Stable Fuzzy Control System by Design - Fuzzy Modeling and Control: Theory and Applications - page 79

4.3.2 Design Algorithm

In this section, an algorithm for the fuzzy controller design based on the Theorem
4.1 is presented. To facilitate the design of the fuzzy controller, the synthesis process
is divided into three phases: an initialization phase ,a design phase , and a final
adjustment phase of the equilibrium state (Andújar et al. 2009 ).

4.3.2.1 Initialization Phase

If there is no information to create the rules of the fuzzy controller, they can be created
from the rules of the fuzzy model of the plant, so that each of the fuzzy controller
outputs shall consist of all the rules of the fuzzy model plant. After defining the
antecedents, they remain constant for the remainder of the design process.
To initialize the regions q involved in Theorem 4.1, the antecedents defined for

the fuzzy controller are used. Thus, the state space is divided into so many regions as
rules own the fuzzy controller. Similarly, to define the membership functions

ϕ q ( x )

r

can be used the degree of activation of the rules of the fuzzy controller, ω

j (

x ) , which

is calculated from ( 4.18 ).
Finally, the matrices P q are initialized as the identity matrix.
The initialization phase can be synthesized by the algorithm shown in Fig. 4.4 .

4.3.2.2 Design Phase

During the design phase, the consequents of the fuzzy controller and the matrices
involving compliance Theorem 4.1 should be obtained. To ensure that the matrices
P q are positive definite using Cholesky decomposition, so that:

T q T q P q = (4.50)

where T q is an upper triangular matrix with all its diagonal elements positive.
Theorem 4.1 must be met continuously in a region around the steady state, how-
ever, from a practical point of view, it is necessary to discretize the space and evaluate
compliance of the theorem in a finite number of points. Nevertheless, the discretiza-
tion will maintain the stability condition only if F

( x )< 0 in a finite and dense enough

set of points and f d x are bounded and locally Lipschitz func-

tions (Johansen 2000 ). Using a minimization algorithm (for example a genetic algo-
rithm) with the objective function given in Fig. 4.5 in a dense enough set of points,
the Theorem 4.1 will be satisfied in the region where the objective function get is
null. Programming routines necessary for the evaluation of the design phase are in
Barragán and Andújar ( 2012 ).

( x ) , ϕ q ( x ) and d ϕ q ( x )/