System Dynamics
edit
Consider the motion of a quadrotor along a horizontal plane alone. The control law is synthesised for roll motion(rotation wrt. X-axis) coupled with linear motion along Y-axis. The equations of motion governing the 2-D motion are given by(taken from the paper provided below):
{
d
2
x
d
t
2
=
−
μ
m
Q
d
x
d
t
−
g
t
a
n
θ
c
o
s
ϕ
;
d
2
y
d
t
2
=
−
μ
m
Q
d
y
d
t
−
g
t
a
n
ϕ
,
{\displaystyle {\begin{aligned}&\quad \quad {\begin{cases}{\frac {d^{2}x}{dt^{2}}}=-{\frac {\mu }{m_{Q}}}{\frac {dx}{dt}}-g{\frac {tan\theta }{cos\phi }};\\{\frac {d^{2}y}{dt^{2}}}=-{\frac {\mu }{m_{Q}}}{\frac {dy}{dt}}-gtan\phi ,\\\end{cases}}\end{aligned}}}
w
h
e
r
e
{\displaystyle where}
x
{\displaystyle x}
and
y
{\displaystyle y}
are the displacements along X & Y axis, respectively.
ϕ
{\displaystyle \phi }
and
θ
{\displaystyle \theta }
are the angular displacement about X & Y axis, respectively.
m
Q
{\displaystyle m_{Q}}
and
μ
{\displaystyle \mu }
are constants that depend on the intrinsic properties of the quadrotor.
The state space representation of The
H
∞
{\displaystyle H_{\infty }}
Quadrotor Guidance is given below,
{
d
X
(
t
)
d
t
=
A
X
(
t
)
+
B
u
u
(
t
)
+
B
d
(
t
)
d
(
t
)
;
Y
=
C
X
(
t
)
,
{\displaystyle {\begin{aligned}&\quad \quad {\begin{cases}{\frac {dX(t)}{dt}}=AX(t)+B_{u}u(t)+B_{d}(t)d(t);\\Y=CX(t),\\\end{cases}}\end{aligned}}}
w
h
e
r
e
:
{\displaystyle where:}
X
=
[
X
1
,
X
2
,
X
3
,
X
4
,
X
5
]
T
=
[
x
,
d
x
d
t
,
y
,
d
y
d
t
,
Δ
Ω
]
T
,
{\displaystyle X=[X_{1},X_{2},X_{3},X_{4},X_{5}]^{T}=[x,{\frac {dx}{dt}},y,{\frac {dy}{dt}},\Delta \Omega ]^{T},}
(where
Δ
Ω
{\displaystyle \Delta \Omega }
is the increment in motor's rotation rate)
A
=
[
0
1
0
0
0
0
−
μ
m
Q
g
0
0
0
0
0
1
0
0
0
0
0
K
1
0
0
0
0
K
2
]
{\displaystyle A={\begin{bmatrix}0&1&0&0&0\\0&-{\frac {\mu }{m_{Q}}}&g&0&0\\0&0&0&1&0\\0&0&0&0&K_{1}\\0&0&0&0&K_{2}\\\end{bmatrix}}}
B
u
=
[
0
1
K
3
0
K
4
]
{\displaystyle B_{u}={\begin{bmatrix}0\\1\\K_{3}\\0\\K_{4}\\\end{bmatrix}}}
B
d
=
[
0
0
0
0
K
3
]
{\displaystyle B_{d}={\begin{bmatrix}0\\0\\0\\0\\K_{3}\\\end{bmatrix}}}
C
=
[
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
]
{\displaystyle C={\begin{bmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\\end{bmatrix}}}
C
2
=
[
I
3
x
3
0
3
x
3
]
{\displaystyle C_{2}={\begin{bmatrix}I_{3x3}&0_{3x3}\end{bmatrix}}}
D
1
=
10
−
3
×
L
1
,
D
2
=
10
−
3
×
L
2
{\displaystyle D_{1}=10^{-3}\times L_{1},D_{2}=10^{-3}\times L_{2}}
The Optimization Problem
edit
For the above-linearized model of the quadrotor, the linear control would be as below:
u
=
−
K
∗
V
(
t
)
=
−
K
∗
C
∗
X
(
t
)
,
{\displaystyle u=-K*V(t)=-K*C*X(t),}
where V is the voltage input,
K
∈
R
1
x
4
{\displaystyle K\in R^{1x4}}
and
C
∈
R
4
x
5
.
{\displaystyle C\in R^{4x5}.}
The cost function of this controller could be defined as:
j
=
∫
0
∞
|
|
z
(
t
)
|
|
2
d
t
=
∫
0
∞
|
|
x
T
Q
x
+
u
T
R
u
|
|
2
d
t
{\displaystyle j=\int _{0}^{\infty }\!||z(t)||^{2}\,\mathrm {d} t=\int _{0}^{\infty }\!||x^{T}Qx+u^{T}Ru||^{2}\,\mathrm {d} t}
,
where,
z
=
[
Q
0
0
R
]
[
x
u
]
.
{\displaystyle z={\begin{bmatrix}{\sqrt {Q}}&0\\0&{\sqrt {R}}\end{bmatrix}}{\begin{bmatrix}x\\u\end{bmatrix}}.}
[
P
A
+
A
T
P
+
Q
P
B
u
P
B
d
L
T
B
u
T
P
−
R
0
0
B
d
T
P
0
−
γ
2
I
0
L
0
0
−
R
]
{\displaystyle {\begin{bmatrix}PA+A^{T}P+Q&PB_{u}&PB_{d}&L^{T}\\B_{u}^{T}P&-R&0&0\\B_{d}^{T}P&0&-\gamma ^{2}I&0\\L&0&0&-R\end{bmatrix}}}
< 0 .
Solving the above LMI gives the value of
γ
{\displaystyle \gamma }
for X>0,
γ
{\displaystyle \gamma }
>0 and Z.
Conclusion
edit
This LMI can be used in a problem and can be solved using the solvers like Yalmip,sedumi,gurobi etc,. This can be used to analyze the state feedback control and path tracking of a quadcopter.