Introduction
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An LMI-based state feedback approach that ensures optimum path tracking and improved steady state performance of a quadrotor in both translational and rotational movements.
Quadcopter Dynamics
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The motion of Quad Copter in 6DOF is controlled by varying the rpm of four rotors individually, thereby changing the vertical, horizontal and rotational forces.
The structure is symmetric, thus the inertia matrices are diagonal.
The center of mass corresponds to the origin of the physical coordinate system.
A quadcopter is a rigid body.
State Space Representation
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x
˙
(
t
)
=
A
x
(
t
)
+
B
u
(
t
)
y
(
t
)
=
C
x
(
t
)
+
D
u
(
t
)
{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)+Du(t)\\\end{aligned}}}
where x(t) is state vector, y(t) is output vector and u(t) is Input or control vector.
A is the system matrix
B is the input matrix
C is the output matrix
D is the feed forward matrix
x
˙
(
t
)
=
d
d
t
x
(
t
)
{\displaystyle {\dot {x}}(t)={\frac {d}{dt}}x(t)}
Quadcopter modelling with 6 degree of freedom
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LMI for stability Analysis
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[
P
A
T
−
W
T
B
T
+
A
P
−
B
W
+
μ
2
δ
−
P
A
T
+
W
T
B
T
−
D
P
I
P
D
T
−
A
P
+
B
W
D
T
P
+
P
D
−
P
I
−
P
−
δ
I
]
<
0
{\displaystyle {\begin{bmatrix}PA^{T}-W^{T}B^{T}+AP-BW+\mu ^{2}\delta &-PA^{T}+W^{T}B^{T}-DP&I\\PD^{T}-AP+BW&D^{T}P+PD&-P\\I&-P&-\delta I\\\end{bmatrix}}<0}
K
=
W
P
−
1
{\displaystyle K=WP^{-}1}
Solving the above LMI yields the unknown coefficients of the feedback control. The system will be then asymptotically stable and path track will be achieved.
LPV Attitude State Feedback Controller Design
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To design a
H
∞
{\displaystyle H\infty }
LPV feedback control scheme for the altitude/attitude stabilization of the quadrotor aircraft,
Output
y
=
[
ϕ
θ
ψ
Z
]
T
{\displaystyle y={\begin{bmatrix}\phi &\theta &\psi &\mathbb {Z} \end{bmatrix}}^{T}}
of the system must track r =
[
ϕ
r
e
f
θ
r
e
f
ψ
r
e
f
Z
r
e
f
]
T
{\displaystyle {\begin{bmatrix}\phi _{ref}&\theta _{ref}&\psi _{ref}&\mathbb {Z} _{ref}\end{bmatrix}}^{T}}
, a reference trajectory.
To achieve the objective outputs of the integrator are considered as extra state variables
x
e
=
[
x
ϕ
x
θ
x
ψ
x
Z
]
T
{\displaystyle x_{e}={\begin{bmatrix}x_{\phi }&x_{\theta }&x_{\psi }&x_{Z}\end{bmatrix}}^{T}}
as
x
ϕ
=
∫
0
t
e
ϕ
(
δ
)
d
δ
d
x
,
e
ϕ
=
ϕ
r
e
f
−
ϕ
{\displaystyle x_{\phi }=\int _{0}^{t}e_{\phi }(\delta )d\delta \,dx,e_{\phi }=\phi _{ref}-\phi }
x
θ
=
∫
0
t
e
θ
(
δ
)
d
δ
d
x
,
e
θ
=
θ
r
e
f
−
θ
{\displaystyle x_{\theta }=\int _{0}^{t}e_{\theta }(\delta )d\delta \,dx,e_{\theta }=\theta _{ref}-\theta }
x
ψ
=
∫
0
t
e
ψ
(
δ
)
d
δ
d
x
,
e
ψ
=
ψ
r
e
f
−
ψ
{\displaystyle x_{\psi }=\int _{0}^{t}e_{\psi }(\delta )d\delta \,dx,e_{\psi }=\psi _{ref}-\psi }
x
z
=
∫
0
t
e
z
(
δ
)
d
δ
d
x
,
e
z
=
z
r
e
f
−
z
{\displaystyle x_{z}=\int _{0}^{t}e_{z}(\delta )d\delta \,dx,e_{z}=z_{ref}-z}
In this case the error signal e=y-r ,for the outputs
U
1
,
U
2
,
U
3
,
U
4
{\displaystyle U_{1},U_{2},U_{3},U_{4}}
,the weight functions
W
u
i
,
i
=
1
,
2
,
3
,
4
{\displaystyle W_{ui},i=1,2,3,4}
are added to the system. The system matrices of weight functions are
A
u
i
,
B
u
i
,
C
u
i
a
n
d
D
u
i
.
