1 Adaptive Neural Network Control of Nonlinear Systems S. Sam Ge Department of Electrical & Computer...

1

Adaptive Neural Network Control of Nonlinear Systems

S. Sam Ge Department of Electrical & Computer Engineering National University of Singapore Singapore 117576

E-mail: [email protected] http://vlab.ee.nus.edu.sg/~sge

2

Work Place

Inaugural IEEE Multi-conference on Systems & Control

16th IEEE Conference on Control Applications (CCA)

22nd IEEE International Symposium on Intelligent Control

F. L. Lewis, Sponsorship Chair

S. Jagannathan, ISIC Program Chair

T. Parisini, Conference Editorial Board Chair

3

Work Place

4

Content

1. Introduction

2. System Descriptions

3. Neural Network Approximation

4. State-Feedback Control for SISO

5. Output-Feedback Control for SISO

6. Simulation Study

7. Conclusion

5

1. Introduction

Neural network control has gone through

• the pioneering works,

• the pains against the skeptics and doubts, and

• the graceful acceptance, and maturity

as a powerful tool for control of nonlinear systems.

Thanks to the many distinguished individuals and their families:

Narendra, Levin, Lewis, Calise, Polycarpou, Hovakimyan, Jagannathan, Slotine, Ge, …

6

1 Intelligent Control

The most intelligent system in nature!

Info. Feedback

Real-time Control

Decision Making

7

1 Adaptive NN Control

Cycling or driving ， we never thinking of the so-called mathematical models ！

Plant

Info. feedback

Adaptation & Learning

Control Law

8


9


10


11


12

1. Adaptive NN Control

•System Modeling is usually more difficult than control system design

•Model based control though rigorous, it depends too much on model building。•Before 90s ： Off-line NN Training

•After 90s ： Combining adaptive control, and NN parametrization, on-line adaptive NN control is investigated.

13

Continuous to Discrete

Owing to different analytical tools used,

results in continuous time are not necessarily hold in discrete time.

14

Discrete-time SISO system

where

2 System Descriptions

1

1

1

( 1) ( ( )) ( ( )) ( ), 1

( 1) ( ( )) ( ( )) ( ) ( ),

( ),

i i i i i i

n n n n i

k

k f k g k k i n

k f k g k u k d k

y k

1 2

1

( ) ( ), ( ),..., ( ) ,

, are unknown functions

is the control, and is the bounded distrubance.

T ii i

i i

k k k k R

f g

u d

15

The sign of ( ) ( 1, , ), are

known and there exist two constants , 0 such

that ( ) , ( )

i

ii

ni i ni

g i n

g g

g g g k R

Assumption 1 :

The desired trajectory ( ) ,

0 is smooth and known, where | ( )

d y

y

y k

k y k

Assumption 2 :

2.1 Assumptions

16

2.2 System Descriptions

1

1, 1, 1, 1, 1, 1, 1 1 11 1 1 1 1 1

1, 1, 1, 11 1 1

, , , , , , 1

, ,

( 1) ( ) ( ) , 1 1

( 1) ( ) ( ) ,

( 1) ( ( )) ( ( )) ( ), 1 1

( 1) ( ( ),j

i i i i i i

n n n

j i j i j i j i j i j i j jj j j j j j

j n j n jj j

k f k g k k i n

k f k g k u k

k f k g k k i n

k f k u

,1 ,

, , , , , , 1

, , 1 ,

,1

( )) ( ( )) ( )

( 1) ( ( )) ( ( )) ( ), 1 1

( 1) ( ( ), ( )) ( ( )) ( ),

( ) ( ), 1

n

j n jj

n i n i n i n i n i n i n nn n n n n n

n n n n n n n nn n n

j j

k g k u k

k f k g k k i n

k f k u k g k u k

y k k j n

Discrete-time MIMO system

17

2.4 System Properties

• The inputs are in triangular form.

• There are both inputs and states interconnections.

• The system cannot be expressed as

(k+1)=F((k))+G((k))U(k)

which makes the feedback linearization method

not applicable.

We have the following observations:

18

3 Neural Network Approximation

In control engineering, different types of neural networks, including LPNN (RBF, HONN) and MLNN, have been used as function approximators over a compact set.

