TransferFunctionModel supports improper transfer functions. When converted to the time domain, an improper transfer function creates a descriptor StateSpaceModel. The proper and polynomial parts of the transfer function can be decoupled in the state-space model with KroneckerModelDecomposition.
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The descriptor matrix is block diagonal with an identity matrix for the states of the proper subsystem and a nilpotent matrix for the states of the polynomial subsystem.
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The solution to the full system is the sum of the solutions to the two parts.