226 CHAPTER 6. NEURAL CONTROL
(a) Test the neural network performance to an input speciΩ fied byu(t)=
0. 2 e−^0.^02 πksin (0. 2 πk) if 0 ≤k≤ 100
0. 2 e−^0.^02 πksin (0. 2 πk)+0. 8 e(−^0.^02 πk)cos (0. 2 πk) if 100 <k≤ 300
(b) Simulate the performance of the trained neural network withu(t)as
specified above, and with additive Gaussian noiseη(t)=N(0, 0 .04).
Discuss the performance.
- A nonlinear process is characterized by a single input and two outputs as
y 1 (t)=f 1 (y 1 (t−1),y 1 (t−2),y 2 (t−1),y 2 (t−2)) +u(t)
y 2 (t)=f 2 (y 1 (t−1),y 1 (t−2),y 2 (t−1),y 2 (t−2)) +u(t)
where
f 1 (y 1 (t−1),y 1 (t−2),y 2 (t−1),y 2 (t−2)) =
≥
1 .85 + 0. 24 e−y^1 (t−1)
2 ¥
y 2 (t−1)−
≥
0 .35 + 0. 34 e−y^1 (t−2)
2 ¥
y 2 (t−2)
f 2 (y 1 (t−1),y 1 (t−2),y 2 (t−1),y 2 (t−2)) =
≥
1 .35 + 0. 28 e−y^2 (t−1)
2 ¥
y 1 (t−1)−
≥
0 .65 + 0. 25 e−y^2 (t−2)
2 ¥
y 1 (t−2)
(a) Determine through simulations the range ofu(t)∈[min,max]for
which the nonlinear system is stable.
(b) Develop a backpropagation neural network that identifiesf 1 andf 2.
(c) Choosing a variable frequency sinusoidal input within the range de-
termined in part (a), discuss the performance of the trained neural
network.
- For the magnetic levitation exercise in Chapter 4, we are required to re-
place the fuzzy controller with a suitable neural network controller. The
equation of motion for the steel ball and the equation governing variation
in the current in the electrical circuit are reproduced here for convenience.
For the steel ball,
M
d^2 z
dt^2
=Mg−F
and for the electrical circuit,
L
di
dt
+Ri=V
whereLis the coil inductance andRis the coil resistance. The coupling
equation where the forceF is related to the currentiis given by
F=km
i^2
z^2
