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JordanModelDecomposition

JordanModelDecomposition[ss]
yields the Jordan decomposition of a StateSpaceModel object ss. The result is a list where s is a similarity matrix and jc is the Jordan canonical form of ss.
  • The state-space model ss can be given as StateSpaceModel, where a, b, c, and d represent the state, input, output, and transmission matrices in either a continuous-time or a discrete-time system:
continuous-time system
discrete-time system
  • The transformation , where is the new state vector and is a similarity matrix that spans the linearly independent eigenvectors of , transforms the system into the Jordan canonical form:
, continuous-time system
,.discrete-time system
  • The new state matrix is the Jordan canonical form of the old state matrix .
Compute the Jordan decomposition of a state-space model:
Compute the Jordan decomposition of a state-space model:
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Click for copyable input
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The Jordan decomposition of a second-order system:
The Jordan decomposition of a discrete-time system:
A system with complex poles:
Repeated poles appear in Jordan blocks:
A system is controllable if and only if the Jordan blocks of have distinct eigenvalues, and the row elements of corresponding to the last row of each Jordan block are not all zero:
A system is observable if and only if the Jordan blocks of have distinct eigenvalues, and the column elements of corresponding to the first row of each Jordan block are not all zero:
In the Jordan canonical form the eigenvalues are along the diagonal of the state matrix:
The Jordan canonical form is related to the original system via the similarity transform:
The Jordan canonical form of a state-space model is the similarity transformation associated with the Jordan decomposition of its state matrix:
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