The state-space model is the most convenient model for describing Multiple-Input Multiple-Output (MIMO) systems. State-space models often are preferable to polynomial models, especially in modern control applications that focus on multivariable systems.

You can estimate both continuous and discrete state-space models.

Continuous

Use partially known model estimation methods to estimate continuous state-space models. You must provide an initial guess for each parameter before conducting estimation. The following equations show the form of the continuous state-space model:

x = A x + B u + K e
y = C x + D u + e

where

  • A is an n × n state matrix of the given system
  • B is an n × m input matrix of the given system
  • C is an r × n output matrix of the given system
  • D is an r × m direct transmission matrix of the given system
  • K is the Kalman gain matrix
  • e is the system disturbance

Discrete

Use the SI Estimate State-Space Model and SI Estimate State-Space Model from FRF VIs to estimate discrete state-space models. The SI Estimate State-Space Model VI supports the following two estimation methods:

  • Deterministic-stochastic subspace method—This method uses principal component analysis to estimate parameters. This method uses both stimulus and response signals to estimate state-space models. This method includes the stochastic parts of the system in the model structure.
  • Realization method—This method uses the impulse response to estimate only the deterministic state-space model. This method does not include stochastic parts of the system in the model structure.

    • The following equations show the form of the discrete state-space model:

    x k + 1 = A x k + B u k + K e k
    y k = C x k + D u k + e k
    Note The equations for the discrete state-space model based on frequency-domain data do not contain Ke(k) and e(k).

    where

    • k is the model sampling time multiplied by the discrete time step, where the discrete time step equals 0, 1, 2, …
    • n is the number of model states
    • m is the number of model inputs
    • r is the number of model outputs
    • x is the model state vector
    • u is the model input vector
    • y is the model output vector
    • e(k) is the system disturbance

    The state-space transfer matrices A, B, C, and D often reflect physical characteristics of a system. The dimension of the state vector x is the only setting you must provide for the state-space model.