Nonlinear Control Sy.. - Free

124 CHAPTER 4. LYAPUNOV STABILITY II: NONAUTONOMOUS SYSTEMS
The importance of Theorem 4.8 is that it shows that the exponential stability is robust
with respect to a class of perturbations. Notice also that g(x, t) need not be known. Only
the bound (iv) is necessary.
Special Case: Consider the system of the form
Ax + g(x, t)
(4.34)
where A E 1Rn"n. Denote by Ai, i = 1, 2, . , n the eigenvalues of A and assume that
t(.\i) < 0, Vi. Assume also that the perturbation term g(x, t) satisfies the bound
119(X,0112 0, Vx E R"-
Theorem 3.10 guarantees that for any Q = QT > 0 there exists a unique matrix P = PT > 0,
that is the solution of the Lyapunov equation
PA+ATP=-Q.
The matrix P has the property that defining V(x) = xTPx and denoting ,\,,,in(P) and
Amax(P) respectively as the minimum and maximum eigenvalues of the matrix P, we have
that
-amin(P)IIx112 < V(x) < amax(P)Ilx112
- aV Ax = -xTQx < -Amin(Q)IIXI12
Thus V (x) is positive definite and it is a Lyapunov function for the linear system k = Ax.
Also
It follows that
IlOx
av
ax
Px+xTP = 2xTP
112xTP112 = 2Amax(P)Ilx112
aAx +
x ax 9(x, t)
< -amin(Q)IIx112 + [2Amax(P)IIx112] [711x112]
< -Amin(Q)Ilxl12+27., (P)Ilx1I2
fl (x) < [-Amin(Q) + 27'\, (P)] IIxI12
It follows that the origin is asymptotically stable if
-Amin(Q) + 2'yA.x(P) < 0