Nonlinear Control Sy.. - Free

4.11. EXERCISES 133
To study the stability of the origin, we consider the (time-independent) Lyapunov function
candidate V(x) = 2xi(k) + 2x1(k)x2(k) + 4x2(k), which can be easily seen to be positive
definite. We need to find AV(x) = V(x(k + 1)) - V(x(k)), we have
V(x(k+1)) = 2xi(k+1)+2x1(k+1)x2(k+1)+4x2(k+1)
2 [xi(k) + x2(k)]2 + 2(x1(k) +
+4[axl(k) + 2x2(k)]2
V(x(k)) = 2x2 + 2xix2 + 4x2.
From here, after some trivial manipulations, we conclude that
x2(k)][ax3
(k) + 2x2(k)]
AV(x) = V(x(k + 1)) - V(x(k)) _ 2x2 + 2ax4 + 6ax3 x2 + 4a2x6.
Therefore we have the following cases of interest:
a < 0. In this case, AV(x) is negative definite in a neighborhood of the origin, and
the origin is locally asymptotically stable (uniformly, since the system is autonomous).
a = 0. In this case AV(x) = V(x(k + 1)) - V(x(k)) = -2x2 < 0, and thus the origin
is stable.
4.11 Exercises
(4.1) Prove Lemma 4.1.
(4.2) Prove Lemma 4.2.
(4.3) Prove Theorem 4.4.
(4.4) Characterize each of the following functions W : 1R2 x R --4 R as: (a) positive definite
or not, (b) decreasing or not, (c) radially unbounded or not.
(i) W1 (X, t) = (xi + x2)
(ii) W2(x, t) = (xl + x2)et.