For this system, the state and output are defined
x
⊤
=
[
˙
θ
ψ
˙
ψ
∫
ψ
]
(2.4)
and
y
=
ψ
Solving for the acceleration terms in the linear equations of motion given in Equation 2.1 we get
¨
θ
=
−
h
J
y
˙
ψ
+
1
J
y
τ
y
¨
ψ
=
h
J
z
˙
θ
Substituting the state in , we obtain the following state-space matrices:
A
=
0
0
−
h
J
y
0
0
0
1
0
h
J
z
0
0
0
0
1
0
0
and
B
=
1
J
y
0
0
0
.
The system output (control variable) is the red gimbal angle which is the second entry in the system state vector i.e.,
ψ
. Based on this, the
C
and
D
matrices in the output equation are
C
=
[
0
1
0
0
]
and
D
= [0]
The velocities of the red and blue gimbal angles can be computed in the digital controller, e.g., by taking the derivative
and filtering the result though a high-pass filter. The integral state can be computed by integrating the red gimbal
angle measurement in the digital controller.
2.2 Control
In Section 2.1, we found a linear state-state space model that represents the 3 DOF Gyroscope system. This
model is used to investigate the stability properties of the system in Section 2.2.1. In Section 2.2.2, the notion of
controllability is introduced. The Linear Quadratic Regulator (LQR) algorithm is a common way to find the control
gain and is discussed in Section 2.2.3. Lastly, Section 2.2.4 describes the state-feedback control used to control the
red gimbal position.
2.2.1 Stability
The stability of a system can be determined from its poles ([3]):
• Stable systems have poles only in the left-hand plane.
• Unstable systems have at least one pole in the right-hand plane and/or poles of multiplicity greater than 1 on
the imaginary axis.
3D GYRO Laboratory Guide
6
