2020 11 new rl for opt ctrl and graph games

Moncrief-O’Donnell Chair, UTA Research Institute (UTARI)The University of Texas at Arlington, USA

F.L. Lewis National Academy of Inventors

Talk available online at http://Lewisgroup.uta.edu

New Developments in Integral Reinforcement Learning:Continuous-time Optimal Control and Games

Invited byXie Shengli

SharonRachel

New Research ResultsIntegral Reinforcement Learning for Online Optimal ControlIRL for Online Solution of Multi-player GamesMulti‐Player Games on Communication Graphs Off‐Policy LearningExperience ReplayBio-inspired Multi-Actor CriticsOutput Synchronization of Heterogeneous MAS

Applications to:MicrogridRoboticsIndustry Process ControlUAV

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

Multi-player Games Occur in:Networked Systems Bandwidth AssignmentEconomicsControl 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

D. Vrabie, K. Vamvoudakis, and F.L. Lewis,Optimal Adaptive Control and DifferentialGames by Reinforcement LearningPrinciples, IET Press,2012.

BooksF.L. Lewis, D. Vrabie, and V. Syrmos,Optimal Control, third edition, John Wiley andSons, New York, 2012.New Chapters on:

Reinforcement LearningDifferential Games

F.L. Lewis and D. Vrabie,“Reinforcement learning and adaptive dynamic programming for feedback control,”IEEE Circuits & Systems Magazine, Invited Feature Article, pp. 32-50, Third Quarter 2009.

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

Multi‐player Game SolutionsIEEE Control Systems Magazine,Dec 2017

How can one do Policy Iteration for Unknown Continuous‐Time Systems?What is Value Iteration for Continuous‐Time systems?How can one do ADP for CT Systems?

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

x x x

)()())(,()),(,( 1 khkhkkkk xVxVxhxrhxVxH Directly leads to temporal difference techniques System dynamics does not occur Two occurrences of value allow APPROXIMATE DYNAMIC PROGRAMMING methods

Leads to off‐line solutions if system dynamics is knownHard to do on‐line learning

Discrete‐Time System Hamiltonian Function

Continuous‐Time System Hamiltonian Function

How to define temporal difference? System dynamics DOES occur Only ONE occurrence of value gradient

RL ADP has been developed for Discrete-Time Systems1 ( , )k k kx f x u

( , )x f x u

Four ADP Methods proposed by Paul Werbos

Heuristic dynamic programming

Dual heuristic programming

AD Heuristic dynamic programming

AD Dual heuristic programming

(Watkins Q Learning)

Critic NN to approximate:

Value

Gradient xV

)( kxV Q function ),( kk uxQ

GradientsuQ

xQ

,

Action NN to approximate the Control

Bertsekas- Neurodynamic Programming

Barto & Bradtke- Q-learning proof (Imposed a settling time)

Discrete-Time SystemsAdaptive (Approximate) Dynamic Programming

Value Iteration

Bertsekas- Neurodynamic Programming

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

Model‐Based SolutionFull 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 HJB equation

Matrix equation

xux Ax Bu

SystemControlK

PBPBRQPAPA TT 10

1 TK R B P

On-line real-timeControl Loop

Off-line Design LoopUsing Riccati equation

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

MATLAB Routine LQR(A,B,Q,R)

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.

Reinforcement Learning turns out to be the key to this!

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

Bring together Optimal Control and Adaptive Control

SystemAdaptiveLearning system

ControlInputs

outputs

environmentTuneactor

Reinforcementsignal

Actor

Critic

Desiredperformance

Reinforcement learningIvan Pavlov 1890s

Actor-Critic Learning

We want OPTIMAL performance- ADP- Approximate Dynamic Programming

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.

Optimality in Biological Systems

1 Tu R B Px Kx

BuAxx

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

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


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 equation10 T T T T T Tx PAx x A Px x Qx x PBR B Px

u Kx Given any stabilizing FB policy

0 ( , , ) ( ) ( )T T TVH x u x A BK P P A BK Q K RK xx

Lyapunov equation

Full system dynamics must be knownOff‐line solution

Riccati equation

PBPBRQPAPA TT 10

1 Tu R B Px Kx

BuAxx

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

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

Policy Iterations for the Linear Quadratic Regulator

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

u Kx

System

Cost


Given any stabilizing FB policy

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

Optimal Control is

Algebraic Riccati equation

LQR Policy iteration = Kleinman algorithm

1. For a given control policy solve for the cost:

2. Improve policy:

If started with a stabilizing control policy the matrix monotonically converges to the unique positive definite solution of the Riccati equation.

Every iteration step will return a stabilizing controller. The system has to be known.

ju K x

0 T Tj j j j j jA P P A Q K RK

11

Tj jK R B P

j jA A BK

0K jP

Kleinman 1968

Bellman eq. = Lyapunov eq.

OFF‐LINE DESIGNMUST SOLVE LYAPUNOV EQUATION AT EACH STEP.

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

CT Systems‐ 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

Problem‐ System dynamics shows up in Hamiltonian

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

xV T

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


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

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

jj j

Vf x h x r x h x

x

(0) 0jV

1121 ( ) ( ) jT

j

Vh x R g x

x

Policy Iteration Solution

Pick stabilizing initial control policy

Policy Evaluation ‐ Find cost, Bellman eq.

Policy improvement ‐ Update control

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= US Patent

Work of Draguna Vrabie 2009Integral Reinforcement Learning

Bad Bellman Equation

Good Bellman Equation

To avoid solving HJB equation

( ( )) ( , ) ( ( ))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 controlAutomatica 2009, Neural Networks 2009

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

Tk k k k

t

V x t Q x u Ru dt V x t T

Approximate value by Weierstrass Approximator Network ( )TV W x

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

T T Tk k k k

t

W x t Q x u Ru dt W x t T

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

T Tk k k

t

W x t x t T Q x u Ru dt

regression vector Reinforcement on time interval [t, t+T]

kWNow use RLS or batch least-squares along the trajectory to get new weights

Then find updated FB1 11 1

2 21 1( ( ))( ) ( ) ( )

( )

TT Tk

k k kV x tu h x R g x R g x Wx x t

Approximate Dynamic Programming Implementation

Direct Optimal Adaptive Control for Partially Unknown CT Systems

Value Function Approximation (VFA) to Solve Bellman Equation– Paul Werbos (ADP), Dimitri Bertsekas (NDP)

