Nonlinear Control Sy.. - Free

98 CHAPTER 3. LYAPUNOV STABILITY I. AUTONOMOUS SYSTEMS
(i) Choose an arbitrary symmetric, positive definite matrix Q.
(ii) Find P that satisfies equation (3.22) and verify that it is positive definite.
Equation (3.22) appears very frequently in the literature and is called Lyapunov equation.
Remarks: There are two important points to notice here. In the first place, the approach
just described may seem unnecessarily complicated. Indeed, it seems to be easier to first
select a positive definite P and use this matrix to find Q, thus eliminating the need for
solving the Lyapunov equation. This approach may however lead to inconclusive results.
Consider, for example, the system with the following A matrix,
taking P = I, we have that
=
0 4
`4 -8 -12
-Q = PA + AT P = [
0 -4
4 -24 ]
and the resulting Q is not positive definite. Therefore no conclusion can be drawn from this
regarding the stability of the origin of this system.
The second point to notice is that clearly the procedure described above for the
stability analysis based on the pair (P, Q) depends on the existence of a unique solution
of the Lyapunov equation for a given matrix A. The following theorem guarantees the
existence of such a solution.
Theorem 3.10 The eigenvalues at of a matrix A E R T satisfy te(A1) < 0 if and only
if for any given symmetric positive definite matrix Q there exists a unique positive definite
symmetric matrix P satisfying the Lyapunov equation (3.22)
Proof: Assume first that given Q > 0, 2P > 0 satisfying (3.22). Thus V = xTPx > 0
and V = -xT Qx < 0 and asymptotic stability follows from Theorem 3.2. For the converse
assume that te(A1) < 0 and given Q, define P as follows:
P = 10
00
eATtQeAt
dt
this P is well defined, given the assumptions on the eigenvalues of A. The matrix P is also
symmetric, since (eATt)T = eAt. We claim that P so defined is positive definite. To see