# Linearized equations of motion

The state-space equation system (9.30) above is nonlinear. To make stability and control problems tractable, these equations are first put into linearized forms. We assume some steady flight in the trim state x0, 80. The trim state has X0 = 0, except for the position rate and heading rate components.

Xe = , ye = v’0 , ze = w’0 , ф = r0

Straight flight has a zero heading rate rg = 0, while steady turning flight has r0 = 0.

In the system equation (9.30), consider small perturbations Ax(t) and AS(t) about the trim state, as shown in Figure 9.3.

 x(t) = xo + A x(t) (9.31) 8(t) = 80 + A8(t) _ df ~ f(x0, <50) + — AX + dx df as.. ,д* (9.32) x 0 + ax = f (xo+Ax, So+As) (9.33)

Since the trim state is physical, it must by itself obey the equations of motion.

Xо = f (xo, so) (9.34)  Subtracting (9.34) from (9.33) gives the linearized equations of motion which govern the small state vector perturbations,

with A and 13 being the system Jacobian matrices which depend on the trim state. actual state later perturbation

x(t) = x о + Ax( t)

xо trim state

Figure 9.3: Aircraft state considered as a small perturbation from a trim state. Instability is indicated if the perturbation grows exponentially from an initial perturbation.