hw3_revB

hw3_revB - ECE194D Homework1 Spring2011 Homework 1 (Due...

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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 Ziegler-Nichols guidelines. In Lecture 5, we reviewed two methods for obtaining an initial set of gains for a proportional-integral-derivation (PID) controller, based only on a set response for the system. The resulting closed-loop system typically requires further fine-tuning; 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 straight-forward 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൅ ൅߬ ݏቁ ൌ ܭ ൅ ቀ ቁ· ൅ ሺܭ߬ ሻݏ Figure 3.1a – Open-loop system step response. Position (top) and velocity (bottom) are both shown. Ziegler-Nichols first method. This method simply requires estimating two values from a unit step response for the open-loop system. To do so, we first draw a tangent line at the inflection point in the S-shaped unit step response for the open-loop 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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ECE 194D 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 , ߬ 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 ൅ 200ݏ ൅ 5݁4ሻ ሺݏ ൅ 500ሻሺݏ ൅ 10ݏ ൅ 125ሻሺݏ ൅ 160ݏ ൅ 8000ሻ i) Simulate and submit a plot of the step response of the closed-loop system, using the nominal PID gains. What is the percent overshoot, approximately?
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hw3_revB - ECE194D Homework1 Spring2011 Homework 1 (Due...

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