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

How to Do Research

Distinguished Foreign Consulting ProfessorChongqing University, China

Supported by :NSF - Paul WerbosAFOSR EuropeONR – Marc SteinbergUS TARDEC

Supported by :China NNSFChina Project 111

Invited by Y.D. Song

“Some are born with knowledge, some derive it from study, and some acquire it only after a painful realization of their ignorance.

But the knowledge being possessed, it comes to the same thing.

Some study with a natural ease, some from a desire for advantages, and some by strenuous effort.

But the achievement being made, it comes to the same thing.”

Pythagoras 500 BC Natural Philosophy, Ethics, and Mathematics

Numbers and the harmony of the spheres

Music, Mathematics, Gymnastics, Astronomy, Medicine

Translate music to mathematical equationsRatios in music

Fire, air, water, earth

The School of Pythagoras esoterikoi and exoterikoi

Mathematikoi - learnersAkoustiki- listeners

Patterns in NatureSerenity and Self-Possession

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 Zi500 BC

Man’s task is to understand patterns innature and society.

Mencius

It is man’s obligation to explore the most difficult questions inthe clearest possible way and use reason and intellect to arriveat the best answer.

Man’s task is to understand patterns in nature and society.

The first task is to understand the individual problem, then toanalyze symptoms and causes, and only then to designtreatment and controls.

Ibn Sina 1002-1042(Avicenna)

Read papers‐ Look for Patterns

Read Papers!Find the Missing Link!

19

Purpose of Publishing Papers• disseminate research result for academic

exchanges• At stake is your reputation • It is not “the more, the better” • Write papers only when you have good results

and when you do, do your best. • Write it, rewrite it, and rewrite it again 止于至善。

Jie Huang, Chinese University of Hong Kong

20

Ingredients of Good Research

• An open problem, or a self‐formulated problem with impact to the field or society. 

• The research is motivated or stimulated from the urge of knowledge discovery or from the need of society. 

• New tools are developed, or existing tools are modified to solve the problem 

• No pain, No‐gain

Jie Huang, Chinese University of Hong Kong

21

Importance of Writing• A good paper is always based on a piece of good research, but 

a good research may not necessarily lead to a good paper. • A good paper can add significant value to a research work• The beginning of  the writing is not the end of the research, 

but a beginning of a deeper phase of the research.  • The process of writing is the process of soul‐searching, and 

may provoke new ideas and  new directions.• 4C’s: Clarity, Concise, Creativity and Courtesy

Jie Huang, Chinese University of Hong Kong

Optimal ControlAdaptive Control Find the Key- Reinforcement learningFunctions of the Brain – multiple oscillations

Game Theory –Sun TzeApplications to Manufacturing

Optimality in Control Systems DesignR. Kalman 1960

Rocket Orbit Injection

http://microsat.sm.bmstu.ru/e-library/Launch/Dnepr_GEO.pdf

FmmmF

rwvv

mF

rrvw

wr

cos

sin2

2

ObjectivesGet to orbit in minimum timeUse minimum fuel

Dynamics

Optimal Control is Effective for:Aircraft AutopilotsVehicle engine controlAerospace VehiclesShip ControlIndustrial Process Control

Multi-player Games Occur in:EconomicsControl Theory disturbance rejectionTeam gamesInternational politicsSports strategy

But, optimal control and game solutions are found byOffline solution of Matrix Design equationsA full dynamical model of the system is needed

Optimality and Games

PBPBRQPAPA TT 10

1 Tu R B Px Kx

BuAxx

( ( )) ( ) ( ) ( )T T T

t

V x t x Qx u Ru d x t Px t

Optimal Control- The Linear Quadratic Regulator (LQR)

System dynamics

Performance Index

Solve the Algebraic Riccati Equation (ARE)

Then the optimal feedback control is

Minimum energy, minimum control effort

The Solution

Result:Let Q and R be symmetric and positive definiteThen the control u=-Kx minimizes V(x)and makes the system stable and robust

