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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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Why does the convolution integral have to be surjectuve?
Submitted by: demetri
Submitted on: October 27, 2003
Identifier:
L5.1
A surjective mapping is one that reaches the entire range space; another word for this is "onto". Thus, if a map is surjective, we can always find a point in the domain that is mapped to a desired point in the range.
When the integral is surjective, viewed as a linear map from the control space to the state space, it means that we can always find an input function (i.e. a point in the domain) which maps to a desired state (i.e. a point in the range).
The reachability matrix discussed in class allows us to check whether the convolution term is surjective in terms of rank of a matrix, which is generally much easier than attempting a direct verification.
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Is there a MATLAB function for checking if a matrix is full-rank?
Submitted by: demetri
Submitted on: October 27, 2003
Identifier:
L5.1
The function "rank()" will return the rank of an arbitrary rectangular matrix. If the rank is equal to the number of columns, it is said to be "full rank".
On a technical aside, this function is not particularly well conditioned numerically (i.e. it can be sensitive to small perturbations in the matrix). In this class we will always (attempt to) give well-conditioned problems, but in reality, you can get matrices which are "technically" full rank, but practically not so.
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If a second order LTI system with states x and x
Submitted by: atiwari
Submitted on: October 27, 2003
Identifier:
L5.1
The correct way of answering this question will be to remodel your system as a third order system with states x,x' and x''. Doing so, will help you write the dynamics of the system in the state space form X' = AX + BU : Y = CX, where the state vector X = [x x' x'']^T. The C matrix here is [0 0 1]. Now you can estimate the full state [x x' x''], if the matrix pair (A,C) is observable. Observability will be covered in the wednesday (10/29/03) lecture.
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