Scalar equation with vector unknowns

Same form as standard System ID problems in Adaptive Control

Optimal ControlandAdaptive Controlcome togetherOn this slide.Because of RL

CT Policy Iteration – How to implement online?Linear Systems Quadratic Cost‐ LQR

1 111 12 11 121 2 1 2

2 212 22 12 22

1 2 1 2

1 2 1 211 12 22 11 12 22

2 2 2 2

( ) ( )

( ) ( )( ) ( ) ( ) ( )

( ) ( )

( ) ( )2 2( ) ( )

( ) ( )t t T

Tk

p p p px t x t Tx t x t x t T x t T

p p p px t x t T

x xp p p x x p p p x x

x x

p x t x t T

Quadratic basis set

( ) ( ) ( )( ) ( ) ( ) ( )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

Policy evaluation‐ solve IRL Bellman Equation

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

T T T Tk k k k

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

( ( )) ( ) ( ) ( )T TV x t x t Px t W x Value function is quadratic

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

T T T Tk k k k

tp t p x t x t T x Q L RL x d

( , )t t T

Same form as standard System ID problems

11 12 22TW p p p

( )x

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

T T Tk k k k

t

W x t W x t x t T Q x u Ru dt t

2

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

T T Tk k k k

t T

W x t T W x t T x t T Q x u Ru dt t T

3

2

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

T T Tk k k k

t T

W x t T W x t T x t T Q x u Ru dt t T

11 12

12 22

p pp p

11 12 22TW p p p

Solving the IRL Bellman Equation

Need data from 3 time intervals to get 3 equations to solve for 3 unknowns

Now solve by Batch least-squares

Or can use Recursive Least-Squares (RLS)

Solve for value function parameters

Put together

( ( )) ( ( )) ( ( 2 )) ( ) ( ) ( 2 )TkW x t x t T x t T t t T t T

t t+T

observe x(t)

apply uk=Kkx

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=Kkx


update P

t+3T

observe x(t+2T)

apply uk=Kkx


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

( ) ( )k ku t K x t

Continuous‐time control with discrete gain updates

t

Kk

k0 1 2 3 4 5

Reinforcement Intervals T need not be the sameThey can be selected on‐line in real time

Gain update (Policy)

Control

T

Interval T can vary

Persistence of Excitation

Regression vector must be PE

Relates to choice of reinforcement interval T

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

T Tk k k

t

W x t x t T Q x u Ru dt

Implementation Policy evaluationNeed to solve online

Add a new state= Integral Reinforcement

T Tx Q x u R u

This is the controller dynamics or memory

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

T T Tk k k

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

Direct Optimal Adaptive Controller

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

Draguna Vrabie

IRL requires aDynamicControlSystemw/ 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

CT time Actor‐Critic Structure

Actor / Critic structure for CT Systems

Theta waves 4-8 Hz

Reinforcement learning

Motor control 200 Hz

Optimal Adaptive IRL for CT systems

A new structure of adaptive controllers

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

k k kt


1121 1( ) ( )T k

k kVu h x R g xx

D. Vrabie, 2009

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 ControlDDO

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


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

Simulation 1‐ F‐16 aircraft pitch rate controller

1.01887 0.90506 0.00215 00.82225 1.07741 0.17555 0

0 0 1 1x x u

Stevens and Lewis 2003

*1 11 12 13 22 23 33[p 2p 2p p 2p p ]

[1.4245 1.1682 -0.1352 1.4349 -0.1501 0.4329]

T

T

W

PBPBRQPAPA TT 10 ARE

Exact solution

,Q I R I

][ eqx

Select quadratic NN basis set for VFA

p. 304

Use short period approximation dynamics

0 0.5 1 1.5 2-0.3

-0.2

-0.1

0Control signal

Time (s)

0 0.5 1 1.5 2-0.4

-0.2

0Controller parameters

Time (s)

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4System states

Time (s)

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

Critic parameters

Time (s)

P(1,1)P(1,2)P(2,2)P(1,1) - optimalP(1,2) - optimalP(2,2) - optimal

Simulations on: F‐16 autopilot

A matrix not needed

Converge to SS Riccati equation soln

Solves ARE online without knowing A

PBPBRQPAPA TT 10

1/ / 0 0 00 1/ 1/ 0 0

,1/ 0 1/ 1/ 1/

0 0 0 0

p p p

T T

G G G G

E

T K TT T

A BRT T T T

K

x Ax Bu

( ) [ ( ) ( ) ( ) ( )]Tg gx t f t P t X t E t

FrequencyGenerator outputGovernor positionIntegral control

Simulation 2: Load Frequency Control of Electric Power system

PBPBRQPAPA TT 10

0.4750 0.4766 0.0601 0.4751

0.4766 0.7831 0.1237 0.3829

0.0601 0.1237 0.0513 0.0298

0.4751 0.3829 0.0298 2.3370

.AREP

ARE

ARE solution using full dynamics model (A,B)

A. Al-Tamimi, D. Vrabie, Youyi Wang

0.4750 0.4766 0.0601 0.4751

0.4766 0.7831 0.1237 0.3829

0.0601 0.1237 0.0513 0.0298

0.4751 0.3829 0.0298 2.3370

.AREP

PBPBRQPAPA TT 10

0 10 20 30 40 50 60-0.5

0

0.5

1

1.5

2

2.5P matrix parameters P(1,1),P(1,3),P(2,4),P(4,4)

Time (s)

P(1,1)P(1,3)P(2,4)P(4,4)

( ), ( ), ( : )x t x t T t t T Fifteen data points

Hence, the value estimate was updated every 1.5s.

IRL period of T= 0.1s.

Solves ARE online without knowing A

0.4802 0.4768 0.0603 0.4754

0.4768 0.7887 0.1239 0.3834

0.0603 0.1239 0.0567 0.0300

0.4754 0.3843 0.0300 2.3433

.critic NNP

0 1 2 3 4 5 6-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1System states

Time (s)

Reinforcement Learning Software is online at

https://lewisgroup.uta.edu/

Software from sponsored research

Linked under

Optimal Control Design Allows a Lot of Design Freedom

Optimal Control for Constrained Input Systems

This is a quasi-norm

2

0

2 ( )u

Tq

u d

Weaker than a norm –homogeneity property is replaced by the weaker symmetry property

qqxx

(Used by Lyshevsky for H2 control)

Control constrained by saturation function tanh(p)

p

1

-1

0 0

( , ) ( ) 2 ( )u

TJ u d Q x d dt

Encode constraint into Value function

Then Is BOUNDED1 ( )T Vu R g xx

Murad Abu-Khalaf

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

k k kt


IRL Policy iteration Initial stabilizing control is needed

Cost update

Control gain update

Policy evaluation‐ IRL Bellman Equation

Policy improvement11

21 1( ) ( )T kk k

Vu h x R g xx

CT PI Bellman eq.= Lyapunov eq.