MATLAB function [K,P]=lqr(A,B,Q,R)

xux Ax Bu

SystemControlK

PBPBRQPAPA TT 10

1 TK R B P

On-line real-timeControl Loop

Off-line Design LoopUsing ARE

Optimal Control- The Linear Quadratic Regulator (LQR)

An Offline Design Procedurethat requires Knowledge of system dynamics model (A,B)

System modeling is expensive, time consuming, and inaccurate

( , )Q R

User prescribed optimization criterion ( ( )) ( )T T

t

V x t x Qx u Ru d

A. Optimal control B. Zero-sum games C. Non zero-sum games

1. System dynamics

2. Value/cost function

3. Bellman equation

4. HJ solution equation (Riccati eq.)

5. Policy iteration – gives the structure we need

Adaptive Control Structures for:

We want to find optimal control solutions Online in real-time Using adaptive control techniquesWithout knowing the full dynamics

For nonlinear systems and general performance indices

Adaptive Control is online and works for unknown systems.Generally not Optimal

Optimal Control is off-line, and needs to know the system dynamics to solve design eqs.

We want ONLINE DIRECT ADAPTIVE OPTIMAL ControlFor any performance cost of our own choosing

Reinforcement Learning turns out to be the key to this!

Apply New Tools

Look in a different disciplineMathComputational Intelligence

Machine LearningThermodynamics Fluid MechanicsPhysics

Find The Key!

Reinforcement Learning

Every living organism improves its control actions based on rewards received from the environment

The resources available to living organisms are usually meager.Nature uses optimal control.

1. Apply a control.  Evaluate the benefit of that control.2. Improve the control policy.

RL finds optimal policies by evaluating the effects of suboptimal policies

Optimality in Biological Systems

Different methods of learning

SystemAdaptiveLearning system

ControlInputs

outputs

environmentTuneactor

Reinforcementsignal

Actor

Critic

Desiredperformance

Reinforcement learningIvan Pavlov 1890s

Actor-Critic Learning

We want OPTIMAL performanceADP- Approximate Dynamic Programming- Paul Werbos

Sutton & Barto book

D. Vrabie, K. Vamvoudakis, and F.L. Lewis, Optimal Adaptive Control and Differential Games by Reinforcement Learning Principles, IET Press, 2012.

BooksF.L. Lewis, D. Vrabie, and V. Syrmos, Optimal Control, third edition, John Wiley and Sons, New York, 2012.

New Chapters on:Reinforcement LearningDifferential Games

F.L. Lewis and D. Vrabie,“Reinforcement learning andadaptive dynamic programmingfor feedback control,”IEEE Circuits & SystemsMagazine, Invited Feature Article,pp. 32-50, Third Quarter 2009.

IEEE Control Systems Magazine,F. Lewis, D. Vrabie, and K.Vamvoudakis,“Reinforcement learning andfeedback Control,” Dec. 2012

1 Tu R B Px Kx

BuAxx

( ( )) ( ) ( ) ( )T T T

t

V x t x Qx u Ru d x t Px t

0 ( , , ) 2 2 ( )T

T T T T T T TV VH x u V x Qx u Ru x x Qx u Ru x P Ax Bu x Qx u Rux x

Derivation of Linear Quadratic Regulator

System 

Cost

Differentiate using Leibniz’ formula

Optimal Control is

Scalar equation

( )T TV x Qx u Ru

Differential equivalent is the Bellman equation

Stationarity condition  0 2 2 TH Ru B Pxu

1 10 2 ( )T T T T Tx P Ax BR B Px x Qx x PBR B Px HJB equation

10 T T T T T Tx PAx x A Px x Qx x PBR B Px

PBPBRQPAPA TT 10

1 Tu R B Px Kx

BuAxx

( ( )) ( ) ( ) ( )T T T

t

V x t x Qx u Ru d x t Px t

Full system dynamics must be knownOff‐line solution

0 ( , , ) 2 ( )T T TVH x u x P Ax Bu x Qx u Rux

Standard Solution for Linear Quadratic Regulator

0 ( ) ( )T TA BK P P A BK Q K RK

u Kx

System 

Cost

Differential equivalent is the Bellman equation

Given any stabilizing FB policy

The cost value is found by solving Lyapunov equation = Bellman equation

Optimal Control is

Algebraic Riccati equation

Scalar equation

Matrix equation

0 ( , , ) 2 ( ) x

( ) ( )

