OBJECTIVES:  

To design a compensator for the given system such that the closed-loop system satisfies a set of frequency-domain performance objectives;

To become familiar with certain characteristics of non-minimum-phase systems.

OVERVIEW:

The system to be controlled is given by the following transfer function. Note that the open-loop zero is in the right-half plane (RHP). The RHP zero is referred to as a "non-minimum-phase" zero. Why?

The plant will be in the normal unity-feedback, closed-loop configuration, with the compensator G_c(s) at the output of the summing junction forming the error signal. A disturbance input appears between the compensator output and the plant input. The block diagram below shows the configuration.

TASKS:

The compensated system must satisfy the following specifications:

• the phase margin must be at least 45 degrees;

• the gain crossover frequency must be approximately 1 r/s; and

• the velocity error constant K_v must be 30.

Design a compensator which accomplishes this. You may use MATLAB or SIMULINK to initially evaluate the design. SIMULINK is required for the last task in the unit.

Assuming that the disturbance D(s) = 0, plot the uncompensated (G_c(s) = 1) closed-loop step and ramp responses. Also plot the compensated closed-loop step and ramp responses (using your compensator design) and compare them with the uncompensated responses. What major changes are there between the uncompensated and compensated responses and how did those changes come about? Be specific. Simulations of the uncompensated system should be run for 10 - 20 seconds, and for the compensated system, the simulations should be run for 50 - 60 seconds. 

Determine the closed-loop step response overshoot and settling time for both the compensated and uncompensated systems. Are there any "strange" characteristics in the step responses? Give your opinion on their origin. Do you see any similarity between any part of the step response and steering a bicycle around a sharp corner?

Let the reference input be zero and the disturbance input be a unit step function. Determine and plot the output response of the compensated system due to the disturbance. What is the steady-state response to the disturbance? Why is this the case? What would have been the steady-state response of the uncompensated system for a step disturbance input?

Compute and plot the closed-loop frequency response magnitudes and phases for both the compensated and uncompensated systems, assuming that D(s) = 0. Define the closed-loop bandwidth to be the frequency where the magnitude = -3 db. Determine the closed-loop bandwidths for the two systems. How do the settling times of the step responses compare with the bandwidths? 

If a pure sinusoidal input signal, with frequency equal to the closed-loop bandwith, was applied at the reference input of the compensated system, with D(s) = 0, what would be the output signal after transient terms had decayed? What does this say about the system's ability to accurately follow a sinusoidal signal at that frequency?

Is your closed-loop system robustly stable to changes in the compensator gain? How do you justify your answer? How much gain change will the system tolerate without becoming unstable? If a pure time delay is placed anywhere inside the loop, what is the maximum value that the delay can have without losing closed-loop stability?

Develop a SIMULINK model for the block diagram above, including both input sources. Simulate the system for 150 seconds with a fixed step size of 0.05 seconds or smaller. Let the reference signal be a unit step applied at t = 0, and let the disturbance signal be a unit step applied at t = 20 seconds. Store values for the output c(t), the error signal e(t), and the control signal u(t). Plot the three signals vs. time, and comment on the final values of the three signals. Are they reasonable based on the given system and specifications? Why?

REPORT:

Write a report which documents your design process.  Include the necessary frequency-domain and time-domain plots to illustrate and justify your work. Provide answers to all the questions posed in the various tasks.