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:

$\begin{array}{cc}\dot{x}(t)& =A(t)x(t)+B_1(t)u(t)+B_2(t)d(t)\\ y(t)& =C(t)x(t)+D_1(t)u(t)+D_2(t)d(t)\end{array}$

where $x(t)=[{\omega }_x\varphi {]}^{\text{T}}$ is the state variable, $u(t)={\delta }_x$ is the control input, and $y=\varphi$ is the output. The parameters ${\omega }_x$, $\varphi$, 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:

$\begin{array}{c}\left[\begin{array}{c}{\dot{\omega }}_x(t)\\ \dot{\varphi }(t)\end{array}\right]=\left[\begin{array}{cc}-c_1(t)& 0\\ 1& 0\end{array}\right]\left[\begin{array}{c}{\omega }_x(t)\\ \varphi (t)\end{array}\right]+\left[\begin{array}{c}-c_3(t)\\ 0\end{array}\right]{\delta }_x(t)\end{array}$

$\begin{array}{c}y(t)=\varphi (t)\end{array}$

which can be represented in state space form as:

$\begin{array}{c}A(t)=\left[\begin{array}{cc}-c_1(t)& 0\\ 1& 0\end{array}\right]\end{array}$

$\begin{array}{c}{B}_1(t)=\left[\begin{array}{c}-c_3(t)\\ 0\end{array}\right],B_2(t)=\left[\begin{array}{c}0\\ 0\end{array}\right]\end{array}$

$\begin{array}{c}C(t)=\left[\begin{array}{cc}0& 1\end{array}\right]\end{array}$

$\begin{array}{c}{D}_1(t)=0,D_2(t)=0\end{array}$

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:

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

is stable.

2. The $H_{\mathrm{\infty }}$ 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:

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

The LMI: LMI for BTT missile attitude control

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

$\begin{array}{cc}& \text{min}\gamma \\ & \text{s.t.}X>0\\ & \left[\begin{array}{ccc}(AX+B_{1}W{)}^T+AX+B_{1}W& B_2& (CX+D_{1}W{)}^T\\ B_2^T& -\gamma I& D_2^T\\ CX+D_{1}W& D_2& -\gamma I\end{array}\right]<0\end{array}$

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:

$K=W{X}^{-1}$

Implementation

A link to MATLAB code for the problem in the GitHub repository:

https://github.com/scarris8/LMI-for-BTT-Missile-Roll-Control

Related LMIs

LMI for Attitude Control of Nonrotating Missles, Pitch Channel

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

External Links

• [1] - LMI in Control Systems Analysis, Design and Applications

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