T T T T

T T T

VH x u x P A BK x Qx x K RKxx

x A BK P P A BK Q K RK x

10 T T T T T Tx PAx x A Px x Qx x PBR B Px HJQ equation

Matrix equation

t

T

t

dtRuuxQdtuxrtxV ))((),())((

( , , ) ( , ) ( , ) ( ) ( ) ( , ) 0T TV V VH x u V r x u x r x u f x g x u r x u

x x x

112( ) ( )T Vu h x R g x

x

dxdVggR

dxdVxQf

dxdV T

TT *1

*

41

*

)(0

, (0) 0V

Nonlinear System dynamics 

Cost/value 

Bellman Equation, in terms of the Hamiltonian function

Stationary Control Policy

HJB equation

Derivation of Nonlinear Optimal Regulator

Off‐line solutionHJB hard to solve.   May not have smooth solution.Dynamics must be known

Stationarity condition 0Hu

( , ) ( ) ( )x f x u f x g x u

Leibniz gives Differential equivalent

To find online methods for optimal control Focus on these two equations

),,(),(),(0 uxVxHuxruxf

xV T

0 ( , ( )) ( , ( ))T

jj j

Vf x h x r x h x

x

(0) 0jV

1121( ) ( ) jT

j

Vh x R g x

x

CT Policy Iteration – a Reinforcement Learning Technique

• Convergence proved by Leake and Liu 1967, Saridis 1979 if Lyapunov eq. solved exactly

• Beard & Saridis used Galerkin Integrals to solve Lyapunov eq.• Abu Khalaf & Lewis used NN to approx. V for nonlinear systems and proved convergence

RuuxQuxr T )(),(Utility 

The cost is given by solving the CT Bellman equation

Policy Iteration Solution

Pick stabilizing initial control policy

Policy Evaluation ‐ Find cost, Bellman eq.

Policy improvement ‐ Update control Full system dynamics must be knownOff‐line solution

dxdVggR

dxdVxQf

dxdV T

TT *1

*

41

*

)(0

Scalar equation

M. Abu-Khalaf, F.L. Lewis, and J. Huang, “Policyiterations on the Hamilton-Jacobi-Isaacs equation for H-infinity state feedback control with input saturation,”IEEE Trans. Automatic Control, vol. 51, no. 12, pp.1989-1995, Dec. 2006.

( ) ( )u x h xGiven any admissible policy

0 ( )h x

Converges to solution of HJB

Can Avoid knowledge of drift term f(x)

Work of Draguna Vrabie

Policy iteration requires repeated solution of the CT Bellman equation

0 ( , ( )) ( , ( )) ( , ( )) ( ) ( , , ( ))T T

TV V VV r x u x x r x u x f x u x Q x u Ru H x u xx x x

This can be done online without knowing f(x)using measurements of x(t), u(t) along the system trajectories

Integral Reinforcement Learning

( ) ( )x f x g x u

D. Vrabie, O. Pastravanu, M. Abu-Khalaf, and F. L. Lewis, “Adaptive optimal control for continuous-time linear systems based on policy iteration,” Automatica, vol. 45, pp. 477-484, 2009.

