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.”