LMI for Attitude Control of BTT Missles, Pitch/Yaw Channel
The dynamic model of a bank-to-burn (BTT) missile can be simplified for practical application. The dynamic model for a BTT missile is given by the same model used for nonrotating missiles. However, in this case we can assume that the missile is axis-symmetrically designed, and thus Jx = Jy. We assume that the roll channel is independent of the pitch and yaw channels.
The state-space representation for the pitch/yaw channel can be written as follows:
x
˙
(
t
)
=
A
(
t
)
x
(
t
)
+
B
(
t
)
u
(
t
)
y
(
t
)
=
C
(
t
)
x
(
t
)
+
D
(
t
)
u
(
t
)
{\displaystyle {\begin{aligned}{\dot {x}}(t)&=A(t)x(t)+B(t)u(t)\\y(t)&=C(t)x(t)+D(t)u(t)\end{aligned}}}
where
x
(
t
)
=
[
ω
z
α
ω
y
β
]
T
{\displaystyle x(t)=[\omega _{z}\quad \alpha \quad \omega _{y}\quad \beta ]^{\text{T}}}
is the state vector,
u
(
t
)
=
[
δ
z
δ
y
]
T
{\displaystyle u(t)=[\delta _{z}\quad \delta _{y}]^{\text{T}}}
is the control input vector, and
y
=
[
n
z
n
y
]
T
{\displaystyle y=[n_{z}\quad n_{y}]^{\text{T}}}
is the output vector. The parameters
ω
z
{\displaystyle \omega _{z}}
,
α
{\displaystyle \alpha }
,
ω
y
{\displaystyle \omega _{y}}
, and
β
{\displaystyle \beta }
refer to the pitch angular velocity, the pitch angle (angle of attack), yaw angular velocity, and yaw angle, respectively. The parameters
δ
z
{\displaystyle \delta _{z}}
and
δ
y
{\displaystyle \delta _{y}}
refer to the elevator and rudder deflections, respectively. Finally, the parameters
n
z
{\displaystyle n_{z}}
and
n
y
{\displaystyle n_{y}}
refer to the overloads on the normal and side directions, respectively.
The model for the pitch/yaw channel is as follows:
[
α
˙
(
t
)
β
˙
(
t
)
ω
˙
z
(
t
)
ω
˙
y
(
t
)
n
y
(
t
)
n
z
(
t
)
]
=
[
ω
z
(
t
)
−
ω
x
(
t
)
β
(
t
)
/
57.3
−
a
4
(
t
)
α
(
t
)
−
a
5
(
t
)
δ
z
(
t
)
ω
y
(
t
)
−
ω
x
(
t
)
α
(
t
)
/
57.3
−
b
4
(
t
)
β
(
t
)
−
b
5
(
t
)
δ
y
(
t
)
−
a
1
(
t
)
ω
z
(
t
)
+
a
1
′
(
t
)
α
˙
(
t
)
−
a
2
(
t
)
α
(
t
)
−
a
3
(
t
)
δ
z
(
t
)
+
(
J
x
−
J
y
)
/
(
57.3
J
z
)
ω
x
(
t
)
ω
y
(
t
)
−
b
1
(
t
)
ω
y
(
t
)
+
b
1
′
(
t
)
β
˙
(
t
)
−
b
2
(
t
)
β
(
t
)
−
b
3
(
t
)
δ
y
(
t
)
+
(
J
z
−
J
x
)
/
(
57.3
J
y
)
ω
x
(
t
)
ω
z
(
t
)
V
(
t
)
/
(
57.3
g
)
(
a
4
(
t
)
α
(
t
)
+
a
5
(
t
)
δ
z
(
t
)
)
−
V
(
t
)
/
(
57.3
g
)
(
b
4
(
t
)
β
(
t
)
+
b
5
(
t
)
δ
y
(
t
)
)
]
{\displaystyle {\begin{aligned}{\begin{bmatrix}{\dot {\alpha }}(t)\\{\dot {\beta }}(t)\\{\dot {\omega }}_{z}(t)\\{\dot {\omega }}_{y}(t)\\n_{y}(t)\\n_{z}(t)\end{bmatrix}}={\begin{bmatrix}\omega _{z}(t)-\omega _{x}(t)\beta (t)/57.3-a_{4}(t)\alpha (t)-a_{5}(t)\delta _{z}(t)\\\omega _{y}(t)-\omega _{x}(t)\alpha (t)/57.3-b_{4}(t)\beta (t)-b_{5}(t)\delta _{y}(t)\\-a_{1}(t)\omega _{z}(t)+a'_{1}(t){\dot {\alpha }}(t)-a_{2}(t)\alpha (t)-a_{3}(t)\delta _{z}(t)+(J_{x}-J_{y})/(57.3J_{z})\omega _{x}(t)\omega _{y}(t)\\-b_{1}(t)\omega _{y}(t)+b'_{1}(t){\dot {\beta }}(t)-b_{2}(t)\beta (t)-b_{3}(t)\delta _{y}(t)+(J_{z}-J_{x})/(57.3J_{y})\omega _{x}(t)\omega _{z}(t)\\V(t)/(57.3g)(a_{4}(t)\alpha (t)+a_{5}(t)\delta _{z}(t))\\-V(t)/(57.3g)(b_{4}(t)\beta (t)+b_{5}(t)\delta _{y}(t))\end{bmatrix}}\end{aligned}}}
which can be represented in state space form as:
A
(
t
,
ω
x
)
=
[
−
A
11
(
t
)
A
12
(
t
,
ω
x
)
A
21
(
t
,