*

z

*

For any smooth function ( ), there exists ideal weights such

that the smooth function ( ) can be approximated by an ideal

NN on a compact set :

( ) ( ) ,

where is called the NN appro

m

Tz

z

z W

z

R

z W S z

ximation error.

LPNN:

19

3 Neural Network Approximator

For clarity of analysis, consider HONNs

1 2

( )

( , ) ( ), and ( ) ,

( ) ( ), ( ),..., ( ) ,

( ) ( ) , 1,2,...,j

i

T l

Tl

d i

i jj I

W z W S z W S z R

S z s z s z s z

s z s z i l

where

1

1 2

,..., , ( ) ,

weight vector, node number,

, ,..., Collection of not-ordered subsets of 1,2,..., ,

( ) non-negative integers.

j j

j j

z zT m

m z j z z

l

j

e ez z z R s z

e eW l

I I I l m

d i

20

3 Neural Network Approximation

The particular choice of NN is used for analysis only, similar results can be obtained for (extended to)

other linearly parametrized networks,

radial basis function networks, polynomials, splines functions, fuzzy systems,

and, the multiple layer neural networks (Nonlinear).

Different choices affect performance though.

21

Part I

22

The non-causality is one of the main problems for strict-feedback nonlinear system through backstepping in discrete time.

4 State-Feedback Control

1. Non-causal Problem,

2. System Transformation,

3. Desired Control,

4. Stable Control System Design

The following issues are in order

23

Consider the discrete SISO system given

4.1 Non-causal Problem

1

1

1

( 1) ( ( )) ( ( )) ( ), 1

( 1) ( ( )) ( ( )) ( ) ( ),

( ),

i i i i i i

n n n n i

k

k f k g k k i n

k f k g k u k d k

y k

11 111

22 222

1

1( 1)( ) ( ( ))

( ( ))

1( ) ( ( ))

( ( ))

...

( 1)

dy kk f kg k

k f kg k

k

Direct application of backstepping, the following ideal fictitious controls are in order:

2( ),... cannot be realised because of future variables.k

24

4.1 n-step Ahead Predictor

1

Consider the original system description as

a .

Through suitable system transf

one-step ahead predictor

step ahead pre

ormation an equivalent

, which is capable of predicting

the future stat

dic

es (

tor

n

2), ( 1), , ( 1).nk n k n k

For the n-step ahead predictor, backstepping desig

without the non

n

can be applie -causal probd lem.

25

Re-examining the system

4.2 System Transformation

1

1

1

( ( )) ( ( )) , 1

( 1) ( ( )) ( ( )) ( ) ( )

( 1) ( )

,

( ),

i i i i

n n n n i

k

i if k g k i n

k f k g k u k d k

y k

k k

1 1 1 1 1

1

After one step, the first 1 equations become

( ( 1)) ( ( 1))

1, , 2

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

( 2) 1)

)

(i i i i

n n n

i

n n n

i

(n - )

f k g k

i n

k f k k

k

k

k

g

26


1 ,1 2

, 1

, ,1 1

, 1

( 1) ( ( ))

Let ... ...

( 1) ( ( ))

where ( ( )) ( ( )) (

( ( ))

( )) ( )

( 1)

, 1, , 1.

cn

ci n i i

cn i i n i i i

ci ii n

i

F k

k f k

k f k

f k f k g k k i

k

n

, 1

1, 2

1 1 1

1, , 12 , 1 2

We know that

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

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

where ( ( )) ( ) ( ( ( ))) ( ( ))

( ( ))

n i

ci n i i

n n n n n n

c c cn i i i

cn ni i i ii i

k f k i n

k F k G k

F

k

f k g F k kkf f

1 1 , 1 1 1 , 1( ( )) ( ( ( ))), ( ( )) ( ( ( )))c cn n n n n n n n n n n nF k f F k G k g F k

27


3,2

1 2,1

2 2 2 3

2,1 1 3,1 1 1 3,1 1

2

( 1) ( ( )),

( 1) ( (

R

)) ( ( )) ( 2)

where ( ( )) ( ( ( ))) (

ecursively, after (n-2) steps, the first two equations bec

( ( ))) (

ome

( ))

