Spring 2003 1661AC22 Longitudinal Dynamics For notational simplicity, let X=Fn, Y= Fu, and Z= F aF Longitudinal equations(1-15 )can be rewritten as mi=X+X2- mg cos(0+△X mli-qUo)= Zuu+ Zww+ Zii+Zaq-mg sin 000+AZc Iyyq= Mau+ Mww+ Mii+Ma+AMc There is no roll/yaw motion, so q=0 The control commands△X≡△F,△Z≡△F,and△Me≡△M have not yet been specified Rewrite in state space form as n XX -mg cos eo △X (m-Zia ZuZu za+mUo -mg sin 00w △Z Mhiw+ly9 Mu Mw Mg 0 q △M 00 1 0 0001[a1「XnXn mg cos o △X 0 m-Zi 0ai Za zu z+mU0- mg sin e0△Z 0 -Mi lm o Mn ma 00010 EX AX+c descriptor state space form X=E-(AX+c)=AX+cSpring 2003 16.61 AC 2–2 Longitudinal Dynamics • For notational simplicity, let X = Fx, Y = Fy, and Z = Fz Xu ≡
∂Fx ∂u ,... • Longitudinal equations (1–15) can be rewritten as: mu˙ = Xuu + Xww − mg cos Θ0θ + ∆Xc m( ˙w − qU0) = Zuu + Zww + Zw˙ w˙ + Zqq − mg sin Θ0θ + ∆Zc Iyyq˙ = Muu + Mww + Mw˙w˙ + Mqq + ∆Mc – There is no roll/yaw motion, so q = ˙ θ. – The control commands ∆Xc ≡ ∆Fc x, ∆Zc ≡ ∆Fc z , and ∆Mc ≡ ∆Mc have not yet been specified. • Rewrite in state space form as         mu˙ (m − Zw˙) ˙w −Mw˙w˙ + Iyyq˙ ˙ θ         =         Xu Xw 0 −mg cos Θ0 Zu Zw Zq + mU0 −mg sin Θ0 Mu Mw Mq 0 00 1 0                 u w q θ         +         ∆Xc ∆Zc ∆Mc 0                 m 0 00 0 m − Zw˙ 0 0 0 −Mw˙ Iyy 0 0 0 01                 u˙ w˙ q˙ ˙ θ         =         Xu Xw 0 −mg cos Θ0 Zu Zw Zq + mU0 −mg sin Θ0 Mu Mw Mq 0 00 1 0                 u w q θ         +         ∆Xc ∆Zc ∆Mc 0         EX˙ = AXˆ + ˆc descriptor state space form X˙ = E−1 (AXˆ + ˆc) = AX + c