CHAPTER 4 Experimental Results
4.2 Example 2: The aircraft attitude control system
The aircraft attitude control system is different from the inverted pendulum system. It is a simplified linear system [17]. The control surfaces of modern aircraft are controlled by electric actuators with electronics controls. Figure 4-10 illustrates the control block diagram of one axis of such a position control system.
Figure 4-10. Block diagram of an attitude-control system of an aircraft
The objective of the system is to have the output of the system,
θ
y( ) t
, follow the unit step inputθ
r( ) t
. The transfer function is defined as( ) ( )
( ) (
4500361.2)
y
r
s K
G s s s s
θ
=
θ
=+ (4-3) Where K is gain parameter of preamplifier. When we have introduced the aircraft attitude control system, we must set the specifications as follows:
1. Steady state error due to unit ramp input ≤ 0.000443 2. Maximum overshoot ≤ 5 percent
3. Rise time ≤ 0.005 sec
4. Settling time ≤ 0.007 sec
Now, we will design the Hopfield NN controller for this problem so that the above specifications can be satisfied. The first specification is different from the other specification, so we discuss it first. It is the specification about steady state error due to unit ramp input. It must be satisfied by using preamplifier
K
. The system transfer function in (4-3) has a term in denominator, so it is a type 1 system. For type 1 system, the steady state error due to unit step input is zero, but it is a constant for unit ramp input. Thus we must choose a proper gain of preamplifiers
K
to reach the first specification. Now, we compute the velocity error constantk
v of transfer function in (4-3): As we get the velocity error constant , we can compute the steady state error due to unit ramp input According to the first specification, we should let the result of (4-5) be smaller than 0.000443.Thus we calculate the minimum value of gain of preamplifier K:
0.08026 1 0.000443 K 181.17
K ≤ ⇒ ≥
(4-6) In order to reach the first specification, we choose the gain of preamplifierK
equal to 181.17. Thus the transfer functionG s becomes to ( )
( ) (
815265361.2)
G s
=s s
+ (4-7) Now we hope the output of the system can reach the specifications in actual real time control case. The overall real control architecture of Hopfield NN controller is shown in Figure 4-3.
The Hopfield NN in this application has two neurons and the first neuron output
x
1 is the control signal.Figure 4-11. Real control architecture of Hopfield NN controller
For training data set, we can generate time response curve as shown in Figure 4-12 so that the above 2, 3, 4 specifications. That is, maximum overshoot = 0.91 percent, rise time = 0.00284 sec and settling time = 0.00424 sec in Figure 4-12.
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Time
Magnitude
Training data of aircraft control system
Figure 4-12. The training data of the aircraft control system
After training by Algorithm 3-2 developed in Chapter 3, the trained weighting factors in
Hopfield NN can be used to generate control signal. The final four weighting factors in this application are:
Weighting factor
w
11w
12w
21w
22 Trained value -47.552 -0.000032 -44.98 -0.048| Resistance | 0.021Ω 31250Ω 0.0222Ω 20.833Ω Table 4-2. Trained weighting factors of aircraft system
The following Figure 4-13 is the output when the Hopfield NN is applied to the closed-loop control of the attitude control of aircraft.
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Time
Magnitude
Actual running of aircraft control system
Figure 4-13. The output of the aircraft control system by Hopfield NN controller
Figure 4-14 shows the response of Figure 4-13 in transient period (0 ~ 0.01 sec). It is obvious from Figure 4-14 that maximum overshoot is 3.77% ≤ 5 percent, rise time is 0.0028sec ≤ 0.005 sec and settling time is 0.0052 sec ≤ 0.007 sec. These data shows that the Hopfield NN controller indeed can be applied as a controller in this linear control system.
Figure 4-14. The plot of Figure 4-13 in transient period Figure 4-15 shows the control signal generated from the Hopfield NN controller.
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
-0.2 0 0.2 0.4 0.6 0.8 1 1.2
The control signal generated from Hopfield neural network
Magnitide
Time
Figure 4-15. The control signal the aircraft control system
In this problem, we have shown that the Hopfield NN controller can be used to control the linear system like the aircraft control system.
CHAPTER 5 Conclusions
The application of Hopfield NN is extended to control systems in this thesis. The problem of deciding weighting factors for continuous Hopfield NN is solved in this thesis using back propagation approach. To improve the convergence rate of the back propagation training of Hopfield NN, a dynamic optimal training algorithm is also proposed to speed up the convergence speed. A new architecture of using Hopfield NN as a real-time controller is also proposed in this thesis. The popular nonlinear inverted pendulum system is first illustrated and controlled by Hopfield NN. An aircraft attitude linear control system is also illustrated and controlled by Hopfield NN. Excellent results have been obtained. In comparison with classical approaches, the Hopfield NN is very easy to implement using simple RC circuit. For higher order systems, it is also suitable for VLSI implementation. The problem of negative resistance in the trained weighting factors can be easily solved by placing a multiplexer in front of every resistance to allow for negative resistance. The following Figure 5-1 shows the inclusion of multiplexer for input signal of Hopfield NN.
Figure 5-1 The inclusion of multiplexer in Hopfield NN
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