A First Course in FUZZY and NEURAL CONTROL

6.6. EXERCISES AND PROJECTS 225

where

ζ(τ)=y(τ)+

Xn

i=1

aiy(τ−i)−

Xm

i=0

biu(τ−i)

is the bias term. The approximate model may thus be interpreted as a lin-
ear model affected by a constant disturbanceζ(τ), depending on the current
operating point.
To apply the instantaneous linearization technique to the design of con-
trollers, a linear model is extracted from a neural network model of the system
at each sampling time and a linear controller is designed. The control design is
based on the certainty equivalence principle ñ that is, the design block assumes
the extracted linear model is a perfect description of the system. One can regard
this as a gain-scheduling controller with an infinite schedule.
One appealing feature of the linearization technique is that essentially any
linear control design can be incorporated in the process. Linearization results
in the bias term have to be compensated for, of course. This can be achieved
by introducing integral action into the controller, in which case other constant
disturbances will be compensated for as well.
Depending on the character of the nonlinearities of the system, and the se-
lected reference trajectory, the linear model can be considered valid only within
a relatively short time period. One must be careful to choose controller designs
not violating the limits of the approximation. Instantaneous linearization can
be applied to deterministic or stochastic models, but we have discussed only
the deterministic case. We refer you to [54] for more details on instantaneous
linearization.

6.6 Exercises and projects

1. A nonlinear system is defined byy(k)=f(y(k−1)) + [u(k−1)]^3 ,where

f(y(k−1)) =

y(k−1)

1+y(k−1)^2

isthesystemtobeidentified. Develop a backpropagation neural network

that can identify the nonlinear system. Examine the performance to the

following input

u(k)=

Ω

2 e−^0.^02 πk if 0 ≤k≤ 50

10 e−^0.^01 πksin (0. 2 πk) if 50 <k≤ 150

Compare the performance of the network using two ofMatlabís training

algorithms, namely,trainlm, the Levenberg-Marquardt optimization, and

traingd, the gradient descent optimization techniques.

2. A nonlinear chemical process is specified asy(t)=f(y(t−1),y(t−2)) +
   0 .1(u(t−1))^2 ,wheref(y(t−1),y(t−2)) = 0. 5 y(t−1)− 0. 1 y(t−1)y(t−2)
   isthesystemtobeidentified. Develop a backpropagation neural network
   that can identify the nonlinear system.