Lemma 1 – Draguna Vrabie

Solves Bellman equation without knowing f(x,u)

( ( )) ( , ) ( ( )), (0) 0t T

t

V x t r x u d V x t T V

0 ( , ) ( , ) ( , , ), (0) 0TV Vf x u r x u H x u V

x x

Allows definition of temporal difference error for CT systems

( ) ( ( )) ( , ) ( ( ))t T

t

e t V x t r x u d V x t T

Integral reinf. form  (IRL) for the CT Bellman eq.Is equivalent to

( ( )) ( , ) ( , ) ( , )t T

t t t T

V x t r x u d r x u d r x u d

value

Key Idea

( ( )) ( , ) ( ( ))t T

k k kt

V x t r x u dt V x t T

IRL Policy iteration

Initial stabilizing control is needed

Cost update

Control gain update

f(x) and g(x) do not appear

g(x) needed for control update

Policy evaluation‐ IRL Bellman Equation

Policy improvement11

21 1( ) ( )T kk k

Vu h x R g xx

),,(),(),(0 uxVxHuxruxf

xV T

Equivalent to

Solves Bellman eq. (nonlinear Lyapunov eq.) without knowing system dynamics

CT Bellman eq.

Integral Reinforcement Learning (IRL)- Draguna Vrabie

D. Vrabie proved convergence to the optimal value and control

Converges to solution to HJB eq.dx

dVggRdx

dVxQfdx

dV TTT *

1*

41

*

)(0

Shi nian shu muBai nian shu ren

Keshi-Wu nian shu xuesheng

十年树木，百年树人

t t+T

observe x(t)

apply uk=Lkx

observe cost integral

update P

Do RLS until convergence to Pk

update control gain

Integral Reinforcement Learning (IRL)

Data set at time [t,t+T)

( ), ( , ), ( )x t t t T x t T

t+2T

observe x(t+T)

apply uk=Lkx

observe cost integral

update P

t+3T

observe x(t+2T)

apply uk=Lkx

observe cost integral

update P

11

Tk kK R B P

( , )t t T ( , 2 )t T t T ( 2 , 3 )t T t T

Or use batch least-squares

(x( )) ( ( )) ( ) ( ) ( ) ( , )t T

T T Tk k k

tW t x t T x Q K RK x d t t T

Solve Bellman Equation - Solves Lyapunov eq. without knowing dynamics

This is a data-based approach that uses measurements of x(t), u(t)Instead of the plant dynamical model.

A is not needed anywhere

Direct Optimal Adaptive Controller

A hybrid continuous/discrete dynamic controller whose internal state is the observed cost over the interval

Draguna Vrabie

DynamicControlSystemw/ MEMORY

xu

V

ZOH T

x Ax Bu System

T Tx Q x u Ru

Critic

ActorK

T T

Run RLS or use batch L.S.To identify value of current control

Update FB gain afterCritic has converged

Reinforcement interval T can be selected on line on the fly – can change

Solves Riccati Equation Online without knowing A matrix

1 TK R B P

PBPBRQPAPA TT 10

xux Ax Bu

SystemControlK On-line Control Loop

On-line Performance Loop

Data-driven Online Adaptive Optimal Control

An Online Supervisory Control Procedurethat requires no Knowledge of system dynamics model A

Automatically tunes the control gains in real time to optimize a user given cost functionUses measured data (u(t),x(t)) along system trajectories

( , )J Q RUser prescribed optimization criterion

( ) ( ) ( )( ) ( ) ( ) ( )t T

T T T Tk k k k

tx t P x t x Q K RK x d x t T P x t T

11

Tk kK R B P

Data set at time [t,t+T)

( ), ( , ), ( )x t t t T x t T

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

Motor control 200 Hz

Oscillation is a fundamental property of neural tissue

Brain has multiple adaptive clocks with different timescales

theta rhythm, Hippocampus, Thalamus, 4-10 Hzsensory processing, memory and voluntary control of movement.

gamma rhythms 30-100 Hz, hippocampus and neocortex high cognitive activity.

• consolidation of memory• spatial mapping of the environment – place cells

The high frequency processing is due to the large amounts of sensorial data to be processed

Spinal cord

D. Vrabie and F.L. Lewis, “Neural network approach to continuous-time direct adaptive optimal control forpartially-unknown nonlinear systems,” Neural Networks, vol. 22, no. 3, pp. 237-246, Apr. 2009.