IRL Value Iteration - Draguna Vrabie

Converges to solution to HJB eq.dx

dVggRdx

dVxQfdx

dV TTT *

1*

41

*

)(0

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

k k kt


IRL Value iteration Initial stabilizing control is NOT needed

Cost update

Control gain update

Value evaluation‐ IRL Bellman Equation

Policy improvement1 11

21 1( ) ( )T kk k

Vu h x R g xx

CT VI Bellman eq.

Converges if T is small enough

Kung Tz 500 BCConfucius

ArcheryChariot driving

MusicRites and Rituals

PoetryMathematics

孔子Man’s relations to

FamilyFriendsSocietyNationEmperorAncestors

Tian xia da tongHarmony under heaven


Theta waves 4-8 Hz

Reinforcement learning


Optimal Adaptive IRL for CT systems


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

k k kt


1121 1( ) ( )T k

k kVu h x R g xx

D. Vrabie, 2009


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


theta rhythms 4-10 HzDeliberativeevaluation

control

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


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

Kenji Doya

86

Synchronous Real‐time Data‐driven Optimal Control


Theta waves 4-8 Hz


Optimal Adaptive Integral Reinforcement Learning for CT systems


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

k k kt


1121 1( ) ( )T k

k kVu h x R g xx

D. Vrabie, 2009

Policy Iteration gives the structure needed for online optimal solution

1 1ˆ( ) ( ) ( )TV x W x x Take VFA as

Then IRL Bellman eq

becomes

Critic Network

1 1̂, ( ) TV x W

Action Network for Control Approximation

111 22

ˆ( ) ( ) ,T Tu x R g x W

Synchronous Online Solution of Optimal Control for Nonlinear SystemsKyriakos Vamvoudakis

( ( )) ( ) ( ( ))t

Tk k

t T

V x t Q x u Ru dt V x t T

1 1

ˆ ˆ( ( )) ( ) ( ( ))t

T T Tk k

t T

W x t T Q x u Ru dt W x t

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

11 2 21ˆ ˆ ˆ( ( )) ( ) 04

tT T

t T

x t W Q x W D W

Define

Bellman eq becomes

Then there exists an N0 such that, for the number of hidden layer units

Theorem (Vamvoudakis & Vrabie)‐ Online Learning of Nonlinear Optimal Control

Let be PE. Tune critic NN weights as

Tune actor NN weights as

0N N

the closed‐loop system state, the critic NN error

and the actor NN error are UUB bounded. 2 1 2ˆW W W

1 1 1̂W W W

Learning the Value

Learning the control policy

Data‐driven Online Synchronous Policy Iteration using IRLDoes not need to know f(x)

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

11 1 1 2 22

( ( )) 1ˆ ˆ ˆ ˆ( ( )) ( )41 ( ( )) ( ( ))

tT T

Tt T

x tW a x t W Q x W D W dx t x t

1 12 2 2 2 1 1 2 2 14 2( ( ))ˆ ˆ ˆ ˆ ˆ( ( )) ( )

1 ( ( )) ( ( ))

TT

T

x tW a F W F x t W a D x W Wx t x t


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

Vamvoudakis & Vrabie

1 11 1 1 2 2 2

1 1( ) ( ) ( ) ( ).2 2

T TL t V x tr W a W tr W a W

Lyapunov energy‐based Proof:

1140 ( )

T TTdV dV dVf Q x gR g

dx dx dx

V(x)= Unknown solution to HJB eq.

2 1 2ˆW W W

1 1 1̂W W W

1 1 1( , , ) ( ) ( )T THH x W u W f gu Q x u Ru

W1= Unknown LS solution to Bellman equation for given N

Guarantees stability

Adaptive Critic structure Reinforcement learning

Two Learning NetworksTune them Simultaneously

Synchronous Online Solution of Optimal Control for Nonlinear Systems

A new form of Adaptive Control with TWO tunable networks


K.G. Vamvoudakis and F.L. Lewis, “Online actor-critic algorithm to solve the continuous-time infinitehorizon optimal control problem,” Automatica, vol. 46, no. 5, pp. 878-888, May 2010.

11 1 1 2 22

( ( )) 1ˆ ˆ ˆ ˆ( ( )) ( )41 ( ( )) ( ( ))

tT T

Tt T

x tW a x t W Q x W D W dx t x t

1 12 2 2 2 1 1 2 2 14 2( ( ))ˆ ˆ ˆ ˆ ˆ( ( )) ( )

1 ( ( )) ( ( ))

TT

T

x tW a F W F x t W a D x W Wx t x t

System

Action network

Policy Evaluation(Critic network)cost

The Adaptive Critic Architecture

Adaptive CriticsValue update- solve Bellman eq.

Control policy update

Critic and Actor tuned simultaneouslyLeads to ONLINE FORWARD-IN-TIME implementation of optimal control

Optimal Adaptive Control

1 1( ) ( )TV x W x

111 22

ˆ( ) ( )T Tu x R g x W

A New Adaptive Control Structure with Multiple Tuned Loops

A New Class of Adaptive Control

Plantcontrol output

Identify the Controller-Direct Adaptive

Identify the system model-Indirect Adaptive

Identify the performance value-Optimal Adaptive

)()( xWxV T

Simulation 1‐ F‐16 aircraft pitch rate controller

1.01887 0.90506 0.00215 00.82225 1.07741 0.17555 0

0 0 1 1x x u

Stevens and Lewis 2003

*1 11 12 13 22 23 33[p 2p 2p p 2p p ]

[1.4245 1.1682 -0.1352 1.4349 -0.1501 0.4329]

T

T

W

PBPBRQPAPA TT 10

Solves ARE online

Exact solution

1̂( ) [1.4279 1.1612 -0.1366 1.4462 -0.1480 0.4317] .TfW t

Algorithm converges to

,Q I R I

Must add probing noise to get PE

111 22

ˆ( ) ( ) ( )T Tu x R g x W n t

2ˆ ( ) [1.4279 1.1612 -0.1366 1.4462 -0.1480 0.4317]TfW t

(exponentially decay n(t))

