7.5. EXERCISES AND PROJECTS 247
- A nonlinear plant is governed by the following difference equation
y(k+1)=0. 3 y(k)+0.6[y(k−1)]^0.^5 +f[r(k)]
wherey(k)andr(k)are the output and input, respectively, at time stepk.
The unknown functionf[r(k)]has the formf(r)=0.5sin(πr)cos(3πr)+
0 .3cos(πr)sin(5r).
(a) Develop a backpropagation neural network to perform system iden-
tification of the unknown plant.
(b) Use ANFIS to identify the unknown system.
(c) How does the performance of the neural network compare with that
of ANFIS?
- The following dynamical equation describes the behavior of a nonlinear
plant
y(k+1)=
y(k)u(k)
1+[|y(k−1)|]^0.^3
−
∑
1 −e−u(k)
1+e−u(k)
∏
wherey(k)andu(k)are the output and input, respectively, at time step
k.
(a) Assuming the control lawu(k)to the system is specified, identify the
nonlinear plant dynamics using ANFIS.
(b) For the desired output of the plant specified by
yd(k)=0.5sin(2πk)cos(2πk)+0.3sin^2 (2πk/15)
how well can ANFIS predict the system behavior?
(c) Compare ANFIS results with that of a neural network-based system
identification.
- Construct an ANFIS that is equivalent to a two-input, two-rule Mamdani
fuzzy model. Describe the function that you use to approximate centroid
defuzzification. Explain how this function is converted into node functions
in the resulting ANFIS.
- Construct an ANFIS that is equivalent to a two-input, two-rule Larsen
fuzzy model. Describe the function that you use to approximate centroid
defuzzification. Explain how this function is converted into node functions
in the resulting ANFIS.
- Approximate the functionf(x 1 ,x 2 )=^12 x^21 +^12 x^22 with a Sugeno fuzzy
inference system (FIS) using Gaussian antecedent membership functions
and linear consequent membership functions.
