Nonlinear Dynamics

, Volume 94, Issue 4, pp 2715–2725 | Cite as

An equilibrium-based learning approach with application to robotic fish

  • Xuefang Li
  • Qinyuan Ren
  • Jian-Xin Xu
Original Paper


In this work, we extend the concept of integral control to equilibrium-based learning control. As far as the plant reaches an equilibrium that deviates from the reference, a learning mechanism will update the control action. The new control action will drive the plant output to reach a new equilibrium that is closer the reference set-point. By applying fixed point theorem, we can prove the convergence of the controlled equilibrium to the reference set-point exponentially, where the plant dynamics can be generically nonlinear and non-affine. The only prior information required is a non-singular input–output gradient of the stabilized plant. As a real-time application, the proposed control method is applied to motion control of a tail-actuated robotic fish. To facilitate the controller design, the dynamical model of the robotic fish is established based on Newton’s second law and Lighthill’s small amplitude model. In the end, both simulations and experiments are conducted to illustrate the effectiveness of the proposed learning approach.


Learning control Robotic fish Motion control Turning control Self-adaption 



Open Research Project of the State Key Laboratory of Industrial Control Technology, Zhejiang University, China (Grant No. ICT170345).


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Electrical and Electronic EngineeringImperial College LondonLondonUK
  2. 2.College of Control Science and EngineeringZhejiang UniversityHangzhouChina
  3. 3.Department of Electrical and Computer EngineeringNational University of SingaporeSingaporeSingapore

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