LMIs in Control/pages/LMI for Attitude Control of BTT Missiles

LMI for Attitude Control of BTT Missles, Roll 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 design, and thus Jx = Jy. We assume that the roll channel is independent of the pitch and yaw channels.

The System

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

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

where $x(t)=[\omega _{x}\quad \phi ]^{\text{T}}$ is the state variable, $u(t)=\delta _{x}$ is the control input, and $y=\phi$ is the output. The parameters $\omega _{x}$ , $\phi$ , and $\delta _{x}$ refer to the roll angular velocity, the roll angle, and the aileron deflection, respectively.

The Data

The system can be described as:

${\begin{aligned}{\begin{bmatrix}{\dot {\omega }}_{x}(t)\\{\dot {\phi }}(t)\end{bmatrix}}={\begin{bmatrix}-c_{1}(t)&0\\1&0\end{bmatrix}}{\begin{bmatrix}\omega _{x}(t)\\\phi (t)\end{bmatrix}}+{\begin{bmatrix}-c_{3}(t)\\0\end{bmatrix}}\delta _{x}(t)\end{aligned}}$

${\begin{aligned}y(t)=\phi (t)\end{aligned}}$

which can be represented in state space form as:

${\begin{aligned}A(t)={\begin{bmatrix}-c_{1}(t)&0\\1&0\end{bmatrix}}\end{aligned}}$

${\begin{aligned}B_{1}(t)={\begin{bmatrix}-c_{3}(t)\\0\end{bmatrix}},\quad B_{2}(t)={\begin{bmatrix}0\\0\end{bmatrix}}\end{aligned}}$

${\begin{aligned}C(t)={\begin{bmatrix}0&1\end{bmatrix}}\end{aligned}}$

${\begin{aligned}D_{1}(t)=0,\quad D_{2}(t)=0\end{aligned}}$

where $c_{1}(t)$ and $c_{3}(t)$ are the system parameters.

The Optimization Problem

The optimization problem is to find a state feedback control law $u=Kx$ such that:

1. The closed-loop system:

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

is stable.

2. The $H_{\infty }$ norm of the transfer function:

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

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

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

The LMI: LMI for BTT missile attitude control

Using Theorem 8.1 in [1], the problem can be equivalently expressed in the following form:

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

Conclusion:

As mentioned, the aim is to attenuate the disturbance on the performance of the missile. The parameter $\gamma$ is the disturbance attenuation level. However, it should be noted that this model for the roll channel for a BTT missile is very simple and easy to handle, there is no disturbance to attenuate. This problem is presented here for completeness when used in a full BTT missile model along with the pitch/yaw channels. When the matrices $W$ and $X$ are determined in the optimization problem, the controller gain matrix can be computed by: