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:

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

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

The Data

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

{\displaystyle {\begin{aligned}{\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{aligned}}}

which can be represented in state space form as:

{\displaystyle {\begin{aligned}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{aligned}}}

with

{\displaystyle {\begin{aligned}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{aligned}}}

{\displaystyle {\begin{aligned}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{aligned}}}

{\displaystyle {\begin{aligned}C(t)={\begin{bmatrix}0&0&0&-b_{4}(t)\\0&a_{4}(t)&0&0\end{bmatrix}}\end{aligned}}}

{\displaystyle {\begin{aligned}D(t)=V(t)/(57.3g){\begin{bmatrix}0&-b_{5}(t)\\a_{5}(t)&0\end{bmatrix}}\end{aligned}}}

where ${\displaystyle a(t)}$ , ${\displaystyle a'(t)}$ , ${\displaystyle b(t)}$ , and ${\displaystyle b'(t)}$  are the system parameters.

The Optimization Problem

The optimization problem is to find a state feedback control law ${\displaystyle u=Kx_{v}(t)}$  with ${\displaystyle v}$  being an external input such that:

the closed-loop system:

{\displaystyle {\begin{aligned}{\dot {x}}&=A_{c}(t,\omega _{x})x(t)+B(t)v(t)\end{aligned}}}

where

{\displaystyle {\begin{aligned}A_{c}(t,\omega _{x})x(t)&=A(t,\omega _{x})+B(t)K\end{aligned}}}

is uniformly asymptotically stable.

The LMI: LMI for BTT missile attitude control

Let ${\displaystyle A_{i}}$ , ${\displaystyle B_{i}}$ , ${\displaystyle i=1,2,...,n}$  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 ${\displaystyle P>0,W}$  which satisfy ${\displaystyle A_{i}P+B_{i}W+PA_{i}^{T}+W^{T}B_{i}^{T}<0,\quad i=1,2,...,n}$

Conclusion:

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

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

Implementation

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