11.5 Donkey Revisited
187
In the simulation, we require the donkey to travel to L = 100 m in T = 20 s and
plot the solutions for weak friction α = 0.1 as a solid line and for ten times stronger
friction as a dashed line. We see that for strong friction it is advantageous to quickly
reach a lower and more constant speed, compared to the situation with weak friction,
which is characterized by an almost constant acceleration until the mid-point and an
equally constant deceleration towards the destination.
With the donkey safely back home in the stable, we now turn to general linear
systems, mentioned at the end of Sect. 11.1. Objective functionals that are based on
integrals over quadratic forms are rather common and we therefore devote the next
section to linear quadratic regulators.
11.6 Linear Quadratic Regulators
The dynamics of state vectors x for linear systems is determined by (11.27)
and (11.28)
˙
x = A(t)x + B(t)u and J [x, u] =
1
2
t f
t 0
x
t Q(t)x + u
t R(t)u
dt
with matrices A(t) and B(t). The matrices Q(t) and R(t) that appear in the objective
functional J [x, u] are positive definite. Based on these definitions, we use (11.40) to
construct the Hamiltonian
H(x, u, p) =
1
2
x
t Qx +
1
2
u
t Ru + p
t
(Ax + Bu)
=
1
2
x
t Qx +
1
2
u
t Ru + x
t A
t p + u
t B
t p .
(11.50)
Applying (11.41) gives us the equations of motion for the state and costate variables,
as well as the constraint condition.
˙
p = −
∂H
∂x
= −Qx − A
t p
˙
x =
∂H
∂p
= Ax + Bu
(11.51)
0 =
∂H
∂u
= Ru + B
t p .
Solving the last equation for the controller u allows us to express it through the
costates p via
u = −R
−1 B
t p ,
(11.52)
187
In the simulation, we require the donkey to travel to L = 100 m in T = 20 s and
plot the solutions for weak friction α = 0.1 as a solid line and for ten times stronger
friction as a dashed line. We see that for strong friction it is advantageous to quickly
reach a lower and more constant speed, compared to the situation with weak friction,
which is characterized by an almost constant acceleration until the mid-point and an
equally constant deceleration towards the destination.
With the donkey safely back home in the stable, we now turn to general linear
systems, mentioned at the end of Sect. 11.1. Objective functionals that are based on
integrals over quadratic forms are rather common and we therefore devote the next
section to linear quadratic regulators.
11.6 Linear Quadratic Regulators
The dynamics of state vectors x for linear systems is determined by (11.27)
and (11.28)
˙
x = A(t)x + B(t)u and J [x, u] =
1
2
t f
t 0
x
t Q(t)x + u
t R(t)u
dt
with matrices A(t) and B(t). The matrices Q(t) and R(t) that appear in the objective
functional J [x, u] are positive definite. Based on these definitions, we use (11.40) to
construct the Hamiltonian
H(x, u, p) =
1
2
x
t Qx +
1
2
u
t Ru + p
t
(Ax + Bu)
=
1
2
x
t Qx +
1
2
u
t Ru + x
t A
t p + u
t B
t p .
(11.50)
Applying (11.41) gives us the equations of motion for the state and costate variables,
as well as the constraint condition.
˙
p = −
∂H
∂x
= −Qx − A
t p
˙
x =
∂H
∂p
= Ax + Bu
(11.51)
0 =
∂H
∂u
= Ru + B
t p .
Solving the last equation for the controller u allows us to express it through the
costates p via
u = −R
−1 B
t p ,
(11.52)