{{Short description|Robust nonlinear control to achieve exponential stabilization}}
The '''robust integral of the sign of the error controllers''' or '''RISE controllers''' constitute a class of continuous robust control algorithms developed for nonlinear, control‐affine systems subject to uncertainties and disturbances. Distinguished by their capability to guarantee asymptotic tracking of reference trajectories even in the presence of bounded modeling errors, RISE controllers can be used where the exact system dynamics are unknown.<ref>{{cite journal |last1=Qu |first1=Z. |last2=Xu |first2=J. X. |year=2002 |title=Model-based learning controls and their comparisons using Lyapunov direct method |url=https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1934-6093.2002.tb00336.x |journal=Asian Journal of Control |volume=4 |issue=1 |pages=99–110|doi=10.1111/j.1934-6093.2002.tb00336.x |url-access=subscription }}</ref><ref>{{Cite journal |last1=Xian |first1=B. |last2=Dawson |first2=D.M. |last3=de Queiroz |first3=M.S. |last4=Chen |first4=J. |date=2004 |title=A continuous asymptotic tracking control strategy for uncertain nonlinear systems |journal=IEEE Transactions on Automatic Control |volume=49 |issue=7 |pages=1206–1211 |doi=10.1109/TAC.2004.831148 |bibcode=2004ITAC...49.1206X |issn=1558-2523}}</ref> Recent theoretical advancements have further extended these results to prove exponential stability under appropriate conditions.<ref name=":0">{{Cite journal |last1=Patil |first1=Omkar Sudhir |last2=Isaly |first2=Axton |last3=Xian |first3=Bin |last4=Dixon |first4=Warren E. |date=2022 |title=Exponential Stability With RISE Controllers |journal=IEEE Control Systems Letters |volume=6 |pages=1592–1597 |doi=10.1109/LCSYS.2021.3127134 |bibcode=2022ICSL....6.1592P |issn=2475-1456}}</ref><ref name=":1">{{Cite book |last1=Patil |first1=Omkar Sudhir |last2=Stubbs |first2=Kimberly J. |last3=Amy |first3=Patrick M. |last4=Dixon |first4=Warren E. |chapter=Exponential Stability with RISE Controllers for Uncertain Nonlinear Systems with Unknown Time-Varying State Delays |date=2022 |title=2022 IEEE 61st Conference on Decision and Control (CDC) |pages=6431–6435 |doi=10.1109/CDC51059.2022.9993171|isbn=978-1-6654-6761-2 }}</ref><ref name=":2">{{Cite journal |last1=Patil |first1=Omkar Sudhir |last2=Kamalapurkar |first2=Rushikesh |last3=Dixon |first3=Warren E. |date=2025 |title=Saturated RISE Controllers With Exponential Stability Guarantees: A Projected Dynamical Systems Approach |journal=IEEE Transactions on Automatic Control |volume=70 |issue=7 |pages=4936–4942 |doi=10.1109/TAC.2025.3543246 |bibcode=2025ITAC...70.4936P |issn=1558-2523}}</ref>
== Introduction == RISE controllers are designed for nonlinear systems that can be expressed in the control‐affine form<ref name=":0" />
<math>\dot{x} = d(x,t) + u</math>
where <math>x</math> represents the system state, <math>d(x,t)</math> encapsulates modeling uncertainties and external disturbances, and <math>u</math> is the control input. The methodology employs a continuous control signal that incorporates an integral of the sign of the tracking error, thereby avoiding the chattering typically associated with conventional sliding mode controllers. The control design is underpinned by a Lyapunov stability analysis that utilizes an auxiliary function, often referred to as the P-function, to establish both asymptotic and exponential stability.
== Theoretical framework ==
=== Control law formulation === For a control‐affine nonlinear system, the RISE control law is formulated as<ref name=":0" /><math display="block">u = \dot{x}_d - \alpha e - \hat{d}</math> where <math>\dot{x}_d</math> is the time derivative of the desired trajectory, <math>e = x - x_d</math> represents the tracking error, and <math>\alpha>0</math> is a constant control gain. In order to compensate for uncertainties, an auxiliary term <math>\hat{d}</math> is dynamically updated according to <math display="block">\dot{\hat{d}} = k\,r + e + \beta\,\operatorname{sgn}(e)</math> in which <math>r=\dot{e}+\alpha e</math> is a filtered version of the tracking error, and <math>k</math> as well as <math>\beta</math> are positive control gains. The signum function, <math>\operatorname{sgn}(e)</math>, is incorporated to ensure robust compensation against disturbances, thereby driving the tracking error toward zero.
=== Lyapunov stability and the P-function === A central element of the RISE controller design is the construction of a Lyapunov function that verifies the stability of the closed-loop system. The P-function, an auxiliary construct employed in the stability analysis, is used to demonstrate that the derivative of the Lyapunov function is negative definite. Early analyses based on the P-function established asymptotic stability, while more recent studies<ref name=":0" /><ref name=":1" /><ref name=":2" /> have refined its design to show that, under suitable gain selection, the closed-loop system achieves exponential stability.
== Applications and extensions == RISE controllers have been applied across a broad spectrum of engineering domains. In robotics, for example, they have been deployed for the precise control of manipulators,<ref>{{Cite journal |last1=Fischer |first1=Nicholas |last2=Kan |first2=Zhen |last3=Kamalapurkar |first3=Rushikesh |last4=Dixon |first4=Warren E. |date=2014 |title=Saturated RISE Feedback Control for a Class of Second-Order Nonlinear Systems |journal=IEEE Transactions on Automatic Control |volume=59 |issue=4 |pages=1094–1099 |doi=10.1109/TAC.2013.2286913 |bibcode=2014ITAC...59.1094F |issn=1558-2523}}</ref> autonomous underwater vehicles,<ref>{{Cite journal |last1=Fischer |first1=Nicholas |last2=Hughes |first2=Devin |last3=Walters |first3=Patrick |last4=Schwartz |first4=Eric M. |last5=Dixon |first5=Warren E. |date=2014 |title=Nonlinear RISE-Based Control of an Autonomous Underwater Vehicle |journal=IEEE Transactions on Robotics |volume=30 |issue=4 |pages=845–852 |doi=10.1109/TRO.2014.2305791 |bibcode=2014ITRob..30..845F |issn=1941-0468}}</ref> and mobile robots,<ref>{{Cite journal |last1=Dierks |first1=Travis |last2=Jagannathan |first2=S. |date=2009 |title=Neural Network Control of Mobile Robot Formations Using RISE Feedback |journal=IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics) |volume=39 |issue=2 |pages=332–347 |doi=10.1109/TSMCB.2008.2005122 |pmid=19095558 |bibcode=2009ITSMB..39..332D |issn=1941-0492}}</ref> where the ability to handle significant uncertainties is critical. The versatility of the RISE methodology has also led to its adoption in state estimation, distributed optimization, aerospace control for unmanned aerial vehicles, and precision control in hydraulic systems. Over time, several extensions to the standard RISE framework have been developed, including adaptive strategies that incorporate classical adaptive control techniques to manage structured uncertainties, neural network-based implementations for enhanced nonlinear function approximation, and modifications designed to address issues such as input saturation<ref name=":2" /> and time delays<ref name=":1" />
== References == <references />
Category:Control theory Category:Control engineering Category:Nonlinear control