February 25, 2011
AAE 421, Spring 2011
Homework Seven
Due: Friday, March 4
Exercise 1 Consider an input-output system with input uand output ydescribed by
2¨p−eθ¨
θ+θ˙
θ2=u
−(1 + θ2)¨p+¨
θ−sin θ=u
y=θ+ (1 + θ) cos(u)
(a) Linearize this system about equilibrium conditions corresponding to ue= 0 and pe=θe=
0.
(b) Obtain the A, B, C, D matrices for a state space representation of the linearization found
in part (a).
Exercise 2 Consider the nonlinear input-output system described by
˙x1=−2x2+x1−sin x1+u1
˙x2=−x1−2x2−sin x2+u2
y1=x1+ sin x2
y2=x2+ sin x1.
Using the trim command in MATLAB, determine the trim values of u2and y2so that this
system has a trim condition with u1= 1 and y1= 1.
Exercise 3 (a) Obtain a SIMULINK model of the Cessna 182 with state variables V, α, θ, q ,
input variables el, th and output variables V, γ.
(b) Using the trim command determine the constant value of el which results in horizontal
steady state flight with th = 100 hp.
(c) Obtain the matrices A, B, C, D in a state space representation of the linearization of the
system about the trim conditions found in part (b).
(f) Compare the behavior of the nonlinear and the linearized models. Simulate both models in
MATLAB and compare the responses of δV and δγ for initial conditions close to equilibrium
and not so close to equilibrium.
Exercise 4 Show that the transfer function of
˙x1=x2
˙x2=−α0x1−α1x2+u
y=β0x1+β1x2
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