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Moncrief-O’Donnell Chair, UTA Research Institute (UTARI)The University of Texas at Arlington, USA
and
F.L. Lewis National Academy of Inventors
Talk available online at http://www.UTA.edu/UTARI/acs
New Developments in Integral Reinforcement Learning:Continuous-time Optimal Control and Games
Qian Ren Consulting Professor, State Key Laboratory of Synthetical Automation for Process Industries
Northeastern University, Shenyang, China
Supported by :ONRUS NSF
Supported by :China NNSFChina Project 111
Invited byManfred MorariKonstantinos GatsisPramod Khargonekar George Pappas
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 Control
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
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
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
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
( , , , )X U P RRL for Markov Decision ProcessesX= states, U= controls
P= Probability of going to state x’ from state xgiven that the control is uR= Expected reward on going to state x’ from state x given that the control is u
R.S. Sutton and A.G. Barto, Reinforcement Learning– An Introduction, MIT Press, Cambridge, Massachusetts, 1998.D.P. Bertsekas and J. N. Tsitsiklis, Neuro‐Dynamic Programming, Athena Scientific, MA, 1996.W.B. Powell, Approximate Dynamic Programming: Solving the Curses of Dimensionality, Wiley, New York, 2009.
,( ) { | } { | }k T
i kk k T k i k
i kV x E J x x E r x x
*( , ) arg min ( ) arg min { | } .k T
i kk i k
i kx u V s E r x x
*( ) min ( ) min { | } .
k Ti k
k k i ki k
V x V x E r x x
( , )x u
determine a policy ( , )x u to minimize the expected future cost Optimal control problem
optimal policy
optimal value
Expected Value of a policy
Policy evaluation by Bellman eq.
Policy Improvement' '
'( ) ( , ) ( ')u u
j j xx xx ju x
V x x u P R V x 1 ' '
'( , ) arg min ( ')u u
j xx xx ju x
x u P R V x .for all x X
.for all x X
Policy Iteration
Policy Evaluation equation is a system of N simultaneous linear equations, one for each state. ' ( ) ( )V x V x Policy Improvement makes
Discrete State
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
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
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
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
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 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
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
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
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
( ( )) ( , ) ( ( ))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
( ( )) ( ) ( ( ))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
( ( )) ( ( )) ( ( )) ( ) ( )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
observe cost integral
update P
t+3T
observe x(t+2T)
apply uk=Kkx
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
( ) ( )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
V x t r x u dt V x t T
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
Data set at time [t,t+T)
( ), ( , ), ( )x t t t T x t T
Optimal Control Design Allows a Lot of Design Freedom
( ( )) ( , ) ( ( ))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
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
V x t r x u dt V x t T
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
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
V x t r x u dt V x t T
1121 1( ) ( )T k
k kVu h x R g xx
D. Vrabie, 2009
Doya, Kimura, Kawato 2001
Limbic system
Motor control 200 Hz
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
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
Kenji Doya
79
Synchronous Real‐time Data‐driven Optimal Control
Actor / Critic structure for CT Systems
Theta waves 4-8 Hz
Motor control 200 Hz
Optimal Adaptive Integral Reinforcement Learning for CT systems
A new structure of adaptive controllers
( ( )) ( , ) ( ( ))t T
k k kt
V x t r x u dt V x t T
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
Data set at time [t,t+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
A new structure of adaptive controllers
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
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
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
Games on Communication Graphs
500 BC
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
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
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.
1u
2u
iu Control action of player i
Value function of player i
New Differential Graphical Game
( )i
i i i i i i ij j jj N
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
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
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
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
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
( , ) ( ) ( ) ( )( )
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
Leibniz gives Differential equivalent
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
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
154
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
On‐policy RL
163
Target and behaviorpolicy
SystemRef.
Target policy: The policy that we are learning about.Behavior policy: The policy that generates actions and behavior
Target policy and behavior policy are the same
Sutton and Barto Book
Off‐Policy Reinforcement Learning
Humans can learn optimal policies while actually playing suboptimal policies
Off‐policy RL
164
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
J x r x u d
[ [ [] [ ]] ]( ( )) ( ( )) ( )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
Yu Jiang & Zhong-Ping Jiang, Automatica 2012
system
value
Must know g(x)
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.
175
Output Synchronization of Heterogeneous MAS
176
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
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 Problem
Augmented Systems
Performance index
Control
Our Solution
180
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
181
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)
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.
