Mathematical Model Definitions (Control Design and Simulation Module)

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Updated2023-03-14
4 minute(s) read

The LabVIEW Control Design and Simulation Module provides tools to study the dynamics of systems described by linear time-invariant (LTI) continuous and discrete models. You can create deterministic state-space, transfer function, and zero-pole-gain models. You also can create stochastic state-space models and the second-order statistics noise models. You can use these forms to describe both single-input single-output (SISO) and multiple-input multiple-output (MIMO) systems.

Continuous transfer function and zero-pole-gain models use the s variable to define time, whereas discrete transfer function and zero-pole-gain models use the z variable to define time. Continuous state-space models use the t variable to define time, whereas discrete state-space models use the k variable to define time.

Deterministic State-Space Model

Continuous
Discrete	x(k + 1) = Ax(k) + Bu(k)
	y(k) = Cx(k) + Du(k)

Stochastic State-Space Model

Continuous
Discrete	x(k + 1) = Ax(k) + Bu(k) + Gw(k)
	y(k) = Cx(k) + Du(k) + Hw(k) + v(k)
Second-Order Statistics Noise Model	Q = E{w . w^T} – E{w} . E^T{w}
	R = E{v . v^T} – E{v} . E^T{v}
	N = E{w . v^T} – E{w} . E^T{v}

where	t is continuous time.
	k is the model sampling time multiplied by the discrete time step, where the discrete time step equals 0, 1, 2, …
	x is the model state vector.
	u is the model input vector.
	y is the model output vector.
	w is the q × 1 process noise vector.
	v is the r × 1 measurement noise vector.
	A is an n × n state matrix of the given model.
	B is an n × m input matrix of the given model.
	C is an r × n output matrix of the given model.
	D is an r × m direct transmission matrix of the given model.
	n is the number of model states.
	m is the number of model inputs.
	r is the number of model outputs.
	q is the length of process noise vector w.
	G is the n × q weighting matrix relating the process noise vector w to the model states.
	H is the r × q weighting matrix relating the process noise vector w to the model outputs.
	Q is the q × q auto-covariance matrix of w.
	R is the r × r auto-covariance matrix of v.
	N is the q × r cross-covariance matrix between w and v.
	E{} denotes the expected value or the mean of the enclosed term(s).

Transfer Function Model

	SISO	MIMO
Continuous
Discrete

Zero-Pole-Gain Model

	SISO	MIMO
Continuous
Discrete

where	s is the Laplace variable and continuous time
	z is discrete time
	m is the order of the numerator polynomial function
	n is the order of the denominator polynomial function
	b_m are the coefficients of the numerator polynomial function
	a_n are the coefficients of the denominator polynomial function
	Z_m are the locations of the model zeros
	P_n are the locations of the model poles
	k is the gain of the model
	H_ij is the transfer function or zero-pole-gain equation at the i^th input and j^th output of a MIMO model

LabVIEW Control Design and Simulation Module

Table of Contents

Mathematical Model Definitions (Control Design and Simulation Module)

Deterministic State-Space Model

Stochastic State-Space Model

Transfer Function Model

Zero-Pole-Gain Model

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