1

2 1

3 11 11 12 2 2

2

3 2

3

2 0 0 1.42790 1.1612

0 0 -0.1366ˆ ( ) 01.44620 2 01-0.148000.43170 0 2

T

T

T

xx xx x

u x R B Px Rxx x

x

][ eqx

Select quadratic NN basis set for VFA

Critic NN parameters‐Converge to ARE solution

System states

Simulation 2. – Nonlinear System

2( ) ( ) ,x f x g x u x R

21 2

1 2

1

1

0.5 0.5 (1 ( )( )

cos(2 ) 2

0 ( )

)

.cos(2 ) 2

x xf x

x

g x

x x

x

,Q I R I

* 2 21 2

1( )2

V x x x Optimal Value

*1 2( ) (cos(2 ) 2) .u x x x Optimal control

2 21 1 1 2 2( ) [ ] ,Tx x x x x Select VFA basis set

1̂( ) [0.5017 -0.0020 1.0008] .TfW t


2ˆ ( ) [0.5017 -0.0020 1.0008] .T

fW t 111

2 2 121

2

2 0 0.50170ˆ ( ) -0.0020

cos(2 ) 20 2 1.0008

TT x

u x R x xx

x

Nevistic V. and Primbs J. A. (1996)Converse optimal

dxdVggR

dxdVxQf

dxdV T

TT *1

*

41

*

)(0

Solves HJB equation online

Critic NN parameters states

Optimal value fn. Value fn. approx. error Control approx error

Data‐driven Online Solution of Differential GamesSynchronous Solution of Multi‐player Non Zero‐sum Games

Multi‐player Game SolutionsIEEE Control Systems Magazine,Dec 2017

Multi‐player Differential Games

Sun Tz bin fa孙子兵法

500 BC

Multi‐player Differential Games

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

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, Automatica 2011

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 Riccati Equations

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

118

112( ) ( ) ,T

i ii i ix R g x V i N

Control policies

Real‐Time Solution of Multi‐Player Games

1 210

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

Ti N i i ij i

j

V x Q x R dt i N

Value functions

Solve Bellman eq.

Policy Iteration Solution:

11

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

k k k T i kN i j j i

j

r x V f x g x V i N

1 112( ) ( ) ,k T k

i ii i ix R g x V i N Policy Update

Non‐Zero Sum Games – Synchronous Policy Iteration Kyriakos Vamvoudakis

Convergence has not been provenHard to solve Hamiltonian equationBut this gives the structure we need for online Synchronous PI Solution

Differential equivalent gives coupled Bellman eqs.

11 1

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

T Ti j ij j i j j i i N

j jQ x u R u V f x g x u H x V u u i N


11 1

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

T Ti j ij j i j j i i N

j jQ x u R u V f x g x u H x V u u i N

Coupled Bellman eqs.

112( ) ( ) ,T

i ii i ix R g x V i N

Control policies

Need to solve online:

Policy Iteration gives the structure needed for online solution

Each player needs 2 NN – a Critic and an Actor

Real‐Time Solution of Multi‐Player Games

N Critic Neural Networks for VFA 1 1 1ˆ ˆ( ) ( ) ,TV x W x 2 2 2

ˆ ˆ( ) ( )TV x W x

11 1 1 1 1 1 11 1 2 12 22

1 1

ˆ ˆ[ ( ) ]( 1)

T T TTW a W Q x u R u u R u

13 3 2 3 1 3 1 1 11 21 11 1 3 2 2 1 3 1 1

1 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ{( ) ( ) ( ) ( ) }4 4

T T T T T TW F W F W g x R R R g x W m W D x W m W

N Actor Neural Networks

On‐Line Learning – for Player 1:

111 11 1 1 32

ˆ( ) ( ) ,T Tu x R g x W

Online Synchronous PI Solution for Multi‐Player Games

Learns Bellman eq. solution

Kyriakos Vamvoudakis

112 22 2 2 42

ˆ( ) ( )T Tu x R g x W

2-player case

Each player needs 2 NN – a Critic and an Actor

Learns control policy

Player 1 Player 2

Convergence is proven using Lyapunov functions

K.G. Vamvoudakis and F.L. Lewis, “Multi-Player Non-Zero Sum Games: Online Adaptive LearningSolution of Coupled Hamilton-Jacobi Equations,” Automatica, vol. 47, pp. 1556-1569, 2011.

1 11 1 1 2 2 2

1 1( ) ( ) ( ) ( ).2 2

T TL t V x tr W a W tr W a W

Lyapunov energy‐based Proof:

1140 ( )

T TTdV dV dVf Q x gR g

dx dx dx

V(x)= Unknown solution to HJB eq.

2 1 2ˆW W W

1 1 1̂W W W

1 1 1( , , ) ( ) ( )T THH x W u W f gu Q x u Ru

W1= Unknown LS solution to Bellman equation for given N

Guarantees stability

Multi‐Agent Learning Guaranteed Convegence Proof

Process i Cost Function Optimization

Process j1 controlupdate

Process i controlupdate

Process j2 controlupdate

Optimal Performance of Each Process Depends on the Control of its Neighbor Processes

Control Policy of Each Process Depends on the Performance of its Neighbor Processes

Process i Cost Function Optimization

Process i controlupdate

Process j2 Cost Function Optimization

Process j1 Cost Function Optimization

Multi-player Games for Multi-Process Optimal ControlData-Driven Optimization (DDO)

Simulation. – Nonlinear System – 2‐player game 2( ) ( ) ( ) ,x f x g x u k x d x

1 2 11 22 12 212 2 , 2 2 , 2 2Q Q I R R I R R I

Optimal Value s*

1 2( ) 2(cos(2 ) 2)u x x x Optimal Policies

Select VFA basis set


2 4ˆ ˆ( ) [0.2514 0.0006 0.5001] ( )T

f fW t W t

2 21 1 12 1 22 4

22

1 2 1

1

4

1

( )cos(2 ) 2 (4 ) 2

0 0 ( ) , ( )

( ) (sin )

(s.

cos(2 ) 2 (in )4 ) 2

xf x

x x x

g x k

x

x x

x x

x

* 2 21 1 2

1( )2

V x x x

* 21 2( ) (sin(4 ) 2)d x x x

2 21 2 1 1 2 2( ) ( ) [ ]x x x x x x

1 3ˆ ˆ( ) [0.5015 0.0007 1.0001] ( )T

f fW t W t

111

11 2 121

2

2 0 0.50150

ˆ( ) 0.0007cos(2 ) 2

0 2 1.0001

TT x

u x R x xx

x

111

22 2 12 21

2

2 0 0.25140ˆ( ) 0.0006sin(4 ) 2 0 2 0.5001

TT x

d x R x xx x

* 2 22 1 2

1 1( )4 2

V x x x

Solves HJB equations online11

21

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

Critic 1 NN parameters Critic 2 NN parameters

Evolution of the States 3D approximation error of control for player 1.

