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APR_E5.txt
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APR_E5.txt
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TECHNISCHE UNIVERSITAT MUNCHEN
LEHRSTUHL FUR INFORMATIONSTECHNISCHE REGELUNG
ORDINARIA: UNIV.-PROF. DR.-ING. SANDRA HIRCHE
Adaptive and Predictive Control (APC)
Dr. Stefan Sosnowski / Denis Cehaji´
SS2014
26.05.2014.
EXERCISE 5
Problem 1: SPR Functions
a) State the three conditions that a strictly positive real (SPR) function has to fulfill.
b) Which transfer functions are SPR?
• G1 (s) = (s+a)(s+b) with a, b, c, d > 0
(s+c)(s+d)
• G2 (s) = e−sT s+a with a, T > 0
• G3 (s) = (s+a)(s+b) with a, b > 0
• G4 (s) =
(s+b)(s+c)
with a, b, c > 0
Problem 2: Gain Scheduling in Ship Steering
In ship control often autopilots are applied. Assume that the ship dynamics, as an approximation, can
be reduced to the motion in the horizontal plane. Then the control input is the rudder angle δ that
has to be chosen appropriately to achieve a desired heading ψ. A common linear model for the ship
dynamics is the Nomoto model, relating heading ψ and rudder angle δ:
Gp (s) =
ψ(s)
δ(s)
s(s + a)
with
b = b0 u 2
a = a0 u.
The parameters a and b of the model thus depend on the ships velocity u. Further a0 and b0 are
constants.
a) What are possible reasons that make adaptive control necessary in ship control?
b) Use the ideal PD control law
Gr (s) = K(1 + sTd )
and determine the damping D for a nominal velocity u = unom . How does damping change for
velocities different from unom ? Use anom = −0.3, bnom = 0.8, Knom = 2.5, and Td,nom = 0.86,
which realizes Dnom = 0.5.
c) Deduce specifications for a gain scheduling controller, if the damping D is desired to be constant
for all velocities u. What does this mean in practice?