Nonlinear Control Sy.. - Free

4.4. PROOF OF THE STABILITY THEOREMS 115
Theorem 4.4 Suppose that all the conditions of theorem 4.2 are satisfied, and in addition
assume that there exist positive constants K1, K2, and K3 such that
KllIxMMP < W(x,t) < K2IIxllP
W(x) < -KsllxI '.
Then the origin is exponentially stable. Moreover, if the conditions hold globally, the x = 0
is globally exponentially stable.
Example 4.6 Consider the following system:
21 = -xl - e-2tx2
{ ±2 = x1 - x2.
To study the stability of the origin for this system, we consider the following Lyapunov
function candidate:
W (X, t) = X2 + (1 + e-2i)x2.
Clearly
Vi (x)
thus, we have that
Then
(x1 + x2) < W(x,t) < (x1 + 2x2) = V2(x)
W (x, t) is positive definite, since Vl (x) < W (x, t), with V1 positive definite in R2.
W (x, t) is decrescent, since W (x, t) > V2 (x), with V2 also positive definite in R2.
W (x' t)
ax
f (x, t) + at
-2[x1 - X1X2 + x2(1 + 2e-2t)]
< -2[x1 - x1x2 + 3x21.
It follows that W (x, t) is negative definite and the origin is globally asymptotically stable.
4.4 Proof of the Stability Theorems
We now elaborate the proofs of theorems 4.1-4.3.
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