MECH 431 Midterm, Spring 2012

Midterm in Control Engineering MECH 435 Spring 2012 Duration: 90 minutes Family name: First name: Student Id: achieved points max. points P1 20 P2 20 P3 20 Total 60 MECH 435 Midterm Spring-2012 / Name/ID: page 2 1 20 Pts (33%) Given is a DC motor control system with cascaded position and velocity feedback: Figure 1: DC motor control block diagram The following parameters are given: , a) What is the steady state error of the system to a step input? b) Determine the velocity error constant. What is the condition on and to ensure a steady state error of of the closed loop to a unit ramp input? c) Determine and such that the settling time is around 4 seconds (for the 2% criterion) and the steady state error is . d) What effect would it have, if was negative? ¡ Km ¿ms + 1 Kv ¡ Kp µr µ 1 s _µ Kp Kv ¿m = 0:5 Km = 1 Kv Kp ess = 0:01 ess = 0:01 Kv MECH 435 Midterm Spring-2012 / Name/ID: page 3 2 20 Pts (33%) Given the plant: Construct the root-locus while following the steps below: a) Locate poles and zeros in the complex plane b) Determine the part of the real axis included in the root-locus c) Construct the asymptotes (if any) and determine their angles and their intersection with the real axis d) Determine the break-away/break-in points (if any) e) Determine the angles of departure/arrival (if any) f) Determine the gain K above which the system becomes unstable (if applicable) g) Determine the intersection of the root-locus with the imaginary axis (if any) G(s) = s2 ¡ s + 1 s(s + 1) MECH 435 Midterm Spring-2012 / Name/ID: page 4 3 20 Pts (33%) In this exercise we would like to control the plant using the controller and a unity feedback. The setup is as shown in Figure 2 below: Figure 2: Control Scheme of the plant G(s) The transfer function of the plant is given by: The root locus of the plant is given in Figure 3. We would like to design a controller that achieves a settling time seconds and a damping ratio . a) Based on the root locus, show that it is not possible to satisfy the requirements stated above using only a gain in the controller, i.e., Figure 3: Root Locus of the plant G(s) G(s) Gc(s) G(s) G(s) = s + 2 s(s + 1)(s + 3) G(s) Ts = 2 » = p (2)=2 Gc(s) = K MECH 435 Midterm Spring-2012 / Name/ID: page 5 b) Design a lead controller that achieves the requirements stated above c) Design a PD controller that achieves these same specifications