User:Margav06/sandbox/Click here to continue/Applications of Linear systems/LMI for Attitude Control of Nonrotating Missiles, Pitch Channel

LMI for Attitude Control of Nonrotating Missles, Pitch Channel

The dynamic model of a missile is very complicated and a simplified model is used. To do so, we consider a simplified attitude system model for the pitch channel in the system. We aim to achieve a non-rotating motion of missiles. It is worthwhile to note that the attitude control design for the pitch channel and the yaw/roll channel can be solved exactly in the same way while representing matrices of the system are different.

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 $$ x = [\alpha \quad w_{z} \quad \delta_{z}]^{\text{T}}$$, $$ u = \delta_{zc} $$, $$ y = [\alpha \quad n_{y}]^{\text{T}}$$, and $$ d = [\beta \quad w_{y}]^{\text{T}}$$ are the state variable, control input, output, and disturbance vectors, respectively. The paprameters $$ \alpha $$, $$ w_{z} $$, $$ \delta_{z} $$, $$ \delta_{zc} $$, $$ n_{y} $$, $$ \beta $$, and $$ w_{y} $$ stand for the attack angle, pitch angular velocity, the elevator deflection, the input actuator deflection, the overload on the side direction, the sideslip angle, and the yaw angular velocity, respectively.

The Data
In the aforementioned pitch channel system, the matrices $$ A(t), B_{1}(t), B_{2}(t), C(t), D_{1}(t),$$ and $$D_{2}(t)$$ are given as:

$$ \begin{align} A(t) = \begin{bmatrix} -a_{4}(t) & 1 & -a_{5}(t) \\ -\acute{a}_{1}(t)a_{4}(t) - a_{2}(t) & \acute{a}_{1}(t) - a_{1}(t) & \acute{a}_{1}(t)a_{5}(t) - a_{3}(t) \\ 0 & 0 & -1/\tau_{z} \end{bmatrix} \end{align} $$

$$ \begin{align} B_{1}(t) = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \quad B_{2}(t) = \frac{w_{x}}{57.3} \begin{bmatrix} -1 & 0 \\ -\acute{a}_{1}(t) & \frac{J_{x}-J_{y}}{J_{z}} \\ 0 & 0 \end{bmatrix} \end{align} $$

$$ \begin{align} C(t) = \frac{w_{x}}{57.3} \begin{bmatrix} 57.3g & 0 & 0 \\ V(t)a_{4}(t) & 0 & V(t)a_{5}(t) \end{bmatrix} \end{align} $$

$$ \begin{align} D_{1}(t) = 0, \quad D_{2}(t) = \frac{1}{57.3g} \begin{bmatrix} 0 & 0 \\ V(t)b_{7}(t) & 0 \end{bmatrix} \end{align} $$

where $$ a_{1}(t) \sim a_{6}(t), \quad b_{1}(t) \sim b_{7}(t), \acute{a}_{1}(t), \acute{b}_{1}(t) $$ and $$ c_{1}(t) \sim c_{4}(t) $$ are the system parameters. Moreover, $$ V $$ is the speed of the missle and $$J_{x}$$, $$J_{y}$$, and $$J_{z}$$ are the rotary inertia of the missle corresponding to the body coordinates.

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 non-rotating missle 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. 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 codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Non-rotating-Missle-Attitude-Control

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

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