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Moncrief-O’Donnell Chair, UTA Research Institute (UTARI)The University of Texas at Arlington, USA
and
F.L. Lewis, NAI
Talk available online at http://www.UTA.edu/UTARI/acs
Lyapunov Design for ControlsSupported by :China Qian Ren Program, NEUChina Education Ministry Project 111 (No.B08015)NSF, ONR
Foreign Professor, Chongqing University, China
Automation & Robotics Research Institute (ARRI)The University of Texas at Arlington
F.L. LewisMoncrief-O’Donnell Endowed Chair
Head, Controls & Sensors Group
Talk available online at http://ARRI.uta.edu/acs
Nonlinear Network Structures for Feedback Control:Part I- Adaptive, Robust, & Neural Net Control
Supported by :NSF - PAUL WERBOSARO – RANDY ZACHERY
David Song Yongduan
He who exerts his mind to the utmost knows nature’s pattern.
The way of learning is none other than finding the lost mind.
Meng Tz500 BC
Man’s task is to understand patterns innature and society.
Mencius
Importance of Feedback Control
Darwin 1850- FB and natural selectionVito Volterra 1890- FB and fish population balanceAdam Smith 1760- FB and international economyJames Watt 1780- FB and the steam engineFB and cell homeostasis
The resources available to most species for their survival are meager and limited
Nature uses Optimal control
Feedback Control Systems
Aircraft autopilotsCar engine controlsShip controllersCompute Hard disk drive controllersIndustry process control – chemical, manufacturingRobot control
Industrial Revolution –Windmill control, British millwrights - 1600sSteam engine and prime movers
James Watt 1769SteamshipSteam Locomotive boiler control
Sputnik 1957Aerospace systems
Newton’s Law
v(t)
x(t)
F(t)m
)()( tumtFx
xmmaF
Industrial Process and Motion Systems (Vehicles, Robots)
τqB)+τq+G(q)+F(q)q(q,+Vq)qM( dm )(
Coriolis/centripetalforce
gravity friction disturbances
Actuatorproblems
inertia
Control Input
Lagrange’s Eqs. Of Motion
Lagrange Dynamical systems
Dynamical System Models
)()()(
xhyuxgxfx
Nonlinear system
Continuous-Time Systems Discrete-Time Systems
)()()(1
kk
kkkk
xhyuxgxfx
Linear system
CxyBuAxx
kk
kkk
CxyBAxx
1
1/s
f(x)
h(x)g(x)
z-1
xx yu
Control Inputs Internal States Measured Outputs
Issues in Feedback Control
system
Feedbackcontroller
Feedforwardcontroller
Measured outputs
Control inputs
Desired trajectories
Sensornoise
Disturbances
StabilityTracking BoundednessRobustness
to disturbancesto unknown dynamics
Unknown Process dynamicsProcess NonlinearitiesUnknown Disturbances
Definitions of System Stability
xe
xe+B
xe-B
Const Bound B
tt0 t0+T
T
x(t)
x(t)
t
x(t)
t
Asymptotic Stability Marginal or Bounded Stability -Stable in the Sense of Lyapunov (SISL)
Uniform Ultimate Boundedness
)()(
1 kk xfxxfx
d
B(d)
Clips are from Nonlinear Control Systems book by Slotine and Li.
21 2
1y us a s a
Example 1. Linear System
1 2y a y a y u
desired to track a reference input ( )dy t
de y y Tracking error
1 2de y a y a y u
r e e Sliding variable
1 2dr e e y e a y a y u
du v y e Auxiliary input
1 2r a y a y v Error dynamics
Of the form ( )r f x v
1 2 1 2( ) ( )Tyf x a y a y a a W x
y
Unknown function
Feedback Linearization
Unknown parameters
Known Regression Vector
Example 2. Nonlinear Lagrange System( , ) ( )y d y y k y u
unknown nonlinear damping term unknown nonlinear friction
( )r f x v
( ) ( , ) ( )f x d y y k y with
Assume Linear in the Parameters (LIP)
1
1
( , )( ) ( )
( , )Td y y
f x D K W xk y y
Known possibly nonlinear regression function
Unknown parameters
desired to track a reference input ( )dy t
de y y Tracking error
( , ) ( )de y d y y k y u
r e e Sliding variable
Error dynamicsdu v y e Auxiliary input
Feedback Linearization
Lagrangian System Appears in:Process controlMechanical systemsRobots
plantr(t)
Feedback Linearization Controller
A dynamic controller
du v y e
u
yy d
d
yy e
e
I
r e e
e v?
controller
The equations give the FB controller structure
Feedforward terms
Tracking Loop
Adaptive Control
Error Dynamics
( )r f x v r(t)= control errorControl input
Unknown nonlinearities
( )Tr W x v
Error Dynamics
Assume: f(x) is known to be of the structure
( ) ( )Tf x W x
LINEAR-IN-THE-PARAMETERS (LIP)
Known basis set= regression vector – DEPENDS ON THE SYSTEM
Unknown parameter vector
Controller
ˆ ˆ( ) ( ) ( )Tv vv f x K r W t x K r
ˆ( ) ( )W t W W t Parameter estimation error
ˆ( ) ( ) ( )T T Tvr W x v W x W x K r
( )Tvr W x K r
closed-loop system becomes
Est. error drives the control error
ˆ ˆ ( ) TdW W F x rdt
Parameter estimate is updated (tuned) using the adaptive tuning law
Adaptive Control
Pos. def. control gain
ESTIMATE of unknown parameters
plantr(t)
Feedback Linearization Adaptive ControllerA dynamic controller
du v y e
u
yy d
d
yy
ee
I
r e e
e
v
The equations give the FB controller structure
ˆ ˆ( ) ( ) ( )Tv vv f x K r W t x K r
vK
ˆ ( ) TW F x rˆ ( ) ( )TW t x
ˆ ( )f x
Tunable inner loop
Tracking Loop
Feedback terms
Neural Networks for Control
F.L. Lewis, S. Jagannathan, and A.Yesildirek,Neural Network Control of RobotManipulators and Nonlinear Systems,Taylor and Francis, London, 1999.
NN control in Chapter 4
Control System Design Approach
( ) ( , ) ( ) ( )m dM q q V q q q F q G q
( ) ( ) ( )de t q t q t
r e e
dm xfrVrM )(
Robot dynamics
Tracking Error definition
Error dynamics
Siding variable
( ) M( ( ) ( ) )dMr M e e q t q t e
( ) ( )( ) ( , )( ) ( ) ( )d m df x M q q e V q q q e F q G q
Where the unknown function is
qdRobot System[I]
qe
PD Tracking Loop
r
dm qFqGqqqVqqM )()(),()( Robot dynamics
?controller
)()()( tqtqte d Tracking error
eer Sliding variable
The equations give the FB controller structure
Kung Tz 500 BCConfucius
ArcheryChariot driving
MusicRites and Rituals
PoetryMathematics
孔子Man’s relations to
FamilyFriendsSocietyNationEmperorAncestors
Tian xia da tongHarmony under heaven
124 BC - Han Imperial University in Chang-an
Handling High-Frequency Dynamics
Actuator Dynamics
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