4/29/24, 3:18 PMexam3s2024: AER E 331 Section 1 (Spring 2024)
https://canvas.iastate.edu/courses/106445/quizzes/5070611/11
exam3s2024
Due Apr 26 at 11:59pm
Points 100
Questions 20
Available Apr 26 at 5pm - May 3 at 11:59pm
Time Limit 120 Minutes
Instructions
Attempt History
AttemptTimeScore
LATESTAttempt 1107 minutes45 out of 100 *
* Some questions not yet graded
Score for this quiz: 45 out of 100 *
* Some questions not yet graded
Submitted Apr 26 at 7:03pm
This attempt took 107 minutes.
Question 1
5 / 5 pts
Correct!
5
Correct Answer
5
This is a 2hr timed exam. Please allocate a time for it that will maintain a secure connection
throughout. Give answers to at least 3-4 decimal places, unless otherwise stated. This exam will
be available from 5pm to 11:59pm on Friday 4/26. You will have only ONE attempt to complete the
exam. Finally, this is not a group exam. Out of respect for your peers and me, please carry it out
ALONE.
Note: A partially completed Matlab code is included at the end of this exam. Should you wish to
use it, copy/paste it into a Matlab script file.
Fro the dynamical system x' = A*x + B*u with A = { 2 , -5 ; 5 -7 } find the coefficient of the s-term for the
associated characteristic polynomial p(s) = det(s*I - A).
4/29/24, 3:18 PMexam3s2024: AER E 331 Section 1 (Spring 2024)
https://canvas.iastate.edu/courses/106445/quizzes/5070612/11
Question 2
5 / 5 pts
Correct!
0.189
Correct Answer
0.189 margin of error +/- 5%
Question 3
5 / 5 pts
Correct!
-4.2
Correct Answer
-4.2 margin of error +/- 2%
Question 4
5 / 5 pts
Correct!
4.5
Correct Answer
4.5 margin of error +/- 2%
Question 5
5 / 5 pts
For the underdamped dynamical system x' = A*x + B*u with A = [ -1 , 9 ; -3 -1 } , compute damping
ratio .
Consider the plant dynamics described by y'' + 4.2*y' +40.4*y = f. Define the state variable x whose
components are x=y' , x=y , and x=Integral(y). Find the sum of the eigenvalues of A in the state
equation x'=Ax+Bu .
123
Consider x' = Ax + Bu and y=Cx where A=[ -1 -7 -2 ; 1 0 0 ; 0 1 0 ]; B=[1;0;0]; , C=[ 0 1 9 ]; D=0.
This is called the controller canonical form for a single-input/single-output transfer function.. Assuming
it is stable, compute the static gain.
Consider the plant TF Gp = 100/((s^2+2*s+100)*(s+1)); Use the 'tf2ss' command to obtain the
[A,B,C,D] matrices. Then use the 'place' command to find the control matrix K that will place closed loop
4/29/24, 3:18 PMexam3s2024: AER E 331 Section 1 (Spring 2024)
https://canvas.iastate.edu/courses/106445/quizzes/5070613/11
Correct!
-63
-63 (with margin: 2)
Question 6
0 / 5 pts
You Answered
0
Between 2.7 and 3.2
Question 7
0 / 5 pts
division by pi
division by 2*pi
You Answered
multiplication by 2*pi
Correct Answer
multiplication by pi
Nothing. They are both the same
poles at p=[-2+1i*2 ; -2-1i*2 ; -10]; Finally, give the sum of the elements of K.
The angular position/torque transfer function of a given satellite is Gp = 1/(10*s^2); [ = (s)/T(s) ] .
Define the state as x = [ x ; x ]= [ ' ; ]. Then A = [0 -0.1 ;1 0]; and B=[0.1;0]; Use the 'place'
command to place closed loop poles at p=[ -1 ; -2]. Note that resulting controller is a PD controller. In
contrast to the root locus method that chooses a K-value for only one CL pole, this method has placed
both. Place this controller in the forward loop, and obtain the CL unit step response. Then use the data
tap to estimate the the time at which the response is 5% above the steady state value.
12
The turbulence psd expressions given in (6.52-6.54) are 1-sided, and they are based on including the
factor 1/2 in the forward Fourier transform. To modify them so that they are consistent with the
definition of a Laplace transform (i.e. 2-sided with the 1/2 in the inverse transform) requires
4/29/24, 3:18 PMexam3s2024: AER E 331 Section 1 (Spring 2024)
https://canvas.iastate.edu/courses/106445/quizzes/5070614/11
Question 8
5 / 5 pts
Correct!
