C++ Neural Networks and Fuzzy Logic: Preface

Lyapunov Function

A Lyapunov function is a function that decreases with time, taking on non−negative values. It is used to

correspond between the state variables of a system and real numbers. The state of the system changes as time

changes, and the function decreases. Thus, the Lyapunov function decreases with each change of state of the

system.

We can construct a simple example of a function with the property of decreasing with each change of state as

follows. Suppose a real number, x, represents the state of a dynamic system at time t. Also suppose that x is

bounded for any t by a positive real number M. That means x is less than M for every value of t.

Then the function,

       f(x,t) = exp(−|x|/(M+|x|+t))

is non−negative and decreases with increasing t.

Local Minimum

A function f(x) is defined to have a local minimum at y, with a value z, if

       f(y) = z, and f(x) e z, for each x, such that there exists

       a positive real number h such that y – h d x d y + h.

In other words, there is no other value of x in a neighborhood of y, where the value of the function is smaller

than z.

There can be more than one local minimum for a function in its domain. A Step function (with a graph

resembling a staircase) is a simple example of a function with an infinite number of points in its domain with

local minima.

Global Minimum

A function f(x) is defined to have a global minimum at y, with a value z, if

       f(y) = z, and f(x) e z, for each x in the domain of

       the function f.

C++ Neural Networks and Fuzzy Logic:Preface

Global Minimum

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In other words, there is no other value of x in the domain of the function f, where the value of the function is

smaller than z. Clearly, a global minimum is also a local minimum, but a local minimum may not be a global

minimum.

There can be more than one global minimum for a function in its domain. The trigonometric function f(x) =

sinx is a simple example of a function with an infinite number of points with global minima. You may recall

that sin(3À/ 2), sin (7À/ 2), and so on are all –1, the smallest value for the sine function.