3D approximation error value for player 1.

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.

,i i i ix Ax B u

0 0x Ax

0( ) ( ),ix t x t i

0( ) ( ),i

i ij i j i ij N

e x x g x x

( ) ,nix t ( ) im

iu t

Graphical GamesSynchronization‐ Cooperative Tracker Problem

Node dynamics

Target generator dynamics

Synchronization problem

Local neighborhood tracking error (Lihua Xie)

x0(t)

( )i


A d g B u e B u

12

0

( (0), , ) ( )i


j N


12

0

( ( ), ( ), ( ))i i i iL t u t u t dt

Local nbhd. tracking error dynamics

Define Local nbhd. performance index

Local agent dynamics driven by neighbors’ controls

Values driven by neighbors’ controls

K.G. Vamvoudakis, F.L. Lewis, and G.R. Hudas, “Multi-Agent Differential Graphical Games: online adaptive learning solution for synchronization with optimality,” Automatica, vol. 48, no. 8, pp. 1598-1611, Aug. 2012.

M. Abouheaf, K. Vamvoudakis, F.L. Lewis, S. Haesaert, and R. Babuska, “Multi-Agent Discrete-Time GraphicalGames and Reinforcement Learning Solutions,” Automatica, Vol. 50, no. 12, pp. 3038-3053, 2014.

OPTIMAL SYNCHRONIZATION

1u

2u

iu Control action of player i

Value function of player i

New Differential Graphical Game

( )i


A d g B u e B u

State dynamics of agent i

Local DynamicsLocal Value Function

Only depends on graph neighbors

12

0

( (0), , ) ( )i


j N


1

N

i ii

z Az B u

12

10

( (0), , ) ( )N

T Ti i i j ij j

j

J z u u z Qz u R u dt

1u

2u

iu Control action of player i

Central Dynamics

Value function of player i

Standard Multi-Agent Differential Game

Central DynamicsLocal Value Functiondepends on ALL

other control actions

* * *( , ) ( , ), ,i j G j i j G jJ u u J u u i j N

New Definition of Nash Equilibrium for Graphical Games

* * *1 2, ,...,u u u

* * * *( , ) ( , ),i i i G i i i G iJ J u u J u u i N

Def: Interactive Nash equilibrium

are in Interactive Nash equilibrium if

2. There exists a policy such that ju

1.

That is, every player can find a policy that changes the value of every other player.

1. They are in Nash equilibrium

2. Interaction Condition

To restore symmetry of Nash Equilibrium

1 1 12 2 2( , , , ) ( ) 0

i i

TT T Ti i

i i i i i i i i i ij j j i ii i i ii i j ij ji i j N j N

V VH u u A d g B u e B u Q u R u u R u

10 ( ) Ti ii i i ii i

i i

H Vu d g R B

u

2 1 2 1 11 1 12 2 2( ) ( ) 0,

i

TT Tj jc T T Ti i i

i i ii i i i i ii i j j j jj ij jj ji i i j jj N

V VV V VA Q d g B R B d g B R R R B i N

2 1 1( ) ( ) ,i

jc T Tii i i i i ii i ij j j j jj j

i jj N

VVA A d g B R B e d g B R B i N

* *( , , , ) 0ii i i i

i

VH u u

12( ( )) ( )

i

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

j Nt

V t Q u R u u R u dt

Value function

Differential equivalent (Leibniz formula) is Bellman’s Equation

Stationarity Condition

1. Coupled HJ equations

where

Graphical Game Solution Equations

Now use Synchronous PI to learn optimal Nash policies online in real‐time as players interact

Distributed Multi‐Agent Learning Proofs

Kyriakos Vamvoudakis

Online Solution of Graphical Games

Use Reinforcement Learning Convergence Results

POLICY ITERATION

Kyriakos VamvoudakisMulti‐agent Learning Convergence proofs

Data‐driven Online Solution of Differential GamesZero‐sum 2‐Player Games and H‐infinity Control

( , ) ( ) ( ) ( )( )

x f x u f x g x u k x dy h x

System

Cost

Online Zero‐Sum Differential Games

22( ( ), , ) ( , , )T T

t t

V x t u d h h u Ru d dt r x u d dt

H-infinity Control

2 players


K.G. Vamvoudakis and F.L. Lewis, “Online solution of nonlinear two-player zero-sum games using synchronous policy iteration,” Int. J. Robust and Nonlinear Control, vol. 22, pp. 1460-1483, 2012.

D. Vrabie and F.L. Lewis, “Adaptive dynamic programming for online solution of a zero-sum differential game,” J Control Theory App., vol. 9, no. 3, pp. 353–360, 2011

Optimal control/dist. policies found by stationarity conditions

Game saddle point solution found from Hamiltonian ‐ ZS Game BELLMAN EQUATION

0 , 0H Hu d

112 ( )Tu R g x V 2

1 ( )2

Td k x V

22( , , , ) ( ( ) ( ) ( ) ) 0TT TVH x u d h h u Ru d V f x g x u k x dx

HJI equation * *

12

0 ( , , , )1 1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )4 4

T T T T T T

H x V u d

h h V x f x V x g x R g x V x V x kk V x

0)0( V

Optimal control/dist. policies found by stationarity conditions

HJI equation

Game saddle point solution found from Hamiltonian ‐ ZS Game BELLMAN EQUATION

(‘Nonlinear Game Riccati’ equation)

0 , 0H Hu d

112 ( )Tu R g x V

21 ( )

2Td k x V

* *

12

0 ( , , , )1 1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )4 4

T T T T T T

H x V u d

h h V x f x V x g x R g x V x V x kk V x

22( , , , ) ( ( ) ( ) ( ) ) 0TT TVH x u d h h u Ru d V f x g x u k x dx

Minimizes H(.)

Maximizes H(.)

