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FAQ (Frequently Asked Questions)
Category:
CDS 110b Winter 2005
Identifiers: H10 H11 H14 H15 H9 L14 midterm review
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Are there matlab m-files to convert controllers to work with the dfan simulink block?
Submitted by: macmardg
Submitted on: January 4, 2005
Identifier:
H9
Yes - download the file runsim.m off of the web site and type help runsim to see instructions; this uses the controllers for x and theta axes, assembles the full controller, calls simulink, and plots and returns the results. Let me know if you have a problem with it that isn't answered in the instructions.
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How does the sign of the plant affect the control design?
Submitted by: macmardg
Submitted on: January 5, 2005
Identifier:
H9
The plant for the x-axis is negative (that is, instead of requiring negative feedback to stabilize it, it requires positive feedback). Note that PC=(-P)*(-C), so that if one designs a compensator for -P, as usual, then changing the sign of the compensator will stabilize P.
Also note that the need to change the sign of the compensator will affect plots such as the root locus. The rule about the locus existing to the left of an odd number of poles/zeroes applies for negative feedback. Changing the sign of the feedback will mean that the root locus will exist to the right of an odd number of poles and zeroes.
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There's a typo in 1(c): "rad/sec" should be "rad"
Submitted by: macmardg
Submitted on: January 5, 2005
Identifier:
H9
You're right. Oops. Also, the text isn't clear that question 1 is to design control for theta, and question 2 is to design the (outer loop) control for x
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I dont understand the actuator saturation
Submitted by: macmardg
Submitted on: January 5, 2005
Identifier:
H9
There is a maximum side force u that the physical ducted fan can produce (i.e. an actuator saturation). The forces are generated by a fan, and the thrust can be vectored through some angle by a set of flaps (controlled by a model airplane servo motor).
To stabilize the fan in hover, the thrust must be set at f_2 = m*gamma = 7N. (Note that the class notes may be different from the problem set. The control u = f1 is the side force produced by vectoring, as opposed to the axial force f2 required to counteract gravity.)
If we assume that the total thrust does not change as the thrust is vectored, then the maximum side force is equal to the nominal thrust times the sin of the vectoring angle of 40 degrees. (This is the behaviour one would get if the vertical control were open-loop, that is, simply set at a fixed thrust. If there were a control loop closed on the vertical position, then a better assumption would be that the vertical component of the thrust remained fixed at 7N. The actual saturation algorithm used in the Simulink is closer to the latter and somewhat more complex.)
Almost any controller you devise will result in an actuator saturation even for very small step inputs in the desired position. This is not necessarily a problem, but does mean that the actual system would not respond in the same way as your linear model does. You do not need to redesign the controller to obtain better performance in the presence of the saturation.
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I'm still confused about the actuator saturation!
Submitted by: mjdunlop
Submitted on: January 8, 2005
Identifier:
H9
The 40 degree angle listed in 1c is the angle of the fan flaps. Note that this is NOT theta, which is the angle the entire fan is tilted. If the maximum thrust generated coming out the back of the fan is 7N you can figure out what component of that is a purely sideways force (u) when the flaps are at a 40 degree angle. This number is the maximum value of u you can have, so compare that to what your plot of u vs. time looks like.
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What do you mean by the terms damping ratio and natural frequency in 1c and 1d ?
Submitted by: vijay
Submitted on: January 17, 2005
Identifier:
H10
The position control system is a second-order system. So for any control law, there would be some natural frequency and damping ratio for the closed loop system (Recall from CDS110a that these terms are defined for any second order system; for a revision of these concepts, you can take a look at ocw.mit.edu/NR/rdonlyres/Mathematics/18-03Spring2004/
B76E6F4F-7B05-4DA0-A5A5-03FA4ACCB6B2/0/sup_13.pdf)
All you have to do in 1c and 1d is to find out the value of these terms for the control law from 1b.
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Minor typo on Prob 3
Submitted by: mjdunlop
Submitted on: January 21, 2005
Identifier:
H11
Problem 3b should read:
...the observer gains of problem 2...
