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final01 - ECE 147A FEEDBACK CONTROL SYSTEMS THEORY AND...

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Unformatted text preview: ECE 147A FEEDBACK CONTROL SYSTEMS - THEORY AND DESIGN F01 Final Exam Closed book and notes. No calculators. Show all work. Name: Problem 1: / 20 Problem 2: / 20 Problem 3: / 20 Problem 4: / 20 Problem 5: / 20 Total: /100 Name: Problem 1 A system has monic numerator and denominator polynomials with the roots shown. Find a controller to stabilize the system. Draw the root locus associated with adjusting the gain of your controller. Be precise about locations where branches meet and where branches cross the imaginary axis. For what range of gains is the closed-loop stable? Name: Problem 2 Give short answers to the following questions: 1. For the nonlinear system é = —6l — :13 + sin(0):1:3 + cos(6)sat(u) 515 = —0.l:'r + sin(0) + sat(u) y = a: what would be the degree of the denominator polynomial in the transfer function corresponding to the linearization around the zero operating point? 2. For a plant that is at least relative degree two and has no poles with positive real part, what is the statement of the Bode sensitivity integral? What are the ramifications for disturbance attenuation? 3. What is the Bode gain/ phase relationship rule of thumb and when does it apply? 4. For the unity negative feedback configuration with prefilter write the output y in terms of the reference r, the disturbance d and the noise n in the Laplace domain. 5. Approximate, in decibels, the velocity constant for a system with the loop gain frequency response shown below. 120 100 1o" 10" to‘ to“ 10‘ Name: 1 Problem 3 For the plant m a certain controller is able to place the closed-loop poles at the roots of the equation 34 + 10.33 + K152 + K23 + K3 = 0 where K1, K2 and K3 are arbitrary. What are the requirements on the values K1, K2 and K3 to guarantee that the closed-loop is stable? Give specific values for K1, K2 and K3 that make the closed-loop stable, if such values exist. Name: Problem 4 Suppose the plant to control is modeled by a single integrator. The controller to be used is a simple proportional control K. There is a time-delay of 1 second between the controller and the plant. What is the range of proportional gains for which the closed—loop (negative unity feedback configuration) is stable? Explain your answer in terms of a Nyquist diagram. (Hint: the transfer function for a time delay of 6 seconds evaluated on the you axis is 6‘3“"5 where the units for w is radians per second.) Name: Problem 5 A certain undamped pendulum has, for small excursions about the down position, the frequency response shown below. Design a controller for this system so that: Bode Diagram 60 40 N O Magnitude (dB) Phase (deg) | (D O —135 Frequency (rad/sec) 1. additive constant disturbances at the input to the plant have no effect on the steady-state output; 2. the magnitude of the closed-loop sensitivity is g 0.1 for all w E [vacozl rad/sec where wco, represents the open—loop crossover frequency; 3. the phase margin of the closed—loop is Z 45 degrees. I will remind you that the formulas 1 T—x/E l — sin(¢mam) 1 + sin(¢mm) wmaw a may have some meaning here. Estimate the peak steady-state magnitude of the closed-loop output when the input disturbance is 0.18in(t). Explain your reasoning. ...
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