Minmax solutions

2

0

2

0

2

0

2

0

2

)(

)(

)(

)(

dttd

dtuhh

dttd

dttz T

System( ) ( ) ( )( )

TT T

x f x g x u k x dy h x

z y u

( )u l x

d

u

z

x control

Performance output disturbance

Find control u(t) so that For all L2 disturbancesAnd a prescribed gain

L2 Gain Problem

H‐Infinity Control Using Reinforcement Learning

Zero‐Sum differential game ‐ Nature as the opposing player

Disturbance Rejection

The game has a unique value (saddle‐point solution) iff the Nash condition holds

1. For a given control policy solve for the value ( )ju x 1( ( ))jV x t

111 12( ) ( )T

j ju x R g x V

Double Policy Iteration Algorithm to Solve HJI

Start with stabilizing initial policy 0 ( )u x

ZS game Bellman eq.

3. Improve policy:

• Convergence proved by Van der Schaft if can solve nonlinear Lyapunov equation exactly

• Abu Khalaf & Lewis used NN to approximate V for nonlinear systems and proved convergence

ONLINE SOLUTION‐ Use IRL to solve the ZS Bellman eq. and update the actors at each step

220 ( )( )T iT i T ij j j jh h V x f gu kd u Ru d

Add inner loop to solve for available storage

12

1 ( )2

i T ijd k x V

2. Set . For i=0,1,… solve for 1( ( )),i ijV x t d 0 0d

On convergence set 1( ) ( )ij jV x V x

Murad AbukhalafDraguna Vrabie

Actor‐Critic structure ‐ three time scales

x

u2 1 0;x Ax B u B w x

System

Controller/Player 1

w( 1)T i kuu B P x 1

1T i

ww B P x

Disturbance/Player 2

CriticLearning procedure

ˆ ˆ, if =1

ˆ ˆ ˆ ˆ, if >1

T T T

T T

V x C Cx u u i

V w w u u i

T T

1 2

1iuP ( 1) 1 ( 1)i k i i k

u u uP P Z

V

iuP

x

( ) 1 ( )i k i i ku u uP P Z

2

1/ / 0 0 00 1/ 1/ 0 0

,1/ 0 1/ 1/ 1/

0 0 0 0

p p p

T T

G G G G

E

T K TT T

A BRT T T T

K

2 1x Ax B u B d

( ) [ ( ) ( ) ( ) ( )]Tg gx t f t P t X t E t

FrequencyGenerator outputGovernor positionIntegral control

-0.0665 8 0 0 0 -3.663 3.663 0

, [0 0 13.7355 0] . -6.86 0 -13.736 -13.736 0.6 0 0 0

TA B

Simulation- H-inf control for Electric Power Plant- LFC

A is unknownB1, B2 are known

1 8 0 0 0 TB

2 2 1 10 ( )T T T TA P PA C C P B B B B P

Load disturbance

Simulation result – Electric Power Plant LFC

• System – Power plant ‐ internally stable system; – system state

(incremental changes of: frequency deviation, generator output, governor position and integral control)

– Player 1 ‐ controller ; Player 2 – load disturbance

• Nash equilibrium solution

• Online learned solution using ADP – after 5 updates of the parameters of Player 2

[ ( ) ( ) ( ) ( )]g gx f t P t X t E t

0.6036 0.7398 0.0609 0.5877 0.7398 1.5438 0.1702 0.5978 0.0609 0.1702 0.0502 0.0357 0.5877 0.5978 0.0357 2.3307

uP

5

0.6036 0.7399 0.0609 0.5877 0.7399 1.5440 0.1702 0.5979 0.0609 0.1702 0.0502 0.0357 0.5877 0.5979 0.0357 2.3307

uP

2 2 1 10 ( )T T T TA P PA C C P B B B B P Solves GARE online without knowing A

Parameters of the critic – ARE Solution elements

– Cost function learning using least squares

– Sampling integration time T=0.1 s

– The policy of Player 1 is updated every 2.5 s

– The policy of Player 2 is updated only when the policy of Player 1 has converged

– Number of updates of Player 1 before an update of Player 2

moments when Player 2 is updated

0 50 100 150 200 250 300

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5Parameters of the cost function of the game

Time (s)

P(1,1)P(1,2)P(2,2)P(3,4)P(4,4)

163

New Developments in IRL for CT SystemsQ Learning for CT SystemsExperience ReplayOff-Policy IRL

IRL with Experience ReplayHumans use memories of past experiences to tune current policies

( ) ( ( )) ( ( )) ( )x t f x t g x t u t= +

1

0( ( )) ( ( )) 2 ( tanh ( ))( )u

T

t

V x t Q x v Rdv dt l l t¥

-= +ò ò1

0( ) 2 tanh ( ) ( ) ( ( ) ( ) ) 0, (0) 0( )

uTTQ x v Rdv V x f x g x u Vl l-+ + + = =ò

1tanh (1 2 ) ( ) ( )( )Tu R g x V xl l* - *= -

1ˆ ˆ( ) ( )TV x W xf=

system

Value

1 1 12

( )ˆ ˆ( ) ( ) ( ) ( )(1 ( ) ( ))

( )T

T

tW t p t t W t

t t

fa f

f fD

= - +D+D D

Modares and Lewis, Automatica 2014Girish Chowdhary‐ concurrent learningSutton and Barto book

Bellman Equation

Action Update

VFA‐ Value Function Approximation

( ( )) ( ( )) ( ( ))x t x t x t Tf f fD = - -

1

0( ) 2 ( tanh ( ))( )

tu

T

t T

p t Q v Rdv dl l t-

-

= +ò ò

i/o Data Measurements

Standard Critic Weight Tuning

IRL Bellman Equation 1

0( ( )) ( ( )) 2 ( tanh ( )) ( ( ))( )

tu

T

t T

V x t T Q x v Rdv d V x tt l l t-

-

- = + +ò ò

110

( ( )) 2 ( tanh ( )) ( ( )) ( )( )t

uT T

B

t T

Q x v Rdv d W x t tt l l t f e-

-

+ + D ºò òBellman Eq gives Linear Equation for Weights

IRL with Experience ReplayHumans use memories of past experiences to tune current policies

1ˆ ˆ( ) ( )TV x W xf=

1 1 1 1 12 2

1

( )ˆ ˆ ˆ( ) ( ) ( ) ( ) ( )(1 ( ) ( )) (1 )

( ) ( )l

jT Tj jT T

j j j

tW t p t t W t p W t

t t

ffa f a f

f f f f=

DD= - +D +D

+D D +D D- å

NN weight tuning uses past samples

Improvements1. Speeds up convergence2. PE condition is milder

Modares and Lewis, Automatica 2014Girish Chowdhary‐ concurrent learningSutton and Barto book

VFA‐ Value Function Approximation

Data from Previous time intervals

( ( )) ( ( )) ( ( ))x t x t x t Tf f fD = - -

1

0( ) 2 ( tanh ( ))( )

tu

T

t T

p t Q v Rdv dl l t-

-

= +ò ò

i/o Data Measurements

Previous data

New Principles

Off-Policy Learning

Off‐policy RL

173

Behavior Policy System

Target policy

Ref.

