BUILT-IN MATHEMATICA SYMBOL

# 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
• 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:

 Out[3]=

The gains for a discrete-time system:

 Out[1]=

The gains for an unobservable system:

 Out[2]//MatrixForm=

Although unobservable, the system is detectable:

 Out[3]=