This document discusses an adaptive neural network controller for strict-feedback nonlinear systems. It begins by introducing the research topic of controlling such nonlinear systems. It then describes using a neural network to approximate unknown functions in the system model. The third part details designing an adaptive controller for a first-order strict-feedback system using Lyapunov stability analysis and updating the neural network weights online. Finally, it applies this approach to control the joints of a three degree-of-freedom robotic arm. Simulation results demonstrate the link trajectories track the desired motions with small errors.

1. Adaptive neural network controller
for strict – feedback nonlinear
systems
Nguyen Cong Dan
Hanoi University of Science and
Technology

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
211/23/2015 Nguyen Cong Dan

3. Part 1: Researching object
• Nonlinear systems written by strict – feedback
form:
3
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 x
x f x x g x x x
x f x x x g x x x u
y x
 
  

  


&
&
&
11/23/2015 Nguyen Cong Dan

4. 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
4
( ) : m
h Z R R
( ) ( )T T
nng Z W S V Z
1
,1
TT m
Z R 
   Z
  ( 1)
1 2, ,... m l
lV v v v R  
 
 1 2, ,...
T l
lW w w w R 
11/23/2015 Nguyen Cong Dan

5. Neural network to approximate
a smooth function.
5
INPUT LAYER 1 LAYER 2 OUTPUT
1
2
3
1
2
l
m+1
V(m+1,l) W(l,1)
11/23/2015 Nguyen Cong Dan

6. Part 3: Designing adaptive neural
network controller
• First order system:
• Define:
• Choose an integral-type Lyapunov function:
6
1 1 1 1 1 1
1
( ) ( )x f x g x u
y 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    
11/23/2015 Nguyen Cong Dan

7. 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.
7
 1 1 1 1
1 1
1
( ) ( )
( )
u k t z h Z
x
  
g
1
1 1 1 1 1 1 1 10
* ( ) ( ) ( ) ( )d dh Z x f x y z y d     &  3
1 1[ , , ]T
d dZ x y y R &
1 1( )h Z
11/23/2015 Nguyen Cong Dan

8. 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 1
1 1
1 ˆ ˆ[ ( ) ( )]
( )
T T
u k t z W S V
x
   Z
g
11/23/2015 Nguyen Cong Dan

9. 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.
9
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 x
x f x x g x x x
x f x x x g x x x u n
y x
 
  

  


&
&
&
11/23/2015 Nguyen Cong Dan

10. Part 4: Apply this method to
control Robot SCARA
• Robot Scara 3 DOF (2 Revolute joints and 1 Prismatic joint)
10
(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[ , , ]T
q q q q
Joint 2
Joint 3
Joint 1
11/23/2015 Nguyen Cong Dan

11. Robot Scara 3 DOF
Dynamic equation:
11
( ) ( , )M q q q q U && &
1
2
T
U T
F
 
   
  
 1
q M U
  &&
11 12 13
21 22 23
31 32 33
( ) ;
m m m
M q m m m
m m m
 
 
  
 
 
1
2
3
( , ) ;
h
q q h
h

 
 
  
 
 
&
11/23/2015 Nguyen Cong Dan

12. Robot Scara 3 DOF
So, we can write dynamic equation for each link:
12
1 1 1
1 11 1 12 2 13 3 1
1 1 1
2 21 1 22 2 23 3 2
1 1 1
3 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
  
  
  
   
   
   
&&
&&
&&
11/23/2015 Nguyen Cong Dan

13. 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)
13
1 1
1 2
1 1 1
2 11 1 12 2 13 3 1
q x
x x
x M h M h M h u  
 



   
&
&
2 2 3 3, , ,q q q q& &1q&&
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14. Simulation
• Model Matlab – Simulink:
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15. Link’s trajectory follows desired
trajectory
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16. Tracking error
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17. 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.
1711/23/2015 Nguyen Cong Dan