Assignment Aim: The aim of the experiment is to provide experience of controller design based the frequency response of a dynamical system and of state feedback controller design.
Course Learning Outcomes: On successful completion of the assignment, students should be able to:
Obtain open-loop frequency response data from a linear system and, from the data obtained, plot the Bode diagram of the system;
Determine a transfer function model of a linear system from its Bode diagram using asymptotic approximation;
Determine the gain and phase margins of a linear system from its Bode diagram and the system bandwidth from the Bode diagram;
Carry out the design of a phase-lead compensator for a system;
Obtain a state-space model of an open-loop system from its transfer function;
Design a state-feedback controller for a system;
Do a comparative evaluation of phase-lead control with observed state feedback control.
Laboratory Based Controller Design Assignment Tasks -
Task 1: First-Order Open-Loop Frequency Response
a) From your knowledge of K and T, and the background theory, plot the asymptotic Bode diagram of the first-order open-loop system with input θi and output ωo on linear-log paper.
b) Plot the experimental Bode diagram of the first-order open-loop system using the data of the open-loop frequency response test on linear-log paper. Hence estimate the value of the system gain K and the system time constant T.
Task 2: Second-Order Open-Loop Frequency Response
a) Determine the Bode diagram of the second-order open-loop system with angular position output using the data of the closed-loop frequency response test.
b) Determine the phase crossover frequency, phase margin and estimate the value of the damping ratio.
Task 3: Step Response of the Second-Order System
a) Using the second-order open-loop transfer function obtained in Task 2a) a model of the unity negative feedback system can be obtained and its step response simulated using MATLAB.
b) Use the simulated step response of the closed-loop system in Task 3a) to justify the need for a controller for the system and an appropriate type of controller.
Task 4: Design via Frequency Response
a) Design a phase-lead compensator for the complete second-order open-loop system with angular position output.
b) Use MATLAB to simulate the step response of the compensated system and verify your design by showing it meets the specifications. Give your evaluation of your compensator design.
Task 5: Design via State Space
a) Derive a state space model of the complete second-order open-loop system with angular position output. Use this model to design a state feedback controller for the system. Explain how the design could be implemented using tacho feedback.
b) Assuming tacho feedback is not available, derive the compensator for observed state feedback using the state feedback controller and evaluate the compensator using MATLAB. Give your comparative evaluation of your observed state feedback design with your phase lead compensator design.
Task 6: Conclusions
Given your conclusions with respect to your designed controllers and any insights gained from carrying out the assignment.
NOTE: all frequency response plotting should be done by hand. You may also confirm your hand-plotted results using the exported plots using Discovery software.
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