( ( ))

cn

n n

c c c cn n n n

n

k n f k

k n F k G k k n

f k f F k g F k f k

F k

2 3,2 2 2 3,2( ( ( ))), ( ( )) ( ( ( )))c cn n nf F k G k g F k

1 1 1 2

1 1 2,1 1 1 2,1

( ) (

After one mor

( )) ( ( )

e step, the firs

) ( 1)

where

t equation beco

( ( )) ( ( ( ))), ( ( )) ( ( ( )))

mes

n n

c cn n n n

k n F k G k k n

F k f F k G k g F k

28

1 1 1 2

1 1 1

1

1

( ) ( ( )) ( ( )) ( 1)

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

( 1) ( ( )) ( ( )) ( ) ( )

( )

with ( ) ( ( )) and ( ) ( ( ))

n n

n n n n n n

n n n n n

k

i i n i i n

k n F k G k k n

k F k G k k

k f k g k u k d k

y k

F k F k G k G k

Through the coordinate transformation, we have


29

*1 1

1

* *2 1 2

2

* *1 2 1

1

* *1

1( 1) ( ) ( )

( )

1( 2) ( 1) ( )

( )

...

1( 1) ( 2) ( )

( )

1( ) ( 1) ( )

( )

d

n n nn

n nn

k n y k n F kG k

k n k n F kG k

k k F kG k

u k k f kg k

The desired (virtual) controls are given by:

1

2 2 1

It is obvious that by recursive substitution, ( 1),

( 2), , ( 1), ( ) can be determined at the

ti The non-causal problem ime instan s solved.t .

k n

k n k u k

k

4.2 Desired Virtual Controls

30

1

1

ˆ( ) ( ( )), 1, ,

ˆ( ) ( ( ))

where

( ) [ ( ), ( )]

( ) [ ( ), ( )] , 2, ,

Ti i i i

Tn n n

T Tn d

T Ti n i

k W S z k i n

u k W S z k

z k k y k n

z k k k i n

The desired controls are functions of unknown functions, thus are not feasible.

As such NN control is called upon to construct a feasible controller.

Let us consider the fictitious controls and the control as:

4.4 Adaptive Neural Control

31

The errors are defined as

1 1

2 2 1

1

( ) ( ) ( )

( ) ( ) ( 1)

( ) ( ) ( 1)

d

n n n

e k k y k

e k k k n

e k k k

ˆ ˆ ˆ( 1) ( ) [ ( ( )) ( 1) ( )]

, 1, 2, ,

T Ti i i i i i i i i i i

i

W k W k S z k e k W k

k k n i i n

Neural network weight update laws are

4.4 Adaptive Neural Control

32

1

*1 1

1 1 1 1 1

2 11 1 1 1 1

1

Define , 1,2,...,

From definitions of ( ), ( ) and ( ), we obtain

( ) ( ) ( ) ( ( ))

Consider the Laypunov function candida

Step

te

1(

1

) ( (

:

) )

i

Tz

T

k k n i i n

k W k e k

e k n G k W k S z k

V k e k W k jg

1

1

1 10

1 1 1

22 21

1 1 1 1 1 1 1 1 1 1 1 11 1

221 *

1 1 1 1 1 1 1 1 1 11

( )

ˆThe first difference of ( ) along and is given by

1 ˆ( 1) ( ) 1 ( ) ,

where

1 ,

n

j

z

W k j

V k e W

V e k e k g W kg g

gl g l W

4.5 Stability Analysis

33

12 1

0 1

As in Step 1, we obtain

( 1) ( ) ( ) ( ( ))

Consider the Laypunov function candidate

1( ) ( ) ( ) ( ) ( )

The first differenc

Step

e

2 1

o

i

i

Ti i i i i z

n i iT

i i i i i i i jj ji

e k n i G k W k S z k

V k e k W k j W k j V kg

n

:

i

2

2 2

0

22*

1

ˆf ( ) along and is given by

1 ˆ( 1) ( ) 1 ( ) ,

where

1 , i

i i i

ij

i j j i i i i i i i i ij j j

i zi i i i i i i i i i i

i

V k e W

V e k e k g W kg g

gl g l W


34

'

' 1

2 1

1

As in Step , we obtain

( 1) ( ) ( ) ( ( ))

( )where which is bounded.

