Lecture 4. Optimal Control and Stabilization of Balance Systems
In this lecture we consider two problems from control theory. The first is an
optimal control problem, leading up to the falling cat theorem while the second is
a stabilization problem.Optimal control for the Heisenberg system. We first illustrate some basic
techniques in a simple example called the Heisenberg system. This example, due to
Brockett [1981], is a prototype for the falling cat problem and a variety of optimal
steering problems.
The Heisenberg system is the following control system in JR^3 :
X = U1
iJ =U2
Z = X U 2 - YU1,
where u 1 and u 2 are control inputs. The system may be written as
q = U1g1 + U2g2
where q = (x , y, z )T,g 1 = (1,0, y)T and g2=(0,1,-x)T. Note that g 1 and g2 are
a set of independent vector fields satisfying the constraint
i =xi; - yx.One verifies that the Jacobi-Lie bracket of the vector fields g 1 and g2 is
[g1) g2] = 2g3where g 3 = (0, 0, 1). In fact, the three vector fields g 1 , g2, g3 span all of !R^3 and,
as a Lie algebra, is just the Heisenberg algebra for the basic operators (up to scale
factors) q , p and the identity, from quantum mechanics.
By general controllability theorems (Chow's theorem) that are closely related
to the Frobenius theorem, one knows that one can, with a suitable choice of controls
u 1 and u 2 , steer trajectories between any two points in IR^3. In p articular, we are
interested in the following optimal steering problem (see Figure 4.1):
Optimal steering problem. Given numbers a> 0 and T > 0, find time depen-
dent controls u 1 , u 2 that steer the trajectory starting at (0 , 0, 0) at time t = 0 to the
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