Midterm 2 EE 3417Linear Systems, Fall 2015
Instructor: Dan Popa, Ph.D.
Your Name (Print Clearly)
Instructions:
- Exam time: Take-home, 5 days (Nov. 53:30 pm to Nov. 10, 12:30 pm). Late sumission policy: grade drops by 25% every 24 hours overdue.
- There are 5 problems, each requiring handwritten answers, or MATLAB code. Please print a copy of your MATLAB code, and your resulting plots, and attach it with your submission. If you use MATLAB to carry out additional tasks (symbolic calculations, etc), you must attach all such code and its output.
- 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 any materials such as books, notes, web materials, etc., is allowed. Any materials found online or in different books than the textbook should be properly referenced as a source.The use of computer tools such as MATLAB is allowed.
- You are required to work alone, meaning that no discussion amongst yourselves, or with anyone else is allowed (including internet bulletin boards, help from friends, etc). Anything you put on the exam solutions should be well understood by yourself and/or properly cited in your submission.
- Failure to observe this honor code will result in severe penalties according to current UTA policies.You may ask the course instructor for clarification questions via email to .
- Start working on the exam as soon as possible and do not wait until the weekend.
- Midterm grades: retrieve your midterm during office hours according to the following schedule – Last name initial A-H – Tuesday, Nov. 17, 8:30 am -10:30 am, initial I-S, Thursday, Nov. 19, 8:30am -10:30 am., T-Z, Tuesday, Nov. 24, 9:30-10:30 am. If you have a scheduling conflict, please let me know to reschedule you before classes end.
- Your grade for midterm 2 will be set to zero unless the exam is picked up directly from the instructor prior to the end of fall semester.
Problem 1(Filter, 20 pts)
Afilter is described by the following differential equation:
where x(t) represents the input of the filter and y(t) is the output.
1)Find the filter’s transfer function.
2)Determine the zeros and poles of the filter, and plot the pole zero map. Based on this, can you determine the type of filter this is?
3)Sketch the filter’s Bode plot by hand (e.g. just show the asymptotes), then verify the result using MATLAB’s command ‘bode’. Based on this plot, can you determine the type of filter this is?
4)Draw a DFII realization diagram for this filter using gains, summers and integrator elements.
5)An input signal is applied to this filter. Using the bode plot, determine the steady-state output of the filter.
Problem 2(Second Order Filter, 25 pts)
Bandpass filters pass a band of frequencies, while attenuating those outside the band. The simplest of such filters is second order, with the following transfer function:
where is the resonant frequency, and ξ is the damping ratio. In electronics, to implement such a filter we can use variants of the Sallen-Key circuit shown in Figure 1.Here you can use (with proper referencing) any circuit material that offers the transfer function based on capacitor and resistor values for the given Sallen-Key topology.
Figure 1: Band-pass filter circuit.
1)Write Kirchoff’s laws for this circuit to show that the transfer function of this circuit is similar to H(s). Identify the value of the resonant frequency and the damping ratio as a function of resistors and capacitor values in the circuit.
2)Draw the Bode plot first by hand, then verify your results using MATLAB.
3)The value of the circuit’s quality factor Q is defined as . Show that this value determines the height of the resonant peak, by performing analytical calculations to find the maximum amplitude of the response.
4)Using analytical calculations, show that the term detrmines the 3 db bandwidth of the circuit, defined as the frequency band in which the filter gain is up to 41% smaller than the peak value.
5)Find the resonant frequency, the bandwidth, and the maximum value of the circuit gain, and verify these values graphically with the Bode plot.
Problem 3 (Closed-loop control performance, 20 pts)
A robotic arm actuated at the elbow is shown in Figure 2, along with its control system.
1)Find a transfer function Gp(s) so that the steady-state error (e.g the input signal to the controller at t=) for a step input is zero.
2)Using this filter, plot the output for a step input with K=1.
3)Is this control system stable for high gain values K=100? Why?
Figure 2. Control Block diagram for a robot arm.
Problem 4 (Filter Design 10 pts)
1) Design a 4th order high-pass Butterworth filter with a High Frequency gain of 10 DB and a cutoff frequency of 10 rad/s.
2) Using MATLAB, plot the frequency response of this filter, and the location of the filter poles.
Problem 5 (LTIC Response 25 pts)
The I/O relationship of a LTIC with input x(t) and output y(t) is given by:
The input is a sine wave x(t)=2sin(2t)u(t), and the initial conditions on the output and its first two derivatives at 0- are 1, 0, 0, respectively.
1) Find the transfer function, poles, zeros and modes of the system.
2) Find the zero input and zero state response of the system using Frequency domain methods.
3) Find the transient and steady-state responses of the system using FD methods.
4) Find the forced and natural responses of the system using FD methods.
5) Find the initial conditions which will result in a zero transient response for the system.
6) Plot the total response y(t) of the system using MATLAB.
7) Find the 0+ conditions for the system’s step response and impulse responses, and verify them using MATLAB.
8) Find and draw the controllable canonical form realization of the system. Specify the matrices A, B, C, D for this realization.