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set5 - Chapter 3 State Space Analysis Part 1 Canonical and...

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a39 a38 a36 a37 Chapter 3 – State Space Analysis Part 1 – Canonical and Condensed Forms P. M. Van Dooren Department of Mathematical Engineering Catholic University of Louvain K. A. Gallivan School of Computational Science Florida State University Spring 2006 1
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a39 a38 a36 a37 State Space Analysis Determine the properties of the continuous time LTI system ˙ x = Ax + Bu y = Cx + Du or the discrete time LTI system x k +1 = Ax k + Bu k y k = Cx k + Du k where there are p inputs, m outputs and n degrees of freedom in the state. 2
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a39 a38 a36 a37 Forms and Freedom Some properties are invariant under similarity, i.e., { A, B, C, D } { T 1 AT, T 1 B, CT, D } are the same with respect to some properties. T is invertible and has n 2 degrees of freedom and one constraint: det ( T ) negationslash = 0 . Need to consider for a given property question form of system desired tranformations used to achieve desired form 3
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a39 a38 a36 a37 Forms We will consider: SISO canonical forms – T is nonsingular. SISO condensed forms – T is unitary (orthogonal). MIMO condensed forms – T is unitary (orthogonal). 4
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a39 a38 a36 a37 SISO Jordan Canonical Form Assume m = p = 1 and n = 5 and for a system with unrepeated poles, i.e., n distinct eigenvalues of A . We have, in general, b A j = T 1 AT has diagonal blocks that are Jordan blocks. 2 4 b b j b A j d b c j 3 5 = 2 6 6 6 6 6 6 6 6 6 6 6 4 × × 0 0 0 0 × 0 × 0 0 0 × 0 0 × 0 0 × 0 0 0 × 0 × 0 0 0 0 × × 1 1 1 1 1 3 7 7 7 7 7 7 7 7 7 7 7 5 , 5
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a39 a38 a36 a37 SISO Controller Canonical Form Assume m = p = 1 and n = 5 . If A is noderogatory, we have, in general, b A c = T 1 AT is in row upper companion matrix form and b b c = e 1 . 2 4 b b c b A c d b c c 3 5 = 2 6 6 6 6 6 6 6 6 6 6 6 4 1 × × × × × 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 × × × × × × 3 7 7 7 7 7 7 7 7 7 7 7 5 , 6
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a39 a38 a36 a37 SISO Observer Canonical Form Assume m = p = 1 and n = 5 . If A is noderogatory, we have, in general, b A o = T 1 AT is in column upper companion matrix form and b c o = e T 1 . 2 4 b b o b A o d b c o 3 5 = 2 6 6 6 6 6 6 6 6 6 6 6 4 × 0 0 0 0 × × 1 0 0 0 × × 0 1 0 0 × × 0 0 1 0 × × 0 0 0 1 × × 0 0 0 0 1 3 7 7 7 7 7 7 7 7 7 7 7 5 . 7
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a39 a38 a36 a37 Nonzero Structure Canonical forms have n 2 elements assigned either to 0 or to 1. An unitary (orthogonal) transform has n ( n 1) / 2 degrees of freedom due to U H U = I n . The best one can hope for is to create n ( n 1) / 2 zeros. The unitary transformation-based forms are called condensed forms 8
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a39 a38 a36 a37 SISO Schur Form Assume m = p = 1 and n = 5 . We have, in general, b A s = U H AU is an upper triangular matrix. 2 4 b b s b A s d b c s 3 5 = 2 6 6 6 6 6 6 6 6 6 6 6 4 × × × × × × × 0 × × × × × 0 0 × × × × 0 0 0 × × × 0 0 0 0 × × × × × × × 3 7 7 7 7 7 7 7 7 7 7 7 5 , 9
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a39 a38 a36 a37 SISO Controller Hessenberg Form Assume m = p = 1 and n = 5 . We have, in general, b A c = U H AU is an upper Hessenberg matrix and b b c = βe 1 .
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