The solution is given in Theorem 2, generalizing the polynomial formulas from  to time-varying systems
. The difference which makes this extension non-obvious is that a time-invariant system is described by two non-commutative polynomials while a time-varying system
requires three polynomials, and the third one has to be incorporated into the analysis.
In order to demonstrate the effectiveness of our proposed algorithm, let the discrete-time linear time-varying system
For the linear time-varying system
(1)-(2), assume the estimate of the state at k - 1 is [??](k - 1 | k - 1), the corresponding estimation error covariance is P(k - 1 | k - 1), and then the fusion estimate of the state at k is
Caption: Figure 5: Fault estimating result in 30th iterations of time-varying system
In the experiment, no external excitation is generated to excite the coupled time-varying system
(the shaker shown in Figure 9 is not in use), and the system is only excited by the motion of the moving mass.
Although the first paper about the commutativity appeared in 1977  which had introduced the commutativity concept for the first time and studied the commutativity of the first-order continuous-time linear time-varying systems
, this paper is important for proving that a time-varying system
can be commutative with another time-varying system
only; further very few classes of systems can be commutative.
Secondly, a dynamic fuzzy logic controller is designed by means of a linear time-varying system
to compensate for the unknown model uncertainties [f.sub.[DELTA]](x) and [g.sub.[DELTA]](x) and disturbance d(t).
Additional simplifying assumptions allow one to describe the time-varying system
with finite set of coefficients.
The state-space model of non-linear time-varying system
is supposed to consist of a discrete-time difference equation.
We first consider the continuous-time nonlinear time-varying system
Hence, many authors devote themselves to studying the stability and many effective methods of the time-varying system
to gain less conservative delay-dependent stability criteria [1-11], which include linear systems with the delay-fraction theory [2, 3] and nonlinear systems Lure systems [6-8].
A moving mass is a proper characteristic of a time-varying system
, which is in general one of the sources of nonstationary signals.