Nonlinear Control Sy.. - Free

4.5. ANALYSIS OF LINEAR TIME-VARYING SYSTEMS 119
4.5 Analysis of Linear Time-Varying Systems
In this section we review the stability of linear time-varying systems using Lyapunov tools.
Consider the system:
± = A(t)x (4.25)
where A(.) is an n x n matrix whose entries are real-valued continuous functions of t E R.
It is a well-established result that the solution of the state equation with initial condition
xo is completely characterized by the state transition matrix 0, Vt > to (4.27)
II'(t, to) II = Iml x II4(t, to)xll.
We now endeavor to prove the existence of a Lyapunov function that guarantees
exponential stability of the origin for the system (4.25). Consider the function
W (x' t) = xT P(t)x (4.28)
where P(t) satisfies the assumptions that it is (i) continuously differentiable, (ii) symmetric,
(iii) bounded, and (iv) positive definite. Under these assumptions, there exist constants
k1, k2 E R+ satisfying
or
kixTx < xTP(t)x < k2xTx Vt > 0,Vx E R"
k1IIx1I2 < W(x,t) < k2IIXII2 Vt > 0,Vx c 1R".
This implies that W (x, t) is positive definite, decrescent, and radially unbounded.
W (x, t) = iT P(t)x + xT Pi + xT P(t)x
= xTATPX+xTPAx+xTP(t)x
= xT[PA+ATP+P(t)]x