Stabilization of a class of underactuated Euler Lagrange system using an approximate model

Yildiz H. A., Goren-Sumer L.

TRANSACTIONS OF THE INSTITUTE OF MEASUREMENT AND CONTROL, vol.44, pp.1569-1578, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 44
  • Publication Date: 2022
  • Doi Number: 10.1177/01423312211058556
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Applied Science & Technology Source, Communication Abstracts, Computer & Applied Sciences, INSPEC, Metadex, DIALNET, Civil Engineering Abstracts
  • Page Numbers: pp.1569-1578
  • Keywords: Approximate model, control systems design, Euler Lagrange, Lyapunov, nonlinear control, underactuated systems, PASSIVITY-BASED CONTROL, MECHANICAL SYSTEMS, INTERCONNECTION, ENERGY
  • Istanbul Technical University Affiliated: Yes


The energy shaping method, Controlled Lagrangian, is a well-known approach to stabilize the underactuated Euler Lagrange (EL) systems. In this approach, to construct a control rule, some nonlinear and nonhomogeneous partial differential equations (PDEs), which are called matching conditions, must be solved. In this paper, a method is proposed to obtain an approximate solution of these matching conditions for a class of underactuated EL systems. To develop this method, the potential energy matching condition is transformed to a set of linear PDEs using an approximation of inertia matrices. Hence, the assignable potential energy function and the controlled inertia matrix both are constructed as a common solution of these PDEs. Subsequently, the gyroscopic and dissipative forces are determined as the solution for kinetic energy matching condition. Conclusively, the control rule is constructed by adding energy shaping rule and additional dissipation injection to provide asymptotic stability. The stability analysis of the closed-loop system which used the control rule derived with the proposed method is also provided. To demonstrate the success of the proposed method, the stability problem of the inverted pendulum on a cart is considered.