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EEL 4652C Control Systems 1 (Spring 2024) Homework 1 Study the “EEL 4652C Midterm E xam” given in Fall 202 3. The exam’s questions and their solutions are posted for your convenience in the Canvas module “Spring 2024 Homework Assignments” and in Canvas Assignments. For each of the three problems there are new tasks given in this homework assignment. Each of the problems (and sub-problems) requires some analytical solution and explanations along with MATLAB and/or Simulink verification or analysis. Both parts of each sub-problem are equally weighted. Problem 1: Transient Response In the exam the unit step response of G(s) is shown. G(s) has poles at {s = -1, s = -6.1, s = -2 ± j30, s = -20} and zero at s = -6. The 5 th order G(s) system can be nicely approximated by a first order model G(s) reduced = 1/(s+1). 1.1 Use MATLAB Control Systems Toolbox (CST) to compare the step response of G(s) to G(s) reduced . Show both step responses on the same LTIVIEW screen. [Comment: In more recent versions of MATLAB the ltiview command has been replaced by LinearSystemAnalyzer working exactly the same as ltiview . Since we are interested only in the step response, you may also use the step command.] 1.2 How many time constants does it take for the step response of G(s) to reach 90% of its steady-state value? 1.3 Remove the pole at s = -1 from G(s). Replace it with a complex pole so that: (i) The new complex pole is dominant (that is, the 6 th order G(s) new can be approximated by the new system G(s) new-reduced that has the complex pole that you chose, (ii) Peak Overshoot of the step response is 7% and (iii) The 5% Settling Time of G(s) new is around 4 [sec]. Problem 2: Closed Loop Stability and Tracking In the given unity feedback loop the process is P(s) = 8/s(s+1).
2.1 Demonstrate the argument between Engineer A and B in parts 2.2 and 2.3 of the exam problem. Use CST. 2.2 Demonstrate in Simulink the solution offered by Engineer C (in part 2.4 of the exam). Observe the ramp input r(t), the output y(t) and the error signal e(t). [Observe r(t) and y(t) on the same scale, and observe e(t) on a separate scope]. 2.3 Process P(s) is replaced by an unstable process P new (s) = 8/s(1-s). The specifications are: (i) Stabilize the loop, and (ii) Zero steady-state error to step input. Engineer D claims that P and PI controls are no longer acceptable. A PD controller C(s) = K P + K D s is suggested. Is she right? Demonstrate your analytic solution in Simulink or CST. Problem 3 DC Motor Control: In the exam we considered a permanent magnet DC motor that has the following parameters: 2 { 4.2[ ], 0.04[ ], 0.25[ / ], 0.25[ / / ], 0.002[ ]} a a T E m L R L H K N m A K V rad s J J Kg m = = = = = = Let’s call this Case A. The load inertia and the motor inertia were matched L m J J = and the connection between the two inertias was by direct drive. The electromechanical time constant was much larger than the electrical time constant: m e  . In the following block diagram G(s) stands for the simplified armature voltage to speed transfer function 1 ( ) ( ) 1 E a m s K U s s = + (in speed control problems, either open-loop or closed-loop).
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