Fa2004 16.33312-4 The next step is to include these new forces and moments in the equations of motion 00 X mg cos eo 0 z;00 Zu zu la t mUo -mg sin eo 0 Mu Mw Mq q 016 00 0 Zs Zi 00 X1-X20 z2 -M -Mu -M 0 0 →Ei=Ax+Bu+B Multiply through by e to get new state space model A r+B, u+B, w which has both control u and disturbance w inputs A similar operation can be performed for the lateral dynamics in terms of the disturbance inputs vg, Pg, and rg Can now compute the response to specific types of gusts, such as a step or sinusoidal function, but usually are far more interested in the response to a stochastic gust field� � Fall 2004 16.333 12–4 • The next step is to include these new forces and moments in the equations of motion ⎡ ⎤ ⎡⎤⎡ ⎤ ⎤⎡ m 0 0 0 u˙ Xu Xw 0 −mg cos Θ0 u ⎢ ⎢ ⎢ ⎣ ⎥ ⎥ ⎥ ⎦ ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎣ ⎢ ⎢ ⎢ ⎣ ⎥ ⎥ ⎥ ⎦ 0 m − Zw˙ 0 0 0 −Mw˙ Iyy 0 0 0 0 1 w˙ q˙ Zu Zw Zq + mU0 −mg sin Θ0 Mu Mw Mq 0 w q = θ ˙ 0 0 1 0 θ ⎡ ⎤ Xδe Xδp Zδe Zδp Mδe Mδp ⎥ ⎥ ⎥ ⎦ δe δt ⎢ ⎢ ⎢ ⎣ + 0 0 −Xu −Xw 0 ⎡ ⎤ ⎡ ⎤ ⎢ ⎢ ⎢ ⎣ ⎣ ⎥ ⎥ ⎥ ⎦ ug −Zu −Zw 0 −Mu −Mw −Mq + w ⎦g qg 0 0 0 Ex ˆ Buu + ˆ ⇒ ˙ = Ax + ˆ Bww • Multiply through by E−1 to get new state space model x˙ = Ax + Buu + Bww which has both control u and disturbance w inputs. – A similar operation can be performed for the lateral dynamics in terms of the disturbance inputs vg, pg, and rg. • Can now compute the response to specific types of gusts, such as a step or sinusoidal function, but usually are far more interested in the response to a stochastic gust field ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎣