Mathematical Model Definitions (Control Design and Simulation Module)
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
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 process noise vector. | |
v is the 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. | |
G is a matrix relating w to the model states. | |
H is a matrix relating w to the model outputs. | |
Q is the auto-covariance matrix of w. | |
R is the auto-covariance matrix of v. | |
N is the 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 |