Final EE 4314 Control Systems, Spring 2011
Your Name (Print Clearly)
Instructions:
· Exam time: 2:00 pm – 4:30 pm (150 mins).
· You are required to work alone, meaning that no discussion amongst yourselves is allowed.
· There are five problems, each requiring handwritten answers. Please write on the blank sheets provided, and staple your sheets to the exam.
· In order to receive full credit, you must clearly explain your assumptions, reasoning, and results (e.g. justify your answers). You will get more credit for having a slightly incorrect answer while showing a sound understanding of the concepts than for posting the right answer without explaining how you arrived at the result.
· Use of cell phones or calculators of any kind during this exam is not allowed, but you can refer to your 5 page, double-sided cheat sheet and the table provided.
· Time management: while it is up to you to manage your time during the exam, you may consider moving on to the next problem after 30 minutes so that you can work on all the problems in the alloted time.
Problem 1 (20 pts) (General)
For questions a)-d), use diagrams, words, and formulas. The questions are worth 4 pts each.
a) Define the Nyquist frequency of a digital control system and discuss criteria for selecting sampling times that offer good performance.
b) Define Gain and Phase margins of a closed loop system, and draw ilustrative plots to visualize them.
c) Explain the impact of the three PID controller constants to the performance of a closed loop system, and list criteria to select them.
d) Define the exponential of a matrix, discuss methods to calculate it, and ways it is used in characterizing the time-domain evolution of state-space models.
Problem 2 (20 pts) (Dynamic Models)
A bridged-T network is often used in AC control systems as a filter. The network is represented by the circuit in figure 1.
a) Select appropriate states for the system and write the state equations describing this system (5 pts).
b) Write the time-domain system equations into standard steady-state form (5 pts).
c) Find the transfer function between input and output for R1=R2=0.5Ω, C=1F (5 pts).
d) If this system is used in closed-loop with a proportional controller, do you expect it to be stable? Why? (5 pts)
Figure 1: The T-bridge network circuit.
Problem 3 (20 pts) (Closed – loop control of unstable plant)
A jump-jet aircraft has a control system with input r(t) (aircraft desired orientation), and output y(t) (aircraft orientation) described by the block-diagram in figure 2, in which K(s) is a simple lead (or lag) compensator.
Figure 2: Closed loop control system diagram for an aircraft
a) Is the open-loop system stable? Why? Is the closed loop system minimum phase? Why? (4 pts)
b) If p=2, find a necessary and sufficient condition that z must satisfy for stability of the closed loop system. Based on this condition, discuss whether the resulting controller type is lead or lag. (8 pts)
c) Convert the closed-loop system with input r(t) and output y(t) into state-space form (8 pts).
Problem 4 (15 pts) (Digital Control)
Consider a unity gain feedback system for a continuous-time plant G(s), which is implemented using a digital controller.
a) Find the associated Z-transform G(z) for a sampling time T=0.1s (5 pts). During conversion, leave the expressions for exponents as they are (e.g. without calculating them).
b) Find the Tustin’s and MMPZ discrete equivalents of G(s) (6 pts).
c) Write down the difference equation describing the I/O relationship associated to the Z-transforms from part b) (4 pts).
Problem 5 (25 pts) (Nyquist Plot and Root Locus)
Consider a feedback system for a continuous-time plant G(s) which is implemented using a controller K(s). Assume that
, , ,
a) For each of the three controllers K(s)=Ki(s), i=1,2,3, sketch the Nyquist contour, draw the Nyquist plot, and the root locus of the system. Be as detailed as possible (15 pts).
b) Based on the plots, discuss (with justification) the closed loop stability of the three systems (10 pts).