Target policy and behavior policy are different

Humans can learn optimal policies while actually applying suboptimal policies

H. Modares, F.L. Lewis, and Z.-P. Jiang, “H-infinity Tracking Control of Completely-unknown Continuous-time Systems via Off-policyReinforcement Learning ,” IEEE Transactions on Neural Networks and Learning Systems, vol. 26, no. 10, pp. 2550-2562, Oct. 2015.

Ruizhuo Song, F.L. Lewis, Qinglai Wei, “Off-Policy Integral Reinforcement Learning Method to Solve Nonlinear Continuous-TimeMulti-Player Non-Zero-Sum Games,” IEEE Transactions on Neural Networks and Learning Systems, vol. 28, no. 3, pp. 704-713, 2017.

Bahare Kiumarsi, Frank L. Lewis, Zhong-Ping Jiang, “H-infinity Control of Linear Discrete-time Systems: Off-policy ReinforcementLearning,” Automatica, vol. 78, pp. 144-152, 2017.

Off-policy IRLHumans can learn optimal policies while actually applying suboptimal policies

( ) ( )x f x g x u

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

TJ x r x u d u dQ x Ru

[ [ [] [ ]] ]( ( )) ( ( )) ( )i i

t T t

t i i

T

t TJ x t J x t T Q x d Ru du

On-policy IRL

[ 1] 1 [ ]12

i T ixu R g J

[ ] [ ]( )i ix f gu g u u Off-policy IRL

[ ] [ ] [ ] [ ] [ 1] [ ]( ( )) ( ( )) ( ) 2 ( )t t tii i

t T t T t T

T i i T iJ x t J x t T Q x d R d du u u R u u

This is a linear equation for andThey can be found simultaneously online using measured data using Kronecker product and VFA

[ ]iJ [ 1]iu

1. Completely unknown system dynamics2. Can use applied u(t) for –

disturbance rejection – Z.P. Jiang ‐ R. Song and Lewis, 2015robust control – Y. Jiang & Z.P. Jiang, IEEE TCS 2012exploring probing noise – without bias ! ‐ J.Y. Lee, J.B. Park, Y.H. Choi 2012

DDO

system

value

Must know g(x)Learned optimaltarget policy

Applied behavior policy

Off‐policy for Multi‐player NZS Games

1( ) ( )N

jjx f x g x u

1 2 1( ( )) ( ( , , , , )) ( ( ) )N T

i i N i j ij jjt tV x t r x u u u d Q x u R u d

[ ] [ ]1 1

( ) ( ) ( )( )N Nk kj j jj j

x f x g x u g x u u

Off‐policy

[ ] [ ] [ ] [ ] [ 1] [ ]1 1

( ( )) ( ( 2 ( ))) ( )t T t T t TN Nkk k T k k T k

j ij j i ii j jji i it t jtu R u u R u uV x t T V x t Q x d d d

1. Solve online using measured data for 2. Completely unknown dynamics3. Add exploring noise with no bias

[ ] [ 1],k ki iV u

DDO

Angela Song and Lewis

On‐policy

Off-Policy Learning for Estimating Malicious Adversaries’ Hidden True Intent

( ) ( )k k k ki i i i i i i i i ix Ax Bu Dv B u u D v v

1 2 112( ( )) ( ( )) ( , , ) ( ) ( ) ( ) ( )

t T t Tk k k k k T k k T k

i i i i i i i i i ii i i i ii i it t

V x t V x t T r x u v dt u R u u v R v v dt

i i i i i ix Ax Bu Dv

MAS H‐infinity control

MA Systems

Cost

Off‐policy IRL

Two Opposing Teams

2 21 12 2( ( )) (x ) ( , , v )

i i

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

t tj j

V x t Q x u R u u R u v T v v T v dt r x u dt

1 12

1 1,2 2

k kk T k Ti ii i i i

i i

V Vu B v D

x x

Optimal Target policiesActual Behavior policiesOff‐Policy Bellman Eq.

LQ Tracker for Continuous‐time Systems – Follow a Leader

System dynamics

Assume the reference trajectory is generated by

Augmented system state

Augmented system

Value function

x Ax Buy C x

d dy F y

( )( )

( )d

x tX t

y t

1A B

X X u T X B uF

00 0

1 2 du K X K x K y= = +

Control

Optimal Tracker Solution by Reinforcement Learning

11 2 1

[ , ] TK K K R B P-= =-

11 1 1 1

0T T TT P TP P C QC PB R B Pg -+ - + - =

Tracker ARE

Optimal Control Solution

Optimal Target policy learnedActual Behavior policy applied

184

Output Synchronization of Heterogeneous MAS

185

Heterogeneous Multi‐Agents

Leader

Output regulation error

leader node x0

i i i i i

i i i

x A x B u

y C x

= +

=

0 0Sz z=

0 0y R z=

0( ) ( ) ( ) 0i it y t y t

Output regulator equations

i i i i i

i i

A B S

C R

P + G = P

P =

Dynamics are different, state dimensions can be differento/p reg eqs capture the common core of all the agents dynamicsAnd define a synchronization manifold

Output Synchronization of Heterogeneous MAS

Standard Solution

i i i i i

i i i

x A x B u

y C x

= +

=

0 0Sz z=

0 0y R z=

MAS Leader

Output regulator equations

i i i i i

i i

A B S

C R

P + G = P

P =Control

01

( ) ( )N

i i ij j i i ij

S c a gz z z z z z=

é ùê ú= + - + -ê úë ûå

1 1 1 1 2( ) ( )

i i i i i i i i i i i i i i i i iu K x K x K K x Kz z z z= -P +G = + G- P º +

Must know your dynamics And leader’s dynamics S,RAnd solve o/p regulator equations

0( ) ( ) 0,iy t y t i- "o/p regulation

,i iA B

Pioneered by Jie Huang

Optimal Output Synchronization of Heterogeneous MAS Using Off-policy IRLNageshrao, Modares, Lopes, Babuska, Lewis

i i i i i

i i i

x A x B u

y C x

= +

=

0 0Sz z=

0 0y R z=

0

( ) ( ) iT

n pT T

iX t x t z +é ù= Îê úë û

1i i i i iX T X B u= + 1

0,

0 0i i

i i

A BT B

S

é ù é ùê ú ê ú= =ê ú ê úê ú ê úë û ë û

( )

