PROBLEMS

$53$

$2\mathrm{.}19$ Consider the simplified model of an aircraft shown in Fig. $2\mathrm{.}25.$ It is assumed that the aircraft is in an equilibrium state at the pitched angle ${}_0,$ elevator angle $u_0,$ altitude $h_0,$ and cruising speed $v_0.$ It is assumed that small deviations of $$ and $u$ from ${}_0$ and $u_0$ generate

forces $f_1=k_1$ and $f_2=k_{2}u$ as shown in the figure. Let $m$ be the mass of the aircraft, I the moment of inertia about the center of gravity $P,b{}^{}$ the aerodynamic damping, and $h$ the deviation of the altitude from $h_0.$ Find a state$-$space equation to describe the system. Show also that the transfer function from $u$ to $h,$ by neglecting the effect of I, is

hat$(g)(s)=\frac{(hat(h))(s)}{(hat(u))(s)}=\frac{k_1{k}_2{l}_2-k_{2}bs}{m{s}^2(bs+k_1{l}_1)}$