= 0.3; β= 0.7
Estimated damping
coefficient = 0.162
Acceleration Response Spectrum
Simulation for a 20-storie highrise building:
- Steel structural model
accounting for shear and
bending deformations.
Natural Period of original
structural model:
- 1st Mode: 1.78 sec
- 2nd Mode: 0.577 sec
- 3rd Mode: 0.310 sec
- Dampers are installed on every
floor.
- Each damper is controlled only
based upon the interstory drift
response velocity.  Autonomousdecentralized control.
- Damper is effective only on shear
deformation.
Autonomous-Decentralized
Control System:
- Each of distributed control systems
is autonomously controlled by its
own local, decentralized
controller, not by only one center
controller.
Building 1: α=β= 0.7
Building 2: α=β= 1.0
20
18
18
16
16
14
14
12
12
s tory
s tory
20
10
α =β=1.0
8
10
8
α =β=0.7
6
uncontrolled
α =β=0.7
4
4
2
2
0
0
1
1.5
2
α =β=1.0
6
2.5
acceleration[m /s 2 ]
(a) Accelerations
3
uncontrolled
0
0.05
0.1
0.15
displacem ent[m ]
(b) displacements
Maximum resoponses
0.2
The presented concept can be put
into practice utilizing an oilhydraulic damper-based device.
- A damper containing an
electromagnetic relief valve is
utilized.
The presented concept can be put
into practice utilizing an oilhydraulic damper-based device.
- A damper containing an
electromagnetic relief valve is
utilized.  This is a kind of
variable-orifice damper.
electromagnetic
relief valve
電磁リリーフ弁
ピストン
piston
orifice
オリフィス
ゴムブッシュ
図13
rubber bush
オイルダンパ
Experimental model of semiactive
variable slip-force level damper
DampingForce[kN]
10
8
6
4
2
0
proposed model
8V
6V
4V
2V
0V
0
5
10
Velocity[cm/s]
Relationship between damper velocity
and electric voltage given to the valve
15
shear force (kN)
shear force (kN)
Experimental results responding to sinusoidal
excitation with increasing amplitudes
Constant
slip-force
level
Variable
slip-force
level
Displacement (mm)
Reference:
A. Nishitani, Y. Nitta and Y. Ikeda
(2003).
“Semiactive structural-control based on
variable slip-force level dampers,”
J. of Structural Engineering, ASCE,
Vol. 129(7), pp.933-940.
■
- Semiactive and smart concept based
schemes have been presented for
structural control of buildings
as well as the full scale implementations
of some of such schemes in Japan.
-
■
- The concept of semiactive variable slipforce level dampers has been presented.
■ 10. Future directions:
- Semiactive and smart strategies, or
smart passive strategies, are expected to
play more and more significant role in
the future stage of structural
engineering, integrating the
autonomous-decentralized concept.
-
■ Optimal control:
LQ control & LQG control:
LQ:
Linear, Quadratic
LQG:
Linear, Quadratic and Gaussian
-
■ LQ control & LQG control:
Two schemes for optimal control:
Response: whether probabilistic
or deterministic?
If the response is probabilistic,
then the control input will be
probabilistic.
 LQG control.
-
■ LQ control & LQG control:
In the case where
the response and control input are
stationary, Gaussian random processes,
 LQG control.
-
The equation of motion of a structural
system with control input:
x  2ς ω0 x  ω x  f  u
2
0
d  x  0
 
2
dt  x   ω0
  x  0 
0 
     u    f
 2ς ω0   x  1
1
1
This equation is rewritten as :
x  Ax  bu  cf
The state equation:
x  Ax  bu  cf
In the above,
f : deterministic, then
x : deterministic, then
u : deterministic.
LQ control.
The state equation:
x  Ax  bu  cf
In the above,
f : stationary Gaussian white - noise
with zero - mean, then
x : stationary Gaussian with zero - mean,
u : stationary Gaussian with zero - mean.
LQG control
J  lim
T 
1
T

T
2

[ x ( t )Qx( t )  ru ( t )] dt
0
If u and x are probabilistic,
J is also probabilistic.

1

E[ J ]  E lim
T  T


T
2


[
x
(
t
)
Qx
(
t
)

ru
(
t
)]
dt
0


u and x are stationary.
E[ J ]  lim
T 
1

T
T
0
( E[ x ( t )Qx( t )]  E[ ru ( t )]) dt
2
u and x are stationary.
E[ J ]  lim
T 
1

T
T
( E[ x ( t )Qx( t )]  E[ ru ( t )]) dt
2
0
E[ J ]  lim ( E[ x Qx]  E[ ru ])
2
T 

T
T
0
E[ J ]  lim ( E[ x Qx]  E[ ru ])
2
T 
1
dt
u and x are stationary.
E[ J ]  lim ( E[ x Qx]  E[ ru ])
2
T 
Control input is given by
Pu( t )  Gx( t )
Feedback gain G is given by
1
G  r bP
satisfying the following equation :
1
PA  AP  Pbr bP  Q  0
Riccati Equation.
LQG control:
LQG control statistically satisfies
the samllest value of E[J].
Epigram:
Little people discuss other people.
Average people discuss events.
Big people discuss ideas.
(M.S. Grewal, A.P. Andrews.
Kalman Filtering: Theory and Practice
Using MATLAB [Second Edition], John
Wiley, 2001)
Thanks for your
attention.