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FAQ (Frequently Asked Questions)
Category:
CDS 101/110 Fall 2003
Identifiers: FN H0 H1 H2 H3 H4 H5 H6 H7 H8 L0.0 L1.1 L1.2 L10.1 L2.1 L2.2 L3.1 L3.2 L4.1 L4.2 L5.1 L5.2 L6.2 L8.1 L9.1 L9.2
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Did we suppose all along the lecture we know exactly the dynamics? What happens if we don't?
Submitted by: waydo
Submitted on: October 29, 2003
Identifier:
L5.2
There was another related question: "If our model is incomplete or wrong (i.e. the mass changes on the Segway), can an estimator compensate about that (since the estimator is based on that model). If yes, drawbacks?
The analysis we did in class does indeed assume we know the dynamics (that is, the A,B,C matrices) exactly. However, if you assume the dynamics are close to these you can follow through the same analysis with, for example, the dynamics matrix for the estimator being A + δA, where δA is small, and find that the errors in your estimate will be small.
In the case of the Segway, the controller rather than the estimator compensates for effects like the different masses of different riders - this is exactly the type of uncertainty that feedback is great at managing. When designing such a system, one would specify a range of expected rider masses and verify that the controller meets its specifications for any of them. If there were an estimator, its performance would be best for the particular dynamics it was designed around, but it could be designed such that the errors due to different riders would be small.
The drawback in having to design to a range of dynamics rather than an exact model is that the achievable performance may be limited - a controller that works adequately for a range of systems will be outperformed by a controller tailored exactly to a particular system when applied to that particular system. Performance and robustness to uncertainty must often be traded against one another when designing a controller.
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What assumptions were made on A, B, C, D in derivations and proof in the lecture?
Submitted by: mreiser
Submitted on: October 29, 2003
Identifier:
L5.2
All of the derivations for the observable canonical form and the theorem stating that observability implies there is arbitrary assignment of the poles of the (A-LC) matrix (and vice versa), were done assuming a single-input single-output (SISO) system. Reachability and observability are well defined concepts for multi-input multi-output (MIMO) systems and the test is the same, just compute the reachability and observability matrices; but the specifics of the canonical forms and the spectral assignment theorems will not be necessary for this class. If you would like to see the SISO case again, the derivation is also carried out in the text by A. Lewis linked from the course home page. Intuitively it should seem reasonable that more inputs can only help your reachability and more outputs can only help observability. Since the test for e.g. reachability is the same for the MIMO and SISO case, then having a 3 state system with one input will give a 3x3 reachability matrix; the same system with 2 inputs will yield a 3x6 reachability matrix. Since we only need rank = 3 for reachability, more inputs can help reachability.
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(CT-1)(TAT-1) = CA ; What happened to the second T-1
Submitted by: atiwari
Submitted on: October 29, 2003
Identifier:
L5.2
It was actually there but was moved to the RHS. The equation read as:
CA = [-a1 1 0 0 .... 0]T
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C = T1
CA = -a1T1 + T2 this will give the first 2 rows of the matrix P in the relation T = P(Observ_Mat), where do you get the other rows from? [Observ_Mat is the observability matrix]
Submitted by: atiwari
Submitted on: October 29, 2003
Identifier:
L5.2
To get the third row of P you should do the following:
(CT-1) (TAT-1) (TAT-1) = [a12 - a2 -a1 1 ....... 0]
This implies that
T3 = CA2 - (a12 - a2) T1 + a1T2
substitute for T1 and T2 and you get a relation for T3. Again multiply TAT-1 on the LHS to get relations for T4 and higher rows.
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What about controllability for multiple inputs and observability for multiple outputs? Do the same results hold in those cases?
Submitted by: atiwari
Submitted on: October 29, 2003
Identifier:
L5.2
Yes, all the results stated in the lecture about eigenvalue placement of L and K matrices, hold in the general case of multiple inputs and outputs.
That is to say
The eigen values of A + BK are freely assignable, by choosing K of appropriate dimensions, iff the pair (A,B) is controllable.
and similarly for A - LC and observability case.
However note that the proof for the multiple input and output case was not covered in the lecture. If you are interested, the proof is given in "A Course in Robust Control Theory - Paganini and Dullerud". I own a copy of the book which I am willing to lend it to you to look at the proofs. Email me at atiwari@caltech.edu if you need the book.
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