• Marginally stable systems have one pole on the imaginary axis and the other poles in the left-hand plane.
The poles are the roots of the system's characteristic equation. From the state-space, the characteristic equation of
the system can be found using
det
(
sI
−
A
) = 0
(2.5)
where
det()
is the determinant function,
s
is the Laplace operator, and
I
the identity matrix. These are the
eigenvalues
of the state-space matrix
A
.
2.2.2 Controllability
If the control input,
u
, of a system can take each state variable,
x
i
where
i
= 1
. . . n
, from an initial state to a final
state then the system is controllable, otherwise it is uncontrollable ([3]).
Rank Test
The system is controllable if the rank of its controllability matrix
T
=
[
B AB A
2
B . . . A
n
B
]
(2.6)
equals the number of states in the system,
rank
(
T
) =
n.
(2.7)
2.2.3 Linear Quadratic Regular (LQR)
If (A,B) are controllable, then the Linear Quadratic Regulator optimization method can be used to find a feedback
control gain. Given the plant model in Equation 2.2, find a control input
u
that minimizes the cost function
J
=
∫
∞
0
x
(
t
)
′
Qx
(
t
) +
u
(
t
)
′
Ru
(
t
)
dt,
(2.8)
where
Q
and
R
are the weighting matrices. The weighting matrices affect how LQR minimizes the function and are,
essentially, tuning variables.
Given the control law
u
=
−
Kx
, the state-space in Equation 2.2 becomes
˙
x
=
Ax
+
B
(
−
Kx
)
=
(
A
−
BK
)
x
2.2.4 Feedback Control
The feedback control loop in Figure 2.2 is designed to stabilize the red gimbal to a desired position,
ψ
d
.
Figure 2.2: State-feedback control loop
The reference state is defined
x
d
=
[
0
ψ
d
0
0
]
3D GYRO Laboratory Guide
v 1.1