control.matlab.dlqe

control.matlab.dlqe(A, G, C, QN, RN[, N])[source]

Linear quadratic estimator design (Kalman filter) for discrete-time systems. Given the system

x[n+1] &= Ax[n] + Bu[n] + Gw[n] \\
y[n] &= Cx[n] + Du[n] + v[n]

with unbiased process noise w and measurement noise v with covariances

E{ww'} = QN,    E{vv'} = RN,    E{wv'} = NN

The dlqe() function computes the observer gain matrix L such that the stationary (non-time-varying) Kalman filter

x_e[n+1] = A x_e[n] + B u[n] + L(y[n] - C x_e[n] - D u[n])

produces a state estimate x_e[n] that minimizes the expected squared error using the sensor measurements y. The noise cross-correlation NN is set to zero when omitted.

Parameters
  • A (2D array_like) – Dynamics and noise input matrices

  • G (2D array_like) – Dynamics and noise input matrices

  • QN (2D array_like) – Process and sensor noise covariance matrices

  • RN (2D array_like) – Process and sensor noise covariance matrices

  • NN (2D array, optional) – Cross covariance matrix (not yet supported)

  • method (str, optional) – Set the method used for computing the result. Current methods are ‘slycot’ and ‘scipy’. If set to None (default), try ‘slycot’ first and then ‘scipy’.

Returns

  • L (2D array (or matrix)) – Kalman estimator gain

  • P (2D array (or matrix)) – Solution to Riccati equation

    A P + P A^T - (P C^T + G N) R^{-1}  (C P + N^T G^T) + G Q G^T = 0

  • E (1D array) – Eigenvalues of estimator poles eig(A - L C)

Notes

The return type for 2D arrays depends on the default class set for state space operations. See use_numpy_matrix().

Examples

>>> L, P, E = dlqe(A, G, C, QN, RN)
>>> L, P, E = dlqe(A, G, C, QN, RN, NN)

See also

dlqr, lqe, lqr