11.2 Control and Feedback
181
contribution is characterized by the function h that depends only on the state variables
x f at the final time t f and is often called end-point constraint.
In many applications the equations of motion and their dependence on the controllers is given by a linear set of equations
˙
x = Ax + Bu
(11.27)
where the matrices A and B can depend on time, but often are constant. Moreover,
in many applications, the objective functional can be phrased as an integral over a
quadratic form, given by
J [x, u] =
t f
t 0
x
t Qx + u
t Ru
dt ,
(11.28)
where Q and R are positive semi-definite, possibly time-dependent, matrices.
Quadratic forms are computationally attractive, because they lead to solutions that
can be found by analytic means, as we shall see in a little while.
Solving an optimal control problem involves determining the controller u(t) given
as a function of time. We thus seek to minimize a functional J [x, u] in order to determine a function u(t), which is vaguely similar to finding the equations of motion from
the action integral S[q] =
t f
t 0
L(q, ˙
q)dt that depends on a Lagrangian L(q, ˙
q). In
Chap. 3 we used methods from variational calculus to determine the Euler-Lagrange
equations. They are the equations of motion, whose solution gives the trajectory
q(t). Here we have a similar problem. The sought controller u takes the role of
the trajectory, the objective J [x, u] takes the role of the action, and g(x, u, t) that
of the Lagrangian. The only difficulty is that here we have (11.25) changing the
state variables simultaneously. It turns out that this difficulty can be overcome by
using a Hamiltonian—instead of a Lagrangian—framework. We therefore briefly
recapitulate the basics of Hamiltonian mechanics [4, 5].
11.3 Hamiltonian Mechanics
In Sect. 3.1 we found that minimizing the action integral S[q] that is defined in terms
of the Lagrangian L(q, ˙
q)
δS[q] = δ
⎡
⎣
t f
t 0
L(q, ˙
q)dt
⎤
⎦ = 0
leads to
0 =
d
dt
∂ L
∂ ˙
q
−
∂ L
∂q
,
(11.29)
which are the Euler-Lagrange equations to describe the equations of motion for the
trajectory q(t). Whereas the Lagrangian L(q, ˙
q), here assumed to be time invariant,
depends on the state variable q and its derivative ˙
q, we want the Hamiltonian to
181
contribution is characterized by the function h that depends only on the state variables
x f at the final time t f and is often called end-point constraint.
In many applications the equations of motion and their dependence on the controllers is given by a linear set of equations
˙
x = Ax + Bu
(11.27)
where the matrices A and B can depend on time, but often are constant. Moreover,
in many applications, the objective functional can be phrased as an integral over a
quadratic form, given by
J [x, u] =
t f
t 0
x
t Qx + u
t Ru
dt ,
(11.28)
where Q and R are positive semi-definite, possibly time-dependent, matrices.
Quadratic forms are computationally attractive, because they lead to solutions that
can be found by analytic means, as we shall see in a little while.
Solving an optimal control problem involves determining the controller u(t) given
as a function of time. We thus seek to minimize a functional J [x, u] in order to determine a function u(t), which is vaguely similar to finding the equations of motion from
the action integral S[q] =
t f
t 0
L(q, ˙
q)dt that depends on a Lagrangian L(q, ˙
q). In
Chap. 3 we used methods from variational calculus to determine the Euler-Lagrange
equations. They are the equations of motion, whose solution gives the trajectory
q(t). Here we have a similar problem. The sought controller u takes the role of
the trajectory, the objective J [x, u] takes the role of the action, and g(x, u, t) that
of the Lagrangian. The only difficulty is that here we have (11.25) changing the
state variables simultaneously. It turns out that this difficulty can be overcome by
using a Hamiltonian—instead of a Lagrangian—framework. We therefore briefly
recapitulate the basics of Hamiltonian mechanics [4, 5].
11.3 Hamiltonian Mechanics
In Sect. 3.1 we found that minimizing the action integral S[q] that is defined in terms
of the Lagrangian L(q, ˙
q)
δS[q] = δ
⎡
⎣
t f
t 0
L(q, ˙
q)dt
⎤
⎦ = 0
leads to
0 =
d
dt
∂ L
∂ ˙
q
−
∂ L
∂q
,
(11.29)
which are the Euler-Lagrange equations to describe the equations of motion for the
trajectory q(t). Whereas the Lagrangian L(q, ˙
q), here assumed to be time invariant,
depends on the state variable q and its derivative ˙
q, we want the Hamiltonian to