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LQEstimatorGains

LQEstimatorGains
gives the optimal estimator gain matrix for the StateSpaceModel object ss with process and measurement noise covariance matrices w and v.
LQEstimatorGains
includes the cross-covariance matrix h.
LQEstimatorGains
specifies sensors as the noisy measurements of ss.
LQEstimatorGains
specifies dinputs as the deterministic inputs of ss.
  • The state-space model ss can be given as StateSpaceModel, 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
  • 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 .
  • 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 non-singular, the pair is detectable, and is stabilizable for any .
The Kalman gain matrix for a continuous-time system:
The gains for a discrete-time system:
The gains for an unobservable system:
Although unobservable, the system is detectable:
The Kalman gain matrix for a continuous-time system:
In[1]:=
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Out[1]=
 
The gains for a discrete-time system:
In[1]:=
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Out[1]=
 
The gains for an unobservable system:
In[1]:=
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Out[1]//MatrixForm=
Although unobservable, the system is detectable:
In[2]:=
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Out[2]=
 
Determine the optimal estimator gains of a continuous-time system:
The gains for a discrete-time system with nonzero cross-covariance:
The Kalman gains for a continuous-time system with cross-correlated noises:
Use the first output as the measurement:
Use the second output as the measurement:
The Kalman gains for a system in which the last four inputs are stochastic disturbances:
Estimator gain for a system with two deterministic inputs and two stochastic inputs:
The poles of the Kalman estimator:
Compute the Kalman gains that smooth the response of a stochastic system:
The response of the system to a sinusoid input in the presence of process and measurement noise:
Filtered response :
Compute the Kalman estimator gains using the underlying Riccati equation:
LQEstimatorGains gives the same result:
The gains for a discrete-time system can be computed using DiscreteRiccatiSolve:
LQEstimatorGains gives the same result:
The optimal estimator gains can be computed as the conjugate transpose of optimal regulator gains of the dual system:
The dual relationship for a discrete-time system:
The measurement noise covariance matrix must be positive definite:
Optimal estimator gain can be computed for an unobservable system only if it is detectable:
The last mode is unstable and unobservable:
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