Bill Goodwine's Courses

You must do all the computations for these problems by hand. You may verify your answers using matlab.

1. Consider
   • 
   1. Plot the root locus plot for G(s).
   2. Consider this system under unity feedback (Figure 9.97 of the course text) for proportional control, which would mean C(s)=k. Using the root locus plot, determine the range of values of k for which the system would have less than 10% overshoot.
   3. Use either simulink or the step() command to verify your answer for the previous part. Use at least two different k values, one very near the limit that should have just about exactly 10% overshoot, and the other that is well less than 10% overshoot.
   4. For the two k values you selected in the previous part, where are the poles for the closed loop system located in the complex plane? You may use either the quadratic formula, or the matlab function pzmap() to determine this. For those pole locations, what are the corresponding rise times and settling times? Are your computations consistent with the step responses you plotted? For this and all the other rise time problems, use the approximation that the rise time is 1.8 divided by the natural frequency.
   5. Sketch the region in the complex plane where the closed loop poles need to be to have a 3% settling time less than 1.5 seconds.
   6. Sketch the region in the complex plane where the closed loop poles need to be to have a rise time less than 0.5 seconds.
   7. Design a lead compensator by hand which has
      • overshoot less than 10%;
      • rise time less than 0.5 seconds; and,
      • 3% settling time less than 1.5 seconds.
      Verify your design using either the step() command or simulink.
   8. Design a lag compensator that satisfies the transient response specifications listed above, but that reduces the steady-state error by a factor of 5.
2. Consider
   • 
   1. Plot the root locus plot for G(s).
   2. Consider this system under unity feedback (Figure 9.97 of the course text) for proportional control, which would mean C(s)=k. Using the root locus plot, determine the range of values of k for which the system would have less than 20% overshoot.
   3. Use either simulink or the step() command to verify your answer for the previous part. Use at least two different k values, one very near the limit that should have just about exactly 10% overshoot, and the other that is well less than 20% overshoot.
   4. For the two k values you selected in the previous part, where are the poles for the closed loop system located in the complex plane? You may use either the quadratic formula, or the matlab function pzmap() to determine this. For those pole locations, what are the corresponding rise times and settling times? Are your computations consistent with the step responses you plotted? For this and all the other rise time problems, use the approximation that the rise time is 1.8 divided by the natural frequency.
   5. Sketch the region in the complex plane where the closed loop poles need to be to have a 2% settling time less than 2.6 seconds.
   6. Sketch the region in the complex plane where the closed loop poles need to be to have a rise time less than 1 second.
   7. Design a lead compensator by hand which has
      • overshoot less than 20%;
      • rise time less than 1 seconds; and,
      • 2% settling time less than 2.6 seconds.
      Verify your design using either the step() command or simulink.
   8. Design a lag compensator that satisfies the transient response specifications listed above, but that reduces the steady-state error by a factor of 3.
3. Consider
   • 
   1. Plot the root locus plot for G(s).
   2. Consider this system under unity feedback (Figure 9.97 of the course text) for proportional control, which would mean C(s)=k. Using the root locus plot, determine the range of values of k for which the closed loop system would be stable.
   3. Use either simulink or the step() command to verify your answer for the previous part. Use at least two different k values, one very near the stability limit and one that is well within the stability margins.
   4. For the two k values you selected in the previous part, where are the poles for the closed loop system located in the complex plane? You may use either the quadratic formula, or the matlab function pzmap() to determine this. For those pole locations, what are the corresponding rise times and settling times? Are your computations consistent with the step responses you plotted? For this and all the other rise time problems, use the approximation that the rise time is 1.8 divided by the natural frequency.
   5. Sketch the region in the complex plane where the closed loop poles need to be to have a rise time less than 0.3 seconds.
   6. Design a lead compensator by hand which is stable and has a rise time less than 0.3 seconds. Don't go crazy with it, but try to make the settling times and overshoot be reasonable too.
      Verify your design using either the step() command or simulink.
   7. Design a lag compensator that satisfies the transient response specifications listed above, but that reduces the steady-state error by a factor of 10.
4. Consider
   • 
   1. Plot the root locus plot for G(s).
   2. Consider this system under unity feedback (Figure 9.97 of the course text) for proportional control, which would mean C(s)=k. Using the root locus plot, determine the range of values of k for which the closed loop system would be stable.
   3. Use either simulink or the step() command to verify your answer for the previous part. Use at least three different k values, one very near the stability limit and one that is well within the stability margins and one that is unstable.
   4. For the two stable k values you selected in the previous part, where are the poles for the closed loop system located in the complex plane? You may use either the quadratic formula, or the matlab function pzmap() to determine this. For those pole locations, what are the corresponding rise times and settling times? Are your computations consistent with the step responses you plotted? For this and all the other rise time problems, use the approximation that the rise time is 1.8 divided by the natural frequency.
   5. Design a lead compensator that makes the root locus for this system go through the point s=-4+4i. What will be the overshoot, rise time and settling time for the system? Verify it as above.
   6. Design a lag compensator that satisfies the transient response specifications listed above, but that reduces the steady-state error by a factor of 2.

