An Introduction to Fuzzy Control - Fuzzy Logic: An Introductory Course for Engineering Students - page 187

¨ + ˙ + =

m y b y ky u

which represents a second order linear system. In this case the state variables are
defined as

x 1 ( t ) = y ( t )

x 2 ( t ) =˙ y ( t )

and so the state space representation is,
˙

x 1 = x 2

k b 1

x 2 =− ˙

m x 1 − m x 2 + m u

The output equation is defined by

y ( t ) = x 1 ( t ).

In a matrix form, the state equation can be rewritten as,
˙

01
−

x 1
x 2

0
1
m

u

x 1
˙

= + ,

k
m

b
m

x 2 −

similarly the output equation is

= 10 x 1
x 2

y .

Both state and output equations form the matrix state space representation of the
system

˙

x = Ax + Bu

y = Cx

with

01
−

0
1
m

= 10 .

A = , B = , C

k
m

b
m

−

8.3.2 Takagi-Sugeno Models for Control of Nonlinear Systems

The so-called Takagi-Sugeno (TS) fuzzy model is particularly useful for the control
of nonlinear systems. The TS fuzzymodel is described by fuzzy “if-then” rules which
represent local linear input-output relations of a nonlinear system in a state-space