1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 4. CONTROL AND STABILIZATION OF BALANCE SYSTEMS 391

Note that the variable s is "shifted" by the one form T and a term quadratic in e

is added. There is a general construction of new Lagrangians of this type based on

changes of connections (velocity shifts) and using Kaluza-Klein ideas.

The variable s is still cyclic. The associated conservation law for the confrolled

Lagrangian is

d d.

dtPs = u := -'Y dt [k(B)B].

We strive to identify the term on the right hand side with the control force (the

force exerted on the cart). A (relatively miraculous) computation shows that thee

equation for the controlled Lagrangian matches with the e equation for the controlled

cart if we choose k(B) = r;; cos e for a real number r;; and if we choose <J = -(3/('Yr;;).

The resulting control law is

u = "fK: (sine e^2 +cos e f ( e, e)) ,

where

. DsinB + B^2 ('

62

+(Jr;;) cosBsinB

f(B, B) = 'Y

a - (~

2

• f3r;;) cos^2 e
  In linear approximation it is a proportional controller, u =constant x e.
  Stabilization. The B dynamics is stabilized if the energy has an extremum at the
  equilibrium (in this case a maximum); this leads to a condition on the "nonlinear
  gain" r;;:
  Cl!"( - (32

K, > f3'Y > o.

Having a maximum and not a minimum is not a problem even if dissipation is
present-simulate negative dissipation by the controller; one then gets asymptotic
stability instead of Liapunov stability.
Summary for the inverted pendulum. We get a stabilizing feedback control
law (an expression for the control force needed as a function of the state of the
pendulum) provided r;; is chosen to satisfy the preceding inequality. Stability is
determined by energy considerations. This procedure allows one to discover the
stabilizing control law as long as one has a rich class of controlled Lagrangians to
work with. Our theory provides such a class.
This approach is attractive because it is done within the context of mechanics;
one can understand the stabilization in terms of the effective creation of an energy
extremum.
One can still ask many questions about this construction, such as: the role
of damping, the swing-up problem, the efficiency and energy consumption of the
method, etc. These issues are not all settled.
For problems with nonholonomic constraints (like a bicycle) there is reason to
believe that a similar construction will work. This is based on recent advances in
the geometry of nonholonomic systems and the associated stability theory.

A rigid body with a symmetric rotor. Now we give another stabilization prob-
lem that is solved by the same technique as the inverted pendulum. This example,
investigated by Bloch, Krishnaprasad, Marsden and Sanchez [1992], provided a
main motivation for the general approach of controlled Lagrangians.