( )

Consider the Laypunov function candidate

1(

Step n

) ( ) ( ) ( )

:

( )

n

n n

Tn n n n n z

z zn

nT

n n n n n jjn

i

e k g k W k S z k

d k

g k

V k e k W k j W k j V kg

1

22 2

0

22*

1


1 ˆ( 1) ( ) 1 ( ) ,

where

1 , n

n n n

nj

n j j n n n n n n n nj j j

n zn n n n n n n n n n n

n

V k e W

V e k e k g W kg g

gl g l W


35

jj

If we choose the design parameters as follows:

1 1, , 1

1 (1 )

then 0 once any one of the errors satisfies

( ) for 1 .

This implies the boundedness of ( ),

jj j j j

n

j j n

n

j nl g l g

V n

e k g j n

V k

1

1

0,

which leads to the boundedness of the tracking errors ( ),..., ( )

because ( ) (0) ( ) .

n

k

n n nj

k

e k e k

V k V V j


36

1 2

Furthermore, the tracking errors will asymptotically converge to

the compact set denoted by ,

where : , ,..., , 1

The adaptation dynamics of is

( 1) ( ) ( ) (

nn

T

n n j i n

j

j j j j j j j

R

g j n

W

W k A k W k S z

*( )) ( )

where ( ) ( ) ( ( )) ( ( )) and

the terms and ( ) are bounded.

ˆTherefore the boundedness of ( ) or equivalently of ( )

are proven.

j

j

j j j z j j j

Tj j j j j j j j j j j j

z j j

j j

k G k W

A k I G k S z k S z k

G k

W k W k


37

1

n

The closed-loop adaptive system consisting of plant,

controller and update law is semi-globally uniformly ultimately

bounded (SGUUB), and has an equilibrium at [ ( ), , ( )] 0,

with (

:

0

Tne k e k

Theorem 1

n

) initialized in , and design parameters appropriately chosen.

This guarantees that all the signals including the states ( ),

ˆthe control input ( ) and NN weight estimates ( ),

(1 ) are SGU

j

k

u k W k

j n

k

UB, subsequently

lim ( ) ( )

where is a small positive constant.

dy k y k


38

Part IIBefor

e

39

Part IIAfter

40

For output feedback control, the strict-feedback form is transformed into a cascade form.

For equivalent transformation of coordinates, it is important to ensure that the transformation map is diffeomorphism.

The following issues will be highlighted:

1. Coordinate Transformation

2. Diffeomorphism

3. Cascade Form

4. Control Design

5 Output-Feedback Control

41

5.1 Coordinate Transformation

1

1

11 1

11 1

Through iterative substitutions of ( 1 ) for 2,...,

into ( ), we obtain

( ) ( ),

where ( ) ( ) ( ) ( ),

( ( )) ( ( ))

( ( ))

i

jn n

n j j mjj

n n

nm

F k G k

F

k n i i n

k n

k n u k

f k G k F k G kk

1

1( ( )) ( ) ( ),

n

nj

n jG g k G kk

1For the convenience of analysis, assume ( ) 0. d k

42

1 2

1 1

2 1

1

Define the new system states ( ) ( ), ( ),..., ( ) ,

( ) ( ),

( ) ( 1),

( ) ( 1),

which

Tn

n

x k x k x k x k

x k k

x k k

x k k n

1 ,1 2 2,1

can be written as ( ) ( ) , where ( ) is a

nonlinear coordinate transformation of the form,

( ) ( ), ( ) ,..., ( ) .

n n

Tc cn n n

x k T k T k

T k k f k f k

5.1 Coordinate Transformation

43

5.2 Diffeomorphism

1

1 1

1

1

Let be an open subset of and let ,..., :

be a smooth map. If the Jacobian Matrix

is nonsingular at some point , or equivalently,

n nn

n

n n

n

U R U R

x xddx

x x

dp U Rank

at some point , then there exists a neighborhood of

such that : ( ) is a diffeomorphism (Marino & Tomei, 1995).

ndx

p U V U p

V V

44

5.2 Diffeomorphism

1

From the definition, it can be seen that ( ) is obviously smooth.

