An Introduction to Fuzzy Control - Fuzzy Logic: An Introductory Course for Engineering Students - page 190

8.3.3 Stability Analysis

TS fuzzy model is a convex combination of linear models. This structure facili-
tates stability analysis and observer design by using effective algorithms based on
Lyapunov functions and linear matrix inequalities (LMI).
Talking about the Lyapunov function V

of a system, is in a way equivalent
to talk about the potential energy of the system. For a system to be stable in the
sense of Lyapunov, it must remain within a bounded region “close-enough” to an
equilibrium point starting from any initial state, and when external input is u

( x )

0.
Similarly, asymptotic stability means that the system, not only remains close, but
must eventually converge to an equilibrium point. Any system in a given initial state
must have a positive potential energy V

( t ) =

0 in an equilibrium point),
and to ensure that it eventually converges to an equilibrium point, the derivative of
this Lyapunov function must be negative

( x ) 0( V ( x ) =

V

d

0. In a sense, it must
be proven that the energy of the system eventually dissipates taking the system to an
equilibrium point, when no external input is acting over the system ( u

( x ) =

dt V

( x )

0).
The basic stability analysis considers the following quadratic Lyapunov function:

( t ) =

x T V ( x ) = ( t ) Px ( t ),

P T

with P 0. If this Lyapunov function is considered, its derivative along the

trajectories of the TS fuzzy subsystem of the type

= >

x ˙ = A i x is

V

x T A i

( ) =˙ ( ) + ˙ ( ) = +

x t P P x t P PA i

and so:
Theorem 8.3.3 The equilibrium of a TS fuzzy system is globally asymptotically
stable if there exists a common positive definite matrix P such that

A i P + PA i < 0 ,

for i r.

That is, a common P has to exist for all subsystems. This theorem reduces to the
Lyapunov stability theorem for continuous linear systems when r

= 1 , 2 ,...,

= 1. If there exists

a P > 0 such that V ( x ( t )) proves the stability of the system, it is also said to be

quadratically stable as V is called a quadratic Lyapunov function.

The theorem presents a sufficient condition for the quadratic stability of the TS
system, and this is a common P problem that can be solved efficiently via convex
optimization techniques for LMIs. For systems and control, the LMI optimization is
particularly useful due to the fact that a wide variety of system and control problems
can be recast as LMI problems. Apart from a few special cases these problems do
not have analytical solutions. However, through the LMI framework they can be
efficiently solved numerically in all cases. Therefore, recasting a control problem as
an LMI problem is equivalent to finding a “solution” to the original problem.

( x ( t ))