2.3
Correct Answer
2.3 margin of error +/- 2%
Question 9
0 / 5 pts
You Answered
10.0278
Correct Answer
18.0278 margin of error +/- 5%
Question 10
5 / 5 pts
Correct!
24
Correct Answer
24 margin of error +/- 2%
Question 11
0 / 5 pts
Suppose that a random process x(t) has the psd S(w)=2*2.3/{2.3^2 + w) } = |G(iw)|. The -3dB
frequency w is the frequency at which S(w)=0.5*S(0). Find the value for w.
22
000
A certain Mil-Spec requires the simulation of turbulence excitation of the short period mode of a given
plane. The short period transfer function is G= 1/(s^2 + 0.5*s +25). The turbulence psd that is the input
to G is specified as S()=c(/1+). It is required that the output from G , with psd S() , have a
value of 2 at =5. Find the required value for c. [Hint: The response psd is S(w)=|G(iw)| S(w).]
sp
spw
22
spR
Rsp
2
w
Consider the simplistic gust model w(t) = 8 for 0 the Laplace transform of any x(t). Then, for x(t)=w(t) carry out the integral to compute the value for W(s=i0). 4/29/24, 3:18 PMexam3s2024: AER E 331 Section 1 (Spring 2024) https://canvas.iastate.edu/courses/106445/quizzes/5070615/11 You Answered 44.2842 Correct Answer 39.76 margin of error +/- 5% Question 12 0 / 5 pts You Answered 0 Correct Answer 1 Question 13 0 / 5 pts You Answered 10.0586 Consider the real-world approximation of the unit impulse given by d(t) = 1/ for 0 Laplace transform of d(t) is D = (1-exp(-s))/(s*) . For = 0.07 find the -3.01dB bandwidth associated with D(i). [Hint: This can be easily achieved using a Bode plot and a data tap.] From entry #12 of Table 8.1 on p.602 , we have the following Laplace/Z Transform pair: . In the s-domain the Final Value Theorem in the s-domain (assuming a final value exists) is . In the z-domain it is . For a=6 and T=2.0 compute the final value for y(kT). From entry #8 of Table 8.1 on p.602 , we have the following Laplace/Z Transform pair: . These can be viewed either as transforms of x(t) and x(kT), or they can be viewed as transfer functions G(s)=Y(s)/X(s) and G(z)=Y(z)/X(z). In this problem view them as transfer functions. It should be clear that the static gain of G(s) is G(0)=1. For a=9.4 and T=0.29 use a Bode plot to obtain the static gain for G(z). Do NOT give your answer in dB. NOTE: Matlab incorporates a factor of T [as it should for a fair comparison to G(s)]. 4/29/24, 3:18 PMexam3s2024: AER E 331 Section 1 (Spring 2024) https://canvas.iastate.edu/courses/106445/quizzes/5070616/11 Correct Answer 2.917 margin of error +/- 5% Question 14 5 / 5 pts Correct! 2.4992 Correct Answer 2.5 margin of error +/- 5% Question 15 0 / 5 pts You Answered 0.1423 0.753 (with margin: 0.01) Question 16 0 / 5 pts where . Obtain a recursion equation of the form given by y = cy + ... + dx + dx + ... . Then for a=6 and T=0.8 compute . k1k-1001k-1 This problem and the next address the pitch rate damper block diagram Compute the rate gyro gain needed to achieve dominant poles having a damping ratio zeta=0.707. For a system with transfer function G = 5/(s+5) and Nyquist frequency w=c*a where c=5 use the 'c2d' command with the 'impulse' flag to obtain Gz. From overlaid Bode plots for G and Gz, use the data tap to determine the amount of magnitude aliasing (in dB) that occurs at the Nyquist frequency. N 4/29/24, 3:18 PMexam3s2024: AER E 331 Section 1 (Spring 2024) https://canvas.iastate.edu/courses/106445/quizzes/5070617/11 You Answered 24.4855 Correct Answer 6.3997 margin of error +/- 10% Question 17 0 / 5 pts You Answered 0.0593 Correct Answer 0.0931 margin of error +/- 10% Question 18 5 / 5 pts Correct! 