Stabilization by an estimated state controller of nonlinear time-varying systems

Abstract

In linear systems, it is well known that any combination of stabilizing state feedback controller and a state observer results in the output feedback controller which asymptotically stabilizes the closed-loop system. This is known as the separation principle, which does not generally hold for nonlinear systems. For nonlinear systems, the output feedback problem is usually pursued by extending the results of observer synthesis, which itself is a challenging problem. In this paper, we investigate the stabilization via an observer configuration problem for some classes of nonlinear time-varying systems by using new integral inequalities of Gronwall type. Given that the sufficient conditions of the controller and observer problem are satisfied, we show that the proposed controller with estimated state feedback from the proposed practical observer will achieve the global practical exponential stabilization. Furthermore, the effectiveness of the proposed scheme is shown through simulation for two numerical examples.

On observer compensator design for non-autonomous control semi-linear evolution equations

“…We observe that the practical Luenberger observer (16) gives a good estimate of the state of (1) provided that the non-autonomous system (3) is detectable.…”

“…Remark 3. Uniform practical exponential stability given in (7) are said with growth constants 𝑐, 𝜔, which it is similar to the definition introduced by [16] in the case of finite-dimensional systems. The inequality (7) indicates that the trajectory will be ultimately bounded, that is the solution is bounded and approaches toward a neighborhood of the origin for sufficiently large 𝑡.…”

“…In finite dimensions, one simple way of designing a compensator is to first construct a state feedback stabilizer and an observer for the system and then combine the two to design a compensator using the feedback of the observer instead of the state. This is the so-called separation principle, see [4,16,18,19]. There are almost no results on observers and compensators of control infinite-dimensional systems with the associated nominal system being linear that depends on the time.…”