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## Q. 7.EX.1

Satellite Attitude Control Model in State-Variable Form
Determine the $\pmb F, \pmb G, \pmb H, J$ matrices in the state-variable form for the satellite attitude control model in Example 2.3 with $M_D = 0.$

## Verified Solution

Define the attitude and the angular velocity of the satellite as the state-variables so that $x \triangleq [θ ω]^T. ^2$ The single second-order equation (2.15) can then be written in an equivalent way as two first-order equations:

$F_c d + M_D = I \ddot{θ} .$            (2.15)

$\dot{θ} = ω,$

$\dot{ω} = \frac{d}{I}F_c.$

These equations are expressed, using Eq. (7.3), $\dot{ \pmb x} = \pmb F \pmb x + \pmb Gu,$ as

$\left[\begin{matrix} \dot\theta \\ \dot w \end{matrix} \right] = \left[\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix} \right] \left[\begin{matrix} \theta \\w \end{matrix} \right] + \left[\begin{matrix} 0 \\ d/I \end{matrix} \right] F_c .$

The output of the system is the satellite attitude, $y = θ.$ Using Eq. (7.4), $y = \pmb {Hx}+Ju,$ this relation is expressed as

$y = \left[\begin{matrix} 1 & 0 \end{matrix} \right] \left[\begin{matrix} \theta \\ w \end{matrix} \right] .$

Therefore, the matrices for the state-variable form are

$\pmb F = \left[\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix} \right] , \ \ \ \pmb G = \left[\begin{matrix} 0 \\ d/I \end{matrix} \right] , \ \ \ \pmb H = \left[\begin{matrix} 1 & 0 \end{matrix} \right] , \ \ \ J = 0 ,$

and the input $u \triangleq F_c.$

For this very simple example, the state-variable form is a more cumbersome way of writing the differential equation than the second-order version in Eq. (2.15). However, the method is not more cumbersome for most systems, and the advantages of having a standard form for use in computer-aided design have led to widespread use of the state-variable form.

$^2$ The symbol $\triangleq$ means “is to be defined.”