For diffeomorphism, we only need to show that Jacobian Matrix

( ) is of full rank, then ( ) must be a diffeomorphism.

From t

n

n

T k

T x k T k

1 1 1 2 1 12 2

1 1 1 1

2 2

2 1 1 3

he definitions, we obtain

( 1) ( ) ( ) ( )( ) ( )

( ) ( )

( ) 1 ( 1)Therefore, , 0

( ) ( ) ( )

k f k x k f x kk t k

g k g x k

t k t kx k g x k x k

45

5.2 Diffeomorphism

2 2 1 2

3 3

2 1 2

3

3 1 2 2 1 2

Likewise, we are also able to obtain,

( 1) ( ), ( )( ) ( ) .

( ), ( )

and

( ) 1.

( ) ( ) ( ), ( )

Continue the process iteratively, we are able to obtain

( ) ( ) ( )

wh

T

T

T

n n n

t k f x k t kk t k

g x k t k

t kx k g x k g x k t k

k t x k t k

1 1

ere

( ) 1.

( ) ( ) ... ( ),..., ( )n

Tn n n n

t kx k g x k g x k t k

46

5.2 Diffeomorphism

1

11 2

The unique transformation ( ) is given by

( ) = x ( ), ( ),..., ( ) and its Jacobian Matrix isn

T x k

T x k k t k t k

1 1 1

1 1

1 0 0

1* 0

( )( )... ... ...( )

1* *

( ) ... ( ),..., ( ) Tn n n

g x kT x k

x k

g x k g x k t k

1 ( )From Assumption 1, Trace 0.

( )

Therefore, coordinate transformation ( ( )), : ,

is a smooth map and a diffeomorphism.

nn x

T x k

x k

T k T R

47

5.3 Cascade Form

1 2

2 3

Accordingly, the original system can be equivlaently transformed

into the cascade system:

( 1) ( ),

( 1) ( ),

x k x k

x k x k

1

1 2

( 1) ( ( )) ( ( )) (k),

( )

where

( ) ( ), ( ),..., ( )

and unkno

n

k

T nn

x k f x k g x k u

y x k

x k x k x k x k R

1

1

wn functions

( ) ( )

( ) ( ) .

f x k F T x k

g x k G T x k

48

5.3 Cascade Form

1 1

1 1 1

1

Define

( ) ,..., ,

= ( 1),..., ( 1), ( )

( 1) ( 1),..., ( 1)

( ) ( ), ( )

Tk n k k

T

T

TT Tk

y k y y y

x k n x k x k

u k u k u k n

z k y k u k

2 1n

z R

49

5.3 Cascade Form

1 2 3

1

1 2

2 3

According to the definition of the new states, we have

( ) ( 1) ...

( 2) ( 1) ( 1)

( ) ( ) ( ( ))

Similarly,

( ) ... ( 3)

k

n k n

k n i

k n

y x k x k

x k n f x k n g x k n u

f y k g y k u z k

y x k x k n f

2

1 2

2 3 2 3

( 1) ( 1)

Substituting ( ( )), we can show that

( ) ( ) ( ) ( ( ))

k n

k

k k n

y k g y k u

y z k

y x k f z k g z k u z k

50

5.3 Cascade Form

1

1 2

0

Recursive substitution leads to

( ) ( ) , 1

Define

( ( )) , ( ) ,..., ( )

Therefore, we obtain

( 1) ( ( )) ( ( )) ( )

= ( )

k i i i

Tn

k n n

y x k z k i n

x z k x k z k z k

y x k f x k g x k u k

f z k

0 ( ) ( )g z k u k

51

5.4 Control Design

*0

0

2

1 1

Therefore, we can design a -step deadbeat control where

1 ( ) ( ( ))

( ( ))

Define

( ) ( ), ( )

where

( ), ,

k d

T nd z

z k d k

n

u y k n f z kg z k

z k z k y k n R

y k u y u

* *

( ) , ( ) ,

The ideal control law can be re-constructed as

( ( )) ( ( )) , ,

u y d y

Tz z

k y k y

u z k W S z k z

52

1 1 1

Choose the control law and the weight updating law as

ˆ( ( )) ( ) ( ( )),

ˆ ˆ ˆ( 1) ( ) ( ( )) ( 1) ( ) ,

T

k d

u z k W k S z k

W k W k S z k y y k W k

1

where

1,

and diagonal gain matrix 0, and 0.T

k k n

5.4 Control Design

53

5.4 Control Design

0 0

*

0

The error dynamics can be rewritten as

ˆ( ) ( ) ( ( )) ( ( )) ( ) ( ( )).

