TVSC_Lecture_3

TVSC_Lecture_3

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Unformatted text preview: ted versions of the block-diagonal matrices 3 (ZA) = ZAZ T Z 2 AZ 2T Z 3 AZ 3T Z 3 = A(1) A(2) A(3) Z 3 . A(1) A(2) A(3) This notional simplification leads to another notational simplification A[n] := A(1) A(2) . . . A(n) , such that we can rewrite the expression for (ZA)n as n (ZA) = A(1) A(2) . . . A(n) Z n = A[n] Z n A[n] Finally, after all these notional modifications we can write out the series expansion −1 (1 − ZA) = 1 + A[1] Z 1 + A[2] Z 2 + A[3] Z 3 + . . . , (7) which looks structurally very similar to the conventional form of the geometric series, in spite of the elements in the series being block-diagonal matrices. 5 Lecture 3 In the following we will use this modified von Neuman expansion (7) to further move on with producing the Toeplitz operator T . The first term in the expansion then looks like A[1] Z 1 = A(1) Z 1 = = .. . A−1 A0 A1 .. = .. . .. . . . .. · .. . 0 1 0 1 0 1 .. .. . . 0 A− 1 0 A0 0 A1 0 .. . .. . Following this recipe we can write up the second term of the series expansion more explicitely as A[2] Z 2 = A(1) A(2) Z 2 = = = .. . A−1 A0 A1 .. .. . .. . · A− 1 A− 2 .. A−2 . A0 A−1 A1 A0 0 0 A0 A−1 0 0 A1 A0 0 0 .. . A−1 .. 0 .. . .. . . A0 .. 0 0 1 0 0 1 0 0 .. . . ... .. . · .. . . ... .. . · .. . . .. . A−1 A−2 = = .. . .. . 0 0 1 0 0 1 0 0 .. . .. . .. . . 6 Lecture 3 Adding up the first three terms of the series expansion produces the lower-triangular tri-diagonal matrix .. . .. . 1 . .. A1 1 [1] 1 [2] 2 1+A Z +A Z = . A2 A1 A2 1 A3 A2 A3 1 .. . A4 A3 A4 .. .. . . Just to make the construction principle more visible I add the forth term matrix .. . . .. 1 .. . A1 1 1 + A[1] Z 1 + A[2] Z 2 + A[3] Z 3 = . . . A2 A1 A2 1 .. . A3 A2 A1 A3 A2 A3 0 A4 A3 A2 A4 A3 .. .. . . Taking this intermediate result we can finally generate the Toeplitz .. . .. . .. 1 . . .. C0 A1 1 . C1 . . A2 A1 A2 1 C2 . .. .. A3 A2 A1 A3 A2 A3 . A4 A3 A2 A1 A4 A3 A2 A4 A3 .. .. . . of the series to produce the 1 A4 .. . 1 .. . .. . . operator according to (4) 1 A4 .. . 1 .. . .. . .. . .. . 0 B −1 0 B0 0 B1 Example: Finite Matrix We now want to use a simple example to demonstrate the inner workings of the state-space computations. To this end, let us consider the example of the finite dimensional 4 × 4 (non-Toeplitz) matrix 1 1/2 1 , T = (8) 1/6 1/3 1 1/24 1/12 1/4 1 which we will interpret as a transfer operator of a simple time-varying system. The matrix corresponds to a time-varying system since the matrix does not exhibit Toeplitz structure and because the matrix is 0 .. . .. . . 7 Lecture 3 finite dimensional. Furthermore, T is lower-triangular, indicating the system to be causal. Here T is a 4 × 4 matrix, however, real systems of interest are significantly larger, typical of the order n ∼ 10, 000. In the following discussion the term signal is used interchangeably with vector, likewise the term system is used for matrix. Similar to our discussion about time-invariant systems, we strive to represent the matrix T in terms of the linear fractional map −1 T = D + C (1 − ZA) ZB ....
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This note was uploaded on 07/07/2013 for the course EI 2012 taught by Professor Tum during the Winter '12 term at TU München.

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