D. Vrabie, F.L. Lewis, D. Levine, “Neural Network-Based Adaptive Optimal Controller- A Continuous-Time Formulation -,” Proc. Int. Conf. Intelligent Control, Shanghai, Sept. 2008.

Doya, Kimura, Kawato 2001

Limbic system

Cerebral cortexMotor areas

ThalamusBasal ganglia

Cerebellum

Brainstem

Spinal cord

Interoceptivereceptors

Exteroceptivereceptors

Muscle contraction and movement

Summary of Motor Control in the Human Nervous System

reflex

Supervisedlearning

ReinforcementLearning- dopamine

(eye movement)inf. olive

Hippocampus

Unsupervisedlearning

Limbic System

Motor control 200 Hz

theta rhythms 4-10 Hz

picture by E. StinguD. Vrabie

Memoryfunctions

Long term

Short term

Hierarchy of multiple parallel loops

gamma rhythms 30-100 Hz

Adaptive Critic structure

Theta waves 4-8 Hz

Reinforcement learning

Motor control 200 Hz

Sun Tz bin fa孙子兵法

500 BC

Manufacturing as the Interactions of Multiple AgentsEach machine has it own dynamics and cost functionNeighboring machines influence each other most stronglyThere are local optimization requirements as well as global necessities

( )i

i i i i i i ij j jj N

A d g B u e B u

12

0

( (0), , ) ( )i

T T Ti i i i i ii i i ii i j ij j

j N

J u u Q u R u u R u dt

Each process has its own dynamics

And cost function

Each process helps other processes achieve optimality and efficiency

F.L. Lewis, H. Zhang, A. Das, K. Hengster-Movric, Cooperative Control of Multi-Agent Systems: Optimal Design and Adaptive Control, Springer-Verlag, 2013

Key Point

Lyapunov Functions and Performance IndicesMust depend on graph topology

Hongwei Zhang, F.L. Lewis, and Abhijit Das“Optimal design for synchronization of cooperative systems: state feedback, observer and output feedback,”IEEE Trans. Automatic Control, vol. 56, no. 8, pp. 1948-1952, August 2011.

H. Zhang, F.L. Lewis, and Z. Qu, "Lyapunov, Adaptive, and Optimal Design Techniques for Cooperative Systems on Directed Communication Graphs," IEEE Trans. Industrial Electronics, vol. 59, no. 7, pp. 3026‐3041, July 2012.

Real‐Time Solution of Multi‐Player NZS Games

1

( ) ( )N

j jj

x f x g x u

Multi‐Player Nonlinear Systems

Optimal control *1 2

10

( (0), , , ) min ( ( ) ) ;i

NT

i N i i ij ij

V x Q x R dt i N

* * * * * * * *1 2 1 1 2( , , ,..., ) ( , , ,..., ),i i i iV V V i N Nash equilibrium

1

1

0 ,N

T T Ti c c i i j j jj ij jj j j

j

P A A P Q P B R R R B P i N

Requires Offline solution of coupled Hamilton‐Jacobi –Bellman  eqs.

Kyriakos Vamvoudakis

Continuous‐time,  N players

112

1

0 ( ) ( ) ( ) ( )N

T Ti j jj j j

j

V f x g x R g x V

114

1

( ) ( ) ( ) , (0) 0N

T T Ti j j jj ij jj j j i

j

Q x V g x R R R g x V V

Linear Quadratic Regulator Case‐ coupled AREs

These are hard to solveIn the nonlinear case, HJB generally cannot be solved

100

112( ) ( ) ,T

i ii i ix R g x V i N

Control policies

Lao Zi

The way that can be told is not the Constant WayThe name that can be named is not the Constant Name

For nameless is the true wayBeyond the myriad experiences of the world

To experience without intention is to sense the world

All experience is an archwherethrough gleams that untravelled landwhose margins fade forever as we move

Dao ke dao feichang daoMing ke ming feichang ming