Systems and Control
Broad in scope, this text shows the multidisciplinary role of dynamics and control, presents neural networks, fuzzy systems, and genetic algorithms, and provides a self-contained introduction to chaotic systems.
TABLE OF CONTENTS Systems and Control
4.3: Stability of Linear Systems
4.3 Stability of Linear Systems
It follows from the discussion in the previous sections that a linear time-invariant dynamical system modeled by the equations
is asymptotically stable if and only if the solution to equation (4.4) decays to zero as t ? ? for any initial state x 0. We can view the vector 
as defining the coordinates of a point moving in an n-dimensional state space 
. In an asymptotically stable system modeled by equation (4.4), this point converges to the origin of 
. We can argue that if a trajectory is converging to the origin of the state space, then it should be possible to find a family of nested surfaces described by V( x 1, x 2, , x n) = c, c ? 0, such that monotonically decreasing values of c give surfaces progressively shrinking in on the origin with the limiting surface V( x 1, x 2, x n) = 0 being the origin x = 0. Furthermore, along every trajectory of (4.4), the parameter c should steadily decrease. We now introduce some notation and definitions that we use in the ensuing discussion.
Let 
be real-valued function and let 
be a compact region containing the origin x = 0 in its interior.
Definition 4.8
We say that the function V = V( x) is positive semidefinite (p.s.d.) in
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