HW3 solns 2020 - Homework 3 solutions PDF

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2020 Fall BME 228/428: Homework #3 Solutions 

1. For each of the transfer functions shown below, find the locations of the poles and zeros, plot them on the s-plane, and then write an expression for the general form of the step response without solving for the inverse Laplace transform. State the nature of each response (overdamped, underdamped, and so on). (a) jw

Pole: -2; c(t) = (A + Be-2t )u(t); first-order response

 -2

(b) jw

Poles: -3, -6; c(t) = (A + Be -3t + Ce-6t)u(t); over-damped response

 -6

-3

(c) jw

Poles: -10, -20; Zero: -7; c(t) = (A + Be-10t + Ce-20t)u(t); over-damped response

-7



-20 -10

(d) jw

Poles: (-3+j315 ), (-3-j315 ); c(t) = [A + e-3t (Bcos(315t) +Csin(315t)]u(t); under-damped

11.6

 -3 -11 .6 6

(e) jw

Poles: j3, -j3; Zero: -2; c(t) = [A + Bcos (3t)+Csin(3t)]u(t); un-damped

3

 -2

-3

(f) jw

Poles: -10, -10; Zero: -5; c(t) = (A + Be-10t + Cte-10t)u(t); critically damped

-10 -5



2. Derive the relationship for damping ratio as a function of percent overshoot.

3. For each of the second-order systems that follow, find n, Ts, Tp and %OS. (a)

(b)

(c)

4.

For each pair of second-order system specifications that follow, find the location of the second-order pair of poles. (a)

(b)

(c)

5.

For the system shown below, do the following: (a)

Find the transfer function G(s)=X(s)/F(s).

(b)

Find n, Ts, Tp and %OS.

6. Industrial robots are used for myriad applications; for example, to move heavy bags from one location to another. Assume a model for the open-loop swivel controller and plant of

where is the Laplace transform of the robot's output swivel velocity and Vi(s) is the voltage applied to the controller. Evaluate percent overshoot, settling time, and peak time of the response of the open-loop swivel velocity to a step-voltage input. Justify all assumptions.

To justify a second-order assumption, the effects of extra poles and zeroes must be small. Here, we see that the complex poles are at -2j2.449, and the third pole is at -10 or 5X times the real part further out from the origin. Thus, the time domain response of the extra pole will settle/decay much faster and the second-order approximation is valid. BME 448. Effective control of insulin injections can result in better lives for diabetic persons. Automatically controlled insulin injection by means of a pump and a sensor that measures blood sugar can be very effective. A pump and injection system has a feedback control as shown, where Y(s)/R(s) = K(s+2)/[s2+(K+1)s+2K]. Calculate the suitable gain (K), so that the overshoot of the step response due to the drug injection is approximately 7%; explain your answer.

Y(s)/R(s) = K(s+2) / [s2+(K+1)s+2K] Therefore, Wn=2K and 2Wn=K+1, hence  = (K+1)/22K  2 = (K+1)2/8K Using the equation that relates  and %OS (prob 2), we find that for a 7% OS,  = 0.65. Plugging in: 0.652 = (K+1)2/8K  K2-1.38K+1=0  K=0.690.72j The required value of K is a complex number, but this is not possible for a realizable system (whose transfer function coefficients must be real). Note that the zero in the transfer function is not much further out than the real part of the complex poles for any value of K, thus the second-order approximation is not valid. The value of K to yield the lowest %OS would have to determined empirically from plots of y(t) as function of K....