When designing a lead compensator, we're trying to pull the root locus through a point that we specify, since we know that the chosen point will satisfy the given specifications (overshoot, rise time, etc.). Do you want us to pull the root locus through a point near the edge of the region that satisfies the conditions or can we just pick any point, well into the valid region?

AL089 wrote:When designing a lead compensator, we're trying to pull the root locus through a point that we specify, since we know that the chosen point will satisfy the given specifications (overshoot, rise time, etc.). Do you want us to pull the root locus through a point near the edge of the region that satisfies the conditions or can we just pick any point, well into the valid region?

Going somewhat into the valid region is best. It isn't an issue when doing fake problems on matlab, but if you go too far that would require larger actuators in a real system.

Professor, while calculating the rise time for one of the problems I obtained much more accurate values when I use the rise time formula in the book and not 1.8 / natural frequency. For example for one problem I got a tr=0.72 s, when the plot in simulink indicated a value of approximately 1.5 s, and with the formula in the book tr= 1.53 s. How should we handle this issue?

elazaroa wrote:Professor, while calculating the rise time for one of the problems I obtained much more accurate values when I use the rise time formula in the book and not 1.8 / natural frequency. For example for one problem I got a tr=0.72 s, when the plot in simulink indicated a value of approximately 1.5 s, and with the formula in the book tr= 1.53 s. How should we handle this issue?

To keep things simple, all you need for this course is the 1.8/omega_n. However, it is only an approximation and the other ways may sometimes be better.

when designing a lead compensator for the third order system in problem 3, do we need to create a new pole and zero? or can we use the pole that is at -10 already and only add a zero? this will change the asymptote angle to +-90 and allow us to pull the root-locus much farther to the left.

tmo3290 wrote:when designing a lead compensator for the third order system in problem 3, do we need to create a new pole and zero? or can we use the pole that is at -10 already and only add a zero? this will change the asymptote angle to +-90 and allow us to pull the root-locus much farther to the left.

A lead compensator component adds both a pole and zero, so you can't just add the zero, unfortunately.

On problem 4, the poles from a k value very close to the stability limit are real, so how should we handle this when calculating the rise time and "settling time". Since the solution is exponential, it seems that the settling time, if even defined for this case, is the same thing as the rise time.

Thanks

mkiener wrote:On problem 4, the poles from a k value very close to the stability limit are real, so how should we handle this when calculating the rise time and "settling time". Since the solution is exponential, it seems that the settling time, if even defined for this case, is the same thing as the rise time.

Thanks

You are right, the settling time really isn't defined, so just ignore it for any parts of problems that have real poles.

Professor I don't know if im missing something, but for problem 2 when determining the gain, k, using equation 9.31 I get a completely different answer than the one Matlab gets. Does this equation only apply when the poles are real? The distances I got where around ~1 and ~3 resulting in a k ~3, but MATLAB tells me its 5.7

frodrig3 wrote:Professor I don't know if im missing something, but for problem 2 when determining the gain, k, using equation 9.31 I get a completely different answer than the one Matlab gets. Does this equation only apply when the poles are real? The distances I got where around ~1 and ~3 resulting in a k ~3, but MATLAB tells me its 5.7

It should work for any combination of poles.