1 1( ( )) ( )

( ) ( )

i t T T Ti i i i i i i it i

Ti i i

V X t e X C Q C K W K X d

X t P X t

g t t¥ - -= +

=ò

1 2 0i i i i i iu K x K K Xz= + =

MAS Leader

Optimal Tracker ProblemAugmented Systems

Performance index

Control

Our Solution

Optimal Tracker Solution by Reinforcement Learning

11 2 1

[ , ] Ti i i i i iK K K W B P-= =-

11 1 1 1

0T T Ti i i i i i i i i i i i i iT P T P P C QC P B W B Pg -+ - + - =

Tracker ARE

Algorithm 1. On-policy IRL State-feedback algorithm

( )

0 0( ) ( ) ( ) ( ) ( ) ( )i i

t tt tT T Ti i i i i i i i it

e X t t P X t t X t P X t e y y Q y y ddg d g tk kd d t

+- - -+ + - = - - -ò

Policy Evaluation‐ Solve IRL Bellman equation

Policy Update‐1 1 1 1

1 2 1[ , ] T

i i i i i iK K K W B Pk k k k+ + + -= =-

Must know 1iB

Bellman equation is solved using RLS or batch LSIt requires a Persistence of Excitation (PE) condition that may be hard to satisfy

Theorem‐ Algorithm 1 converges to the solution to the ARE

1 2 0i i i i i iu K x K K Xz= + =Control

189

1 1 1

( ) ( ) ( )i i i i i i i i i i i i i i iX T B K X B u K X T X B u K Xk k k= + + - º + -

Off‐Policy RL

1i i i i i

X TX B u= +

Tracker dynamics

Rewrite as

( )

0 0

( ) 1

( ) ( ) ( ) ( ) ( ) ( )

2 ( )

i i

i

t tt tT T Ti i i i i i i i it

t t t Ti i i i i it

e X t t P X t t X t P X t e y y Q y y d

e u K X W K X d

dg d g tk k

d g t k k

d d t

t

+- - -

+ - - +

+ + - = - - -

+ -

ò

ò

Now the Bellman equation becomes

Extra term containing 1iKk+

Algorithm 2. Off-policy IRL Data-based algorithm

Iterate on this equation and solve for simultaneously at each step 1,i iP Kk k+

Note about probing noise If then ( )i i i i i iu K X e u K X ek k= + - =

Do not have to know any dynamics i i i i i

i i i

x A x B u

y C x

= +

=0 0Sz z=

0 0y R z=agent Or leader

190

11 1 1 1

0T T Ti i i i i i i i i i i i i iT P T P P C QC P B W B Pg -+ - + - =

Theorem‐ Off‐policy Algorithm 2 converges to the solution to the ARE

Then the solution to the output regulator equations i i i i i

i i

A B S

C R

P + G = P

P =

Is given by 1

11 12( )i i

iP P-P = -

Let 11 12

21 22

i i

i ii

P PP

P P

é ùê ú= ê úê úë û

Theorem‐ o/p reg eq solution

1

2 1 11 12( )i i

i i iK K P P-G = -

Do not have to know the Agent dynamics or the leader’s dynamics (S,R)

191

01

ˆ ( ) ( )N

i i i ij j i i ij

S c a gz z z z z z=

é ùê ú= + - + -ê úë ûå

vec 0

1

ˆ ( ) ( ) ( )N

i Si q i ij j i i ij

S I a gz z z z z=

é ùê ú= -G Ä - + -ê úë ûå

1 2ˆ i

i i i i i i i ii

xu K x K K X Kz

z

é ùê ú= + º º ê úê úë û

1 2 0i i i i i iu K x K K Xz= + =

To avoid knowledge of leader’s state in

Use adaptive observer for leader’s state

Then use control

Note that may not converge to actual leader’s matrix SˆiS

Do not have to know the leader’s dynamics (S,R)

New Principles

There Appear to be Multiple Reinforcement Learning Loops in the Brain

Multiple Actor-Critic Learning Structures

Narendra MMAC - Multiple Model Adaptive Control

Applications of Reinforcement Learning

Microgrid Control Human‐Robot Interactive LearningIndustrial process control‐Mineral grinding in Gansu, ChinaResilient Control to Cyber‐Attacks in Networked Multi‐agent SystemsDecision & Control for Heterogeneous MAS (different dynamics)

Intelligent Operational Control for Complex Industrial Processes

Professor Chai Tianyou

State Key Laboratory of Synthetical Automation for Process Industries

Northeastern UniversityMay 20, 2013

Jinliang Ding

1. Jinliang Ding, H. Modares, Tianyou Chai, and F.L. Lewis, "Data-based Multi-objective Plant-widePerformance Optimization of Industrial Processes under Dynamic Environments,” IEEE Trans. IndustrialInformatics, vol.12, no. 2, pp. 454-465, April 2016.

2. Xinglong Lu, B. Kiumarsi, Tianyou Chai, and F.L. Lewis, “Data-driven Optimal Control of OperationalIndices for a Class of Industrial Processes,” IET Control Theory & Applications, vol. 10, no. 12, pp. 1348-1356, 2016.

Production line for mineral processing plant

Mineral Processing Plant in Gansu China

RL for Human-Robot Interaction (HRI)1. H. Modares, I. Ranatunga, F.L. Lewis, and D.O. Popa, “Optimized Assistive Human-robot

Interaction using Reinforcement Learning,” IEEE Transactions on Cybernetics, vol. 46, no. 3,pp. 655-667, 2016.

2. I. Ranatunga, F.L. Lewis, D.O. Popa, and S.M. Tousif, "Adaptive Admittance Control forHuman-Robot Interaction Using Model Reference Design and Adaptive Inverse Filtering" IEEETransactions on Control Systems Technology, vol. 25, no. 1, pp. 278-285, Jan. 2017.

3. B. AlQaudi, H. Modares, I. Ranatunga, S.M. Tousif, F.L. Lewis, and D.O. Popa, “Modelreference adaptive impedance control for physical human robot interaction,” Control Theory andTechnology, vol. 14, no. 1, pp. 1-15, Feb. 2016.

PR2 meets Isura

2020 11 new rl for opt ctrl and graph games

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