ω
x
)
A
22
(
t
)
]
{\displaystyle {\begin{aligned}A(t,\omega _{x})={\begin{bmatrix}-A_{11}(t)&A_{12}(t,\omega _{x})\\A_{21}(t,\omega _{x})&A_{22}(t)\end{bmatrix}}\end{aligned}}}
with
A
11
(
t
)
=
[
a
1
′
(
t
)
−
a
1
(
t
)
−
a
1
′
(
t
)
a
4
(
t
)
−
a
2
(
t
)
1
−
a
4
(
t
)
]
,
A
22
(
t
)
=
[
−
b
1
(
t
)
−
b
1
′
(
t
)
b
1
′
(
t
)
b
4
(
t
)
−
b
2
(
t
)
1
−
b
4
(
t
)
]
,
A
12
(
t
,
ω
x
)
=
ω
x
(
t
)
/
57.3
[
(
J
x
−
J
y
)
/
J
z
−
a
1
′
(
t
)
0
1
]
,
A
21
(
t
,
ω
x
)
=
ω
x
(
t
)
/
57.3
[
(
J
z
−
J
x
)
/
J
y
−
b
1
′
(
t
)
0
1
]
{\displaystyle {\begin{aligned}A_{11}(t)={\begin{bmatrix}a'_{1}(t)-a_{1}(t)&-a'_{1}(t)a_{4}(t)-a_{2}(t)\\1&-a_{4}(t)\end{bmatrix}},\quad A_{22}(t)={\begin{bmatrix}-b_{1}(t)-b'_{1}(t)&b'_{1}(t)b_{4}(t)-b_{2}(t)\\1&-b_{4}(t)\end{bmatrix}},\quad A_{12}(t,\omega _{x})=\omega _{x}(t)/57.3{\begin{bmatrix}(J_{x}-J_{y})/J_{z}&-a'_{1}(t)\\0&1\end{bmatrix}},\quad A_{21}(t,\omega _{x})=\omega _{x}(t)/57.3{\begin{bmatrix}(J_{z}-J_{x})/J_{y}&-b'_{1}(t)\\0&1\end{bmatrix}}\end{aligned}}}
B
(
t
)
=
[
−
a
1
(
t
)
a
5
(
t
)
0
−
a
5
(
t
)
0
0
−
b
1
′
(
t
)
b
5
(
t
)
−
b
3
(
t
)
0
−
b
5
(
t
)
]
{\displaystyle {\begin{aligned}B(t)={\begin{bmatrix}-a_{1}(t)a_{5}(t)&0\\-a_{5}(t)&0\\0&-b'_{1}(t)b_{5}(t)-b_{3}(t)\\0&-b_{5}(t)\end{bmatrix}}\end{aligned}}}
C
(
t
)
=
[
0
0
0
−
b
4
(
t
)
0
a
4
(
t
)
0
0
]
{\displaystyle {\begin{aligned}C(t)={\begin{bmatrix}0&0&0&-b_{4}(t)\\0&a_{4}(t)&0&0\end{bmatrix}}\end{aligned}}}
D
(
t
)
=
V
(
t
)
/
(
57.3
g
)
[
0
−
b
5
(
t
)
a
5
(
t
)
0
]
{\displaystyle {\begin{aligned}D(t)=V(t)/(57.3g){\begin{bmatrix}0&-b_{5}(t)\\a_{5}(t)&0\end{bmatrix}}\end{aligned}}}
where
a
(
t
)
{\displaystyle a(t)}
,
a
′
(
t
)
{\displaystyle a'(t)}
,
b
(
t
)
{\displaystyle b(t)}
, and
b
′
(
t
)
{\displaystyle b'(t)}
are the system parameters.
The Optimization Problem
edit
The optimization problem is to find a state feedback control law
u
=
K
x
v
(
t
)
{\displaystyle u=Kx_{v}(t)}
with
v
{\displaystyle v}
being an external input such that:
the closed-loop system:
x
˙
=
A
c
(
t
,
ω
x
)
x
(
t
)
+
B
(
t
)
v
(
t
)
{\displaystyle {\begin{aligned}{\dot {x}}&=A_{c}(t,\omega _{x})x(t)+B(t)v(t)\end{aligned}}}
where
A
c
(
t
,
ω
x
)
x
(
t
)
=
A
(
t
,
ω
x
)
+
B
(
t
)
K
{\displaystyle {\begin{aligned}A_{c}(t,\omega _{x})x(t)&=A(t,\omega _{x})+B(t)K\end{aligned}}}
is uniformly asymptotically stable.
The LMI: LMI for BTT missile attitude control
edit
Let
A
i
{\displaystyle A_{i}}
,
B
i
{\displaystyle B_{i}}
,
i
=
1
,
2
,
.
.
.
,
n
{\displaystyle i=1,2,...,n}
be defined by the set of extremes of the uncertain parameters of the system.
Using Theorem 7.8 in [1], the problem can be equivalently expressed in the following form:
There exist
P
>
0
,
W
{\displaystyle P>0,W}
which satisfy
A
i
P
+
B
i
W
+
P
A
i
T
+
W
T
B
i
T
<
0
,
i
=
1
,
2
,
.
.
.
,
n
{\displaystyle A_{i}P+B_{i}W+PA_{i}^{T}+W^{T}B_{i}^{T}<0,\quad i=1,2,...,n}
The goal of this LMI is to find a controller that can quadratically stabilize the missile at all operating points. When the matrices
W
{\displaystyle W}
and
P
{\displaystyle P}
are determined in the optimization problem, the controller gain matrix can be computed by:
K
=
W
P
−
1
{\displaystyle K=WP^{-1}}
[1] - LMI in Control Systems Analysis, Design and Applications