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How do I plot responses to an initial condition?
Submitted by: mjdunlop
Submitted on: January 23, 2005
Identifier:
H11
Matlab's "inital" command is what you want.
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What is a "Butterworth pattern"?
Submitted by: mjdunlop
Submitted on: January 23, 2005
Identifier:
H11
The Butterworth pattern is just a way of evenly spacing poles in the left half plane. Figure 6.7 in Friedland should clarify how it works.
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Block diagram confusion
Submitted by: mjdunlop
Submitted on: January 23, 2005
Identifier:
H11
Having a mental picture of the block diagram for the controller/observer and plant will help you figure out the appropriate transfer functions you need to use in problems 2 and 3. Figure 8.1 in Friedland lays this out nicely, but beware that Friedland's K is Prof. MacMynowski's L and Friedland's G is Prof. MacMynowski's K.
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What do you mean by the observer state response in problem 2?
Submitted by: haomiao
Submitted on: January 23, 2005
Identifier:
H11
We're actually looking for the error, not x_hat. The point of this problem is to show you what happens to the error in your observer over time.
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Spectral factorization errors
Submitted by: macmardg
Submitted on: January 27, 2005
Identifier:
L14
A pole or zero at p gives a term in the spectrum of (omega^2+p^2). Therefore, if we take the spectrum, substitute lambda=-omega^2, we will have terms (lambda-alpha) where alpha=p^2. Thus while Friedland (p. 394) has a sign error in step 1, step 3 is correct. The factor.pdf on the web page has been corrected.
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Correction from midterm review lecture
Submitted by: mjdunlop
Submitted on: February 4, 2005
Identifier:
midterm review
The 3rd condition from Pontryagin's Maximum Principle in the bang-bang control example should have been u = -sign(lambda^T B). I left out the minus sign in class.
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real, rational stable matrix???
Submitted by: macmardg
Submitted on: February 21, 2005
Identifier:
H14
Everywhere you see something like "real, rational stable matrix", think "stable transfer function" or "stable transfer function matrix" depending on the implied dimension. We typically choose W_2 and the nominal plant to have rational representations whose coefficients are real so that one can (ultimately) use a state space representation. And Delta, which is unknown but fixed, does not need to be rational, but can always be approximated with arbitrary accuracy and so it is sufficient to consider it to also be within the set of rational stable transfer functions with real coefficients. Sorry for the confusion...
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In #2, what is the relationship between G(s) and H(s)?
Submitted by: macmardg
Submitted on: February 27, 2005
Identifier:
H15
Sorry, I mis-typed - the phase criterion should refer to G(s) rather than H(s).
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Note error in wording in #2(b)
Submitted by: macmardg
Submitted on: February 27, 2005
Identifier:
H15
Question 2(b) should be re-worded as follows: "If H_2 is positive real but otherwise unknown, then the condition that H_1 be strictly positive real is also necessary to guarantee stability of the feedback interconnection."
Note that the intent is to be able to guarantee stability for any H_1 that is positive real but otherwise unknown; the original wording (without the word guarantee) is not correct. (Thanks to Phil for pointing this out.)
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I've been having some problems with problem 1b
Submitted by: macmardg
Submitted on: February 28, 2005
Identifier:
H15
That would be because, as written, its essentially unsolvable. For this combination of uncertainty and performance, only a necessary condition can be easily derived and not a necessary and sufficient condition. (Oops.)
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I'm still confused about 1b
Submitted by: macmardg
Submitted on: March 1, 2005
Identifier:
H15
The intent is to come up with a necessary condition that guarantees robust performance in the presence of the unknown delta. (I.e., robust performance with respect to the uncertainty delta.) Since delta is unknown, the condition should not depend on delta, but only on P, S, T, and the weights. This particular combination of uncertainty and performance metric is in the "messy" category described in DFT Table 4.2; there is an example given in Section 4.4 that walks through the development of necessary conditions for a similar "messy" problem.
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