Nonlinear Control Sy.. - Free

3.6. POSITIVE DEFINITE FUNCTIONS REVISITED 81
Example 3.9 Let V (x) = xT Px, where P is a symmetric matrix. This function is positive
definite if and only if the eigenvalues of the symmetric matrix P are strictly positive. Denote
)m%n(P) and ),ax(P) the minimum and maximum eigenvalues of P, respectively. It then
follows that
Am.m(P)Ilxll2 < xTPx < Amax(P)Ilxll2
.m,n(P)Ilxll2 < V(x) < Amax(P)Ilxll2.
Thus, al, a2 : [0, oc) --> R+, and are defined by
al(x) = .min(P)Ilxl12
a2(x) = \max(P)Ilxll2.
For completeness, we now show that it is possible to re state the stability definition
in terms of class 1C of functions.
Lemma 3.2 The equilibrium xe of the system (3.1) is stable if and only if there exists a
class IC function a(.) and a constant a such that
Proof: See the Appendix.
1x(0) - xell < a = lx(t) - xell 0. (3.4)
A stronger class of functions is needed in the definition of asymptotic stability.
Definition 3.10 A continuous function 3 : (0, a) x pg+ -+ R+ is said to be in the class KL
if
(i) For fixed s, 33(r, s) is in the class IC with respect to r.
(ii) For fixed r, 3(r, s) is decreasing with respect to s.
(iii) 0(r,s)---*0 ass -oc.
Lemma 3.3 The equilibrium xe of the system (3.1) is asymptotically stable if and only if
there exists a class ICL function and a constant e such that
Proof: See the Appendix.
1x(0) - xell < S = 1x(t) - xell 0. (3.5)