TransferFunctionModel
SystemsModelExtract
BodePlot

Control Systems
Engineering
System Modeling

Model an Aircraft's Longitudinal Dynamics

Use a transfer-function model of an aircraft's longitudinal dynamics to assess its frequency response to an elevator deflection

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The longitudinal dynamics are parametrized by the velocity , angle of attack α and pitch θ:

In[1]:=

{,α,θ}={0.0005(s-70)(s+0.5),-0.02(s+80)(

+0.0065+0.006),-1.4(s+0.02)(s+0.4)};

The elevator deflection angle δ has two quadratic terms indicating two possible damping ratios:

In[2]:=

δ=0.0084+0.013s+1.411

+1.005

//Factor

Out[2]=

1.(0.006+0.005s+1.

)(1.4+1.s+1.

)

The values of the damping ratios reveal a short-period and a long-period mode:

In[3]:=

{p1,p2}=Table[Solve[quad0,s],{quad,{0.006+0.005s+

,1.4+s+

}}];{s1,s2}=Table[s/.p,{p,{p1,p2}}]

Out[3]=

{{-0.0025-0.0774193,-0.0025+0.0774193},{-0.5-1.07238,-0.5+1.07238}}

In[4]:=

{zeta1,zeta2}=Table[-Re[s]/Sqrt[Re[s]^2+Im[s]^2],{s,{s1,s2}}]

Out[4]=

{{0.0322749,0.0322749},{0.422577,0.422577}}

A single-input, three-output (SIMO) transfer-function matrix model of the system:

In[5]:=

tfm=TransferFunctionModel{{{},{α},{θ}},δ},s,

labels



Out[5]=

	δ
	0.0005(-70+s)(0.5+s) (0.006+0.005s+1. 2 s )(1.4+1.s+1. 2 s )
α	- 0.02(80+s)(0.0125+ 2 s ) (0.006+0.005s+1. 2 s )(1.4+1.s+1. 2 s )
θ	- 1.4(0.02+s)(0.4+s) (0.006+0.005s+1. 2 s )(1.4+1.s+1. 2 s )

Use

SystemsModelExtract

to obtain the individual transfer-functions for , α, θ:

In[6]:=

{tfm,tfmα,tfmθ}=Table[SystemsModelExtract[tfm,1,i],{i,3}]

Out[6]=



	δ
	0.0005(-70+s)(0.5+s) (0.006+0.005s+1. 2 s )(1.4+1.s+1. 2 s )

	δ
α	- 0.02(80+s)(0.0125+ 2 s ) (0.006+0.005s+1. 2 s )(1.4+1.s+1. 2 s )

	δ
θ	- 1.4(0.02+s)(0.4+s) (0.006+0.005s+1. 2 s )(1.4+1.s+1. 2 s )



The distinct modes can be observed in each transfer-function as breaks in the slope:

In[7]:=

Table[BodePlot[SystemsModelExtract[tfm,1,1],PlotLayout"Magnitude"],{tfm,{tfm,tfmα,tfmθ}}]

Out[7]=





Source Metadata

Citation: Tewari 2002, Modern Control Design With MATLAB and SIMULINK.

Related Symbols

TransferFunctionModel
SystemsModelExtract
BodePlot

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