The solution is given in Theorem 2, generalizing the polynomial formulas from [4] 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 as follows.

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 [12] 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 with delay

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.