1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 4. CONTROL AND STABILIZATION OF BALANCE SYSTEMS 393

where k is a gain parameter. The system retains the 51 symmetry and Pk =

[3 - kl1 3 is a new conserved quantity. The closed loop equations (eliminating the

rotor variable) are

These equations are still Hamiltonian with

H=~(Ili+m+((l-k)Il3-Pk)

2

)+~ Pf

2 >-1 >-2 (1-k)J 3 2J 3 (l-k)'

using the Lie-Poisson (rigid body) Poisson structure on so(3)*.

Noteworthy special cases are

l. k = 0, the uncontrolled case,

2. k = J3f >.3, the driven case, where a= constant.

How this fits into the general scheme of controlled Lagrangians. Start

with the free Lagrangian:

1 2 2 1 2 1 ·2

Lo= 2(>- 1 f21 + >-202) + 2hr23 + 2h(0 3 +a).

Consider the conserved quantity

Po = h ( 03 + a) = l3

associated with the 51 action (rotor symmetry). Choose a horizontal one form to

change the connection:

T = rfl3

for a suitable constant r. Now construct a new Lagrangian obtained by replacing

a by a + TQ and modifying the metrics on the horizontal and vertical spaces using

two scalars u and p to produce a controlled Lagrangian Lr,<7,p·

With suitable r, u, p , the momentum conjugate to a for this Lagrangian is Pk

(up to a factor) and the res ulting Euler-Poincare equations give the feedback con-

trolled system! Thus, our construction explains the otherwise "strange" Lagrangian

and Hamiltonian structures found earlier by hand. This construction also works for

the stabilization problem for underwater vehicle dynamics, discussed in Lecture 2.

For details, see Bloch, Leonard and Marsden [1998].

Stabilization. With the control problem in Hamiltonian form, we can use the

energy-Casimir method for stability. Let P = 0 and consider the equilibrium

(0, M, 0). For k > 1 - h/ >-2, this equilibrium is stable. This is proved by con-

sidering H + C where C = cp(llITll^2 ). Pick cp so that

8(H + C)l(o,M,o) = 0.

One computes that 82 ( H + C) is negative definite if k > 1-h/ >. 2 and cp" ( M^2 ) < 0.

The phase portrait is that of the standard rigid body for the uncontrolled case

where k = 0 (see Lecture 1). The feedback control in effect modifies the Lagrangian

to interchange the moments of inertia of the system. The stabilization that takes