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

**LMI for Attitude Control of BTT Missles, Pitch/Yaw 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**

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

where is the state vector, is the control input vector, and is the output vector. The parameters , , , and refer to the pitch angular velocity, the pitch angle (angle of attack), yaw angular velocity, and yaw angle, respectively. The parameters and refer to the elevator and rudder deflections, respectively. Finally, the parameters and refer to the overloads on the normal and side directions, respectively.

**The Data**

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The model for the pitch/yaw channel is as follows:

which can be represented in state space form as:

with

where , , , and are the system parameters.

**The Optimization Problem**

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The optimization problem is to find a state feedback control law with being an external input such that:

the closed-loop system:

where

is uniformly asymptotically stable.

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

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Let , , be defined by the set of extremes of the uncertain parameters of the system.

Using Theorem 7.8 in [1], the problem can be equivalently expressed in the following form:

There exist which satisfy

**Conclusion:**

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The goal of this LMI is to find a controller that can quadratically stabilize the missile at all operating points. When the matrices and are determined in the optimization problem, the controller gain matrix can be computed by:

**Implementation**

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

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

**Related LMIs**

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LMI for Attitude Control of BTT Missles, Roll Channel

LMI for Attitude Control of Nonrotating Missles, Pitch Channel

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

**External Links**

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