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