Using the definition of ( ), ( ) and , we obtain

( ) ( ( )) ( ) ( ( )) ,

Choose the following Ly

Ty d

k

Ty z

e k n y k n f z k g z k W k S z k

u k W k u

e k n g z k W k S z k

12

1 10

22 2

1

21

1

apunov function candidate:

1( ) ( ) ( )


1 ˆ( ) ( 1) ( ) 1 ( ) ,

where 1 ,

nT

yj

y

y y

z

V k e W k j W k jg

V k e W

V k e k e k g W kg g

gl g l W

2*

54

20

If we choose the design parameters as follows:

1 1, ,

1 (1 )

then 0 once the tracking error ( ) .

This implies the boundedness of ( ) for 0,

which leads to the boundedness of .

Sim

y

y

l gl g

V e k g

V k k

e

ilarly, we can show that the ( ) stays in a small compact set

and the control signal . We k

W k

u L

5.4 Control Design

55

0 0 0

*

Finally, if we initialize state , (0) ,

and suitable parameters and , to make small enough,

there exists a constant such that all tracking errors

asymptotically converge to and

y Wy W

k

W

*

( )

asymptotically converges to for all .

This implies that the closed-loop system is SGUUB.

ˆThen and ( ) will hold for all 0.

We

k y

k

k k

y W k L k

5.4 Control Design

56

5.4 Control Design

* * *y0 y W0 W

Consider the closed-loop system consisting of

system, controller and adaptation law. There exist compact sets

, and positive constants , and

such that if

(i) All the assump

l

Theorem 2 :

y0 W0

*

* *

tions are satisfied and the initial condition

(0) and (0) , and

(ii) the design parameters are suitably chosen such that ,

, with being the largest eigenvalue of

then

y W

l l

the closed-loop system is SGUUB. The tracking error

can be made arbitrarily small by increasing the approximation

accuracy of the NNs.

57

Part III

58

6 Simulation Study

Consider a nonlinear discrete-time SISO plant

1 1 1 2

2 2 2

1

1 0.3 ,

1 ,

( )k

k f k k

k f k u k d k

y k

where

21

1 1 21

12 2 2 2

1 2

1

1.4,

1

,1

0.1cos 0.05 cos .

kf k

k

kf k

k k

d k k k

59

3: Simulation Studies (cont.)6.1 State Feedback Control

1 2 1 2

1 2 1 2

Design parameters:

diag{0.08}, diag{0.08}, 22,

ˆ ˆInitial conditions: (0) (0) 0, (0) (0) 0.

l l

x x W W

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3: Simulation Studies (cont.)6.2 Direct Output Feedback Control

1

1 2 1 2

Design parameters:

diag{0.15}, 29,

ˆ ˆInitial conditions: (0) (0) 0, (0) (0) 0.

l

x x W W

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8. Conclusion

1. Adaptive full state feedback NN control has been presented via backstepping for a class of nonlinear unknown discrete-time SISO systems in strict-feedback form.

2. By transforming the system to sequential decrease cascade form, the non-causal problem was solved.

3. High order neural networks are used as the emulators of desired virtual and practical controls, which avoids possible control singularity problem.

4. By transforming the system into cascade form, an adaptive direct output feedback control scheme has also been presented using neural networks.

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Research“ … I seem to have been only like a boy playing on the seashore, diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, while the great ocean of truth lay all undiscovered before me.”

Isaac Newton

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With Appreciation

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1 Adaptive Neural Network Control of Nonlinear Systems S. Sam Ge Department of Electrical & Computer...

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