# LMIs in Control/pages/LMI for Attitude Control of Nonrotating Missiles

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:

{\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 ${\displaystyle x=[\alpha \quad w_{z}\quad \delta _{z}]^{\text{T}}}$ , ${\displaystyle u=\delta _{zc}}$  , ${\displaystyle y=[\alpha \quad n_{y}]^{\text{T}}}$ , and ${\displaystyle d=[\beta \quad w_{y}]^{\text{T}}}$  are the state variable, control input, output, and disturbance vectors, respectively. The paprameters ${\displaystyle \alpha }$ , ${\displaystyle w_{z}}$ , ${\displaystyle \delta _{z}}$ , ${\displaystyle \delta _{zc}}$ , ${\displaystyle n_{y}}$ , ${\displaystyle \beta }$ , and ${\displaystyle w_{y}}$  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 ${\displaystyle A(t),B_{1}(t),B_{2}(t),C(t),D_{1}(t),}$  and ${\displaystyle D_{2}(t)}$  are given as:

{\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}}}

{\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}}}

{\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}}}

{\displaystyle {\begin{aligned}D_{1}(t)=0,\quad D_{2}(t)={\frac {1}{57.3g}}{\begin{bmatrix}0&0\\V(t)b_{7}(t)&0\end{bmatrix}}\end{aligned}}}

where ${\displaystyle a_{1}(t)\sim a_{6}(t),\quad b_{1}(t)\sim b_{7}(t),{\acute {a}}_{1}(t),{\acute {b}}_{1}(t)}$  and ${\displaystyle c_{1}(t)\sim c_{4}(t)}$  are the system parameters. Moreover, ${\displaystyle V}$  is the speed of the missle and ${\displaystyle J_{x}}$ , ${\displaystyle J_{y}}$ , and ${\displaystyle J_{z}}$  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 ${\displaystyle u=Kx}$  such that:

1. The closed-loop system:

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

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

is less than a positive scalar value, ${\displaystyle \gamma }$ . Thus:

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

## 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:

{\displaystyle {\begin{aligned}&{\text{min}}\quad \gamma \\&{\text{s.t.}}\quad X>0\\&{\begin{bmatrix}(AX+B_{1}W)^{T}+AX+B_{1}W&B_{2}&(CX+D_{1}W)^{T}\\B_{2}^{T}&-\gamma I&D_{2}^{T}\\CX+D_{1}W&D_{2}&-\gamma I\end{bmatrix}}<0\end{aligned}}}

## Conclusion:

As mentioned, the aim is to attenuate the disturbance on the performance of the missile. The parameter ${\displaystyle \gamma }$  is the disturbance attenuation level. When the matrices ${\displaystyle W}$  and ${\displaystyle X}$  are determined in the optimization problem, the controller gain matrix can be computed by:

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

## Implementation

A link to Matlab codes for this problem in the Github repository: