Adaptive neural network controller Presentation

Adaptive neural network controller for strict – feedback nonlinear

systems

Nguyen Cong DanHanoi University of Science and

Technology

05/01/2023 Nguyen Cong Dan 2

The main content Researching object

Using neural networks to approximate a smooth function

Designing an adaptive controller for strict-feedback nonlinear systems

Applying this controller to Robot SCARA

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Part 1: Researching object

• Nonlinear systems written by strict – feedback form:

1 1 1 1 1 2

2 2 1 2 2 1 2 3

1 2 1 2

1

( ) ( )( , ) ( , )

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

x f x g x xx f x x g x x x

x f x x x g x x x uy x

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Part 2: Neural network to approximate

a smooth function.• With a smooth unknown function:

• Constructing a neural network having structure :

Where : : Input vector

: the first layer weight

: the second layer weight

( ) : mh Z R R

( ) ( )T Tnng Z W S V Z

1,1TT mZ R Z

( 1)1 2, ,... m l

lV v v v R

1 2, ,... T llW w w w R

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Neural network to approximate a smooth function.

INPUT LAYER 1 LAYER 2 OUTPUT

1

2

3

1

2

l

m+1

V(m+1,l) W(l,1)

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Part 3: Designing adaptive neural network controller

• First order system:

• Define: • Choose an integral-type Lyapunov function:

1 1 1 1 1 1

1

( ) ( )x f x g x uy x

1 1 ;dz x y 1 1 1 1 1 1( ) ( ) / ( )x x g x g

1

1 10( ) 0

z

z dV y d

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Designing adaptive neural network controller

• We can choose a controller for this system:

Where:

• Following Lyapunov theory, control function u1 can make this system stability.

• Problem: f1(.) and g1(.) are uncertain, thus we can’t define exactly as well as u1.

1 1 1 11 1

1 ( ) ( )( )

u k t z h Zx

g

1

1 1 1 1 1 1 1 10* ( ) ( ) ( ) ( )d dh Z x f x y z y d 3

1 1[ , , ]Td dZ x y y R

1 1( )h Z

Nguyen Cong Dan

05/01/2023 Nguyen Cong Dan

Designing adaptive neural network controller

• Apply the neural network designed to approximate

• And layer weights are updated during training process by proper laws.

8

1 1( )h Z

1 1 1 1 1 1 11 1

1 ˆ ˆ[ ( ) ( )]( )

T Tu k t z W S Vx

Zg

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N-dimensional nonlinear system

• Using backstepping design we can find adaptive NN controller for n-dimensional system. We need n steps, each step uses an intermediate control function.

• By viewing x2 as a virtual control input for (1); x3 as a virtual control input for (2)… along the similar process with the first order system. We can design intermediate control functions, train Neural network weights, and offer final controller.

1 1 1 1 1 2

2 2 1 2 2 1 2 3

1 2 1 2

1

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

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

x f x g x xx f x x g x x x

x f x x x g x x x u ny x

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Part 4: Apply this method to control Robot SCARA

• Robot Scara 3 DOF (2 Revolute joints and 1 Prismatic joint)

(1)(2)

(3)

- mi and ai : are the mass and the length of link i, (i =1,2,3)- qi is the joint variable in joint i

1 1 2 2 3 3; ; q q q d

1 2 3[ , , ]Tq q q q

Joint 2

Joint 3 Joint 1

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Robot Scara 3 DOF

Dynamic equation:

( ) ( , )M q q q q U

1

2

TU T

F

1 q M U

11 12 13

21 22 23

31 32 33

( ) ;m m m

M q m m mm m m

1

2

3

( , ) ;h

q q hh

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Robot Scara 3 DOF

So, we can write dynamic equation for each link:

1 1 11 11 1 12 2 13 3 1

1 1 12 21 1 22 2 23 3 2

1 1 13 31 1 32 2 33 3 3

(1)

(2)

(3)

q M h M h M h u

q M h M h M h u

q M h M h M h u

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Robot Scara 3 DOF

• Dynamic equation (1) written by strict-feedback form

• Although depend on we can view them as uncertain ingredients. Design adaptive independent controller for q1 like Part 3.

• Similar procedure with equation (2) and (3)

1 1

1 2

1 1 12 11 1 12 2 13 3 1

q xx x

x M h M h M h u

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

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Simulation

• Model Matlab – Simulink:

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Link’s trajectory follows desired trajectory

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Tracking error

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Conclusion

• The report has presented a method about adaptive NN control for strict-feedback nonlinear systems using backstepping design.

• Apply successfully this method to robotic arm, we can design independent controllers for each link of robot SCARA.