The region of attraction of the Lyapunov asymptotic stability at the origin is defined to be a ball centered at the origin, which is clearly simply connected and also bounded in the local case. In this article, the concept of substability is proposed, which allows gapsand holesin the region of attraction of the Lyapunov exponential stability, and also allows the origin to be a boundary point of the region of attraction. The concept is meaningful and useful in many practical applications, but is particularly made so with the control of single-and multi-order subfully actuated systems. Specifically, the singular set of a sub-FAS is first defined, and a substabilizing controller is then designed such that the closed-loop system is a constant linear one with an arbitrarily assignable eigen-polynomial, but with its initial values restricted within a so-called region of exponential attraction (ROEA). Consequently, the substabilizing controller drives all the state trajectories starting from the ROEA exponentially to the origin. The introduced concept of substabilization is of great importance because, on the one side, it is often practically useful since the designed ROEA is often large enough for certain applications, while on the other side, Lyapunov asymptotically stabilizing controllers can be further easily established based on substabilization. Several examples are given to demonstrate the proposed theories.

Substability and Substabilization: Control of Subfully Actuated Systems.

A general time-varying nonlinear uncertain system with time-varying delays is proposed, which is composed of two subsystems: one is an uncertain fully actuated subsystem representing the controllable part in the system, the other is an isolated globally uniformly asymptotically (GUA) stable autonomous subsystem which represents the uncontrollable part in the system. Both the single-order fully actuated system (FAS) and the multiorder FAS representations of the controllable subsystem are introduced. The problem of robust stabilization of such a compound system with full-state feedback can be converted into an input-to-state GUA stabilization problem of the fully actuated subsystem, which is solved for both types of single-and multiorder FASs with very general assumptions on the uncertain perturbed functions. Under certain conditions, the solution reduces to that for robust stabilization with partial-state feedback, and naturally reduce to that for robust stabilization of FASs. Two illustrative examples demonstrate both the effect and the application procedure of the proposed robust stabilization approach.

Robust Stabilization of Time-Varying Nonlinear Systems With Time-Varying Delays: A Fully Actuated System Approach.

Brockett’s First Example: An FAS Approach Treatment

Stabilization via Fully Actuated System Approach: A Case Study

In this paper, the fully actuated system (FAS) approaches for continuous-time systems with time-varying state delays and a constant input delay are presented. Two types of continuous-time high-order FASs are proposed: single-order FASs with both state and input delays and multi-order FASs with both state and input delays. Controllers for both types of time-delay FASs are designed based on the full-actuation features of the systems. Unlike the case of FASs with state delays only, a prediction scheme is required and constructed for both types of FASs with input delays. Similar to the case of FASs with state delays only, constant linear closed-loop systems with arbitrarily assignable eigenstructures are also developed. Illustrative examples are provided to demonstrate the effect of the proposed theories. 

https://link.springer.com/content/pdf/10.1007/s11432-021-3460-y.pdf

Fully actuated system approaches for continuous-time delay systems: part 2. Systems with input delays

Fully Actuated System Approach for 6DOF Spacecraft Control Based on Extended State Observer

Fully actuated system approach to attitude control of flexible spacecraft with nonlinear time-varying inertia

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