7.9 MATLAB Experiment on Continuous-Time Systems
Purpose: In this experiment, we analyze time responses of a real physical system using
MATLAB. We study impulse, step, and sinusoidal responses of the yaw rate dynamics
under the influence of the rudder for a commercial aircraft, and draw some useful
conclusions about the aircraft’s dynamic behavior. In addition, by examining the location
of the aircraft eigenvalues and poles we make conclusions about its internal and BIBO
stability. By performing this experiment, students will realize how simple and easy it is
to analyze higher-order linear continuous-time dynamic systems using MATLAB.
A mathematical model that describes the lateral dynamics of a commercial aircraft
is given by a fourth-order differential equation [16]. Using the numerical data from [16],
the corresponding differential equation is given by
!"$#%& $'
(*)
+ ,'!-
)
+ #.,/-
)
""0/-
1%&
)
+ !.,2-3
where
4
is the yaw rate and
-35
stands for the changes in the rudder.
Part 1. Using the MATLAB function impulse, find the impulse response, that is,
observe the yaw rate changes due to an impulse delta signal disturbance acting on the
rudder,
-35
(76
. Plot the impulse time response in the range
98;: <#>=
seconds.
Find the impulse response analytically by using the methodology from Section 7.4. Plot
the obtained analytical result using MATLAB, and compare it with the obtained simulation
result (using the MATLAB function impulse).
Part 2. Find the step response using the MATLAB function step. Plot the step
response during the first 200 seconds. Comment on the physical meaning of the obtained
results. Do you expect that the aircraft moves to the right when the rudder moves to
the left and vice versa? Check the obtained steady state value for the yaw rate by using
formulas (4.39–40).
Part 3. Assume that the rudder is under wind disturbances that can be approximated
by a sinusoidal function,
-
(@?ACB
for the first 200 seconds. Find and plot the aircraft
yaw rate dynamics during that time interval. Estimate the maximal yaw rate change due
to a sinusoidal disturbance whose maximal magnitude is equal to 1. (Hint: Use the
MATLAB function lsim(num,den,f,t) with t=0:0.1:200 and f=sin(t).)
Part 4. Examine the aircraft’s internal stability by finding its eigenvalues. Use
the MATLAB function roots. Find the aircraft’s transfer function zeros and poles and
check whether common factors of the transfer function numerator and denominator can
be cancelled. Comment on the aircraft’s bounded-input BIBO stability.
Comment: Examining system stability should be the first task in any analysis of a
linear system. If the system is found to be unstable, the system should not be subjected
to any input; instead, stabilization techniques should be first applied (for example, by
introducing a feedback loop as presented in Section 4.4), and then the system can be
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