Control and stabilization of systems with homoclinic orbits

In this paper we consider the control of two physical systems, the near wall region of a turbulent boundary layer and the rigid body, using techniques from the theory of nonlinear dynamical systems. Both these systems have saddle points linked by heteroclinic orbits. In the fluid system we show how the structure of the phase space can be used to keep the system near an (unstable) saddle. For the rigid body system we discuss passage along the orbit as a possible control manouver, and show how the Energy-Casimir method can be used to analyze stabilization of the system about the saddles

Matching and stabilization by the method of controlled Lagrangians

We describe a class of mechanical systems for which the “method of controlled Lagrangians” provides a family of control laws that stabilize an unstable (relative) equilibrium. The controlled Lagrangian approach involves making modifications to the Lagrangian for the uncontrolled system such that the Euler-Lagrange equations derived from the modified or “controlled” Lagrangian describe the closed-loop system. For the closed-loop equations to be consistent with available control inputs, the modifications to the Lagrangian must satisfy “matching” conditions. Our matching and stabilizability conditions are constructive; they provide the form of the controlled Lagrangian, the control law and, in some cases, conditions on the control gain(s) to ensure stability. The method is applied to stabilization of an inverted spherical pendulum on a cart and to stabilization of steady rotation of a rigid spacecraft about its unstable intermediate axis using an internal rotor

Stabilization of the pendulum on a rotor arm by the method of controlled Lagrangians

Obtains feedback stabilization of an inverted pendulum on a rotor arm by the “method of controlled Lagrangians”. This approach involves modifying the Lagrangian for the uncontrolled system so that the Euler-Lagrange equations derived from the modified or “controlled” Lagrangian describe the closed-loop system. For the closed-loop equations to be consistent with available control inputs, the modifications to the Lagrangian must satisfy “matching” conditions. The pendulum on a rotor arm requires an interesting generalization of our earlier approach which was used for systems such as a pendulum on a cart

The energy–momentum method for the stability of non-holonomic systems

In this paper we analyze the stability of relative equilibria of nonholonomic systems (that is, mechanical systems with nonintegrable constraints such as rolling constraints). In the absence of external dissipation, such systems conserve energy, but nonetheless can exhibit both neutrally stable and asymptotically stable, as well as linearly unstable relative equilibria. To carry out the stability analysis, we use a generalization of the energy-momentum method combined with the Lyapunov-Malkin theorem and the center manifold theorem. While this approach is consistent with the energy-momentum method for holonomic systems, it extends it in substantial ways. The theory is illustrated with several examples, including the the rolling disk, the roller racer, and the rattleback top

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

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. We use kinetic shaping to preserve symmetry and only stabilize systems module the symmetry group. 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

An Optimal Control Formulation for Inviscid Incompressible Ideal Fluid Flow

In this paper we consider the Hamiltonian formulation of the equations of incompressible ideal fluid flow from the point of view of optimal control theory. The equations are compared to the finite symmetric rigid body equations analyzed earlier by the authors. We discuss various aspects of the Hamiltonian structure of the Euler equations and show in particular that the optimal control approach leads to a standard formulation of the Euler equations -- the so-called impulse equations in their Lagrangian form. We discuss various other aspects of the Euler equations from a pedagogical point of view. We show that the Hamiltonian in the maximum principle is given by the pairing of the Eulerian impulse density with the velocity. We provide a comparative discussion of the flow equations in their Eulerian and Lagrangian form and describe how these forms occur naturally in the context of optimal control. We demonstrate that the extremal equations corresponding to the optimal control problem for the flow have a natural canonical symplectic structure.Comment: 6 pages, no figures. To appear in Proceedings of the 39th IEEEE Conference on Decision and Contro

Nonholonomic Dynamics

Nonholonomic systems are, roughly speaking, mechanical systems with constraints on their velocity that are not derivable from position constraints. They arise, for instance, in mechanical systems that have rolling contact (for example, the rolling of wheels without slipping) or certain kinds of sliding contact (such as the sliding of skates). They are a remarkable generalization of classical Lagrangian and Hamiltonian systems in which one allows position constraints only. There are some fascinating differences between nonholonomic systems and classical Hamiltonian or Lagrangian systems. Among other things: nonholonomic systems are nonvariational—they arise from the Lagrange-d’Alembert principle and not from Hamilton’s principle; while energy is preserved for nonholonomic systems, momentum is not always preserved for systems with symmetry (i.e., there is nontrivial dynamics associated with the nonholonomic generalization of Noether’s theorem); nonholonomic systems are almost Poisson but not Poisson (i.e., there is a bracket that together with the energy on the phase space defines the motion, but the bracket generally does not satisfy the Jacobi identity); and finally, unlike the Hamiltonian setting, volume may not be preserved in the phase space, leading to interesting asymptotic stability in some cases, despite energy conservation. The purpose of this article is to engage the reader’s interest by highlighting some of these differences along with some current research in the area. There has been some confusion in the literature for quite some time over issues such as the variational character of nonholonomic systems, so it is appropriate that we begin with a brief review of the history of the subject

Stabilization of mechanical systems using controlled Lagrangians

We propose an algorithmic approach to stabilization of Lagrangian systems. The first step involves making admissible modifications to the Lagrangian for the uncontrolled system, thereby constructing what we call the controlled Lagrangian. The Euler-Lagrange equations derived from the controlled Lagrangian describe the closed-loop system where new terms 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. The procedure is demonstrated for the problem of stabilization of an inverted pendulum on a cart and for the problem of stabilization of rotation of a rigid spacecraft about its unstable intermediate axis using a single internal rotor. Similar results hold for the dynamics of an underwater vehicle

Flat Nonholonomic Matching

In this paper we extend the matching technique to a class of nonholonomic systems with symmetries. Assuming that the momentum equation defines an integrable distribution, we introduce a family of reduced systems. The method of controlled Lagrangians is then applied to these systems resulting in a smooth stabilizing controller

Controlled Lagrangian Methods and Tracking of Accelerated Motions

Matching techniques are applied to the problem of stabilization of uniformly accelerated motions of mechanical systems with symmetry. The theory is illustrated with a simple model-a wheel and pendulum system