# LMIs in Control/pages/LMI for Nonrotating Missiles Attitude Control yaw roll channel

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

Deriving the exact dynamic modeling of a missile is a very complicated procedure. Thus, a simplified model is used to model the missile dynamics. To do so, we consider a simplified attitude system model for the yaw/roll channel of the system. We aim to achieve a non-rotating motion of missiles. Note that the attitude control design for the yaw/roll channel and the pitch channel can be solved exactly in the same way except for different representing matrices of the system.

## The System

The state-space representation for the yaw/roll 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 ${\displaystyle x=[\beta \quad w_{y}\quad w_{x}\quad \delta _{x}\quad \delta _{y}]^{\text{T}}}$, ${\displaystyle u=[\delta _{xc}\quad \delta _{yc}]^{\text{T}}}$ , ${\displaystyle y=[n_{z}\quad w_{x}]^{\text{T}}}$, and ${\displaystyle d=\delta _{z}}$ 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 yaw/roll 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_{11}(t)&A_{12}(t)\\0&A_{22}(t)\end{bmatrix}}\end{aligned}}}

where

{\displaystyle {\begin{aligned}A_{11}(t)={\begin{bmatrix}-b_{4}(t)&1&{\frac {\alpha (t)}{57.3}}\\-{\acute {b}}_{1}(t)b_{4}(t)-b_{2}(t)&-{\acute {b}}_{1}(t)-{\acute {b}}_{1}(t)&{\frac {J_{y}-J_{z}}{57.3J_{x}}}{w}_{z}(t)-{\frac {{\acute {b}}_{1}\alpha (t)}{57.3}}\\c_{2}(t)&{\frac {J_{y}-J_{z}}{57.3J_{x}}}{w}_{z}(t)&-c_{1}(t)\end{bmatrix}}\end{aligned}}}

{\displaystyle {\begin{aligned}A_{12}(t)={\begin{bmatrix}0&-b_{5}(t)\\0&-{\acute {b}}_{1}(t){\acute {b}}_{5}-b_{3}(t)\\-c_{3}(t)&c_{4}(t)\end{bmatrix}}\end{aligned}}}

{\displaystyle {\begin{aligned}A_{22}(t)=-{\frac {1}{\tau _{x}\tau _{y}}}{\begin{bmatrix}\tau _{y}&0\\0&\tau _{x}\end{bmatrix}}\end{aligned}}}

and

{\displaystyle {\begin{aligned}B_{1}(t)={\begin{bmatrix}0&0\\0&0\\0&0\\{\frac {1}{\tau _{x}}}&0\\0&{\frac {1}{\tau _{y}}}\end{bmatrix}},\quad B_{2}(t)={\begin{bmatrix}a_{6}(t)\\-{\acute {b}}_{1}(t)a_{6}(t)\\0\\0\\0\end{bmatrix}}\end{aligned}}}

{\displaystyle {\begin{aligned}C(t)=-{\frac {1}{57.3g}}{\begin{bmatrix}V(t)b_{4}(t)&0&0&0&V(t)b_{5}(t)\\0&0&-57.3g&0&0\end{bmatrix}}\end{aligned}}}

{\displaystyle {\begin{aligned}D_{1}(t)=0,\quad D_{2}(t)=-{\frac {V(t)}{57.3g}}{\begin{bmatrix}b_{6}(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: