**6. Conclusion**

The implementation are coded in Matlab using the initial states: *x*<sup>1</sup>

**5**. **Figure 4(a)** and **(b)** shows that the observer states *x*

18 Robust Control - Theoretical Models and Case Studies

resetting propagation law Γ(*t*) and the estimated *ς*(*t*), that is, *ς*

are depicted in **Figure 5(c)** and **(d)**. It is seen clearly that the Γ(*t*) and *ς*

**Figure 5.** (a) is the values of Γ(*t*). (b) demonstrates the least-squares estimated results of *ς*

of the observer dynamics e(t). (d) computes the 2-norm value of time-varying matrix function *F* (*t*) with *b* = 0.

*x* ^ 2

time.

(0)=0.1, *ς*

(0)=0.5, *x*<sup>2</sup>

^ cohere with the plant states *x*. It is,

^(*t*) are shown in **Figure 5(a)** and

^(0)=0.1, Γ(0)=*k*<sup>0</sup> =2, and *k*<sup>1</sup> =0.3. The simulation results are depicted in **Figures 4** and

therefore, seen that the observer (38) being driven by time-varying term *e*(*t*) can actually trace the plant (37). The control input *u*(*t*) to the system is shown in **Figure 4**(c). The covariance

**(b)**. The observer driving force *e*(*t*) and 2-norm value of the time-varying matrix function *F*(*t*)

accommodate the time-varying effects that driving the observer dynamics as the closed-loop system is approaching equilibrium point. The driving force to the observer dynamics *e*(*t*) shows the same results. **Figure 5(c)** depicts that the time-varying matrix *F*(*t*) is indeed varying with

(0)= −0.6, *x*

^ 1 (0)=0.0,

^(*t*) are adjusted to

^(*t*). (c) is the driving force

This paper has developed the modified time-invariant observer control for a class of timevarying systems. The control scheme is suitable for the time-varying system that can be characterized by the multiplicative type of time-invariant and time-varying parts. The timeinvariant observer is constructed directly from the time-invariant part of the system with additional adaptation forces that are prepared to account for time-varying effects coming from the measured output feeding into the modified observer. The derivation of adaptation forces is based on the least squares algorithms in which the minimization of the cost of error dynamics considers as the criteria. It is seen from the illustrative application that the closed-loop systems are showing exponentially stable with system states being asymptotically approached by the modified observer. Finally, the LMI process has been demonstrated for the synthesis of control and observer gains and their implementation on a mass-spring-damper system proves the effectiveness of the design.
