A separation principle for the control of a class of nonlinear systems

Abstract

In this paper we give a general formulation of a number of control problems. This formulation considers a wide class of systems and any globally bounded state feedback controller that renders a certain compact set positively invariant and asymptotically attractive. We develop a converse Lyapunov theorem, and we prove that, by implementing the control law using a high-gain observer, we can recover asymptotic stability of the attractive set, its region of attraction, and trajectories.

Peaking-Free Output-Feedback Adaptive Neural Control Under a Nonseparation Principle

“…It was proved in [2] that the combination of high-gain observers with control saturation enables output-feedback control to recover the trajectories of its state-feedback counterpart if the observer gain is sufficiently high. For nonlinear systems with functional uncertainties, output-feedback approximation-based adaptive control (AAC) using high-gain observers and fuzzy systems/neural networks (NNs) has also been intensively studied [3]- [8] to overcome the limitations of state-feedback AAC [9]- [15].…”

Robust output feedback control of incompletely observable nonlinear systems without input dynamic extension

“…Thus, Theorem 3 implies that the original uniformly asymptotically stable positively invariant set under state feedback is recovered, in the limit as E -+ 0, under output feedback. Furthermore, note that the implication of Theorem 3 is essentially identical to one idea found in [6]. This indicates that using an input derivative observer is advantageous in that, besides simplifying the controller design, in the presence of disturbance it yields the same stability results as when dynamic extension is employed at the input side of the system.…”

“…This modification results in a simplification of the control design problem in that the original state feedback controller can be directly employed by the output feedback controller (i.e., , we eliminate the need to design a control law for the higher-order "extended system" in point 1 above). We prove that, in the presence of disturbances, this approach guarantees uniform ultimate boundedness (UUB) of the closed-loop system trajectories with respect to n/ (in complete analogy with the result of Theorem 4 in [6], but here we do not investigate conditions needed on the disturbance so that asymptotic stability is recovered). When no disturbance affects the system, it is shown that a design that is based on a sliding mode observer recovers the asymptotic stability of the closed-loop system with respect to N , while the high-gain observer achieves UUB.…”

“…In [6] the authors use a similar assumption, with the additional requirement that f, and h, be globally bounded functions of x, a requirement that is not restrictive since it can be fulfilled by using saturation functions. On the other hand, owing to the structure of our output feedback controller, this restriction is not needed in our framework.…”

“…On the other hand, owing to the structure of our output feedback controller, this restriction is not needed in our framework. Notice that, while in [6] d ( t ) is a measurable exogenous signal, we assume d ( t ) to be an unmeasurable disturbance (indeed we do not include d ( t ) as an input to the control law (4)), and hence our control objective is somewhat different.…”