# American Institute of Mathematical Sciences

September  2015, 5(3): 585-611. doi: 10.3934/mcrf.2015.5.585

## Generalized homogeneous systems with applications to nonlinear control: A survey

 1 Department of Electrical and Computer Engineering, The University of Texas at San Antonio, San Antonio, TX 78249, United States 2 Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH 44106, United States 3 School of Automation, Southeast University, Nanjing, Jiangsu 210096, China

Received  November 2014 Revised  May 2015 Published  July 2015

This survey provides a unified homogeneous perspective on recent advances in the global stabilization of various nonlinear systems with uncertainty. We first review definitions and properties of homogeneous systems and illustrate how the homogeneous system theory can yield elegant feedback stabilizers for certain homogeneous systems. By taking advantage of homogeneity, we then present the so-called Adding a Power Integrator (AAPI) technique and discuss how it can be employed to recursively construct smooth state feedback stabilizers for uncertain nonlinear systems with uncontrollable linearizations. Based on the AAPI technique, a non-smooth version as well as a generalized version of AAPI approaches can be further developed from a homogeneous viewpoint, resulting in solutions to the global stabilization of genuinely nonlinear systems that may not be controlled, even locally, by any smooth state feedback. In the case of output feedback control, we demonstrate in this survey why the homogeneity is the key in developing a homogeneous domination approach, which has been successful in solving some difficult nonlinear control problems including, for instance, the global stabilization of systems with higher-order nonlinearities via output feedback. Finally, we show how the notion of Homogeneity with Monotone Degrees (HWMD) plays a decisive role in unifying smooth and non-smooth AAPI methods under one framework. Other applications of HWMD will be also summarized and discussed in this paper, along the directions of constructing smooth stabilizers for nonlinear systems in special forms and low-gain'' controllers for a class of general upper-triangular systems.
Citation: Chunjiang Qian, Wei Lin, Wenting Zha. Generalized homogeneous systems with applications to nonlinear control: A survey. Mathematical Control & Related Fields, 2015, 5 (3) : 585-611. doi: 10.3934/mcrf.2015.5.585
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##### References:
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