[<< wikibooks] LMIs in Control/pages/LMI for Attitude Control of Nonrotating Missles
LMI for Attitude Control of Nonrotating Missles, Pitch Channel
The dynamic model of a missile is very complicated and a simplified model is used. To do so, we consider a simplified attitude system model for the pitch channel in the system. We aim to achieve a non-rotating motion of missiles. It is worthwhile to note that the attitude control design for the pitch channel and the yaw/roll channel can be solved exactly in the same way while representing matrices of the system are different.

== The System ==
The state-space representation for the pitch channel can be written as follows:

x
˙

(
t
)

=
A
(
t
)
x
(
t
)
+

B

1

(
t
)
u
(
t
)
+

B

2

(
t
)
d
(
t
)

y
(
t
)

=
C
(
t
)
x
(
t
)
+

D

1

(
t
)
u
(
t
)
+

D

2

(
t
)
d
(
t
)

{\displaystyle {\begin{aligned}{\dot {x}}(t)&=A(t)x(t)+B_{1}(t)u(t)+B_{2}(t)d(t)\\y(t)&=C(t)x(t)+D_{1}(t)u(t)+D_{2}(t)d(t)\end{aligned}}}

where

x
=
[
α

w

z

δ

z

]

T

,

u
=

δ

z
c

{\displaystyle u=\delta _{zc}}
,

y
=
[
α

n

y

]

T

, and

d
=
[
β

w

y

]

T

are the state variable, control input, output, and disturbance vectors, respectively. The paprameters

α

{\displaystyle \alpha }
,

w

z

{\displaystyle w_{z}}
,

δ

z

{\displaystyle \delta _{z}}
,

δ

z
c

{\displaystyle \delta _{zc}}
,

n

y

{\displaystyle n_{y}}
,

β

{\displaystyle \beta }
, and

w

y

{\displaystyle w_{y}}
stand for the attack angle, pitch angular velocity, the elevator deflection, the input actuator deflection, the overload on the side direction, the sideslip angle, and the yaw angular velocity, respectively.

== The Data ==
In the aforementioned pitch channel system, the matrices

A
(
t
)
,

B

1

(
t
)
,

B

2

(
t
)
,
C
(
t
)
,

D

1

(
t
)
,

{\displaystyle A(t),B_{1}(t),B_{2}(t),C(t),D_{1}(t),}
and

D

2

(
t
)

{\displaystyle D_{2}(t)}
are given as:

A
(
t
)
=

[

−

a

4

(
t
)

1

−

a

5

(
t
)

−

a
´

1

(
t
)

a

4

(
t
)
−

a

2

(
t
)

a
´

1

(
t
)
−

a

1

(
t
)

a
´

1

(
t
)

a

5

(
t
)
−

a

3

(
t
)

0

0

−
1

/

τ

z

]

{\displaystyle {\begin{aligned}A(t)={\begin{bmatrix}-a_{4}(t)&1&-a_{5}(t)\\-{\acute {a}}_{1}(t)a_{4}(t)-a_{2}(t)&{\acute {a}}_{1}(t)-a_{1}(t)&{\acute {a}}_{1}(t)a_{5}(t)-a_{3}(t)\\0&0&-1/\tau _{z}\end{bmatrix}}\end{aligned}}}

B

1

(
t
)
=

[

0

0

1

]

,

B

2

(
t
)
=

w

x

57.3

[

−
1

0

−

a
´

1

(
t
)

J

x

−

J

y

J

z

0

0

]

{\displaystyle {\begin{aligned}B_{1}(t)={\begin{bmatrix}0\\0\\1\end{bmatrix}},\quad B_{2}(t)={\frac {w_{x}}{57.3}}{\begin{bmatrix}-1&0\\-{\acute {a}}_{1}(t)&{\frac {J_{x}-J_{y}}{J_{z}}}\\0&0\end{bmatrix}}\end{aligned}}}

C
(
t
)
=

w

x

57.3

[

57.3
g

0

0

V
(
t
)

a

4

(
t
)

0

V
(
t
)

a

5

(
t
)

]

{\displaystyle {\begin{aligned}C(t)={\frac {w_{x}}{57.3}}{\begin{bmatrix}57.3g&0&0\\V(t)a_{4}(t)&0&V(t)a_{5}(t)\end{bmatrix}}\end{aligned}}}

D

1

(
t
)
=
0
,

D

2

(
t
)
=

1

57.3
g

[

0

0

V
(
t
)

b

7

(
t
)

0

]

where

a

1

(
t
)
∼

a

6

(
t
)
,

b

1

(
t
)
∼

b

7

(
t
)
,

a
´

1

(
t
)
,

b
´

1

(
t
)

{\displaystyle a_{1}(t)\sim a_{6}(t),\quad b_{1}(t)\sim b_{7}(t),{\acute {a}}_{1}(t),{\acute {b}}_{1}(t)}
and

c

1

(
t
)
∼

c

4

(
t
)

{\displaystyle c_{1}(t)\sim c_{4}(t)}
are the system parameters. Moreover,

V

{\displaystyle V}
is the speed of the missle and

J

x

{\displaystyle J_{x}}
,

J

y

{\displaystyle J_{y}}
, and

J

z

{\displaystyle J_{z}}
are the rotary inertia of the missle corresponding to the body coordinates.

== The Optimization Problem ==
The optimization problem is to find a state feedback control law

u
=
K
x

{\displaystyle u=Kx}
such that:
1. The closed-loop system:

x
˙

=
(
A
+

B

1

K
)
x
+

B

2

d

z

=
(
C
+

D

1

K
)
x
+

D

2

d

{\displaystyle {\begin{aligned}{\dot {x}}&=(A+B_{1}K)x+B_{2}d\\z&=(C+D_{1}K)x+D_{2}d\end{aligned}}}

is stable.
2. The

H

∞

{\displaystyle H_{\infty }}
norm of the transfer function:

G

z
d

(
s
)
=
(
C
+

D

1

K
)
(
s
I
−
(
A
+

B

1

K
)

)

−
1

B

2

+

D

2

{\displaystyle G_{zd}(s)=(C+D_{1}K)(sI-(A+B_{1}K))^{-1}B_{2}+D_{2}}

is less than a positive scalar value,

γ

{\displaystyle \gamma }
. Thus:

|

|

G

z
d

(
s
)

|

|

∞

<
γ

{\displaystyle ||G_{zd}(s)||_{\infty }<\gamma }

== The LMI: LMI for non-rotating missle attitude control ==
Using Theorem 8.1 in [1], the problem can be equivalently expressed in the following form:

min

γ

s.t.

X
>
0

[

(
A
X
+

B

1

W

)

T

+
A
X
+

B

1

W

B

2

(
C
X
+

D

1

W

)

T

B

2

T

−
γ
I

D

2

T

C
X
+

D

1

W

D

2

−
γ
I

]

<
0

== Conclusion: ==
As mentioned, the aim is to attenuate the disturbance on the performance of the missile. The parameter

γ

{\displaystyle \gamma }
is the disturbance attenuation level. When the matrices

W

{\displaystyle W}
and

X

{\displaystyle X}
are determined in the optimization problem, the controller gain matrix can be computed by:

K
=
W

X

−
1

{\displaystyle K=WX^{-1}}

== Implementation ==
A link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Non-rotating-Missle-Attitude-Control

== Related LMIs ==
LMI for Attitude Control of Nonrotating Missles, Yaw/Roll Channel

LMIs in Control/Tools