11.6 Linear Quadratic Regulators
189
In the following section we apply these methods to stabilize the Robinson Crusoe
economy close to its equilibrium.
11.7 Controlling the Robinson-Crusoe Economy
Let us now use the optimal feedback controller from the previous section to stabilize
the profits of the Robinson-Crusoe economy under stady-state conditions, should the
system be perturbed. This is the task of a chief executive officer of a company in
order to ensure a reliable and constant dividend paid to the share holders. But here
we develop a strategy to do this automatically. Recall that the steady-state conditions
of the Robinson-Crusoe economy are given in (11.11). Moreover, Fig. 11.2 shows
their dependence on the deprecation δ and on the technological level λ. In order to
adapt the model to the continuous-time framework, used in the previous section, we
use the dynamics described by (11.17), reproduced here
˙
k = −δ
k + i
and
p
= λ
k
− i
,
where the prime denotes the per-unit-time quantities, introduced just before (11.17).
Eliminating the investment i
from the two equations leads to
˙
k = −δ
k + λ
k
− p
.
(11.59)
In this equation we recognize the capital k as the state variable. Furthermore, we can
control the capital by adjusting the profits p
suitably. Unfortunately, this system is
non-linear, but we still attempt to stabilize it, once it has reached its steady state,
which is defined by ˙ ¯
k = 0 and leads to the condition ¯
p
= λ
¯
k
− δ
¯
k. Expanding
both k = ¯
k + x and p
= ¯
p
+ u and inserting in (11.59), we obtain
˙
x =
λ
¯
k
−1
− δ
x − u .
(11.60)
With the definitions a = λ
¯
k
−1
− δ
and b = −1 this equation can be cast into
the canonical form, defined by (11.27), of a control problem ˙
x = ax + bu.
A suitable objective functional for this problem is given by the one-dimensional
version of (11.28)
J [x, u] =
1
2
t f
t 0
qx
2
+ ru
2
dt
(11.61)
with weights q and r for the state variable x and the controller u, respectively. If
we are eager to return to the steady state as quickly as possible, we assign a greater
weight to the first term q r . Conversely, if we want to keep the share holders
happy and maintain a constant level of profits, we use q r , which, however, will
lead to a longer time to reach the stady state.
189
In the following section we apply these methods to stabilize the Robinson Crusoe
economy close to its equilibrium.
11.7 Controlling the Robinson-Crusoe Economy
Let us now use the optimal feedback controller from the previous section to stabilize
the profits of the Robinson-Crusoe economy under stady-state conditions, should the
system be perturbed. This is the task of a chief executive officer of a company in
order to ensure a reliable and constant dividend paid to the share holders. But here
we develop a strategy to do this automatically. Recall that the steady-state conditions
of the Robinson-Crusoe economy are given in (11.11). Moreover, Fig. 11.2 shows
their dependence on the deprecation δ and on the technological level λ. In order to
adapt the model to the continuous-time framework, used in the previous section, we
use the dynamics described by (11.17), reproduced here
˙
k = −δ
k + i
and
p
= λ
k
− i
,
where the prime denotes the per-unit-time quantities, introduced just before (11.17).
Eliminating the investment i
from the two equations leads to
˙
k = −δ
k + λ
k
− p
.
(11.59)
In this equation we recognize the capital k as the state variable. Furthermore, we can
control the capital by adjusting the profits p
suitably. Unfortunately, this system is
non-linear, but we still attempt to stabilize it, once it has reached its steady state,
which is defined by ˙ ¯
k = 0 and leads to the condition ¯
p
= λ
¯
k
− δ
¯
k. Expanding
both k = ¯
k + x and p
= ¯
p
+ u and inserting in (11.59), we obtain
˙
x =
λ
¯
k
−1
− δ
x − u .
(11.60)
With the definitions a = λ
¯
k
−1
− δ
and b = −1 this equation can be cast into
the canonical form, defined by (11.27), of a control problem ˙
x = ax + bu.
A suitable objective functional for this problem is given by the one-dimensional
version of (11.28)
J [x, u] =
1
2
t f
t 0
qx
2
+ ru
2
dt
(11.61)
with weights q and r for the state variable x and the controller u, respectively. If
we are eager to return to the steady state as quickly as possible, we assign a greater
weight to the first term q r . Conversely, if we want to keep the share holders
happy and maintain a constant level of profits, we use q r , which, however, will
lead to a longer time to reach the stady state.