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11 Optimal Control Theory
Using the formalism from the previous section we solve the steady-state Riccati
equation, given by (11.58), and find the factor K , which is done in Exercise 2.
Subsequently inserting it in (11.52) leads to the control law
u = −
a
b
1 ±
1 +
qb 2
ra 2
x .
(11.62)
Choosing the positive sign for the root and inserting into ˙
x = ax + bu leads to
˙
x = −a
1 +
qb 2
ra 2 x = −
1
τ
x
(11.63)
which shows that the feedback causes perturbations to damp with an exponential
time scale τ given by 1/τ = a
1 + qb 2 /ra 2
Figure 11.6 shows two simulations of the perturbed steady-state of a Robinson
Crusoe economy with parameters λ = 1, = 0.7, and δ = 0.2. The system starts
from a perturbed state with x = k − ¯
k = −0.1 and then evolves while applying the
controller u = p
− ¯
p
, calculated by the control law specified in (11.62). The upper
plot shows the x and u as a function of time t for q = 5 and r = 1 chosen as weights in
the objective functional from (11.61). Here the controller tries to return to the steady
state x = 0 and u = 0 very quickly, but penalizes large values of the controller u five
times less. Indeed, we see that deviation from the stady-state profits u, shown as the
Fig. 11.6 Using a linear quadratic regulator on the Robinson Crusoe economy with λ = 1, = 0.7,
and δ = 0.2. In the upper plot a quick return to the stady state is favored by choosing q = 5 and
r = 1 in (11.61). In the lower plot a small deviation of the profits is favored by choosing q = 1 and
r = 5, which causes the recovery time τ to be much longer
11 Optimal Control Theory
Using the formalism from the previous section we solve the steady-state Riccati
equation, given by (11.58), and find the factor K , which is done in Exercise 2.
Subsequently inserting it in (11.52) leads to the control law
u = −
a
b
1 ±
1 +
qb 2
ra 2
x .
(11.62)
Choosing the positive sign for the root and inserting into ˙
x = ax + bu leads to
˙
x = −a
1 +
qb 2
ra 2 x = −
1
τ
x
(11.63)
which shows that the feedback causes perturbations to damp with an exponential
time scale τ given by 1/τ = a
1 + qb 2 /ra 2
Figure 11.6 shows two simulations of the perturbed steady-state of a Robinson
Crusoe economy with parameters λ = 1, = 0.7, and δ = 0.2. The system starts
from a perturbed state with x = k − ¯
k = −0.1 and then evolves while applying the
controller u = p
− ¯
p
, calculated by the control law specified in (11.62). The upper
plot shows the x and u as a function of time t for q = 5 and r = 1 chosen as weights in
the objective functional from (11.61). Here the controller tries to return to the steady
state x = 0 and u = 0 very quickly, but penalizes large values of the controller u five
times less. Indeed, we see that deviation from the stady-state profits u, shown as the
Fig. 11.6 Using a linear quadratic regulator on the Robinson Crusoe economy with λ = 1, = 0.7,
and δ = 0.2. In the upper plot a quick return to the stady state is favored by choosing q = 5 and
r = 1 in (11.61). In the lower plot a small deviation of the profits is favored by choosing q = 1 and
r = 5, which causes the recovery time τ to be much longer