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

$$ \begin{align} \dot{x}(t) &= A(t)x(t) + B(t)u(t) \\ y(t) &= C(t)x(t) + D(t)u(t) \end{align} $$

where $$ x(t) = [\omega_{z} \quad \alpha \quad \omega_{y} \quad \beta]^{\text{T}}$$ is the state vector, $$ u(t) = [\delta_{z} \quad \delta_{y}]^{\text{T}} $$ is the control input vector, and $$ y = [n_{z} \quad n_{y}]^{\text{T}} $$ is the output vector. The parameters $$ \omega_{z} $$, $$ \alpha $$, $$ \omega_{y} $$, and $$ \beta $$ refer to the pitch angular velocity, the pitch angle (angle of attack), yaw angular velocity, and yaw angle, respectively. The parameters $$ \delta_{z} $$ and $$ \delta_{y} $$ refer to the elevator and rudder deflections, respectively. Finally, the parameters $$ n_{z} $$ and $$ n_{y} $$ refer to the overloads on the normal and side directions, respectively.

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

$$ \begin{align} \begin{bmatrix} \dot{\alpha}(t) \\ \dot{\beta}(t) \\ \dot{\omega}_z(t) \\ \dot{\omega}_y(t) \\ n_y(t) \\ n_z(t) \end{bmatrix} = \begin{bmatrix} \omega_z(t)-\omega_x(t)\beta(t)/57.3-a_4(t)\alpha(t)-a_5(t)\delta_z(t) \\ \omega_y(t)-\omega_x(t)\alpha(t)/57.3-b_4(t)\beta(t)-b_5(t)\delta_y(t) \\ -a_1(t)\omega_z(t)+a'_1(t)\dot{\alpha}(t)-a_2(t)\alpha(t)-a_3(t)\delta_z(t)+(J_x-J_y)/(57.3J_z)\omega_x(t)\omega_y(t) \\ -b_1(t)\omega_y(t)+b'_1(t)\dot{\beta}(t)-b_2(t)\beta(t)-b_3(t)\delta_y(t)+(J_z-J_x)/(57.3J_y)\omega_x(t)\omega_z(t) \\ V(t)/(57.3g)(a_4(t)\alpha(t)+a_5(t)\delta_z(t)) \\ -V(t)/(57.3g)(b_4(t)\beta(t)+b_5(t)\delta_y(t)) \end{bmatrix} \end{align} $$

which can be represented in state space form as:

$$ \begin{align} A(t,\omega_x) = \begin{bmatrix} -A_{11}(t) & A_{12}(t,\omega_x) \\ A_{21}(t,\omega_x) & A_{22}(t) \end{bmatrix} \end{align} $$

with

$$ \begin{align} A_{11}(t) = \begin{bmatrix} a'_1(t)-a_1(t) & -a'_1(t)a_4(t)-a_2(t) \\ 1 & -a_4(t) \end{bmatrix} , \quad A_{22}(t) = \begin{bmatrix} -b_1(t)-b'_1(t) & b'_1(t)b_4(t)-b_2(t) \\ 1 & -b_4(t) \end{bmatrix} , \quad A_{12}(t,\omega_x) = \omega_x(t)/57.3 \begin{bmatrix} (J_x-J_y)/J_z & -a'_1(t) \\ 0 & 1 \end{bmatrix} , \quad A_{21}(t,\omega_x) = \omega_x(t)/57.3 \begin{bmatrix} (J_z-J_x)/J_y & -b'_1(t) \\ 0 & 1 \end{bmatrix} \end{align} $$

$$ \begin{align} B(t) = \begin{bmatrix} -a_{1}(t)a_5(t) & 0 \\ -a_5(t) & 0 \\ 0 & -b'_1(t)b_5(t)-b_3(t) \\ 0 & -b_5(t) \end{bmatrix} \end{align} $$

$$ \begin{align} C(t) = \begin{bmatrix} 0 & 0 & 0 & -b_4(t) \\ 0 & a_4(t) & 0 & 0 \end{bmatrix} \end{align} $$

$$ \begin{align} D(t) = V(t)/(57.3g) \begin{bmatrix} 0 & -b_5(t) \\ a_5(t) & 0 \end{bmatrix} \end{align} $$

where $$ a(t) $$, $$ a'(t) $$, $$ b(t) $$, and $$ b'(t) $$ are the system parameters.

The Optimization Problem
The optimization problem is to find a state feedback control law $$ u=Kx_v(t)$$ with $$ v $$ being an external input such that:

the closed-loop system:

$$ \begin{align} \dot{x} &= A_c(t,\omega_x)x(t) + B(t)v(t) \end{align} $$

where

$$ \begin{align} A_c(t,\omega_x)x(t) &= A(t,\omega_x) + B(t)K \end{align} $$

is uniformly asymptotically stable.

The LMI: LMI for BTT missile attitude control
Let $$ A_i $$, $$ B_i $$, $$ i = 1,2,...,n $$ 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 $$ P > 0, W $$ which satisfy $$ A_iP+B_{i}W + PA_i^{T} + W^{T}B_{i}^{T} < 0, \quad i = 1,2,...,n $$

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

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

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

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

Related LMIs
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

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