Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem

Anthony M. Bloch, Naomi Ehrich Leonard, Jerrold E. Marsden

Research output: Contribution to journalArticlepeer-review

416 Scopus citations


We develop a method for the stabilization of mechanical systems with symmetry based on the technique of controlled Lagrangians. The procedure involves making structured modifications to the Lagrangian for the uncontrolled system, thereby constructing the controlled Lagrangian. The Euler-Lagrange equations derived from the controlled Lagrangian describe the closed-loop system, where new terms in these equations are identified with control forces. Since the controlled system is Lagrangian by construction, energy methods can be used to find control gains that yield closed-loop stability. In this paper we use kinetic shaping to preserve symmetry and only stabilize systems modulo the symmetry group. In the sequel to this paper (Part II), we extend the technique to include potential shaping and we achieve stabilization in the full phase space. The procedure is demonstrated for several underactuated balance problems, including the stabilization of an inverted planar pendulum on a cart moving on a line and an inverted spherical pendulum on a cart moving in the plane.

Original languageEnglish (US)
Pages (from-to)2253-2270
Number of pages18
JournalIEEE Transactions on Automatic Control
Issue number12
StatePublished - 2000
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering


  • Lagrangian mechanics
  • Nonlinear systems
  • Stabilization


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