LQEstimatorGains

LQEstimatorGains[ssm,{w,v}]
gives the optimal estimator gain matrix for the StateSpaceModel ssm, with process and measurement noise covariance matrices w and v.

LQEstimatorGains[ssm,{w,v,h}]
includes the cross-covariance matrix h.

LQEstimatorGains[{ssm,sensors},{}]
specifies sensors as the noisy measurements of ssm.

LQEstimatorGains[{ssm,sensors,dinputs},{}]
specifies dinputs as the deterministic inputs of ssm.

Details and OptionsDetails and Options

  • The standard state-space model ssm can be given as StateSpaceModel[{a,b,c,d}] in either continuous time or discrete time:
  • continuous-time system
    discrete-time system
  • The descriptor state-space model ssm can be given as StateSpaceModel[{a,b,c,d,e}] in either continuous time or discrete time:
  • continuous-time system
    discrete-time system
  • LQEstimatorGains also accepts nonlinear systems specified by AffineStateSpaceModel and NonlinearStateSpaceModel.
  • For nonlinear systems, the operating values of state and input variables are taken into consideration, and the gains are computed based on the approximate Taylor linearization.
  • The input can include the process noise , as well as deterministic inputs .
  • The argument dinputs is a list of integers specifying the positions of in .
  • The output consists of the noisy measurements as well as other outputs.
  • The argument sensors is a list of integers specifying the positions of in .
  • LQEstimatorGains[ssm,{}] is equivalent to LQEstimatorGains[{ssm,All,None},{}].
  • The noisy measurements are modeled as , where and are the submatrices of and associated with , and is the noise.
  • The process and measurement noises are assumed to be white and Gaussian:
  • , process noise
    , measurement noise
  • The cross-covariance between the process and measurement noises is given by .
  • If omitted, h is assumed to be a zero matrix.
  • The estimator with the optimal gain minimizes , where is the estimated state vector.
  • For continuous-time systems, the optimal gain is computed as , where is the solution of the continuous algebraic Riccati equation . The matrix is the submatrix of associated with the process noise.
  • For discrete-time systems, the optimal gain is computed as , where is the solution of the discrete Riccati equation .
  • The optimal estimator is asymptotically stable if is nonsingular, the pair is detectable, and is stabilizable for any .

ExamplesExamplesopen allclose all

Basic Examples  (3)Basic Examples  (3)

The Kalman gain matrix for a continuous-time system:

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The gains for a discrete-time system:

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The gains for an unobservable system:

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Although unobservable, the system is detectable:

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Introduced in 2010
(8.0)
| Updated in 2014
(10.0)