# LMIs in Control/pages/LMI for Attitude Control of BTT Missiles Roll Channel

**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 designed, and thus Jx = Jy. We assume that the roll channel is independent of the pitch and yaw channels.

**The System**
edit

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

where is the state variable, is the control input, and is the output. The parameters , , and refer to the roll angular velocity, the roll angle, and the aileron deflection, respectively.

**The Data**
edit

The system can be described as:

which can be represented in state space form as:

where and are the system parameters.

**The Optimization Problem**
edit

The optimization problem is to find a state feedback control law such that:

1. The closed-loop system:

is stable.

2. The norm of the transfer function:

is less than a positive scalar value, . Thus:

**The LMI:** LMI for BTT missile attitude control
edit

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

**Conclusion:**
edit

As mentioned, the aim is to attenuate the disturbance on the performance of the missile. The parameter 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 and are determined in the optimization problem, the controller gain matrix can be computed by:

**Implementation**
edit

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

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

**Related LMIs**
edit

LMI for Attitude Control of BTT Missiles, Pitch/Yaw Channel

LMI for Attitude Control of Nonrotating Missiles, Pitch Channel

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

**External Links**
edit

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