{\displaystyle A_{ui},B_{ui},C_{ui}andD_{ui}.}
The dynamic of all the weight functions
W
u
1
,
W
u
2
,
W
u
3
,
W
u
4
{\displaystyle W_{u1},W_{u2},W_{u3},W_{u4}}
can be constituted as,
x
˙
u
=
A
u
x
u
+
B
u
u
y
u
=
C
u
x
u
+
D
u
u
{\displaystyle {\begin{aligned}{\dot {x}}_{u}=A_{u}x_{u}+B_{u}u\\y_{u}=C_{u}x_{u}+D_{u}u\end{aligned}}}
where
x
u
=
[
x
u
1
x
u
2
x
u
3
x
u
4
]
T
{\displaystyle x_{u}={\begin{bmatrix}x_{u1}&x_{u2}&x_{u3}&x_{u4}\end{bmatrix}}^{T}}
is the state,
u
=
[
U
1
U
2
U
3
U
4
]
T
{\displaystyle u={\begin{bmatrix}U_{1}&U_{2}&U_{3}&U_{4}\end{bmatrix}}^{T}}
represents the input
y
u
=
[
Z
1
Z
2
Z
3
Z
4
]
T
{\displaystyle y_{u}={\begin{bmatrix}Z_{1}&Z_{2}&Z_{3}&Z_{4}\end{bmatrix}}^{T}}
The system matrices of the weight function can be deducted as,
Δ
u
=
[
Δ
u
1
0
0
0
0
Δ
u
2
0
0
0
0
Δ
u
3
0
0
0
Δ
u
4
]
{\displaystyle \Delta _{u}={\begin{bmatrix}\Delta _{u1}&0&0&0\\0&\Delta _{u2}&0&0\\0&0&\Delta _{u3}\\0&0&0&\Delta _{u4}\end{bmatrix}}}
,
Δ
∈
A
,
B
,
C
,
D
{\displaystyle \Delta \in {A,B,C,D}}
w
=
[
r
d
]
T
{\displaystyle w={\begin{bmatrix}r&d\end{bmatrix}}^{T}}
,
z
=
[
y
u
e
]
T
{\displaystyle z={\begin{bmatrix}y_{u}&e\end{bmatrix}}^{T}}
,and
x
¯
=
[
x
x
e
x
u
]
T
{\displaystyle {\bar {x}}={\begin{bmatrix}x&x_{e}&x_{u}\end{bmatrix}}^{T}}
are the exogenous input and exogenous output respectively.
The system differential equations with augumented ststes and weight functions are,
x
¯
˙
=
∑
i
=
1
16
μ
i
(
A
¯
i
x
¯
+
B
¯
1
i
w
+
B
¯
2
u
)
z
=
C
x
¯
+
D
11
w
+
D
12
u
{\displaystyle {\begin{aligned}{\dot {\bar {x}}}=\sum _{i=1}^{16}\mu _{i}({\bar {A}}_{i}{\bar {x}}+{\bar {B}}1_{i}w+{\bar {B}}_{2}u)\\z=C_{\bar {x}}+D11w+D12u\end{aligned}}}
where,
A
¯
i
=
[
A
¯
i
0
0
0
−
C
0
0
0
0
A
u
]
{\displaystyle {\bar {A}}_{i}={\begin{bmatrix}{\bar {A}}_{i}&0&0&0\\-C&0&0\\0&0&A_{u}\end{bmatrix}}}
;
B
¯
1
=
[
0
E
¯
i
−
I
4
0
0
0
]
{\displaystyle {\bar {B}}_{1}={\begin{bmatrix}0&{\bar {E}}_{i}\\-I_{4}&0\\0&0\end{bmatrix}}}
B
¯
2
=
[
B
¯
i
0
B
u
]
{\displaystyle {\bar {B}}_{2}={\begin{bmatrix}{\bar {B}}_{i}\\0\\B_{u}\end{bmatrix}}}
C
1
=
[
0
0
C
u
C
0
0
]
{\displaystyle C_{1}={\begin{bmatrix}0&0&C_{u}\\C&0&0\end{bmatrix}}}
D
11
=
[
0
0
−
I
4
0
]
{\displaystyle D_{11}={\begin{bmatrix}0&0\\-I_{4}&0\end{bmatrix}}}
D
12
=
[
D
u
0
]
{\displaystyle D_{12}={\begin{bmatrix}D_{u}\\0\end{bmatrix}}}
Making the closed loop system,
d
o
t
x
¯
=
∑
i
=
1
16
μ
i
(
(
A
¯
i
+
B
¯
2
k
i
)
x
¯
(
t
)
+
B
¯
1
w
)
{\displaystyle dot{\bar {x}}=\sum _{i=1}^{16}\mu _{i}(({\bar {A}}_{i}+{\bar {B}}_{2}ki){\bar {x}}(t)+{\bar {B}}_{1}w)}
LMI for H∞ optimal state-feedback
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X
=
X
T
>
0
{\displaystyle X=X^{T}>0}
[
H
e
(
A
¯
i
X
+
B
¯
2
Y
i
(
∗
)
T
(
∗
)
T
B
¯
1
i
T
−
γ
I
(
∗
)
T
C
¯
1
i
X
+
D
12
Y
i
D
¯
11
−
γ
I
]
<
0
{\displaystyle {\begin{bmatrix}He({\bar {A}}_{i}X+{\bar {B}}_{2}Y_{i}&(*)^{T}&(*)^{T}\\{\bar {B}}_{1i}^{T}&-\gamma I&(*)^{T}\\{\bar {C}}_{1i}X+D_{12}Y_{i}&{\bar {D}}_{11}&-\gamma I\end{bmatrix}}<0}
H
e
(
A
¯
i
X
+
B
¯
2
Y
i
+
2
α
X
<
0
{\displaystyle He({\bar {A}}_{i}X+{\bar {B}}_{2}Y_{i}+2\alpha X<0}
; i=1 to 16
By solving the LMI,the optimal H\inf state feed-back controller with the smallest attenuation level \gamma >0 for the attitude/altitude subsystem of the mass-varying quadcopter is
K
(
ρ
)
=
∑
i
=
1
16
μ
i
Y
i
X
−
1
{\displaystyle K(\rho )=\sum _{i=1}^{16}\mu _{i}Y_{i}X^{-}1}
Conclusion
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This LMI can be used to analyze the state feedback control and path tracking of a quadcopter.
Implementation
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This LMI can be used in a problem and can be solved using the solvers like Yalmip,sedumi,gurobi etc,.
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