gives the estimator gain matrix for the StateSpaceModel ssm, such that the poles of the estimator are pi.


specifies the measured outputs outi to use.

Details and Options

  • EstimatorGains is also known as observer gains or observer pole placement.
  • The state-space model ssm can be given as StateSpaceModel[{a,b,c,d}], where a, b, c, and d represent the state, input, output, and transmission matrices in either a continuous-time or a discrete-time system:
  • continuous-time system
    discrete-time system
  • If ssm is observable, the eigenvalues of will be {p1,p2,,pn}, where is the computed estimator gain matrix.
  • For a descriptor system StateSpaceModel[{a,b,c,d,e}], the number of poles that can be specified is determined by the rank of e and the observability of the system.
  • EstimatorGains 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.
  • EstimatorGains[{ssm,{out1,}},] is equivalent to EstimatorGains[ssm1,], where ssm1SystemsModelExtract[ssm,All,{out1,}].
  • The observer dynamics are given by:
  • continuous-time system
    discrete-time system
  • In the case of a square nonsingular matrix , the state vector can be computed as x=TemplateBox[{c}, Inverse].(y-d.u).
  • EstimatorGains accepts a Method option with settings given by:
  • Automaticautomatic method selection
    "Ackermann"Ackermann method
    "KNVD"KautskyNicholsVan Dooren method
  • The estimator gains are computed as the state feedback gains of the dual system.


open all close all

Basic Examples  (3)

Compute estimator gains for a continuous-time system:

Click for copyable input

A discrete-time system:

Click for copyable input

Estimator gains for a two-output system with only the second output measured:

Click for copyable input

Scope  (6)

Options  (6)

Applications  (2)

Properties & Relations  (7)

Possible Issues  (4)

Introduced in 2010
Updated in 2014