CD Separate Stable Model VI

Owning Palette: Model Reduction VIs

Requires: Control Design and Simulation Module

Separates the stable and unstable terms of a model. Continuous systems have unstable terms if real parts > tolerance. Discrete systems have unstable terms if magnitudes > 1-tolerance. Wire data to the Input State-Space Model input to determine the polymorphic instance to use or manually select the instance.

Details  

CD Separate Stable Model (State-Space)

	Input State-Space Model contains a mathematical representation of and information about the system on which this VI operates.
	Method specifies the error model reduction method. 0 Additive (default)—Separates stable and unstable terms as described in the Details section. 1 Multiplicative—Separates stable poles and zeros from unstable poles and zeros as described in the Details section. This method supports only SISO models.
	error in describes error conditions that occur before this node runs. This input provides standard error in functionality.
	Stable State-Space Term returns the stable term of the state-space model.
	Unstable State-Space Term returns the unstable term of the state-space model.
	error out contains error information. This output provides standard error out functionality.

CD Separate Stable Model (Transfer Function)

	Input Transfer Function Model contains a mathematical representation of and information about the system on which this VI operates.
	Method specifies the error model reduction method. 0 Additive (default)—Separates stable and unstable terms as described in the Details section. 1 Multiplicative—Separates stable poles and zeros from unstable poles and zeros as described in the Details section. This method supports only SISO models.
	error in describes error conditions that occur before this node runs. This input provides standard error in functionality.
	Stable Transfer Function Term returns the stable term of the transfer function model.
	Unstable Transfer Function Term returns the stable term of the transfer function model.
	error out contains error information. This output provides standard error out functionality.

CD Separate Stable Model (Zero-Pole-Gain)

	Input ZPK Model contains a mathematical representation of and information about the system on which this VI operates.
	Method specifies the error model reduction method. 0 Additive (default)—Separates stable and unstable terms as described in the Details section. 1 Multiplicative—Separates stable poles and zeros from unstable poles and zeros as described in the Details section. This method supports only SISO models.
	error in describes error conditions that occur before this node runs. This input provides standard error in functionality.
	Stable ZPK Term returns the stable term of the zero-pole-gain model.
	Unstable ZPK Term returns the unstable term of the zero-pole-gain model.
	error out contains error information. This output provides standard error out functionality.

CD Separate Stable Model Details

The Method input allows you to choose between two algorithms that separate the stable and unstable terms of a model.

Additive Method Details

The Additive method of separating stable and unstable terms decomposes the input model such that input_model = stable_terms + unstable_terms. The algorithm transforms the A matrix to Schur form, and then counts the number of stable and unstable eigenvalues, together with those for which classification is doubtful. Stable eigenvalues are those Re[s] < 0 for continuous time and |z| < 1 for discrete time. Doubtful eigenvalues are those for which the real part of the eigenvalue has magnitude less than or equal to tolerance for continuous-time, or eigenvalue magnitude within the following range for discrete time: 1 - tolerance, 1 + tolerance.

The algorithm then computes a real ordered Schur decomposition of A so that after the transformation, the following is true:

where the eigenvalues of A_S and A_U are respectively stable and unstable. The algorithm then determines a matrix X satisfying -A_SX + XA_U + A_SU = 0. The eigenvalue properties of AS and AU guarantee that X exists. If doubtful eigenvalues are present, this VI assigns them to the unstable part.

Multiplicative Method Details

The Multiplicative method of separating stable and unstable terms decomposes the input model such that input_model = stable_terms * unstable_terms. During this process, this VI converts the model to zero-pole-gain form. Unstable terms have the poles or zeros with Re[s] > 0 for continuous time and |z| > 1 for discrete time. Otherwise, poles are considered stable. This algorithm moves any delay information to the unstable term and moves the gain of the model to the stable term.