Nonlinear Control Sy.. - Free

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CHAPTER 3. LYAPUNOV STABILITY I. AUTONOMOUS SYSTEMS
(ii) V(x) > 0, Vx in D - {0}.
V : D -4 IR is said to be positive definite in D if condition (ii) is replaced by (ii')
(ii') V(x) > 0 in D - {0}.
Finally, V : D -+ R is said to be negative definite (semi definite)in D if -V is positive
definite (semi definite).
We will often abuse the notation slightly and write V > 0, V > 0, and V < 0 in D to
indicate that V is positive definite, semi definite, and negative definite in D, respectively.
Example 3.2 The simplest and perhaps more important class of positive definite function
is defined as follow:s,
V(x):IR"->1R=xTQx, QEIRnxn,
Q=QT.
In this case, defines a quadratic form. Since by assumption, Q is symmetric (i.e.,
Q = QT), we know that its eigenvalues Ai, i = 1, n, are all real. Thus we have that
positive definite
positive semi definite
V(.) negative definite
negative semi definite
Thus, for example:
Vi (x) : 1R2 --> R = axi + bx2 = [x1, x2] [
xTQx>O,dx0O b A,>O,di=1,..., n
xTQx>0,Vx#0 = A,>0,Vi=1,..., n
xTQx < 0, Vx # 0 b At < 0, Vi = 1, ... , n
xTQx0, b'a, b > 0
> 0, Va > 0.
is not positive definite since for any x2 # 0, any x of the form x' = [0,x2]T # 0;
however, V2(x*) = 0.
Positive definite functions (PDFs) constitute the basic building block of the Lyapunov
theory. PDFs can be seen as an abstraction of the total "energy" stored in a system, as we
will see. All of the Lyapunov stability theorems focus on the study of the time derivative of a
positive definite function along the trajectories of 3.1. In other words, given an autonomous