BABS Controlled by HNN Controller Trained by Nonlinear Reference

We use the following examples to examine the abilities of HNN controllers to control IPS with initial state different from the initial state of training the HNN controller by the nonlinear reference controller. We let the values ofw₁₁, w₁₂, w , ₁₃

w14, w₂₁, w₂₂, w , ₂₃ w₂₄, w , ₃₁ w , ₃₂ w , ₃₃ w , ₃₄ w₄₁, w₄₂, w and ₄₃ w₄₄ be the same with section 4.4.3 as the equation (4-41):

⎥⎥

is fixed for the initial state of BABS: the initial position of the ball=0.2 (meter),the initial velocity of the ball=0 (meter/sec),the initial angle of the beam=10 (degree), and the initial angular speed of the beam of the BABS=0 (degree/sec) in the training phase. And then, we will examine the following sets (initial position (meter), initial velocity (meter/sec), initial angle (degree), initial angular speed (degree/sec)) of the initial state of BABS: (0.1, 0, 5, 0) and (0.4, 0, 20, 0) in the working phase as the following figures:

Fig-4.48. The control torques of BABS with initial ball’s position=0.1 meter, initial beam’s angle=5 degree: the nonlinear reference controller (dash line) and HNN

controller (solid line)

Fig-4.49. The ball’s positions of BABS with initial ball’s position=0.1 meter, initial beam’s angle=5 degree: the nonlinear reference controller (dash line) and HNN

controller (solid line)

Fig-4.50. The ball’s velocities of BABS with initial ball’s position=0.1 meter, initial beam’s angle=5 degree: the nonlinear reference controller (dash line) and HNN

controller (solid line)

Fig-4.51. The beam’s angles of BABS with initial ball’s position=0.1 meter, initial beam’s angle=5 degree: the nonlinear reference controller (dash line) and HNN

controller (solid line)

Fig-4.52. The beam’s angular speeds of BABS with initial ball’s position=0.1 meter, initial beam’s angle=5 degree: the nonlinear reference controller (dash line) and

HNN controller (solid line)

Fig-4.53. The node 1 (2) (3) (4) voltage v₁ (v₂) (v ) (₃ v₄) of the HNN circuit

Fig-4.54. The control torques of BABS with initial ball’s position=0.4 meter, initial beam’s angle=20 degree: the nonlinear reference controller (dash line) and HNN

controller (solid line)

Fig-4.55. The ball’s positions of BABS with initial ball’s position=0.4 meter, initial beam’s angle=20 degree: the nonlinear reference controller (dash line) and HNN

controller (solid line)

Fig-4.56. The ball’s velocities of BABS with initial ball’s position=0.4 meter, initial beam’s angle=20 degree: the nonlinear reference controller (dash line) and HNN

controller (solid line)

Fig-4.57. The beam’s angles of BABS with initial ball’s position=0.4 meter, initial beam’s angle=20 degree: the nonlinear reference controller (dash line) and HNN

controller (solid line)

Fig-4.58. The beam’s angular speeds of BABS with initial ball’s position=0.4 meter, initial beam’s angle=20 degree: the nonlinear reference controller (dash line) and

HNN controller (solid line)

Fig-4.59. The node 1 (2) (3) (4) voltage v₁ (v₂) (v ) (₃ v₄) of the HNN circuit

Chapter 5

Discussion and Conclusion

5.1 Discussion of Parameters Setting

The resistance R of the HNN controller by the way that larger resistance causes larger node voltagev. Because a large voltage is not preferred, therefore a small R should be chosen. The time constant τ can be express as

=RC

τ , (5-1) which is the product of the resistance R and the capacitanceC. τ cannot be chosen too large because it leads to slow response. The amplification constant c_amp is also an important parameter of the HNN controller. According to the voltage amplifierϕ(•)=tanh(•), the output of a neuron of the HNN is limited between the values -1 to +1, we express as the following inequality.

1 ) tanh(

1≤ = ≤

− x_j v_j (5-2) From the inequality (5-2), we can show that the output control signal of HNN controller is limited as

amp n

j j amp

amp u c x nc

nc ≤ = ≤

−

∑

=

∧

1

, (5-3) where n is the number of neurons in the HNN. That is the absolute value of the output control signal of the HNN controller is limited by the product of the amplification constant c_amp and the number of neuronsn.

The learning rate η must be a positive as we discussed in the section 2.3. The large η is not preferred because that will contradict (2-14) [7]. However η cannot be chosen too small or the weightings convergence will be too slow, so we use a proper value of the learning rate η to let the simulation process well.

The simulation time is chosen long enough so that the regulation state of the controlled system can be near the desired point.

The time interval is set to be a small enough value to get the required accuracy of the differential equations of the system controlled by the HNN controller.

In many cases one epoch is enough to achieve favorable training performance.

More training epoch is unnecessary unless one epoch training cannot achieve the preferred performance.

If we do not have prior knowledge of the proper weighting vector, we can just

simple initialize W from zero vector. Notice that the values of the elements of W are the same in the same column. This can be briefly explained as following. We note the equations (3-22) and (3-24), they are very similar. We write them down in the simple form to show the difference. For (3-22), the equation is w₁₁(k+1)=w₁₁(k)−...×{1−[tanh(v₁)]²} , while for (3-24), the equation isw₂₁(k+1)=w₂₁(k)−...×{1−[tanh(v₂)]²}. And we note the equations (3-10) and (3-11), we find if w₁₁ =w₂₁ andw₁₂ =w₂₂, then we can get i₁ =i₂ by equation (3-11), and furthermore, we can get v₁ =v₂ by equation (3-10). So it is interesting that because we set the initial values of all the weighting factors zeros, so

0

At last, the important facts are obtained as following:

n

It should keep in mind that the equation (5-4) is satisfied on the premise of the following:

After training the weighting factors of HNN controller, we find that the output (control signal) of HNN controller is approximated to the output (control signal) of the well-designed controller and the outputs of the plant controlled by the HNN controller are approximated to the outputs of the plant controlled by the well-designed controller. So, the trained HNN controller can be a good model of the well-designed controller with the controlled plant with the same initial state with HNN trained. Furthermore, although the weighting factors of HNN controller are trained by well-designed controller with the plant with initial state different from the initial state of the plant controlled by the HNN controller, the HNN controller can still control the plant well, and the output (control signal) of HNN controller is approximated to the output (control signal) of the well-designed controller and the outputs of the plant controlled by the HNN controller are still approximated to the outputs of the plant controlled by the well-designed controller. So, the trained HNN controller not only can “memorize” the output (control signal) of the well-designed

controller with the controlled plant in the same initial state, but it can also “simulate”

the output (control signal) of the well-designed controller with the controlled plant in the different initial state. So, the HNN controller has ability more than just to memorize the training data, and this property of HNN controller is important for applications.

Faults due to the aging of a controller for a control system are very common; once they happen, the controller is quite difficult to be repaired for some reasons. We proposed an HNN controller for a control system to solve this problem. After discussing the two examples of the nonlinear systems controlled by the HNN controllers, we understand that the HNN has the potential to be a good controller. The key point of the HNN controller is the parameters, especially the weighting factors between each two neurons of one HNN. To design an HNN controller for some specified nonlinear system is still a challenge. In this thesis, we trained the weighting factors of the HNN controller to mimic the existing controller. Then, the trained HNN controller is used to replace the existing controller. Can we control the system by an HNN controller trained online without a reference controller? We will focus our research interests on exploring the potential of this interesting question.

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