CD Calculate Integrals with Matrix Exponential VI

Owning Palette: Solvers VIs

Requires: Control Design and Simulation Module

Calculates integrals involving matrix exponentials. These integrals are called Van Loan integrals.

Details  

	Ac specifies an n x n state matrix, where n is the number of states.
	Bc specifies an n x m input matrix, where m is the number of inputs.
	Qc is an n x n matrix specifying the state weight matrix. Q must be symmetric and positive semi-definite.
	Rc is an m x m matrix specifying the input weight matrix. R must be symmetric and positive definite. The default is an identity matrix of appropriate dimensions.
	Nc is an n x m matrix which specifies the state-input cross weight matrix. The value of Nc must be such that the matrix (Qc–Nc.inv(Rc).Nc') is positive semi-definite. The default value of Nc is a matrix of zeros of appropriate dimensions.
	Ts specifies the upper limit of the integral of the matrix this VI calculates. Refer to the Details section for the matrix this VI calculates.
	error in describes error conditions that occur before this node runs. This input provides standard error in functionality.
	Qd returns the 1 x 1 block of the matrix this VI calculates. Refer to the Details section for the matrix this VI calculates.
	Rd returns the 2 x 2 block of the matrix this VI calculates. Refer to the Details section for the matrix this VI calculates.
	Nd returns the 1 x 2 block of the matrix this VI calculates. Refer to the Details section for the matrix this VI calculates.
	error out contains error information. This output provides standard error out functionality.

CD Calculate Integrals with Matrix Exponential Details

This VI calculates the outputs according to the following equations:

where

If you interpret the inputs to this VI as describing information about a continuous system model, you can interpret the outputs of this VI as discretized versions of these inputs, where Ts is the sampling time.

For example, if you make the following assumptions:

• Ac represents the continuous system matrix that describes the states of the system.
• Bc represents the continuous input matrix that relates the inputs to the states.
• Qc represents the continuous cost matrix penalizing the system states. This matrix also can represent the continuous process noise spectral intensity matrix.
• Rc represents the continuous cost matrix penalizing the system inputs. This matrix also can represent the continuous measurement noise spectral intensity matrix.
• Nc represents the continuous cost matrix penalizing the cross product between the system states and the system inputs. This matrix also can represent the continuous cross-spectral intensity matrix between the process noise and the measurement noise.
• Ts represents the sampling time this VI uses to discretize the continuous matrices Ac, Bc, Qc, Rc, and Nc.

then the following statements are true:

• Qd is the discrete equivalent of Qc.
• Rd is the discrete equivalent of Rc.
• Nd is the discrete equivalent of Nc.