Nonlinear Control Sy.. - Free

120 CHAPTER 4. LYAPUNOV STABILITY II: NONAUTONOMOUS SYSTEMS
or
W(x,t) = -xTQ(t)xT
where we notice that Q(t) is symmetric by construction. According to theorem 4.2, if Q(t)
is positive definite, then the origin is uniformly asymptotically stable. This is the case if
there exist k3, k4 E 1(P+ such that
k3xTx < xTQ(t)x < k4xTx Vt > 0,Vx E IRT.
Moreover, if these conditions are satisfied, then
k3IIxII2 < Q(t) < k4IIxII2 Vt > 0,Vx E Rn
and then the origin is exponentially stable by Theorem 4.4.
Theorem 4.6 Consider the system (4.25). The equilibrium state x = 0 is exponentially
stable if and only if for any given symmetric, positive definite, continuous, and bounded matrix
Q(t), there exist a symmetric, positive definite, continuously differentiable, and bounded
matrix P(t) such that
Proof: See the Appendix.
4.5.1 The Linearization Principle
-Q(t) = P(t)A(t) + AT(t)p(t) + P(t). (4.29)
Linear time-varying system often arise as a consequence of linearizing the equations of a
nonlinear system. Indeed, given a nonlinear system of the form
x = f (x, t) f : D x [0, oo) -> R' (4.30)
with f (x, t) having continuous partial derivatives of all orders with respect to x, then it is
possible to expand f (x, t) using Taylor's series about the equilibrium point x = 0 to obtain
X = f (x, t) = f (0, t) + of Ix-o x + HOT
where HOT= higher-order terms. Given that x = 0 is an equilibrium point, f (0, t) = 0,
thus we can write
or
f(x,t) N Of x=O
x = A(t)x + HOT (4.31)