ECE 194D
Homework 1
Spring 2011
Prof. Katie Byl
Page 1 of 8
Due 5/17/2011
Homework 1
(Due Tuesday, May 17, at 5pm)
1) PID controller tuning via ZieglerNichols guidelines.
In Lecture 5, we reviewed two
methods for obtaining an initial set of gains for a proportionalintegralderivation (PID)
controller, based only on a set response for the system.
The resulting closedloop system
typically requires further finetuning; however, there are often advantages in finding approximate
control gains rapidly. In particular, this method can be useful when a model for the plant does
not exist, and/or when further system ID (to obtain a frequency response plot) requires the
implementation of a “good” controller.
Although this problem looks long, it should be relatively straightforward to complete rapidly.
Parts a and b each use the “first method” of Ziegler and Nichols, which they proposed in a paper
back in 1942.
(Wow, that’s old!)
Part c uses a “second method”. In all cases, we will assume the
following form for the PID controller:
ܥሺݏሻൌ ܭቀ1
ଵ
ఛ
௦
߬
ௗ
ݏቁ ൌ ܭ ቀ
ఛ
ቁ·
ଵ
௦
ሺܭ߬
ௗ
ሻݏ
Figure 3.1a – Openloop system step response. Position (top) and velocity (bottom) are both shown.
ZieglerNichols first method.
This method simply requires estimating two values from a unit
step response for the openloop system.
To do so, we first draw a tangent line at the inflection
point in the Sshaped unit step response for the openloop system, as shown by the dashed line in
Figure 3.1a. Note that the inflection point is the location with the peak slope (i.e., highest
velocity). Next, estimate the following time parameters: 1) The location, “L”, in which a tangent
line drawn at the inflection point intersects the time axis. 2) The time between the intersection of
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Homework 1
Spring 2011
Prof. Katie Byl
Page 2 of 8
Due 5/17/2011
the tangent line with y=0 and with y=1, which we will call “T”.
Note also that T=1/R (with units
adjusted appropriately), where R is the peak velocity (i.e., the velocity at the inflection point).
Once L and T have been estimated (this has been done for you in Figure 3.1a…), the suggested
gains for the PID controller are then:
ܭൌ1
.2
்
ൌ
ଵ.ଶ
ோ
,
߬
ൌ2ܮ
,
߬
ௗ
ൌ
ଶ
a)
For part a, L and T are clearly labeled in Figure 3.1a.
For simulation purposes, assume that
the open loop plant for Figure 3.1a is given by the following transfer function:
ܩ
ሺݏሻ ൌ
1݁4ሺݏ
ଶ
200ݏ 5݁4ሻ
ሺݏ 500ሻሺݏ
ଶ
10ݏ 125ሻሺݏ
ଶ
160ݏ 8000ሻ
i) Simulate and submit a plot of the step response of the closedloop system, using the nominal
PID gains. What is the percent overshoot, approximately?
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 Fall '09
 Prof. Katie Byl

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