New Concepts for the Estimation of Takagi-Sugeno Model Based on Extended Kalman Filter - Fuzzy Modeling and Control: Theory and Applications - page 18

Fig. 1.2 Inverted pendulum system

In order that this iterative method becomes equivalent to the non iterative one
based on weighting of parameters mentioned before in Sect. 1.4 , we should taken
into consideration that the factor

of the non iterative method weights the norm
of the parameters in front of the quadratic sum of errors resulted for the modeling
of samples. On the other hand, the factor weights the norm of parameters infront
of each one of the errors of samples. If m is the number of used samples, then the
equivalence is produced supposing that

γ

δ = γ/ m and in general δ γ should be

fulfilled.

1.6 Illustrative Example

In this section the proposed estimation method is illustrated by an example of an
inverted pendulum (see Fig. 1.2 ).
The inverted pendulum can be represented as follows:

ml θ

2 sen

u + θ

g sen θ − cos θ( )

θ =

M + m

(1.99)

m cos 2 4 θ

l (

3 −

)

M + m

where denotes the angular position (in radians) deviated from the equilibrium

position (vertical axis) of the pendulum and

θ

θ

is the angular velocity, g(gravity

9.8 m acceleration) =

s 2 , M(mass) of the cart

= 1kg, m(mass) of the pole = 0.1kg, l is

the distance from the center of the mass (m) of the pole to the cart = 0.5m.

= θ Assuming that x 1 = θ and x 2 , then ( 1.99 ) can be rewritten in state space

form as follows: