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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:

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

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:

$$ \begin{align} \begin{bmatrix} \dot{\omega}_x(t) \\ \dot{\phi}(t) \end{bmatrix} = \begin{bmatrix} -c_1(t) & 0 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} \omega_x(t) \\ \phi(t) \end{bmatrix} + \begin{bmatrix} -c_3(t) \\ 0 \end{bmatrix} \delta_x(t) \end{align} $$

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

which can be represented in state space form as:

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

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

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

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

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:

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

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:

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

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:

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

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

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