1.3462 Correct Answer 1.3462 margin of error +/- 5% Question 19 0 / 5 pts You Answered 0.0007 A lowpass filter G(s)=1.9/(s+1.9) is to be replaced by a digital filter. Find the value of the sampling period, T, such that the Nyquist frequency will be 25dB below the filter static gain. An analog dynamical system is to be converted to a digital one by using a backward difference approximation of For a=7 b=10, and T=0.6 compute the sum that is associated with the recursion . Consider a PD controller of the form . Even though Gc(z;tustin) is excellent at matching the phase of Gc(s), it performs poorly in matching the magnitude at higher frequencies. Define . For a=8 and 0.07 overlay Bode plots of Gc(s) and Gc(z;tustin). Then compute the dB difference that is the dB value of the Tustin magnitude minus the dB value for Gc(s) at the frequency where r=0.6. Finally, let K=6. 4/29/24, 3:18 PMexam3s2024: AER E 331 Section 1 (Spring 2024) https://canvas.iastate.edu/courses/106445/quizzes/5070618/11 Correct Answer 3.2583 margin of error +/- 5% Question 20 0 / 5 pts You Answered 0.0015 0.028 (with margin: 0.004) Question 21 0 / 0 pts The plant that uses a controller in a unity feedback configuration is to have the analog controller replaced by a digital controller arrived at by using Tustin's method. The controller sample period is T=0.05 sec. Overlay the analog and digital CL unit step responses. Then use data taps to compute the magnitude of the difference between the responses at the time when analog response is a maximum. %exam3_331_S24_Partial.m s=tf('s'); %Q1: %================================================ %Q2: A=*****; %============================================== %Q3: A=******; %=============================================== %Q4: %============================================== %Q5: Gp = 100/((s^2+2*s+100)*(s+1)); [b,a]=tfdata(Gp,'v'); %Obtain num and den coefficient arrays [A,B,C,D]=tf2ss(b,a); p1=-2+1i*2; p2=conj(p1); p3=-10; %Specified CL poles p=[p1;p2;p3]; K=*****; ANS=sum(K); %============================================== %Q6: 4/29/24, 3:18 PMexam3s2024: AER E 331 Section 1 (Spring 2024) https://canvas.iastate.edu/courses/106445/quizzes/5070619/11 Gp=1/(10*s^2); A=[0 -0.1;1 0]; B=[0.1; 0]; p=[-1;-2]; K=*****; Gc=K(1)*s+K(2); W= *****; figure(6) step(W) grid %=============================================== %Q7: %============================================== %Q8: %============================================== %Q9: Gsp=1/(s^2+0.5*s+25); Gsp_si5=*****; %========================================= %Q10: %============================================== %Q11: For your use if you wish: d=***; G=(1-exp(-d*s))/s; figure(11) bode(G) %============================================== %Q12: %============================================== %Q13: a=*****; T=*****; G=a/(s+a); Gz=c2d(G,T,'***'); figure(13) bode(G,Gz) grid ANSdB=*****; ANS=*****; %============================================= %Q14: 4/29/24, 3:18 PMexam3s2024: AER E 331 Section 1 (Spring 2024) https://canvas.iastate.edu/courses/106445/quizzes/50706110/11 %=========================================== %Q15: G1=-10/(s+10; Gp=-2*(s+0.3)/(s^2+0.65*s+2.15); figure(15) rlocus(***) ANS=***; %========================================== %Q16: a=****; G=a/(s+a); Gz=c2d(G,T,'imp'); figure(15) bode(G,Gz) ANS=***; %========================================= %Q17: %========================================= %Q18: %========================================= %Q19: K=****; a=*****; ra=****; wN=****; T=*****; r=*****; w0=*****; G=K*(s+a); Gz=c2d(G,T,'tus'); figure(19) bode(G) hold on bode(Gz) ANS=*****; %=================================== %Q20: Gp=1/(s^2+s+25); Gc=6.5*(s+0.5); W=feedback(*****); T=0.05; 4/29/24, 3:18 PMexam3s2024: AER E 331 Section 1 (Spring 2024) https://canvas.iastate.edu/courses/106445/quizzes/50706111/11 Your Answer: Quiz Score: 45 out of 100 * Some questions not yet graded Gpz=c2d(Gp,T,*****); Gcz=c2d(Gc,T,'tus'); Wz=feedback(*****); figure(20) step(W) hold on step(Wz) ANS=***** 1
